SYSTEM AND TABLES
OF
Life Insurance.
A TREATISE DEVELOPED FROM THE EXPERIENCE AND RECORDS OF
THIRTY AMERICAN LIFE OFFICES, UNDER THE DIRECTION
OF A COMMITTEE OF ACTUARIES.
BY
LEVI W. MEECH,
ACTUARY IN CHARGE.
PUBLISHED BY SUBSCRIPTION.
FOR ADDITIONAL COPIES, ADDRESS THE ACTUARY IN CHARGE,
NORWICH, CONN. Entered according to Act of Congress, in the year 1881,
By LEVI W. MEECH, in Charge,
In the Office of the Librarian of Congress, at Washington.
Smith A McDougal, Electrotypers,
82 Beckman St., N. Y. PREFACE.
THE origin of the present collection of statistics, dates from the re-organization of
the Chamber of Life Insurance in 1873. Among the leading objects, it was
provided that, “there shall be a standing Committee on Mortality Experience,
appointed by the Executive Committee, immediately after the Annual Election
the Chamber, in each year, and consisting of five members, to whom shall be
ferred the work of collecting the statistics of Mortality Experience, classifying,
arranging, and tabulating them for practical use, with power to employ skilled
assistance.”
In entering upon the duties assigned them, this Committee, after some prelimi-
nary consultation, decided to invite a more full expression of opinions in a general
meeting of Actuaries, which was held in New York, Nov. 19, 1874. Among those
present and participating in the deliberations, besides the Committee, were S. C.
Chandler, Jr., D. Parks Fackler, C. M. Hibbard, Sheppard Homans, Jas. Weir Mason,
Prof. C. F. McCay, Emory McClintock, George W. Phillips, William D. Whiting, and
Charlton T. Lewis, Secretary of the Chamber. Besides these, a considerable number
communicated their views by letter. An enlarged plan of operations having been
generally called for, the Committee, in concurrence, subsequently issued the following
Circular, under date of March 10, 1875 ; copies of which were addressed to all Life
Insurance Companies in the United States:
Sir :
The Chamber of Life Insurance has, through its Executive Committee, appointed the under-
signed a Committee upon Mortality Experience, with instructions to procure from every American
Company willing to furnish it, their experience in full up to a recent period. In accordance therewith,
we respectfully and most earnestly request you to favor us with the data of your company to Decem-
ber 31, 1874, as indicated below. While the amount of labor to be performed both by the companies
and the Committee will be considerable, the importance of this work, if well done, to American
companies, cannot be over estimated. Tables made from such experience can justly be regarded as of
higher authority, and will probably more nearly approximate the true average of mortality among
insured American lives than any now in use. And should similar examinations be made hereafter, as
we trust may be the case, in which the results of the work now in hand may be merged, by such
periodic examinations, and the aggregation of experience, we shall be able measurably to trace any
general improvement in longevity, the effect of causes operating specially within the period under
consideration, the effect of public sanitary measures, and probably many other points of interest
incident to the development and improvement of a new and growing country. We hope that the
present undertaking is but the beginning of a systematic and profound study of the mortality among
insured American lives.
In settling the details of this work, we have had in view certain important classifications,leaving
many minor though interesting ones to be wrought out by the companies themselves; and the saving
of all but indispensable labor to the companies contributing their data,
The points agreed upon by this Committee for consideration are:
First, The general mortality among insured lives ;
Second, The mortality by sexes ;
Third, The mortality by localities, taking the State as a basis, except in certain States where
the County is to be designated ; 4
PREFACE.
Fourth, The mortality by yeai;s of insurance, in order to show the effect and duration of
medical selection ;
Fifth, The ratio of financial loss to sums exposed to risk ;
Sixth, A classification of the causes of death, in general, by locality, and such other special
relations as may be deemed advisable hereafter.
For these purposes the Committee have adopted the enclosed card, as enabling the companies to
furnish the necessary data with the least possible labor. The instructions upon another page will
enable a clerk to make correct and rapid use of it.
Companies which have already arranged their data for their own examination in a manner to
supply the Committee what is called for by the card, need only furnish their cards or sheets already
prepared or their classifications if these correspond to those above stated, and any such material will
be returned after use; and, if desired, all cards will be returned. No tabulation or exhibit will be
made in any manner of the separate mortality of any company, and the data will be held as confiden-
tially given. It is our desire that the data may be given with all reasonable speed, and in view of the
great interest and importance attaching to the work, we trust all will make this matter one of special
care and attention, and employ in it sufficiently skilled labor to insure neat and accurate execution.
The cards should be carefully verified before being sent to the Committee, which will have no means
of correcting errors made at the offices of the companies.
The results of this labor will be reported to the Chamber at the earliest practicable moment, and
by it published so as to be accessible to all desiring it.
We believe it to be the general conviction among American and foreign Insurance officials and
actuaries that the material now capable of being supplied here will give results of peculiar value and
importance, and that American companies owe it to the business at large and to their own interests
that these should be collated and properly classified; and that it would be a cause of regret should
any Company fail to join in this work, which is not undertaken for the benefit of the Chamber alone,
or of the companies belonging to it, but for that of all companies doing business in this country. We
promise our best endeavors to make the character of the work worthy of its importance.
Please indicate at once to the Chairman at Hartford, Conn., the number of cards you may need,
and the probable time at which they can be completed.
Respectfully,
Jacob L. Greene, Chairman.
Edwin W. Bryant,
Robert A. Granniss,
Levi W. Meech,
Howell W. St. John.
The way had been opened for this investigation, by the London Institute of
Actuaries, which had collected the records of twenty Life Offices in England and
Scotland. But the strongest incentive, lay in the recent remarkable increase of Life
Insurance in the United States, creating a demand for a more systematic exposition of
insured experience. The following summary from the current Massachusetts and New
York State Reports, comprising the greater part of the Companies in the Union, may
serve to illustrate the magnitude of the business at this epoch.
In a general statement, the fifty thousand policies of 1859 had, in four years or
in 1863, increased to a hundred thousand. Then the rate of increase was temporarily
doubled, and within two years more, at the end of the year 1865, the number exceeded
two hundred thousand. In another two years, or in 1867, it exceeded four hundred
thousand. And in 1872, the number of policies had risen to eight hundred thousand.
By this time, the annual discontinuances began to balance the diminished number of
entrants; and at the epoch of the experience, Dec. 31, 1874, the total number exposed
to risk has been nearly stationary for five or six years, with about eight hundred
thousand policies in force.
The preceding Circular was accompanied by the adopted form of policy cards, and
instructions, which are given hereafter. In the final result, twenty-three Companies
returned the cards, filled out from their office records ; and seven gave the results of
their own investigations, brought down to the beginning of the year 1874. The latter
returns were accompanied by collateral explanations and results sufficient for verifying PREFA CF.
5
and resolving them into the original elements, in uniformity with the card reductions.
In this manner the entire returns were united ; and it was found that, of all the valid
experience, recorded upon the office registers in the United States, so favorable had
the enterprise been regarded, fully three-fourths had been contributed for this collection.
The following are the names of the Companies whose combined experience is given
in the succeeding pages, with the year of organization and aggregate number of policies:
1. Aetna Life, Connecticut, 1850
2. American Mutual, Connecticut, . . 1848
3. Brooklyn, New York, 1864
4. Charter Oak, Connecticut, .... 1850
5. Connecticut General, Connecticut, . 1865
6. Connecticut Mutual, Connecticut, . 1846
7. Continental, New York, 1866
8. Covenant, Missouri, 1853
9. Equitable, Iowa 1867
10. Equitable, New York, 1859
11. Germania, New York, 1860
12. Globe, New York, 1864
13. John Hancock, Massachusetts, . . . 1862
14. Life Association, Missouri, .... 1868
15. Massachusetts Hospital, Mass., . . 1823
Companies.
Organization.
16. Massachusetts Mutual, Massachusetts, 1851
17. Metropolitan, New York 1867
18. Mutual Lire, New York, 1843
19. Mutual Benefit, New Jersey, . . . 1845
20. National of United States, Illinois, 1868
21. New England Mutual, Massachusetts, 1844
22. Northwestern, Wisconsin 1858
23. Penn Mutual, Pennsylvania, .... 1847
24. St. Louis Mutual, Missouri, .... 1858
25. Travelers, Connecticut, 1866
26. Union Central Ohio, ...... 1867
27. Union Mutual, Maine, 1849
28. United States, New York, .... 1850
29. Washington, New York, 1860
30. Western New York 1868
Companies.
Organization.
Summary in 1874. Net Number op Policies or Lines.
Amount Insured.
Existing or not terminated,
549,418
$1,367,217,000
Discontinued, ....
431,568
1,159,867,000
Died,
46,548
130,224,000
Whole Number, .
. 1,027,529
$2,657,308,000
Exposed te Risk one Year,
. 4,504,797
$11,839,005,500
These figures represent the net numbers after more than a hundred thousand
secondary and other policies have been omitted or transferred according to the rules of
reduction. Thus the total Experience of the Thirty Life Offices has a net basis of
more than a million of policies, or lives, insured for more than twenty-six hundred
millions of dollars.
The arrangement of so extensive a series of tables has been guided, as far as possi-
ble, by the best established precedents, and mere innovation has been carefully avoided.
In several particulars, however, the system of Life Tables as described in that great
repository, the Journal of the Institute of Actuaries, was found to be still in a state of
progress. Different courses were indicated for determining the proper resultant between
the effects of the medical selection and the counter selection of the insured, such as the
omission of the first three or five years experience. Also shorter methods were demand-
ed for changing the standard life table to higher or lower grades of mortality. And
another desideratum was the condensation within a convenient compass of the volumi-
nous tables for term policies, endowments, and joint lives, to give monthly as well as
annual values. It has been much easier to describe, than to supply what is needed.
That an advance might be made to meet these difficulties, the actuary in charge has
given them all the consideration which other duties have permitted ; with what success
the examination of the following pages prepared by him, must determine. Among the
improvenemts to meet the necessities of the work, may be named, the principles of
transformation of Office data, the system of final series, the life table graded for different 6
PREFACE.
climates, the proof of the logarithmic law of mortality, the general valuation tables, the
arrangement of joint lives and survivorships, with new investigations described hereafter.
The first reductions of the Mortality Experience, were commenced in November,
1875 ; but the principal part of the returns arrived in the following summer, when the
number of clerks engaged in their reduction, was augmented from three to ten. Other
returns came in at later dates; and the last was received at the close of the year 1877.
With an adequate clerical force, the reductions were pressed forward as fast as the
returns arrived; and in March 1878, the first fruits appeared in the issue of copies of
the Climatic Tables for examination.
It may here be mentioned, that after the preliminary arrangements had been
initiated and were fairly in progress, two of the Committee withdrew from membership
though not from interest in the work, and their places were supplied in succession by
the election of Emory McClintock and George W. Phillips. The former actuary had
especially advocated the importance of the returns of losses and amounts insured ; while
the latter actuary, by elaborate office reductions, had prepared the way for the present
climatic statistics. For the position of chairman of the committee, Levi W. Meech was
chosen, who had at an early stage been designated to take charge of the returns, and to
make the proper reductions at the expense of the Chamber. And in May, 1878, after
the dissolution of the Chamber, John M. Taylor, Secretary of the Connecticut Mutual
Life Office, accepted the position of Treasurer.
In the medical department, the classification of diseases given by the cards, required
the services of a physician long conversant with office examinations, and the current
applications from different parts of the United States. Such an examiner was found in
J. C. Jackson, M.D., of Hartford, by whom the numerous returns of diseases were faith-
fully examined during several months, till the medical statistics were classified under
the most favorable conditions of accuracy and authority. It should also be mentioned
that the returns from two Western Life Companies, were generously furnished by the
late William E. Harvey, Actuary, after his own reductions. And in this connection,
for maintaining the highest attainable accuracy, thanks and commendation are due to
our clerical force, especially to M. C. Spring, S. E. Warner, and S. M. Hayward, assist-
ants. From the amateur collection courteously furnished by Mr. J. Downes, of Wash-
ington, D. C., we have copied the accurate table of anti-logarithms, reduced from the
eleven place values of Dodson, compared with Vega; and have added proportional parts
for the middle of each division, computed with three extra decimals.
To give confidence in employing the vital statistics, it may be proper to state, that
all the original returns had been copied under responsible direction, from regular Office
records, alike free from objection. The subsequent reductions by the Committee, have
been verified by two independent operations; and a further check was obtained by
comparison of the parallel columns of Lives and Amounts. The work has been nearly
six years in progress, under a single direction of large experience; and with favorable
conditions, has thus resulted in an authentic standard for reference.
Levi W. Meech, in Charge.
Edwin W. Bryant,
Emory McClintock,
George W. Phillips,
Howell W. St. John. CONTENTS.
FART FIRST.
ELEMENTARY OBSERVATIONS AND TABLES.
DIVISION FIRST. Mortality and Loss Experience.
SECTION PAGE
1. Instructions and Sample Cards 13
II. Adjustments of Cards. Duration 21
Method for Joint Lives, Transfers, Original Death Claims 22
III. Classifications of Data 23
IY. Calendar Years and Policy Years 24
V. Notation and Annual Equations - 25
VI. Transformation of Data 26
VII. Exposed to Risk 28
VIII. Comparative Results. Ratios of Mortality and of Loss to 1874 29
Influence of Medical Examinations and the Amounts at Risk 30
Insured Females 34
IX. Continuation of the Experience of 1874 to Final Series 35
X. Notes on Graduation. Short Method for the Expectation of Life 38
Graduation by Formula and by Divided Differences 40
Loss relative to Term and other Species of Insurances 42
DIVISION SECOND. Climatic Tables.
I. Computation op Probable Loss, with Tabular Factors 43
Change of Probable Loss from the standard of one Life Table to another 46
II. Classification op Counties. Miscellaneous Statistics 46
Climate Extra 48
III. The Life Table Graded for Different Climates 49
Climatic Formulas and Special Ratios for Tables 50 8
CONTENTS.
SECTION PAGE
I. Description of the Tables 52
II. Statistics of Consumption Compared 54
Sanitary Survey of the United States 56
Table of the Relation of Height and Weight 57
III. The Germ Theory of Malarial Diseases 57
Table of Sickness and Recovery in Adult Life 58
DIVISION THIRD. Medical Statistics.
DIVISION FOURTH. Time and Monetary Tables.
I. The System of Interest and Annuities 59
II. Time and Interest Tables for Days 60
Copy Multiplication 61
III. Compound Interest and Discount Tables. For Twelve Rates of Interest
CONTINUED TO 100 YEARS 64
Connection of the Tables 65
IV. To find the Rate of Interest of an Annuity 65
V. To find the Rate of Interest on the Price of Funded Stocks or Bonds 67
Table of the Auxiliary Factor 68
VI. Interest Compounded oftener than Once a Year 69
Correction of the Present Value of Half-Yearly and Quarterly Annuities 70
Illustrations of Compound Interest 71
P5 .A. P£, T SECOND.
LIFE ANNUITIES AND INSURANCES.
I. First Principles of Probability for Reference 233
Mathematical and Moral Expectation 235
II. The Life Table. Law of Mortality Established 237
Construction of Life Tables from the Final Series of Tables III and IV 237
Formula and Constants of the Life Table for Males 237
Fluctuating periods of health or Climacterics 238
Expressions for the relative Force and the Rate of Mortality 239
III. Present Value of Life Annuities 240
Temporary and Deferred Annuities. Joint Lives ... 241
Simpson’s Rule. Half-Yearly and Quarterly Annuities 242
Annuities payable to the day of Death 242
IV. Expectation of Life and Probable Life 242
The Curtate and the Even Expectation 243
Probable Life in Europe 244 CONTENTS.
9
SECTION
PAGE
V. Commutation Tables 244
Methods of Construction 246
VI. Single and Annual Premiums of Insurance 248
1. Formulas for Single Premium 249
2. Annual Premium 249
3. Five, Ten, or Limited Annual Premiums 250
4. Temporary Insurances 250
5. Pure Endowments 251
6. Endowment Insurances. 251
7. Deferred Insurances 252
8. Return Premiums 252
9. Increasing and Decreasing Scales of Premium 253
10. Increasing and Decreasing Insurances 254
11. Price of Annuities secured by Life Insurance 255
12. Insurance on Female Life 256
13. Insurance on Joint Lives 256
Method for Three or Four Joint Lives 258
Equalization of Annuities. Tables for Simpson’s Rule 259
14. Survivorship Insurances. Makeham’s General Solution 260
15. Insurances payable at Death. 16. Semi-Annual and Quarterly Premiums 264
VII. Reserve or Valuation of Policies 264
Months and Days. Reserve on Paid-up Policies 265
Valuation of Annual Premium Policies. Prospective Formulas 265
Valuation of Gross Premiums 266
Valuation of Annual Premium Policies. Retrospective Formulas 267
Elementary Formula of Valuation 268
Application to Return Premium and other Species of Policy 268
General Valuation Table. Adjustment for Months 269
Valuation Table for Joint Lives by Single Entry 270
Reserve on Survivorship Insurances, new process 271
Short Method for the equivalent Sum Insured by Paid-up Policy 271
Non-Forfeiture. Annual Valuation of an Insurance Office 272
Reference to Methods for the Distribution of Surplus 273
Illustrations of the Contribution Formula 274
Tontine Dividends 295
VIII. Life Annuities. Survivorship and Reversionary Annuities. Successive Lives. 275
Present Value of Single Life Annuities of different Species 275
Reserve or Valuation of Annuity Policies 276
Annuity extended to an extra payment; also to the day of Death 277
Present Value of an Annuity on the longest Life or last Survivor 277
Annuity on the last n Survivors of m Lives 278
Reversionary Annuities. Verification 279
Partition of Annuities. Conditional Periods 281
Survivorship Problems. Successive Lives 283
IX. Derivative Formulas. The Continuous Method 285
Annual Derivation. Tables of Addition and Subtraction Logarithms...: 286
Summation Formula applied to Annuities and Instalments 288
Change from Ordinary to Continuous Annuities and Premiums 290
Applications of the Differential and Integral Calculus 291
Total Probability of Survivorship between 'Two Lives 294
Remarkable Simplification of the Survivorship Problems 295
Notation 296
Interpolation. Constants 298 TABLES IIST FART FIRST.
DIVISION FIRST. Mortality and Loss Experience.
TABLE PAGE
I. Observations on Insured Male Life 77-123
Summary (A). According to Ages of Exposure 74
Summary (B). According to Years of Insurance 76
II. Observations on Insured Female Life 127-158
Summary (A). According to Ages of Exposure 124
Summary (B). According to Years of Insurance 126
III. Male Life (A). Mortality and Death Claims per cent, by Ages 159
Male Life (B). Mortality and Discontinued per cent, by Durations 161
Male Life (C). Logarithm of Multiplier for Final Series 162
Average Sum Insured 159-161
IV. Female Life (A). Mortality and Death Claims per cent, by Ages 164
Female Life (B). Mortality and Discontinued per cent, by Durations 166
Female Life (C). Logarithm of Multiplier for Final Series 166
Average Sum Insured 164-166
Mean Percentage of Mortality by the Year of Insurance 30
Male Life. Deaths and Claims per cent, to 1874 .. 31
Female Life. Death and Claims per cent, to 1874 32
Comparison from several Tables of Mortality 34
Comparisons of Experience to 1874 with Final Series 36-38
Short Method for Expectation of Life 39
Loss relative to Species and to Amount of Policy 42
DIVISION SECOND. Climatic Tables.
V. Probable Loss and Actual Loss by States and Territories 167
VI. Order and Variability of Climatic Ratios in thirty-five States 168
VII. Probable Loss and Actual Loss by Counties in twelve States 169-179
VIII. Climatic Commutation Table. Interest 4 per cent 180
Multiplier to give the Probable Loss 44
Classification of Counties 46
Age of Mean Rate of Probable Loss 47
Comparison of Northampton and Carlisle Annuities 49
DIVISION THIRD. Medical Statistics.
IX. General Table of Diseases and Deaths with Variability 182-185
X, (A) Males. (B) Females. Diseases and Durations of Policy at Death 186
XI. .(A), (B). Diseases and Ages at Death 188
XII. (A), (B). Diseases and Deaths by States and Territories 190
XIII. (A), (B). Ratio of Deaths to Exposed, by Duration of Policy 192
XIV. (A), (B). Ratio of Deaths to Living by Ages of Exposure 194
XV. (A), (B). Percentage Distribution of Disease by Duration of Policy 196
XVI. (B), (A). Percentage Distribution of Diseases by Ages of Exposure 197
Note.—The Formula of Computation is generally given on the first Page of each Table. TABLES.
11
TABLE
PAGE
XVII. (A), (B). Climatic Distribution of Diseases among 100 Deaths. Results of the Census 199
(C). Diseases and Deaths among 10,000 Exposed in each Group of States 201
A Sanitary Survey of the United States 56
Table of Relative Height and Weight 57
Table of Sickness and Recovery in adult Life 58
DIVISION FOURTH. Time and Monetary Tables.
XVIII. Calendar and Decimal Parts of a Year 202
XIX. Simple Interest for Days at 5 per cent 204-207
Copy Multiplication 62
XX. Amount of $1 at the End of any Number of Years 208-211
XXI. Present Value of $1 due at the End of any Number of Years 212-215
XXII. Amount of $1 per annum at the End of any Number of Years 216-219
XXIII. Present Value of $1 per annum for any Number of Years 220-223
XXIV. Annuity which $1 will purchase for any Number of Years. 224-227
XXV. Logarithm of the Present Value of $1 due at the End of any Number of Years.. . 228-231
Table of the Auxiliary Factor 68
Illustrations of Compound Interest 71
TABLES IN SECOND.
THIRTY AMERICAN OFFICES’ EXPERIENCE.
XXVI. Male Life Table. Expectation of Life and Probable Life *2
XXVII. Common Logarithms of the preceding Life Table *4
XXVIII. Life Annuities. Present Value (a) *6
XXIX. Annual Premium to insure $1000 for Life *8
XXX. Single Premium to insure $1000 for Life *10
XXXI. Reversion or Life Insurance equivalent to Single Premium of $1 *12
XXXII. Life Annuity which $1000 will purchase *13
XXXIII. Commutation Table and Logarithms, 3 per cent *14
XXXIV. Commutation Table and Logarithms, per cent *18
XXXV. Commutation Table and Logarithms, 4 per cent *22
XXXVI. Commutation Table and Logarithms, per cent *26
XXXVII. Commutation Table and Logarithms, 5 per cent *30
XXXVIII. Commutation Table and Logarithms, 6 per cent *34
XXXIX. Commutation Table and Logarithms, 7 per cent *38
XL. Commutation Table and Logarithms, 8 per cent *42
XLI. Commutation Table, 9 per cent *44
XLII. Commutation Table, 10 per cent , *46
XLIII. Common Logarithms for Special Formulas *48
XLIV. Life and Temporary Annuities, or Log (l + a^~1), 4 per cent *50
XLV. Life and Temporary Annuities, or Log (1 + «”_1), 41 per cent *61
XLVI. Life and Temporary Annuities, or Log 1 + r,r£~*), 5 per cent *72 12
TABLES.
TABLE FAGE
XLVII. Life and Temporary Annuities, or Log (1 46 per cent *83
XLVIII. Conversion Table from Annuities to Annual Premiums *94
XLIX. Conversion Table from Annuities to Single Premiums *99
L. Single Premium for Life or Reserve on 1000 by Months *104
LI. Logarithm of Single Premium by Months on $1 Insured *108
LII. Single Premium or Reserve for Term Insurance of 1000 *112
LIII. Annual Premium for Term Insurance with Valuation Factor *118
LIV. Single Premium or Reserve for Endowment Insurance *122
LV. Continued Annual Premium for Endowment Insurance of 1000 payable at Death
or Age 30, 35, , 75. Also X (P—7t) *135
LVI. Five Annual Premium for Endowment Insurance as above *138
LVII. Ten Annual Premium for Endowment Insurance as above. *141
LVIII. Continued Annual Premium for Endowment Insurance of 1000, payable at Death,
or at the end of 5, 10, ...., 40 Years *144
LIX. Five Annual Premium for Endowment Insurance as above *146
LX. Ten Annual Premium for Endowment Insurance as above *148
LXI. Limited Annual Premiums for Life Policies *150
LXII. Net Value or Reserve on Ten Payment Life Policies *151
LXIII. General Valuation Table for Annual Premium Policies *154
LXIV. Value of f and f for every Year and Month of Age *200
JOINT LIVES MALE.
LXV. Correction for reducing Two Lives or Three to the case of Two Equal Ages *204
LXVI. D and N Columns for Two Equal Ages, 3 per cent *200
LXVII. D and N Columns for Two Equal Ages, 3f per cent *207
LXVIII. D and N Columns for Two Equal Ages, 4 per cent *208
LXIX. D and N Columns for Two Equal Ages, 44 per cent. *209
LXX. D and N Columns for Two Equal Ages, 5 per cent *210
LXXI. D and N Columns for Two Equal Ages, 6 per cent *211
LXXII. Annuities on the Joint Continuance of Two Lives *212
LXXIII. Annual Premium for Insurance on Two Joint Lives *213
LXXIV. Single Premium or Reserve on Two Joint Lives *214
LXXV. Ten Payment Premiums with Valuation Factors *215
LXXVI. General Valuation of Annual Premium Policies on Joint Lives *216
LXXVII. Female Life Table. Expectation of Life *224
LXXVIII. Female Life. D, N, a, and Correction of Age. 4 per cent *226
MISCELLANEOUS.
LXXIX. Comparison of different Life Tables *228
LXXX. Expectation of Life by various Tables *230
Probable Life in Europe Text, 244
LXXXI. Percentage of Mortality by different Life Tables *232
LXXXII. Anti-Logarithms, or Numbers to Logarithms *234 PART FIRST.
ELEMENTARY OBSERVATIONS AND TABLES.
The Elementary Observations and Tables of Part First are classed into four
Divisions, as follows:
First. Mortality and Loss Experience, - - Tables I-IY.
Second. Climatic Tables, ------- “ Y-VIII.
Third. Medical Statistics, - -- -- -- « IX-XVI.
Fourth. Time and Monetary Tables - - - “ XYII-XXV.
DIVISION FIRST.
MORTALITY AND LOSS EXPERIENCE.
The received method of reduction presupposes the new business of different
calendar years to be superimposed or brought together, as if the entrant insurances
were all effected in one calendar year. Thus the entrants at the age of thirty, for
example, are all classified together, as entering in the same initial year. From this
origin, the annual deaths and discontinuances are traced forward and noted year by
year, through the whole course of insured experience. Such are the elementary
observations given in Tables I-1V, for male, and for female life. Let us first glance
at the methods from the beginning.
SECTION I.
Instructions and Cards.
The following Instructions of the Committee in March, 1875, were published
and attached to folio card-boards, for constant reference in filling out the blank cards,
from the Office records.
INSTRUCTIONS.
Let each figure and word be plainly written with a pointed
pencil of medium hardness.
The first blank, in the left hand upper corner, is for the con-
venience of the Committee in separating the cards, and will contain
their private mark for the several companies, and need not be
regarded by you,—nor the second, which contains the policy-num-
ber,—inserted consecutively by the printer’s numbering machine.
Private mark and
number of Policy. 14
MORTALITY AND LOSS EXPERIENCE.
Residence.
The third blank, “residence,” is to be filled with the name of
the State only, except that in the following States the County is
also to be inserted, viz.: Virginia, North Carolina, South Caro-
lina, Georgia, Florida, Alabama, Tennessee, Arkansas, Mississippi,
Louisiana and Texas.
In case the residence be in any foreign country, draw a line
through the word “State” and write the name of the country in
the same space.
The series of blanks relating to the amount, kind, and duration
of the policy are to be treated as follows:
In the top space, under “Amount,” write the amount insured,
thousands at the left hand, and hundreds at the right hand of the
light perpendicular line, using no ciphers, where even hundreds or
thousands are entered.
On the same horizontal line under “ Kind,” indicate the kind
of policy by L for Life, E for Endowment, and T for Term
Policies.
On the same horizontal line, under “Cal. Year of Entry,”
enter the calendar year in which the policy was issued.
If the policy be still in force, make a dash after the figures
“18” in the same horizontal line, under the “ Cal. Year of Exit.”
If it has ceased to be in force for any reason except death, enter
upon that line and under the “ Cal. Year of Exit,” the calendar
year in which it ceased, and make a dash in the blank “ Cause of
Death.”
If it ceased by death, make the same entries, hut carefully write
D after the year of exit.
If any changes have occurred in the amount or kind, the
original number, age and date being retained, other entries must
be made as follows :
The original amount, kind and year of entry having been
entered as above, the calendar year in which the change was made
is to be taken as the year of exit of the original policy, and entered
in the top line under that head.
Then an entry of the changed amount or kind, as the case may
be, is to be made under the proper head on the second line, and
the date of the change is to be taken as the calendar year of entry
of the changed policy, which will of course correspond with the
calendar year of exit of the original policy.
The termination by death or otherwise of the second or changed
policy is to be treated in the same manner as has been pointed out
for the original.
Should the changed policy have been still further changed, the
proper entries are to be made upon the third line, under the several
heads, in the same manner as required for the changes from the
original.
If more than two changes have taken place, the subsequent
ones may be entered in the bottom blank for memoranda.
Amount and Kind.
Tears of Entry and
Exit.
Exit by Death.
Changes in Amount
and Kind.
Exit by Death under
CHANGED Policies.
Second Changes.
Third and subse-
quent changes. MORTALITY AND LOSS EXPERIENCE.
15
If the policy in which change has been made, be in force, make
a dash after the figures “ 18 ” under the “ Cal. Year of Exit,” in
the horizontal line occupied by that policy.
If a policy has lapsed and has subsequently been revived under
the same number, date, age, etc., and the revival has taken place
in the calendar year in which the lapse occurred no entry of these
facts need be made.
If the revival has taken place in a later calendar year than
that in which the lapse occurred, then the calendar year of lapse is
to be entered as the year of exit of the original policy, and the
revival is to be treated as a change, and the proper entries are to
be made upon the second line, the calendar year of revival being
taken as the year of entry of the changed policy, and write_ the
letter E. in the memorandum blank at the bottom.
If revival or change has been made by a policy under a new
number, without a new medical examination, let the card for the
new policy contain in the memorandum blank the number of the
original policy also, with an 0 written before it.
In case policies after lapse remain good by their own terms for
a fixed or “pro rata” amount of the original sum insured, the
year of lapse is to be taken as the year of exit of the original, and
also as the year of entry of the changed or “pro rata” amount.
In case such policies, after continuing for a time for the “pro
rata ” amount, are revived under the same number, date, etc., such
revival is to be treated as a second change and entered accordingly.
The blank “Age at Entry,” is to be filled with the office age, or
that upon which the premium is written.
When for any special reason, the age has been rated higher or
lower than the true age, the true age at nearest birthday should
be stated in the memorandum blank with the words “ true age ”
before it.
The blank following “Age of Entry,” upon the same line, may
be disregarded, being for the use of the Committee only.
The blank “Sex” is to be marked with a dash for a male life,
and with a capital E for a female life. '
In the blank “ Cause of Death ” write the cause stated upon the
Company’s register.
The following special cases must be carefully noted:
Omit all policies “ not taken ” or not actually put in force.
Omit all policies of re-insurance, or for the benefit of other
companies.
In case of a “joint life policy” make a card in full for each
life, using a separate card for the second, but writing in the
number.
When changed Pol-
icy is in force.
Revival of lapsed
I'olicies.
Revival or change
WITHOUT new med-
ical examination.
Policies lapsed and
good for “ PRO RA-
TA ” amounts.
Revival of “ PRO
RATA" Policies.
Age of Entry.
“ Rated Age."
“ True Age."
Sex.
Cause of Death.
“Not Taken" Pol’s.
“ Re-insur." Pol’s.
Joint Life Policies. 16
MORTALITY AND LOSS EXPERIENCE.
Joint Life Policies.
Mark each card in the memorandum blank with a capital J,
and place them both in an envelope also marked J.
It may sometimes happen that a “joint life policy” is discon-
tinued as to one of the lives, and so adjusted as to continue upon
the other life singly.
The card representing the life upon which the insurance is dis-
continued will show the amount, kind, calendar year of entry and
calendar year of exit, in the usual manner of a terminated policy.
The card representing the life upon which the insurance is
continued will show the change in the same manner as other
changes are indicated, with the addition of the letter S in the
memorandum blank, signifying “ Single Life.”
No special designation need be made of policies with a limited
number of premium payments, whether completed or otherwise.
When a policy has terminated for any cause other than death,
the cause need not be stated, but be careful to make a dash in the
blank “Cause of Death.”
Limited premium
Policies.
Policies terminated
by causes other than
Death.
SAMPLE CARDS.
A. A A. \ x
State, (y'b'i&at'yie. State. dt'yie..
RESIDENCE. —————— RESIDENCE. —
County. County.
a . xt- . Cal. Year Cal. Year . , t-- j Cal. Year Cal. Year
Amount. Kind. of Entry. of Exit Amount. Kind. of Entry of Exit.
/0 2? 18 &4 18— /0 £? 18 64 18—
____________ _____I
18 18 18 18
| 18 18 18 18
Age at Entry. SO /0 Age at Entry. SO SO
Sex. Sex.
Cause of Death, Cause of Death. MORTALITY AND LOSS EXPERIENCE.
17
B.
State.
RESIDENCE. ——
County. c^Q-edd-t-tcc.
. , rr- a Cal. Year I Cal. Year
Amount. Kind. of Entry | of Exit
y 3 <§ V8640 I 1803
3 1803 I 18—
18 18
Age at Entry. S?
__ ■■ i
Sex.
Cause of Death.
°. \.
State. cJi& add.
RESIDENCE.
County.
. , - a Cal. Year Cal. Year
Amount. Kind. | of Entry of Exit>.
3 1833 186-4
3 j 1804 180^
/ j S 18—
Age at Entry. 23
Sex.
Cause of Death.
D. \
State. Cyf'htdd.
RESIDENCE. -
County. L/Jad^-uad.
. , rr- A Cal. Year Cal. Year
Amount. Rind. of Entry of Exit.
SO (§ 1803 18
18 18
18 18
Age at Entry. 42
Sex.
Cause of Death.
E.
State. S2/(Q)€i'ndfz<3.
RESIDENCE. —
County.
» * T7-- A Cal. Year Cal. Year
Amount. Kind. | of Entry of Exit-
.
6 1800 1803
s is is 72
18 18
Age at Entry. 34
Sex.
Cause of Death. (^di-earn-cdt-ta.
I 18
MORTALITY AND LOSS EXPERIENCE.
| F
State. CstffQtZ'i'U, 3a w c/.
RESIDENCE. ' " |
County.
A . xr- a Cal. Year j Cal. Year i
ot Entry. | of Exit.
3 i!/ 18/3(3 18(3(3
3 3 (§ 18 (3(3 18(3$
3 18 <33 18/3^
Age at Entry. 29
Sex.
Cause of Death. CjfgemX
Gr.
State.
RESIDENCE.
County.
* . tt. j Cal. Year Cal. Year
Amount. Kind. , of Entry of Exit>
/o g 18/3 187^
18 18
18 18
*
Age at Entry. 33
Sex.
Cause of Death. —
j H. 332 $3''X
State.
RESIDENCE.
County.
» . tt« a Cal. Year Cal. Year
Amount. Kind. ()f Entry. of Exit. j
3 3 j 18—
18 18
18 i 18
Age at Entry. 33
Sex.
Cause of Death.
/
IIII. 332Q3
: State.
RESIDENCE. ——
County.
. , Tt. , Cal. Year Cal. Year
Amount. Kind. of Ent of Exit.
1
3 3 183$ 18—
18 18
18 18
Age at Entry. 33
Sex.
Cause of Death.
| / MORTALITY AND LOSS EXPERIENCE.
19
I.
State, cddettt 1
RESIDENCE. —
County.
. . tr- a Cal. Year Cal. Year
Amount. Kind. of Entry | of Exit
/ I J dd \Z4# j 1877
18 18
18 I 18
—L—i
Age at Entry. c/7
Sex.
Cause of Death.
u
II.
State.
RESIDENCE.
County.
, Tr- j Cal. Year i Cal. Year
Amount. Kind. of Entiy | of Exit.
7 s dd \&4
idd idd ?y i.
RESIDENCE. 1
; County. JLcL on.
a , rr- a Cal. Year Cal. Year
Amount. Kind. Qf Entry of Exit.
18/J> 18
18 18
18 18
Age at Entry. S?
Sex.
Cause of Death.
(Q—vgyes
DESCRIPTION OF SAMPLE CARDS.
Each of the sample cards is marked with a letter of the alphabet—a single letter
for male lives and a double letter for female lives—the single and double letter cards
each showing the same data.
Thus, A shows a policy on the life of a resident of Maine, insured at age 30 for
$10,000; Life Policy; year 18G4; still in force; male life. AA shows precisely the
same data for a female life.
B shows a policy on the life of a resident of Virginia, Henrico County, for
$7,500; Endowment; issued 1860; changed in 1863 to a $3,000 Life Policy; still in
force; age at entry 37.
C shows a policy on the life of a resident of Massachusetts for $500; Life ; issued
in 1855 ; changed in 1864 to $3,000 Life; changed in 1869 to $7,500 Life; still in
force; age at entry 25.
D shows a policy on a resident of Bolivar County, Mississippi, for $10,000;
Endowment; issued in 1865 ; terminated by death in 1870; age at entry 42; cause of
death, Typhoid Fever.
E shows a policy upon a resident of Kansas for $5,000; Endowment; issued in
1860; changed in 1865 to a Life policy for the same amount; terminated in 1872 by
death from Pneumonia; age at entry 34.
F shows a policy upon a resident of Maryland for $3,000 ; Life ; issued in 1860;
changed in 1866 to $2,500; Endowment; and changed again in 1868 to $800 Endow-
ment ; terminated in 1873 by death from Remittent Fever; age at entry 29. MORTALITY AND LOSS EXPERIENCE.
21
0 shows a policy upon a resident of Vermont; issued for $10,000 ; Life ; m 1870;
terminated in 1872 for any cause other than death; age at entry 38.
H and HH are the proper cards for a Joint Life Policy upon residents of Michigan ;
male and female ; for $3,500 Life ; issued in 1859 ; still in force ; age of the male 34;
age of the female 32.
I and II are the cards of a Joint Life Policy upon residents of New York, issued
for $7,500; Life; in 1849; terminated by death of female in 1870,from Cerebro Spinal
Meningitis; age at entry of male 3G; age of female 30.
K and KK are cards representing Joint Lives; insured for $5,000 in 1860 ; and
from which the female life was dropped in 1870; the policy adjusted to continue as a
single insurance upon the male from that date.
L shows a policy upon the life of a resident of New Jersey for $5,000; Life;
issued in 1860; at age 25 ; lapsed in 1865; revived in 1866, and still in force.
M shows a policy upon the life of a resident of New Hampshire for $5,000; Life ;
issued in 1860; at age 30 : lapsed in 1865 ; continuing good by its own terms for
$2,500, and revived for the original amount in 1868 ; still in force.
N shows a life Policy still in force, upon the life of a resident of Jackson County,
Mississippi, issued for $5,000, in 1872, at age 37, without medical examination, to revive
or take the place of Policy No. 19,263, upon the same life, lapsed.
SECTION II.
Adjustment of Cards. Durations. Joint Lives. Transfers.
After an interval of four years, during which these Instructions have been followed
to their proper completion, the following additional remarks and specifications may he
noted.
The general plan has been guided by the adoption of two important precepts.
First, that every insurance must commence with a Medical Examination. Second,
that after the Medical Examination at the beginning, the method of reduction must
correctly trace the counter selections of the insured, which are continued from year to
year, by changes of the Amount Insured, that is, by increase or by decrease, by taking
out additional policies, or by total lapse, surrender, or decease. In this aspect, the
investigation might not inappropriately be termed the Problem of the Selections.
To accord with the latter precept, the English method of Lives, on the plan of
a census of living and deaths only, without the sums insured, has been supplemented
by parallel columns of the Amounts, treating dollars as lives. The present statistics,
therefore, are two-fold; and have required more than double the usual labor in their
preparation.
The cards, on reception, were first examined from beginning to end, to select out
all the secondary insurances, such as the preceding sample card N, marked 0 under-
neath. This particular entry 0—19,263 simply refers from the secondary to the
original insurance noted on card No. 19,263. The secondary or “O-cards” so called,
were next copied on the middle line of the originals, which were sought out for this pur-
pose, before the regular order of the cards by number was broken up. The O-cards,
being about one-tenth of the whole, together with Reversionary Dividends, were laid
aside as supernumerary. 22
MORTALITY AND LOSS EXPERIENCE.
After this and other preparations, presently described, the next step was to note
the Duration on each card of single insurance, which was found by subtracting the
first “Cal. Year of Entry” from the last “Cal. Year of Exit,” or (in case of Noil-
terminated or Existing policies) from the Epoch 1874, when the experience was inven-
toried. Thus on the. preceding cards A, and AA, the difference of 1864 and 1874, that
is, Duration 10, is noted opposite the age. We may observe that the sum of the age
and duration, or 30 and 10 on the card A, gives the advanced age of the insured,
40 years, in 1874. A more full account of the connection of Age and Duration will be
given hereafter.
With respect to cards having two or more successive insurances, the Durations are
noted, after preparation, in the following manner. To refer to a few instances:—Where
the amounts are alike as on the preceding card E, the entries may be regarded as two
parts of one experience, or 15,000 exposed to risk through a duration of 12 years, like
a single policy. Where the amounts are different, as shown on the preceding card F,
the above rules require its resolution into three new separately written cards, all
commencing with the medical examination of 1860; that is, (1) a Death card of the
last amount $800 for the whole Duration 13 years; (2) a Discontinued card of the
difference 2.500-800, or $1,700 from 1860 to 1868 or Duration 8 years ; and (3) another
Discontinued card of the difference 3,000-2,500, or $500 from 1860 to 1866 or Duration
6 years.
All the other similar cards B, C, M, can be superseded by single cards on the same
principle. Indeed, experience seems to indicate that separate cards are more convenient
in practice, than the three ruled lines of the present arrangement. Thus the card B is
to be resolved (1) into an Existing card of the last amount $3,000 for Duration
14 years, and (2) a Discontinued card of the difference 7,500-3,000 or $4,500 for
3 years.
On cards C, M, the lower amount always subtracted from the next preceding
amount over it, will give a negative difference. The corresponding cards of this
species are to be reserved for future subtraction from positive results of the same age
and duration. Thus, card C is to be superseded by (t) an Existing card of the last
amount $7,500 from 1855 to 1874 or 19 years; (2) a Discontinued card for the negative
difference—$4,500 from 1855 to 1869 or Duration 14 years; and (3) another Discon-
tinued card for the negative difference—$2,500 from 1855 to 1864 or Duration 9 years.
These cards (2), (3) are to be reserved for future subtraction as above mentioned.
These reductions will be best understood by clerks who are familiar with algebraic
operations.
Heretofore, the cards of Joint Lives have usually been reduced as single lives. But
the death of the first of a couple, is occasionally followed by the death of the second in
the same year; and such second deaths, although noted on the single life plan, escape
the record of joint lives. To correct for this omission, let q, q' or 1—p, 1—p denote
the probabilities that two persons insured for %a each will die within one year. As
single lives, the Expectation of Loss in one year will be a(q + q'). As joint lives, the
Expectation of Loss will be a(l—pp), or a—«(1—q) (1—q'), that is, a{q-\-q' — qq);
which may evidently take the form
aq{l—\q')+aq'{\—\q).
The last expression of Probable Loss may also represent the Actual Loss on Joint
Lives. Hence if each Actual Loss is divided by the six months probability (1 — of
the survivor, the results will correspond to the experience of single lives. Since the
average value of lq' is about \ of 1 per cent, we may for every 200 Death cards of
Joint Lives, select and change into a Death card one Discontinued card of the same MORTALITY AND LOSS EXPERIENCE.
23
sex, and of similar age, duration and amount. After this simple correction, first
devised in the present investigation, the pairs of cards of joint lives can be separated,
and the whole series united with the other cards for single lives.
Another practical artifice was employed to change a few thousand policies from
the plan of next birthday to the American custom of nearest birthday. Instead of
dating back each card of the whole series half a year, each alternate card was dated
back a ivliole year; since all the policies were for ordinary amounts, and opposite errors
would compensate each other. Another large collection of data with ages according to
last birthday, was dated forward on the same convenient principle.
There were also several thousand Transferred Policies, or policies which on the
winding-up of the Company that issued them, had been transferred to another Com-
pany. The transfer was accompanied with the original ages or years of Entry of the
Existing policies, but the corresponding Discontinuances and Deaths up to the time of
transfer, were not stated. Thus, suppose 100 or more persons to have entered the
Cadmus Life Company at the age of 30 years. Five years after, the Cadmus is closed
up, and the survivors at the age of 35, are transferred to a second Company. The valid
experiences commencing at age 35, or five years after the medical examination, should
enter Tables I and II, not on the first line, but on the fifth line below, at Duration 5.
This object can be accomplished by writing two cards for each transfer; the card
(1) beginning with the medical examination at the age of 30, and extending forward
into the returns of the second Compauy, as usual; and the card (2) beginning at the
same age of 30, to be discontinued after a Duration of 5 years. The latter is to be
reserved for future subtraction from positive Discontinuances of the same age and
duration. In this way, the first five years exposure of the card (1) will finally be can-
celled by the card (2), as was proposed.
In relation to Endowment Insurances, composed of two parts ; that is, of a
Temporary Insurance, and a Pure Endowment for the same amount; since the
maturity of the Endowments was not specified on our cards, their effect on the whole
experience is that of Temporary Insurance only.
Death Claims reduced by compromise or by litigation have been restored to the
original amount, as the uniform standard of this investigation. In the case of a single
Company where the amount actually paid, had been given in lieu of the normal Death
Claims, a small correction of about 2|- per cent, was applied for compromised and
litigated claims, taken together.
Having thus represented the whole series by single cards, each commencing with a
medical examination, and having noted on each of them its proper Duration, we apply
the advantages of the card system as follows.
SECTION III.
Classifications of Data.
The next operation was to separate the cards of Male Life from those of Female
Life. Each of these groups was then divided into the three classes of Existing,
Discontinued, and Died.
Each of these classes was then sub-classified according to the Durations of Policy,
0, 1, 2, 3, 4, And these were next sub-divided into smaller packages of cards
corresponding to the Age of Entry. The counting of cards in these packages, gave
the Existing, Discontinued, and Died, of Lives in Table I and II. And the addition
of the corresponding Amounts, by the same packages, gave the parallel columns of 24
MORTALITY AND LOSS EXPERIENCE.
Existing, Discontinued and Death Claims. The latter were first added in full, and then
changed to the nearest Thousands of Dollars, omitting the last three residual figures.
For every Age of Entry, the sum of the Existing, Discontinued, and Died, expresses
the total Number of Entrants at the head of the columns. For example, at the age
of 27 years, on Page 90; 36,221 had entered, and were insured for the initial amount of
$87,030,000. Of these policies, 18,328 were still in force, or existing at the epoch in
1874; also 16,790 had been discontinued; and 1,103 had died, as noted at the foot of
the columns. Of the Deaths, the largest number in any one year, and more than
one-third of the Discontinuances had occurred in the second year, shown opposite
Duration 1. This feature is common to all the Companies. The first ten or fifteen
years Duration comprise the greatest part of the business; and the whole is practically
limited to thirty years; although a few cases are of older date.
SECTION IV.
Calendar Years and Policy Years.
For greater convenience, as before mentioned, the new business of different cal-
endar years is superimposed, for this investigation, as if all the insured entered the
Company uniformly during one initial year. On the average they enter at the middle
of this year; and so the average are exposed for only the latter half of the first calendar
year. The Exits as well as the Entrances of subsequent years, are likewise assumed
to occur uniformly during each separate calendar year. Average Policy Years thus
begin and end with the middle of Calendar Years ; and the average birthday is taken at
the middle of the initial year.
The ratio of mortality for the first year has been a subject of discussion in the
Journal of the Institute ; but, as there stated, “it is difficult to devise a better plan
and in practice, the regular formula for subsequent years has been generally applied
to the first year, as will be presently exhibited.
Throughout the Tables I and II, the first column headed “Years of Insurance,”
0, 1, 2, 3, etc. expresses the Durations found by subtracting the Year of Entry on the
cards from the Year of Exit or termination. For future reference, the more definite
scale, 0—1| —2 J-, etc. is also given in the right hand columns of Summary
A and B, at the beginning of Tables I and II. As before described, assuming the
entrances to occur uniformly during the twelve months of the initial calendar year,
the average date of insurance or entry is taken at the middle of the year, when the
“Years of Insurance ” begin, as follows. The title “Years of Insurance” is synony-
mous with “ Durations” and with “Policy Years.”
Calendar
Years.
Years of
Insurance.
Existing, Discontinued, Died.
1 Beginning
■] Middle
Beginning
0
Insurances commence.
Average date of Entrance.
Middle
i
First Year’s “Existing” record closes.
f Middle
End
1
Average date of Deaths and Discontinued.
■j Middle
Beginning
1
Survivors enter on second Policy Year.
Middle
1J
Second Year’s “Existing” record closes.
( Middle
End
2
Average date of Deaths and Discontinued.
■j Middle
Beginning
2
Survivors enter on third Policy Year.
( Etc
Etc
2$
Third Year’s “ Existing ” record closes. MORTALITY AND LOSS EXPERIENCE.
25
SECTION V.
Notation and Annual Equations.
Let N = number of admissions during any year of Age.
D = “ “ Deaths during any year of Age.
r = “ “ Discontinuances during any year of Age.
e — “ “ survivors entering on any year of Age.
R = “ “ Existing at the end of the year.
E], D„ ru £„ R[ = the like numbers for the next higher year of age, etc. These
are here adopted from the short method of W. S. B. Woolhouse in the Journal of the
Institute, Yol. 13, page 10.
M = e + p-_*(r + D) = number exposed at the middle of the year, if the changes
occur uniformly. M can also denote the middle altitude of a trapezoid, whose base is
unity or one year, and whose area represents the total exposure of the year. Assuming
half the annual decrement to occur before, and the other half to occur after, the
middle of the year, we have,
p = = the proportion that survive one year’s exposure to Death.
-l,'f’
p’ = ——~ = the proportion that survive one year’s exposure to Discontinuance.
p = . —fj-—jy- = = the proportion that survive one years exposure to
+ 2(; + ) £ both Death and Discontinuance.
q = = —P ~ ProPortion that die in one year.
0 = M+-|D=e + £N'—\r~ Exposed to Risk [of Death]. The last denominator
M + |D or 0 has been named the “ Exposed to Risk,”
although it differs from the mean exposure M shown
above.
a = ——R—r, for the initial year of Entry or Duration 0.
1 — J — qr?—7— = «—i — proportion that discontinue in one year.
L M + £r 0 + l-r—i D 1 1 J
T D T -f- D
1 — p" = —-7 = 75—5— = proportion of both deaths and discontinuances
1 M+i(r + D) 0 + \
’ m one year.
For Duration 0,
ii U 1
“ “ 2,
“ “ 3,
etc.
e0 = N= Sum of (R + D + r),
ei = N—(R0 4- D0 + r0),
e2 = ei — (Ri 4- D, + ?’l),
e3 — e2 — (Da + D2 r2),
etc.
0O ?’0),
01 — - ei
02 = £2—
03 - ez—\r3,
etc.
By eliminating e, and by transposition, or substitution,
0i = 2/30—R0—Do jri,
02 — 0i — Rl — Dj jrl’i—jV2,
03 = 02—Do—Do -|t2—-§r3,
etc. etc. 26
MORTALITY AND LOSS EXPERIENCE.
Ro — Wo—Pi— Dy—\r 1,
Ri = Pi—P2—l)i—jtr1—hr2>
etc. etc.
M0 = -|-N—|(r0 + D0),
M, = 2M0—R0—i(ri + Dj),
M2 = Mj—Ri + rj)—^(D2-f-r2),
M3 = M2—R2—^(D2 -f- ?’2)
etc. etc.
A method for carrying forward the Existing to their proportional terminations in
the columns of Discontinued and Died, will be described hereafter in Section IX.
SECTION VI.
Transformation of Data.
The foregoing equations have been particularly serviceable in transforming the
results, which had been wrought out by different Actuaries and Companies, to a
uniform system. A brief memorandum of some of the operations may be useful for
future reference. The Experience of one, designated as Company A, gave the values
of N, p, and D for Lives at every age, with but an abstract of the values of r. After
the latter had been interpolated for all ages, the Existing were then found from the
above equations of R.
Having now obtained the elementary data for Lives, the omitted Amounts were
next supplied by proportions wrought out for the Existing, Discontinued, and Died,
separately at each age. Thus, taking the first two terms from the general experience
of the other similar Companies, we have the simple statement, as R of the Lives is to
R of the Amounts, so is R of the Lives in Company A, to the corresponding Amount
at the same age.
The returns from another, which may be designated as Company B, gave the
Deaths and Exposed to Risk by Policy years, accompanied by the following explanation.
“The classification is by Calendar Years of Issue and Policy Years after Issue. In
regard to fractions of years, a policy for $10,000 lapsed or discontinued at the end of
years stands pro rata in the fourth year as $5,000, etc. If a policy was terminated
by Death, it was entered as at risk for the whole year, and entered as a death for the
whole amount.” In order to change to the forms of the card system, the data P, D
in thousands of dollars were first added as follows. Here P denotes the Amount
Exposed to risk in one Policy year, and D the corresponding Death Claims.
Age
of
Entry.
Values of P. 1st Policy Tear.
1809
’70
’71
’72
’73
’74A
P.
14
1
6
7
15
5
1
6
16
17
5
5
1
1
12
18
6
1
8
15
19
5
3
7
15
Age
of
Entry.
P.
2d Policy Year.
1869
’70
’71
’72
’73 A
P.
14
1
6
7
IS
16
S
5
17
1
1
18
S
1
8
14
19
3
3
8 MORTALITY AND LOSS EXPERIENCE.
27
For any one Age of Entry, as 18 years, let D'„ P„ denote the Deaths and the
Exposure of the 1st Policy Year; D'2, P2, the like quantities for the 2d Policy Year;
and so on. Let us assume the entrants to begin, on an average, in the middle of the
initial calendar year. Then if the deaths occur uniformly in each year, and D0, D1?
D2, . .. denote the Deaths on the card system, we have
D0 — i; Dj — ! + -|D 2; D2 — 2 -)- 4D 3; etc.
In like manner if R0, Rl5 etc. denote the Existing on the card system, at the end of
Calendar years or the middle of Policy years, and Deaths were included as above
described, we shall have by the middle ordinate of the trapezoid denoting the exposure
of the policy year, restricted to the last columns denoted by ’74A, ’73A, ’72A, ...
R0 = Pi—fD'i; Ri = Po—£D'S; Ro = P3—IP's; etc.
Lastly if r0, rx, r2, denote the Discontinued on the card system, r0 only is to
be estimated, being about one-half, or rather 0.45 of rx. It is evident that discontin-
uances and lapses will occur mostly at the end of policy years, that is, at the middle of
calendar years, on the average. We also note that by adding and subtracting 41)j, the
middle ordinate Pt— £D\ becomes P„ at the beginning, and P, — D', at the end of the
policy year. Consequently P2 is the value at the beginning of the next year; and the
Existing R0 were withdrawn (at the end of the calendar year) six months previous.
Taking the difference, we have
rx = Pj — D'j—R0—P2; r2 = P2—D'2—Ri—P3; etc.
After transformation of the Amounts given by Company B, in accordance with
these three series of equations, investigated for the purpose, the corresponding normal
number of Lives or Policies was then obtained conversely from the principle of pro-
portions employed for Company A.
The records of a third Company, denoted as Company C, gave the Deaths and
Exposed for both Lives and Amounts, according to Calendar years of exposure, with
correction for fractional parts of the year. The age nearest to Jan. 1st had been taken
as the Age of Entry. After the first Policy on any life, the subsequent policies or
surrenders, had been regarded merely as increase or decrease of the first Policy. By
aid of further statistics, the stated Loss actually paid, was corrected to the Loss as
insured. The lives were assumed to enter the Company at the middle of the initial
calendar year, on an average, and in this sense, at their real ages. The results annually
wrought out by the Office, had been given in the form of summation tables of the
quantities N, M, D, R, which, besides determining the final ratios of mortality, were
sufficiently extensive for other purposes. In order to unite with the returns of other
Companies, as here presented, it became necessary to resolve the results into their
original or annual elements. A single example will illustrate the special process
devised for this object.
Years of
Probable
Ages.
Insurance.
Deaths
34
35
36
37
38
0
1 3 6.33
1
258.8 6
2
2 3 0.55
3
212.97
4
198.11
5
1 79.33
Probable Deaths.
2,283.48
5 3 50
55.37
57.83
60.04
61.33 28
MORTALITY AND LOSS EXPERIENCE.
The horizontal and vertical sums were thus given, and it was required to fill out
the series of squares, with such compatible numbers as should make up these sums.
Instead of resorting to equations of condition for five year periods, the squares were
first filled out approximately, from the Experience of another similar Company. Their
Vertical sums, when compared with the true sums, gave a multiplier for each column,
such that the vertical sums of products agreed with the true sums, shown at the foot of
the columns. Next, the same products were added horizontally ; their sums when
compared with the actual sums in the left hand column, gave a new set of multipliers,
such that the horizontal sums of the new products agreed with those stated in the left
hand column respectively. The same products were next added vertically, to get new
multipliers and products as first described, and so on. It proved to be a converging
process, which terminated when the last multipliers became virtually 1; and the joint
conditions of horizontal and of vertical summation were satisfactorily fulfilled. The
resulting Probable Losses when divided by the same tabular probabilities previously
used as multipliers to determine the Probable Loss, gave the corresponding values of
M as required.
After this leading determination, different courses were suggested. Approximate
values of D could readily be found by applying the known ratios of mortality to the
values of M ; and approximate values of R and r could then be derived from the
foregoing Equations for M (page 26), by assuming the ratio of R to r to be the same
as in the general experience of other similar Companies, at the same age and duration.
Each equation would then have only one unknown quantity. And these approximate
values could lastly be made to accord with the given sums by slight percentage
corrections. But the work being already performed, the present sketch sufficiently
indicates the methods of interpolation. The new converging process, which opened
the way as above described, could be applied more extensively, as where the sums of the
columns had been taken in the two directions vertically and diagonally.
SECTION VII.
Exposed to Risk.
After the original data had been reduced to the three forms of Existing, Discon-
tinued, and Died, shown in Tables I and II, the next process was to fill out the
columns of Exposed to Risk [of Death]. This peculiar term has already been defined
under Notation, by the quantity (3, or M -f 41).
To explain the application,—in Table I Male Life, Age at Entry 33 years, for
example, the Number of Entrants 40,498 at the head of the first columns, was found
by adding the first three sums at the foot of the columns. And the Entrant Amount
Insured $104,474,000 was found in like manner. Then by the preceding equations
for fi (page 25), half the number of entrants less half of 1,872 the discontinuances of
the year 0, gives 19,313 years of exposure in the year 0. And the number of entrants
N diminished by the sum of Existing, Discontinued, and Died of the Year 0, gave the
surviving entrants *ej, on the year 1. And e, diminished by half the discontinued
6,484 of the year 1, gave j3, or 32,927 Exposed to risk during the year 1. After
the first year, the process is uniform, as follows. MORTALITY AND LOSS EXPERIENCE.
29
FROM TABLE I. AGE AT ENTRY 33 YEARS.
Year of
Insurance.
Entrants less Existing,
Discontinued, and Died.
Surviving
Entrants.
Subtract
£ Disc’d.
Exposed
to .Risk.
0
4(40,498)
£(1,872)
19,313
1
40,498—(2,349 + 1,872 +108)
36,169
1(6,484)
32,927
2
36,169—(2,181 + 6,484 + 219)
27,285
to
00
Oi
25,8624
3
27,285-11,776 + 2,845 + 218)
22,446
£(1,856)
21,518
In the columns of Amounts, the Exposed to Risk [of Death] are found in the
same manner. Thence, the ratios of mortality, and of discontinuance, or of both
combined, can be determined by the preceding formulas. The collective data will be
found unusually complete for these, as well as other inquiries.
The preceding principles require that, after passing from Deaths to the Ratios of
mortality, or to the life table, the current Ages on the plan of “nearest birthday” are
to be diminished by \ year, as if changing the Ages from the middle to the beginning
of the year of the ratio. Thus current is changed to 29 years. The English
custom of age “next birthday” would require a similar, final change of one year,
reducing the current age 30 to 29, for example, as noted in the first volume of the
Institute Experience, page 18.
SECTION VIII.
Comparative Results. Ratios of Mortality and of Loss
to the Year 1874.
The collection of the Thirty Life Offices represents an aggregate of about 1,177,000
original policies. But under the rules of reduction, before described,these were adjusted
to the net number stated below, comprising above a million of regular observations. It
will be seen that the number of female entrants was relatively small; the total propor-
tion being twenty-two males to one female insured. The ages of entry group themselves
above and below the mean age ol‘ thirty-live, the half of “ three-score years and ten.”
Numbers.
Males.
Females.
Total.
Number of Entrants
982,734
44,795
1,027,529
Whole number of Deaths
44,485
2,058
46,543
Average Age of Entry for Lives, vears
35.23
34.46
35.20
“ “ “ “ Amounts, years. ...
36.27
34.87
36.23
Proportions :
Existing.
Discon-
tinued.
Died.
Total.
Of 982,734 Male Entrants
53.7
41.8
4.5
100.0
“ 44,795 Female Entrants
49.7
45.7
4.6
100.0
“ 1,027,529 Male and Female Entrants
53.5
42.0
4.5
100.0 30
MORTALITY AND LOSS EXPERIENCE.
Dividing now the corrected number “ Exposed to Risk ” by the whole number of
Entrants, we find at the epoch of the Experience in 1874, the Mean Duration of
Policies was 4.36 years. Adding this to the Average Age of Entry 35.23 years, we find
that the Lives Insured had attained the Average Age of 39.6 years at the epoch. In
the older Experience of the Twenty British Offices, the Mean Duration of Policies at
the epoch in 1863 was 9.12 years; the average Age of Entry was 35.3 years; and the
average age attained by all the lives insured, was 44.4 years. Resuming the Experience
of the American Offices, we have next the following results.
MEAN PERCENTAGE OF MORTALITY ACCORDING TO THE YEAR OF INSURANCE.
ALL AGES. TO 1874.
Year
of
Insurance.
Male
Life.
Female Life.
Year
of
Insurance.
Deaths
Per cent.
Claims
Per cent.
Deaths
Per cent.
Claims
Per cent.
0
0.63
0.63
0.86
0.93
0
1
0.81
0.85
1.08
1.12
1
2
0.92
1.00
1.06
1.09
2
3
1.00
1.06
1.14
1.27
3
4
1.09
1.19
1.16
1.32
4
5
1.13
1.23
1.28
1.43
5
6
1.17
1.27
1.40
1.54
6
7
1.22
1.34
1.33
1.35
7
8
1.20
1.27
1.06
1.16
8
9
1.23
1.25
1.73
1.54
9
Total,
1.03
1.10
1.16
1.23
Total.
The preceding abstract, from Tables III and IV (B), shows how many die out of
100 Exposed to one year’s mortality, the average being above one per cent. In the
“Year of Insurance” 0, as before explained, the observations extend, on the average,
over half a year from the day of entry, although the percentage is the annual rate.
The “Year of Insurance” 1 extends from £ a year to years from entry; and so on.
The principal effect of the medical selection is more correctly exhibited on Pages
31-33, for Equal Ages of Exposure ; the average rate being generally attained within
2|- years. At the same time, the counter selections of the insured are proceeding by
changes of amount, so that the final claims are generally about four per cent, in excess
of what would be the loss for equal Policies or amounts insured.—Compare p. 192.
The computation of the following summary for Male Life may be thus illustrated.
For “Years of Insurance” 0, for example, we add the Deaths and Exposed in Table I
for Ages of Exposure 30, 31, 32, 33, 34, 35 Years, which give 89, 90, 95, 108, 99, 94, or
a total of 575 Deaths, and in like manner 115,458.5 Exposed to Risk. These Deaths
divided by the Exposed give 0.498 per cent, as stated in the Table on next page. The
six middle Ages current,strictly represent the period 29|-35-|; the central five years
of which are 30-35 years; the fraction being excluded only to facilitate future
comparisons. From the true middle age 32£, dropping £ year leaves the middle initial
age 32, as before specified. It will be seen that the ages bordering on the five-year
periods, as 25, 30, etc. are added twice. The ratios of Death Claims were found in the
same manner, subject in case of future graduation to a small correction shown in
Section X, 3. MORTALITY AND LOSS EXPERIENCE.
31
For the summary of Female Life, a parallel method was employed; the data being
first added in regular five-year periods, such as 30, 31, 32, 33, 34; then 35, 36, 37, 38,
39, etc., or 29-|— 34£; 34-|-39£; etc. To advance these periods \ year, that is, to
30-35; 35-40, etc., one-tenth of the increase from one quinquennial to the next, was
added to the former ratio. Thus for the periods (0) 29?,—34-?,-, the ratios of
Deaths per cent, are 27-f-3956 and 36-T-4258 or .6825 aud .8455. One-tenth of their
difference .0163 added to the former gives .6988 or .699 as stated in the abstract on next
page under 25-30 at the head of the column. As a method of adjustment, there
appears little or no ground of preference between this and the former operation for
Male Life, with a fair measure of regularity in the data.
MALE LIFE. DEATHS AND CLAIMS PER CENT. TO 1874.
Years
oe
Ages of- Exposure.
15-
-20
20-
-25
25-
-30
30-
-35
Insurance.
Deaths
Claims
Deaths
Claims
Deaths
Claims
Deaths
Claims
PER CENT.
PER CENT.
PER CENT.
PER CENT.
PER CENT.
PER CENT.
PER CENT.
PER CENT.
0
.735
.597
.571
.446
.481
.442
.498
.485
1
.673
.625
.677
.620
.607
.603
.617
.613
2
.802
.746
.782
.846
.691
.735
.691
.764
3
.034
.544
.685
.626
.777
.737
.744
.758
4
.590
.776
.740
.719
.811
.868
.833
.854
5
.400
.421
.899
1.038
.796
.885
.827
.914
6
1.288
.305
1.223
.650
.879
.905
.801
.882
7
2.116
1.227
1.003
1.209
.860
.879
.837
.827
8
.306
.548
.812
.897
.799
.856
9
1.022
.337
.561
.459
.801
.841
0—4
.709
.636
.672
.620
.647
.645
.666
.683
5-9
.725
.432
.967
.910
.823
.879
.817
.879
10-14
1.890
1.572
1.295
.850
1.037
1.194
15-19
.862
.386
1.243
1.634
20-24
2.778
1.105
Total,
.709
.627
.689
.634
.671
.673
.705
.732
35-
-40
40-
-45
45-
-50
50
55
0
.520
.528
.663
.666
.840
.802
1.186
1.354
1
.709
.831
.845
.839
1.034
1.031
1.322
1.309
2
.785
.871
.931
.942
1.126
1.107
1.442
1.495
3
.852
.884
.958
.950
1.096
1.118
1.403
1.525
4
.920
1.000
1.019
1.108
1.184
1.252
1.469
1.512
5
.865
.956
1.045
1.124
1.185
1.206
1.392
1.424
d
.914
.963
1.028
1.071
1.145
1.241
1.557
1.489
7
.946
1.003
1.007
1.038
1.127
1.291
1.506
1.611
8
.789
.843
1.013
1.020
1.099
1.102
1.525
1.517
9
.869
.918
.900
.961
1.036
1.014
1.568
1.398
0—4
.758
.829
.892
.905
1.069
1.074
1.379
1.442
5—9
.883
.948
1.013
1.061
1.133
1.194
1.497
1.488
10-14
.933
.936
1.007
1.015
1.185
1.309
1.423
1.467
15-19
.929
1.006
.903
1.031
1.134
1.212
1.379
1.453
20-24
2.201
2.010
1.135
1.566
1.019
1.067
1.460
1.567
25-29
3.150
6.349
1.393
2.534
1.828
2.653
Total,
.802
.866
.938
.962
1.102
1.138
1.426
1.469 32
MORTALITY AND LOSS EXPERIENCE.
MALE LIFE. DEATHS AND CLAIMS PER CENT. TO 1874.
Years
Ages of Exposure.
of
55-
O
CD
1
O
CD
65
65-
70
70-75
Insurance.
Deaths
Claims
Deaths
Claims
Deaths
Claims
Deaths
Claims
PER CENT.
PER CENT.
PER CENT.
PER CENT.
PER CENT.
PER CENT.
PER CENT.
PER CENT.
0
1.550
1.739
2.425
2.429
3.200
5.010
4.138
3.003
1
1.851
1.806
2.721
2.703
3.142
2.713
4.902
5.066
2
1.887
2.071
2.644
2.423
3.460
3.821
3.663
4.484
3
2.021
2.113
2.867
2.886
3.610
4.169
3.338
2.193
4
2.158
2.018
2.913
3.164
3.689
3.899
3.464
3.136
5
2.077
2.117
3.008
2.820
3.811
3.797
6.452
4.626
6
1.992
2.121
2.761
3.402
3.706
4.341
6.656
6.849
7
2.079
2.121
2.903
3.153
4.631
4.285
4.928
4.937
8
2.121
2.179
2.585
2.727
3.819
3.339
5.405
6.438
9
2.074
2.060
2.724
3.074
3.537
4.140
7.355
7.901
0—4
1.926
1.968
2.756
2.767
3.482
3.809
3.726
3.437
5—9
2.064
2.122
2.820
3.043
3.921
3.998
6.109
6.027
10-14
1.824
1.920
2.619
2.583
3.794
3.819
6.014
5.531
15-19
1.875
2.037
2.681
3.005
4.211
4.624
5.037
4.934
20-24
1.790
2.053
2.775
3.214
4.523
5.494
6.131
6.396
25-29
1.727
1.708
2.871
2.940
3.727
4.098
6.061
6.145
Total,
1.942
2.017
2.756
2.899
3.934
4.236
5.631
5.550
75-
O
00
80-85
iO
GO
CD
O
All Ages.
0—4
5.096
5.150
14.286
9.524
.881
.937
5—9
8.480
8.066
3.636
1.550
1.179
1.267
10-14
9.707
8.808
9.677
9.160
1.171
1.433
15-19
7.942
7.119
11.834
13.598
4.445
5.333
1.679
1.798
20-24
8.284
6.943
12.012
9.825
21.239
25.478
2.339
2.582
25-29
10.336
9.848
11.282
9.773
20.779
18.065
2.943
3.173
Total,
8.483
7.724
11.332
10.342
16.667
17.757
1.028
1.096
FEMALE LIFE. DEATHS AND CLAIMS PER CENT. TO 1874.
Ages of
Exposure.
Years
of
Insurance.
15-
20
to
o
-25
25-
CO
O
CO
O
CO
Or
Deaths
Claims
Deaths
Claims
Deaths
Claims
Deaths
Claims
PER CENT.
PER CENT.
PER CENT.
PER CENT.
PER CENT.
PER CENT.
PER CENT.
PER CENT.
0
.808
.784
1.062
1.042
.699
.621
.840
1.104
1
.585
.463
.952
.944
1.056
1.004
1.076
1.148
2
.600
.396
1.049
1.031
1.019
1.091
.946
.814
3
1.415
2.307
1.122
1.383
1.023
1.113
1.139
1.292
4
.993
.526
1.047
1.561
1.018
1.085
5
.380
.455
1.175
1.054
1.010
.882
6
1.894
.924
1.065
1.231
1.372
1.491
7
1.020
1.193
1.184
.959
1.023
.713
8
1.916
1.674
2.083
6.180
1.250
1.206
9
2.071
1.388
0—4
.714
.707
1.024
1.017
.971
1.015
1.009
1.080
5—9
.991
.764
1.180
1.419
1.199
1.081
10-14
.949
2.110
1.013
.948
15-19
3.310
3.124
Total,
.676
.671
1.108
.999
.997
1.065
1.052
1.084 MORTALITY AND LOSS EXPERIENCE.
33
FEMALE LIFE. DEATHS AND CLAIMS PER CENT. TO 1874.
Years
Ages of
Exposure
of
Insurance.
35-
-40
40-
-45
45-
-50
50-
55
Deaths
Claims
Deaths
Claims
Deaths
Claims
Deaths
Claims
PER CENT.
PER CENT.
PER CENT.
PER CENT.
PER CENT.
PER CENT.
PER CENT.
PER CENT.
0
.795
.946
.838
.955
| .975
1.013
1.193
.842
1
1.157
1.322
1.101
1.007
.899
.763
1.064
1.545
2
.917
1.080
1.009
1.027
.806
.724
1.496
1.570
3
.916
.785
1.108
1.159
1.239
1.399
1.179
1.470
4
.789
.780
1.145
1.197
1.124
1.563
1.550
1.524
5
1.086
1.145
1.090
1.390
1.393
1.533
1.804
1.943
6
1.264
1.383
1.097
1.027
1.287
1.656
1.601
1.627
7
.937
1.078
.909
1.118
1.497
1.332
1.427
1.470
8
1.509
1.531
.410
.479
.351
.233
.954
.976
9
.913
.868
1.456
1.646
1.257
.963
1.941
1.481
0—4
.946
1.030
1.051
1.064
.997
1.052
1.295
1.441
5—9
1.142
1.221
1.000
1.147
1.239
1.312
1.568
1.594
10-14
.843
.839
1.294
1.435
1.027
.994
1.189
1.153
15-19
2.440
3.128
1.031
1.252
.830
.766
1.299
1.509
20-24
.940
.678
1.116
.986
.826
.590
25-29
2.564
1.786
Total,
1.002
1.082
1.049
1.104
1.059
1.103
1.341
1.432
55-
-60
60-
65
65-
70
70-75
0
1.388
1.932
28.571
18.182
1
1.492
1.964
1.957
1.541
4.032
2.899
2
1.862
2.149
3.434
3.301
4.496
6.344
6.250
5.000
3
1.351
1.962
2.125
2.438
4.445
8.375
4
1.898
1.743
3.173
4.035
5.489
8.595
3.279
1.482
5
2.067
2.521
3.479
5.128
3.509
4.301
6
1.936
2.411
1.651
2.953
5.368
4.989
10.127
14.595
7
2.867
2.842
3.042
3.603
6.017
6.049
2.899
5.042
8
1.145
0.878
2.406
1.989
3.429
1.250
9
2.035
1.242
2.160
2.316
7.429
8.154
17.168
10.222
0—4
1.601
1.950
2.418
2.590
4.211
6.038
4.407
3.100
5—9
2.057
2.215
2.650
3.585
4.799
4.722
5.458
6.171
10-14
1.621
1.819
1.847
2.100
4.533
4.643
5.262
4.905
15-19
2.372
1.877
1.207
2.561
2.367
2.124
3.112
6.316
20-24
2.251
2.093
3.070
2.758
3.292
4.466
5.377
7.292
25-29
1.695
1.064
Total,
1.831
2.015
2.385
2.822
4.039
4.937
4.783
5.617
75-
-80
80-85
85-
-90
All Ages.
0—4
14.909
14.107
40.000
28.571
1.060
1.135
5—9
8.207
8.523
23.000
16.250
50.000
50.000
1.329
1.423
10-14
25.352
17.391
1.373
1.439
15-19
6.203
8.516
9.091
24.000
1.545
1.876
20-24
21.536
23.482
31.666
35.952
16.667
16.667
2.179
2.270
25-29
0.717
0.431
Total,
14.587
14.428
20.128
22.013
16.282
21.363
1.158
1.234 MORTALITY AND LOSS EXPERIENCE.
34
Such are some of the effects in the aggregate. But a more exact impression will
be given by glancing down the columns of these tables, and noting the incessant
fluctuations of the rates. For instance, in the periods 50-55, 65-70 the claims per cent,
of the first year have exceeded those of the second year for Male Life; and in Female
Life, the same feature prevails as often as its opposite, for all ages. It should be noted
that in each vertical column, the percentages refer to different groups of persons
insured. The percentages for the same group proceed forward and downward in a
diagonal direction, for five-year periods. Thus for the same class of females entering
at the Ages 25-30, the successive Claims per cent, in five-year periods are given as
1.015, 1.081, 0.839, 1.252, 0.986,
It may be proper to observe that in the national census, the mortality of females,
in the great mass, is less than that of the male population. In life insurance statistics,
the same feature would undoubtedly prevail, if a fair representation of the female class
in the community were insured. But in practice, the class that actually insure have
so many unhealthy lives, especially in the period of maternity, that the total claims
per cent, of females (1.23) have very considerably exceeded the rate (1.10) of insured
males. The principal causes of this difference will be developed hereafter in the
Medical Statistics. Table XIY (B) indicates especially the risk incident to the birth
of the first child, which is well known to be far greater than at any subsequent
delivery, with compensations in the future conditions of life. Compare also results from
the Census of Scotland in the Journal of the Institute of' Actuaries, Vol. 22, page 233.
COMPARISON OF DEATHS PER CENT. FROM SEVERAL TABLES OF MORTALITY.
Ages.
30 American Offices.
20 British Offices.
American
Life, 1858.
Carlisle
Table.
Males.
Females.
Diff. 1
Males.
Females.
Diff.
M. AND F.
M. AND F.
15-20
0.71
0.68
+ .03
0.47
0.76
-.29
1.31
0.68
20-25
0.69
1.10
-.41
0.69
0.85
-.16
1.07
0.70
25-30
0.67
1.00
-.33
0.69
1.18
-.49
0.78
0.82
30-35
0.71
1.05
-.34
0.82
1.13
-.31
1.04
1.01
35-40
0.80
1.00
-.20
0.95
1.21
-.26
0.98
1.09
40-45
0.94
1.05
-.11
1.07
1.28
-.21
0.83
1.41
45-50
1.10
1.06
+ .04
1.36
1.39
-.03
1.13
1.44
50-55
1.43
1.34
+ .09
1.74
1.57
+ .17
1.55
1.52
55-60
1.94
1.83
+ .11
2.40
2.02
+ .38
1.87
2.20
60-65
2.76
2.39
+ .37
3.48
2.86
+ .62
4.34
3.68
65-70
3.93
4.04
-.11
5.02
4.37
+ .65
4.15
4.45
70-75
5.63
4.78
+ .85
7.33
6.84
+ .49
5.19
6.97
75-80
8.48
14.59
-6.11
11.00
10.66
+ .34
8.48
10.54
80-85
11.33
20.13
-8.80
16.52
12.51
+ 4.01
13.86
85-90
16.67
16.28
+ .39
22.35
22.82
-.47
19.92
90-95
32.73
22.68
+ 10.05
28.61
Deaths,
44,485
2,058
20,521
3,335
750
1,840
Entrants,
982 734
44,795
130,243
16,604
19,725
This Table represents the percentages from original observations simply. Gradu-
ation would require a small correction described in Section X, 3. In the columns of
the Thirty Offices, under the age of forty-five, the mortality of females will be seen to
exceed that of males by at least one-third part, for a considerable extent; from the age
of forty-five to sixty-five, the order is reversed, and the mortality of males exceeds that MORTALITY AND LOSS EXPERIENCE.
35
of females by a much smaller difference relatively. Above sixty-five, the female
experience is small in numbers, and fluctuating in value. Yet in the larger experience
of British females from sixty-five to eighty, the excess of male mortality is still contin-
ued, though reversed afterward.
The last column but one, is derived from the published Report of the Experience
of the Mutual Life Insurance Company of New York to Feb. 1, 1858, pages 10, 11.
From its middle ages, between thirty and sixty, Mr. Sheppard Homans, the Company’s
Actuary, constructed the common American Experience Table of 1858, with some
modifications.
Passing now to the last column, we note that all the percentages of the Carlisle
Table exceed those of male life in the Thirty Offices Experience. And the percentages
of the Twenty British Offices, male as well as female, with one or two exceptions at
the beginning, show a similar excess over that of American insured males.
SECTION IX.
Continuation of the Experience of 1874 to Final Series.
The record of a Life Office noted upon the card returns will now be regarded as
an inventory of the state of its business up to that particular time; from which the
average for a longer series of years remains to be determined.
The applications of the Life Table to the current business of Insurance, whether
to the surrender or change of policies, or to reserves, all presuppose that such Life-
Table is based on terminated experience. In the analogous problem of the theory of
probabilities, where the game is stopped prematurely, it is proved that the stakes
should he divided among the players, in proportion to their probabilities of winning.
Although the game stops, the determination of the several portions evidently depends
on a knowledge of the chances, or law of the game when played through.
In the present case, the returns of 1874 comprise 549,418 Existing policies, of
which the record closes before their future experience is determined. The column of
Existing in Tables I and II now represents only first experiences, cut into dissimilar
portions by the end of the 1st, the 2d, and other years after the medical examination,
according to the year chosen for making out the cards or returns. Had the canvass of
the office registers been deferred fifteen or twenty years, the most of the policies or
lives would evidently have passed to their termination, and would have been returned
as Discontinued or Died. And this mature experience, including the present returns,
would have furnished the genuine elements for constructing the Life Table.
But not to wait for such supplementary experience by natural termination, we
have found that it can be approximately supplied by carrying forward the Existing to
their proportional terminations in the columns of Discontinued and Died; and the
same for Amounts Insured and Claims. The present experience down to 1874 will
give the requisite multipliers, and the products so found, may be conveniently termed
Final Series.
Let us here investigate the weights or multipliers for this purpose. On page 25 is
a formula assigning the probability p" of surviving one year’s exposure to both death
and discontinuance. Hence from any age of Entry in Tables I and II we can
determine the surviving entrants, N, Npf, Upfpf, .... of Final Series denoted by
N, e\, e'2, .... as they would be, if the present Existing, all proceeded to their termina-
tions in Discontinuance or Death. And the annual differences, 1ST — e/, e/—e2', etc. are 36
MORTALITY AND LOSS EXPERIENCE.
evidently multipliers of the corresponding sums of discontinued and died on the same
line of Table I. Hence their respective ratios will give the required weights or multi-
pliers (w) of Tables III and IV.
Thus in the accompanying illustration, the first columns of Existing, Discon-
tinued and Claims are simply the sums of the corresponding Amounts from Table I in
Thousands of Dollars under Ages of Entry 36 and 37, (mean age 36|). The product
of these (except Existing), by the Multiplier opposite in the last column, gives the
Discontinued Amounts and Claims, (which now include the Existing carried forward),
under Einal Series.
AMOUNTS FROM TABLE I. AGES AT ENTRY 86 AND 37 YEARS.
Years
Experience ending
1874.
Final Series.
op
Insur-
Existing
Discon-
Entrants
Entrants
Discon-
Death
Exposed to
Multiplier
ANCE.
(R).
TINUED.
ClA1M8.
(«)•
(«')•
TINUED.
Claims.
Risk.
H-
0
11,763
7,443
457
196,462
196,462
7,443
457
94,509.5
1.0000
1
10,058
27,369
1,743
176,799
188,562
29,181
1,859
173,971.5
1.0665
2
8,433
15,564
1,250
137,637
157,522
17,813
1,431
148,615.5
1.1445
3
7,692
10,655
959
112,390
138,278
13,109
1,180
131,723.5
1.2304
4
8,705
7,753
916
93,084
123,989
10,327
1,220
118; 825.5
1.3320
5
9,533
5,665
922
75,710
112,442
8,414
1,369
108,235
1.4852
6
9,629
3,433
665
59,590
102,659
5,914
1,146
99,702
1.7228
7
8,991
2,617
470
45,863
95,599
5,455
980
92,871.5
2.0845
8
7,915
1,508
366
33,785
89,164
3,980
966
87,174
2.6392
9
5,659
707
217
23,996
84,218
2,481
762
82,977.5
3.5098
10
3,659
496
179
17,413
80,975
2,307
832
79,821.5
4.6505
11
2,427
330
224
13,079
77.836
1,964
1,333
76,854
5.9515
12
1,107
191
149
10,098
74,539
1,410
1,100
73,834
7.3819
13
852
167
113
8,651
72,029
1,391
941
71,333.5
8.3264
14
937
142
124
7,519
69,697
1,316
1,149
69,039
9.2700
15
730
78
78
6,316
67,232
830
830
66,817
10.645
16
596
57
98
5,430
65,572
688
1,184
65,228
12.076
17
339
44
77
4,679
63,700
599
1,050
03,400
13.615
18
341
37
59
4,219
62,051
544
868
61,779
14.709
19
390
40
55
3,782
00,639
041
881
60,318.5
16.035
20
273
45
49
3,297
59,117
807
879
58,713.5
17.931
21
196
16
53
2,930
57,431
313
1,037
57,274.5
19.602
22
198
31
74
2,665
56,081
652
1,557
55,755
21.044
23
264
17
56
2,362
53,872
388
1,278
53,678
22.808
24
347
6
69
2,025
52,206
155
1,783
52,128.5
25.781
25
477
3
48
1,603
50,268
94
1,504
50,221
31.362
26
377
20
30
1,075
48,670
903
1,354
48,218.5
45.277
27
345
3
33
648
46,413
215
2,361
46,305.5
71.618 1
28
160
11
267
29
35
1
96
30
56
1
60
32
1
38
2
102,484
84,429
9,549
The first column of Entrants (e) is readily computed by the formula of page 25,
preparatory to obtaining the values of p" by the formula there given.
s
w = -•
p" = —til.
e
litt
e1 = ep .
r' = wr.
D' = wD.
Here et denotes the entrants on the line or space next below that of the existing R and
entrants e. Multiplying N or 196,402 in this example by the first value of p" gives MORTALITY AND LOSS EXPERIENCE.
37
188,502, that is, the second value of e'; and this by the next value of p" gives the next
value of d; and so on. The entrants d are here computed through 27 years, omitting
the rest on account of the smallness and irregularity of the last data. Now the first
difference of column s' or 7,900 divided by the sum of the first Discontinued and
Claims 7,443+457 gives 1, the first multiplier in column (w). The second difference
of d or 31,040 divided by 27,361 + 1,743 gives 1.0665 the second multiplier in column
(w), and so on.
Another Solution. Since Entrants (e) always commence where Existing (R) end,
at the beginning of any one year, we have first the proportion ; Entrants : Discon-
tinued in the following twelve month :: Existing : proportional Discontinued. The
sum of the second and fourth terms gives the Discontinued of Final Series, on the
same line of the Table. And the Deaths of Final Series are found in like manner.
Adding the third term to the Existing at the beginning of the next year, and then
subtracting the two-fourth terms of Discontinued and Deaths just found, will give the
entire surviving Existing from previous years, which will be the third term of the
proportions for the next year, and so on.
Exposed to Risk. After applying either solution, the Discontinued and Deaths
so found, will give the Exposed to Risk, in Final Series through the usual formula on
page 25, by making the Existing (R) to be zero. Or in a different way, the Exposed
to Risk may be found by multiplying the Exposed to Risk in the first series of 1874,
by the same multiplier (w) that is used to change the Discontinued and Died on the
same line to Final Series, from Tables I or II. This implies that the percentages of
Death and Discontinuance, but not their absolute numbers, are virtually alike on the
same line of the series of 187f and of Final Series. This agreement is shown in the
following examples, computed with five-place logarithms, from Amounts Insured.
ELEMENTARY CLAIMS PER CENT.
Years
of
Insur-
ance.
Ages of Entry
36 and 37.
Ages of Entry
56 and 57.
Years
OF
Insur-
ance.
Ages of Entry
36 and 37.
Ages of Entry
56 and 57.
(1874).
Final
Series.
(1874).
Final
Series,
(1874).
Final
Series.
(1874).
Final
Series.
0
0.484
0.484
1.352
1.352
13
1.319
1.319
7.816
7.817
1
1.069
1.069
1.852
1 .850
14
1.685
1.664
4.847
4.850
2
0.963
0.963
2.206
2.206
15
1.243
1.242
3.170
3.175
3
0.896
0.896
2.436
2.436
16
1.814
1.815
4.248
4.249
4
1.027
1.027
3.125
3.123
17
1.653
1.656
11.007
11.012
5
1.265
1.265
2.761
2.760
18
1.405
1.405
0.899
0.899
6
1.149
1.149
4.243
4.240
19
1.482
1.461
9.848
9.852
7
1.055
1.055
3.716
3.718
20
1.496
1.497
10.687
10.674
8
1.108
1.108
3.200
3.198
21
1.814
1.811
5.066
5.072
9
0.918
0.918
3.237
3.235
22
2.793
2.793
19.954
19.914
10
1.043
1.042
4.200
4.201
23
2.380
2.381
7.233
7.229
11
1.735
1.734
5.478
5.488
24
3.413
3.420
4.396
4.412
12
1.490
1.490
5.071
5.071
25
2.997
2.995
13.123
13.115
Weights. The foregoing system of multipliers (w) will be seen to coincide with
the method of weights, used in the process of average. It is evident that the weights
{w) have the same relation to a life table designed for questions of population, without
interest,—that discounted weights (w. vn), where n denotes the ‘‘Years of Insurance,”
have to a life table designed to give (he well known Commutation Columns, for
different rates of interest. 38
MORTALITY AND LOSS EXPERIENCE.
Accordingly, discounted weights (w. vn), assuming v to be Ig-1.04, have been
preferred in Tables III and IV, for determining the Claims of Final Series. In
consequence of large irregularities in the statistics of Female Life, only the mean
weights, (tv) and (tv. vn) were tabulated by their logarithms in Table IV (C). Deferring
further explanations, and the construction of life tables to Part II, we subjoin an
example of the preparatory operation for a single Age of Exposure of Male Life; the
weights of Table III being changed from (w) to (tv. vn).
AGGREGATE CLAIMS PER CENT. EXAMPLE. AGE OF EXPOSURE 24£ TO 25|.
Years of
Insurance.
Age of
Entry.
Log’m
Weight.
(1874).
Final
Series.
Claims.
Exposed.
Claims.
Exposed.
0
25
0.0000
168.0
36,205.0
168.0
36,205.0
1
24
0.0142
305.0
50,185.0
315.2
51,856.0
2
28
0.0318
257.0
30,560.5
276.5
32,885.0
3
22
0.0510
111.0
17,851.0
124.8
20,077.0
4
21
0.0908
68.0
9,781.0
83.8
12,056.0
5
20
0.1348
52.0
4,374.5
70.9
5,966.2
6
19
0.1938
18.0
2,079.5
28.1
3,249.4
7
18
0.3009
14.0
951.0
28.0
1,901.5
8
17
0.4288
6.0
448.0
16.1
1,202.5
9
16
0.5670
1.0
227.5
3.7
839.7
10
15
0.6271
1.0
93.0
4.2
394.1
11
14
0.6781
1.0
67.0
4.8
319.3
12
13
0.6875
8.0
38.9
13
12
0.6559
11.5
—
52.1
Sum
1,002.0
152,842.5
1,124.1
167,042.7
Claims per cent
0.656
0.673
SECTION X.
Notes on G-raduation. Short Method for the Expectation of
Life. Graduation by Formula and by Divided Differ-
ences. Loss relative to Term and other species of
Insurances.
In further preparation for the construction of life tables in Part II, and other
applications, the following notes of investigation are here placed on record. To correct
for minor irregularities in the data, the observations may be grouped in five or ten-year
periods. And from these five or ten-year sums, the annual values are again derived in
a more regular and continuous series, by the process of graduation. This may be
done in several ways.
1. The Graphic Method. On a base line representing years, or one of the two
dimensions, set off distances corresponding to tbe given periods of age. Each of these
distances is made the base of a rectangle, whose altitude is the average obtained by
dividing the respective group of observations by its number of years. Having plotted
this series of rectangles, whose upper contour forms a broken line, draw with a free MORTALITY AND LOSS EXPERIENCE.
39
hand, a mean curve to supersede the broken line. This curved line must be drawn as
little curved as the conditions admit of, and never change its direction abruptly. It
will generally cut off a triangular space from one corner of each rectangle, and by
compensation take from without, an equal space above the next corner of the rectangle.
A preliminary curve is often drawn in pencil, and then successively adjusted by
measuring, till the spaces exterior and interior to each rectangle are made equal.
Finally, the ordinates of the curve measured year by year, give the graduated series
required.
The living and the deaths for the Carlisle Table were graduated separately by this
method, as described in Milne on Annuities and Insurances, Vol. I, page 100, (London,
1815). In respect to the vital statistics of the city of Carlisle and other places, he
observes, it is very desirable that the exact numbers be given monthly for the first
three months after birth; quarterly for the remaining three quarters of the first year
of age ; and after that, separately for the 2d, 3d, 4th and 5th years of age. And then
intervals of five years to the extremity of life may do very well. And if the number
of inhabitants be very considerable, intervals of ten years may be employed after the
age of twenty or thirty.
2. Short Method. When life tables have once been formed by the graphic or
other correct method, they may furnish corrections to shorten future applications.
From the three life tables constructed by Milne, Vol. II, pages 404, 405, 564, 534, 566,
we have derived for future reference, the following method for passing directly from
the original data, to an outline of the Expectations of Life. And the same process
might easily be adapted to Life Annuities. At the age of 80 years, the Expectation of
Life is generally 5 years, more or less; and the deviations of the Expectations, and
the Logarithms of the probabilities of life, from their mean values, appear very nearly
proportional to each other, in different Tables. On this clue, are founded the equations
below, which give a close approximation to the Relative and to the True Expectation
at the age of 80. From the Relative Expectation at 80, we derive the Relative values
at 70, 60, 50 .... 5 years, to which a small correction is to be added to give the true
Expectations.
EXAMPLE FROM THE ORIGINAL DATA OF THE CARLISLE TABLE.
Period
n.
Living
L.
Deaths
D.
„L—*D
Age.
Expectation.
Difference.
Cor.
nxXL + $V
Calc’d.
True.
Carl.
S. M.
S. F.
0- 5
9,868.5
812
1.816225
0
38.72
...
5- 10
8,703
89
1.977795
5
51.26
51.25
-.01
-.02
+ .02
-.04
10- 15
6,853.5
34
1.989230
10
48.85
48.82
-.03
+ .01
+ .01
-.01
15- 20
6,471
44
1.985235
15
45.02
45.00
-.02
00
+ .02
0 1
20- 30
12,730.5
96
1.96725
20
41.49
41.46
-.03
00
+ .03
0
30- 40
8,406
89
1.95402
30
34.35
34.34
-.01
-.01
+ .01
0
40- 50
8,226
118
1.93770
40
27.63
27.61
-.02
00
+ .03
0
50- GO
5,638.5
103
1.92067
50
21.13
21.11
-.02
-.01
+ .02
tA
GO- 70
4,194
173
1.82083
60
14.37
14.34
-. 03
+ .01
+ .03
A c
70- 80
1,831.5
152
1.63936
70
9.22
9.18
— .04
+ .05
00
tV C
80- 90
558
98
1.23529
80
5.46
5.51
+ .05
+ .01
-.05
c
90-100
94.5
28
2.70366
90
3.28
100-105
18
4
100
2.28
73,593
1,840
Xpn. 40
MORTALITY AND LOSS EXPERIENCE.
The third column shows the Deaths in the city of Carlisle during nine years; and
the second, the corresponding Population exposed to risk, as adjusted from two cen-
suses. The third column is derived from these by the formula at the head of the
column, where L and D correspond to the period n, that is, 5 or 10 years, on the same
line. It expresses the common logarithm of an approximate probability of surviving
the period, as noted at the foot of the column.
Let s denote the sum of the Logarithms of the three given probabilities from the
age of 50 to 80. In the present example s is 1.38086 or —0.61914.
Also let e'x, ex, denote the Relative and the True Expectation at the age 80, as
defined by the following equations :
e'80 = 10.844 + 10.315$, e80 = 7.833 + 3.833s.
The multipliers of C to give the correction, Cor. are shown in the lower part of the
last column of the accompanying Table. From e' at 80, the decennial values of e' are
derived one from another successively down to the age of 20, by the relative formula,
e'x-,0 = 5+p'°(e'x + 5).
ex = e’x + Oor.
C = eao—e'8Q.
Under 20 years of age, the formula for five-year intervals will be
e x—5 — 2.5 -j-p5(e * + 2.5).
The probability^10 orp* is obtained at once from the fourth column of the Table,
corresponding to the given interval of age. The first equations applied to the Carlisle
data give e'80 = 4.46 and c80 = 5.46 ; whence their difference C = 1.00. And the cor-
rections to be added to e are known from the last column of the Table. From the age
of 80 down to 5, the Expectations calculated by this simple process, differ but slightly
from the standard results of the graphic method, as shown in the last column but three.
And the next two columns headed S. M. and S. F. derived from the data and Tables
of Swedish Males and Swedish Females found in Milne’s treatise, show even smaller
residuals. ,
3. Correction. A plausible but erroneous assumption is occasionally made, that
when the sum of the Deaths from 60 to 65 years of age, for example, is divided by the
sum of the Exposed to Risk at the same ages, the resulting ratio of mortality will be
that of the middle age, 62-63. In respect to the values of l, so found in the life table,
a correction will be required, which is additive up to about the age of 72, and sub-
tractive after that age. Mr. Woolhouse finds the required correction to be the central
second difference plus one-fifth of the fourth difference (of the five values of l in the
period), to be applied with the sign just described. In the same communication will
be found Woolhouse’s Method of Final Adjustment, after a previous graduation.
(Journal, Yol. XXI, p. 58, and Yol. XIII, p. 98.)
Jf. Graduation by Formula. Among various formulas presented for choice, let us
here employ a modified geometric progression, with three constants to be determined
from the data. For example, the observations of three five-year periods of age A, B, C
are given, to be resolved into the series for single years. Let ux denote the number for
the age x; then for annual and for five-year periods :
ux = a + bcx, u0 = a + b, ux = a + Sc1, w2 = a+Jc2, ....
A = 5 a + bs, B = 5a + bsc5, C = 5a + Z>sc10; MORTALITY AND LOSS EXPERIENCE.
41
s = l + c+o' + + <‘ = eF-±, C‘ = §3|,
5, e °~B h — (B—A) (c 1) t
5 <*—l> b~ (e6—l)2 ’
u6 = a + be6, Ut = a + bc\ u6 — a + be6.
Thus from A, B, C may be found u6, w7, u8, the three middle terms of B. From
B, 0, D may be similarly found the three middle terms of C, and so on. Then u9, uw
the two omitted'terms between these triplets can be supplied by divided differences, as
presently described. And the six other initial or final terms of the whole series, can be
regularly computed by the local process above. In case the sum of any five computed
terms differ from the original given sum, one-fifth of the excess can be apportioned to
each term. Also instead of five-year periods, the same solution applied to ten-year
periods will give five sums of two terms each, which can afterwards be separated into
single terms, by applying series. And instead of interpolating Deaths D, and Living L,
separately, it may be preferable to take the combinations L—£D, and for
greater regularity.
Conversely, it will sometimes be required to find x corresponding to a value of u
or ux, intermediate to u0, ul} u9, noted above. In this case, logarithms can be applied
to either of the results of elimination below, involving x. After solution, if the
suffixes of u, instead of 0, 1, 2, are changed to n, n + h, n + 2h, for instance, then x
should be changed to n -(-xh.
c = u!=l! e-= 1
Ml—M0 Ml — M0
u*—uo , s(s-l) n , x(x-l){x-2)
*+ 1.2 (C-1) + 073 (c_1) + •"•
5. Interpolation by Divided Differences. It has just 'been shown in 3. how far
the divided sum or average- of five terms differs from the middle term. But to sub-
stitute the divided difference of the extremes at the middle point, for the middle
difference, is much more accurate, since it differs from the truth by only half the
central third difference. Although often insensible, yet provision will be made for
correcting even this slight residual, whenever it appears. Embarrassment is sometimes
experienced in bringing together portions of the life table interpolated separately, so
that the differences at the junctions shall be free from break or abrupt changes. The
new artifice here suggested, is that of substituting divided differences at the middle
points.
For one of the simpler applications, let us first suppose that two terms d, e,
interpolated by different formulas are to be reconstructed so as to present a regular
continuity. Let the series be
...a b c (d,e) f g A.. 3 y = (f-c)~ (A^+^Tp+^5^)-
, f—a , , , q—b „ h—c
d = c+y-\-J e = f=e+yH——.
Thus, at first canceling the terms d, e, which include the junction, we find a slight
correction y, such that the three divided differences each corrected by y, shall make up
the proper quantity f—c. Then d, e are computed; also /, to prove the operation,
by coinciding with the previous value of /, as above indicated. 42
MORTALITY AND LOSS EXPERIENCE.
For the reconstruction of four terms f, g, li, i by five differences, let the adjacent
terms be
...abode (/ g, h i) j k l rn n ....
„ , . v (j—a , k—b l—c m—d n—e\
5,= c,-«)-(V+-r+-r+—+-r)-
. j—a „ k—b , l—c ,
f=e+y + —$-> ff=f+y + -Q-f h=g+y+—, etc.
The same principle might evidently be extended to the omission and re-computation
of second or third differences; so that it offers facilities of unusual precision and
importance.
6. Loss Experience on different Species of Policies. An inspection of the ratios
below, may give the most correct impression. The experience of Term Policies is fluc-
tuating, and appears generally dependent on particular circumstances in connection
with other classes of Insurance. Besides ten Companies classified in regard to Species,
the statistics of six Companies in the general collection, were classified in respect to the
Amount of Policy. A difference of opinion exists in respect to the value of such minor
classifications, which might be indefinitely extended to Paid-up Life, Ten Payment
Life, Paid-up Endowment Policies, and others.
LOSS ON DIFFERENT SPECIES OF POLICIES.
Number op
Life Policies.
Endowment Insurances.
Term Policies.
Companies.
Probable L.
Actual L.
Ratio.
Probable L.
Actual L.
Ratio.
Probable L.
Actual L.
o
s
<
PH
$
$
100:
$
$
100:
$
$
100:
3
3,661,757
3,799,225
104
645,702
533,013
88
169,204
165,500
98
2
7,076,232
7,068,489
99
1,119,234
859,350
77
408,101
448,530
110
2
3,611,112
3,766,759
104
302,716
264,537
87
55,419
84,067
152
3
2,997,823
3,614,288
120
399,790
391,172
98
77,285
42,900
56
10
17,346,924
18,248,761
105
2,467,442
2,048,072
83
710,009
740,997
105
LOSS RELATIVE TO THE AMOUNT OF POLICY.
Species of
Policies.
Under $4,000.
$4,000 to $10,000.
Over $10,000.
Probable L.
Actual L.
Ratio.
Probable L.
Actual L.
Ratio.
Probable L.
Actual L.
Ratio.
$
$
100:
$
$
100:
$
$
100:
Life Policies . . .
1,994,890
1,915,905
96
2,718,160
2,874,100
106
33,440
35,000
105
Endowment Ins.
258,228
222,699
86
261,060
190,000
73
1,575
20,000
Term Policies. .
88,717
101,967
115
82,126
74,000
90
4,750
7 Companies...
3,827,440
3,672,003
96
4,393,274
4,679,100
106
119,157
85,000
71 CLIMATIC TABLES.
43
DIVISION SECOND.
CLIMATIC TABLES.
Or thirty life insurance Companies included in the general collection, twenty-
seven have returned the locality of the insured, by States or Counties. The results.are
given in Tables V, VI and VII.
In Table VI, the Order and Variability of Ratios is shown in detail, according to
twelve Companies whose mean ratios are less than 94, six Companies whose mean
ratios were between 94 and 100, and nine Companies whose mean ratios were greater
than 100. The last six columns exhibit the Probable and the Actual Loss in Thousands
of Dollars; that is, 870 there signifies $870,000, and so on. For any State or group
of adjacent States, the variability may be estimated by comparing the mean with the
associated values horizontally.
SECTION I.
Computations of Probable Loss.
For future applications, as well as to illustrate the method of computing the
columns of Probable Loss, we here insert the two accompanying Tables of multipliers
(A), (B), to three decimals, based on the Thirty Offices Experience. A policy of $5,000
entered at the Age of 37, for example, has the Duration or Years of Insurance 10
and is in force or Existing at the epoch of Office investigation. The tabular prob-
ability of dying in this interval is .101, which multiplied by 5,000 gives $505, the total
Probable Loss since the day of Entry. Had the same policy been Discontinued or
Died, the other Table (B) would have given the multiplier .095, and the product of
.095 by 5,000 or $475 would be the Probable Loss for ten years. The result differs
from the former for Existing, by half a year’s exposure, according to the general system
of reduction described in Division I, page 24. Accordingly, the values (A) are simply
the means of the adjacent values (B). Thus the first Table gives Multipliers for the
sum of Probable Losses for 1|, 2|-, 3£, .... years Existing, and the second Table for
0, 1, 2, 3, ... . years till Death or Discontinuance.
A modification to give the annual values separately, has been practiced by several
Life Offices, that compute the Probable Loss at the end of every business year. For
this object, Table XXYI will give the annual multiplier q for every age attained,
denoted by x-\-n. A policy of $0,000 exposed for the half or third of a year, is
accounted as $3,000 or $2,000 respectively exposed for the whole year; and so for other
fractional parts. A joint-life policy terminates at the first death. To shorten the
labor, sometimes, the tabular multipliers represent only the two nearest decimals ; and
a compensation of positive and negative errors is assumed in the final sum. 44
CLIMATIC TABLES.
THIRTY OFFICES EXPERIENCE.
(A).
Multiplier for Existing to give the whole Probable Loss.
Dura-
tion
Age of Entry
X.
Dura-
n + k-
10-14
15-19
20-24
25-29
30-34
35-39
40-44
45^9
50-54
55-59
60-64
65-69
n + k•
o*
.008
.003
.003
.003
.004
.004
.005
.006
.008
.011
.016
.023
o*
lk
.010
.010
.011
.011
.012
.013
.015
.019
.024
.033
.047
.069
1*
2 k
.017
.017
.017
.018
.020
.022
.026
.032
.041
.056
.079
.116
2*
3*
.023
.024
.025
.026
.027
.031
.037
.045
.059
.080
.114
.165
3*
4k
.029
.030
.032
.033
.035
.040
.047
.059
.077
.105
.149
.216
4*
5k
.036
.037
.038
.040
.044
.050
.059
.074
.096
.132
.186
.267
5*
6k
.042
.044
.045
.048
.052
.059
.070
.089
.117
.160
.225
.319
6*
7k
.049
.050
.053
.055
.061
.069
.083
.104
.138
.189
.264
.372
7*
8k
.055
.057
.059
.063
.069
.079
.096
.121
.159
.219
.305
.425
8*
9k
.062
.064
.066
.071
.079
.090
.109
.138
.183
.250
.347
.477
9k
10k
.068
.070
.074
.079
.088
.101
.122
.156
.207
.283
.389
.529
10*
Ilk
.074
.077
.081
.087
.097
.112
.137
.175
.232
.317
.433
.580
11*
12k
.081
.084
.088
.095
.106
.124
.152
.194
.259
.352
.477
.630
12*
13|
.087
.091
.096
.103
.116
.136
.167
.215
.286
.388
.521
.677
13*
14*
.094
.098
.103
.112
.126
.149
.184
.237
.315
.425
.565
.722
14*
15k
.100
.104
.110
.121
.137
.162
.201
.259
.345
.462
.608
.765
15*
16k
.107
.111
.118
.130
.148
.175
.218
.283
.376
.500
.650
.804
161
17k
.113
.118
.126
.139
.159
.189
.237
.308
.408
.539
.692
.838
171
18k
.119
.126
.134
.148
.170
.205
.256
.333
.440
.578
.731
.870
18*
19k
.127
.133
.143
.158
.182
.220
.277
.360
.474
.616
.769
.898
19*
20k
.133
.140
.151
.168
.195
.236
.298
.388
.509
.654
.804
.921
201
21k
.139
.148
.159
.178
.208
.253
.321
.417
.544
.692
.837
.941
21*
22k
.147
.155
.169
.189
.221
.271
.344
.446
.579
.728
.866
.957
22*
23 k
.154
.163
.178
.200
.236
.289
.369
.477
.614
.763
.892
.970
231
24 k
.161
.171
.187
.212
.251
.309
.394
.509
.650
.796
.915
.981
24k
25 k
.168
.179
.196
.224
.266
.330
.420
.541
.685
.827
.934
.988
25*
26k
.175
.187
.207
.237
.283
.351
.447
.574
.719
.856
.951
.993
26*
27k
.183
.196
.217
.250
.300
.374
.476
.607
.752
.882
.964
.996
27*
28k
.190
.205
.228
.264
.318
.397
.505
.640
.784
.905
.975
.998
28*
29k
.198
.214
.239
.278
.337
.421
.535
.673
.814
.925
.984
.999
29*
30k
.205
.223
.251
.293
.356
.446
.565
.705
.842
.942
.989
.999
301
31k
.213
.233
.263
.309
.377
.472
.596
.737
.868
.957
.994
.999
31*
32k
.222
.243
.276
.326
.398
.499
.627
.768
.892
.969
.997
32*
331
.231
.254
.289
.343
.421
.527
.659
.798
.913
.978
.999
33*
34*
.240
.265
.303
.361
.444
.555
.690
.826
.932
.986
.999
34*
35k
.249
.276
.318
.379
.468
.585
.721
.853
.947
.991
.999
35*
36 k
.258
.287
.333
.399
.493
.614
.751
.877
.960
.995
.999
361
37k
.267
.300
.349
.420
.519
.644
.781
.898
.971
.997
37*
38*
.278
.313
.366
.441
.546
.674
.809
.919
.980
.999
38*
39*
.288
.326
.383
.463
.573
.704
.835
.936
.987
.999
39*
40*
.298
.340
.401
.487
.601
.733
.860
.951
.992
.999
40*
41*
.310
.355
.420
.512
.630
.762
.883
.963
.995
.999
41*
42 *
.322
.370
.440
.537
.658
.790
.904
.973
.997
421
43*
.335
.386
.461
.562
.687
.817
.923
.982
.999
43*
44*
.349
.403
.483
.588
.716
.842
.940
.988
44*
10-14
15-19
20-24
25-29
30-34
35-39
40-44
45-49
50-54
55-59
60-64
65-69
Formula
Multiplier =
J CLIMATIC TABLES.
45
THIRTY OFFICES’ EXPERIENCE.
(B).
Multiplier for Discontinued and Died to give the whole Probable Loss.
Dura-
tion
Age of Entry
X.
Dura-
tion
n.
10-14
15-19
20-24
25-29
30-34
35-39
40^44
45^9
50-54
55-59
60-64
65-69
n.
0
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
0
1
.006
.007
.007
.007
.008
.008
.010
.012
.016
.022
.031
.045
1
2
.013
.013
.014
.014
.016
.017
.020
.025
.032
.044
.063
.092
2
3
.020
.020
.021
.022
.023
.026
.031
.038
.050
.068
.096
.140
3
4
.026
.027
.028
.029
.031
.035
.042
.052
.068
.092
.131
.190
4
5
.032
.033
.035
.036
.039
.045
.053
.066
.086
.118
.167
.241
5
6
.039
.040
.041
.044
.048
.054
.064
.081
.106
.146
.205
.293
6
7
.045
.047
.049
.051
.056
.064
.076
.096
.127
.174
.244
.345
7
8
.052
.053
.056
.059
.065
.074
.089
.112
.148
.203
.284
.398
8
9
.058
.060
.062
.067
.074
.084
.102
.129
.171
.234
.326
.451
9
10
.065
.067
.070
.074
.083
.095
.115
.147
.194
.266
.368
.503
10
11
.071
.073
.077
.083
.092
.106
.129
.165
.219
.299
.411
.555
11
12
.077
.080
.084
.091
.101
.118
.144
.184
.245
.334
.455
.605
12
13
.084
.087
.092
.099
.111
.130
.159
.204
.272
.369
.499
.654
13
14
.090
.094
.099
.107
.121
.142
.175
.225
.300
.406
.543
.700
14
15
.097
.101
.106
.116
.131
.155
.192
.248
.329
.443
.586
.744
15
16
.103
.107
.114
.125
.142
.168
.209
.271
.360
.481
.629
.785
16
17
.110
.114
.122
.134
.153
.182
.227
.295
.391
.519
.671
.822
17
18
.116
.122
.130
.143
.164
.197
.246
.320
.424
.558
.712
.855
18
19
.123
.129
.138
.153
.176
.212
.266
.346
.457
.597
.751
.885
19
20
.130
.136
.147
.163
.188
.228
.287
.374
.491
.635
.787
.910
20
21
.136
.144
.155
.173
.201
.244
.309
.402
.526
.673
.821
.932
21
22
.143
.151
.164
.183
.214
.262
.332
.431
.561
.710
.852
.949
22
23
.150
.159
.173
.194
.228
.280
.356
.461
.596
.746
.879
.964
23
24
.157
.167
.182
.206
.243
.299
.381
.493
.632
.780
.904
.976
24
25
.164
.175
.191
.218
.258
.319
.407
.524
.667
.812
.925
.985
25
26
.171
.183
.201
.230
.274
.340
.433
.557
.702
.842
.943
.991
26
27
.179
.191
.212
.243
.291
.362
.461
.590
.735
.869
.958
.995
27
28
.186
.200
.222
.256
.308
.385
.490
.623
.768
.894
.970
.997
28
29
.194
.209
.233
.271
.327
.408
.519
.656
.799
.915
.980
.999
29
30
.201
.218
.245
.285
.346
.433
.550
.689
.828
.934
.987
.999
30
31
.209
.228
.257
.301
.366
.459
.580
.721
.856
.950
.992
.999
31
32
.217
.238
.269
.317
.387
.485
.611
.753
.880
.963
.995
.999
32
33
.226
.248
.282
.334
.409
.513
.643
.783
.903
.974
.998
33
34
.235
.259
.296
.351
.432
.541
.674
.812
.923
.982
.999
34
35
.244
.270
.310
.370
.456
.570
.705
.840
.940
.989
.999
35
36
.253
.281
.325
.389
.480
.599
.736
.865
.954
.993
.999
36
37
.262
.293
.340
.409
.506
.629
.766
.888
.966
.996
.999
37
38
.272
.306
.357
.430
.532
.659
.795
.909
.976
.998
38
39
.283
.319
.374
.452
.559
.689
.822
.928
.984
.999
39
40
.293
.333
.392
.475
.587
.718
.848
.944
.990
.999
40
41
.304
.347
.410
.499
.615
.748
.872
.957
.994
.999
41
42
.316
.362
.430
.524
.644
.776
.894
.968
.996
.999
42
43
.328
.378
.450
.549
.672
.804
.914
.978
.998
43
44
.341
.395
.471
.575
.701
.830
.932
.985
.999
44
10-14
15-19
20-24
25-29
30-34
35-39
40-44
45-49
50-54
55-59
60-64
65-69
Formula, Multiplier =
_ j
ux 46
CLIMATIC TABLES.
The climatic Tables being already in use, are retained in the original form, based
on multipliers from the common American Table of 1858. Parallel computations of
the total, lead to the following precepts: To change the Probable Loss Am. 1858, in
Tables V, VI, or VII, to the standard of—
The Thirty American Offices Table ; Deduct part.
“ Twenty British “ “ HM-; Add per cent.
“ Seventeen British “ “ Add 8f per cent.
The corresponding factors are 0.9667, 1.0519, 1.0837, and their logarithms 1.98530,
0.02198, 0.03492. The columns of Actual Loss of course remain unchanged. By
these precepts, a present Probable Loss of $1,204,521 to be represented on the basis
of the Thirty Offices Table, is changed to $1,164,370. And the tabular ratio of 100 :134
becomes 96f : 134, or 100 : 139 nearly.
SECTION II.
Classification of Comities. Miscellaneous Statistics.
Preparatory to the classification of Counties in respect to Mortality, the simple
division shown on the common Geological maps of the United States may he noted.
First, the lowland or Alluvial Counties commencing with the eastern shores of Maryland
and Virginia, and following the line of tide-water on the Atlantic and Gulf coasts to
the borders of Mexico, and up the valley of the Mississippi to the mouth of the Ohio
river. These alluvial Counties are denoted in Table VII by one dot annexed. Between
these and the line of 400 feet elevation, are the Middle Counties, of which the tertiary
portion are here denoted by three dots, and the few cretaceous by two dots. The
remaining or Upland Counties above the line of 400 feet elevation, according to
Prof. Guyot, are here indicated by the absence of dots. The Counties of Indiana admit
of separate consideration. Also the present object may be found to require a modifica-
tion of these divisions.
States.
Alluvial Count’s.
Middle Counties.
Upland Counties.
Total
Counties.
Prob. Loss.
Act. Loss.
Prob. Loss.
Act. Loss.
Prob. Loss.
Act. Loss.
Prob. Loss.
Act. Loss.
Alabama ..
390,23G
544,453
175,977
194,580
523,007
729,014
1,204,521
1,618,048
Arkansas..
98,449
221,995
125,202
196,084
46,238
80,040
295,413
508,119
Florida....
133,347
224,596
28,758
32,550
181,635
303,852
Georgia....
2,117
7,500
524,221
603,540
355,738
257,860
950,653
911,536
Louisiana..
1,252,180
2,194,255
23,901
30,660
1,363,520
2,400,915
Mississippi
301,087
552,260
416,626
585,344
13,491
15,660
795,476
1,308,632
N. Carolina
225,053
353,555
390,924
442,633
178,804
201,721
843,815
1,069,409
S. Carolina.
374,487
446,438
192,307
182,193
179,009
227,004
753,060
864,635
Tennesee.
708,436
1,488,636
181,460
209,380
467,280
544,055
1,383,648
2,262,121
Texas
214,043
511,467
332,901
572,267
184,655
264,238
778,713
1,358,972
Virginia...
3,567
13,000
385,727
349,100
141,204
202,750
646,188
673,350
Total
3,709,002
6,558,155
2,778,004
3,398,331
2,089,426
2,522,342
9,196,642
13,279,589
Ratio (Am. 1858),
100:177
100
122
100
121
100
: 144
By 80 Offices Table,
100:183
100
127
100
125
100
: 149
By 20 Offices Table, TO 100 :168
100
116
100
115
100
: 137
By 17 Offices Table,
100:163
100
113
100
111
100
: 133
CLASSIFICATION OF COUNTIES CLIMATIC TABLES.
47
A singular feature is presented in the close agreement of the climatic Ratios for
Middle and Upland Counties, indicating that these may be united; or that hereafter,
two classes, Alluvial and Upland, will generally be sufficient. A wide contrast is shown
between their characteristic Ratios, 100 : 183 and 100 : 126, by the Thirty Offices Table.
It will also be noted that the Upland Counties of Georgia at the southern termination
of the Alleghany mountains are exceptionally healthy in the ratio of 100 : 72.
In the State of Indiana, the experience of eleven smaller Companies gave a
Probable Loss of $563,770 and an Actual Loss of $743,866; Ratio 100 : 132. By the
accession of other Companies, the total ratio was materially altered, as shown at the
end of Table VII. The less healthy Counties are reported to lie in the north-eastern
part of the State, and along the Wabash and Ohio rivers.
The following additional data pertaining to the Southern States, are arranged from
the statements of particular Companies, more especially to show in the last column the
Age corresponding to the mean Rate of Probable Loss divided by the number Exposed
to Risk. The results of the 27 Companies in Table V, and the 3 other Companies, for
the whole United States, are subjoined.
AGE OF MEAN RATE OF PROBABLE LOSS.
Companies.
Deaths
or
Loss.
Exposed
to
Risk.
Probable
Deaths
or Loss.
Rate
op Prob-
able.
Age
op
Rate.
Table
employed.
Pros. : Act.
As 100:
.ZEtna Life (III)
1,366.3
74,270
966.90
1.3019
46.27
17 Offices
141
St. Louis Mutual
677
37,940
442.96
1.1675
46.26
Am. 1858
153
Covenant “
243.3
17,594
203.95
1.1592
46.07
“ “
119
Continental, N. Y
459.8
20,937
249.31
1.1910
44.56
20 Offices
184
Scottish Amicable ..
139
5,966
79.94
1.3400
46.60
“ “
174
27 Companies, U. S
192,049.0
9,327,400
104,531.5
1.1209
45.11
Am. 1858
98
3 “ “
35,961
2,909,037
37,580.7
1.2919
48.69
“ “
85
It may here be noted that the Experience of the iEtna, Covenant and Continental,
also of the 27 Companies and 3 Companies, is stated in thousands of dollars. Three-
fourths of the business of the Covenant of St. Louis was located in the State of
Missouri. The Experience of the Scottish Amicable here represents only the portion
located in the British colonies of the West Indies; where the climatic ratio by the
20 Offices Table, 100 : 174 so far exceeds the like ratio 100 : 168 before found for
Alluvial counties in the Southern States. With the omission of the Continental, 1ST. Y.,
whose business appears somewhat exceptional, the Age of the mean Rate of Probable
Loss relative to exposure, differs little from 46.3 years, in the Southern or tropical
experience of the other four Companies.
With regard to the comparative Moi'tality of insured Females, the Southern
experience of the St. Louis Mutual, comprising 628 deaths of Males, and 48 of
Females, gave the ratio of Probable to Actual Loss (Am. 1858) for Males 1.51, and
for Females 1.70; thus indicating greater mortality on the part of Females.
In relation to the Mortality by Policy years, the total experience of the St. Louis 48
CLIMATIC TABLES.
Mutual, comprising 1486 deaths, North and South, gave the following ratios of
Probable Loss, denoted by 1, to Actual Loss (Am. 1858):
ST. LOUIS MUTUAL.
Policy Years....
1st.
2d.
3d.
4th.
5 th.
6tli.
7 th.
8th.
9th.
10th.
Total.
Deaths
3G0
330
232
180
146
87
73
40
20
7..
1486
Ratios
1.18
1.35
1.55
1.26
1.68
1.21
1.52
1.81
1.34
1.36..
1.33
SCOTTISH AMICABLE IN THE WEST INDIES.
Deaths
9
10
13
12
14
G
After Gtli year, 75
139
Eatios
1.59
0.97
1.47
1.59
2.12
1.03
(Am. 1858), 1.91
1.65
The mean rate 1.33 of the former Company is reached in the second year; that of
the latter, nearly in the first year, hut the ratios are irregular from paucity of data.
For information upon other portions of the globe, reference may be made especially to
the following articles in the Journal of the Institute of Actuaries:
On the rates of extra premium for foreign travelling and residence. By Arthur
H. Baily, Esq. Actuary of the London Assurance Coi'poration. Yol. 15, pp.
77-94, 18G9.
On the Additional Premium required for Residence in Foreign Climates. By
the Honorary President, James Meikle, Fellow, etc. Yol. 19, p. 268, 1876.
On the Death Rate among Assured Lives in the West Indies, being the Experience
of the Scottish Amicable Life Assurance Society during Thirty Years, 18J/.6-76. By
John Stott, late Manager of the Society, Fellow, etc. Yol. 21, p. 153, 1878.
In proceeding from the temperate to the torrid zone, it is generally admitted that,
“other things being equal,—height, soil, and river drainage,—the mortality will increase
as we go South; and that heat stimulates the vital functions, so that maturity is
reached earlier, and decay commences earlier.”
Climate Extra. The net extra to meet the additional risk, is roughly found
by dividing the excess of the Actual above the Probable Loss, by the Exposure. Thus
among the examples given by the distinguished actuary and manager T. B. Sprague, in
the Journal of the Institute, Yol. 19, page 297, the years of Life exposed in India were
229, Actual Deaths 11, Probable Deaths, HM, 3.499. And (11— = .03276;
or the Climate Extra is 3.276 per cent, of the sum insured. By a more extensive
method, Mr. A. H. Baily now President of the Institute, in his memoir above quoted,
first constructed a life table from the foreign experience, and thence a scale of annual
premiums; the difference between this and the home scale, gave the climate extras for
the several ages of exposure. Besides the ordinary extra, which averaged one per cent,
in the West Indies, the difference between unacclimated and acclimated lives led the
Scottish Amicable to charge an additional extra of one per cent, of the sum insured,
during the first three years of foreign residence. On returning to reside in Great
Britain, with ordinary health, all extras are omitted. CLIMATIC TABLES.
49
SECTION III.
The Life Table graded for different Climates.
Let us now consider the application of Table VI to permanent life insurances for
different grades of mortality. In a State, for example, where the ratio of Probable to
Actual Loss by Table VI corrected to the Thirty Offices Table, has been 100 : 139 in
the mean Experience of 27 Companies; and the ratios were 100 : 112, J00 : 153, and
100 : 161 in the three component groups of Companies, the most experienced actuaries
and managers could best determine the single resultant or probable cliviatic ratio for
Office premiums. Such examination would be influenced by the probable extent of
new business, the medical and counter selections, prevailing causes of death,—and
south of the latitude of 36° 30', also extending north of it, the alluvial Counties
distinguished in Table VII. For convenience, the net ratio thus conventionally adopted
may be denoted by 100 : lOOp, that is 1 : p.
The Thirty Offices collection defines the experience of insured mortality along a
wide belt of the temperate zone, the middle line centering nearly on the latitude of
the city of New York. The resulting standard, Table XXVI has the advantage of
unusually large data, critical accuracy and improved construction. To construct other
life Tables, from the small numbers of more distant climates, North or South, would
involve great labor and complexity. The difficulties are marvellously reduced by the
discovery of a double gradation. That is, Table XXVI being graduated in the direction
of the scale of ages, is virtually graded in another direction also, by its commutation
columns, which represent life tables for so many different climates, or climatic ratios.
And the values for intermediate ratios, can be found by simple interpolation.
This improvement was first suggested by the entire agreement of Table XXVI
with the well-known law of Gompertz, modified by Makeham, observing that in previous
comparisons of the annual rate or (log p), change of the first term (log s) of Make-
ham’s formula, corresponded with changes of mortality due to local or climatic causes;
the other term or power of (q) being nearly constant for a long series of years or ages.
Again, from a different source, but on the same principle, the experience of the
Scottish Amicable in the West Indies, was stated by the late manager and actuary to
be represented by the Northampton table better than any other of the current life
tables. Yet our examination indicates, that by a simple change of the rate of interest,
the Carlisle Table might be substituted, as shown by the following comparison of life
annuities; and the same might be extended to life premiums, with a simple modification
presently described.
LIFE ANNUITIES.
Age, Years
25
35
45
55
65
75
Northampton 4 per cent
15.44
14.04
12.28
10.20
7.76
4.96
Carlisle 5 per cent
15.30
14.13
12.65
10.35
7.77
4.99
Northampton 5 per cent
13.57
12.50
11.11
9.38
7.28
4.74
Carlisle 6 per cent
13.46
12.57
11.43
9.52
7.28
4.76 50
CLIMATIC TABLES.
In the first publication of the climatic Tables, it was provisionally assumed that,
above the age of twenty or some lower limit, the different grades of mortality (y) are
proportional to those of the standard Table. The preceding comparison of annuities
as well as other reasons, afterwards suggested to the writer the more convenient and
exact hypothesis, to which Mr. McClintock appears to have arrived in a different way,
that the probabilities of survival (p) for different grades of mortality, are proportional
to those of the standard Life Table. In another expression, the rates of survival and
of interest may be commuted. That is, so far as relates to life annuities, an increase
of the rate of mortality, may be represented by an increase in the rate of interest;
and conversely.
For illustration, let us distinguish the Northampton values by an accent, and write
the well-known formulas below ; then since annuities are assumed to be equal:
Northampton 4 per cent.
a'x = v'p'x{ 1 + a'a+i).
Carlisle 5 per cent.
ax = + «a>+i).
Relation.
v'p'x — vpx.
Consequently
,
l + a * X)ra- — Dx~ 1’‘aQ>
NV_Nj,_N, D»
D'a: ~ Dx “ Dy X D*
n'x = 1000 j — (!—»')
'*=i°00|rTi-(1-')|-
Ordinary Life, n’x = 713.4-1000^'—v).
Reserve = 1 _i±£*±» = j_i±f£±i.
1 “h ® a; 1 +
Temporary Annuity, o'" = o" = a;+1 J ''
Premium of Term Insurance, P' = y'——P' = P + 1000(y'—y).
x — x+n
Hence it appears that with very simple modifications, the Northampton values at
4 per cent, can all be derived virtually from the 5 per cent. Carlisle values.
In like manner, the values corresponding to different grades of climatic mortality,
can be conveniently derived from the columns of the Thirty Offices Table. The age of
the mean ratio of mortality has been determined on page 47 to be about 46 years. For
the present purpose, this determination will supply the place of another element, the
mean ratio of probable loss to the number exposed to risk, in the climatic district.
In a given locality, let the probable loss as computed by the Thirty Offices Table, be
to the actual loss as 1 : p. That is, let p denote the climatic ratio ; and let an accent
distinguish the values of v, q, p for the locality. Then by the Relation preceding,
and by page 47,
v'pu = VpK, or v'(l —= VPto-
q'46 = P?46-
Tin- • , V' — Vpis
Eliminating q 46, p = —7-—•
746
Let the value of v' depend constantly on 4 per cent, interest; that of pK, qiS on
the Thirty Offices Table. Then if v depends on 4, 4£ or 5 per cent, interest, the
values of p thus found will be 1, 1.405, 1.805 respectively; which may be termed
Special Climatic Ratios. CLIMA TIC TABLES.
51
For the First Special Ratio 1.405, we have p x — y’ nvvPa:- From this, we might
construct a special life table, and 4 per cent, commutation columns, which would give
the net life premium on 1000, at the age of 30 years for example, 19.50. But the
same can be found much more easily by the preceding formula, n'x = (nx, 4-| per cent.)
+ 1000(F—v); where ——— = 0.00460066, to be multiplied by 1000. And so
for other similar formulas.
1.04
For the Second Special Ratio 1.805, we have p'x=—-—px.
1.05
Also tt'x = (nx, 5 per cent.) -f 1000(V — v) ; where —-j——jr— = 0.00915751 to be mul-
1. U4 _L. UO
tiplied by 1000.
Therefore for the D, X, S columns according to 4 per cent, interest and the
Climatic Ratio 1.405, we have in Table VIII substituted the 44 per cent, columns of
the Thirty Offices standard. And for the D, N, S columns corresponding to 4 per cent,
and the Climatic Ratio 1.805, Table VIII gives the equivalent 5 per cent, columns of
the standard. But it is needful to observe, that this substitution could not be extended
to the C, M, R columns, without modification. Nor is it necessary, since all the formulas
can be expressed in terms of D, N, S, v and v.
In conclusion, the Climatic Ratios given in Tables V, VI and VII, being first
changed to the basis of the Thirty Offices Table, the corresponding 4 per cent,
premiums and annuities may be found from those of the Special Ratios in Table VIII
by proportion or interpolation. And the preceding formulas may be extended to other
species of Premiums and Reserves. The relation v'p' = vp first occurs in the treatise
on Policy Life Lines, by James Meikle, Edinburgh, 1870. 52
MEDICAL STATISTICS.
DIVISION THIRD.
MEDICAL STATISTICS.
Statistics of Diseases or Causes of Death have been returned by twenty-seven of
the thirty Coinj)anies included in the general collection. These twenty-seven are the
same which gave the Climatic Statistics, with one exception. The total number of
Deaths from All Causes was 37,624; of which 35,442 were Deaths of Males, and 2,182
of Females; also there were 8,919 Deaths additional in the three Companies not
specifying diseases. Under experienced medical advice and direction, the results are
given in the most approved forms for reference in Tables IX-XVII. The classification
of diseases accords nearly with the last plan of Dr. Farr, as recently applied in the
Medical Beport of the Mutual Life of New York.
An examination of medical treatises, including special works on Medical Examina-
tion for Life Insurance, like Brinton or Allen, will suggest many inquiries, to which
the present Tables will often give an instructive answer. What diseases increase, and
what diminish as age advances? What diseases or causes of death are beyond the
cognizance, and what are most easily recognized by medical examination, and at what
ages? What is the distribution of diseases in different parts of the United States?
On these and kindred topics, the statistics give valuable information, and will suggest,
in lieu of extended commentary, a ready reference to the Tables themselves.
In examining these records, we are reminded of the more numerous unrecorded
cases, in which, the innate vital force has reverted from morbid or diseased conditions
into the old channels of health. The struggle to rise from a diseased condition, may
be short and decisive, terminating in death or in convalescence, or it may be longer
prolonged with varying fortunes. The contending forces often appear so equally
balanced, that a very little will turn the scale. The struggle between the tendency to
health and the tendency to disease, acute or chronic, may last from a day to months,
and even years. Thus medical statistics illustrate the nature of the mysterious vital
force, by its final developments under varied conditions of age, duration, sex and climate.
SECTION I.
Description of the Tables.
The first of the series, Table IX, is a general Table giving firstly the Number of
Deaths of insured males, and similarly of females, by 100 Diseases, in seven classes,
commencing with zymotic. The next three columns give the results for two, for five,
and for twenty Companies arranged to include about one-tliird of the whole, or nearly
12,541 in each group. By glancing across the three columns horizontally, the
Variability by each disease is ordinarily indicated. For example, the deaths by
Typhoid Fever, at the head of the list, appear to have been quite uniformly distributed. MEDICAL STATISTICS.
53
Iii the last column, the 37,624 deaths are represented on the percentage scale of
100.00 distributed proportionally to the number of deaths by each disease. The prin-
cipal or leading diseases in order, are Typhoid Fever 5.99 per cent., other Fevers 4.35,
Consumption 18.31, Apoplexy 4.70, Paralysis and Disease of Brain 4.34, Heart Disease
3.61, Pneumonia or Inflammation of the Lungs 7.68, Accidents and Injuries 7.21 per
cent, of all. These and other results may be deemed as correct as can possibly be
obtained under existing circumstances. Some allowance is to be made for the inexact-
ness of medical science, with gradual changes and transposition of medical terms.
Also when one disease is accounted as the producing cause of another, the last or fatal
disease is the one noted in these statistics, from Office registers.
Table X, XI, XII contain only the absolute numbers of Deaths, which constitute
one of the two terms of ratio or comparison in the Tables following, with separate
columns for males (A), and Females (B).
Tables XIII and XV are a further development of Table X with respect to
Duration of Insurance. After the seven general classes, the statistics are subjoined for
thirty-nine leading Diseases. The corresponding Exposure of the living is given at
the foot of the columns in Table XIII. For example, in the first column, 2,445
Deaths divided by 356,210 years Exposed gives 0.00686, that is, 0.686 Deaths per cent,
or 68.6 in 10,000 Exposed to Risk, as shown at the head of the column. It will be
noticed that during the first two or three years of Male Life, the deaths from Zymotic
diseases, and from Accidents and Injuries, have generally exceeded the average, thus
showing this part of the mortality to be uncontrolled by the Medical examination.
And the same may be remarked in Female Life, including Childbirth and Puerperal
Diseases. On the other hand the efficient diagnoses of Constitutional and Heart
diseases, and, in general, of “ All Causes,” is very evident from the inspection.
Tables XIV and XVI give, according to the Ages of Life or Exposure, in extension
from Table XI, the ratio of deaths to the Living, and the ratio to total Deaths. The
Exposed to Risk by Ages are inserted at the foot of Table XIV; and the number of
diseases is further reduced in Tables XV (B) and XVI (B) for female life. All these
results by Ages of Life, will be found not less valuable than those from the different
point of view of the Durations of insurance.
Finally, Tables XII, and XVII (A), (B), (C), indicate the Climatic Distribution
of Diseases in the United States. These figures relate to Actual Deaths, while the
Climatic Tables V, VI, and VII are based on Amounts Insured; the one class may
supplement the other. In comparing the deaths of 703 Foreign Males, Table XVII (A),
with an equal number of the native-born insured, the greater exemption of foreigners
from Zymotic and Constitutional Diseases is counterbalanced by an excess of deaths
from Apoplexy (chiefly of Germans in one Company), from Abscess, Hemorrhage etc.
of Lungs, and Accidents and Injuries. About two-thirds of this experience, however,
occurred in British America, and the remaining numbers are too small for final
conclusions.
Again, comparing insured Male Deaths with those of the U. S. Census of 1870,
shown in Table XVII, for equal totals, the 'proportions dying from three of the seven
classes, that is, from Circulatory, Respiratory and Miscellaneous Diseases, are nearly
equal. And the decreased mortality of insured lives from Constitutional Diseases is
counterbalanced by a marked excess of deaths from Nervous and Digestive Diseases,
and a small excess from Zymotic Diseases. In respect to Female life the differences of
distribution evidently follow a different order.
Representing the total deaths by 100, in each of the seven groups of States, we
have in Table XVII (B) the relative proportions of insured males dying from thirty- 54
MEDICAL STATISTICS.
seven principal diseases. It is the comparison of a part with the whole, that is, deaths
with total deaths, since the living or Exposed to Risk are here omitted.
But the last Table, XVII (C) gives a sanitary survey of the United States, according
to the deaths and diseases among 10,000 living insured in each group of States. To
arrive at the living, the following process was employed. First, as verified in the mean
of two of the largest Life Companies, the ratio of Probable to Actual Claims in Table V
was assumed to be the same as the like ratio for Deaths; and the Age of mean Probable
Deaths, to be 46 years, as already indicated, page 47. That is, in any group of States,
Probable Deaths Probable Claims # Probable Deaths
Actual Deaths Actual Claims ’ Exposed 46 ’ 01
Actual Deaths Actual Claims Tabular Deaths
Exposed 46 X Probable Claims 10,000 Living ’
Here the second equation is divided by the first, to give the third. By changing
the middle member to a decimal, then multiplying both numerator and 1 the denom-
inator by 10,000 we obtain the last member, as shown at the head of Table (C). The
value of q or 0.011562 corresponds to the Actual and Probable Claims found in
Table V. The rest of the values in Table C were then readily found by multiplying
the head of the column into the proportional parts of Table XVII (B). By this
technical process, an additional and very important Table is obtained.
SECTION II.
Statistics of Consumption Compared. Sanitary Survey of
the United States.
To illustrate the relation of the statistics to any specified disease, let us take for a
first example, Consumption, the most fatal of all. Firstly in Table IX, the deaths
2,283, 2,231, and 2,372 in three groups, prove tlie deaths from Consumption to have
occurred at a nearly uniform rate of frequency in all the Life Insurance Companies.
Again, the 18.31 per cent, of total deaths shown in the last column, when increased, as
it probably should be, by one-thirteenth of the deaths by Pneumonia (chronic cases), and
by deaths registered from Abscess of Lungs, Hemorrhage of Lungs and Disease of
Lungs, becomes 20. C7 per cent., or more than one-fifth of the whole mortality.
Table XIII confirmed by XV, shows the effect of medical selection exhausted and
the average mortality of consumptive cases reached in the third year of Duration of
the Insurance. Table XIV confirmed by XVI, shows the largest mortality by con-
sumption to occur between the Ages of 20 and 30 years. But after this period, the
proportion of consumptive deaths of Males, bears a two-fold aspect. That is, when
compared with the number of living exposed (Table XIV), it slowly decreases to a
minimum rate between 50 and 60 years, and then gradually rises again, with a heavy
mortality all the while, in every period of age.
FROM TABLE XIV (A).
Ratio of Consumptive Deaths, to 10,000 Living at each Age.
Age
Under 20
20-30
30-40
40-50 50-60
60-70
70-80
All Ages.
Ratio
12.0
22.7
19. G
17.4 15.8
17.6
19.1
18.6 MEDIC A L ST A TISTICS.
55
But when compared with the total contemporary deaths (Table XVI), the propor-
tion from consumption continually decreases, as the complementary proportion from
other causes, increases above the age of thirty years. For Female Life, with more
extensive data, the general statement would doubtless, be similar to that before given
for males.
FROM TABLE XVI (A).
Ratio of Consumptive Deaths, to 100 total Deaths at each Age.
Age 20-30 30-40 40-50 50-60 60-70 70-80 All Ages.
Ratio 30.3 25.8 17.8 10.8 6.5 3.5 18.3
Passing now to the Climatic Distribution in Table XVII (A) we observe that
North of latitude 36° 30', the deaths from consumption are 18.7 per cent, of the total
Deaths for male life, and 19.3 for females. While south of 36° 30', the percentages are
12.3 and 9.0 respectively. The apparent deficiency from consumption in the latter,
proceeds only from comparison with a larger total, including an excess from malarial
or other diseases.
Compared with the United States Census of 1870, the percentage of consumptive
deaths 18.3 in a total of 100 deaths of insured, is less than the result of the Census,
which is represented by 26.2 for males, and by 29.7 for females. Thus the rejection of
lives uninsurable from incipient consumption is manifest.
Table XVII (B) exhibits in more full detail, the percentage of insured males
dying from consumption regarded as a part of a total of 100 deaths in each group of
States. In the composition of 100 deaths, the consumptive clement will be seen to
decrease in the following order of groups, iii, i, iv, ii, v, vii, vi: as the numbers 21, 20,
18, 17, 16, 15, 12. Thus in proceeding towards the tropical regions, the ratio of deaths
by consumption to those from other diseases, gradually diminishes. But further statis-
tics in the next Table (C) will prove such change of ratio to proceed, not from absolute
decrease of consumptive deaths, but entirely from increased mortality by other causes.
The practical question, what part of the United States is most favorable to con-
sumptive invalids, can now be satisfactorily answered, so far as average yearly results
are concerned, from Table XVII (C).
Group of States Ill V VI I IV II VII
Ratio 22.2 21.5 21.0 20.8 18.5 16.9 16.9
Ratio of Consumptive Deaths, to 10,000 Living.
Group III comprises New Jersey and Pennsylvania; V, Delaware, Maryland, Dist.
of Columbia, Virginia, Kentucky and Missouri; VI, States south of lat. 36° 30';
I, New England and New York; IV, Ohio, Indiana, Illinois, Iowa and Kansas;
II, Michigan, Wisconsin, Minnesota and Nebraska; VII, Washington, Oregon, Cali-
fornia, Utah, Dacotah and New Mexico.
By an obvious generalization, the first four groups, comprising the Atlantic and
Gulf Slates, from Maine to Florida and from Florida to the borders of Mexico, have
very nearly the same rate of consumptive deaths, or 21 annual deaths to 10,000 living.
The Western States show a decrease of the consumptive rate to 18£ in group IV;
while the groups II, and VII, or the Northwestern and Pacific States agree in the 56
MEDICAL STATISTICS.
The bold figures refer to the seven groups of States in the margin. The respective annexed figures are arranged in
ascending order, to show the Proportional Deaths by each disease, among 10,000 Insured Males, living in each group of
States-r From Table XVII (C).
A SANITARY SURVEY OF THE UNITED STATES.
Diseases.
Least Mortality.
Greatest Mortality.
Mean.
Gkoups.
1.
All Causes.
«
2
97.7
4
101.5
1
105.3
3
107.1
7
112 2
5
130.5
6
170.5
118.3
New England.
7
15.9
3
16.6
22.1
I
17.6
23.0
9
18.5
26.3
4
18.7
26.4
5
27.5
27 3
6
3
48.4
27.9
23.3
24.9
New York.
Constitutional Diseases.
2
21.4
7
4
6
1
5
PH
Nervous Diseases
2
11.5
4
14.4
1
15.4
3
15.4
7
18.4
5
20.1
6
22.3
16.8
2.
a
Circulatory Diseases
2
4.4
4
4.6
1
6.6
5
6.7
6
6.8
3
7.3
7
9.2
6.5
Respiratory Diseases....
3
12.1
1
13.3
2
14.6
7
14.6
4
16.6
5
18.1
6
21.5
15.8
Northwest.
1
8.6
7
9.1
2
10.0
3
11.2
4
11.2
5
11.9
6
£2.0
12.0
Michigan.
Miscellanous Diseases...
4
10.0
3
16.6
2
17.3
1
17.4
5
18.9
7
22.9
6
23.2
18.9
Wisconsin.
Minnesota.
Nebraska.
Typhoid and Typhus 1
Fever j
6
4.8
7
5.2
4
67
3
6.8
5
6.9
1
7.6
2
8.5
6.6
Malarial Fever
1
1.7
3
1.8
7
1.9
2
2.3
4
3.3
5
3.8
6
11.8
3.8
1
1.0
3
1.0
9
1.3
/]
1.3
5
1.3
6
1.5
7
2.2
1.4
3.
3
.9
2
1 4
4
1 4
7
1.5
1
3
1.7
1 0
5
3.6
1.6
2.4
.5
6.3
(1
5.8
2.3
tSJ
7
.2
2
.7
.9
4
8
1
.9
5
6
3.6
1.3
New Jersey.
2
.8
1
7
.9
.3
3.4
3
1 1
4
2.0
.4
6
5
3.5
1.7
Pennsylvania.
6
.3
1
.3
2
3
.3
4
5
7
.0
.4
Other Zymotic Diseases.
4
2.8
2
3.2
7
1
3.6
3
3.7
5
6
18.2
5.9
4.
.
Ohio.
2
1.6
1
1.8
4
1.9
7
1.9
6
2.2
5
2.2
7
2.6
2.0
Indiana.
6
1.1
2
1.5
4
1.6
7
1.9
1
2.1
3
2.1
5
2.2
1.8
TIT
■VO
7
16.9
9
10.9
4
18.5
1
20.8
0
21.0
5
21.5
T
22.2
19.7
Iowa.
§
a
Other Constitutional )
Diseases .. (
3
1.0
4
1.0
2
1.4
5
1.4
7
1.4
1
1.7
6
2.0
1.4
Kansas.
Apoplexy
2
3.8
4
4.2
3
5.0
1
5.2
7
6.8
5
7.2
6
8.2
5.8
5.
Cc
3
O
Congestion of Brain
Paralysis, Softening, |
etc. of Brain (
1
2
1.5
5.2
2
4
1.5
6.2
7
3
1.7
6.5
3
1
2.1
7.2
4
7
2.5
7.3
5
6
29
7.5
6
5
4.9
7.8
2.4
6.8
Delaware.
Maryland.
§
Epilepsy and Convul- (
sions \
1
.3
2
.3
4
.4
5
.5
3
.6
6
.7
7
1.5
.6
Dist. Columbia.
Other Nervous Diseases.
2
.7
6
1.0
7
1.1
4
1.1
3
1.2
1
1.2
5
1.7
1.1
Virginia.
Kentucky.
Missouri.
0
4.1
4
4 4
1
6.1
5
6.3
6
5
6 6
3
7.1
7
7.4
6.0
V
S
Other Circulatory Dis?-1
cases j
6
.2
4
9
3
.2
2
.3
.4
1
.5
7
1.8
.5
6.
3
6 6
1
7 3
9
8.5
1.7
7
9.0
4
9.8
1.8
5
10.8
fi
12 6
9.2
South of 36° 30'.
>>
&
Congestion of Lungs
7
.9
3
1.2
2
4
1.7
1
5
2.2
6
2.2
i.7
North Carolina.
Bronchitis and Pleurisy.
3
1.5
5
1.8
1
1.8
2
2.0
4
2.0
7
2.2
6
3.4
2.1
South Carolina.
I
Abscess, Hemorrhage, |
etc. of Lungs )
7
1.8
6
1.9
1
1.9
3
2.0
2
2.0
5
2.4
4
2.4
2.1
Tennessee.
Other Respiratory Dis- j
1
.4
2
.4
7
.7
4
.7
3
.8
5
.9
6
1.4
.8
Georgia.
Florida.
Alabama.
Diseases of Stomach
7
1.3
1
1.6
4
1.9
3
2.0
5
2.1
2
2.3
6
5.1
2.3
Mississippi.
JO
Diseases of Bowels
7
1.6
1
1.9
2
2.2
3
2.5
4
2.5
5
2.5
6
5.8
2.7
Arkansas.
7
.3
6
.5
1
.7
5
.7
2
.8
4
.8
3
1.0
.7
Louisiana,
Diseases of Liver
2
2.4
1
2.7
3
3.7
4
3.7
5
3.9
7
4.2
6
4.8
3.6
Texas.
Other Digestive Dis- |
eases . \
7
1.7
1
1.7
3
2.0
2
2.3
4
2.3
5
2.7
6
5.8
2.6
6
.3
4
.4
1.9
3
.5
2.1
2
.5
1
.5
7
.6
5
.8
.5
7.
Diseases of Kidneys
2
1.4
4
5
6
2.4
7
2.6
3
3.1
1
3.5
2.4
Pacific, etc.
i
Other Urinary Diseases.
4
.5
7
.7
2
.8
1
.9
5
.9
3
1.1
6
1.2
.9
Washington.
§
j>>
Abscess, Hemorrhage,)
Old Age f
Debility, Exhaustion, )
Prostration j
7
4
.3
.7
2
2
.8
1.0
6
1
1.0
1.0
3
6
1.1
1.1
4
7
1.1
1.2
1
5
1.2
1.3
5
3
1.3
2.4
1.0
1.2
Oregon.
California.
Utah.
1
Accidents and Injuries..
3
.6
1
7.2
4
9.0
5
9.3
2
9.7
7
12.8
6
13.3
9.6
Dakotah.
1
1.3
3
1.3
4
1.3
6
1.4
5
1.8
2
2.1
7
3.3
1.8
New Mexico.
| Unknown Causes
2
1.0
3
1.1
4
1.1
7
1.3
5
1.4
1
1.8
6
2.5
i.5 MEDIC A L ST A TIS TICS.
57
more favorable rate of 17 consumptive deaths to 10,000 living. In establishing these
important conclusions, the statistics entirely concur without discordance. Invalids
experiencing this or any other disease will be interested to note the effect of climatic
conditions shown in the accompanying Sanitary Survey of the United States.
Relation of Height and Weight.
For convenient reference, the following Standard Table is introduced, showing the
average or most healthful Weight relative to a given Height or stature. The medium
circumference around the chest is added, although of less importance. “As a rule,
twenty per cent, or one-fifth is almost the maximum variation from this within the
limits of health.” The Table is especially applicable to the medical examination of
males between 30 and 60 years of age, relative to consumption.
IIetght.
Weight.
Medium Chest.
5 feet 1 inch Should weigh 120 lbs. 34.06 inches.
5 “ 2 “ “ “ 125 “ 35.13
5 “ 3 “ “ “ 130 “ 35.70 “
5 “ 4 “ “ “ 135 “ 36.26
5 “ 5 “ “ “ 140 “ 36.83 “
5 “ 6 “ “ “ 143 “ 37.50 “
5 “ 7 “ “ “ 145 “ 38.16 “
5 “ 8 “ “ “ 148 “ 38.53 “
5 “ 9 “ “ <£ 155 “ 39.10 ££
5 “ 10 “ <£ <£ 160 a 39.66 ££
5 “ 11 “ ££ a 165 “ 40.23
6 “ 0 “ K ££ 170 " 40.80 “
SECTION III.
The Germ Theory of Malarial Diseases. Table of Sickness
and Recovery in Adult Life.
The statistics of Fever in Table IX comprise nine or ten kindred varieties, all of
which are included in the general class of Zymotic diseases, of which the family of
fevers constitutes about three-fifths. Malarial Fever is used to designate yellow fever,
remittent fever, and intermittent fever or fever and ague; to which some would add
one or two other varieties, with a part of the cases of fever unspecified. Table
XVII (C) will show that Zymotic diseases including fevers, have been the most
prevalent in the groups of States, V and VI. And the more extended Table VII will
suggest the separation of Alluvial counties, as a class, in which the greatest mortality
has occurred. It may also be observed, that a large portion of the insurances in
group V are located in the chief cities, including Baltimore, Washington and St. Louis.
The space already given to medical Statistics, will here forbid a more extended
discussion of fevers and other diseases. Yet the growing importance of the germ
theory of miasm, and its favorable acceptance among physicians, may claim a passing
notice.
Our knowledge of the phenomena of fevers, is announced to be especially pro-
moted by discoveries with the microscope. In examining the organs and secretions
of persons who have died of fever, the membrane lining the stomach was almost 58
MEDIC A L S TA TIST1CS.
invariably covered with a multitude of microscopic plants. These often covered the
whole intestinal tract; some were found on the surface of the lungs, and some were
detected in the blood. On living patients, these parasites have been found in the
Substance, which is formed at an advanced stage of the fever, at the corners of the
mouth and eyes. In the secretions of persons in full health they cannot be detected.
Water taken up and placed in the rays of the sun till stagnant, became coated with
a green superficial film, on which the microscope showed plants closely resembling
those found in the human body. Some of these parasites showed cell-articulations;
others were hollow and others showed nuclei and spores.
A writer on the causes of Ague remarks that of these “ague plants” is formed
the greyish film wherever black damp earth is turned up and exposed to the sun.
Thence issue innumerable atomic seeds or spores which rise into the air, carrying
pestilence with them. The danger from their growth is greatest in a hot dry season
following a wet one.
By day, the atmosphere is shown by the microscope to be free from those organ-
isms. After sunset, the spores of the “ague plants” rise with the evening dews,
recalling “ the pestilence that walketh in darkness.”
In different parts of the world, these cryptogamic spores rise in the night mists,
to definite heights. In the United States, they seldom rise more than thirty-five to
sixty-five feet above the low levels; in England, not more than from fifteen to thirty feet.
But the spores of the “ague plants” having risen and become entangled in the mists,
may all he carried by the wind, far perhaps from the place of germination.
“Around the entire continent of Africa, except where the headlands jut out into
the sea, there extends a belt of low malarious country from twenty to sixty miles wide.
Within this belt it is unsafe for an unacclimated person to spend a night, although, as
the malaria does not extend higher than about twelve feet from the ground, it is
believed that it greatly conduces to health to sleep in the second or third story.
Travelers visiting Africa should at once go to the higher lands, not allowing themselves
to sleep a single night on the coast.”
According to this theory, the therapeutical effect of cinchona bark and its alka-
loids, in malarial fevers, is due to a process of withering or destroying the fast growth
of the poisonous fungoids in the human system. Recently, Pasteur has extended inocula-
tion with “cultivated ” germs of disease, to protect life against the more malignant types.
TABLE OF SICKNESS, RECOVERY AND MORTALITY.*
Precise
At the Beginning or the Yr. of Age.
In the ensuing Twelve Months.
Weeks of
Sickness.
Age.
Number
Living.
Number
Healthy.
Number
Sick.
Cases of
Sickness.
Cases of
Recovery.
Died.
Average.
25
1000
981.3
18.7
222.4
214.2
7.0
.969
(1872.)
0.78
30
1000
980.5
19.5
206.6
197.1
8.5
.998
0.86
35
1000
980.7
19.2
206.6
197.5
8.5
1.024
1.01
40
1000
977.7
22.3
209.4
198.0
11.4
1.160
1.17
45
1000
975.3
24.7
212.4
199.8
12.0
1.304
1.48
50
1000
968.8
31.2
227.1
211.6
14.8
1.644
1.98
55
1000
962.2
37.8
244.4
224.6
19.8
1.973
2.72
60
1000
950.4
496
265.1
233.8
27.6
2.684
4.03
65
1000
927.6
72.4
282.6
242.9
34.4
3.913
6.30
70
1000
885.4
114.6
303.7
236.1
55.1
6.306
10.08
75
1000
811.8
188.2
307.8
231.8
78.2
9.760
15.07
80
1000
723.6
276.4
298.6
183.6
135.4
13.664
20 71
* Abstract of 800,000 observations in Friendly Societies, from Assurance Magazine, Vol. 16, p. 413. To show Healthy
Life, cases of chronic sickness and the returns from sailors, miners and colliers were left out by Mr. Finlaison. Only the
last column, from Mr. Ratcliffe’s collection in (1872), includes All Classes. TIME AND MONETARY TABLES.
59
DIVISION FOURTH.
TIME AND MONETARY TABLES.
Eon Interest and Annuities Certain.
SECTION I.
The System of Interest and Annuities.
By general custom, interest is understood to increase uniformly, by equal incre-
ments of interest in equal times, during twelve months or the period of conversion, and
is then added to the principal, to form a new principal. When the interest is added at
the end of each successive year, it is said to be compounded annually; when the interest
is added to the principal at the end of every six months, it is said to he compounded
semi-annually, and so on.
When the rate of interest is 6 per cent, for example, the increase on $1 principal is
$0,005 monthly, which at the end of twelve months or a year, amounts to $1.06. Or
omitting the character $ or £ the yearly interest on 1 unit of money is in this case
0.06. Let i denote .04, .05, .06 or any other rate of interest on 1 for one year; then
will 1-f-i denote the amount at the end of a year. By proportion, 1 : l + i: : new
principal 1 -\-i : (l + «)2, the amount at the end of two years. In like manner (1 + t)3
will he the amount of 1 at the end of three years. And in general, (1 + i)n expresses
the amount of 1 at the end of n years, with compound interest.
The present value of 1 payable at the end of a year, is such a sum v as with
its interest, will exactly amount to 1. By proportion 1 : 1 + i :: v : 1; whence
v = —. = (1-f-f)—l. Thus defined, v is die present value of 1 receivable at the end
J- T ®
of a year; and vn is the present value of 1 payable at the end of n years.
Annuities. On this principle, the whole amount of 1 per annum, payable at the
end of each year, with compound interest, will at the end of any number n of years,
evidently be the sum of the geometric series,
(1+i)«-i + (1 + »)*-* + .... + (1+i)+1 = (l + lYi___1 _ A
In like manner, the whole 'present value of 1 per annum, payable and discounted
at the end of each year, will be, for n years,
o O V — Vn+i 1 —(l + i)—71 T,
v -f- v3 -f v3 + .... + v'n = — = — — = V.
1—v i
It may be proper to observe, that the sum of the preceding series has long since
been demonstrated without geometrical progression. Thus at the rate of interest i, 60
TIME AND MONETARY TABLES.
the annual income from the so-called perpetuity \ will evidently be 1, to continue for-
ever. Deducting from this, the present value of the perpetuity deferred n years, that is
(l-i-i)~n
V-Vl—} we have the expression equal to V above, or the present value of n annual
payments of 1. And multiplying this whole present value by (1 + i)n, we reproduce the
above amount A for the end of n years.
In practice, these formulas are seldom computed; since time and labor are saved
by recourse to the standard Monetary Tables of present values and amounts, calculated
for various rates of interest. And generally, $1 and its tabular result, or the latter
alone when multiplied by the proposed principal or number of dollars, gives the value
required, in practical business.—For the Value of an Increasing Annuity, see page 276.
Table XVIII is a Time Table giving in triple columns the day of the month and
the day of the year, also the corresponding decimal of a year of 365 days. The star (*)
designates Monday, or one day in the week (as shown in the last column), from which
any adjacent day of the week is readily noted. Also from any given date, the interval
to the end of the year is found by subtracting the day of the year from 365, in com-
mon years, or the corresponding decimal of a year from 1; to be increased for any
extension into the following year.
SECTION II.
Interest Tables for Days.
Required the interest on $4500 from May 27 to Sept. 8 at 6 per cent. The
difference of the days of the year in the Time Table, 147 and 251, gives the interval 104
days ; which guides in the same Table, to the fraction of a year 0.28493. The interest
for a whole year is 4500 x.06; that is 270; and the product of 270 by the preceding
fraction gives $76.93, the interest required. But the next Table is designed to give
the interest by a more convenient process.
Table XIX. Simple Interest for Days at 5 per cent. The whole year is accounted
to be 365 days ; but if required to change the resulting interest to the basis of 360 days
to the year, add fa part to the tabular result.
It will be seen that the interest for da}Ts is a circulating decimal of eight places,
which can be continued as a repetend to any degree of accuracy. The new arrange-
ment here tabulated for the first time, was suggested from the previous construction
of the Copy Multiplication Table to be presently described. Both give the products
at sight, to be copied and added as in “ contracted multiplication.”
The given interval of Days, guides first to the Reference in the upper part of the
Table; and the Reference guides to the required multiples of interest id the body of
the Table. For illustration, the interval of 111 Days gives the Ref. 104 near the top
of the page; and 104 guides below to the line which shows the interest on 0, 1, 2,
3, ...., 9. It is needful to observe that after the first three horizontal figures in each
column, the other final figures are to be read vertically downward to the drawn line
below, then recommencing at the drawn line above to read down again. Indeed these
eight final figures between drawn lines, are a repetend or period of eight figures, which
may be repeated to any extent. Thus for 111 days (Ref 104), the interest on $1 is
$.015 or the horizontal figures to which are annexed the vertical figures 205479 .. .
making the interest on $1 to be .01'52054794'52054 ... In like manner, the interest TIME AND MONETARY TABLES.
61
on $5, by column 5, is .07'60273972'60.. .; and on $9, it is .13’68493150'68. .. The
decimal point thus occurs on the left of the tabular interest when the dollars are 9 or
a less integer. And the accents merely indicate the part continually repeated.
For any other rate, the interest is first found by the Table at 5 per cent.; and this
increased by one-fifth part gives the interest at 6 per cent., by two-fifths gives the interest
at 7 per cent. etc. From the interest at 5 per cent, subtract one-tenth part to give the
interest at 4£ per cent., or add one-tenth part to give the interest at 5|- per cent. etc.
Example 1. Required the interest on $85,762.00 for 141 days at 7 per cent.
$85762.00
1545.205 8
096.575 5
13.520 7
1.158 6
.038 2
Interest at 5 per cent. - -- -- -- - 1656.496
Add | or - - - - 662.60
Interest at 7 per cent. - $2319.10
The right hand figure of DAYS occurs along the head of
Table XIX, Ref. 110. Each product is copied to begin one place
to the right of its digit in the multiplicand, to bring all the deci-
mal points under each other.
Example 2. What is the amount of $593,704.75 from May 15 to August 28, at
5 per cent, per annum ?
$593,704,750
07,191.780 5
Interval 105 Days, 1,294.520 9
043.150 3
Table XIX, Ref. 132. 10.068 7
.057 4
10 7
, ? 5
Amount $602,244,335
Copy Multiplication Table. In this Table, from which the Interest Table origi-
nated, the carrying figure is added to the next product figure, as usual, and the sum
given by inspection. It presents within a convenient compass, a generalization of the
common process, and was suggested to the writer, by the early design of “Napier’s
rods or bones,” by the first Table of Crelle, the Tables of Laundy, and others. For
large numbers, or “contracted multiplication,” it appears specially adapted, as well as
for “proof,” or correction of the common method, and various other objects.
The bold types or figures of the Table, taken singly, represent every possible
multiplicand, however extended. These are repeated immediately over their product
figures, which may be copied rapidly in any order, as from left to right. In a few
places, the construction of the Table required 10 to be tabulated as a single figure,
denoted by X or x, and so Avritten. But after the usual addition, this character will
disappear in the final product. Also by practice, an easy habit may be acquired of
reading off two figures at a time. Thus on reading 94 (nine, four) of the multiplicand,
look first under 9 only, to read off its product figure 7, passing immediately under 4 to
find the other figure 1, and copy 71. The same routine must be strictly followed in
every example, as follows:
1. Prefix a small cipher to the multiplier to distinguish it, and to be used as its
first figure. On account of “carrying figures” in the construction of the Table, all
single ciphers, A\dien effective, are to be treated like significant figures. 62
TIME AND MONETARY TABLES.
MULTIPLICATION.
0
100
200
300
400
nn
01
23
45
67
89
01
23
45
67
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uu
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00
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00
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00
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01
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00
11
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83
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61
DU
CQ
01
23
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00
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72
71
bo
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01
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89
01
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01
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04
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01
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00
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94
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UU
0
100
200
300
400
For intermediate numbers in the side column, use the upper or less. Thus, for 25, 26, 27, or 28, use the line of 25. TIME AND MONETARY TABLES.
63
MULTIPLICATION.
500
600
700
800
9G0
on
01
23
45
67
89
01
23
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67
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01
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05
05
05
05
05
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28
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07
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63
08
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05
05
05
05
06
06
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07
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01
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05
05
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01
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1
05
05
05
06
16
06
28
40
63
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07
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74
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53
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32
ID
1 7
01
23
45
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01
23
45
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01
23
45
67
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01
23
45
67
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23
45
67
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05
05
05
16
16
06
28
40
73
95
07
41
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3x
74
08
64
20
97
53
09
87
65
54
32
17
20
01
23
45
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01
23
45
67
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01
23
45
67
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01
23
45
67
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01
23
45
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05
05
06
16
16
06
28
41
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07
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30
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21
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53
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32
.
The Present Value of 1 due n years hence, with interest compounded m times in
the year, is
(i \ —mn
In the case of Annuities payable half-yearly, quarterly, or generally m times in
the year, let us firstly assume the interest to be compounded in the same way, or m times
in the year, and that the annuity or yearly payment of 1 is divided into m payments of
each, at equal intervals. Then by substitution as shown by Dr. Price:
1 m V / i\mn
Amount of Annuity for n years = — . - 14— —1 .
J mi |_\ mj
lp / i \ ~mn
Present Value of Annuity for n years — j 1 —(1 + —j
When payable yearly, w = 1; half-yearly, m quarterly, m = 4, 70
TIME AND MONETARY TABLES.
Secondly and more generally, let interest be convertible m times m a year, and the
annuity be payable m! times, each payment being In this case, the discount factor
/ £ \ —m
to give the present value for one year, is (1 4-—J i or for t years, or a fraction, the
/ j \ —mt
factor is II4- —j . Now dividing one year by the number of its payments of
annuity, we have = t, the time or interval from one payment to another. Hence
the present value of the mn payments in n years, will be the sum of the geometric
progression ; since mt = 1:
1 r L i\-mt L i \-2mt L
m \_\ mj \ mj \ m)
/ » \ —mt / j \ —nrtn—mt / / \ —mn
i -K) x-L1+iI_
m'x u(i+ip M/X (1+4p-i'
When m and in' are equal, this manifestly coincides with the last formula on
page 69.
In the denominator, developing by the binomial theorem, since = t,
t i\mt_i_± fl l~ Pi / 1 1 \ / 1 2 W2
\ m'_ \m' m/2\m' m/\m' m/1.2.3
Substituting the series in the above denominator, and dividing,
„ , _ri r1 i/i i\ /1 i \ %m) n r,
Present Value = + - 7 — —» rJ.—fs—-+ • • • x 1— 1-1—
_i 2\m m J \m2 m2) 12 J |_ \ m/ _
In tlie case of annual interest, or m = 1, we have,
Present Value = v + ~^). [ 1 — (1 -pi)-”], nearly,
1 — (1+ m' — 1 . TT m—1
=—i— +-^’ =v»+^--
When the annuity is half-yearly, or m'= 2, the last term becomes 0.25; which
agrees with the usual approximate correction for Life Annuities. More accurately,
when i = .04, the third term of the series above, when united to the second, gives
0.24755. This correction from Annuities Certain, strikingly agrees with those com-
puted by Mr. Sprague from Life Annuities at ages where the initial rate of mortality
plus that of interest equals 4 per cent. Compare Journal of the Institute, Vol. 13,
pp. 212, 308.
Finally for illustration, the annexed Table from David Jones on Annuities will
indicate the effect of compounding oftener than once a year to increase the yearly rate
of interest. In long periods of years, the difference becomes more material; TIME AND MONETARY TABLES.
71
AMOUNT OF fl IN ONE YEAR, AND ITS LOGARITHM.
Annual
RATE OP
Interest.
Pay-
able.
Amount in
one Year.
Logarithm.
Annual
RATE OP
Interest.
Pay-
able.
Amount in
one Year.
Logarithm.
2
Y
1.020000
.00860 01718
5
Y
1.050000
.02118 92991
per cent.
H
1.020100
.00864 27476
per cent.
H
1.050625
.02144 77:08
Q
1.020150
.00866 42470
Q
1.050946
.02158 01275
M
1.020201
.00868 58896
M
1.051271
.02171 41241
21
Y
1.025000
.01072 38654
6
Y
1.060000
.02530 58653
per cent.
H
1.025156
.01079 00638
per cent.
H
1.060900
.02567 44494
Q
1.025235
.01082 35735
Q
1.061364
.02586 41690
M
1.025315
.01085 73620
M
1.061837
.02605 76689
3
Y
1.030000
.01283 72247
7
Y
1.070000
.02938 37777
per cent.
H
1.030225
.01293 20845
per cent.
H
1.071225
.02988 06996
Q
1.030339
.01298 02193
Q
1.071859
.03013 76716
M
1.030454
.01302 88345
M
1.072508
.03040 06137
Y
1.035000
.01494 03498
8
Y
1.080000
.03342 37555
per cent.
H
1.035306
.01506 88358
per cent.
H
1.081600
.03406 66786
Q
1.035462
.01513 41909
Q
1.082432
.03440 06870
M
1.035620
.01520 03069
M
1.083287
.03474 35586
4
Y
1.040000
.01703 33393
9
Y
1.090000
.03742 64979
per cent.
H
1.040400
.01720 03435
per cent.
H
1.092025
.03823 25809
Q
1.040604
.01728 54951
Q
1.093083
.03865 32667
M
1.040811
.0173717793
M
1.094175
.03908 65034
Note.—The period required for a given principal to double itself at compound interest, is
defined by the well-known equation for n years,
(1 + i)n = 2, n — , = —.— + 0.35 nearly. Also (l + —) = 2.
v log.(l + i) i J \ m)
Thus, dividing 70 by the rate of interest as an integer, will give a close approximation, easily
remembered. In the last formula making m = 1, 2, 4, and infinite as before, we find the following
more exact series.
Doubling Period in Years and Decimals.
Interest compounded yearly:
Rate per cent. 2 3 4 5 6 7 8
Doubling Period. 35.004 23.450 17.673 14.207 11.896 10.245 9.00G
Interest compounded semi-annually:
Rate per cent. 2 3 4 5 6 7
Doubling Period. 34.830 23.278 17.502 14.036 11.725 10.075 8.837
Interest compounded quarterly:
Rate per cent. 2 3 4 5 6 7 8
Doubling Period. 34.743 23.191 17.415 13.946 11.639 9.989 8.751
Interest compounded momently:
Rate per cent. 2 3 4 5 6 7 8
Doubling Period. 34.657 23.105 17.329 13.863 11.552 9.902 8.665 TABLES IA PART FIRST.
MORTALITY AND LOSS EXPERIENCE, Tables I-IY. Pages 74-166
CLIMATIC TABLES, - - “ V-YIII. “ 167-181
MEDICAL STATISTICS, “ IX-XVI. “ 182-198
TIME AND MONETARY TABLES, - - - “ XVII-XXV. “ 199-231 74
TABLE I.
MALE LIFE. SUMMARY (A). OBSERVATIONS CLASSIFIED ACCORDING TO THE AGE
OF EXIT OR EXPOSURE.
Age
at
Exit.
Whole
Number of Entrants, 988,734.
2,568,856 Thousand Dollars Insured.
Age at
Exit or
Exposure.
Existing.
Discon-
tinued.
Died.
Exposed to
Risk.
Amounts
Existing.
Amounts
Discontin’d.
Death
Claims.
Exposed to Risk.
0
1
3.5
1
6.5
o- 4
1
9
1
34
18
2
66
4- H\
2
8
7
69
19
9
144.5
n- H
3
6
8
86
13
26
186
24- 34
1+
5
8
99
11
15
211.5
Si- 1+i
5
10
12
107
19
25
230
1+i- 51
6
8
9
105.5
16
19
230.5
5i- 6}
7
10
10
2
110
20
21
6
241
04- 7i-
8
14
15
108.5
31
24
237
71- 84-
9
7
11
1
107.5
23
25
1
228.5
8i- 94-
10
20
14
139
49
20
289.5
94-104
11
20
28
174
42
43
361.5
104-114
12
32
35
239
58
59
490.5
u*-m
13
34
46
2
361.5
76
77
3
741.5
m-isi
u
74
97
3
732.5
129
199
3
1,499
m-ui
15
96
212
3
1,405
226
388
4
2,863
m-m
16
224
394
11
2,375.5
411
764
17
4,632.5
154-164
17
353
615
24
3,693
667
1,132
36
7,134
16i-17i
18
639
972
32
5,736.5
1,188
1,817
55
11,035.5
174-184
19
940
1,782
68
8,857.5
1,666
3,261
118
16,774.5
m-m
20
1,536
2,845
116
13,773.5
2,783
5,308
200
26,120
194-204
21
2,383
4,323
158
21,040.5
4,376
8,210
266
41,378.5
204-214
22
3,246
6,335
200
30,596
6,297
12,672
333
62,732
214-224
23
4,452
8,488
312
42,003
8,711
17,543
609
89,221.5
224-234
21f
6,044
10,333
326
54,367.5
12,056
22,187
707
119,483
234~24i
25
7,309
11,753
468
67,546.5
14,780
26,663
1,002
152,867.5
21+4-254
26
8,879
13,346
530
80,629
18,739
30,876
1,229
186,725
254-264
27
10,221
13,891
639
92,781
21,831
33,441
1,452
218,231.5
264-274
28
11,799
14,782
758
106,513
25,843
35,733
1,734
250,743.5
274-284
29
12,876
15,751
784
118,901
28,841
38,245
1,925
281,450
284-294
30
17,480
16,082
819
129,868
32,578
40,227
2,093
312,842.5
294-304
31
15,092
16,571
961
136,354
34,273
42,664
2,410
339,736
304-314
32
15,991
16,130
979
143,956.5
37,781
42,288
2,444
361,484
31i-32i
33
16,618
16,148
1,057
151,141
38,901
42,554
2,788
381,961
324-334
31+
17,784
16,351
1,193
157,277.5
42,133
43,599
3,213
400,764.5
SSi-Slti
35
17,712
16,286
1,196
161,673.5
43,697
43,855
3,234
414,914.5
31+4-354
36
18,444
16,477
1,218
165,280.5
46,172
44,175
3,380
427,076
354-364
37
18,806
15,497
1,299
166,330
47,066
43,243
4,028
432,046
364-374
38
18,850
14,888
1,403
166,100
45,893
41,408
3,865
432,402
374-384
39
19,317
14,325
1,398
164,729.5
47,916
40,246
3,931
432,874
384-394
i+o
19,351
13,675
1,401
162,619
48,845
38,935
3,823
430,914.5
394—I+O4
1+1
19,683
13,234
1,430
158,904.5
49,273
37,094
4,091
425,161.5
I+O4-I+I4
J+2
18,465
12,134
1,401
152,323
46,579
35,982
4,023
413,268
i+H-m
1+3
17,165
11,438
1,456
146,097
44,676
33,604
3,941
401,060
l+lii—1+8i
u
17,305
10,671
1,446
139,838
44,846
31,433
4,055
387,327
1+34-1+1+4
1+5
16,605
9,847
1,244
133,033.5
43,646
29,657
3,453
372,103
i+i+4~1^4
1+6
16,518
9,235
1,339
126,308.5
43,846
27,922
4,012
356,838.5
1+54—1+64
1+7
15,230
8,397
1,325
117,740
41,080
25,782
3,827
336,969
1+04-1+74
1+8
13,811
7,536
1,280
109,560.5
37,684
24,249
3,905
317,001
1+74-1+84
1+9
13,191
6,891
1,157
102,105
36,291
21,180
3,565
298,249.5
1*84—1+04 table i .
75
MALE LI'E. SUMMARY (A). OBSERVATIONS CLASSIFIED ACCORDING TO THE AGE
OF EXIT OR EXPOSURE.
Age
at
Exit.
Whole Number op Entrants, 982,734.
2,568,856 Thousand Dollars Insured.
Age at
Exit or
Exposure.
Existing.
Discon-
tinued.
Died.
Exposed to
Risk.
Amounts
Existing.
Amounts
Discontin’d.
Death
Claims.
Exposed to Eisk.
50
12,511
6,243
1,192
94,951.5
34,761
19,532
3,554
280,485
49^-504
51
11,365
5,780
1,187
87,291
32,004
18,619
3,606
260,386
504-514
52
10,643
4,935
1,108
79,336.5
30,397
16,366
3,487
238,306.5
514-524
53
9,686
4,413
1,032
71,744
27,339
14,529
3,227
216,122
521-534
54
9,480
3,849
988
64,574.5
27,483
13,619
2,975
195,270
534-544
55
8,335
3,294
982
57,089.5
25,161
11,550
3,183
172,957.5
544—554
56
7,582
2,824
876
50,184
22,555
9,552
2,744
151,660
554-564
57
6,519
2,360
813
43,729
19,400
8,194
2,718
132,555
564-574
58
5,612
1,922
756
38,164.5
16,246
6,837
2,313
115,575
574-584
59
4,985
1,640
721
33,284.5
14,851
5,756
2,300
101,140.5
554-594
60
4,178
1,348
733
28,892
12,630
4,891
2,102
87,621
594-604
61
3,637
1,060
649
24,959.5
11,274
3,784
2,130
75,419.5
6O4-6I4
62
3,313
798
531
21,189
9,528
2,915
1,624
63,379.5
6I4-624
63
2,849
605
509
17,725
8,712
2,275
1,686
53,175
624-684
61+
2,333
499
457
14,678.5
6,748
1,907
1,336
43,325
684—644
65
2,056
407
416
12,116.5
6,023
1,258
1,517
35,712
644—654
66
1,634
311
368
9,732
4,797
1,204
1,080
28,233
654-664
67
1,395
246
271
7,716.5
3,867
798
815
22,082.5
60i-67i
68
1,071
150
259
6,042.5
3,176
601
792
17,188.5
671-684
69
837
130
224
4,694
2,610
431
600
13,025.5
681-694
70
586
95
189
3,595.5
1,602
334
528
9,625
694-701
71
469
68
134
2,783.5
1,289
211
353
7,322
701-711
72
365
52
136
2,148
1,043
134
348
5,576.5
73
300
31
93
1,627
749
77
236
4,131.5
721-731
n
216
24
80
1,221.5
512
73
176
3,104.5
734-744
75
146
20
60
914.5
426
37
143
2,382
744-754
76
94
7
53
701.5
305
11
129
1,800
754-764
77
77
10
54
547.5
244
20
96
1,353
764-77i
78
51
8
47
410
135
16
102
998.5
774-784
79
39
7
29
306.5
86
14
86
749
784-791
80
39
6
21
232
109
13
50
563.5
794-8O4
81
28
3
18
168.5
77
7
36
395.5
8O4-8I4
82
16
7
21
119
46
15
38
273
8H-824
83
12
7
79
43
20
182
824-834
8J.
3
1
7
59.5
11
1
13
118.5
834-844
85
12
2
6
48
29
4
11
92
8J,i-85i
86
3
5
29
10
11
50
854-864
87
1
4
21
3
5
29
864-874
88
4
2
16.5
5
2
21.5
874-884
89
1
3
11
1
7
15
884-894
90
1
2
6.5
1
2
6.5
894-904
91
4
4
904-914
92
1
1
4
1
1
4
911-921
93
1
2
1
2
921-931
94
1
1
934-944
95
1
1
1
1
544—654
527,157
411,092
44,485
4,327,086.0
1,328,4041,114,487
125,965
11,493,967.5 76
TABLE I.
MALE LIFE. SUMMARY (B). OBSERVATIONS CLASSIFIED ACCORDING TO YEARS OF
INSURANCE.
Years of
Insurance.
Whole Number op Entrants, 982,734.
2,568,856 Thousand Dollars Insured.
Duration
of
Insurance.
Existing.
Discon-
tinued.
Died.
Exposed to
Risk.
Amounts
Existing.
Amounts
Discontin’d.
Death
Claims.
Exposed to Risk.
0
60,190
46,913
2,943
467,910.5
153,950
99,417
7,787
1,234,719.5
0 - *
1
51,189
158,722
6,428
793,327
132,847
378,590
18,131
2,118,407
4 4
2
49,307
67,850
5,733
622,424
113,202
208,799
16,668
1,673,734.5
4- 2*
3
45,000
44,573
5,114
511,172.5
105,929
140,830
14,531
1,369,050
24 4
4
48,697
30,418
4,623
423,563
117,147
98,064
13,409
1,129,143
34 H
5
48,237
21,452
3,911
344,308
126,106
69,899
11,213
914,605.5
H- H
6
48,663
14,043
3,205
274,412.5
127,146
44,140
9,166
720,267
54 H
7
45,160
11,563
2,569
209,741.5
117,454
30,606
7,311
546,582
64 7*
8
38,200
5,352
1,837
153,555
100,312
15,703
5,045
398,662.5
74 H
9
28,037
3,231
1,344
109,226.5
69,922
8,937
3,525
280,985.5
®4 9*
10
18,210
2,146
982
77,157
43,800
6,006
2,619
200,067
9f-104
11
10,977
1,359
751
56,212.5
26,698
3,688
2,153
148,801
10414
12
5,120
830
647
43,390
13,301
2,349
1,805
116,931.5
14-14
13
3,779
586
519
36,915
10,656
1,730
1,472
99,786
12413*
14
4,123
574
469
32,037
11,331
1,624
1,290
85,981
13414*
15
3,110
304
399
27,006
8,506
843
1,080
72,126.5
14415*
16
2,254
261
401
23,214.5
6,229
734
1,201
61,752
15416*
17
1,916
180
349
20,339
4,706
528
987
53,691
16417*
18
1,707
157
298
17,905.5
4,452
390
951
47,539
174-18*
19
1,377
124
303
15,760
3,914
325
758
41,778.5
18*-19*
20
1,148
110
270
13,963
3,114
337
758
36,775.5
19*-20*
21
950
95
269
12,442.5
2,499
261
771
32,604.5
204214
22
1,019
77
274
11,137.5
2,529
230
837
29,089
21422|
23
1,541
61
261
9,775.5
3,790
184
768
25,516
22423^
24
2,140
35
218
7,925.5
4,977
82
605
20,825
23424*
25
1,946
24
155
5,538
4,679
53
490
15,175.5
24425*
26
1,420
28
105
3,411
4,094
85
329
9,937.5
25426*
27
930
10
54
1,867
2,838
23
161
5,460.5
264-27*
28
467
6
31
875
1,391
9
83
2,445.5
27*-28*
29
189
5
10
371.5
453
17
15
958.5
28*-29*
30
154
1
8
169.5
432
3
31
480.5
294-304
31
7
16
304-31*
32
1
7
1
16
31432*
33
1
6
5
15
32433*
34
2
1
4
1
2
9.5
33434*
35
2
7
34*-35*
36
2
7
354-36*
37
2
7
36J-37*
38
1
2
2
7
37438*
39
1
1
5
5
384-39*
527,157
411,092
44,485
4,327,086.0
1,328,404
1,114,487
125,965
11,493,967.5 TABLE I.
77
OBSERVATIONS ON MALE LIFE.
Excepting the first Aggregates, the Amounts Insured are stated in Thousand Dollars in the
columns $, where 1 denotes $1000, etc.
60 60
t-* O
h-4 ►—4 H-1 k-4 k-4
COCOMQOt
K-4 k-4 >—4 ►—4 ►—4
CO W H O CDCOM (DOi
4- CO 62 I-* ©
►—4 k—4 K-*
£0 h-4 0
CO CO "-5 05 cn
it- CO 62 I—1 O
Years of
Insurance.
39
62 1—1
62 62 if- 62 4-
1—‘ CO Ox 4- CO
3
p
Of
h-4 r-4
62
Existing.
3
O
18
h-i 62
4- CO 02
—L
d
tel
aq
62
l—i h-i
Discon-
tinued.
O
d
tel
S
*4
w
CTQ
21
H
w
CD
P
1
Died.
P_
253.5
h-4
hi
i—*
h-4
1—4
h—4
I—4
M
4
3
1
1—‘ I—1
02 GC GO 4— —3
bx
21
62 CO CO hf- 62
Ip. O *—*1
GO H 0 CO
<¥*
CO
CO CO
62
h- €*■
Existing.
O
Kj
CD
P
28
f—1
62 CO
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0
O
O
1 Yea
it-
OO H
Discon-
tinued.
CO
0
£
Death
Claims.
21
479.0
M■
Or
h-4
h-4
h-4
I—4
h-4
M
I
9
L
h- M H 62 W
1—‘
bx
4- Cn -3 GO Cn
Cn O CO 02 4-
Ox Cn
d
w
w
0
91.5
CO CO 02
02 02 02 02 GO
1—1 l—i
CO O M 62 p®
Cn bx bx
Exposed
to Risk.
d
g
£
d
OQ
tS tQ
to
27
18
19
20
16
<4, K*
Cn-fc- c© ts> >-< Q^cooo^
02
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kvA k<
Ka.
CO Oo 05 O,
■fc- Ca to Ki Ci
Age at Exit,
►t- CO to t—‘ o
10
CO CO 05 Cn
4- CO 62 t-* O
►—4 ►—4 ►—4 h-4
CO JO M 0
CO 00 -5 05 cn
►12* CO 62 1-1 0
Years of
Insurance.
62 H-
h-4
3
O
J—4
k—4
k^
1—‘ CO 1—‘ h-1 62
k-4 I—4
%
O
62
CO
HP 1—1 CO if—
l—i it- — 4-
Existing.
p
I—*-
2
2
3
d
w
0q
CD
P
h-4
Ol
l-i 62
H-i
1
2
5
3
d
w
K**
Oq
CD
P
h-t
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1—1 62
62 H- CO 02
Discon-
tinued.
0
d
tel
aq
CD
P_
a
H
w
*3
w
►
h-4
h-4
Died.
21
H
5d
1—‘ 62 62 62 1—*
02 o 4- os 4-
Cn bx
21
t-3
CO
iO
00
d
p
c+-
ro
p
k—
k-4
h^
b\
h-4
l—i
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bx
t—1■ 1—'■ l—i 62 t—1
62 4- CO CO 62
Ox Ox Ox Cn
21
H
5°
w>
05
d
ft-
co
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1G4.0
h-4
k-4 h-4
h-4
M Of Oi 0 Q
Cn
i—i 62 62 CO l—i
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bx bx bx
Exposed
to Risk.
21
H
OD
CO
50
d
2
C"k
4- 62
62
10
Of
62
CO -3 62 CO Cn
k-4 JNC
53
OO
h-4
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62 CO l—i O
Existing.
62 Cn 4- CO
05
8
37
Ox CO
h—4
CO GO CO h-4
js
s
4-
O
CO CO
62
4- 62 02 62
Discon-
tinued.
0
8
a
Kj
HH
g
►—1
2;
CD
P
t/2
h~4
1—1
Death
Claims.
O
P
52.5
49
42.5
35
28
d
w
£
0
62
GC
©X
Ox
62
1—> 1—1 1—‘ 62
Ox 62 4-- CO CO
bx bx
CO
•cn
CO 4- Cn CO
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Ox Of
W
d
373.0
CO
3
3
3
1—1 1—1 62 CO
CO 1—1 CO 62 02
Cn bx
Cn Cn -3 42- „
62 if- CO CO w
Exposed
to Risk.
d
-
£
d
OQ
Oo -3 <35 CJi-fc»
2-5
'Cl Co Oo
*
OCS
CD
js
C3
t—4
M M fcS M
H
t>
CK3
t-i CO
Discon-
tinued.
M
s
W
H4
i—i
2
H
~
S-
Died.
3
g
SB
80.0
1—‘ 1—i
M■ 03 05 05
bi bi bi
9.5
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h3
CD
lO
p
bd
P
r+-
CD
*1
79.0
i—11—1
62 CO 00 Or -3
Or
I—i t—1
03 CO 03 Oi CD
Oi Oi Or
5
H
m
oo
tel
r+-
165.0
h-1 1—1 62
I—i I—i I—i
<3 CD O CO 05
bi bi
Exposed
to Risk.
5
>3
CD
8
tel
c+-
05
3
4
3
Oi
4
5
62
T
*v2
OO
62
62 CO 62
62 t—i 62 Oi
'-<5
or
Kj
GO
Oi
62 CO rf*.
-3 Oi 62 CO 62
Existing.
<¥*
l
2
3
6
2
1
h-4
c^
M4
CO 62 -3 62
03
62
O
h-i Oi
Discon-
tinued.
H
5?
CD
g
Or
Or
£
CD
S3
l-S
Death
Claims.
S'
CD
CD
e
185.0
h-4
w^o
36
33.5
25
21
15
15
19.5
d
a
a
«
GO
•<£
Oi
o
62 62 62
i—i t-i
-3 O rfn-
bi
H-i 62 CO CO 62
» 0q fcfc)
CO Oo S Cj O,
Ka Ka
VJ 03 CJi4>-
Ka
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Age at Exit.
62 60 62 60
CO 60 t—1 o
H4 H4 h-4 H4 h-4
CD CO «
GO
f—4
CO t—i 62 CO Oi
h-4
H4 6S
&
Existing.
JS
>—4
CO
H4
l—i 62 rf*. CO -3 t—i
62
30
Oi 62 -3 t—*
Discon-
tinued.
K)
2
CD
CD
SB
i-J
H4
t-i
Death
Claims.
w
CD
g
245.5
o a o
bi
H^
-3 -3 -3 h-1
15
14
13.5
13
13
I—i 62 62 I—i
CD 62 CO -3
Oi Oi Oi
5d
S
p
02
i—i
CD
CD
O
H4 H4
t-i 62 62
00 CO O 03 03
bi bi
Exposed
to Risk.
3
B
a
Cg Ca 6a
S-* CO Co Oo
Ot> 4>-
is is is
io C3 CO Oo ~3
12
13
U
15
16
8
9
10
11
8^1
03 Oi-t~ Os 6©
Ki
k G 'O Oq
Age at Exit. TABLE I.
79
OBSERVATIONS ON MALE LIFE.
to to to to to i—i—* •—» i—•»—»»—*i—»»—»
►P* CO tO h-* O CD0DM05OI ►£* CO tO h-* O
CO OO —J 05 CJt rfi CO to H* o
15
k-4 k-4 k-4 k-4 k-4
rf^COWHO
CO 00 -3 05 on
►£» CO tO h-i O
Years of
Insurance.
h-4
iO to
to
h-4
1—1 to 05 05 1+*- 05 Cn 05
O
k-4
Ox
1—1
Mto M
1—1 05 1—1 l—i h-1
h-i h-i l—i
Existing.
21
?
05
05
iO 1—i
h"4
62 OOQQMM
O
*=d
ft
crq
CD
H-i
On
h-1 h—1 h-4
l—i -3 0*5
Discon-
tinued.
O
ft
OQ
CD
P_
«
M
p
C+-
Died.
%
H
w
05
•3
05
h-4
£0 rfx rfx a Oi Q —1
-3 co to GO on
Exposed
5
H
CD
bn
On
Or
On On on
-1
bn
on on on on
on on on
g
c+-
CO
Kl
oo
iO
05 to
to
h-4
h-i 1—i l—i
l—i 05 05 tS 05 -3 05 CD
1—1
►f-
k^
to
M GO H
to on on to l—i
h-1 tO h-1 &
Existing.
**
a
8
Oi
h-i
05
1—1 1—1 1—1 to
1—1 OOncOPM
—1
0
k-4
k?
(_i 0*5 h-i l—i
to on th
Discon-
tinued.
S'
Kl
CD
P
Death
Claims.
&
CD
CD
P
05
00
-3
h-i h-i H M M H
C5C500COCO 00X0000 © O h-i 05
h-4
Ci
h—1
H H iO W On 05 CO iO —3
. C© iS
iS iS Ki 3*
3*. CO Oq 3?
ki kl
c© c© c© c© c© c© c© c©
Oo 3? 05 'On -k- c© ts 3*
C© is is is is
00 Oo • 05
iS iS iS iS iS
Cn-k- C© is Ki
kl kl k| k*
co Oo 0^
k k k k k
k<
Age at Exit. 80
TABLE I.
OBSERVATIONS ON MALE LIFE.
to to to to ‘ t—1 ‘ ‘ t—» t—' 1—1 1—1 1—‘I—*
u to H‘ © ® OB-JffiOl CO tO h-* © CO tO h-* ©
Age at Entry, 13 Years. *
to to to h-1 h-1 h-1 h-1 h-1
to h-1 O © 00 —3 © Ox
Years of
Insurance.
Age at Entry, 12 Years.
l—i
o
Or
1—1 1—1■ h-i h-i |—i l—i
l—i l—i 1—1 iOiO^OlM‘W ‘00 05
No. of Entrants, 235.
00
CO
h-i
Existing.
No. of Entrants, 155.
120
1—1 1—1 t-1 l-J. rfi.
to ktOHCSO C5 05 00 0 CO
05
CD
h-i
Discon-
tinued.
10
I_1 1—1 1—1 to 1—1 1—1 to l—i
CO
h-i h-i
Died.
865.5
1—1 1—1 1—1 l—l
M l-i tS rfn- 05
l_i|_i|_i|-i H—1 to CO OO CO CO rfk Ox CO h—1 GO 05 tO CO © Ox to CO
a a br
613.0
h-i h-i h-i tO tO JO CO t|k
bx
Exposed
to Risk.
192
l-i|—‘h-1 to 00 to to 00
CO CO h-1 to OtOM^iji.
$458,000 Insured.
175
CO «>
Existing.
$296,000 Insured.
to
00
M to to CO CO 00 f—1
rfk 1—1 (fi. 01 05 tf*. tOOOOrtOkJ
118
h-i tj>
Discon-
tinued.
GO
1—1 Ox 1—‘COM■ rf*. h-1 to
CO
h-i h-i
Death
Claims.
1,653.0
h-1 h-1 to to CO to
h-1 h-1 tOCOOxOOh-1 05 tO GO 05 tO
CO CO CO CO CO k{ kj kf kj 00 GO CD k? COOXtOtOGO 1—iCOOrf^O
a in bx a
1,430.0
l—i h-i h-i rfk Ox ©
bx
Exposed
to Risk.
oa Oa Os Oa Cs c& c© te ta ta ta ta ta ta ta t>a ta h-, k, k k k. 0
<35 an-is- c© ta ki o> co Oo ~-t 05 ax-k, c© ta k ca-co Oo kj 05 a,-k- c©
c© c© c© c© c© ta ta ta
-k- c© ta kocoookj
Age at Exit.
to to to to to to to to to to h-1 h-1 h-1 h-1 h-1 h-1 h-1 h-1 h-1 h-1
CO 00 k| © Ox ►£» CO tO 1—* O CD CO k! 05 Ox i^CdtOHO
CD CO kl 05 CJi 03 to M o
Years of
Insurance.
Age at Entry, 14 Years.
375
M 62 W CO Oi 4— GO 60 60
h-1 h-1 60 h-1 60 h-1 h-1 f_i 60 GO 00 -■ 4>- G>§ Oq Co 0$ Go Go Go Go Go Go 6o 6s 6s 6s 6§ 60 6§ 6§ 6§ 6§ k< k* k*
Oq^O^G^^-
Age at Exit, TABLE T.
81
OBSERVATIONS ON MALE LIFE.
Age at Entry, 15 Years.
•Mg'
°rt
Number of Entrants, 1,086.
$2,088,000 Insured.
Age
w d
at
d d
1,182
1,197
76
9,237.0
2,149
2,418
125
17,590.0
Age at Entry, 18 Years.
Number of
Entrants, 3,973.
$7,466,000 Insured.
0
288
291
9
1,841
563
466
14
3,500
18
1
225
832
21
2,969
386
1,413
36
5,716.5
19
2
181
329
21
2,142.5
341
813
42
4,181.5
20
3
175
232
9
1,660 1
263
463
14
3,160.5
21
4
178
139
7
1,290.5
292
303
16
2,500.5
22
5
174
113
11
979.5
322
253
14
1,914.5
23
6
198
47
5
714.5
362
107
7
1,398.5
24
7
123
30
5
473
279
50
14
951
25
8
111
20
1
320
194
32
2
617
26
9
66
1
2
197.5
129
6
6
402
27
10
39
6
2
126
74
11
8
258.5
28
11
16
4
80
35
11
165.5
29
12
3
1
61.5
5
3
123.5
30
13
5
58
9
117
31
14
9
1
1
52.5
21
2
1
107
32
15
7
42
10
84
33
16
3
2
35
2
7
74
n
17
5
1
30
4
4
65
So
18
1
24
1
57
36
19
5
23
10
56
37
20
2
18
3
46
38
21
1
16
6
43
39
22
2
15
3
37
40
23
2
1
13
6
4
34
41
24
3
10
8
24
4%
25
2
7
3
16
48
26
2
1
5
3
4
13
44
27
1
2
4
6
48 !
28
1
1
2
2
46
1,827
2,046
100
13,206.5
3,336
3,933
197
25,670.5 84
TABLE I.
OBSERVATIONS ON MALE LIFE.
Age at Entry, 19 Years.
>~g
° S
Number of Entrants, 6,365.
$11,576,000 Insured.
Age
ce 2
a> ca
>*&
Existing.
Discon-
tinued.
Died.
Exposed to
Eisk.
Amounts
Existing.
Amounts
Discontin’ed.
Death
Claims.
Exposed to
Risk.
Exit.
$
$
$
$
0
387
559
25
2,903
713
875
37
5,350.5
19
1
363
1,451
39
4,668.5
625
2,497
78
8,702.5
20
2
294
526
26
3,278
521
1,024
65
6,239
21
3
277
353
13
2,518.5
435
825
24
4,728.5
22
4
273
191
15
1,956.5
419
456
34
3,629
23
5
298
121
10
1,512.5
478
256
39
2,820
24
6
285
80
20
1,104
498
191
18
2,079.5
25
7
219
48
11
735
353
91
30
1,422.5
26
8
146
16
2
473
304
39
4
974.5
27
9
102
8
1
313
218
18
1
638
28
10
48
7
2
202.5
94
24
3
398
29
11
34
5
1
146.5
65
10
2
284
30
12
10
3
1
107.5
24
7
3
208.5
31
13
11
1
2
94.5
26
4
1
176
32
14
6
4
1
79
12
5
3
144.5
33
15
8
1
69.5
14
2
126
34
16
7
1
3
60.5
9
5
8
108.5
35
17
5
50
5
89
36
18
4
45
8
84
37
19
6
1
41
5
4
76
38
20
6
1
2
33.5
8
4
5
65
39
21
5
1
25
7
3
50
40
22
5
1
1
18.5
4
1
5
39.5
41
23
6
12
20
30
42
24
2
6
4
10
43
25
2
4
2
6
44
26
2
2
4
4
45
2,811
3,377
177
20,459.0
4,875
6,334
367
38,483.0
Age at Entry, 20 Years.
Number of Entrants, 10,110.
$19,252,000 Insured.
0
704
817
38
4,646.5
1,324
1,415
58
8,918.5
20
1
562
2,297
59
7,402.5
1,085
4,232
112
14,339
21
2
464
790
43
5,238
800
1,769
64
10,141.5
22
3
438
589
31
4,041.5
678
1,359
51
7,713.5
23
4
450
368
18
3,094
721
954
45
5,828
24
5
420
191
26
2,346.5
713
421
52
4,374.5
25
6
414
131
18
1,739.5
697
291
33
3,253.5
26
7
361
81
8
1,201.5
671
141
19
2,307.5
27
8
245
27
7
778.5
455
58
21
1,518
28
9
164
22
2
502
297
39
4
993.5
29 TABLE I.
85
OBSERVATIONS ON MALE LIFE.
Age at Entry, 20 Years.
«W O
Number op Entrants, 10,110.
$19,252,000 Insured.
Age
u £
cd £
O CO
Existing.
Discon-
tinued.
Died.
Exposed to
Risk.
Amounts
Existing.
Amounts
Discontiu’ed.
Death
Claims.
Exposed to
Risk.
Exit.
$
$
$
$
10
77
11
3
319.5
161
22
3
662
SO
11
36
6
5
231
66
21
13
476.5
SI
12
25
6
3
184
52
12
11
381
32
13
9
1
3
152.5
12
6
3
309
S3
14
16
2
139
43
4
289
34
15
27
122
45
244
35
16
14
1
94.5
31
4
197
36
17
15
80
20
164
37
18
2
3
63.5
2
6
141
38
19
10
1
59.5
17
1
135.5
39
20
6
49
13
118
JfO
21
5
2
42
15
3
103.5
U
22
5
1
35.5
7
1
86.5
W
23
10
30
31
79
43
24
4
2
1
19
11
2
4
47
44
25
6
13
8
31
45
26
4
7
16
23
46
27
2
3
4
7
47
28
1
3
48
29
1
1
3
3
49
4,496
5,349
265
32,636.5
7,998 '
10,761
493
62,887.5
Age at Entry, 21 Years.
Number op Entrants
14,896.
$30,749,000 Insured.
0
1,058
1,094
49
6,901
2,018
2,125
62
14,312
21
1
794
3,382
68
11,004
1,628
6,571
115
23,258.5
22
2
672
1,183
70
7,859.5
1,428
2,948
184
16,756
23
3
662
852
42
6,100
1,194
1,994
89
12,673
24
4
697
495
40
4,722.5
1,170
1,224
68
9,781
25
5
622
318
24
3,579
1,254
833
40
7,514.5
26
6
638
176
25
2,686
1,179
428
66
5,590
27
7
503
133
26
1,868.5
1,086
252
54
4,005
28
8
360
48
13
1,249
777
113
31
2,682.5
29
9
265
21
5
841.5
546
61
7
1,787.5
30
10
108
18
2
552
236
44
4
1,182
31
11
79
8
2
429
154
22
8
909
32
12
28
4
6
342
66
11
12
730.5
33
13
40
8
2
302
95
26
1
634
34
14
33
3
3
254.5
56
6
8
522
35
15
28
2
2
216
62
4
3
453
36
16
35
2
1
184
78
2
1
385
37
17
26
2
1
146
41
6
4
302
38
18
15
1
1
117.5
29
3
2
252.5
39
19
23
1
2
100.5
62
1
5
219.5
40 86
TABLE I.
OBSERVATIONS ON MALE LIFE.
Age at Entry, 21 Years.
° s
Number of Entrants
, 14,896.
$30,749,000 Insured.
Age
at
Exit.
cd &
CO
Existing.
Discon-
tinued.
Died.
Exposed to
Risk.
Amounts
Existing.
Amounts
Discontin'ed.
Death
Claims.
Exposed to
Risk.
$
$
$
$
20
13
1
75
15
4
152
41
21
4
1
61
8
1
133
Jp
22
10
1
56
23
1
124
43
23
6
1
45
14
1
100
u
24
21
38
44
85
45
25
6
17
14
41
U6
26
4
2
10
9
1
26.5
47
27
2
5
8
17
48
28
1
3
3
9
49
29
1
1
1.5
3
3
4.5
50
6,754
7,754
388
49,766.0
13,300
16,678
771
104,641.5
Age at Entry, 22 Years.
i
Number of Entrants, 19,955.
$42,124,000 Insured.
0
1,263
1,554
57
9,200.5
2,690
2,945
89
19,589.5
22
1
1,019
4,603
114
14,779.5
2,004
9,035
193
31,882.5
28
2
913
1,627
75
10,531.5
1,745
4,058
158
23,139
24
3
812
1,057
58
8,201.5
1,452
2,712
111
17,851
25
4
895
594
48
6,506
1,749
1,482
128
14,191
26
5
827
437
44
5,047.5
1,684
1,020
96
11,063
27
6
869
244
32
3,836
1,830
640
72
8,453
28
7
700
192
24
2,717
1,364
376
55
6,043
29
8
515
73
16
1,860.5
1,240
177
31
4,347.5
30
9
332
28
12
1,279
786
80
28
2,948
31
10
218
31
8
905.5
532
72
19
2,058
32
11
132
16
6
656
282
30
14
1,456
33
12
41
12
4
504
94
33
9
1,128.5
34
13
54
6
7
450
130
12
22
1,003
35
14
47
9
381.5
96
15
837.5
36
15
56
5
3
327.5
113
7
3
730.5
37
16
31
3
3
264.5
73
5
2
608.5
38
17
40
1
2
228.5
74
4
1
529
39
18
40
2
3
185
90
11
5
446.5
40
19
13
1
1
140.5
23
1
5
345.5
41
20
17
2
125
40
11
311.5
42
21
10
1
107
19
3
266
48
22
7
2
95
16
7
240.5
44
23
28
1
86.5
76
4
219
45
24
23
1
1
57.5
56
3
2
139.5
46
25
16
33
32
80
47
26
6
1
17
17
5
48
48
27
4
10
12
26
49
28
2
6
6
14
50
29
1
4
3
8
51
30
3
3
5
5
52
8,934
10,501
520
68,546.0
18,333
22,740
1,051
150,008.0 TABLE I.
87
OBSERVATIONS ON MALE LIFE.
Age at Entry, 23 Years.
-i 2
°§
Number of Entrants, 24,574.
$54,330,000 Insured.
Age
m d
at
d d
o m
Discon-
Exposed to
Amounts
Amounts
Death
Exposed to
Exit.
>*&
Existing.
tinued.
Died.
Risk.
Existing.
Discontin’ed.
Claims.
Risk.
$
$
$
$
0
1,629
1,748
62
11,413
3,459
3,349
122
25,490.5
23
1
1,379
5,393
113
18,438.5
2,926
10,977
253
41,911.5
n
2
1,105
1,897
96
13,301.5
2,135
5,367
257
30,560.5
25
3
1,025
1,319
85
10,492.5
2,008
3,475
176
23,747.5
26
4
1,195
792
55
8,327
2,348
2,172
149
18,740
27
5
1,093
497
52
6,432.5
2,283
1,305
136
14,504.5
28
6
968
325
42
4,876.5
2,001
816
112
11,025
29
7
829
243
17
3,582.5
1,756
495
30
8,256.5
30
8
689
83
29
2,573.5
1,587
206
66
6,120
31
9
450
49
16
1,789.5
1,008
117
53
4,305.5
32
10
280
34
14
1,282
683
103
37
3,134.5
33
11
150
31
12
955.5
362
81
50
2,322.5
34
12
113
13
13
771.5
254
29
44
1,855.5
35
13
62
9
4
634.5
153
14
7
1,536
36
14
79
7
4
560.5
244
11
9
1,363.5
37
15
55
8
2
470
122
17
3
1,096.5
38
16
54
7
405.5
144
11
957.5
39
17
48
2
7
347
91
5
35
805.5
40
18
36
3
4
289.5
59
4
8
675
47
19
33
3
3
246.5
78
6
5
603
4%
20
34
3
209
101
7
517
43
21
14
1
3
171.5
30
5
9
406.5
44
22
21
2
4
153
32
3
16
363.5
43
23
33
1
126.5
99
4
312
46
24
33
2
93
85
7
211
47
25
28
1
58
52
5
119
48
26
10
29
18
62
49
27
10
19
31
44
50
28
7
9
11
13
51
29
2
2
2
2
52
11,464
12,467
643
88,059.5
24,162
28,572
1,596
201,061.0
Age at Entry, 24 Years.
Number of Entrants
28,504.
$64,563,000 Insured.
0
1,967
1,895
57
13,304.5
4,230
3,793
103
30,385
24
1
1,546
5,871
135
21,649.5
3,321
12,504
305
50,185
25
2
1,388
2,225
119
15,920.5
3,113
6,330
296
37,142
26
3
1,225
1,400
108
12,601
2,323
3,768
225
28,684
27
4
1,344
890
74
10,123
2,502
2,456
172
23,024
28
5
1,278
599
65
7,960.5
2,645
1,608
150
18,318
29
6
1,246
369
42
6,133.5
2,686
981
103
14,228.5
30
7
1,012
308
27
4,507
2,167
580
59
10,659
31
8
856
124
15
3,252
2,111
292
44
7,997
32
9
580
63
12
2,287.5
1,325
140
33
5,626
S3 T ABLE I.
OBSERVATIONS ON MALE LIFE.
Age at Entry, 24 Years.
-hS
°S
Number of Entrants, 28,504.
$64,563,000 Insured.
Age
at
17
58
5
2
425.5
126
14
7
1,027
18
47
2
1
362
132
5
3
884.5
W
19
46
1
3
312.5
129
4
9
745
43
20
27
2
2
262
64
2
10
604
u
21
20
1
3
231.5
45
2
10
528
45
22
25
1
207.5
54
3
470.5
46
23
31
2
4
181
72
7
7
411.5
47
24
38
3
145
76
5
329
48
25
36
1
104
94
2
248
49
26
31
2
66
69
8
148
50
27
17
1
34
40
5
75
51
28
6
1
16
14
4
30
52
29
2
9
5
12
53
30
7
7
7
7
54
13,852
13,912
740
106,494.0
29,977
32,861
1,725
247,751.0
Age at Entry, 25 Years.
Number of Entrants
32,680
$76,582,000 Insured.
0
2,209
2,120
84
15,280
4,998
4,172
168
36,205
'25
1
1,840
6,496
137
25,019
4,184
14,301
338
60,093.5
26
2
1,585
2,396
136
18,596
3,388
6,966
299
44,938
27
3
1,513
1,606
127
14,874
3,182
4,307
276
35,614.5
28
4
1,575
964
110
11,949
3,479
2,834
293
28,586
29
5
1,562
688
66
9,438
3,422
1,927
218
22,433.5
30
6
1,427
427
56
7,252.5
3,066
1,209
157
17,225.5
31
7
1,261
341
47
5,385.5
2,900
777
116
13,009.5
32
8
972
149
34
3,832.5
2,240
380
88
9,415
33
9
618
82
20
2,711
1,747
197
49
6,798.5
34
10
443
49
18
2,007.5
1,034
124
56
4,842
35
11
239
32
11
1,506
605
66
32
3,657
36
12
137
28
14
1,226
348
53
33
2,960.5
37
13
116
22
13
1,050
321
35
27
2,535.5
38
14
116
19
8
900.5
304
45
33
2,147.5
39
15
102
15
9
759.5
281
30
23
1,773
40
16
77
7
9
637.5
189
23
24
1,442.5
U
17
70
4
3
546
167
17
8
1,209.5
42
18
46
4
10
469
67
14
29
1,019
43
19
49
3
4
409.5
112
7
13
912.5
44 TABLE I.
89
OBSERVATIONS ON MALE LIFE.
Age at Entry, 25 Years.
*-S
° s
Number op Entrants, 32,680.
$76,582,000 Insured.
Age
£ £
at
Exit.
cd &
o m
**
Existing.
Discon-
tinued.
Died.
Exposed to
Risk.
Amounts
Existing.
Amounts
Discontin'ed
Death
Claims.
Exposed to
Risk.
$
$
$
$
20
26
1
2
354.5
79
5
7
781.5
45
21
31
1
3
325.5
77
1
5
692.5
46
22
27
1
3
290.5
61
8
4
606
47
23
44
1
3
259.5
142
1
4
536.5
48
24
46
1
1
211.5
90
4
1
388
49
25
66
4
164
72
16
295
50 j
26
47
4
94
95
17
207
51
27
22
1
43
58
5
95
52
28
15
20
19
32
53
29
2
5
4
13
54
30
3
3
9
9
55 !
16,286
15,457
937
125,619.5
36,740
37,503
2,339
300,473.5
Age at Entry, 26 Years.
Number op Entrants, 34,138.
$80,236,000 Insured.
0
2,265
2,182
82
15,978
5,018
4,022
182
38,107
26
1
1,864
6,541
175
26,338.5
4,258
14,556
394
63,736
27
2
1,592
2,560
142
19,749
3,498
7,161
348
48,225.5
28
3
1,530
1,647
111
15,911.5
3,244
4,517
277
38,540.5
29
4
1,561
1,111
115
12,891.5
3,538
2,985
280
31,268.5
30
5
1,506
776
94
10,272
3,391
2,194
216
24,861
31
6
1,451
488
70
8,040
3,213
1,353
169
19.480.5
32
7
1,304
396
61
6,077
3,074
931
174
14,956.5
83
8
1,045
181
37
4,423.5
2,523
462
115
11,012
34
9
750
100
32
3,201
1,813
234
78
8,026
35
10
498
66
17
2,336
1,213
194
36
5,921
36
11
318
41
12
1,767.5
755
102
32
4,524
37
12
126
29
16
1,402.5
327
74
29
3,649
38
13
124
22
11
1,235
329
57
25
3,227.5
39
14
170
20
12
1,079
426
63
29
2,813.5
40
15
130
10
2
882
359
18
5
2,318
Jfi
16
72
9
8
740.5
184
31
23
1,929.5
42
17
90
6
3
653
208
13
9
1,700.5
43
18
51
3
3
555.5
126
8
10
1,473
44
19
65
2
2
499
209
7
2
1,329.5
45
20
44
4
2
429
111
14
4
1,108
21
31
2
5
380
69
3
17
984.5
47
22
32
2
6
342
75
3
17
895.5
48
23
50
2
4
302
128
10
4
797
49
24
65
3
4
245.5
179
5
12
657.5
50 90
TABLE I.
OBSERVATIONS ON MALE LIFE.
Age at Entry, 26 Years.
-S
° (S
Number op Entrants, 34,138.
$80,230,000 Insured.
Age
£ f*
ci pS
cu m
X*
Existing.
Discon-
tinued.
Died.
Exposed to
Risk.
Amounts
Existing.
Amounts
Discontin’cd.
Death
Claims.
Exposed to
Risk.
Exit.
$
$
$
$
25
63
3
175
135
18
464
51
26
44
1
1
108.5
123
1
5
310.5
52
27
29
1
1
62.5
76
4
4
180
53
28
17
1
32
46
5
98
H
29
9
14
24
47
55
30
5
5
23
23
56
16,901
16,205
1,032
136,127.5
38,695
39,022
2,519
332,664.0
Age at Entry, 27 Years.
Number op Entrants, 36,221.
$87,030,000 Insured.
0
2,274
2,050
81
17,085.5
5,478
4,344
189
41,343
27
1
2,019
6,753
201
28,439.5
4,678
15,314
479
69,362
28
2
1,660
2,877
150
21,404.5
3,687
8,413
367
52,341.5
29
3
1,689
1,672
128
17,320
3,604
4,741
306
41,710.5
30
4
1,703
1,168
126
14,083
3,634
3,360
307
33,750
31
5
1,651
776
90
11,282
3,578
2,224
247
27,017
32
6
1,658
473
63
8,916.5
3,781
1,270
188
21,445
33
7
1,413
431
53
6,743.5
3,225
984
118
16,349
34
8
1,213
201
36
4,961.5
2,883
489
94
12,269.5
35
9
873
120
32
3jDo2
2,133
300
77
8,898
36
10
559
77
15
2,548.5
1,343
222
50
6,427
37
11
347
45
22
1,913.5
814
118
63
4,864
38
12
150
35
19
1,504.5
371
81
43
3,887.5
39
13
140
33
15
1,301.5
381
86
31
3,390
40
14
161
23
12
1,118.5
410
74
31
2,898
41
15
99
11
8
928.5
239
20
22
2,410
42
16
78
11
7
810.5
196
24
17
2,127
43
17
74
4
7
718
154
5
24
1,899.5
44
18
63
11
5
629.5
179
28
8
1,705
45
19
55
3
3
554.5
158
11
13
1,498.5
46
20
42
5
1
492.5
132
17
2
1,313.5
47
21
41
3
5
445.5
100
5
10
1,168.5
48
22
41
3
4
396.5
104
6
10
1,053
49
23
47
3
4
348.5
111
15
11
928.5
50
• 24
79
1
2
295.5
174
4
7
797
51
25
71
1
4
213.5
188
3
9
612.5
52
26
58
3
138
164
2
414
53 1
27
36
4
. 77
118
17
248
54
28
16
1
37
38
1
113
55
29
9
20
24
74
56
30
9
2
11
44
6
50
57
18,328
16,790
1,103
148,290.5
42,123
42,158
2,749
362,364.0 TABLE I.
91
OBSERVATIONS ON MALE LIFE.
Age at Entry, 28 Years.
m
Existing.
Discon-
tinued.'
Died.
Exposed to
Risk.
Amounts
Existing.
Amounts
Discontin’ed
Death
Claims.
Exposed to
Risk.
$
%
$
$
10
805
118
33
3,554
1,868
293
90
9,076.5
Jfi
11
547
60
21
2,627
1,284
150
52
6,897
41
12
215
41
19
2,008.5
544
122
40
5,425
42
13
179
25
14
1,741.5
460
69
43
4,745.5
43
14
182
33
20
1,519.5
486
90
50
4,163
44
15
ICO
21
12
1,290.5
468
66
37
3,549
45
16
102
10
12
1,103
302
36
52
2,993
46
17
95
12
9
978
263
40
35
2,601
47
18
69
8
7
864
175
19
10
2,273.5
48
19
72
5
7
781.5
174
12
19
2,073
49
20
65
4
4
698
163
22
13
1,863
50
21
49
2
4
626
139
4
9
1,674
51
22
57
2
5
571
138
7
23
1,520.5
52
23
98
4
8
506
243
6
23
1,353
53
24
116
2
6
397
286
11
12
1,078.5
54
25
93
1
6
273.5
241
4
23
773
55
26
73
3
174
219
10
507
56
27
45
1
1
97.5
152
4
1
276
57
28
29
2
50
67
2
120
58
29
13
20
33
52
59
30
7
7
19
19
60
22? 4o2
18,472
1,376
186,831.0
53,319
48,933
3,726
474,416.5
Age at Entry, 31 Years.
Number of Entrants, 39,923.
$100,042,000 Insured.
0
2,450
1,857
90
19,033
6,132
4,048
253
47,997
31
1
2,197
6,386
196
32,333
5,549
14,806
476
82,206
32
2
1,870
2,747
175
25,373.5
4,235
8,270
517
64,643
83
3
1,872
1,851
162
21,029.5
4,198
5,522
447
52,995
34
4
1,813
1,276
149
17,432
4,300
3,858
438
43,660
35
5
1,867
917
125
14,373.5
4,657
2,842
348
35,572
36
6
1,862
622
105
11,612
4,499
1,818
225
28,237
37
7
1,625
532
84
9,068
3,908
1,331
216
21,938.5
38
8
1,578
206
52
6,990
3,696
646
150
16,826
39
9
1,229
132
36
5,191
2,938
338
114
12,488
40
10
1,142
103
28
3,808.5
2,040
273
81
9,130.5
41
11
479
63
24
2,555.5
1,139
155
58
6,795.5
42
12
228
38
21
2,002
591
104
51
5,469
43
13
152
24
26
1,722
540
67
77
4,741.5
44
14
207
27
15
1,518.5
613
75
35
4,053.5
45
15
150
18
15
1,274
374
39
40
3,348.5
46
16
119
18
17
1,091
334
40
50
2,895
47
17
99
8
15
942
234
29
49
2,476.5
48
18
88
7
8
820.5
222
14
22
2,172
49
19
63
8
10
717
165
14
25
1,914
50 94
TABLE I.
OBSERVATIONS ON MALE LIFE.
1 .
Age at Entry, 31 Years.
'«S
°§
Number of Entrants, 39,923.
$100,042,000 Insured.
Age
co c3
U 5h
at
CD CO
_ . ,
niscon-
. ,
Exposed to
Amounts
Amounts
Death
Exposed to
tinued.
itisk.
Existing.
Discontin’ed.
Claims.
Risk.
$
$
$
$
20
66
4
4
638
199
8
14
1,713
51
21
44
2
11
565
124
4
26
1,494
52
22
42
5
10
506.5
111
9
27
1,337.5
53
23
85
4
8
450
205
11
32
1,189.5
51>
24
95
9
355
230
26
947
55
25
82
1
2
250.5
201
3
10
689.5
56
26
80
2
166
214
5
477
57
27
43
3
84
122
5
258
58
28
20
38
95
131
59
29
11
18
26
36
60
30
7
7
10
10
61
21,665
16,856
1,402
181,964.5
51,901
44,324
3,817
457,841.0
Age at Entry, 33 Years.
Number of Entrants, 40,089.
$101,772,000 Insured.
0
2,428
1,868
95
19,110.5
6,080
4,061
244
48,855.5
32
1
2,037
6,409
187
32,493.5
5,163
15,640
459
83,567
33
2
1,916
2,813
189
25,658.5
4,583
8,484
543
65,883
35
3
1,805
1,776
168
21,259
4,000
5,326
434
53,852
35
4
1,943
1,316
139
17,740
4,488
3,985
368
44,762.5
36
5
1,856
905
118
14,547.5
4,742
2,556
288
36,636
37
6
1,860
589
125
11,826.5
4,435
1,742
339
29,457
38
7
1,841
486
100
9,304
4,332
1,273
297
23,175.5
39
8
1,582
230
61
7,005
4,084
644
156
17,588
50
9
1,288
161
48
5,166.5
2,997
410
133
12,821
51
10
833
91
30
3,704.5
2,055
224
87
9,374
52
11
504
87
23
2,752.5
1,104
159
77
7,040.5
53
12
261
38
21
2,163
626
115
56
5,722.5
55
13
187
24
19
1,850
456
72
57
4,947
55
14
197
28
16
1,618
494
81
36
4,357.5
56
15
145
13
14
1,384.5
389
33
37
3,770.5
51
16
105
9
16
1,214.5
262
7
44
3,324.5
58
17
90
12
18
1,083
237
43
32
2,993.5
59
18
82
10
11
964
252
31
33
2,687.5
50
19
61
9
8
861.5
159
24
27
2,375
51
20
93
3
9
786.5
313
21
36
2,166.5
52
21
47
6
12
680
127
11
30
1,801.5
53
22
59
3
10
616.5
162
8
20
1,635
55
23
65
1
13
545.5
130
4
39
1,447
55
24
117
2
3
466
310
6
6
1,273
56
25
129
1
3
344.5
299
1
10
953.5
57
26
99
2
4
211
279
10
14
639
58
27
52
1
106.5
176
3
339.5
59
28
25
1
54
84
4
162
60
29
11
28
23
74
61
30
17
17
51
51
62
21,735
16,893
1,461
185,562.0
52,892
44,974
3,906
473,732.0 TABLE I.
95
OBSERVATIONS ON MALE LIFE.
Age at Entry, 33 Years.
CO
Existing.
Discon-
tinued.
Died.
Exposed to
Risk.
Amounts
Existing.
Amounts
Discontin’ed.
Death
Claims.
Exposed to
Risk.
$
$
$
$
0
1,384
937
90
11,732.5
3,928
2,170
235
33,841
43
1
1,340
S', 382
187
20,300
3,731
8,782
548
59,128
44-
2
1,244
1,529
172
16,317.5
3,234
5,404
497
47,756
45
3
1,205
1,030
166
13,622
3,171
3,511
503
39,567.5
46
4
1,244
723
146
11,374.5
3,378
2,691
424
32,792.5
47
5
1,266
518
136
9,364
3,589
1,880
434
26,705
48
6
1,259
392
85
7,507
3,444
1,304
293
21,090
49
7
1,329
281
92
5,826.5
3,629
858
289
16,272
50
8
1,012
152
60
4,189
2,835
523
155
11,663.5
51
9
806
88
49
2,997
2,157
264
107
8,280
52
10
466
49
32
2,073.5
1,196
170
91
5,799
53
11
359
32
23
1,535
976
125
77
4,364.5
51f.
12
134
26
19
1,124
385
68
43
3,215
55
13
105
14
15
951
333
58
48
2,724
56
14
108
8
16
820
352
38
48
2,295
57
15
85
6
13
689
225
12
39
1,870
58
16
43
7
10
584.5
121
30
22
1,585
59
17
41
6
8
525
128
15
27
1,419.5
60
18
40
5
7
470.5
115
16
20
1,249
61
19
23
3
15
419.5
49
8
43
1,102
62
20
24
6
8
377
74
15
22
998.5
63
21
23
3
15
340.5
48
8
48
891
64
22
30
8
301
73
35
791
65
23
50
1
13
262.5
138
3
35
681.5
66
24
62
1
8
198.5
149
1
26
506.5
67
25
31
2
5
127
77
4
17
329
68
26
33
1
3
89.5
85
6
8
230
69
27
24
2
53
69
3
134
70
28
13
2
3
26
40
1
9
61.5
71
29
2
1
1
8.5
6
3
1
10.5
72
30
5
5
2
2
73
13,790
9,205
1,407
114,210.5
37,737
27,968
4,147
327,354.0
Age at Entry, 44 Years.
Number of Entrants
1 JO
CO
$64,953,000 Insured.
0
1,374
830
73
10,800.5
3,956
1,911
187
31,521
44
1
1,076
3,072
163
18,618
2,999
7,802
432
54,998
45
2
1,083
1,477
186
15,104.5
2,921
4,943
563
45,194.5
46
3
1,135
987
150
12,603.5
2,793
3,582
445
37,448
47
4
1,202
679
119
10,485.5
3,155
2,531
404
31,153.5
48
5
1,124
535
103
8,557.5
3,291
1,676
305
25,491
49
6
1,256
354
102
6,886
3,822
1,250
294
20,432
50
7
1,097
284
72
5,209
3,409
811
207
15,285.5
51
8
933
132
59
3,832
2,765
415
186
11,056.5
52
9
691
78
47
2,735
1,887
271
130
7,762.5
53 104
TABLE I.
OBSERVATIONS ON MALE LIFE.
Age at Entry, 44 Years.
<3 8
° c
Number of Entrants, 22,431.
$64,953,000 Insured.
Age
si
at
Discon-
tinued.
CD in
*£
Existing.
Died.
Exposed to
Risk.
Amounts
Existing.
Amounts
Discontin’cd.
Death
Claims.
Exposed to
Risk.
Exit.
$
$
$
$
10
447
56
33
1,930
1,198
179
90
5,520.5
5U
11
266
34
28
1,405
686
93
88
4,096.5
55
12
141
28
17
1,080
438
73
52
3,239.5
56
13
106
10
16
903
320
35
58
2,695.5
57
14
101
8
17
772
292
19
46
2,290.5
58
15
77
4
13
648
218
32
43
1,927
59
16
51
7
20
552.5
153
19
56
1,640.5
60
17
33
4
13
476
93
4
54
1,420
61
18
28
4
13
426
76
8
33
1,267
62
19
30
2
10
382
84
24
30
1,142
63
20
20
2
9
340
55
7
49
1,012.5
61f
21
26
1
14
309.5
91
3
51
903.5
65
22
15
13
269
37
92
760
66
23
61
3
16
239.5
159
7
42
627.5
67
24
50
11
161
117
25
423
68
25
38
5
100
98
19
281
69
26
25
1
2
56.5
58
3
10
162.5
70
27
19
29
64
93
71
28
7
10
16
29
72
29
1
3
3
13
73
30
2
2
10
10
n
12,515
8,592
1,324
104,925.5
35,264
25,698
3,991
309,896.5
Age at Entry, 45 Years.
Number of Entrants, 21,980.
$63,491,000 Insured.
0
1,241
859
69
10,560.5
3,441
1,947
222
30,772
45
1
1,137
3,043
180
18,289.5
3,319
8,169
537
53,796.5
46
2
1,076
1,361
170
14,770.5
2,724
4,349
455
43,681.5
47
3
1,055
906
158
12,391
3,034
3,277
450
36,689.5
48-
4
1,217
657
131
10,396.5
3,136
2,223
418
30,455.5
49
5
1,135
482
96
8,479
3,329
1,624
286
24,978
50
6
1,193
339
104
6,837.5
3,635
1,145
279
19,978.5
51
7
1,101
274
79
5,234
3,312
897
246
15,043.5
52
8
949
113
55
3,860.5
2,732
406
169
10,834
53
9
745
95
42
2,752.5
2,054
330
91
7,565
51f
10
490
56
30
1,890
1,266
176
73
5,167
55
11
289
38
16
1,323
773
104
37
3,688
56
12
112
17
13
990.5
336
44
43
2,804
57
13
90
6
20
854
250
35
52
2,385.5
58
14
81
14
23
734
252
35
59
2,048.5
59
15
58
6
20
620
164
14
46
1,713
60
16
38
6
11
536
120
17
38
1,487.5
61
17
45
3
10
482.5
99
10
30
1,316
62
18
44
5
12
423.5
120
6
34
1,179
63
19
15
4
7
363
39
11
13
1,016.5
64 TABLE I.
105
OBSERVATIONS ON MALE LIFE.
Age at Entry, 45 Years.
««§
° a
Number of Entrants, 21,980.
$63,491,000 Insured.
Age
£ f-i
at
cS 2
0) t/3
fcn rt
HH
Existing.
Discon-
tinued.
Died.
Exposed to
Risk.
Amounts
Existing.
Amounts
Discontin’ed
Death
Claims.
Exposed to
Risk.
Exit.
30
$
$
$
$
19
3
11
337.5
68
4
23
957
65
21
22
4
14
304
54
11
35
858.5
66
22
31
1
11
265.5
68
8
38
760
67
23
32
2
6
222
86
6
50
647
68
24
52
13
183
146
32
508
69
25
42
8
118
104
15
330
70
26
21
2
68
75
13
211
71
27
25
2
45
71
4
123
72
28
11
1
1
17.5
33
3
3
46.5
73
29
2
5
6
9
n
30
3
3
3
3
75
12,371
8,295
1,314
103,356.5
34,849
24,851
3,791
301.052.0
Age at Entry, 46 Years.
Number of Entrants, 19,350.
$57,757,000 Insured.
0
1,135
711
67
9,319.5
3,449
1,638
155
28,059.5
56
1
959
2,613
156
16,130.5
3,146
6,791
481
49,119.5
4?
2
900
1,221
137
13,098.5
2,507
4,299
486
39,947.5
1,8
3
937
816
122
11,043
2,607
2,843
421
33,383.5
59
4
1,002
573
126
9,289.5
2,825
2,202
385
27,833
50
5
1,053
417
107
7,666.5
3,102
1,554
327
22,745
51
6
1,093
282
89
6,157
3,301
1,082
256
17,998
52
7
1,002
259
78
4,704.5
3,054
807
247
13,496.5
53
8
891
125
54
3,432.5
2,547
432
141
9,576
55
9
711
76
41
2,387
1,837
245
108
6,549.5
55
10
408
51
26
1,571.5
1,106
134
79
4,415
56
11
262
24
11
1,100
638
59
37
3,133.5
57
12
123
18
16
806
352
49
70
2,404.5
58
13
67
10
11
653
210
60
63
1,928
59
14
69
8
17
566
236
22
45
1,614
60
15
62
2
5
475
187
6
22
1,319
61
16
28
5
6
404.5
73
21
30
1,096.5
62
17
43
2
11
367
128
10
12
978
63
18
15
1
15
311.5
62
4
40
831
65
19
19
o
O
11
279.5
41
6
22
724
65
20
20
1
12
247.5
46
8
35
654
66
21
14
3
8
213.5
50
10
24
564
67
22
30
1
9
189.5
75
3
31
483.5
68
23
27
2
6
149
57
11
15
370.5
69
24
30
2
3
114
68
4
8
291
70
25
26
5
80
58
21
213
71
26
20
3
49
61
7
134
72
27
16
2
26
27
2
66
73
28
4
1
8
21
3
37
n
29
2
3
10
13
75
30
1
1
3
3
76
10,969
7,226
1,155
90.843.0
31,884
22,300
3,573
269,980.5 106
TABLE I.
OBSERVATIONS OX MALE LIFE.
Age at Entry, 47 Years.
<~8
°S
Number of Entrants, 16,859.
$51,934,000 Insured.
Age
m c3
d d
0) GO
Discon-
Exposed to
Amounts
Amounts
Death
Exposed to
Existing.
tinned.
Died.
Risk.
Existing.
Diseontin’cd.
Claims.
Risk.
$
$
$
$
0
955
634
61
8,112.5
2,704
1,528
178
25,198
47
1
788
2,246
173
14,086
2,394
6,252
546
44,388
48
2
781
1,082
133
11,461
2,185
3,860
369
36,392
49
3
853
703
96
9,654.5
2,231
2,608
317
30,604
50
4
863
479
120
8,114.5
2,450
1,903
358
25,800.5
51
5
953
368
66
6,708
2,829
1,409
214
21,336.5
52
6
1,019
252
77
5,379
3,137
950
210
17,114
53
7
938
206
65
4,054
2,936
644
210
12,970
54
8
765
118
53
2,889
3,341
391
150
9,306.5
55
9
572
64
43
1,980
1,528
160
94
5,540
56
10
341
35
22
1,315.5
937
102
87
3,787
57
11
187
27
14
921.5
454
60
39
2,682
58
12
107
9
17
702.5
275
53
54
2,132.5
59
13
56
8
7
570
168
25
18
1,764.5
60
14
45
8
10
499
119
21
32
1,555.5
61
15
41
1
8
439.5
136
2
17
1,393
62
16
37
3
13
388.5
113
10
55
1,234
63
17
23
3
11
335.5
47
17
20
052.5
64
18
32
3
8
298.5
123
7
119
973.5
65
19
18
2
12
256
116
4
35
726
66
20
13
2
7
224
25
8
23
569
67
21
14
1
9
202.5
48
3
20
515.5
68
22
13
1
7
178.5
37
4
15
444
69
23
20
2
6
157
53
4
17
388
70
24
31
6
130
73
14
316
71
25
27
1
4
92.5
69
3
17
227.5
72
26
28
2
61
66
3
140
73
27
18
4
31
43
7
71
74
28
2
2
9
7
7
21
75
29
2
1
5
2
1
7
76
30
2
2
4
4
77
9,544
6,258
1,057
79,257.5
28,650
20,028
3,246
248,653.0
Age at Entry, 48 Years.
Number of Entrants
15,325.
$17,985,000 Insured.
0
905
582
72
7,621.5
2,555
1,775
175
23,105
48
1
785
1,986
152
13,273
2,466
5,366
535
40,797
49
2
810
987
121
10,849.5
2,426
3,274
358
33,476
50
3
758
720
112
9,065
1,888
2,844
362
27,633
51
4
849
431
108
7,619.5
2,342
1,729
373
23,096.5
52
5
838
381
95
6,256.5
2,634
1,499
313
18,767.5
53
6
920
251
80
5,007.5
2,795
879
202
14,631.5
54
7
854
163
48
3,800.5
2,576
540
154
10,925
55
8
801
93
58
2,770.5
2,412
297
203
7,776.5
56
9
540
49
31
1,840.5
1,442
149
80
4,938.5
57 From the Spectator, New York, Oct. 2, 1890.
Correction of an error found in the “System and
Tables of Life Insurance.”
Levi W. Meech.
OINCE the publication in The Spectator, page
\j 57 (July 31), of the error in the 3£ Per Cent.
Tables, so thoroughly examined by Prof. Kendall,
this error has been traced to a clerk copying 0 in
the antique form, employed by some writers, which
was afterwards mistaken for 8. Fortunately the
unusual rate of 3| per cent, is alone affected by it,
and only a portion of its tabular values. The D, C,
M, 11 commutation columns and their tabular loga-
rithms are all right, except at the age of 59, where D
itself, after the verified computation, was copied as
8,803.10, instead of the true value 8,003.10, as before
described.
In the “System and Tables” this erratum is easily’
rectified in D ; and for all the other columns or por-
tions of them, full correction is made in the colum-
nar figures here subjoined. For desirable accuracy
these can be cut in strips to be attached over the
book figures indicated. To facilitate this object,
extra copies have been printed separately. Al-
though the greatest care was used in all the reductions
and results of the Thirty Offices’ Experience, con-
tinued for six years, to insure the highest precision,
yet the fundamental Insurance Tables are the most
important. Accordingly a reward of $10 in a copy of
the “System and Tables” is offered for the discovery
of any further error in this class, to be communica-
ted to the actuary in charge for publication. The
present offer is limited to six years, and to the Stand-
ard Tables 26 to 42, with terminal figures as common-
ly computed. The table of Anti-logarithms is also
included. AGE.
Table 28.
3 A- per ct.
Table 29.
per ct.
Table 32.
per ct.
Table 34.
N.
Hundreds.
Table 34.
S.
Hundreds.
Table 34.
A N.
Table 34.
A S.
10
5905
500
32
1,4
058,3
4523
2179
1
4920
440
53
0,5
456,9
4860
9187
3897
470
75
2,5
926,3
5108
6014
3
2835
598
99
7,2
463,7
5265
2652
4
1734
827
23
4,5
066,5
5324
9095
15
0594
158
49
4,3
732,0
5283
5335
6
9411
601
75
6,5
457,7
5135
1365
1
8183
166
03
1,1
241,1
4876
7178
8
G912
852
33
7,9
080,0
4501
2764
9
5595
668
64
6,8
18,972,1
4002
8116
20
4230
625
96
7,8
915,2
3376
3224
1
2817
724
31
0,7
907,4
2616
8079
2
1355
974
66
5,6
946,7
1714
2672
3
19.9841
388
50.04
2,2
031.1
0664
6991
4
8274
972
44
0,7
158,8
9458
1027
25
6G55
733
85
0,8
328,0
8089
4768
0
4980
689
29
2,5
537,2
6548
8202
7
3250
842
75
5.8
784,6
4826
1318
8
1462
206
23
0,6
068,8
2915
4101
9
9G16
797
74
6,8
388,1
0805
6540
30
7713
621
27
4.4
741,3
8484
8618
1
5747
699
84
3,3
126,8
5943
0323
2
3721
8042
43
3,5
543,5
3170
1637
3
1632
3670
55.06
4,9
7,989,9
0152
2544
4
9481
9593
72
7,5
464,9
6877
3028
35
7266
5836
41
1,3
967,3
3330
3068
G
4989
2407
15
6,1
496,0
5.649496
2648
ty
(
2646
9342
92
2,0
049,8
5361
1746
8
0237
6661
74
8,9
627,8
0907
0340
9
7765
4378
61
6,8
228,8
6117
8409
40
5227
2526
52
5,6
852,0
0972
5927
1
2625
1126
•49
5,3
496,4
5452
2871
2
15.9957
5.0220
52
5,8
61,1
5.499536
19213
3
7227
5.9825
60
7,3
45,2
73200
84925
4
4432
6.9989
75
279,5
47,9
46421
49978
45
1576
8.0738
97
2,5
68,4
19172
14339
G
8659
9.2118
27
6,2
05,9
91426
77975
7
5682
0.4173
64
0,7
59,6
63152
40850
8
2647
1.6942
70.10
5,9
28,8
34319
02927
9
3.9556
3.0482
1.66
1,7
12,9
04894
64164
50
3.6411
4.4845
3.31
8,2
11,1
74839
24519
1
3.3216
6.0085
5.07
5,4
22,8
44114
83946
2
2.9971
7.6268
6.94
3,1
47,4
12679
42398
3
2.6682
9.3463
8.94
1,5
84,2
80487
6.199820
4
2.3350
1.1744
1.07
0,4
32,7
47491
56158
55
1.9978
3.1199
3.35
29,9
2,2
13637
11353
6
1.6573
5.1894
5.78
19,9
2,3
1 78868
65341
7
• 1.3137
7.3939
8.39
10,4
2,4
43126
• 18055
8
0.9677
9.7418
1.18
01,4
2,0
06342
69420
9
0.6195
2.2457
4.17
92,9
0,5
68446
19359 TABLE I.
107
OBSERVATIONS ON MALE LIFE.
Age at Entry, 48 Years.
<«g
° *j
Number op Entrants, 15,835.
$47,985,000 Insured.
Age
£ £
o ca
Discon-
Exposed to
Amounts
Amounts
Death
Exposed to
h-i
Existing.
tinned.
Risk.
Existing.
Discontin’ed.
Claims.
Risk.
$
$
$
$
10
278
31
22
1,229.5
646
115
68
3,284.5
58
11
227
19
13
904.5
575
68
40
2,479
59
12
83
13
16
648.5
208
37
56
1,811.5
60
13
60
10
10
538
157
31
27
1,513.5
61
14
62
9
8
458.5
173
23
31
1,302.5
62
15
50
3
11
382.5
184
8
40
1,083
63
16
22
3
8
318.5
62
11
23
849.5
64-
17
19
8
287
63
19
759
65
18
23
3
9
258.5
65
4
20
675
66
19
18
3
11
223.5
48
5
31
585.5
67
20
9
3
11
191.5
27
8
33
500
68
21
11
1
8
169.5
33
5
18
433.5
69
22
17
1
7
149.5
47
3
23
378.5
70
23
26
1
5
124.5
51
2
8
306
71
24
37
11
93
81
35
246
72
25
13
3
45
28
9
130
73
26
10
4
29
29
11
93
n
27
6
15
14
53
75
28
4
9
21
39
76
29
2
5
11
18
77
30
2
3
2
7
78
31
1
5
79
32
1
5
80
33
1
1
5
5
81
9,039
5,743
1,043
74,190.0
26,061
18,672
3,252
221,708.5
Age at Entry, 49 Years.
Number op Entrants
, 13,873.
$43,119,000 Insured.
0
795
456
57
6,708.5
.2,405
1,090
181
21,014.5
49
1
686
1,776
128
11,677
2,155
4,937
356
36,974.5
50
2
680
841
125
9,554.5
2,071
2,887
383
30,551.5
51
3
725
614
118
8,022
1,973
2,204
383
25,552
52
4
758
394
90
6,675
1,907
1,594
277
21,297
53
5
854
291
96
5,484.5
2,594
1,993
289
17,319.5
54
6
826
219
86
4,279.5
2,545
851
295
13,014.5
55
7
774
148
63
3,184
2,350
481
173
9,508.5
56
8
588
82
38
2,232
1,874
296
120
6,597
57
9
411
54
38
1,538
1,163
150
120
4,380
58
10
261
30
32
1,047
715
72
81
2,986
59
11
146
18
17
730
419
53
54
2,127.5
60
12
71
7
10
554.5
206
35
36
1,610.5
61
13
45
8
10
466
134
29
26
1,336.5
62
14
51
4
10
405
166
9
25
1,157.5
63 108
TABLE I.
OBSERVATIONS ON MALE LIFE.
Age at Entry, 49 Years.
Number of Entrants, 13,873.
143,119,000 Insured.
Age
co d
H H
at
d d
_ .
A
Existing.
Discon-
tinned.
Died.
Risk.
A. ii io lili Is
Existing.
Discontin’ed.
i ii
Claims.
Risk.
Exit.
$
$
$
$
\ 15
39
2
11
341
107
10
39
957
64
16
32
18
290
114
55
806
65
17
10
1
16
239.5
30
1
40
636.5
66
18
21
2
6
212
67
3
15
564.5
67
19
10
2
11
183
25
3
22
479.5
68
20
17
1
9
160.5
53
11
17
425.5
69
21
9
1
8
133.5
18
1
34
349.5
70
22
14
12
116
30
28
297
71
23
9
6
90
19
12
239
72
24
17
6
75
43
28
208
73
25
15
2
4
51
28
1
9
136.5
n
26
13
3
31
50
6
99
75
27
4
1
15
10
4
43
76
28
5
10
14
29
77
29
2
5
4
15
78
30
1
1
1
2.5
3
3
5
9.5
79
7,889
4,954
1,030
64,512.5
23,292
16,714
3,113
200,721.5
Age at Entry, 50 Years.
Number of Entrants, 13,650.
$41,77(5,000 Insured.
0
857
491
85
6,579.5
2,445
1,110
280
20,333
50
1
667
1,796
163
11,319
1,949
4,860
581
35,511
51
2
652
836
147
9,173
1,893
3,108
487
28,997
52
3
673
575
132
7,668.5
1,875
1,986
467
24,070
53
4
831
382
104
6,385
2,266
1,447
340
20,011.5
54
5
823
252
93
5,133
2,707
1,032
298
16,166
55
6
768
177
62
4,002.5
2,393
615
209
12,337.5
56
7
784
150
72
3,009
2,541
558
287
9,149
57
8
575
66
44
2,045
1,725
254
131
5,915
58
9
364
36
29
1,375
1,020
100
86
3,882
59
10
236
38
26
945
603
188
59
2,632
60
11
120
14
17
657
320
43
54
1,854.5
61
12
63
5
16
510.5
142
29
57
1,444.5
62
13
52
8
11
425
176
19
19
1,221.5
63
14
53
6
9
355
153
23
25
1,005.5
64
15
34
4
12
288
119
12
28
810
65
16
19
3
6
238.5
56
16
11
649
66
17
18
2
6
211
53
5
22
571.5
67
18
15
1
10
185.5
33
1
55
493.5
68
19
11
1
4
159.5
26
4
13
403
69
20
16
1
10
143.5
49
6
27
359
70
21
6
1
3
116.5
17
3
7
278.5
71
22
13
1
7
106.5
30
1
19
252.5
72
23
10
3
86
21
8
203
73
24
19
5
73
42
8
174
74 TABLE I.
109
OBSERVATIONS ON MALE LIFE.
Age at Entry, 50 Years.
Years of
Insurance.
Number op Entrants, 13,650.
$41,776,000 Insured.
Age
at
Exit.
Existing.
Discon-
tinued.
Died.
Exposed to
Risk.
Amounts
Existing.
Amounts
Discontin’ed.
Death
Claims.
Exposed to
Risk.
$
$
$
$
25
17
4
49
46
10
124
75
26
7
2
3
27
15
2
7
67
76
27
8
1
16
18
1
44
77
28
3
1
7
8
2
25
78
29
3
15
79
30
3
3
15
15
80
7,717
4,848
1,085
61,295.0
22,756
15,422
3,598
189,014.0
Age at Entry, 51 Years.
Number op Entrants, 10,458.
$32,807,000 Insured.
0
619
406
56
5,026
1,703
1,067
177
15,870
51
1
546
1,381
135
8,686.5
1,565
3,899
445
27,910.5
52
2
556
607
95
7,011.5
1,642
2,373
362
22,764.5
53
3
513
423
90
5,845.5
1,510
1,641
359
18,753.5
54
4
626
317
83
4,872.5
1,847
1,266
285
15,431
55
5
607
225
83
3,892.5
2,035
918
260
12,207
56
6
591
155
61
3,012.5
1,924
554
215
9,176
57
7
557
127
41
2,219.5
1,683
426
132
6,547
58
8
398
44
49
1,536
1,149
140
141
4,449
59
9
254
31
29
1,051.5
837
159
95
3,009.5
60
10
171
12
25
747
422
33
55
1,981.5
61
11
91
9
22
540.5
234
31
63
1,472.5
62
12
58
5
7
420.5
162
18
12
1,151
63
13
35
5
11
350.5
126
18
29
959
64
14
47
3
6
300.5
134
6
16
792
65
15
31
1
8
245.5
109
8
18
635
66
16
26
1
10
205.5
70
3
17
502.5
67
17
22
3
3
167.5
67
8
7
410
68
18
17
2
7
140
64
7
9
328.5
69
19
7
1
4
114.5
16
2
8
251
70
20
3
1
5
102.5
5
2
9
225
71
21
8
1
4
93.5
6
1
10
209.5
72
22
19
6
81
22
11
193
73
23
12
9
56
28
23
160
74
24
14
2
35
33
1
109
75
25
5
4
19
48
9
75
76
26
1
10
5
18
77
27
2
2
9
4
1
13
78
28
3
5
6
8
79
29
2
2
2
2
80
5,841
3,760
857
46,799.0
17,458
12,580
2,769
145,613.5 110
TABLE I.
OBSERVATIONS ON MALE LIFE.
Age at Entry, 52 Years.
° a
Number or Entrants, 9,452.
$29,239,000 Insured.
Age
at
CD c/3
„ . , .
Discon-
,
Exposed to
Amounts
Amounts
Death
Exposed to
tinued.
Risk.
Existing.
Discontin’ed.
Claims.
Risk.
$
$
$
$
0
566
299
42
4,576.5
1,745
694
162
14,272.5
52
1
476
1,222
104
7,934
1,372
3,329
286
24,973.5
53
2
529
551
113
6,467.5
1,507
1,959
325
20,671.5
55
3
497
354
92
5,373
1,375
1,531
303
17,094.5
55
4
566
263
83
4,475.5
1,719
1,025
246
14,138.5
56
5
561
178
62
3,606
1,736
730
199
11,296
57
6
546
100
65
2,844
1,747
405
223
8,793.5
58
7
503
90
75
2,138
1,741
272
181
6,485
59
8
444
44
33
1,493
1,336
208
91
4,323
60
9
266
19
21
984.5
833
62
67
2,761
61
10
166
11
21
682.5
464
45
58
1,807.5
62
11
111
8
18
486
303
17
47
1,254.5
63
12
44
5
10
350.5
75
14
25
889
65
13
28
1
5
293.5
75
6
18
779
65
14
25
2
6
259
48
21
17
672.5
66
15
31
3
7
225.5
72
9
26
592.5
67
16
25
1
7
185.5
92
3
19
488.5
68
17
13
2
6
152
46
5
10
373.5
69
18
2
1
10
131.5
6
2
22
314
70
19
9
1
4
118.5
25
2
4
284
71
20
6
8
105
16
16
254
72
21
10
1
3
90.5
14
3
5
220.5
73
22
5
1
4
76.5
11
2
9
199
75
23
14
1
7
66.5
39
1
20
177.5
75
24
13
3
45
28
8
118
76
25
10
1
6
28.5
23
3
18
80.5
77
26
3
4
12
7
16
38
78
27
3
5
10
15
79
28
2
5
80
29
2
5
81
30
1
2
3
5
82
31
1
2
83
32
1
2
85
33
1
2
85
34
1
1
' 2
2
86
5,473
3,159
820
43,215.5
16,468
10,348
2,423
133,389.5
Age at Entry, 53 Years.
Number of Entrants, 8,213.
$25,055,000 Insured.
0
537
301
32
3,956
1,306
579
130
12,238
53
1
477
1,061
74
6,812.5
1,336
3,145
218
21,467.5
55
2
412
493
99
5,484.5
1,070
1,963
384
17,359.5
55
3
415
306
93
4,574
1,187
1,222
294
14,313
56
4
432
219
89
3,803.5
1.276
835
273
11,803.5
57 TABLE I.
111
OBSERVATIONS ON MALE LIFE.
Age at Entry, 53 Years.
«fH O
° rt
Number of Entrants, 8,213.
$25,055,000 Insured.
Age
£ f-i
m
Existing.
Died.
Exposed to
Risk.
Amounts
Amounts
Death
Exposed to
Risk.
Exit.
tinued.
Existing.
Discontin’cd.
Claims.
$
$
$
$
0
90
53
20
815
219
117
77
2,561
61
1
74
187
24
1,426.5
207
591
80
4,530.5
62
2
73
68
29
1,201
277
257
89
3,819.5
63
3
121
54
32
1,038
291
266
90
3,192
H
4
128
46
27
835
342
228
125
2,564
65
5
109
35
23
639.5
250
171
77
1,897.5
66
6
118
8
15
486
401
30
40
1,470
67
7
101
15
20
341.5
306
28
67
1,000
68
8
61
5
10
210.5
180
9
35
608.5
69
9
30
3
10
135.5
72
11
37
383.5
70
10
18
1
6
93.5
53
1
12
268.5
71
11
11
2
4
68
58
9
12
198.5
72
12
4
2
1
51
4
6
5
121
73
13
4
1
2
44.5
11
2
1
108
n
14
4
2
38
8
8
95
75
15
1
2
32
1
5
79
76
16
1
2
29
4
1
73
77
17
2
1
5
25.5
5
3
12
66.5
78
18
2
18
12
48
79
19
1
16
3
36
80
20
1
15
3
33
81
21
1
1
1
13.5
2
2
3
29
82
22
1
11
3
23
83
23
1
1
10
4
1
20
85
24
3
8
4
15
85
25
2
1
5
6
2
11
86
26
1
2
2
3
87
27
1
1
88
28
1
1
89
29
1
1
1
1
90
957
482
244
7,611.5
2,705
1,731
803
23,257.0
Age at Entry, 62 Years.
Number of Entrants, 1,206.
$4,188,000 Insured.
0
89
27
20
589.5
341
71
38
2,058.5
62
1
52
118
33
1,011
213
333
115
3,571.5
63
2
55
42
21
846
185
181
71
2,986.5
65
3
64
48
29
725
157
161
114
2,559.5
65
4
95
25
27
595.5
305
186
79
2,115
66
5
62
16
18
453
239
155
33
1,560.5
67
6
89
6
11
362
330
19
28
1,201.5
68
7
78
13
19
252.5 j
329
33
55
817.5
69
8
46
4
6
147
131
8
16
413
70
9
28
1
6
92.5
76
14
12
255
71 118
TABLE I.
OBSERVATIONS ON MALE LIFE.
Age at Entry, 62 Years.
v;S
° s
Number of Entrants, 1,206.
$4,188,000 Insured.
Age
U £
a> w
Discon-
Exposed to
Amounts
Amounts
Death
Exposed to
r HH
Existing.
tinued.
Risk.
Existing.
Discontin’ed.
Claims.
Risk.
$
$
$
$
10
5
2
58
20
3
160
72
11
8
1
4
50.5
23
3
6
135.5
73
12
3
1
1
37.5
5
1
5
104.5
n
13
1
2
33
6
6
94
75
14
1
1
30
3
1
82
76
15
2
1
4
27.5
9
2
9
77
77
16
1
3
21
10
7
58
78
17
1
17
1
41
79
18
1
1
16
5
1
40
80
19
1
2
14
2
3
34
81
20
1
11
4
29
82
21
1
10
3
25
83
22
1
1
9
6
3
22
84
23
4
7
9
13
85
24
3
4
86
25
3
4
87
26
1
1
3
2
1
4
88
27
1
1
1
1
89
687
303
216
5,425.5
2,406
1,167
615
18,466.5
Age at Entry, 63 Years.
Number of Entrants, 957.
$2,897,000 Insured.
0
52
24
9
466.5
159
62
24
1,417.5
63
1
47
116
23
814
171
316
60
2,494
64
2
56
52
24
660
192
143
79
2,033.5
65
3
37
37
18
535.5
92
107
74
1,637.5
66
4
64
32
15
446
181
106
42
1,365
67
5
62
15
12
343.5
161
144
48
1,017
68
6
54
10
14
257
172
30
43
721
69
7
47
6
6
181
166
17
16
482.5
70
8
46
2
5
124
99
8
13
288
71
9
15
2
6
71
31
5
14
169.5
72
10
9
1
3
48.5
34
1
5
121.5
73
11
5
1
4
35.5
6
3
6
80.5
n
12
3
1
4
25.5
10
2
2
66
75
13
4
1
1
17.5
22
1
5
52.5
76
14
1
2
12
1
2
25
77
15
1
9
1
22
78
16
8
21
79
17
1
8
5
21
80
18
1
1
7
6
3
16
81
19
1
4.5
3
5.5
82 T A B L E T
119
OBSERVATIONS ON MALE LIFE.
Age at Entry, 63 Years.
s’
° 'a
Number of Entrants, 957.
$2,897,000 Insured.
Age
at
C& £
0 S
w
° s
W
Y1 c<3
*
bi)
a
•73
P 0>
314
bi)
p
QQ
73 .
£44
ci
fc*C
P
ii,
73 .
£44
fcb
B
• 73*
5 a
S3
GO
*3 .
oJ
® m
I^M
i
O 53
O a
3*
O)
3
S.S
hS
1
; H
||
$■1
GO
WZ
wS
(3
be
<
® §
‘3
11
ft®
73
0)
5
Is
hS
M
«
9 p
E.S
O
be
<
$
$
$
$
$
$
$
$
0
46
17
6
287.5
53
22
36
852
65
15
3
7
5
12
81
1
20
66
11
490
37
192
32
1,519
66
16
1
4
2
7
82
2
24
42
12
405
71
110
43
1,299
67
17
3
5
83
3
27
17
15
339.5
77
79
41
1,090.5
68
18
3
5
85
4
34
11
10
283.5
97
43
27
911.5
69
19
1
2.5
3
3.5
85
5
44
14
13
227
127
123
39
704.5
70
20
2
2
86
6
31
8
10
159
95
39
27
457.5
71
21
1
2
1
2
87
7
45
6
5
111
111
31
9
300.5
72
22
1
1
88
8
21
5
58
56
23
165
73
23
1
1
89
9
12
1
4
31.5
33
9
12
81.5
n
24
1
1
90
10
3
15
12
32
75
25
1
1
91
11
12
20
76
26
l
1
1
1
92
12
13
1
1
1
12
11
3
1
2
20
19
77
78
130
94
77
1,363.5
397
259
202
4,215.5
14
2
9
2
14
79
15
1
7
1
12
80
Age at Entry,
67
Years.
16
1
1
5.5
1
2
10.5
81
17
18
4
4
8
8
82
83
No. of Entrants, 229.
$597,000 Insured.
19
4
8
81,
0
11
8
8
110.5
22
14
45
291.5
67
20
1
1
4
3
1
8
85
1
8
22
8
191
12
64
25
484
68
21
1
2
3
4
86
2
4
13
7
157.5
11
22
25
404
69
22
1
1
87
3
8
11
3
134.5
17
44
9
335
70
23
1
1
88
4
18
8
3
114
47
12
9
281
71
24
1
1
89
5
11
6
4
86
25
15
7
211.5
72
25
1
1
1
1
90
6
18
3
4
66.5
66
5
6
169.5
73
309
183
100
2,485.5
775
649
302
7,548.5
7
9
2
■2
42
13
2
4
94
75
8
9
3
28.5
39
74
75
Age at Entry, 66 Years.
9
6
18
12
33
76
10
2
1
12
7
1
21
13
77
78
No. of Entrants, 301.
$858,000 Insured.
11
1
9
1
12
8
12
79
0
9
11
5
145
37
25
11
416.5
66
13
1
8
1
12
80
1
23
29
8
261.5
5
69
14
750.5
67
14
7
11
81
2
11
20
11
206
29
58
29
668
68
15
1
7
1
11
82
3
7
17
8
165.5
42
50
32
556
69
16
1
6
2
10
83
4
10
3
8
140.5
44
4
25
455
70
17
1
5
1
8
85
5
21
6
9
118
64
28
16
370
71
18
4
7
85
6
12
2
5
84
47
8
18
272
72
19
1
4
4
7
86
7
13
3
2
64.5
36
2
4
202
73
20
1
3
1
3
87
8
13
1
2
47.5
32
11
4
155.5
n
21
1
2
1
2
88
9
6
1
2
31.5
20
1
9
113.5
75
22
1
1
89
10
2
3
23
20
7
84
76
23
1
1
90
11
2
3
18
20
3
57
77
24
1
1
91
12
13
3
3
13
10
6
16
34
28
78
79
25
1
1
1
1
92
48
14
7
12
80
105
76
1,027.5
275
182
140
2,498.5 TABLE I.
121
OBSERVATIONS ON MALE LIFE.
Age at Entry, 68 Years.
Age at Entry, 70 Years.
a
No. of Entrants, 151.
$378,000 Insured.
4-3
‘S
No. of Entrants, 58.
$120,000 Insured.
4->
'm
Years o
Insurant
w
d
oi
o
bo
<
Existing.
Discon-
tinued.
Died.
Exposed
to Risk.
Existing.
Discon-
tinued.
Death
Claims.
Exposed
to Risk.
u 03
Is
| Existing.
Discon-
tinued.
j Died.
! Exposed
to Risk.
Existing.
Discon-
tinued.
[ Death
1 Claims.
Exposed
to Risk.
$
$
$
$
$
$
$
$
0
6
5
1
73
6
2
1
188
68
0
2
7
1
25.5
4
8
3
56
70
1
3
16
2
131
17
24
6
357
69
1
1
9
2
43.5
1
11
5
99.5
71
2
1
6
5
115
5
3
20
320.5
70
2
1
3
2
34.5
1
9
4
83.5
72
3
5
7
102.5
14
13
287.5
71
3
3
1
2
29.5
10
5
1
71.5
73
4
6
6
1
91
19
10
1
262
72
4
4
24
15
58
n
5
16
3
8
79.5
50
7
12
233.5
73
5
3
1
1
19.5
7
1
1
42.5
75
6
11
1
5
53.5
22
6
16
165
n
6
5
1
15
10
6
34
76
7
4
2
3
36
15
7
8
120.5
75
7
1
9
2
18
77
8
8
3
28
36
15
94
76
8
3
8
6
16
78
9
3
1
2
16.5
16
2
2
42
77
9
1
1
5
1
1
10
79
10
1
1
2
10.5
2
5
5
20.5
78
10
3
8
80
11
1
6.5
2
10
79
11
3
8
81
12
1
6
1
9
80
12
3
8
82
13
1
1
4.5
1
3
7.5
81
13
3
8
83
14
1
3
1
4
82
14
1
1
2.5
1
5
7.5
84
15
2
3
83
15
1
2
85
16
2
3
84
16
1
2
86
17
2
3
85
17
1
2
87
18
2
3
86
18
1
2
88
19
2
3
87
19
1
1
2
2
89
20
2
2
3
88
23
22
13
233.0
55
35
30
538.5
21
1
2
3
89
22
1
.5
1
.5
90
Age at Entry, 71 Years.
64
51
36
771.0
93
2,142.5
1
Age at Entry, 69 Years.
0
1
15.5
2
39.5
71
1
1
5
2
27.5
1
8
9
73
72
No. of Entrants, 92.
$264,000 Insured.
2
4
2
2
21
1
3
8
57.5
73
14
11
47
n
6
3
1
1
0
1
46
1
132
69
4
1
1
10
1
1
35
75
1
3
6
5
88
8
24
12
251
70
8
33
76
2
1
7
1
73.5
1
34
1
202
71
5
1
1
6
3
4
5
5
65.5
9
14
6
176
72
6
2
1
6
20
5
29
77
4
6
1
2
53.5
17
3
3
152.5
73
7
3
1
3.5
1
20.5
78
14
14
10
7
105.5
37
18
24
334.5
5
9
3
2
43.5
36
3
124
7%
6
8
2
5
30
19
2
10
77
75
7
6
1
16
15
4
47
76
Age at Entry, 72 Years.
8
1
9
10
28
77
9
2
7
4
16
78
No. of Entrants, 24.
$59,000 Insured.
10
2
1
6
2
1
14
79
0
1
1
1
11.5
2
1
1
29
72
11
2
3
10
11
1
1
80
81
82
1
3
1
1
20.5
7
1
1
54.5
73
12
13
1
1
1
1
2
3
1
1
1
1
15.5
13.5
3
1
3
4
45.5
40.5
74
75
42
26
1
1
443.0
127
95
42
1,232.5
4
1
1
1
11.5
5
1
4
34.5
76 122
TABLE I.
OBSERVATIONS ON MALE LIFE.
Age at Entry, 72 Years.
Age at Entry, 75 Years.
s'
No. of Entrants, 24.
$59,000 Insured.
’m
SH s’
No. of Entrants, 11.
$19,000 Insured.
’3
w
® 0
w
bi)
£
gb
£ as
' T3
73 .
D M
do
g
• t3
Death
Claims.
.
ce
S««
' "d
72 .
QQ
73 .
-4J
'R
O 3
g a
s13
"d
0>
5
O-S
o-M
M a
H-S
1
O r-
o 5
o-2
c-M
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CD
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8.3
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73
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5
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Deatl
Claim
&
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bD
<1
$
$
$
$
$
$
$
$
5
1
i
8.5
2
2
24
77
0
5.5
9.5
75
6
1
i
6.5
2
5
20
78
1
i
11
2
19
75
7
1
i
5
i
5
14
79
2
10
17
77
8
1
i
3
2
1
8
80
3
1
2
9
1
16.5
75
9
1
5
81
4
<
15
79
10
1
5
82
5
4
1
6.5
10
1
14.5
59
11
1
5
83
6
1
2
3
4
81
12
1
5
8J+
7
1
1
82
13
1
1
5
5
85
8
1
1
83
9
7
8
100.5
25
11
23
295.0
9
1
1
54
10
1
1
55
Age at Entry, 73 Years.
11
1
1
86
12
1
1
87
88
No. of Entrants, 19.
$44,000 Insured.
13
1
1
14
1
1
89
0
1
2
1
9.5
3
1
22
73
15
1
1
90
1
2
16
9
35.5
74
16
1
1
91
2
3
1
13.5
2
8
27
75
17
1
1
92
3
1
9.5
5
1
18.5
76
18
1
1
93
4
1
8
1
15
77
19
1
1
n
5
6
2
2
7
5
1
2
14
13
8
78
79
80
81
20
i
1
1
1
95
7
8
1
2
2
1
5
6
6
3
2
65.0
14
2
3
109.5
5
4
8
7
71.5
10
28
6
158.0
Age at Entry, 76 Years.
Age at Entry, 74 Years.
No. of Entrants, 2.
$3,000 Insured.
No. of Entrants, 11.
$22,000 Insured.
0
1
1.5
76
i
2
3
77
0
1
1
5
1
1
10.5
74
2
2
3
75
1
1
8.5
1
19.5
75
3
2
3
79
2
1
7.5
1
18.5
76
4
1
2
2
3
80
3
4
3
1
7
7
6
4
18
18
77
78
5
6
1
1
.5
1
1
.5
81
82
5
3
8
79
1
1
10.5
2
1
15.0
6
3
8
80
7
1
1
2.5
2
5
5.5
81
8
1
1
82
Age at Entry
, 77 Years.
9
1
1
83
10
1
1
84
No. of Entrants, 1.
$2,010 Insured.
11
1
1
85
0
12
1
1
86
.5
1
77
13
1
1
87
1
1
2
78
14
1
1
1
1
88
2
1
1
2
2
79
6
4
1
50.5
10
8
4
113.0
1
2.5
2
5.0 TABLE I.
123
OBSERVATIONS ON MALE LIFE.
©
Age at Entry, 81 Years.
Ages at Entry, 79 and 80 Years.
►£» CO to t-» O
Years of
Insurance.
i
Age at Entry, 78 Years.
No. op Entrants, 2.
No. of Entrants, 0.
j Existing.
No. of Entrants, 4.
h-1
h-L
oo
Discon-
tinued.
1 1.5
h-A
bx
h-1
Died.
8.5
h—1 h-1 OO 62
b» On on
Exposed
to Risk.
$2,000 Insured.
$0,000 Insured.
Existing.
1
$5,000 Insured.
h-L
h-1
„ j „ „ Discon-
1 i— <* tinued.
I—1
, 1 .. Death
1—1 1 Claims.
1.5
h-1
bx
12.5
i Y—* £0 4^-
Or Oi Ox
Exposed
to Risk.
Oo Oo
Oo Oo Oo
i© Ki PD CO Oc
Age at Exit.
Or rfi W K) H O
Age at Entry, 88 Years.
CfQ
a
crt~
tel
c+-
00
Kj5°
CD
e oo
cri
00
Or
00
oo
p
3
00
•kl
o
Years of
Insurance.
Age at Entry, 82 Years.
No. of Entrants, 1.
No. of Entrants, 0.
Existing.
j No. of Entrants, 1.
Discon-
tinued.
h-1
Died.
OR
or
h-* MMHH
Cx
Or
Exposed
to Risk.
$1,000 Insured.
$0,000 Insured.
|_a m
Existing.
$1,000 Insured.
„ ; Discon-
** 1 tinued.
i—1
„ Death
' Claims.
5.5
f—1 MHHH
Or
bx
Exposed
to Risk.
OC CO CO S
d
M
bb
.5
w
do
85
pr
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a>
5
73 .
1
hf)
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' 73
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5 a
§‘3
PO
73 .
II
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d £
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M
tub
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M
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A 73
3 §
gfl
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73
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3
73 .
n
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5^
5 a
PO
73 .
§.2
0
1
2
3
4
i
1
2.5
4.5
4
4
4
$
2
$
1
$
$
3
5.5
5
5
5
i
4
5
0
1
2
3
4
i
i
1
1.5
1
1
1
$
$
i
i
$
$
i
1.5
1
1
1
7
S
5
1
2.5
1
2.5
0
5
i
.5
i
.5
0
6
7
2
2
2
2
7
3
6.0
3
6.0
8
9
i
2
1
1
2
1
9
10
Age at Entry, 5 Years.
10
i
1
1
1
11
No. of Entrants, 1.
$1,000 Insured.
3
2
29.5
2
34.0
4
0
.5
.5
5
Age at Entry, 2 Years.
1
2
1
1
.5
1
1
.5
7
No. or Entrants, 2.
$2,000 Insured.
1
2.0
1
2.0
0
1
1
1
1.5
1
1
1.5
2
3
Age at Entry, 6 Years.
2
3
1
1
1
1
4
5
No. of Entrants, 1.
$1,000 Insured.
4
5
1
1
.5
1
1
.5
6
7
0
1
2
3
.5
1
1
1
.5
1
1
1
6
7
8
9
2
6.0
2
6.0
Age at Entry, 3 Years.
4
5
1
1
1
1
10
11
No. of Entrants, 3.
$3,000 Insured.
6
7
1
1
1
1
12
13
0
1
2
1.5
3
2.5
1.5
3
2.5
3
4
5
8
9
1
1
.5
1
1
.5
u
15
1
1
1
1
9.0
1
9.0
3
4
1
1
2
1
1
2
1
6
7
Age at Entry, 1
Years.
5
.5
.5
8
No. of Entrants, 4.
$4,000 Insured.
1
2
10.5
1
2
10.5
0
1
2
3
4
2
3.5
3
2.5
2 I
2
3.5
3
2.5
2
1
1
1
1
7
8
9
10
11 128
TABLE II.
OBSERVATIONS ON FEMALE LIFE.
Age at Entry, 7 Years.
Age at Entry,
9 Years.
No. of Entrants, 4.
$4,000 Insured.
4-3
’3
No. of Entrants, 7.
$21,000 Insured.
4-3
W
0 S
W
02 c<3
bp
A'd
•
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fio
S.W
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CD OD --
b
W
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i-i
to
Discon-
tinued.
o
i>
CfQ
CD
l—i
25
W
P
c+*
1—1
Died.
►3
»
S-
I—1
iO IO tO IO JO to o: CO 03 03 03 03 tP* C3> C3
bn
MHMMH
t-i 03 00 tfc* 03
On On
25
b
«5
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tel
P
r+-
b
108.0
l—i M
MKHI-iH
h-i tO iO 03
bn
ht4- On On
Exposed
to Risk.
25
§
*o
pi
tel
B
C"t~
►b
On M■ OO
h"4 h"4
h-4 h-4 OO O O
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1—1
to
to
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to «»
Existing.
V*
►—4
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jo
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Discon-
tinued.
1
t-1
h-4
5*
an
k5
CD
Pi
to
Death
Claims.
i
Kj
CD
P
i-i
CfJ
t—1 H-1 I—1
tO iO tO iO iO tO -3 ~CH *3 O 00
bn
|—i tO 03 Ot 03
OjO^OH
Or On
a
§
B
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GO
157.0
l-i k-i
l-i k-i h-i M M
J-* iO iO t£“- CD ©
On
$
8
8
6
6
Exposed
to Risk.
b
a
B
u
os os ta ts ta ta *a ta ta i>a ta *a i"* t-* k*
OS Oi-k- Os fca k C toOoS
k< K*. K*. K*. K*.
Os
Os Os
to k*
Os to ts ts to
C)<£i0okt03
tOtOtStStQ k* k kt
On-fc- Os tO k* C3 CO Oo V? Oi
Age at Exit.
to to to to to to to to to H M H H H ►—* h- ►—4 1—* ►—*
co s 03 o* to m o cc oo 03 oi ©a
5
6
7
rf* CO IO h-i ©
to to to to
WtOHO
Years of
Insurance.
h-A
«
k-i l-i
to to I-1 to
HHMOSH
a
o
£**■
crq
CD
h-4
t-i t-i
Existing.
2
o
h-4
O
to
iO 03 03
**
h-4
*o
Discon-
tinued.
o
tel
t>
crq
CD
25
b
W
P
c+-
I—4
Died.
2!
►3
»
p
ct-
146.5
h-4
h-4 h-4 h-4 fcD *0> *nS *0 *0 *0 *0 *0 fcS *>2 fcS fcS
MMKtOH
iO tfn- GO £"• 03
On On On
25
b
03
is
p
e+-
b
136.5
t-1 1-1 to to
Exposed
to Risk.
5
4
J®
IO
r1
tel
y
crte
•b
to
00
h-i i-i
M tO iO ©
H-*
03
03
t—1 i—1 -e%
Existing.
05
JO
t-i
1—4
1—1
HQO^
JO
CO
to
On
Discon-
tinued.
*0
Kj
CD
P
KJ
CD
P
l-i
Death
Claims.
M
g
1—*
-3
I—1
h—4
H-4 h-4 M *o fcD *0> *0 M) *o *0 fcS *0 W W Oi o
HMtSOSM
03 rfx 1—i © k-1
b
a
b
GO
to
On
©
1—1 t-1 to to
Exposed
to Risk.
b
w
«
«
•b
GO
©
On
bn
bn
-fc-.-k-.OsCs Os c© Os Os Os Os Os Os ta ?a ta *a £a ta ta
k. O Oo S 03 Cn-fc- C19 S 03 0>n-fc- Os t© CO Oq
K< K* ki
-3 C*s
Cs C»s Os Os
Oh 4>» Os
Age at Exit. TABLE II.
130
OBSERVATIONS ON FEMALE LIFE.
to to JO to to
if* W tO h-‘ ©
HHHHH ►—1 h-4 h-4 h-1 h-4
CD 00 <1 Oi CD JO H O
© 00 -3 © CJI
l^COtOH©
HHHPH
© 00 ~3 © Ot
if* M tO h* ©
Years of
Insurance.
h^
h-i
05 Oi Oi -3 Oi
Ox 1—1 GO © ©
3
o
©
©
h^
h-A h-^
to h-1 © © to
to if* to *3 It-
Existing.
2
O
h-1
Oi tO if*
h-1 1—1 ©
h* © 1—‘ © Oi
o
d
ft
>
crq
CD
Oi
h-A
h-A
hh to © © if*
©
h*
*3 -3 *3 ©
Discon-
tinued.
O
d
ag
CD
to
to
H
w
to
h-A h-A
Died.
H
w
to to to to to
to to to to CO CO CO
CO CO CO
bi
-3 tO © tO ©
<21
1—1
o< © 00 © ©
to it* O 00 pi
bi bi bi bi
55
s
P
cd-
feJ
crh
326.5
h* tO tO if* if*
bi
h* to to
03 © If* © ©
bi b< ox
© if* Oi -3 If*
h-l- 1—1 h*■ © h*l
bi bt bx bx bx
Exposed
to Kisk.
5
H
ZP
a
p
c-h
Cd
P
C"h
Oi
Oi
h^
Or Q Oi JD Q
1—1 1—1 1—1
l—i l—i 05 05 CO
<3*
Ci
h-A
h—1
h-i h-*
h-^
Oi JnD <2 *0
3
2
81
8
$
Existing.
h-
i-i
©to lf^
HHMCO
© © CO © if*
"o
8
fi?
L5 Years.
©
—3
h-'1
JO JD JO JD M
Oi
$
3
13
17
9
Discon-
tinued.
CO
0
0
§
to
-<>
to
1—11—1
Death
Claims.
CD
P
O O O 0 O
1—1 h* h-‘ h-‘ h* h-1
© © © © if* if* if* if* if* if*
bx
to to © Oi ©
© © © © m
1—1 1—1 (—1 1—1
OO 1—1 if* -3 ©
—3 H-1 © If* tO
bi bi bi
§
B
u
o»
21
©
bi
1—1 tO tO rfh. 1th-
bi
to to © © if*
© If* © ©
bx
h*
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bi bi bi bi bi
Exposed
to Risk.
d
»
B
d
cc
Ca Ca Ca Ca c©
to Qo N <31 Cl
Ca Ca Ca Ca Ca
-is- Ca tc>
tO to to to to
too Qo • <*3 tor.
to to ta ts> to
-fc- c-e tto s-, ©>
h«< h^
Oo Cr,
to t© tc ta
Co Oi
ta to tc tc Ki
C® tS h* to Co
K4. h-4. H-4. 1—1.
Oo -3 Oi tor, 3?-
Age at Exit.
(MMIOMO
to to to to to K—1 ‘ 1—Iipp (—* t—i (—* l—* l—*
rfiCOtOHO © CO -3 © Oi i^UtOHO
© QO
3 © Oi
if* M to h-1 ©
to to
© Oi
Years of
Insurance.
| -a if* © © ©
3
O
~3
-3
r-A
©
1—i to
h-1
Oi O Oi OO Oi
h-1
QHHC5
©
*
o
54
h-A
Existing.
p
HHOlO
©> CO h-1 h-1 to
w
i>
CTq
CD
00
CO
to to © if*
h* If*
GO © ©
If*
O
d
w
h-
CTD
CD
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00
Discon-
tinued.
0
d
ft
>
crq ■
CD
to l—i to
H
to
it*
h-A
1—1
h-i h-*
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w
it*
Died.
H
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h* 1—1
OOOMOiO
l—i Cu © Oi GO
bi bi bi oi
25
s
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c+-
bd
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664.5
1—1 (—1 1—1
1—1 1—1 tototototo OlOiOi
© 00
h* to © if* Ol
if* Oi h* © Oi
bi
h-i
© *3 © OO OO
—3 C31 © © ©
bi bi
a
H
p
P
c+-
td
b
c+-
520.0
h* h*
Exposed
to Risk.
25
Ui
CO
05
P
crh
bd
p
rt-
h^
tO if* © 00 ©
©
to
CO
to
Oi OO
h* i-i
M■ -3 © Or if*
7
13
1
j 12
h-A
M
Oi
if*
Existing.
O
jX3
"o
I—1 CO £0 Oi h-1
Oi £0 Cn r-1 CD
8
j?5
■vj
123
tO tO —3 CO
h—1 h* h—l C5l
© h-i GO © ©
to
i°
05
104
Discon-
tinued.
Or
KJ
CD
P
(-1 Oi CO
fi?
Hi
CD
£3
if*
h-A
l—i
h-i h-i
M
25
Hh
CD
P
©
Death
Claims.
51
h* l-i tO tO l—i
if* -3 h-i -3 ©
tO Oi O M O
bi bi bi
d
w
B
d
m
977.0
COCOCOCOCO if* if* if* if* if* ©©©
II
61
tO © if* © -3
h-> © >f* © Oi
bi
|—l |_1 (—1 h-i l—i
© l—i if* OO h-i
O 05 OI -3 M
bi bi bi
d
w
B
d
CQ
995.0
if* it*^
Exposed
to Risk.
d
M
B
d
qq
tQ t© Ki h-» Ki
Ki toi <55 00 ~-3
Cq Co Cq C=) C© Cq Cq
ta ta ta ta ta
tor, is- Ca ta h*
17
18
19
20
16
Ki C5
Age at Exit. TABLE II.
131
OBSERVATIONS ON FEMALE LIFE.
r
Age
at Entry,
17
Years.
Age
at Entry, 18 Years.
<♦_ ©
No. of Entrants, 209.
$336,000 Insured.
g
No. of Entrants, 372.
| $651,000 Insured.
+->
o“
w
H
ot ce
h *-*
® CO
hb
a
M
' T3
§§
O S
S’c
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hi:
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S 3
nd
1
6
7
26
11
7
6
78.5
46
33
16
11
7
106.5
64.5
25
26
90
112
7
768.5
142
178
16
1,359.5
8
9
3
30.5
11
3
43 5
27
9
7
1
19.5
8
1
30.5
28
Age at Entry, 18 Years.
10
11
1
3
10.5
8
1
4
20
17
29
30
No. of Entrants, 372
$651,000 Insured.
12
8
i
17
31
13
1
1
7.5
5
1
16.5
32
0
1G
31
3
170.5
28
32
4
309.5
18
14
6
1 .
11
33
1
11
109
1
267.5
11
230
1
472
19
15
1
6
11
34
2
21
40
181
27
72
309
20
|3
3
12
26
3
127
18
57
IV
(
217.5
21
16
5
5
8
35
4
8
12
1
93
16
16
1
156
22
17
i
8
36
18
1
5
1
•i
8
37
5
11
11
1
72.5
12
23
2
119.5
23
19
4
7
38
6
LG
5
2
52.5
21
6
1
91
24
20
2
39
7
8
1
4
30
25
4
64
25
1
4
7
8
2
19
1
8
33
26
21
1
3
2
5
40
9
4
1
16.5
5
1
27.5
27
22
2.5
1
4
41
23
1
2
3
42
10
11
12
2
12
4
22
18
10
28
29
30
24
1
1
2
2
43
2
1
10
7
3
5
189
298
9
1,560.0
266
547
18
2,487.0
13
7
10
31
14
1
7
1
10
32
*
15
1
6
1
9
33
16
5
8
34
17
5
8
35
18
5
8
36
19
5
8
37 132
TABLE II.
OBSERVATIONS ON FEMALE LIFE.
30
to to to to so
CO CD —I 05 Ot
to to to to to
4* CO tO t—* O
h-4 h-* h-» h-*
CO CD <1 05
h-4 h-4 h-4 H-4 ►—4
CaD fcO H-4 O
CO CO —3 05 cn
4* CO to 4-1 0
Years of
Insurance.
GC
CD
M-
65
M M I—1 H—1
4-1 4-1 CO
4-1 65 65 65
Ox CD O CO CO
4- co co co 4^
65 65 4- GO 65
Existing.
No.
4-
Ot
65
4-i 4-i
65 CO —3 —3
h-i
CO CO —3 05 05
—3 —3 4— —3 OO
Discon-
tin ued.
§
gj>
Cfq
CD
05
h-1
i—i i-i
65 65 4-i
CO 65 65 CD CO
Died.
»
J\S
'co
00
o
©
h-*
4-i
M• 4-*
I—1 CO
h-1
Or 00 00 CD O
h-11—11—11—11—i
OMHHti
1-1 p p p |U
60 £0 CO GO
4-i 4-i
iOl4 4JMOl
Ot GO CO CO 65
bt bx bt
65 65 CO Or CO
65 CD 00 4*- CO
p at ts oo 00
bt bt bt at bt
Exposed
to Risk.
*-3
«
P
OT
p
c-4
bd
C3
c+-
4^
CD
-3
4^
1—1 CO CO M-
M1 h-4
41- 4*- 00 4-
GOOi6SOOOi
0 4- at -3 05«,
Existing.
b1
to
0
756
CO
4-i 4-i 65
65 05 CD CO --3
4-i 65 4-i
1—i
O O CO (4^
HMMHM
Ol Oi Oi D
4-i 4-i 4-1 4-i 65
Oi Oa •'-{ GO Oj
bi
M M (O
CO 00 4i- CD 05
-3 -3 4*- CD
bx bx bx
-Co at 05 cd at
GO 0 GO GO CD „
0 CD 65 65 w
bt at at
Exposed
to Risk.
OQ
d
w
M
0
50
•k- 4s- -is- -is- -fc-.
co Oo -3 <35 Cn
4s- 4s- 4s- 4s- 4s-
4s— Ka
c© Cs Cq cq
CO Oo Oi c^
c© c© c© c© c©
-fc. Ce iS k O
ts 6a t© to to
to Oo Cl Cr,
*$>
4s— G»a £0 *—4.
Age at Exit.
to to
05 Ot
to to to to to
CO to •—1 O
h-4 h-4 h-4 h-» H-4
COGDMOOl
HHHHH
CO CO -3 05 01
CO tO 4-i O
Years of
Insurance.
CO
Ot
CD
65 4^
65 65 1—‘ 4-i
4-1 4-1
4-1 4-1 CO —3
4-i CO CO CO
05 CD Ox 65 05
C0C0 4»-4i-4;-
0 cd co at
Existing.
2
O
4-
CD
4-i
h"4
CO 4-1
65
65 Ox -3 00 4-i
65
CO 05 —3 65 Ot
0 0 65 at 05
Discon-
tinued.
O
*1
l>
aq
CD
CO
o
4-1
4-i 65 65 h-1 1—1
4-i at go at co
Died.
H
«
CO
o
CO
Ox
65 05
l—i 1—i l_i t—i
OO 65 Ox 05
4-14-14-14-11—1
05 - tto *to
to to Qo s to
*0 *0
Gji 4s- Go fc©
Age at Exit. TABLE II.
133
Observations on female life.
Age at Entry, 22 Years.
Years of
Insurance,
Number or Entrants, 1,141.
$1,986,000 Insured
Age
at
Exit.
Existing.
Discon-
tinued.
Died.
Exposed to
Risk.
Amounts
Existing.
Amounts
Discontin’ed.
Death
Claims.
Exposed to
Risk.
$
$
$
$
0
56
75
7
533
88
116
16
935
22
1
45
278
8
864
79
482
23
1,525
23
2
60
121
9
611.5
94
262
14
1,051
n
3
60
61
4
451.5
68
145
5
739.5
25
4
44
39
6
337.5
74
60
21
564
26
5
34
25
2
255.5
56
57
5
410.5
27
6
35
11
3
201.5
40
14
7
314
28
7
33
16
1
150
50
44
1
238
29
8
29
19
1
98.5
46
29
2
150.5
30
9
11
2
1
58
11
6
1
85
31
10
14
2
45
21
4
70
32
11
5
1
28.5
10
1
44.5
33
12
3
23
2
34
Si
13
1
19.5
3
30.5
35
14
1
19
5
29
36
15
1
17.5
1
23.5
37
16
17
23
38
17
2
2
17
2
2
23
39
18
13
19
40
19
2
13
1
19
Jfi
20
11
18
i2
21
11
18
4S
22
2
11
3
18
U
23
2
9
3
15
45
24
4
7
8
12
46
25
2
3
2
4
47
26
1
1
2
2
48
445
650
46
3,826.5
665
1,220
101
6,415.0
Age at Entry, 23 Years.
Number of Entrants, 1,264.
$2,268,000 Insured.
0
68
81
9
591.5
114
130
22
1,069
23
1
59
300
7
956
92
536
11
1,734
24
2
82
104
8
688
114
221
11
1,252.5
25
3
44
73
7
509.5
60
176
16
929
26
4
57
37
5
403.5
84
87
15
721.5
27
5
51
29
5
308.5
66
60
9
549
28
6
52
14
3
231
80
27
5
430.5
29
7
55
18
160
106
18
323
30
8
30
2
95
67
4
206
31
9
10
2
63
26
4
135
32
10
7
52
11
107
33
11
6
1
1
44.5
13
2
1
95
34 !
12
3
2
36
5
1
79.5
35
13
32
74
36
14
2
1
31.5
8
2
73
37 134
TABLE II.
OBSERVATIONS ON FEMALE LIFE.
Age at Entry, 23 Years.
*&
|
Existing.
Discon-
tinued.
Died.
Exposed to
Risk.
Amounts
Existing.
Amounts
Discontin’ed.
Death
Claims.
Exposed to
Risk.
$
$
$
$
20
2
1
30
2
1
38
48
21
4
27
4
35
49
22
3
23
5
31
50
23
7
20
10
26
51
24
2
1
13
2
1
16
52
25
8
10
10
13
53
26
1
2
1
3
H
27
1
2
55
28
1
1
2
2
56
897
886
69
7,255.0
1,526
1,820
116
13,549.0
Age at Entry, 29 Years.
Number of Entrants, 1,713.
$3,520,000 Insured.
0
78
89
6
812
150
139
12
1,690.5
29
1
63
323
18
1,378.5
107
599
54
2,919.5
30
2
79
133
16
1,069.5
130
354
31
2,282
31
3
81
104
10
856
125
309
25
1,789.5
32
4
93
57
8
684.5
146
208
18
1,381
33
5
89
47
5
531.5
152
124
8
1,051
34
6
90
23
5
402.5
186
60
8
799
35
7
79
27
3
282.5
160
60
8
545
36
8
55
8
3
183
94
21
6
336.5
37
9
38
3
1
119.5
77
9
2
221.5
38
10
15
6
3
76
24
10
7
133
39
11
5
3
1
53.5
12
5
2
94.5
40
12
1
46
2
78
41
13
1
1
3
44.5
2
1
13
75.5
42
14
3
40
9
60
43
15
3
1
1
36.5
2
1
3
50.5
44
16
3
32
11
45
45
17
1
1
28.5
1
1
33.5
46
18
1
1
27
1
2
32
47
19
25
29
48
20
2
25
3
29
49
21
1
23
1
26
50
22
2
22
3
25
51
23
5
20
6
22
52
24
8
15
9
16
53
25
4
7
4
7
54
26
3
3
3
3
55
803
826
84
6,843.5
1,420
1,901
199
13,774.5 TABLE II.
139
OBSERVATIONS ON FEMALE LIFE.
Age at Entry, 30 Years.
<~8
° pi
Number of Entrants, 1,944.
$3,837,000 Insured.
Age
w d
d d
17
1
40
1
59
47
18
1
2
1
38
1
3
1
56.5
48
19
35
2
53
49
20
1
33
1
51
50
21
1
32
1
50
51
'22
1
2
1
30
2
4
1
47
52
23
10
1
27
13
1
42
53
24
8
16
16
28
54
25
4
8
3
12
55
26
4
4
9
9
56
968
884
92
7,943.0
1,589
2,050
198
15,357.5
Age at Entry, 31 Years.
Number of Entrants, 1,674.
$3,563,000 Insured.
0
75
64
4
805
116
108
18
1,727.5
31
1
93
313
13
1,374.5
158
628
15
3,007
32
2
107
125
10
1,049.5
135
368
17
2,336
33
3
85
98
6
821
141
306
25
1,847
SJf
4
91
75
5
643.5
148
232
8
1,412
35
5
79
39
6
490.5
134
225
11
1,027.5
36
6
85
18
3
377
215
46
4
747
37
7
80
18
5
271
144
27
12
491.5
38
8
o
53
on
6
K
4
174
111 K
77
54.
13
9
10
1
315.5
217.5
39
AO 140
TABLE II.
OBSERVATIONS ON FEMALE LIFE
Age at Entry, 31 Years.
<13
<«"
° a
Number or Entrants, 1,674.
$3,563,000 Insured.
Age
H H
'
at
a a
03 CO
„ . . .
Discon-
. ,
Exposed to
Amounts
Amounts
Death
Exposed to
CiXisluig.
tinued.
Risk.
Existing.
Discontin’cd.
Claims.
Risk.
$
$
$
$
10
15
4
2
77
24
12
1
152
41
11
11
5
1
55.5
19
7
5
117.5
4%
12
3
3
39.5
7
3
88.5
48
13
1
35
1
80
44
14
4
1
33.5
14
2
78
45
15
2
2
28 *
5
3
61.5
46
16
2
25
4
55
47
17
2
1
23
8
1
51
48
18
20
42
49
19
20
42
50
20
2
20
11
42
51
21
1
18
2
31
52
22
1
16.5
1
28.5
53
23
2
1
15.5
5
2
27
54
24
5
2
13
13
2
21
. 55
25
4
6
4
6
56
26
2
2
2
2
57
833
778
63
6,565.0
1,441
1,992
130
14,053.5
■ ' i
Age at Entry, 32 Years.
Number of Entrants, 1,763.
$3,671,000 Insured.
0
81
60
8
851.5
139
100
17
1,785.5
32
1
65
383
14
1,422.5
122
755
23
3,037.5
33
2
87
139
10
1,082.5
193
392
21
2,319
8^
3
78
83
10
874.5
120
256
19
1,781
35
4
101
62
5
714
158
150
7
1,439
36
5
84
34
8
560
162
145
22
1,126.5
37
6
103
35
4
433.5
183
85
6
827.5
38
7
87
27
2
295.5
174
50
6
571
39
8
53
6
1
190
106
18
1
357
40
9
30
7
1
129.5
60
15
5
233.5
41
10
19
95
22
161
42
11
14
3
74.5
27
3
137.5
43
12
1
1
1
58.5
2
2
1
108
44
! 13
1
56
3
104
45
14
2
3
1
53.5
4
5
2
98.5
46
15
4
1
48.5
4
1
89.5
47
16
2
1
44
16
3
85
48
17
1
40.5
2
65
49
18
1
40
1
64
50
19
2
1
39
5
1
63
51 T A B L E I I.
141
OBSERVATIONS ON FEMALE LIFE.
Age at Entry, 32 Years.
Number of Entrants, 1,763.
$3,671,000 Insured.
Age
to c3
Jh fH
at
oS &
*&
Existing.
tinued.
Died.
Risk.
Existing.
Discontin'ed
Claims.
Risk.
$
$
$
$
20
1
21
1
30
59
21
2
1
20
3
1
29
60
22
17
25
61
23
3
2
17
4
2
25
62
24
5
12
6
19
63
25
4
7
5
13
64-
26
1
3
4
8
65
27
2
2
4
4
66
697
574
59
5,422.0
1,267
1,397
118
11,068.5
Age at Entry, 40 Years.
Number of
Entrants, 1,444.
$2,888,000 Insured.
0
73
63
5
690.5
129
108
8
1,390
40
1
67
242
17
1,182
119
527
33
2,379.5
41
2
94
110
5
922
174
243
19
1,842.5
42
3
62
77
10
729.5
88
204
20
1,426
43
4
83
65
9
586.5
164
184
21
1,124
44
5
84
22
7
451
133
64
17
815
45
6
87
18
5
340
160
34
10
616
46
7
81
16
3
231
147
33
4
412.5
45
8
45
4
137
87
9
240.5
48
9
25
2
1
89
37
6
1
146
49
10
16
2
61
20
3
103.5
50
11
7
2
43
14
3
80.5
51
12
1
2
34
1
7
61.5
52
13
2
1
31.5
3
5
54.5
53
14
2
1
28.5
10
3
47.5
54
15
1
26
2
36
55
16
25
34
56
17
1
1
1
24.5
1
2
2
33
57
18
22
*
29
58
19
1
22
2
29
59
20
21
27
60
21
2
21
1
27
61
22
1
1
1
18.5
1
1
1
25.5
62
23
4
16
7
23
63
24
7
1
11.5
9
2
15
64
25
4
4
5
5
65
747
630
67
5,768.0
1,311
1,438
139
11,023.0 TABLE II.
147
OBSERVATIONS ON FEMALE LIFE.
Age at Entry, 41 Years.
*-s
° S
Number op Entrants, 1,077.
$2,277,000 Insured.
Age
u £
at
Exit.
c3 £
CD in
*8
Existing.
Discon-
tinued.
Died.
Exposed to
Risk.
Amounts
Existing.
Amounts
Discontin’ed.
Death
Claims.
Exposed to
JEtisk.
$
$
$
$
0
50
35
3
521
84
67
8
1,105
47
1
43
182
10
898
97
374
16
1,931
42
2
58
74
10
717
122
192
20
1,535
JfS
3
53
62
5
581
107
168
14
1,213
44
4
74
43
4
470.5
139
117
8
949.5
49
5
68
26
5
358
111
61
12
713.5
46
6
63
12
4
266
116
37
9
541.5
47
7
64
19
2
183.5
138
60
3
368
45
8
46
3
1
106.5
78
12
1
191
49
9
15
2
57
21
5
103.5
50
10
8
6
38
11
25
67
51
11
2
27
4
44
52
12
2
1
25
2
1
40
53
13
1
22
1
37
64
14
3
21
8
36
55
15
18
28
56
16
1
18
5
28
57
17
2
1
17
3
2
23
58
18
1
1
14
1
1
18
59
19
12
16
60
20
1
12
1
16
61
21
11
15
62
22
1
11
1
15
63
23
1
1
1
9.5
1
1
1
13.5
64
24
4
7
5
11
65
25
1
3
3
6
66
26
2
2
3
3
67
564
465
48
4,426.0
1,062
1,119
96
9,067.5
Age at Entry, 42 Years.
Number op Entrants, 1,101.
$2,242,000 Insured.
0
64
42
4
529.5
124
71
9
1,085.5
42
1
39
176
10
903
56
348
27
1,864
43
2
65
74
10
729
119
174
23
1,520
44
3
69
55
8
589.5
92
162
29
1,210
49
4
64
35
4 '
467.5
137
81
9
967.5
46
5
69
22
7
371
134
67
14
747.5
47
6
65
25
5
271.5
131
58
7
537
48
7
54
11
2
183.5
100
23
2
358.5
49
8
43
5
119.5
86
10
240
50
9
17
2
1
73
42
3
1
147.5
51 148
TABLE II.
OBSERVATIONS ON FEMALE LIFE.
Age at Entry, 42 Years.
«M§
°d
Number or Entrants, 1,101.
$2,242,000 Insured.
Age
m ci
cfl pi
16
10
13
62
2
58
56
4
619
153
178
4
1,207
47
17
10
13
63
3
45
50
5
504
70
114
10
904
58
18
1
10
3
13
65
4
67
25
5
416.5
105
54
19
740
4-9
19
1
8.5
i
9.5
65
5
65
26
5
319
120
50
16
564
50
20
i
8
2
2
9
66
6
47
8
3
232
66
16
7
395
51
21
1
7
7
67
7
61
8
2
174
76
14
10
307
52
22
6
5
68
8
38
5
3
104.5
71
10
7
209
53
23
2
1
6
1
1
5
69
9
19
1
3
60.5
35
3
5
124.5
55
24
2
3
2
3
70
10
12
2
37
26
1
82.5
55
25
1
1
1
1
71
11
2
2
23
2
11
50.5
56
429
321
34
3,155.5
792
758
58
6,267.0
12
13
14
1
1
20
17
16
1
1
43
57
2
4
39
37
58
59
Age at Entry, 47
Years.
15
4
1
1
15.5
6
1
10
36.5
60
No. of Entrants, 653.
$1,299,000 Insured.
16
10
20
61
17
18
19
10
20
20
20
62
0
32
21
2
316
63
32
9
633.5
57
10
bo
65
1
36
82
6
557
61
163
8
1,113.5
58
10
2
43
52
2
448
81
102
3
912
59
20
1
10
2
20
65
3
36
34
2
360
54
92
2
731
50
21
3
9
6
18
66
4
44
28
4
291
92
89
6
584.5
51
22
6
6
5
2
4
12
12
10
67
68
69
5
35
12
4
223
49
45
8
419.5
52
23
24
1
3
6
7
45
38
4
2
3
2
176
125
74
70
11
7
7
2
334.5
244.5
53
55
25
2
2
6
6
70
8
29
4
1
82
48.5
57
10
1
2
164
98.5
46
55
56
57
514
365
42
3,847.0
906
798
100|7,283.0
9
21
3
2
45
5
10
7
1
23.5
17
6
Age at Entry,
46
Years.
11
5
1
15.5
11
2
25
58
12
10
13
59
No. of Entrants, 784.
$1,608,000 Insured.
13
10
13
60
14
1
10
1
13
61
0
53
29
4
377.5
93
46
11
781
56
15
9
1.2
62
1
30
128
6
634
77
273
13
1,321.5
57
16
9
12
63
2
48
47
3
510.5
73
123
6
1,033.5
58
17
9
12
65
3
45
40
8
416
75
107
12
839.5
59
18
9
12
65
4
50
25
2
330.5
100
59
3
669.5
50
19
1
9
1
12
66
5
45
23
2
254.5
93
55
2
509.5
51
20
1
8
2
11
67
6
35
9
3
191.5
72
32
1
371
52
21
7
9
68
7
46
6
1
146
79
14
1
275
53
22
1
1
7
1
1
9
69
8
43
6
93
70
30
173
55
23
3
5
3
7
70
9
15
5
44.5
31
7
84.5
55
24
1
2
2
4
71
10
6
2
2
1
26
9
11
44.5
56
25
1
1
2
2
72
11
12
4
2
19
13
8
3
5
1
30
17
57
58
380
244
29
2,770.5
686
564
49
5,447.5
13
10
13
59
14
10
13
60 TABLE II.
151
OBSERVATIONS ON FEMALE LIFE.
H-4 H-4 h-4 H-4 ►—4
co to ►—4 ©
CO CO *-3 OS CJX
xP* CO tO P-* O
to to
OS ox
to to to to to
Xp5> CO to X—* 0
HHHHH
CDGOMQOT
HMHHH
►f* co to k-* ©
CD OD OS CSX
CO tO P-i 0
Years of
Insurance.
P-i tO GO XP-
C5^-iOOO
CO CO xp- tO CO
CO-3HCO-3
0
435
1—1
1-1 to 1-1
1—1
i-i 1-1
1—1
1-1 1—1 Ox
1—1 to CO Ox Ox
00 05 CO 1—1
CO 05 Ox CO
CO 00 O —3 O
Existing.
3
0
P-i tO
CO 62 Ox CO -3
k—4 CO tO H—4
Oi -3 CD OD
tel
ft
crq
CD
304
1—1
1-1
h-4 h-4
CO Ox CD
p^
to Ox Ox 0 CO
to CO 0 CO 0
Discon-
tinued.
0
ft
h“4
CO CO xf^
-3 05 Ox CO CO
H
W
39
1-1
1-1 to -3
05 -3 CD to
Died.
w
crq
CD
P-i P-1
-3 -3 I—1‘ 00
I—* M■
tO -3 tO -3
05 CO C5 xp*. 05
to CO CO to
CO 0 00 05 -3
W tO H Ox H
as
H
CT
-o
tel
za
0
00
0
X—1 1—1
l—i tO xp- Ox 05
- 3 -3 -3 -3 -3
00 cb
M 1-1 to
1—1 CO 00
HMiO
(P *3 H -3 Ox
tO h-i to 00 IP*
CO xp*. OX 05 CO
tO O tO 05 -3
05 -3 XP-
Exposed
to Risk.
*5
g
$
90
p
crP
tel
bx
Ox
Q<
Ox Ox
bx bx bx
85
p
crb
b
bx
Ox
Ox Ox Ox Ox
bx bx
fc!
r+-
-3
to XP* Ox 05
O^OMO
CO Ox -3 to
tO l—i —3 00 Ox
CD
Kj
0
50
-3
cd
co
1-1
P-1 IP- 1-1
Ox
1-1 i-i
h-i
to tO h-i
l—i
l—i 05 -3 tO 00
“3 l—i CO O 00
p-i l
h-3C0-3Ox« Existing.
05 -3 05 GO !
**
P-i CO
17
2
5
31
15
h-i
Ox CO -3 CO tO
Ox CO tO CO Ox
4^-
UT
8
O
-3
Ox
to
CO l—i
to 05 xP*
00 0 to xP*
1-1 P-1 to
-3 M W M Ox«
Ox 05 (—1 CO CD
Discon-
tinued.
05
19
O
00
t0>
05
h-4
CD h-4
h-4 h-4 h-4
00 GO CO ■
h-4
to
1—1 to -3
P-1 to
CD CD O 00 05®
Death
Claims.
S
Kj
CD
P
P-i 1—1 1—‘ 1—1 tO
O O O xM- Ox
Or bx
M M tO
CO 05 to CO CO
00 05 M 00 JO
bx bx bx bx
Ox —3 CO Ox
xP*. GO 05 tO 05
ox xp- to x—* 0
bx bx bx
w
d
g
«
0
JG
6,394.0
h-i 1—1
1—1 to C5 00 CD
l—i l—i l—i l—i l—i
ip- xP- xP- xM- xp-
M l—i tO tO CO
Ox —3 x—* xp- Ox
bx bx
1—1 to CO Ox
Ox l—i O 05 l—i
tO -3 Ox 00 05
P-l
—3 CD 1—‘bo -3
OOxPOOO®
O tO P-i CD X—i
bx bx bx bx
Exposed
to Risk.
d
1
0
$
os cs os cs C\
Oa t© k* cs co
Ox Ox Ox Ox o,
Oo CS Ox-pi-
Ox Ox Cjx txx-te-
c© tc p-. cs Cjx Ox
•~3 os Ca
Ox C?t Cjx+>- -t—
is P-4. 05 CO Oo
Age at Exit.
to to to to to 1—» 1—» 1—* 1—» »—» M-‘ P-* P-* I—‘ P-*
xP» CO tO P-* O CD OO
00
CD
198
h-1
Discon-
tinued.
0
tel
ft
>
crq
CD
xp^
CO
h-4
—i.
t-i
to 1—1 Ox CO CO
CO to 05 CD 05
H
w
CO
05
H—4
Died.
H
w
2,356.5
M to CO CO Ox 0x05-3-3-0! —3 —3 CO
bx
17.5
12.5
1—1 1—1
CO Ox CD CD
H CD iO M Oi
bx bx bx
to CO CO xP*. to
05 x—i CD 00 00
05 X—i CO tO
bx bx bx
§
H
OD
UT
88
P
ct~
teJ
p
d
DO
T-4
CO CO XP* 05
(—4 |_a (-a
Exposed
to Risk.
CD
te
U
Ki
GO
bx
Ox
bx bx bx
bx bx bx
b
bx
-3 • Cs Cs Cs Cs
CO00-3CS
05 Os Cs OS os Cs
Ox -fc- C© to k* <05
Ox Ox Ox Cn CJx
CO c© Os Ox
4>- C© Ki
v? -3 Cs
ki c 0
cs Cs cs cs Cs
Oo cs <0,4^.
Age at Exit. 152
TABLE II.
OBSERVATIONS ON FEMALE LIFE.
►—4 H-* * h-4 >—4
O®
CT?
CD
h-i
62 <20
-3 20 h-* 62 1—i
H
W
62
05
4-1
4-1
4-1
4_1 20 4-1 (4-
4-i 62 -3 OO 4-i
Died.
H
RS
44 i |_l (_l 6-*
44 62 62 -3
b*
<—<1—1
<20 -3 O <20
l—i 20 O 20 00
b* b*
l—i 62 62 20 62
O 4 CD C5 H
62 CO O* 00 20
b* b*
p*
z
CO
go
C"4
tel
P
C+“
4-1
00
4-1
05
b
4-1 62 00 4* 44 <2* O*
6
6
6
5
05
05
M-
Ci <2 00 C5
4-i
62 20 05 CD (4-
62 CO h4- 00 4-
b* b* b*
62 62 © © 62
O 64 © 00 62
© 64 p* © 62
b*
Exposed
to Risk.
5
CD
£>-
8
P
e4
C3
trb
© 46
62 M CB
23* 20 GO l—i O
05 CO* O* 00 05
400000
■€£
8
M
4-
-3
O
62 4-i 20 4-i
10
4-i 20 20 05 05
00-6 0*00*
© 64 27* 25 OO
00 © 62 OO -3 m
Existing.
y, 51 Yea
i—i
cd
h-1
CD CD -<£
1-4
2*0*0 0
204OH00
*0
Kj
CD
PS
386
O*
62 4-i 62
4-1 O 14- 05
4-i
64 © © © © m
46 00 © © 20
Discon-
tinued.
905,000 Insured.
20
05 00
h-1
5*
49
62
4-1
4-1
M H
MOlMCDM^
Death
Claims.
M H M tO 62
25 rfx tfs. O -3
Or
72.5
55
1—1 1—1 62
62 00 C5
CO Ol ©
bi bi o<
20 41* 05 —3 6p-
-3 tO O -3 4*
to GO 00 4* —3
b* 0*
d
g
B
d
GO
-<*
Or
62 20 05 -3 -3 CD
CD
10
10
10
9
O
4-1 4-1 4-1 4-1 62
O O 4-i 20 05
b*
4-i 62 20
4 ® 20 4 O
420 650-3
b*
64 Ol © 46 64
4-1 © 4-1 © ©
63 20 00 © © w
b*
Exposed
to Risk.
V2
CiOiCi OiC^
©D ©t *■£>■ Os fc©
32 02 23* 23* <2,
C) * 23* 23* 23*
<35 23*-fc- 2© 6a
*■<£ Oil C*s
4? 05 05 O5 05
Q> 2o 0q —3 <33
65
05 <35 05 05
-fc- ce ta 4,
<35 <3* 23* <2, <2*
CO Oo 4? <35
25* 23* 23* 2?, 23*
<2*-4- 2a 6a 4a.
Age at Exit.
6363606063 HMHI-iH
20 63 I—* O CO CD © O*
13
14
10
11
19.
CO OO -3 05 O*
4* CO 60 4-i O
'
63 63 63 63 63
>4 20 63 4-i ©
h-» h-* H-‘ h-1
CDCDMQOi
Years of
Insurance.
f—1
00
h-1 h-1
4-1
444
4-1 4-1 62 4-i
4i* 05 62 <2* 22*
to to 4 62 4-1
4 4 —3 03 22*
a
p
62
4?
20
4-i 4-i 62 4-i
4-i 62 4-i
Existing.
3
p
h-*
h-^
CD
M
4-1 62 62 62 OO
CD
462 44
4-i 62 23* ©
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ft
!>
crq
CD
138
! Discon-
j tinued.
ft
tz{
a
crq
CD
62
O
M
h-A
4-1
62 62 20
f—1
62 62 20 62
►3
a
22
h-l
Died.
1,275.5
1-162202020
b*
O* 05 -3 CC
4-1
20
4-1
4-1 20 O* 00 O
-3 O* 4 C U
b*
133.5
152
264.5
202
166.5
Z
H
CO
p
ct-
bd
p
c+-
1,827.5
4-1 62 O* O*
4-i 4-i
© © 46 © 4-1
Exposed
to Risk.
S
p
C4
bd
p
c4
4J
CJ*
nd
CD
P
g
m
312
62 l-i
>—1
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4-1 62 62 20 62
20 00 20 4^-
20 62 4* 05. 4-i
14-344(4
CJ*
CO
o*
20
4-i
62 4-i 20 4-i
h-1 h-^
Existing.
■€»
62
-3
00
62
20
4-i 4-i 05 62 4-i
62
62 O* O 20
O 4-i 00 <2*
f
321
Discon-
tinued.
|
20
h-L
h-1
i—1
62 62 05
h-i
OD CD GO JO
5
CD
d
g
-
«
CD
P
50
62 4-1
Death
Claims.
5*
2,301.5
62 4^0*0505-3
f—'
•<2 GO CD H
4-i
00
4-i 4-i
20 Ol tO 62 -3
63 © © CD O*
b* b* b*
236.5
299
528
369.5
287
►d
GQ
3,580.5
62 20 © 45 —3
4-i 44 4-1
00 00 © 62 © **
Exposed
to Risk.
d
a
5
a
-6 46 4J "3 46 -3 S l-i tO CO Ol -Mi O CO G O 03 Ol
OtOlOlOtOl C3 -3 M? tG 03 CO M » G O p tG -MJ GO ifk
oi bi bibibibi bibibi bi
Exposed
to Risk.
1—t GO tO GO GO
h-1 M Oi O Ol M OJ O CO tG 00 GO
$038,000 Insured.
CO
CO
Oi
M-i h-i CO tO tM rf^tOrf^GOrf^^
O GO tO CD tG G GO GO tO O GO G ®
Existing.
M H iO Oi CO i*
M00 MMOOM C5 m o;
to I )_i
P MM MMMtOCC^
Ol Ol M tOCOiOMM
Discon-
tinued.
Death
Claims.
Exposed
to Risk.
o
a
o
g
a
CO
d
g
S
«
f—4
h-4 h-4 fcD h-4 h-4 M1 tsD O OitsDh-^OO
50
Oi Oi h-4 Oi Or Oi OO^D^h-4hf^^’
MMM M CO M Cn CO
M M1 M to CO ~3 O CO O Ol CO M CO O
OS-5COHOO 0 S fc© Mi C, Co Go Mi 03 0s t© M, CO p CcMi^Cn-jN
Age at Exit.
MJOMtO t-‘M-‘M-‘M-‘»-‘ M-*M-‘K-‘M-*h-‘
CO to M o CD CD 0{ M> M* GO -M? 00 GO CD ti Mi CO M CO
GO GO GO GO GO GO GO GO GO GOOOtOtO M M CD 00 M CD p Ol Ol p
Or On On On
2,615.0
|-i 1—1 M (_) i—i
k-i CD M M G G G®
Exposed
to Risk.
s MJ v; "3 "Mi 'O "3 ~CJ 03 03 03 03 03 03 03 03 03 03 O
h-A h-A K-A h-A h-A
CD GO CD cn
h-A h-A h-A h-A H-A
GO to h-A 0
CO 00 -3 CD oe
►P* GO IO t—* ©
Years of
Insurance.
i-i i-i
MiS OS 05
14
11
6
00
CO Oe
10
11
O
1-1
£
h^
h-A
h-A l—A
CD O to CO
h-A h-A h-A h-A
CD tO Ol h-A
Existing.
3
O
62 62 33
l—i )—i 62
(fi O 62 Oi OS
o
W
a
H
w
>
05
1—A
62
1—1 ips (Oe
62
62 CO CO OS CJe
Discon-
tinued.
O
a
H
sd
h
h-1
62 H-i 62
OS 62 OS
62
O'
62
CTQ
to
1—A
h-A
l-i I—*
ip
h-A
3
5
3
1
Died.
crq
0
OlOlOtQKi
1—i t-i
-3 CO CO l—i OS 00 >f- CD O
be
l—i r—i i—i
00 O 62 Or GO
05 OS Or CO
be
a
H
®
QO
88
c+-
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b
00
n-
62
be
62 62
tO
bx
rfi*- Oe 05 05
l—i 62 OS Oe *3
OS O OS l—i 62
be be
1—1 1—1 1—1
CO 1—1 OS 05 ©
05 Jps 1—1 CO ~3
be be be be
Exposed
to Risk.
a
H
g
p
88
c-t-
tei
b
h-i 62
l—i 62
62 1—1 Ifi- is O to
1—i 1—i h-‘ 62 62
iSOOOOiSSS
OS rfi O! CD
OS On CO l—i
bi be
151
H 62 62 OS l—i
00 62 00 OI CO
“3 CD Oe 62 00
be
d
to
&
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1—1
4o
0
05
be
os OS
OS 03 OS CD CD
be
1-11—11—1
CSCDOMH
100.5
65.5
38
20
150.5
62
OS
CO
be
224
388
316.5
280.5
Exposed
to Risk.
d
§
H
d
~3 -3 -3 -3 -3
3{ 05 0\-k- c©
-3 3? 3* 05 05
63 3* <©> CO Oo
05 05 05 05 05
3J 05 Oe-k>. Oa
05 05 05 Oe <0e
63 3, <©> CO Oo
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Oo 13
3? -3 *3 *3 3?
05 Oe-P- Oa is
3J *3 05 05 05
K. 05 CO Oo
05 05 05 05 05
05 Ovfcs Oa is
05 05 Oe Cji Gjx
3 S CO Oo 3
Age at Exit.
© CD *3 CD CJe
ifiOSbOHO
20
h-* h-* h-A h-» h-
COODMOC
a
rr
h-* h-A h-A h-» h-*
CO OO *3 CD CJe
OS 62 l—i O
63 62 62 62
OS 62 I—i ©
Years of
Insurance.
►p*. 05 05 00 05
00 if*. 00 05 -3
3
o
“3
Oe
-
h-A
h-A
-a i__t
h-A h-A
CO hA 0 0 to
h-A
oo 0 w
3
p
89
-
Existing.
izj
0
iS M M 00 M
62 OS CO GO 62
**
t6*-
Cfq
CD
39
62 lP-
h-A
WO
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a
a
crq
CD
CD
Ol
Discon-
tinued.
0
>
crq
O
OS OS
62 P-* Ip- OS
S
62
O
i—i
h-i
62 OS OS 62
OS 62 l—i 62
H
to
27
I—i 62
Died.
H
w
M■ t—i 62 lP< Oe
be be be
l—i
05 -3 CO O 05 ‘
05 OS H CD O
be be
a
H
m
i
P
cf-
tel
b
e+-
•-S
05
O
631.5
h—1
1-1 OS OS (P*- rfn
lP- if*. Oe 05 -3
HMiSt6 0e
0^-3001
1-1
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H OS H M Oe
be Oe Oe
a
H
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88
erh
tel
ch-
841.0
1-* 62 IPs ips
Exposed
to Risk.
a
h
cn
00
P
ch-
tei
rh
t—* t—i i—* i—i
Or 62 05 lP>- 62
h-i 62
00 00 I—i ~3 62
h-A
h-A
00
os
1-1
6SMOe
62 I—i I—*
62 62 IP- © ©
62 l-i
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Or
CD
189
62
Existing.
at
00
Ki
CD
P
iS M H CO M
iO f—i
OS OS OS -3 IP-
j*
h-A
o
7
11
62 i4^
00 Oe -3 Oe -3
Or
CO
O
s
S'
147
Discon-
tinued.
Q
S
HH
a
Oe
05 OS OS
£
03
u
Kj
CD
P
►-1
QQ
62
Oe
1-1
Or
1—1 62 ip. 62
OS h-i 62
CD
P
05
05
1—1 Oe
Death
Claims.
M OS tfi. co
05 05 05 05 05
Oo 3J 05 Oe-fc"
05 05 05 05 Oe
Ce is is 0 co
c<5 Oo
Is 0 Co Qo
Age at Exit. TABLE II.
155
OBSERVATIONS ON FEMALE LIFE.
Age at Entry, 60
Years.
Age at Entry, 62 Years.
o g
No. of Entrants, 122.
$215,000 Insured.
-4->
'R
w
No. of Entrants, 54.
$111,000 Insured.
m
H
bb
G
r6
1U'
•4-*
Bg
eS 3
bi
S
'G .
tJO
G
I'd
m
TJ .
c6
s s
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PS
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M
E=J
6 £
5 s
T3
H A
H-l
16 to 19
20, 21
22, 23
24, 25
26, 21
28, 29
30, 31
32, 33
0
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0
1
.03167
.03251
.03076
.03125
.02975
.02950
.02886
.02756
1
2
.06848
.07103
.06595
.06651
.06416
.06304
.06108
.05870
2
3
.11133
.11282
.10211
.10598
.09944
.09678
.09379
.09200
3
4
.15424
.15897
.14333
.14795
.14100
.13500
.13075
.12865
4
5
0.21081
0.22007
0.20526
0.20511
0.19508
0.18536
0.17896
0.17856
5
6
.29608
.30419
.28311
.27934
.26141
.24597
.24303
.24077
6
7
.42017
.41418
.38348
.37144
.34663
.32425
.32288
.31871
7
8
.56518
.56338
.49498
.48051
.44877
.42394
.41637
.41552
8
9
.72039
.72029
.63628
.60940
.56752
.53940
.52961
.53480
9
10
0.90595
0.88140
0.76332
0.73569
0.68621
0.65580
0.64871
0.65832
10
11
1.03608
.98935
.88288
.85079
.78971
.76447
.75669
.76347
11
12
.13428
1.06705
.96695
.93098
.87108
.85081
.84310
.84680
12
i 13
.18797
.11745
1.02247
.98875
.91419
.90053
.89185
.89522
13
u
.24499
.17096
.07471
1.04638
.96450
.94898
.94126
.93811
14
15
1.29687
1.22848
1.14839
1.10994
1.03495
1.01171
1.00502
0.99613
15
16
.34188
.30143
.20883
.17915
.09429
.06871
.06272
1.04468
16
17
.38040
.39219
.27406
.24687
.13776
.11044
.11365
.08012
17
18
.40808
.45412
.33330
.30876
.18436
.15738
.15951
.11317
18
19
.44173
.49038
.39645
.35785
.2287L
.20863
.20061
.16153
19
20
1.44609
1.42734
1.28963
1.26198
1.23978
1.20003
20
21
.52848
.47540
.33599
.29878
.28672
.25242
21
22
.56209
.52180
.37204
.33082
.32485
.28933
22
23
.59941
.57132
.41461
.37611
.36530
.32815
23
24
.77465
.68447
.48049
.44200
.45187
.38124
24
34, 35
36, 37
38, 39
40,41
42, 43
44, 45
40,47
48, 49
0
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0
1
.02736
.02797
.02612
.02682
.02637
.02667
.02592
.02522
1
2
.05863
.05861
.05685
.05787
.05685
.05506
.05485
.05414
2
3
.09011
.09003
.08992
.08450
.08871
.08557
.08437
.18785
3
4
.12711
.12451
.12542
.11724
.12499
.12342
.12055
.12281
4
5
0.17563
0.17178
0.17432
0.16137
0.17407
0.17291
0.16813
0.16904
5
6
.24009
.23623
.23728
.23311
.23927
.23703
.23415
.24216
6
7
.32266
.31900
.32222
.32714
.32248
.33015
.32644
.34079
7
8
.42283
.42148
.43019
.43705
.43852
.44456
.44387
.46653
8
1 9
.53758
.54528
.55948
.57120
.56882
.57546
.61384
.63870
9
10
0.65714
0.66750
0.68624
0.69490
‘ 0.70469
0.70987
0.76135
0.77771
10
11
.76034
.77463
.79305
.81533
.81133
.82799
.89096
.88884
! ii
12
.83609
.86817
.87863
.91130
.91464
.92110
.98369
.99857
12
13
.89591
.92046
.93184
.96849
.97285
.98229
1.05109
1.05690
13
14
.94054
.96708
.98309
1.02026
i
1.02834
1.03559
.09971
.10516
14 TABLE III.
163
MALE LIFE (C.) LOGARITHM OF THE MULTIPLIER {W) TO CARRY FORWARD THE
AMOUNTS EXISTING TO THEIR PROPORTIONAL TERMINATION INCLUSIVE
IN THE FOLLOWING COLUMNS OF TABLE I, THAT IS, TO
FINAL SERIES.
Rheumatism
169
11
180
52
75
53
.48
3
o
Gangrene
51
0
51
18
13
20
.14
o
Tubercular meningitis
10
1
11 I
3
3
5
.03
Lumbar abscess
11
0
11
6
3
2
.03 TABLE LX.
183
GENERAL TABLE OF DISEASES AND DEATHS IN TWENTY-SEVEN LIFE INSURANCE
COMPANIES FROM THEIR ORGANIZATION TO THE YEAR 1874.
The columns headed (2), (5), (20) record the Deaths in so many Companies.
!
Number of Deaths.
Variability. M.
and F,
Per Cent.
DISEASES.
Males.
Females.
Total.
(2)
(5)
(20)
of Total.
All Causes.
35,442
2,182
37,624
12,845
12,456
12,323
100.00
*3
Scrofula
25
6
31
15
10
6
.08
o
Tabes Mesenterica
88
5
93
31
36
26
.25
*->
Morbus Coxae
12
0
12
5
3
4
.03
02
Consumption
G,474
412
6,886
2,283
2,231
2,372
18.31
O
o
Other Constitutional
10
1
11
6
4
1
.03
—
Apoplexy
Congestion of brain
1,705
61
1,766
653
599
514
4.70
655
14
669
234
239
196
1.78
Softening of brain
399
9
408
177
93
138
1.09
Paralysis
Disease of brain
841
32
873
360
250
263
2.32
721
37
758
239
251
268
2.02
Convuls’ns and epilepsy
130
8
138
52
50
36
.37
Insanity
Anxiety
Fright
140
2
1
6
0
0
146
2
1
63
2
1
41
0
0
42
0
0
.39
.01
.00
Encephalitis
277
10
287
109
101
77
.76
s
Cerebro-spinal sclerosis
1
0
1
1
0
0
.00
>
u
Cerebral embolism
1
0
1
1
0
0
.00
CD
Anaemia of brain
8
0
8
4
2
2
.02
Effusion on brain
48
3
51
10
23
18
.14
Neuralgia
17
1
18
5
8
5
.05
Progressive muscular 1
3
0
3
2
1
0
.01
atrophy j
Tetanus
47
4
51
10
24
17
.14
Inflam’t’n of spinal cord
18
0
18
10
5
3
.05
Disease of spinal cord
41
2
43
21
10
12
.11
Congest’n of spinal cord
3
0
3
2
0
1
.01
Other Nervous Diseases
48
6
54
21
18
15
.14
Disease of heart
1,297
62
1,359
534
418
407
3.61
Peri- and endo-carditis
104
9
113
29
38
46
.30
Hypertrophy of heart
Valvular disease of heart
100
98
4
6
104
104
28
14
47
55
29
35
.28
.28
Fatty degen’t’n of heart
Dropsy of heart
Rheumatism of heart
42
56
66
3
6
6
45
62
72
21
27
30
15
13
17
0
12
0
17
14
9
22
25
0
13
4
.12
.16
.19
.01
.07
01
£
O
■4J
Atrophy of heart
Paralysis of heart
4
27
0
1
4
28
4
3
i
2
o
.61
o
Abscess of heart
Angina pectoris
Aneurism of aorta
5
79
51
0
4
1
0
83
52
46
25
6
20
13
A
.22
.14
.05
Rupture of aorta
16
1
17
i
4
Embolus of pulmo-)
5
1
6
1
3
2
.02
nary artery )
Phlebitis
Other Circulatory
18
18
0
2
18
20
4
12
9
3
5
5
.05
.05 184
TABLE IX.
GENERAL TABLE OF DISEASES AND DEATHS.
DISEASES.
Number of Deaths.
Variability. M,
and F.
Per Cent.
Males.
Females.
Total.
(2)
(5)
(20)
of Total.
All Causes.
35,442
2,182
37,624
12,845
12,456
12,323
100.00
Epistaxis
8
0
8
2
3
3
.02
Disease of Larynx
112
3
115
30
44
41
.31
Bronchitis
437
21
458
144
144
170
1.21
Pleurisy
172
7
179
75
64
40
.48
>>
f-t
Congestion of Lungs
568
29
597
187
198
212
1.59
Pneumonia
2,713
176
2,889
990
899
1,000
7.68
c<3
Abscess of Lungs
78
6
84
25
36
23
.22
Hemorrhage of Lungs
283
7
290
93
90
107
.77
Disease of Stomach
and Bowels \
127
7
134
31
41
62
.36
02
Hemorrhage
9
0
9
4
3
2
.02
bj)
Strangulated Hernia
43
12
55
18
17
20
.15
p
Colic, Tympanites and )
Constipation f
69
3
72
22
28
22
.19
Dyspepsia
43
3
46
13
24
9
.12
Gangrene of Tongue
3
0
3
1
2
0
.01
Stricture of (Esophagus
11
0
11
1
8
2
.03
Fistula in Ano
12
0
12
3
5
4
.03
Disease of Spleen
22
0
22
14
4
4
.06
Leucocythaemia
7
1
8
3
0
5
.02
Ascites
41
8
49
15
20
14
.13
Abdominal Tumor
35
3
38
11
20
7
.10
Undefined Diseases, )
Abdominal Organs \
11
0
11
7
0
4
.03
Jaundice
75
2
77
37
21
19
.20
Inflammation of Liver
268
15
283
80
93
110
.75
Cirrhosis of Liver
104
4
108
38
33
37
.29
Abscess of Liver
79
7
86
28
32
26
.23 TABLE IX.
185
GENERAL TABLE OF DISEASES AND DEATHS.
DISEASES.
Number of Deaths.
Variability. M. and F.
Per Cent.
Males.
Females.
Total.
(2)
(5)
(20)
of Total.
All Causes.
35,442
2,182
37,624
12,845
12,456
12,323
100.00
Diseases of Liver
448
19
467
153
180
134
1.24
Congestion of Liver
63
3
66
29
22
15
.18
Hypertrophy of Liver
36
2
38
11
14
13
.10
?h
Ph
Abscess, Hemorrhage)_
etc., of Lungs )
46
102
89
159
200
75
12
683
Other Respiratory
13
33
30
39
53
22
8
198
.
Diseases of Stomach
36
84
91
139
160
95
23
628
>
Diseases of Bowels
62
118
99
163
194
85
18
739
S
o
Dysentery
7
1
5
G
12
3
1
35
&
Diarrhoea
4
7
3
4
3
1
22
N
Cholera
4
2
3
4
2
15
Alchoholism
1
1
Other Zymotic Diseases
9
15
3
ii
8
G
1
53
Dropsy
2
8
7
8
14
G
45
Cancer
2
4
9
18
8
11
3
55
O
Consumption
14
G8
78
85
119
41
8
413
Other Constitutional
1
G
7
7
7
7
1
3G
Apoplexy
7
10
8
1G
13
5
2
61
CO
Congestion of Brain
2
2
4
2
3
2
1
16
2
O
>
Paralysis, Softening }
etc., of Brain f
G
13
15
19
14
11
8
86
Epilepsy, Convulsions
1
4
3
2
10
Other Nervous
1
4
4
6
2
3
1
21
►»
Diseases of Heart
7
11
10
23
32
9
9
101
s
Other Circulatory
2
2
1
5
Pneumonia
19
37
24
43
37
12
5
177
>>
Jh
Congestion of Lungs
5
5
4
9
3
2
28
ce
Bronchitis and Pleurisy
i
5
5
10
3
2
2
28
■p-
Abscess, Hemorrhage)
2
15
8
6
13
1
1
46
Diseases of Bowels
7
19
10
9
17
G
2
70
in
Peritonitis
5
7
9
13
4
3
41
bp
Diseases of Liver
2
8
G
13
15
4
48
p
Other Digestive
8
7
G
10
17
G
2
56
Diabetes
3 '
2
3
Diseases of Kidneys
3
4
4
4
9
1
27
Other Urinary
3
2
3
1
9
Childbirth and Puer-)
30
51
25
45
40
G
197
in
peral Diseases )
o
CD
Diseases of Breast and}
G
23
20
25
28
G
2
110
c6
Uterus f
o
Abscess, Hemorrhage)
2
3
G
5
3
G
3
28
CO
Old Age j
a
Debility, Exhaustion, }
2
3
4
3
9
G
2
29
Prostration, etc. j
34
Accidents and Injuries
5
7
4
7
8
1
2
Suicides
2
1
1
2
1
7
Unknown Causes
4
4
3
3
4
4
1
24
Note. The Total, 2,182, includes 1 more unknown. 188
TABLE XI.
(A.) DISEASES AND AGES AT DEATH OF 35,442 INSURED MALES.
DISEASES.
Ages at Death.
9J-19i
19f29 i
29f39|
39i-49|
491-591
591-691
69^-791
791-891
Total.
All Causes.
133
3,476
9,321
10,840
7,576
3,357
647
92
35,442
Zymotic
54
903
1,866
1,943
1,137
390
58
5
6,356
&H
Constitutional
22
1,117
2,691
2,441
1,342
484
76
2
8,175
Ph
Nervous
9
238
997
1,576
1,410
733
135
8
5,107
8
Circulatory
3
61
328
537
582
389
81
5
1,986
a
Respiratory
11
370
1,168
1,524
1,117
484
88
9
4,770
§
Digestive
12
216
758
1,134
811
361
45
7
3,344
Miscellaneous
22
571
1,513
1,685
1,177
516
164
56
5,704
Typhoid,Typhus
28
472
702
620
361
116
7
2,306
Malarial Fever
4
72
252
286
154
58
5
831
o
Erysipelas
27
86
132
85
37
6
1
374
+-»
Dysentery
7
62
165
160
136
44
12
1
587
s
Diarrhoea
3
27
77
98
74
37
10
2
328
N
Cholera
5
55
134
141
73
21
2
431
Alcoholism
6
41
55
14
1
117
Other Zymotic
7
182
409
451
240
76
16
1
1,382
Dropsy
19
103
191
183
99
15
1
611
Cancer
8
72
186
239
107
20
632
o
Consumption
21
1,052
2,406
1,933
819
219
23
i
6,474
Other Const’l
1
38
110
131
101
59
18
458
Apoplexy
1
42
238
500
598
279
46
l
1,705
CO
CongestionBrain
3
53
190
231
130
43
5
655
O
>
Paralysis, Soft- /
ening Brain 1
4
109
431
663
572
355
79
6
2,236
£
Epilepsy and [
Convulsions )
10
48
48
25
13
2
130
Other Nervous
1
24
90
134
85
43
3
1
381
►>
Diseasesof Heart
3
59
307
488
553
362
78
5
1,855
•Ja
o
Other Circulat’y
2
21
49
29
27
3
131
Pneumonia
G
183
594
882
689
305
49
3
2,711
>>
U
Congest’n Lungs
2
41
132
189
130
57
16
1
568
cd
Bronchitis and}
Pleurisy J
2
42
152
202
132
62
15
4
610
CO
a>
Abscess, Hem- [
or’age Lungs \
1
90
240
192
112
42
6
683
Other Respirat’y
14
50
59
54
18
2
1
7 98
.
Dis. of Stomach
3
24
124
226
159
75
16
1
628
j>
Dis. of Bowels
2
72
219
206
159
73
5
3
739
to
a>
Peritonitis
2
35
73
77
42
15
1
1
246
be
Dis. of Liver
2
37
209
410
266
101
11
2
1,038
Other Digestive
3
48
133
215
185
97
12
693
Diabetes
5
32
58
26
33
4
158
Dis. of Kidneys
4
59
160
291
239
107
25
i
88G
Other Urinary
Childbirth,Pu- )
* t
11
31
53
78
82
25
2
282
CO
erperal Dis. [
o
Dis. Breast [
cd
and Uterus )
Abscess, Hem-)
55
37
339
o
00
or’age,Old Age f
1
16
77
77
52
25
s
Debility, Ex- )
haustion, etc. (
21
60
88
102
79
25
2
378
A ccid’ts, Injuries
16
365
894
820
449
119
15
*,678
Suicides
1
49
126
173
106
18
2
475
U nknownCauses
45
133
125
122
41
16
26
508 TABLE XI.
189
(B.) DISEASES AND AGES AT DEATH OF 2,182 INSURED FEMALES.
DISEASES.
Ages at Death.
19f29|
29f39i 39| 49T
49|-59|
59£-69| 69|-79!
79£-89i
Total.
All Causes.
18
360
698
563
317
164
51
11
2,182
SUMMARY. |
Zymotic
Constitutional
Nervous
Circulatory
Respiratory
Digestive
Miscellaneous
G
5
3
4
48
99
26
5
36
33
113
98
195
45
22
78
89
171
75
158
44
34
78
73
101
51
70
40
22
48
47
39
17
20
29
18
34
24
22
6
2
9
5
12
5
12
2
0
2
1
6
303
549
194
106
293
269
468
Zymotic.
Typhoid,Typhus
Malarial Fever
Erysipelas
Dysentery
Diarrhoea
Cholera
Alcoholism
Other Zymotic
2
1
i
l
i
25
2
1
4
4
4
8
33
15
2
11
14
5
18
32
15
1
7
2
4
14
21
9
5
5
1
1
1
8
5
5
1
5
i
1
3
2
1
1
118
49
10
35
22
15
1
53
4^
CO
G
o
o
Dropsy
Cancer
Consumption
Other Const’l
3
2
2
1
91
5
12
5
165
13
15
24
105
14
15
16
38
1
4
4
10
2
2
45
55
413
36
Cir’y. i Nervous.
Apoplexy
CongestionBrain
Paralysis, Soft- i
ening Brain i
Epilepsy and l
Convulsions }
Other Nervous
4
3
12
1
6
10
8
21
1
5
14
2
22
3
3
19
2
15
2
2
9
2
12
1
5
3
6
1
1
61
16
86
10
21
Diseasesof Heart
Other Circulat’y
5
21
1
31
3
22
17
1
5
101
5
Respiratory.
Pneumonia
Congest’n Lungs
Bronchitis and)
Pleurisy f
Abscess, Hem- i
or’age Lungs }
Other Respirat’y
18
4
4
8
2
45
8
9
13
3
53
6
6
11
2
26
4
5
8
5
27
1
4
2
7
3
1
1
i
177
28
28
46
14
Digestive. 1
Dis. of Stomach
Dis. of Bowels
Peritonitis
Dis. of Liver
Other Digestive
1
1
1
6
9
9
3
6
19
27
17
16
10
11
15
7
20
20
5
14
5
9
14
7
5
2
3
7
5
54
70
41
48
56
Miscellaneous.
Diabetes
Dis. of Kidneys
Other Urinary
Childbirth,Pu- 1
erperal Dis. )
Dis. Breast )
and Uterus )
Abscess, Hem- \
or’age.Old Age f
Debility, Ex- )
haustion, etc. j
Accid’ts,Injuries
Suicides
U nknownCauses
3
1
5
1
66
22
1
4
6
3
5
9
3
94
34
7
7
13
4
2
8
34
33
5
4
6
2
7
1
4
4
15
3
6
2
4
1
4
5
8
3
1
i
l
5
1
1
3
4
2
3
27
9
197
110
28
29
34
7
24 190
TABLE XII.
(A) MALE LIFE. DISEASES AND NUMBER OF DEATHS BY STATES AND TERRITORIES.
Typhoid and
Typhus.
Other Zymotic
Consumption.
Other Constitu-
tional.
Apoplexy.
O
t
CD
t-
©
o
Diseases of Heart.
Pneumonia.
Other Kespiratory.
Digestive System.
Diseases of Kidneys.
Accidents and
Injuries.
Suicides.
All other Diseases.
All Causes.
Alabama
10
50
26
7
10
23
12
20
22
40
5
26
3
6
2G0
Arkansas
21
G
3
2
8
4
11
1
10
11
5
82
California
38
78
121
38
47
84
67
63
38
65
20
92
24
30
805
Colorado
1
1
3
1
2
1
1
2
2
2
6
1
1
24
Connecticut
179
1G5
314
96
67
148
131
108
71
122
42
103
19
83
1648
Delaware
4
11
10
4
3
3
2
7
2
11
3
60
Dist. of Col’bia
7
23
50
7
9
21
9
19
13
17
4
8
4
12
203
Florida
1
10
10
4
6
3
3
2
7
2
48
Georgia
8
45
20
6
i2
20
7
16
14
25
2
14
1
7
197
Idaho
1
1
4
6
Illinois
151
270
398
106
92
233
105
235
134
261
46
206
25
76
2338
Indiana
G3
135
181
37
24
92
38
126
68
98
11
83
12
28
996
Iowa
33
G5
97
28
19
73
36
46
38
62
11
39
9
22
578
Kansas
11
21
19
7
1
14
3
13
5
13
4
22
4
4
141
Kentucky
19
73
93
24
33
49
26
53
33
49
8
57
19
21
557
Louisiana
9
112
54
12
18
44
13
15
14
50
10
31
2
14
398
Maine
74
114
158
47
18
79
34
48
50
64
12
73
12
33
816
i Maryland
54
104
176
57
59
113
66
82
57
91
22
49
6
43
979
Massachusetts
297
392
857
229
182
379
244
245
207
275
136
359
51
255
4108
Michigan
G9
105
162
41
40
77
51
91
55
95
13
88
21
37
945
Minnesota
43
20
70
9
21
29
20
30
25
30
9
40
2
8
356
Mississippi
7
54
26
9
13
14
9
20
10
24
2
19
1
12
220
Missouri
75
277
178
49
66
122
49
101
71
120
16
99
17
62
1302
Mon.Neb.Nev.
9
13
11
5
2
14
2
10
6
14
1
16
1
4
108
New Hamps’re
58
GO
110
22
33
65
23
40
40
56
13
41
5
34
GOO
New Jersey
54
97
249
44
49
97
74
73
46
95
32
52
11
56
1029
New Mexico
1
3
1
1
6
New York
449
643
1431
395
424
768
474
572
473
661
290
422
95
346
7443
N. Carolina
7
39
25
7
6
14
8
17
11
29
3
9
6
181
Ohio
140
232
408
91
111
203
91
166
160
233
38
185
33
95
2186
Oregon
2
3
8
3
3
1
1
3
3
3
5
1
36
Pennsylvania
199
2G4
575
167
135
291
195
174
157
326
85
174
35
170
2947
Rhode Island
15
38
80
24
26
42
29
30
11
35
10
16
3
18
377
S. Carolina
3
13
14
6
10
10
4
9
7
17
3
2
3
101
Tennessee
8
160
63
8
24
22
16
27
16
46
4
27
6
14
441
Texas
7
51
20
10
9
18
6
20
13
34
1
21
5
10
225
Utah
1
3
1
1
2
1
9
Vermont
25
33
58
8
13
24
i7
23
15
' 29
1
21
2
12
281
Virginia
15
33
34
5
13
17
17
14
9
17
3
20
1
7
205
Washington
1
1
1
1
2
6
West Virginia
5
6
13
1
10
10
2
7
5
15
5
1
4
84
Wisconsin
121
142
227
71
46
106
51
114
86
145
17
139
33
64
1362
Unknown
1
5
2
1
4
1
1
1
4
1
24
45
Brit’h America
29
37
73
9
23
45
24
35
45
39
10
46
5
19
439
Other Foreign
i
6
33
41
11
25
21
17
16
17
22
2
33
3
17
264
Total
2307
4049
6472
1705
1703
3403
1987
2711
2056
3345
887
2678474
1665
35442 TABLE XII.
191
(B) FEMALE LIFE. DISEASES AND NUMBER OF DEATHS BY STATES AND TERRITORIES.
Typhoid and
Typhus.
Ollier Zymotic.
Consumption.
Other Constitu-
tional.
Apoplexy.
Other Nervous.
Diseases of Heart.
Pneumonia.
Other Respiratory.
Digestive System.
Diseases of Kidneys.
Childbirth and
Puerperal Diseases.
Diseases of Breast
and Uterus.
I All other Diseases.
All Causes.
Alabama
2
3
3
3
2
1
3
3
1
21
Arkansas
1
1
1
3
California
2
4
2
2
2
1
1
5
6
2
6
33
Colorado
Connecticut
14
16
36
10
6
19
7
7
10
20
2
9
5
13
174
Delaware
1
2
1
1
2
1
• .
8
Dist. of Col’bia
2
3
1
2
2
1
1
12
Florida
1
1
Georgia
7
2
1
1
3
4
• •
2
20
Idaho
Illinois
11
15
35
7
2
7
9
7
10
23
19
8
16
169
Indiana
11
12
9
2
4
3
4
20
5
13
10
5
5
103
Iowa
5
5
11
2
1
3
4
3
4
8
1
5
3
55
Kansas
1
3
1
2
2
1
1
2
13
Kentucky
6
3
1
4
3
3
1
4
3
28
Louisiana
2
1
1
1
1
1
2
2
11
Maine
1
5
1
1
2
1
2
13
Maryland
2
4
16
5
4
7
4
4
2
11
4
3
3
3
72
Massachusetts
7
15
52
17
4
12
12
14
8
23
3
29
8
16
220
Michigan
4
8
19
11
3
4
4
7
5
8
1
6
7
6
93
Minnesota
1
1
1
3
3
6
15
Mississippi
3
3
1
1
1
1
2
3
1
1
1
18
Missouri
4
6
6
4
2
2
1
7
3
10
8
2
3
58
Mon.Neb.Nev.
1
1
1
1
4
New Iiamps’re
5
15
5
1
3
2
5
3
9
1
6
1
2
58
New Jersey
5
14
3
5
3
4
4
5
5
2
5
2
5
62
New Mexico
..
New York
17
33
68
31
7
28
24
33
27
45
8
21
24
19
385
N. Carolina
5
2
3
2
1
1
3
1
1
19
Oh io
6
14
31
10
7
11
5
21
12
16
1
18
9
10
171
Oregon
1
1
Pennsylvania
8
7
30
9
4
11
10
8
4
23
9
9
8
140
Rhode Island
1
6
2
3
1
2
4
1
1
21
S. Carolina
1
1
1
1
4
2
1
11
Tennessee
1
3
3
1
2
1
3
4
1
19
Texas
1
1
1
1
1
1
3
1
10
Utah
Vermont
1
1
9
2
1
1
2
1
3
3
1
2
27
Virginia
2
1
i
1
1
6
Washington
West Virginia
i
2
1
1
5
Wisconsin
7
3
13
4
1
4
1
5
1
8
2
9
5
3
66
Unknown
1
1
1
3
Brit’h America
2
6
1
1
3
1
4
3
3
2
26
Other Foreign
i
3
i
2
1
8
Total
118
184
409
138
63
132
105
176
116
273
27
198
110
133
2182 TABLE XI IT.
192
MALE LIFE (A). RATIO OF DEATHS BY EACH DISEASE TO 10,000 LIVING EXPOSED TO
RISK, AN EQUAL NUMBER 10,000 IN EACH STATED PERIOD OF INSURANCE.
Duration or Years of Insurance.
DISEASES.
ill
9i-19J
19f29£
Total.
Act’ID'th
All Causes.
68.6
80.4
93.1
102.5
114.5
140.9
213.5
102.0
35,442
Ph
<
s
S
p
XJl
Zymotic
Constitutional
Nervous
Circulatory
Respiratory
Digestive
Miscellaneous
23.7
6.3
8.3
2.1
8.9
6.0
13.3
19.0
15.3
9.6
3.0
11.9
7.6
14.0
18.8
22.5
11.8
4.0
12.6
9.2
14.2
17.1
27.5
14.1
5.4
13.6
9.3
15.5
16.3
30.0
17.2
7.3
15.0
10.7
18.0
17.5
32.3
25.6
11.3
19.1
13.8
21.3
21.1
32.4
43.8
25.0
29.4
18.7
43.1
18.3
23.5
14.7
5.7
13.7
9.6
. 16.5
6,356
8,175
5,107
1,986
4,770
3,344
5,704
Zymotic.
Typhoid,Typhus
Malarial Fever
Erysipelas
dysentery
Diarrhoea
Cholera
Alcoholism
Other Zymotic
8.9
3.0
.6
2.6
.8
2.8
.2
4.8
7.4
2.5
.6
1.6
.8
1.5
.4
4.2
7.2
2.4
1.1
1.5
1.0
1.3
.3
4.0
6.2
2.3
1.0
1.6
.9
1.0
.4
3.7
5.7
2.1
1.3
1.4
.8
.8
.3
3.9
5.6
2.1
1.8
1.8
1.6
.7
.6
3.3
7-2
2.3
2.1
2.7
2.1
.4
4.3
6.6
2.4
1.1
1.7
1.0
1.2
.3
4.0
2,306
831
374
587
328
431
117
1,382
Dropsy
.7
.4
1.1
.9
1.3
1.0
1.9
2.0
2.0
2.5
3.3
3.3
4.1
7.8
1.8
1.8
611
632
Consumption
4.3
12.4
19.0
22.3
24.1
23.7
15.9
18.6
6,474
o
Other Const’l
.9
.9
1.2
1.3
1.4
2.0
4.6
1.3
458
Apoplexy
3.2
3.3
4.1
4.7
5.3
8.6
17.0
4.9
1,705
CongestionBrain
1.8
1.6
1.7
1.9
2.0
2.2
2.7
1.9
655
o
Paralysis, Soft-}
2.8
3.9
4.7
5.9
8.2
12.2
19.9
6.4
2,236
ening Brain )
Epilepsy and \
1
2
.4
.5
.4
.6
.5
.4
130
Convulsions \
Other Nervous
.4
.6
.9
1.1
1.3
2.0
3.7
1.1
381
►»
Diseasesof Heart
1.8
2.7
3.8
5.1
6.8
10.7
23.4
5.3
1,855
3
Other Circulat’y
.3
.3
.2
.3
.5
.6
1.6
.4
131
Pneumonia
5.4
7.0
7.2
7.6
8.6
10.7
15.9
7.8
2,711
J-t
Congest’n Lungs
1.2
1.4
1.6
1.4
1.6
2.4
4.4
1.6
568
-M
c$
Bronchitis and )
6.
1.4
1.4
2.0
1.9
2,6
5.6
1.7
610
'p<
0)
Pleurisy j
Abscess, Hem-)
1.3
1.6
1.8
2.1
2.3
2.6
2.1
2.0
683
Ph
or’age Lungs \
Other Respirat’y
.4
.5
.6
.5
.6
.8
1.4
.6
198
Dis. of Stomach
1.0
1.3
1.8
1.8
1.8
3.3
4.1
1.8
628
Dis. of Bowels
1.7
1.8
2.0
2.1
2.2
2.9
3.2
2.1
739
l/l
Peritonitis
.5
.8
.7
.6
.7
.6
1.4
.7
246
bfi
Dis. of Liver
1.2
2.1
3.0
3.0
3.7
4.2
5.2
3.0
1,038
Other Digestive
1.6
1.6
1.7
1.8
2.3
2.8
4.8
2.0
693
Diabetes
.1
.3
.2
.4
.7
.7
2.5
.5
158
Dis. of Kidneys
.9
1.3
2.0
2.3
3.4
4.8
9.0
2.5
886
Other Urinary
.2
.4
.5
.7
1.0
1.9
5.0
.8
282
CO
P
o
CD
P
r2
'S
Childbirth, Pu- j
erperal Dis. j
Dis. Breast }
and Uterus j
Abscess, Hem- \
.3
.7
.6
.8
.9
1.7
11.8
1.0
339
CO
s
or’age,01dAge J
Debility, Ex- \
.2
.6
.7
1.0
1.4
24,
6.8
1.1
378
haustion, etc. \
Accid’ts, Injuries
9.4
8.3
7.7
7.6
7.4
ei i
5.1
7.7
2,678
Suicides
1.0
1.3
1.4
1.3
1.6
1.3
2.0
1.4
475
UnknownCauses
1.2
1.1
1.1
1.4
1.6
2.0
.9
1.5
508
Actual Deaths
2,445
5,161
4,651
7,803
10,062
4,089
1,197
35,442
35,442
Exposed to Risk
356,210 640,900
499,870
757,120
876,050
290,150
56,035
3,476,335 3,476,335 TABLE XIII.
193
(B.) FEMALE LIFE. RATIO OF DEATHS BY EACH DISEASE TO 10,000 LIVING EXPOSED
TO RISK, AN EQUAL NUMBER 10,000 IN EACH STATED PERIOD OF INSURANCE.
DISEASES.
Duration or Years of Insurance.
Hi
4f94
9J-19-1. 19!-29£
Total.
Actual
Deaths,
All Causes.
85.9
103.9
114.1
109.2
123.8
148.3
175.8
114.9
2,182
Zymotic
21.2
15.6
15.3
14.3
13.8
16.8
5.7
16.1
302
Constitutional
8.0
22.4
34.2
28.0
35.8
51.7
31.6
28.8
547
M
Nervous
7.3
7.5
10.5
10.1
8.4
18.3
28.8
10.2
195
S
Circulatory
4.0
2.9
4.1
5.4
7.7
7.2
25.9
5.5
105
s
Respiratory
9.3
16.2
15.2
15.4
16.0
15.1
31.8
15.3
292
p
CO
Digestive
12.3
13.8
12.4
13.3
16.2
15.2
26.0
14.3
273
Miscellaneous
23.8
25.5
22.4
22.7
25.9
24.0
26.0
24.7
468
Typhoid,Typhus
7.0
6.3
7.0
6.2
5.3
5.6
5.7~~
(43
118
Malarial Fever
3.5
3.1
3.7
1.6
1.5
4.0
2.6
49
o
Erysipelas
. .
.7
1.0
.5
10
+-»
o
Dysentery
3.1
, #
i.6
1.4
2.9
2.4
1.9
35
a
Diarrhoea
1.8
1.8
1.0
.9
.7
1.2
22
N
Cholera
1.8
.5
1.0
.9
.5
.8
15
Alcoholism
.1
1
Other Zymotic
4.0
3.9
i.o
2.6
1.9
4.8
2.7
52
Dropsy
.9
2.1
2.4
1.9
3.4
4.8
2.6
49
Cancer
.9
1.0
3.0
4.3
1.9
8.7
8.6
2.7
51
PJ
o
Consumption
G.2
17.7
26.4
20.1
28.8
32.6
23.0
21.5
409
Other Const’1
1.6
2.4
1.7
1.7
5.6
2.0
38
Apoplexy
3.1
2.6
2.7
2.8
3.1
4.0
5.8
3.3
63
in
CongestionBrai n
.9
.5
1.3
.5
.7
1.6
.9
16
2
O
>
u
Paralysis, Soft-)
ening Brain j
3.3
3.4
5.1
4.5
3.4
8.7
23.0
4.5
86
Epilepsy and [
Convulsions j
.9
.7
1.6
.4
9
Other Nervous
1.0
1.4
1.4
.5
2.4
1.1
21
►»
Diseasesof Heart
3.1
2.9
3.4
5.4
7.7
7.2
25.9
5.2
100
O
Other Circulat’y
.9
.7
.3
5
Pneumonia
8.4
9.7
8.1
10.2
9.0
9.5
14.4
9.3
176
>>
Congest’n Lungs
. ,
1.3
1.7
.9
2.2
2.4
5.8
1.5
30
•M
o3
Jh
P4
Bronchitis and)
Pleurisy f
1.3
1.7
2.4
.7
1.6
5.8
1.6
31
in
CD
tf
Abscess, Hem-)
or’age Lungs)
.9
3.9
2.7
1.4
3.1
2.2
42
Other Respirat’y
1.0
.5
1.0
1.6
5.8
.7
13
.
Dis. of Stomach
2.6
3.1
2.0
2.6
3.4
2.4
5.8
2.7
52
j*
Dis. of Bowels
3.1
5.0
3.4
2.1
4.1
4.8
5.8
3.6
69
>
Diseasesof Heart
1.7
1.3
2.5
4.4
10.7
29.2
64.9
75.5
5.3
s
Other Circulat’y
.2
.4
.6
2.2
2.5
.4
>>
f-i
o
Pneumonia
3.4
3.9
4.8
7.9
13.3
24.6
40.8
45.3
7.8
Congest’n Lungs
1.1
.9
1.1
1.7
2,5
4.6
13.3
15.1
1.6
os
Bronchitis and )
1.1
.9
1.2
1.8
2.6
5.0
12.5
60.4
1.7
• i-4
Ph
in
CD
«
Pleurisy j
Abscess, Hem-)
or’age Lungs f
.6
1.9
2.0
.4
2.2
3.4
1.5
5.0
2.0
.6
1.7
Other Respirat’y
.3
.5
1.0
1.7
Dis. of Stomach
1.7
.5
1.0
2.0
3.1
6.0
13.3
15.1
1.8
£
Dis. of Bowels
1.1
1.5
1.8
1.9
3.1
5.9
4.2
15.3
2.1
4->
CO
u
ening Brain f
Epilepsy and (
3
.2
.6
.9
1.6
.4
9
Convulsions j
21
Other Nervous
1.7
.7
.6
.9
8.0
1.1
Diseasesof Heart
1.4
3.1
5.9
9.4
27.2
61.8
5.2
100
S
Other Circulat’y
.2
.6
1.6
.3
5
Pneumonia
5.2
6.5
10.0
11.0
43.2
86.6
9.3
176
u
o
Congest’n Lungs
1.2
1.2
1.1
1.7
1.6
37.1
1.5
30
g3
Bronchitis and {
1.2
1.3
1.1
2.1
6.4
12.3
1.6
31
'&
Pleurisy j
tu
Ph
Abscess, Hem-)
2.3
.6
1.9
2.1
3.4
3.2
. ,
2.2
42
or’age Lungs f
Other Respirat’y
.4
.4
2.1
12.3
.7
13
Dis. of Stomach
1.7
2.7
2.1
2.1
11.2
61.8
2.7
52
►
Dis. of Bowels
2.6
3.9
2.8
5.9
8.0
3.6
69
03
Peritonitis
2.6
2.5
1.4
2.1
3.3
2.1
41
bp
Dis. of Liver
.9
2.3
3.8
3.9
4.8
..
2.8
53
p
Other Digestive
1.7
1.5
3.7
5.9
11.2
3.1
58
Diabetes
.4
.4
12.3
.2
3
Dis. of Kidneys
Other Urinary
i.4
.3
1.3
.4
1.6
1.7
1.7
1.6
1.5
.5
27
9
Childbirth,Pu- {
19.3
13.6
6.3
, ,
10.4
198
erperal Dis. j
o
f-i
Diseases of Bowels
2.5
2.3
2.1
2.1
1.9
2.1
1.5
2.1
CO
Jh
Diseases of Bowels
2.1
2.4
1.9
2.1
2.2
.8
2.1
CO
o
CO
Abscess, Hemorrhage, )
Old Age y
.5
.8
.7
.7
1.1
8.0
1.0
a
Debility, Exhaustion, 1
.6
.7
.8
1.3
2.4
3.9
Prostration, etc. \
1.1
Accidents and Injuries
10.5
9.6
7.6
5.9
3.5
2.3
7.5
Suicides
1.4
1.4
1.6
1.4
.5
.3
1.3
Unknown Causes
1.3
1.4
1.2
• 1.6
1.2
2.5
1.4
Actual Deaths
3,476
9,321
' 10,840
7,576
3,357
647
35,442 TABLE XYII
199
(A). CLIMATIC DISTRIBUTION OF DEATHS AND DISEASES AMONG 100 DEATHS OF
INSURED MALES AND FEMALES.
Ages, 20—75.
Residence at Death.
North of 36°30'
1 South 36° 30'.
Foreign
United States.
U. S. Census 1870.
SEX.
Male.
Fern.
Male.
Fem.
Male,
Male.
Fem.
Male.
Fem,
Deaths from all Causes.
100.0 jlOO.O
100.0
100.0
100.0
100.0
100.0
100.0
100.0
Zymotic Diseases
17.3
13.3
28.4
22.5
15.0
17.8
14.0
15.9
14.5
Constitutional Diseases
23.6
25.6
15.4
16.5
19.1
23.1
24.9
32.1
37.4
Ph
<
Nervous Diseases
14.5
8.9
13.1
11.3
16.1
14.5
8.9
9.5
8.1
S
Circulatory Diseases
5.8
5.0
4.0
3.8
5.8
5.6
4.8
5.9
5.9
s
Respiratory Diseases
13.4
13.4
12.6
12.8
16.1
13.4
13.4
13.3
11.4
CO
Digestive Diseases
9.2
12.2
12.9
14.3
8.7
9.5
12.6
6.5
6.6
Miscellaneous Diseases
16.2
21.6
13.6
18.8
19.2
16.1
21.4
16.8
16.1
Typhoid, Typhus Fever
6.8
5.5
2.8
4.5
5.0
6.5
5.4
6.8
6.6
Malarial Fever
2.1
1.8
6.9
9.0
1.1
2.3
2.3
2.5
2.4
d
Erysipelas
1.1
.5
.9
.9
1.1
.5
.6
.6
O
Dysentery
1.5
1.5
3.4
2.2
1.7
1.6
1.6
.9
.9
Diarrhoea
.9
1.0
2.1
1.5
1.0
.9
1.0
1.7
1.4
Cholera
1.2
.7
1.4
.6
1.2
.7
.1
.1
Alcoholism
.3
.1
.2
.6
.3
.1
1.1
.3
Other Zymotic Diseases
3.4
2.2
10.7
5.3
4.1
3.9
2.4
2.2
2.2
.
Dropsy
1.8
2.2
1.3
3.0
1.3
1.8
2.2
2.1
3.0
m
Cancer
1.8
2.4
.7
2.3
.7
1.7
2.3
1.7
2.5
a
g
Consumption
18.7
19.3
12.3
9.0
16.2
18.3
18.7
26.2
29.7
Other Constitutional
1.3
1.7
1.1
2,2
.9
1.3
1.7
2.1
2.2
Apoplexy
4.8
2.8
4.8
4.5
6.8
4.8
2.9
2.2
1.8
03
Congestion of Brain
1.8
.6
2,9
3.0
1.7
1.9
.7
.3
.3
O
►
M
O
Paralysis, Softening)
etc., of Brain )
6.4
4.0
4.4
3.0
5.8
6.3
3.9
4.0
3.4
Sz;
Epilepsy, Convulsions
.4
.5
.4
.7
.4
.4
.7
.8
Other Nervous
1.1
1.0
.6
.8
1.1
1.1
1.0
2.3
1.8
Diseases of Heart
Other Circulatory
5.4
.4
03 -31
3.9
.1
3.8
4.5
1.3
5.2
.4
4.6
.2
J- 5.9
5.9
>>
o
Pneumonia
Congestion of Lungs
7.7
1.6
7.9
1.5
7.4
1.3
9.8
7.3
1.6
7.6
1.6
8.1
1.4
| 10.2
8.3
cti
u
Bronchitis and Pleurisy
1.7
1.4
2.0
1.5
2.1
1.7
1.4
1.1
1.0
‘p<
Abscess, Hemorrhage)
1.9
2.0
1.1
.8
4.3
1.9
1.9
} 2.0
2.1
03
etc., of Lungs f
Other Respiratory
.5
.6
.8
.7
.8
.6
.6
)
Diseases of Stomach
1.7
2.4
3.0
1.5
1.6
1.8
2.4
1.0
1.1
Diseases of Bowels
2.0
3.0
3.4
5.3
1.7
2.1
3.2
2.6
2.4
in
03
Peritonitis
.7
1.9
.3
.8
1.0
.7
1.9
.2
.5
bf)
Diseases of Liver
2.9
2.2
2.8
4.5
2.8
2.9
2.4
2.0
1.7
Other Digestive
1.9
2.7
3.4
2.2
1.6
2.0
2.7
.7
.9
Diabetes
.5
.2
.1
.3
.5
.1
.4
.1
Diseases of Kidneys
2.6
1.3
1.4
1.7
2.5
1.2
1.9
.8
Other Urinary
.8
.4
.7
.3
.8
.4
.4
.1
03
Childbirth and Puer- j
peral Diseases [
9.0
9.0
9.1
6.5
o
03
0
ce
Diseases of Breast and )
Uterus j
5.0
5.3
5.0
2.4
03
o
03
Abscess, Hemorrhage, |
Old Age )
1.0
1.4
.6
1.0
1.0
1.3
| 2.7
2.7
£
Debility, Exhaustion, )
Prostration, etc. )
1.1
1.4
.7
1.5
.7
1.1
1.4
Accidents and Injuries
7.5
1.5
7.8
1.5
11.2
7.5
1.5
9.1
1.4
Suicides
1.4
.4
.8
1.1
1.3
.3
.9
.3
Unknown Causes
1.3
1.0
1.5
1.5
2.9
1.4
1.1
1.4
1.8
Actual Deaths
32,544
2,012
2,153
133
703
35,442
2,182
104,531
91,898 200
TABLE XVII.
(B). PROPORTIONAL DEATHS AND DISEASES TO 100 TOTAL DEATHS OF
INSURED MALES IN EACH GROUP OF STATES.
Group of States.
L*
11.
III.
IV.
V.
VI.
VII.
Mean
Groups.
All Causes.
100.0
100.0
100.0
100.0
100.0 100.0
100.0
Group.
Zymotic
16.7
19.0
15.5
17.9
21.1
28.4
14.2
19.0
I.*
Constitutional
25.0
21.9
26.0
22.0
20.9
15.4
19.8
21.6
Nervous
14.7
11.8
14.4
13.8
15.4
13.1
16.4
14.2
New England.
S
Circulatory
6.3
4.5
6.8
4.4
5.1
4.0
8.2
5.6
New York.
S
Respiratory
12.6
14.9
11.3
15.9
13.9
12.6
13.0
13.5
II.
GQ
Digestive
8.2
10.2
10.5
10.7
9.1
12.9
8.1
10.0
Miscellaneous
16.5
17.7
15.5
15.3
14.5
13.6
20.3
16.2
Northwest.
Typhoid,Typhus
7.2
8.8
6.4
6.4
5.3
2.8
4.6
5.9
Michigan.
Malarial Fever
1.5
2.4
1.7
3.2
2.9
6.9
1.7
2.9
Wisconsin.
o
Erysipelas
1.0
1.3
.9
1.2
1.0
.9
2.0
1.2
Minnesota.
o
Dysentery
1.6
1.4
.8
1.4
2.8
3.4
1.3
1.8
Nebraska.
H
Diarrhoea
.9
.7
.9
.7
1.2
2.1
.2
1.0
III.
N
Cholera
.8
.8
1.0
1.9
2.7
1.4
.8
1.3
Alcoholism
.3
.3
.3
.4
.4
.2
.5
.3
New Jersey.
Other Zymotic
3.4
3.3
3.5
2.7
4.8
10.7
3.1
4.5
Pennsylvania.
Dropsy
1.7
1.7
2.4
1.8
1.7
1.3
1.7
1.8
IV.
Cancer
2.0
1.6
2.0
1.5
1.7
.7
1.7
1.6
a
o
Consumption
19.7
17.3
20.7
17.7
16.4
12.3
15.1
17.0
Ohio.
O
Other Const’l
1.6
1.3
.9
1.0
1.1
1.1
1.3
1.2
Indiana.
Apoplexy
5.0
4.0
4.6
4.0
5.5
4.8
6.1
4.9
Illinois.
CongestionBrain
1.4
1.5
2.0
2,4
2.2
2.9
1.5
2.0
Iowa.
2
o
>
Jh
Paralysis, Soft-)
ening Brain j
6.9
5.3
6.1
5.9
6.0
4.4
6.5
5.9
Kansas.
Epilepsy and [
Convulsions }
.3
.3
.6
.4
.4
.4
1.3
.5
V.
Other Nervous
1.1
.7
1.1
1.1
1.3
.6
1.0
1.0
Delaware.
Maryland.
5.8
4.2
>>
Diseasesof Heart
6.6
4.2
4.8
3.9
6.6
5.2
Dist. Columbia.
o
Other Circulat’y
.5
.3
.2
.2
.3
.1
1.6
.5
Virginia.
Pneumonia
7.0
8.7
6.2
9.4
8.3
7.4
8.0
7.9
Kentucky.
>»
Congest’n Lungs
1.7
1.7
1.1
1.6
1.7
1.3
.8
1.4
Missouri.
o
ce
Bronchitis and /
Pleurisy f
1.7
2.0
1.4
1.9
1.4
2.0
2.0
1.8
VI.
a>
Dis. of Bowels
1.8
2.2
2.3
2.4
1.9
3.4
1.4
2.2
Georgia.
m
o
CO
Abscess, Hem-}
or’age.OldAge j
1.1
.8
1.0
1.0
1.0
.6
.3
.8
Washington.
Oregon.
3
Debility, Ex- |
haustion, etc. j
1.0
1.0
2.2
.7
1.0
.7
1.1
1.1
California.
Utah.
Accid’ts, Injuries
6.8
10.1
5.7
8.6
7.1
7.8
11.4
8.2
Dakota.
Suicides
1.2
2.1
1.2
1.3
1 1.4
.8
2.9
1.6
New Mexico.
U nknownCauses
1.7
1.0
1.0
1.0
1.1
1.5
1.2
1.2
Actual Deaths
15,273
2,716
3,976 6,239
|3,306
2,153
863 TABLE XVII.
201
(C). PROPORTIONAL DEATHS AND DISEASES TO 10,000 MALES LIVING IN EACH
GROUP OF STATES.
Group of States.
I*
11.
III.
IV.
V.
VI.
VII.
Mean
Groups.
All Causes.
105.3
97.7
107.1
104.5
130.5
170.5
112.2
Group.
Zymotic
17.6
18.5
16.6
18.7
27.5
48.4
15.9
23.3
I.*
Constitutional
26.4
21.4
27.9
23.0
27.3
26.3
22.1
24.9
Nervous
15.4
11.5
15.4
14.4
20.1
22.3
18.4
16.8
New England.
Circulatory
6.6
4.4
7.3
4.6
6.7
6.8
9.2
6.5
i New York.
Respiratory
13.3
14.6
12.1
16.6
18.1
21.5
14.6
15.8
II.
|
CO
Digestive
8.6
10.0
11.2
11.2
11.9
22.0
9.1
12.0
Miscellaneous
17.4
17.3
16.6
16.0
18.9
23.2
22.9
18.9
Northwest.
Typhoid,Typhus
7.6
8.5
6.8
6.7
6.9
4.8
5.2
6.6
Michigan.
Malarial Fever
1.7
2,3
1.8
3.3
3.8
11.8
1.9
3.8
Wisconsin.
o
Erysipelas
1.0
1.3
1.0
1.3
1.3
1.5
2.2
1.4
Minnesota.
o
Dysentery
1.7
1.4
.9
1.4
3.6
5.8
1.5
2.3
Nebraska.
a
Diarrhoea
.9
.7
1.0
.8
1.6
3.6
.2
1.3
III.
N
Cholera
.9
.8
1.1
2.0
3.5
2.4
.9
1.7
Alcoholism
.3
.3
.3
.4
.5
.3
.6
.4
New Jersey.
Other Zymotic
3.6
3.2
3.7
2.8
6.3
18.2
3.4
5.9
Pennsylvania.
Dropsy
1.8
1.6
2.6
1.9
2.2
2.2
1.9
2.0
IV.
Cancer
2.1
1.5
2.1
1.6
2.2
1.1
1.9
1.8
o
Consumption
20.8
16.9
22.2
18.5
21.5
21.0
16.9
19.7
Ohio.
o
Other Const’l
1.7
1.4
1.0
1.0
1.4
2.0
1.4
1.4
Indiana.
Apoplexy
5.2
3.8
5.0
4.2
7.2
8.2
6.8
5.8
Illinois.
C/3
2
O
>
Jh
Congesti onB rai n
Paralysis, Soft-1
ening Brain [
1.5
7.2
1.5
5.2
2.1
6.5
2,5
6.2
2.9
7.8
4.9
7.5
1.7
7.3
2.4
6.8
Iowa.
Kansas.
Epilepsy and )
3
3
.6
A
.5
.7
1.5
.6
V.
Convulsions J
Delaware.
Other Nervous
1.2
.7
1.2
1.1
1.7
1.0
1.1
1.1
i Maryland.
6.1
4.1
►>
Diseasesof Heart
7.1
4.4
6.3
6.6
7.4
6.0
Dist. Columbia.
s
Other Circulat’y
.5
.3
.2
.2
.4
.2
1.8
.5
Virginia.
Pneumonia
7.3
8.5
6.6
9.8
10.8
12.6
9.0
9.2
Kentucky.
>>
Congest’n Lungs
1.8
1.7
1.2
1.7
2.2
2.2
.9
1.7
Missouri.
o
•M
cfl
Jh
Bronchitis and)
Pleurisy f
1.8
2.0
1.5
2.0
1.8
3.4
2.2
2.1
VI.
Ph
CO
CD
Ph
Abscess, Hem-)
or’age Lungs \
1.9
2.0
2.0
2.4
2.4
1.9
1.8
2.1
South of 36° 30'.
Other Respirat’y
.4
.4
.8
7
9
1.4
.7
.8
North Carolina.
South Carolina.
Tennessee.
Dis. of Stomach
1.6
2.3
2.0
1.9
2.1
5.1
1.3
2.3
_>
Dis. of Bowels
1.9
2.2
2.5
2.5
2.5
5.8
1.6
2.7
Georgia.
to
CD
Peritonitis
.7
.8
1.0
.8
.7
.5
.3
.7
Florida.
bo
Dis. of Liver
2.7
2.4
3.7
3.7
3.9
4.8
4.2
3.6
Alabama.
Other Digestive
1.7
2.3
2.0
2.3
2.7
5.8
1.7
2.6
Mississippi.
Diabetes
.5
.5
.5
.4
.8
.3
.6
• 5 |
Arkansas.
Dis. of Kidneys
3.5
1.4
3.1
1.9
2.1
2.4
2.6
2.4
Louisiana.
Other Urinary
Childbirth,Pu- /
.9
.8
1.1
.5
.9
1.2
.7
.9
Texas.
VII.
CO
erperal Dis. f
* *
* *
* *
o
CD
Dis. Breast }
Pacific, etc.
P
and Uterus ]
* *
' *
* *
* *
* *
CD
O
CO
Abscess, Hem-)
or’age,01dAge f
1.2
.8
1.1
1.1
1.3
1.0
.3
1.0
Washington.
Oregon.
s
Debility, Ex- 1
haustion, etc. f
1.0
1.0
2.4
.7
1.3
1.1
1.2
1.2
California.
Utah.
Accid’ts, Injuries
7.2
9.7
6.0
9.0
9.3
13.3
12.8
9.6
Dakota.
Suicides
1.3
2.1
1.3
1.3
1.8
1.4
3.3
1.8
New Mexico.
U nknownCauses
1.8
1.0
1.1
1.1
1.4
2.5
1.3
1.5
Actual Deaths
15,278
2,716
3,976
6,239
3,306
2,153
863 202
TABLE XVIII.
CALENDAR AND DECIMAL PARTS OF A YEAR.
Days,
Cal.
Decimal.
Days.
Cal.
Decimal.
Days.
1
Cal.
Decimal.
Days.
Cal.
Decimal.
0
Jan. 0
.00000000
50
Fb*19
.13698630
100
Apr.10
.27397260
150
May 30
.41095890
1
4=1
.00273973
51
20
.13972603
101
11.27671233
151
31
.41369863
2
2
.00547945
52
21
.14246575
102
12.27945205
152
June 1
.41643836
3
3
.00821918
53
22
.14520548
103
13
.28219178
153
2
.41917808
4
4
.01095890
54
23
.14794521
104
14
.28493151
154
3
.42191781
5
5
.01369863
55
24
.15068493
105
15
.28767123
155
*4
.42465753
6
6
.01643836
56
25
.15342466
106
*16
.29041096
1156
5
.42739726
7
7
.01917808
57
*26
.15616438
107
17
.29315068
157
6
.43013699
8
4=8
.02191781
58
27
.15890411
108
18
.29589041
1158
7
.43287671
9
9
.02465753
59
28
.16164384
109
19
.29863014
,159
8
.43561644
10
10
.02739726
60
Mar. 1
.16438356
110
20
.30136986
160
9
.43835616
11
11
.03013699
61
2
.16712329
111
21
.30410959
1161
10
.44109589
12
12
.03287671
62
3
.16986301
112
22
.30684932
1162
*11
.44383562
13
13
.03561644
63
4
.17260274
113
*23
.30958904
163
12
.44657534
14
14
.03835616
64
*5
.17534247
114
24
.31232877
164
13
.44931507
15
*15
.04109589
65
6
.17808219
115
25
.31506849
165
14
.45205479
16
16
.04383562
66
7
.18082192
116
26
.31780822
166
15
.45479452
17
17
.04657534
67
8
.18356164
117
27
.32054795
167
16
.45753425
18
18
.04931507
68
9
.18630137
118
28
.32328767
168
17
.46027397
19
19
.05205479
69
10
.18904110
119
29
.32602740
169
*18
.46301370
20
20
.05479452
70
11
.19178082
120
*30
.32876712
170
19
.46575342
21
21
.05753425
71
*12
.19452055
121
May 1
.33150685
171
20
.46849315
22
*22
.06027397
72
13
.19726027
122
2
.33424658
172
21
.47123288
23
23
.06301370
73
14
.20000000
123
3
.33698630
173
22
.47397260
24
24
.06575342
74
15
.20273973
124
4
.33972603
174
23
.47671233
25
25
.06849315
75
16
.20547945
125
5
.34246575
175
24
.47945205
26
26
.07123288
76
17
.20821918
126
6
.34520548
176
*25
.48219178
27
27
.07397260
77
18
.21095890
127
*7
.34794521
177
26
.48493151
28
28
.07671233
78
*19
.21369863
128
8
.35068493
178
27
.48767123
29
*29
.07945205
79
20
.21643836
129
9
.35342466
179
28
.49041096
30
30
.08219178
80
21
.21917808
130
10
.35616438
180
29
.49315068
31
31
.08493151
81
22
.22191781
131
11
.35890411
181
30
.49589041
32
Feb. 1
.08767123
82
23
.22465753
132
12
.36164384
182
July l
.49863014
33
2
.09041096
83
24
.22739726
133
13
.36438356
183
*2
.50136986
34
3
.09315068
84
25
.23013699
134
*14
.36712329
184
3
.50410959
35
4
.09589041
85
*26
.23287671
135
15
.36986301
185
4
.50684932
36
*5
.09863014
86
27
.23561644
136
16
.37260274
186
5
.50958904
37
6
.10136986
87
28
.23835616
137
17
.37534247
187
6
.51232877
38
7
.10410959
88
29
.24109589
138
18
.37808219
188
7
.51506849
39
8
.10684932
89
30
.24383562
139
19
.38082192
189
8
.51780822
40
9
.10958904
90
31
.24657534
140
20
.38356164
190
*9
.52054795
41
10
.11232877
91
April 1
.24931507
141
*21
.38630137
191
10
.52328767
42
11
.11506849
92
*2
.25205479
142
22
.38904110
192
11
.52602740
43
*12
.11780822
93
3
.25479452]
143
23
.39178082
193
12
.52876712
44
13
.12054795
94
4
.25753425
144
24
.39452055
194
13
.53150685
45
14
.12328767
95
5
.26027397
145
25
.39726027
195
14
.53424658
46
15
.12602740
96
6
.26301370
146
26
.40000000
196
15
.53698630
47
16
.12876712
97
7
.26575342
147
27
.40273973
197
*16
.53972603
48
17
.13150685
98
8
.26849315
148
*28
.40547945
198
17
.54246575
49
18
.13424658
99
*9
.27123288
149
|
29
.40821918
199
18
.54520548 TABLE X Y ill.
203
CALENDAR AND DECIMAL PARTS OF A TEAR.
Days
. Cal.
Decimal.
Days
Cal.
Decimal.
Days
Cal.
Decimal.
Days,
Cal.
Decimal.
200
Julyr.
1.54794521
250
Sept. 'J
68493151
300
Oct. 27
.82191781
350
Dec. 16
.95890411
201
2(
1.55008493
251
£
.68707123
301
28
.82465753
351
*17
.96164384
202
21
..55342400
252
c
.09041096
302
*29
.82739726
352
18
.96438356
203
2k
.55610438
253
*1C
.69315068
303
30
.83013699
353
19
.96712329
204
*23
.55890411
254
11
.69589041
304
31
.83287671
354
20
.96986301
205
24
-.56104384
255
12
.69863014
305
Nov. 1
.83561644
355
21
.97260274
206
25
,50438350
256
13
.70136980
306
2
.83835616
356
22
.97534247
207
26
.50712329
257
14
.70410959
307
3
.84109589
357
23
.97808219
208
27
.50986301
258
15
.70684932
308
4
.84383562
358
*24
.98082192
209
28
.57260274
259
16
.70958904
309
*5
.84657534
359
25
.98356164
210
29
.57534247
260
*17
.71232877
310
6
.84931507
360
26
.98630137
211
*30
.57808219
261
18
.71506849
311
7
.85205479
361
27
.98904110
212
31
.58082192
262
19
.71780822
312
8
.85479452
362
28
.99178082
213
Aug. 1
.58356104
263
20
.72054795
313
9
.85753425
363
29
.99452055
214
2
.58630137
264
21
.72328767
314
10
.86027397
364
.30
.99720027
215
3
.58904110
265
22
.72602740
315
11
.86301370
365
*31
1.00000000
216
4
.59178082
266
23
.72870712
316
*12
.86575342
—
217
5
.59452055
267
*24
.73150685
317
13
.86849315
YEAR AND DAY OF THE
218
*6
.59720027
268
25
.73424658
318
14
.87123288
WEEK DENOTED BY *.
219
7
.60000000
269
20
.73698630
319
15
.87397260
—
220
8
.00273973
270
27
.73972603
320
16
.87671233
1881
Saturday,
221
9
.00547945
271
28
.74240575
321
17
.87945205
1882
Sunday.
222
10
.60821918
272
29
.74520548
322
18
.88219178
1883
Monday.
223
11
.01095890
273
30
.74794521
323
*19
.88493151
1884
Wedu.
224
12
.01369863
274
Oct. *1
.75008493
324
20
.88767123
1885
Thursday.
225
*13
.01643836
275
2
.75342460
1325
21
.89041096
1886
Friday.
226
14
.61917808
276
3
.75616438
326
22
.89315068
1887
Saturday.
227
15
.62191781
277
4
.75890411
327
23
.89589041
1888
*Sunday. Mond.
228
10
.02405753
278
5
.76164384
328
24
.89863014
1889
Tuesday.
229
17
.02739720
279
0
.76438356
329
25
.90136986
1890
Wednesday.
230
18
.03013699
280
7
.76712329
330
*26
.90410959
1891
Thursday.
231
19
.63287071
281
*8
.76986301
331
27
.90684932
1892
*Friday. Satur.
232
*20
.63561044
282
9
.77260274
332
28
.90958904
1893
Sunday.
233
21
.03835616
283
10
.77534247
333
29
.91232877
1894
Monday.
234
22
.64109589
284
11
778082L9
334
30
.91506849
1895
Tuesday.
235
23
.04383502
285
12
.78082192
335
Dec. 1
.91780822
1896
*Wedn. Thurs.
236
241
.04057534
286
13
.78356164
336
2
.92054795
1897
Friday.
237
25
.64931507
287
14
.78630137
337
*3
.92328767
1898
Saturday.
238
26
.65205479
288
*15
.78904110
338
4
.92602740
1899
Sunday.
239
*27
.05479452
289
16
79178082
339
5
.928767121
1900
Monday.
240
28
05753425
290
17
79452055
340
6
.93150685
1901
Tuesday.
241
29
60027397
291
18
79726027
341
7
93424658
1902
Wednesday.
242
30
66301370
292
19
80000000
342
8
.93698630
1903
Thursday.
243
31
06575342
293
20
80273973
343
9
.93972603
1904
*Friday. Satur.
244
Sept. 1
06849315
294
21
80547945
344
*10
.94246575
1905
Sunday.
245
2
07123288
295
*22
80821918
345
11
94520548
1906
Monday.
246
*3
67397200
296
23
81095890
346
12
94794521
1907
Tuesday.
247
4
07671233
297
24
81369863
347
13
95068493
1908
*Wedn. Thurs.
248
5
07945205
298
25
81643836
348
14
95342466
1909
Friday.
249
6
68219178
299
20
81917808
349
15
95616438
1910
Saturday.
* In 1884 and other leap years, the first star day applies only in January and February, and the last
applies only in the remaining ten months. 204
TABLE XIX.
Days.
0
1
2
3
4
5
6
7
8
9
3ays.
0
1
2
3
4
5
6
7
8
9
0
1
9
17
25
33
41
20
19
49
5
34
4
56
14
11
69
54
47
28
48
1
2
39
57
46
44
32
43
50
65
15
6
42
61
35
6
53
23
24
45
37
29
fri
2
10
52
8
38
51
71
62
3
60
12
7
21
13
5
73
74
82
90
98
106
114
3
18
72
59
30
66
27
36
40
31
70
8
93
92
122
75
112
130
119
117
105
116
4
26
63
68
22
16
64
7
58
67
55
9
123
138
88
S3
125
81
111
124
144
135
Rep.
0
1
2
3
4
5
6
.7
8
9
Ref.
0
1
2
3
4
5
6
7
8
9
1
0
000
000
ooo'ooo
000
000
000
001
001
49
0
001
002
003
004
006
007
008
009
Oil
2
0
001
002
004
005
006
008
009
010
012
50
0
002
004
006
009
Oil
013
016
018
020
3
0
003
007
Oil
014
018
022
025
029
033
51
0
003
006
009
013
016
019
023
026
029
4
0
006
013
020
027
034
041
048
055
062
52
0
002
005
008
Oil
014
017
020
023
025
5
0
009
019
029
039
049
059
069
078
088
53
0
008
017
026
035
043
052
061
070
078
6
0
008
017
025
034
043
051
060
069
077
54
0
007
015
023
030
038
046
053
061
069
7
0
006
012
018
025
031
037
044
050
056
55
0
006
013
020
026
033
040
046
053
060
8
0
003
006
009
012
015
018
021
024
027
56
0
007
014
021
028
035
042
059
056
064
9
0
000
000
000
001
001
001
001
002
002
57
0
001
003
004
006
008
009
Oil
013
014
10
0
002
005
008
010
013
016
019
021
024
58
0
006
012
019
025
032
038
045
051
057
11
0
007
014
022
029
036
044
051
059
066
59
0
004
008
013
017
021
026
030
035
039
12
0
003
007
Oil
015
019
023
027
031
035
60
0
003
007
Oil
015
019
023
026
030
034
13
0
009
019
029
038
048
058
068
077
087
61
0
008
016
025
033
041
050
058
066
075
14
0
007
014
021
029
036
043
050
058
065
62
0
003
007
010
014
017
021
024
028
032
15
0
002
005
007
010
013
015
018
020
023
63
0
005
Oil
016
022
028
033
039
044
050
16
0
006
012
018
024
030
036
042
048
054
64
0
006
012
018
024
030
036
043
049
055
17
0
000
000
001
001
002
002
002
003
003
65
0
002
004
007
009
012
014
017
019
022
18
0
00-1
008
012
016
020
024
028
032
036
66
0
004
009
013
018
023
027
032
037
041
19
0
001
002
003
004
005
006
007
008
009
67
0
006
013
019
026
032
039
046
052
059
20
0
000
001
002
003
004
005
006
007
008
68
0
005
Oil
017
023
028
034
040
046
051
21
0
009
019
028
038
047
057
067
076
086
69
0
007
015
022
030
037
045
052
060
067
22
0
005
Oil
017
023
029
035
041
047
053
70
0
005
010
016
021
026
032
037
042
048
23
0
008
017
026
035
044
053
062
071
080
71
0
003
006
010
013
017
020
023
027
030
24
0
009
018
027
036
045
054
063
072
081
72
0
004
008
012
016
021
025
029
033
038
25
0
000
001
001
002
002
003
003
004
004
*73 '
0
01
02
03
04
05
06
07
08
09
26
0
005
010
016
021
027
032
038
043
049
74
0
010
020
030
040
050
060
0
70
081
091
27
0
004
009
014
019
023
028
033
038
043
75
0
Oil
022
034
045
056
068
079
090
102
28
0
007
015
023
031
039
047
055
063
071
76
0
013
027
041
054
068
082
095
109
123
29
0
009
018
028
037
047
056
066
075
085
77
0
016
033
050
067
084
101
118
135
152
30
0
004
009
013
018
022
027
031
036
040
78
0
019
039
059
079
099
119
139
158
178
31
32
0
0
005
002
010
004
015
006
020
008
026
010
031
012
036
014
041
016
046
018
79
80
0
0
018
016
037
032
055 074
048065
093
081
065
111
097
078
130
114
149
130
104
167
146
117
33
0
000
001
002
002
003
004
004
005
006
81
0
013
026
039052
091
34
35
36
0
0
0
006
008
004
013
016
009
020
025
014
027
033
019
034
042
024
041
050
029
047
059
034
054
067
039
061
076
044
82
83
84
0
0
0
010
012
017
020
025
034
030
038
052
041
050
069
051
063
086
061
076
104
071
089
121
082
101
139
092
114
156
37
0
009
018
027
037
046
055
065
074
083
85
0
013
027
041
055
069
083
097
111
125
38
0
003
006
009
012
015
018
022
025 028
86
0
019
039
059
078
098
118
138
157
177
39
0
001
003
004
006
007
009
010
012 013
87
0
017
034
051
069
086
103
120
138
155
40
0
005
010
015 020
025
030
035
040 045
88
0
012
025
037
050
063
0
75
088
100
113
144
41
0
000
001 002
003
004
004
005
006
007
89
0
016
032
048
064
080
096
112
128
42
0
008
016 024
032
041
049
057
065
073
90
0
010
020
031
041
052
062
072
083
093
43
0
002
004 006
008
010
013
015
017
019
1 91
0
014
028
042
056
070
084
098
112
126
44
0
001
003
005
007
009
Oil
013
015
017
92
0
Oil
022
033
044
055
066
077
088
099
45
0
009
018
027
036
045
055
064
073
082
93
0010
021
032
043
054
065
076
087
098
46
0
001003
005
007
008
010
012
014
016
94
0019
039
058
078
097
117
137
156
176
47
0
007 015
023
031
039
046
054
0lJ2
070
95
0 015
031
047
063
079
095
111
127
143
48
0
008 016
024
032
040
048
056
064
072
96
0018
037
056
075
094
113
132
151
170
0
1
1 2
3
4
5
6
7
8
9
97
0 019
038
057
076
095
114
133
152
171
SIMPLE INTEREST FOR DAYS AT 5 PER CENT. 365 DAYS TO THE YEAR.
* The interest is exact on the liues of 73, 146, 219, 292 and 365. TABLE XIX.
205
SIMPLE INTEREST FOR DAYS AT 5 PER CENT. 365 DAYS TO THE YEAR.
Days.
0
1
2
3
4
5
6
7
8
9
Days.
0
1
2
3
4
5
6
7
8
9
10
76
133
85
91
145
132
103
139
LOO
109
15
171
179
187
166
165
195
148
185
203
192
11
113
104
143
99
136
141
95
89
L37
80
16
190
178
189
196
211
161
156
198
154
184
12
131
140
128
107
77
129
87
84
L42
127
17
197
217
208
149
206
158
164
218
205
176
13
120
101
L21
L15
134
108
79
126
96
97
18
212
173
182
186
177
216
172
209
214
168
14
118
L10
102
94
86
78
146
147
L55
163
19
162
210
153
204
213
201
180
150
202
160
Ref.
0
1
2
3
4
5
6
7
8
9
Ref.
0
1
2
3
4
5
6
7
8
9
98
0
010
021
031
042
052
063
073
084
094
147
0
020
040
060
080
100
120
140
161
181
99
0
045
030
046
061
077
092
108
123
139
148
0
021
042
064
085
106
128
149
1
70
192
100
0
014
029
044
059
073
088
103
118
133
149
0
023
047
071
094
118
142
165
189
213
101
0
017
035
053 071
089
107
125
143
161
150
0
026
053
080
107
134
1
61
188
215
242
102
0
019
038
058077
097
116
136
155
175
151
0
029
059
089
119
149
179
209
238
268
103
0
014
029
043 058
072
087
101
116
130
152
0
028
057
085
114
143
171
200
229
257
104
0
015
030
045 060
076
091
106
121
136
153
0
026
052
078
105
131
157
184
210
236
105
0
012
024
03 61048
060
072
084
096
108
154
0
023
046
069
092
115
138
161
184
207
106
0
010
021
032
042
053
064
074
085
096
155
0
020
040
060
081
101
121
141
162
182
107
0
016
033
050
067
084
101
117
134
151
156
0
022
045
068
090
113
136
159
181
204
108
0
018
036
055
073
092
110
129
147
166
157
0
027
054
082
109
136
164
191
219
246
109
0
014
029
044
059
074
089
104
119
134
158
0
023
047071
095
119
143
167
191
215
110
0
019
038
057
077
096
115
135
154
173
159
0
029
059089
118
148
178
208
237
267
111
0
013
026
039
052
065
078
092
105
118
160
0
027
054
081
109
136
163
190
218
245
112
0
Oil
023
034
046
057
069
080
092
103
161
0
022
045
067
090
113
135
158
180
203
113
0
015
030
045
060
075
090
105
120
135
162
0
026
052
0
78
104
130
156
182
208
234
114
0
010
021
032
043
054
064
075
086
097
163
0
020
040
061
081
102
122
142
163
183
115
0
018
036
054
072
091
109
127
145
163
164
0
024
048
072
096
120
144
168
192
216
116
0
012
024
036
048
060
073
085
097
109
165
0
021
042
063,084
105
126
147
168
189
117
0
Oil
023
035
047
059
071
083
095
107
166
0
020
041
062
083
104
125
146
167
188
118
0019
038
057
076
095
115
134
153
172
167
0
029
059
088
118
147
177
207
236
266
119
0011
023
035
047
058
070
082
094
106
168
0
025
051
077
103
129
155
181
207
233
120
0
017
035
053
071
089
106
124
142
160
169
0
028
057
086
115
144
1
73
202
231
260
121
0
018
036
054
072
090
108
126
144
162
170
0
029
058
087
116
145
174
203
232
261
122
0
Oil
022
033
044
056
067
078
089
101
171
0
020
041
061
082
102
123
143
164
184
123
0
012
024
036
049
061
073
086
098
110
172
0
025
050
076
101
127
152
178
203
229
124
0
013
026
039
053
066
079
093
106
119
173
0
024
049
074
099
123
148
173
198
223
125
0
012
025
038
051
064
077
090
103
115
174
0
027
055
083
111
139
167
195
223
251
126
0
018
037
056
075
093
112
131
150
168
175
0
029
058
088
117
147
176
206
235
265
127
0
017
035
053
070
088
106
123
141
159
176
0
024
049
073
098
122
147
171
196
220
128
0
016
033
050
066
083
100
116
133
150
177
0
025
050
0
75
100
126
151
176
201
226
129
0
017
034
051
068
085
102
129
136
154
178
0
022
044
066
088
110
132
154
176
198
130
0
Oil
023
034
046
058
069
081
093
104
179
0
020
041
062
082
103
124
144
165
186
131
0
016
032
049
065
082
098
115
131
147
180
0
026
053
080
107
134
161
187
214
241
132
0
014
028
043
057
071
086
100
115
129
181
0
028
056
085
113
142
1
70
199
227
256
133
0
013
027
041
055
069
083
096
110
124
182
0
024
049
074
099
124
149
174
199
224
134
0
018
036
055
073
091
110
128
146
165
183
0
029
058
087
117
146
175
205
234
263
135
0
013
027
040
054
067
081
094
108
122
184
0
023
046
069
092
115
138
162
185
208
136
0
015
031
046
062
078
093
109
124
140
185
0
021
043
064
086
107
129
150
172
193
137
0
016
032
048
064
080
096
113
129
145
186
0
025
050
075
100
125
150
175
200
225
138
0
012
024
037
049
062
074
087
099 112
187
0
020
041
062
083
104
124
145
166
187
139
0
014
029
043
058
073
087
102
117 131
188
0
028
056
084
112
141
169
197
225
253
140
0
016
033
049
066
082
099
116
132 149
189
0
022
044
066
088
110
133
155
177
199
141
0
015
031
047
,063
078
094
110
126 141
190
0
021
043
065
087
109
131
153
175
197
142
0
017
035
052
070
087
105
122
140 157
191
0
029
058
087
116
145
175
204
233
262
143
0
015
030
046
061
076
092
107
122
138
192
0021
043
065
087
108
130
152
174
196
144
0
013
026
040
053
06
080
093
107 120
193
0
027
055 083
111
139
166
194
222
250
145
0
014
1028
042
056
07
085
099
113 128
194
0
028
056 084
112
140
168
196
224
252
*146
002
|04
06
08
i 10
12
14
16
18
0
1
2
3
4
5
6
7
8
9 206
TABLE XIX.
SIMPLE INTEREST FOR DAYS AT 5 PER CENT. 365 DAYS TO THE YEAR.
Days.
0
1
2
3
4
5
6
7
8
9
Days.
0
1
2
3
4
5
6
7
8
9
20
157
215
200
193
174
194:
L88
207
181
152
25
291
278
249
285
246
255
259
250
289
245
21
199
169
170
191
183
175
L67
159
151
>19
26
282
287
241
235
283
226
277
286
274
253
22
220
228
236
244
252
260;
239
238
268
221
27
223
275
233
230
288
273
266
247
267
261
23
258
276
265
263
251
262;
269
284
234
229
28
280
254
22o
272
242
243
264
256
248
240
24
271
227
257
270
290
281
222
279
231
237
29
232
224
292
293
301
309
317
325
333
312
Ref.
0
1
2
3
4
5
6
7
8
9
Ref.
0
1
2
3
4
5
6
7
8
9
195
0
021
042
063
084
106
127
148
169
191
244
0
030
061
091
122
152
183
213
244
274
196
0
022
044066
089
111
133
156
178
200
245
0
035
070
106
141
177
212
248
283
319
197
0
023
046
069
093
116
139
163
186
209
246
0
034
069
104
139
173
208
243
278
313
198
0
022
045
068
091
114
137
160
183
205
247
0
037
075
113
151
189
227
265
303
341
199
0
028
057
086
115
143
172
201
230
258
248
0
039
078
118
157
197
236
276
315
355
200
0
027
055
083
110
138
166
193
221
249
249
0
034
069
103
138
172
207
241
2
76
310
201
0
026
053
080
106
133
160
186
213
240
250
0
035
070
105
140
176
211
246
281
3
16
202
0
027
054
081
108
135
162
199
216
244
251
0
032
064096
128
160
192
224
256
288
203
0
021
043
064
086
108
129
151
173
194
252
0
030
061 092
122
153
184
214
245
2
76
204
0
026
052
079
105
132
158
185
211
237
253
0
036
073
110
147
184
221
257
294
331
205
0
024
048
073
097
121
146
170
195
219
254
0
038
076
115
153
192
230
269
307
340
206
0
023
047
071
095
119
143
166
190
214
255
0
034
069
104
139
1
74
209
244
279
314
207
0
028
056
085
113
141
170
198
226
255
256
0
039
078
117
157
196
235
2
75
314
353
208
0
023
047
070
094
117
141
164
188
212
257
0
033
066
099
132
165
198
232
205
298
209
0
025
051
076
102
128
153
179
204
230
258
0
031
063
094
126
157
189
220
252
283
210
0
026
052
078
104
130
156
183
209
235
259
0
035
070
105
140
175
210
245
280
315
211
0
022
044
067
089
112
134
157
179
202
260
0030
061
092
123
154
184
215
246
277
212
0
024
049
073
098
123
147
172
197
221
261
0
038
076
114
152
191
229 267
305
343
213
0
026
053
079
106
132
159
186
212
239
262
0
032
064
096
128
160
193 225
257
289
214
0
025
051
077
103
128
154
180
206
231
263
0
031
063
095
127
159
191 223
255
287
215
0
027
055
082
110
137
165
192
220
247
264
0
039
078
117
156
195
235
2
74
313
352
216
0
025
050
076
101
126
152
177
202
228
265
0031
063
095
127
158
190 222
254
286
217
0
023
046
070
093
117
140
163
187
210
266
0 037
075
113
151
189
226 264
302
340
218
0
024
048
072
096
121
145
169
193
218
267
0 038
076
114
152
190
228266
304
342
*219
0
03
06
09
12
15
18
21
24
27
268
0031
062
093
124
156
187
218
249
281
220
0
030
060
090
120
150
180
210
241
271
269
0032
064
096
129
161
193
226
258
290
221
0
031
062
094
125
156
188
219
250
282
270
0 033
066
099
133
166
199
233
266
299
222
0
033
067
101
134
168
202
235
269
303
271
0 032
065
098
131
164
197
230
263
295
223
0
036
073
110
147
184
221
258
295
332
272
0
038
077
116
155
193
232
271
310
348
224
0
039
079
119
159
199
239
279
318
358
273
(i
037
075
113
150
188
226
203
301
339
225
0
038
077
115
154
193
231
270
309
347
274
0
036
073
110
146
183
220
256
293
330
226
0
036
072
108
145
181
217
254
290
326
275
0
037
074
111
148
185
222
269
296
334
227
0
033
066
099
132
165
198
231
264
297
276
0
031
063
094
126
158
189
221
253
284
228
0
030
060
090
121
151
181
211
242
272
277
0
036
072
109
145
182
218
255
291
327
229
0
032
065
098
130
163
196
229
261
294
278
0
034
068
103
137
171
206
240
275
309
230
0
037
074
112
149
186
224
261
299
336
279
0
033
067
101
135
169
203
236
2
70
304
231
0
033
067
101
135
169
203
237
271
305
280
0
038
076
115
153
191
230
268
300
345
232
0
039
079
119
158
198
238
278
317
357
281
0
033
067
100
134
167
201
234
268
302
233
0
037
074
111
149
186
223
260
298
335
282
0
035
071
106
142
178
213
249
284
320
234
0
032
065
097
130
163
195
228
260
293
283
0
036
072
108
144
180
216
253
289
325
235
0
036
072
108
144
180
216
252
288
324
284
0
032
064
097
129
162
194
227
259
292
236
0
030
060
091
121
152
182
212
243
273
285
0
034
069
103
138
173
207
242
277
311
237
0
034
068
102
136
170
204
238
272
306
286
0
036
073
109
146
182
219
256
292
329
238
0
031
062
093
124
155
186
217
248
279
287
0
035
071
107
143
178
214
250
286
321
239
0
030
061
092
123
154
185
216
247
278
288
0
037
075
112
150
187
225
262
300
337
240
0
039
079
118
158
197
237
277
316
356
289
0
035
070
106
141
176
212
247
282
318
241
0
035
071
107
143
179
215
251
287
323
290
0
033
066
100
133
167
200
233
267
300
242
0
038077
116
155
194
233
272
311
350
291
0 034
068
102
136
171
205
239
273
308
243
0
039078
117
156
195
234
2731312
351
*292
o|04
08
12
16
20
24
28
32
36 TABLE XIX.
207
SIMPLE INTEREST FOR DAYS AT 5 PER CENT. 365 DAYS TO THE YEAR.
Days.
0
1
2
3
4
5
6
7
8
9
Days.
0
1
2
3
4
5
6
7
8
9
30
311
341
294
331
349
338
336;
324
335
342
35
320
340
334
353
327
298
345
315
316
337
31
357
307
302
344!
300
330
343
363
354
195
36
329
321
313
305
297
365
w
32
352
304
310
364
351
322
358!
319,
328
332
33
323
362
318
355
360
314
308;
356
399
350
34
359
347
326
396
348
306
303:
361
346
339
Ref.
0
1
2
3
4
5
6
7
8
9
Ref.
0
1
2
3
4
5
6
7
8
9
293
0
040
080
120
160
200
240
280
321
361
341
0
041
082
123
164
206
247
288
329
371
294
01041
082
124
165
206
248
2891330
372
342
0
042
084
126
169
211
2
53
296
338
380
295
0 043
087
131
174
218
262
305349393
343
0
043
086
129
173
216
259
303
346
389
296
0
046
093
140
187
234
281
3283751422
344
0
042
085
128
171
214
257
300
343
385
297
0
049
099 149
199
249
299
349
398|448
345
0
048
097
146
195
243
292
341
390
438
298
0
048
097 145
194
243
291
340
3891437
346
0
047
095
143
190
238
286
333
381
429
299
0
046
092
138
185
231
277
324 370 416
347
0
046
093
140
186
233
280
326
3
73
420
300
0
043
086
129
172
215
258
301
344
387
348
0
047
094
141
188
235
282
339
376
424
301
0
040
080
120
161
201
241
281
322
362
349
0
041
083
124
166
208
249
291
333
374
302
0
042
085
128
170
213
256
299
341
384
350
0
046
092
139
185
232
278
325
371
417
303
0
047
094
142
189
236
284
331
379
426
351
0
044
088
133
177
221
266
310
355
399
304
0
043
087
131
175
219
263
307
351
395
352
0
043
087
131
175
219
263
306
350
394
305
0
049
099
149
198
248
298
348
397
447
353
0
048
096
145
193
241
290
338
386
435
306
0
047
094
141
189
236
283
330
378
425
354
0
043
087
130
174
217
261
304
348
392
307
0
042
085
127
170
213
255
298
340
383
355
0
045
091
136
182
228
273
319
364
410
308
0
046
092
138
184
230
276
322
368 414
356
0
046
092
138
184
230
276
323
369
415
309
0
040
080
121
161
202
242
282
323
363
357
0
042
084
127
169
212
254
297
339
382
310
0
044
088
132
176
220
264
308
352
396
358
0
044
089
133
178
223
267
312
357
401
311
0
041
082
123
164
205
246
287
328
369
359
0
046
093
139
186
232
279
326
372
419
312
0
040
081
122
163
204
245
286
327
368
360
0
045
091
137
183
228
274
320
366
411
313
0
049
099
148
198
247
297
347
396
446
361
0
047
095
142
190
237
285
332
380
427
314
0
045
091
137
183
229
275
321
367
413
362
0
045
090
136
181
226
272
317
362
408
315
0
048
097
146
195
244
293
342
391
440
363
0
043
086
130
173
217
260
303
347
390
316
0
049
098
147
196
245
294
343
392
441
364
0
044
088
132
176
221
265
309
353
398
317
0
040
081
121
162
202
243
283
324
364
*365
0
05
10
15
20
25
30
35
40
45
318
0
045
090
136
181
227
272
318
363
409
0
1
0
a
3.
5
ft
7
ft
Q
319
0
044
089
134
179
o.oq
268
313
358
403
320
0
047
095
143
191
239
287
335
383
431
Example.
Required the
interest on
$475,289.37
321
0
049
098
148
197
247
296
346
395
445
for 317 days at 5 per cent.
In the upper part of the Table, 317 days give at
the
322
0
044
089
133
178
222
267
311
356
400
junction,
363
as Reference. And
this 363 guides to
323
0
045
090
135
180
226
271
316
361
406
the line 363 (near the end of the Table in
this
in-
324
0
042
084
126
168
210
252
294
336
378
stance) giving the multiples of interest for “ contracted
—
—
111 ul tip liCEi tioii.
325
0
040 081
122
162
203
244
284
325
366
On this
line of Kef.
the interest on
$5
is
$.217
326
327
0
0
046093
048096
140
145
187
193
234
242
281
290
327
339
374
387
421
436
nearly, or $ 21712328767123 more nearly, which fig-
ures (after the first three) are the final figures in that
column of eight spaces, commencing at 7 and reading
328
0
044
089
134
179
224
269
314
359
404
down to the drawn line below, then recommencing at
329
0
049
098
147
197
246
295
345
394
443
the line drawn above ;
and
so
generally.
Ref
(363)
4
330
331
0
0
043
041
086
083
129
124
172
166
215
207
258
249
302
290
345 388
332 373
The decimal points of prin-
einal and interest are made to
$475,289.37
17,369.863
332
0
045
090
135
180
225
270
315
360 405
agree, by
commencing
each
3,039.726
7
333
0
040
081
122
163
204
244
285
326
367
tabular product one place to
the rie-lit of its figure in the
217.123
08.684
5
2
334
0
048
096
144
192
241
289
337
385
433
multiplicand.
3.473
8
335
0
042
084
126
168
210
253
295
337
379
.390
9
336
0
041
083
125
167
209
251
293
335
377
13
3
3
7
337
0
049
098
147
196
245
295
344
393
442
-
338
0
041
083
125
167
208
250
292
334
376
Interest 317 days at 5 per cent.
Note. For 4 per cent, subtract
$29,639,270
1: for 6 per
cent.
339
0
047
095
143
191
239
286
334
382
430
add
4; for 4
per
cent.
subtract A ;
etc.,
from
the
340
0
048
096
144
192
240
288
336
384
432
tabular result.
2
8
in an
■ases
, to oiianffe irom ine oasis 01 ooo to
0
1
3
4
5
6
7
9
of 360 days to the year
, adc
fa part to the interest, 208
TABLE XX.
THE AMOUNT OF $1 AT THE END OF ANY NUMBER OF YEARS.
Years.
2
Per Cent.
a*
Per Cent.
3
Per Cent.
3|
Per Cent.
4
Per Cent.
4|
Per Cent.
Years.
1
1.02000000
1.02500000
1.03000000
1.03500000
1.04000000
1.04500000
1
2
1.04040000
1.05062500
1.06090000
1.07122500
1.08160000
1.09202500
2
3
1.06120800
1.07689063
1.09272700
1.10871787
1.12486400
1.14116613
3
4
1.08243216
1.10381289
1.12550881
1.14752300
1.16985850
1.19251860
4
5
1.10408080
1.13140821
1.15927407
1.18768631
1.21665290
1.24618194
5
6
1.12616242
1.15969342
1.19405230
1.22925533
1.26531902
1.30226012
6
7
1.14868567
1.18868575
1.22987387
1.27227926
1.31593178
1.36086183
7
8
1.17165938
1.21840290
1.26677008
1.31680904
1.36856905
1.42210061
8
9
1.19509257
1.24886297
1.30477318
1.36289735
1.42331181
1.48609514
9
10
1.21899442
1.28008454
1.34391638
1.41059876
1.48024428
1.55296942
10
11
1.24337431
1.31208666
1.38423387
1.45996972
1.53945400
1.62285305
11
12
1.26824179
1.34488882
1.42576089
1.51106866
1.60103222
1.69588143
12
13
1.29360663
1.37851104
1.46853371
1.56395606
1.06507351
1.77219610
13
14
1.31947876
1.41297382
1.51258972
1.61869452
1.73167645
1.85194492
14
15
1.34586834
1.44829817
1.55796742
1.67534883
1.80094351
1.93528244
15
16
1.37278571
1.48450562
1.60470644
1.73398604
1.87298125
2.02237015
16
17
1.40024142
1.52161826
1.65284763
1.79467555
1.94790050
2.11337681
17
18
1.42824625
1.55965872
1.70243306
1.85748920
2.02581052
2,20847877
18
19
1.45681117
1.59865019
1.75350605
1.92250132
2.10084918
2.30786031
19
20
1.48594740
1.63861644
1.80611123
1.98978886
2.19112314
2.41171402
20
21
1.51566634
1.67958185
1.86029457
2.05943147
2.27876807
2.52024116
21
22
1.54597967
1.72157140
1.91610341
2.13151158
2.36991879
2.63365201
22
23
1.57689926
1.76461068
1.97358651
2.20611448
2.40471554
2.75216635
23
24
1.60843725
1.80872595
2.03279411
2.28332849
2.50330410
2.87601383
24
25
1.64060599
1.85394410
2.09377793
2.36324498
2.66583633
3.00543446
25
26
1.67341811
1.90029270
2.15659127
2.44595856
2.77240978
3.14067901
26
27
1.70688648
1.94780002
2.22128901
2.53156711
2.88330858
3.28200956
27
28
1.74102421
1.99649502
2.28792768
2.62017196
2.99870332
3.42969999
28
29
1.77584469
2.04640739
2.35656551
2.71187798
3.11865145
3.58403649
29
30
1.81136158
2.09756758
2.42726247
2.80679370
3.24339751
3.74531813
30
31
1.84758882
2.15000677
2.50008035
2.90503148
3.37313341
3.91385745
31
32
1.88454059
2.20375694
2.57508276
3.00070759
3.50805875
4.08998104
32
33
1.92223140
2.25885086
2.65233524
3.11194235
3.64838110
4.27403018
33
34
1.96067603
2.31532213
2.73190530
3.22086033
3.79431634
4.46636154
34
35
1.99988955
2.37320519
2.81386245
3.33359045
3.94608899
4.66734781
35
36
2.03988734
2.43253532
2.89827833
3.45026611
4.10393255
4.87737846
36
37
2.08068509
2.49334870
2.98522668
3.57102542
4.26808980
5.09686049
37
38
2.12229879
2.55568242
3.07478348
3.09601131
4.43881345
5.32621921
38
39
2.16474477
2.61957448
3.16702698
3.82537171
4.61636599
5.56589908
39
40
2.20803966
2.68506384
3.26203779
3.95925972
4.80102063
5.81636454
40
41
2.25220046
2.75219043
3.35989893
4.09783381
4.99306145
6.07810094
41
42
2.29724447
2.82099520
3.46069589
4.24125799
5.19278391
6.35161548
42
43
2.34318936
2.89152008
3.56451677
4.38970202
5.40049527
6.63743818
43
44
2.39005314
2.96380808
3.67145227
4.54334100
5.61651508
6.93612290
44
45
2.43785421
3.03790328
3.78159584
4.70235855
5.84117568
7.24824843
45
46
2.48661129
3.11385086
3.89504372
4.86694110
0.07482271
7.57441961
46
47
2.53634352
3.19169713
4.01189503
5.03728404
6.31781562
7.91526849
47
48
2.58707039
3.27148956
4.13225188
5.21358898
0.57052824
8.27145557
48
49
2.63881179
3.35327680
4.25621944
5.39606459
6.83334937
8.64367107
49
50
2.69158803
3.43710872
4.38390602
5.58492086
7.10668335
9.03263627
50
Formula, Amount = (1 + i)n. TABLE XX.
209
THE AMOUNT OF $1 AT THE END OF ANY NUMBER OF YEARS.
Years.
5
Per Cent.
6
Per Cent.
7
Per Cent.
8
Per Cent.
9
Per Cent.
10
Per Cent.
| Years.
1
1.05000000
1.06000000
1.07000000
1.08000000
1.09000000
1.10000000
1
2
1.10250000
1.12360000
1.14490000
1.16640000
1.18810000
1.21000000
2
3
1.15762500
1.19101600
1.22504300
1.25971200
1.29502900
1.33100000
3
4
1.21550625
1.26247696
1.31079601
1.36048896
1.41158161
1.40410000
4
5
1.27628156
1.33822558
1.40255173
1.46932808
1.53862395
1.61051000
5
6
1.34009564
1.41851911
1.50073035
1.58687432
1.67710011
1.77156100
6
7
1.40710042
1.50363026
1.60578148
1.71382427
1.82803912
1.94871710
7 |
8
1.47745544
1.59384807
1.71818618
1.85093021
1.99256264
2.14358881
8
9
1.55132822
1.68947896
1.838459211
1.99900463
2.17189328
2.35794769
9
10
1.62889463
1.79084770
1.96715136
2.15892500
2.36736367
2.59374246
10
11
1.71033936
1.89829856
2.10485195
2.33163900
2.58042641
2.85311671
11
12
1.79585633
2.01219647
2.25219159
2.51817012
2.81266478
3.13842838
12
13
1.88564914
2.13292826
2.40984500
2.71962373
3.06580461
3.45227121
13
14
1.97993160
2.26090396
2.57853415
2.93719362
3.34172703
3.79749834
14
15
2.07892818
2.39655819
2.75903154
3.17216911
3.64248246
4.17724817
15
16
2.18287459
2.54035168
2.95216375
3.42594264
3.97030588
4.59497299
16
17
2.29201832
2.69277279
3.15881521
3.70001805
4.32763341
5.05447028
17
18
2.40661923
2.85433915
3.37993228
3.99601950
4.71712042
5.55991731
18
19
2.52695020
3.02559950
3.61652753
4.31570106
5.14166125
6.11590904
19
20
2.65329771
3.20713547
3.86968446
4.66095714
5.60441077
6.72749995
20
21
2.78596259
3.39956360
4.14056237
5.03383372
6.10880774
7.40024994
21
22
2.92526072
3.60353742
4.43040174
5.43654041
0.65860043
8.14027494
22
23
3.07152376
3.81974966
4.74052986
5.87146365
7.25787447
8.95430243
23
24
3.22509994
4.04893464
5.07236695
6.34118074
7.91108317
9.84973268
24
25
3.38635494
4.29187072
5.42743264
6.84847520
8.62308060
10.83470594
25
26
3.55567269
4.54938296
5.80735292
7.39635321
9.39915792
11.91817654
26
27
3.73345632
4.82234594
6.21386763
7.98806141
10.24508213
13.10999419
27
28
3.92012914
5.11168670
6.64883836
8.62710639
11.16713952
14.42099361
28
29
4.11613560
5.41838790
7.11425705
9.31727490
12.17218208
15.86309297
29
30
4.32194238
5.74349117
7.61225504
10.06265689
13.26767847
17.44940227
30
31
4.53803949
6.08810064
8.14511290
10.86766944
14.46176953
19.19434250
31
32
4.76494147
6.45338668
8.71527080
11.73708300
15.76332879
21.11377675
32
33
5.00318854
6.84058988
9.32533975
12.67604963
17.18202838
23.22515442
33
34
5.25334797
7.25102528
9.97811354
13.69013361
18.72841093
25.54766986
34
35
5.51601537
7.68608679
10.67658148
14.78534429
20.41396792
28.10243685
35
36
5.79181614
8.14725290
11.42394219
15.96817184
22.25122503
30.91268053
36
37
6.08140694
8.63608712
12.22361814
17.24562558
24.25383528
34.00394859
37
38
6.38547729
9.15425235
13.07927141
18.62527563
26.43668040
37.40434344
38
39
6.70475115
9.70350749
13.99482041
20.11529768
28.81598170
41.14477779
39
40
7.03998871
10.28571794
14.97445784
21.72452150
31.40942005
45.25925557
40
41
7.39198815
10.90286101
16.02266989
23.46248322
34.23626786
49.78518112
41
42
7.76158756
LI.55703267
17.14425678
25.33948187
37.31753197
54.76369924
42
43
8.14966693
12.25045463
18.34435475
27.36664042
40.67610984
60.24006916
43
44
8.55715028
12.98548191
19.62845959
29.55597166
44.33095973
66.26407608
44
45
8.98500779
13.76461083
21.00245176
31.92044939
48.32728610
72.89048369
45
46
9.43425818
14.59048748
22.47262338
34.47408534
52.67674185
80.17953205
46
47
9.90597109
15.46591673
24.04570702
37.23201217
57.41764862
88.19748526
47
48
10.40126965
16.39387173
25.72890651
40.21057314
62.58523700
97.01723378
48
49
10.92133313
17.37750403
27.52992997
43.42741899
68.21790833
106.71895716
49
50
11.46739979|18.42015427
29.45702506
46.90161251
74.35752008
117.39085288
50 210
T A BLE XX.
THE AMOUNT OF $1 AT THE END OF ANY NUMBER OF YEARS.
Years.
2
Per Cent.
Per Cent.
3
Per Cent.
Per Cent.
4
Per Cent.
4i
Per Cent.
YearB.
51
2.74541979
3.52303644
4.51542320
5.78039930
7.39095068
9.43910490
51
52
2.80032819
3.61111235
4.65088590
5.98271327
7.68658871
9.86386463
52
53
2.85633475
3.70139016
4.79041247
6.19210823
7.99405226
10.30773853!
53
54
2.91346144
3.79392491
4.93412485
6.40883202
8.31381435
10.77158677
54
55
2.97173067
3.88877303
5.08214859
6.63314114
8.64636692
11.25630817
55
56
3.03116529
3.98599236
5.23461305
6.86530108
8.9922216011.76284204
56
57
3.09178859
4.08564217
5.39165144
7.10558662
9.35191046
12.29216993
57
58
3.15362436
4.18778322
5.55340098
7.35428215
9.72598688
12.84531758
58
59
3.21669685
4.29247780
5.72000301
7.61168203
10.11502635
13.42335687
59
60
3.28103079
4.39978975
5.89160310
7.87809090
10.51962741
14.02740793
60
61
3.34665140
4.50978449
6.06835120
8.15382408
10.94041250
14.65864129
61'
62
3.41358443
4.62252910
6.25040173
8.43920792
11.37802900
15.31828014
62
63
3.48185612
4.73809233
6.43791378
8.73458020
11.83315016
16.00760275
63
64
3.55149324
4.85654464
6.63105120
9.04029051
12.30647617
16.72794487
64
65
3.62252311
4.97795826
6.82998273
9.35670068
12.79873522
17.48070239
65
66
3.69497357
5.10240721
7.03488222
9.68418520
13.31068463
18.26733400
66
67
3.76887304
5.22996739
7.24592868
10.02313168
13.84311201
19.08936403
67
68
3.84425050
5.36071658
7.46330654
10.37394129
14.39683649
19.94838541
68
69
3.92113551
5.49473449
7.68720574
10.73702924
14.97270995
20.84606276
69
70
3.99955822
5.63210286
7.91782191
11.11282526
15.57161835
21.78413558
70
71
4.07954939
5.77290543
8.15535657
11.50177414
16.19448308
22.76442168
71
72
4.16114038
5.91722806
8.40001727
11.90433624
16.84226241
23.78882066
72
73
4.24436318
6.06515876
8.65201778
12.32098801
17.51595290
24.85931759
73
74
4.32925045
6.21678773
8.91157832
12.75222259
18.21659102
25.97798688
74
75
4.41583546
6.37220743
9.17892567
13.19855038
18.94525466
27.14699629
75
76
4.50415216
6.53151261
9.45429344
13.66049964
19.70306485
28.36861112 76
77
4.59423521
6.69480043
9.73792224
14.13861713
20.49118744
29.64519862
77
78
4.68611991
6.86217044 10.03005991
14.63346873
21.31083494
30.97923256
78
79
4.77984231
7.0337247040.33090170
15.14564013
22.16326833
32.37329802
79
80
4.87543916
7.20956782 10.64089056
15.67573754
23.04979907
33.83009643
80
81
4.97294794
7.38980701 10.96011727
16.22438835
23.97179103
35.35245077
81
82
5.07240690
7.57455219
11.28892079
16.79224194
24.93066267
36.94331106
82
83
5.17385504
7.76391599
11.62758841
17.37997041
25.92788918
38.60576006
83
84
5.27733214
7.95801389
11.97641607
17.98826938
26.96500475
40.34301926
84
85
5.38287878
8.15696424
12.33570855
18.61785880
28.04360494
42.15845513
85
86
5.49053636
8.36088834
12.70577981
19.26948386
29.16534913
44.05558561
86
87
5.60034708
8.56991055
13.08695320
19.94391580
30.33196310
46.03808696
87
88
5.71235402
8.78415832
13:47956180
20.64195285
31.54524162
48.10980087
88
89
5.82660110
9.00376228
13.88394865
21.36442120
32.80705129
50.27474191
89
90
5.94313313
9.22885633 14.30046711
22.11217594
34.11933334
52.53710530
90
91
6.06199579
9.4595777414.72948112
22.88610210
35.48410667
54.90127503
91
92
6.18323570
9.69606718 15.17136556
23.68711567
36.90347094
57.37183241
92
93
6.30690042
9.9384688615.62650652
24.51616472
38.37960978
59.95356487
93
94
6.43303843
10.18693058116.09530172
25.-37423049
39.91479417
62.65147529
94
95
6.56169920
10.44160385 16.57816077
26.26232856
41.51138594
65.47079168
95
96
6.69293318
10.70264395 17.07550559
27.18151005
43.17184137
68.41697730
96
97
6.82679184
10.97021004
17.58777076
.28.13286291
44.89871503
71.49574128
97
98
6.96332768
11.24446530
18.11540388
29.11751311
46.69466363
74.71304964
98
99
7.10259423
11.52557693
18.65886600
30.13662607
48.56245017
78.07513687
99
100
7.24464612
11.81371635 19.21863198,31.19140798
1 li
50.50494818
81.58851803
100 TABLE XX.
211
THE AMOUNT OF $1 AT THE END OF ANY NUMBER OF YEARS.
Years.
5
Per Cent.
6
Per Cent.
7
Per Cent.
8
Per Cent.
10
Per Cent.
Years.
51
12.04076978
19.52536353
31.51901682
50.65374151
129.12993817
51
52
12.04280821
20.69688534
33.72534799
54.70604083
142.04293198
52
53
13.27494868
21.93869841
36.08012235
59.08252410
156.24722518
53
54
13.93869611
23.25502037
38.01215092
63.80912003
171.87194770
54
55
14.63563092
24.65032159
41.31500148
68.91385611
189.05914247
55
56
15.36741246
26.12934089
44.20705159
74.42696460
207.96505672
56
57
16.13578309
27.69710134
47.30154520
80.38112177
228.76156239
57
58
16.94257224
29.35892742
50.61265336
86.81101151
251.63771863
58
59
17.78970085
31.12046307
54.15553910
93.75654043
276.80149049
59
60
18.67918589
32.98769085
57.94642083
101.25706307
304.48163954
60
61
19.61314519
34.96695230
62.00267671
109.35762870
334.92980350
61
62
20.59380245
37.06496944
66.34286408
118.10623906
368.42278385
62
63
21.62349257
39.28886761
70.98680457
127.55473819
405.26506223
63
64
22.70466720
41.64619966
75.95594509
137.75911724
445.79156845
64
65
23.83990056
44.14497164
81.27286124
148.77984662
490.37072530
65
66
25.03189559
46.79366994
86.90196153
100.68223435
539.40779783
66
67
26.28349037
49.60129014
93.04929884
173.53681310
593.34857761
67
68
27.59766488
52.57736755
99.56274976
187.41975814
652.68343537
68
69
28.97754813
55.73200960
100.53214224
202.41333880
717.95177891
69
70
30.42642554
59.07593018
113.98939220
218.00640590
789.74695680
70
71
31.94774681
62.62048599
121.90864965
230.09491837
868.72105248
71
72
33.54513415
66.37771515
130.50045513
254.98251184
955.59381773
72
73
35.22239086
70.36037806
139.64190099
275.38111279
1,051.15319950
73
74
36.98351040
74.58200074
149.41684047
297.41160181
1,156.26851945
74
75
38.83268592
79.05692078
159.87601931
321.20452996
1,271.89537140
75
76
40.77432022
83.80033603
171.06734060
346.90089235
1,399.08490853
76
77
42.81303623
88.82835619
183.04205450
374.05290374
1,538.99339939
77
78
44.95368804
94.15805756
195.85499832
404.62520084
1,092.89273933
78
79
47.20137244
99.80754102
209.56484820
436.99521691
1,862.18201326
79
80
49.56144107
105.79599348
224.23438758
471.95483426
2,048.40021459
80
81
52.03951312
112.14375309
239.93079471
509.71122100
2,253.24023604
81
82
54.64148878
118.87237827
256.72595034
550.48811868
2,478.50425965
82
83
57.37356322
126.00472097
274.69070686
594.52716818
2,726.42068561
83
84
60.24224138
133.56500423
293.92554054
642.08934163
2,999.06275418
84
85
63.25435344
141.57890448
314.50032838
093.45648896
3,298.90902959
85
86
66.41707112
150.07363875
336.51535136
748.93300808
3,628.80593255
86
87
69.73792467
159.07805708
360.07142596
808.84764873
3,991.75252581
87
88
73.22482091
168.62274050
385.27642578
873.55546062
4,390.92777839
88
89
76.88606195
178.74010493
412,24577558
943.43989747
4,830.02055623
89
90
80.73036505
189.46451123
441.10297987
1,018.91508927
5,313.02261185
90
91
84.76688330
200.83238190
471.98018840'
1,100.42829641
5,844.32487303
91
92
89.00522747
212.88232481
505.01880160
1,188.46256013
6,428.75736034
92
93
93.45548884
225.65526430
540.37011777
1,283.53956494
7,071.63309637
93
94
98.12826328
239.19458010
578.19602602
1,386.22273013
7,778.79640601
94
95
103.03467645
253.54025497
618.66974784
1,497.12054854
8,556.07604661
95
96
108.18641027
268.75903027
061.97063019
1,616.89019242
9,412.34365127
96
97
113.59573078
284.88457208
708.31499430
1,746.24140782
10,353.57801640
97
98
119.27551732
301.97764041
757.89704390
1,885.94072044
11,388.93581804
98
99
125.23929319
320.09630519
810.94983697
2,036.81597808
12,527.82939984
99
100
131.50125785
339.30208351
867.71632556
2,199.76125632
13,780.61233982
100 212
TABLE XXI.
THE PRESENT VALUE OF $1 DUE AT THE END OF ANY NUMBER OF YEARS.
Years.
2
Per Cent.
Per Cent.
3
Per Cent.
3^
Per Cent.
4
Per Cent.
4.1
*2
Per Cent.
Years.
1
.98039216
.97560976
.97087379
.96618357
.96153846
.95693780
i!
2
.96116878
.95181440
.94259591
.93351070
.92455621
.91572995
2
3
.94232233
.92859941
.91514166
.90194270
.88899636
.87629660
3
4
.92384543
.90595064
.88848705
.87144223
.85480419
.83856134
4
5
.90573081
.88385429
.86260878
.84197317
.82192711
.80245105
5
6
.88797138
.86229687
.83748426
.81350064
.79031453
.76789574
6
7
.87056018
.84126524
.81309151
.78599096
.75991781
.73482846
7
8
.85349037
.82074657
.78940923
.75941155
.73069020
.70318513
8
9
.83675527
.80072836
.76641673
.73373097
.70258674
.67290443
9
10
.82034830
.78119840
.74409391
.70891881
.67556417
.64392768
10
11
.80426304
.76214478
.72242127
.68494571
.64958093
.61619874
11
12
.78849318
.74355589
.70137988
.66178330
.62459705
.58966386
12
13
.77303253
•72542038
.68095134
.63940415
.60057409
.56427164
13
14
.75787502
.70772720
.66111781
.61778179
.57747508
.53997286
14
15
.74301473
.69046556
.64186195
.59689062
.55526450
.51672044
15
16
.72844581
.67362493
.62316694
.57670591
.53390818
.49446932
16
17
.71416256
.65719506
.60501645
.55720378
.51337325
.47317639
17
18
.70015937
.64116591
.58739461
.53836114
.49362812
.45280037
18
19
.68643076
.62552772
.57028603
.52015569
.47464242
.43330179
19
20
.67297133
.61027094
.55367576
.50256588
.45638695
.41464286
20
21
.65977582
.59538629
.53754928
.48557090
.43883360
.39678743
21
22
.64683904
.58086467
.52189250
.46915063
.42195539
.37970089
22
23
.63415592
.56669724
.50669175
.45328563
.40572633
.36335013
23
24
.62172149
.55287535
.49193374
.43795713
.39012147
.34770347
24
25
.60953087
.53939059
.47760557
.42314699
.37511680
.33273060
25
26
.59757928
.52623472
.46369473
.40883767
.36068923
.31840248
26
27
.58586204
.51339973
.45018906
.39501224
,34681657
.30469137
27
28
.57437455
.50087778
.43707675
.38165434
.33347747
.29157069
28
29
.56311231
.48866125
.42434636
.36874816
.32065141
.27901502
29
30
.55207089
.47674269
.41198676
.35627841
.30831867
.26700002
30
31
.54124597
.46511481
.39998715
.34423035
.29646026
.25550241
31
32
.53063330
.45377055
.38833703
.33258971
.28505794
.24449991
32
33
.52022873
.44270298
.37702625
.32134271
.27409417
.23397121
33
34
.51002817
.43190534
.36604490
.31047605
.26355209
.22389589
34
35
.50002761
.42137107
.35538340
.29997686
.25341547
.21425444
35
36
.49022315
.41109372
.34503243
.28983272
.24366872
.20502817
36
37
.48061093
.40106705
.33498294
.28003161
.23429685
.19619921
37
38
.47118719
.39128492
.32522615 1
.27056194
.22528543
.18775044
38
39
.46194822
.38174139
.31575355
.26141251
.21662061
.17966549
39
40
.45289042
.37243062
.30655684
.25257247
.20828904
.17192870
40
41
.44401021
.36334695
.29762800
.24403137
.20027793
.16452507
41
42
.43530413
.35448483.
.28895922
.23577910
.19257493
.15744026
42
43
.42676875
.34583886
.28054294
,22780590
.18516820
.15066054
43
44
.41840074
.33740376
.27237178
.22010232
.17804635
.14417276
44
45
.41019680
.32917440
.26443862
•21265924
.17119841
.13796437
45
46
.40215373
.32114576
.25673653
; .20546787
.16461386
.13202332
46
47
.39426836
.31331294
.24925876
.19851968
.15828256
.12633810
47
48
.38653761
.30567116
.24199880
.19180645
.15219476
.12089771
48
49
.37895844
.29821576
.23495029
.18532024
.14634112
.11569158
49
50
.37152788
.29094221
.22810708
.17905338
.14071261
.11070965
50
Formula, Present Value = -—r— = vn.
(! + *)“ TABLE XXI.
213
THE PRESENT VALUE OF $1 DUE AT THE END OF ANY NUMBER OF YEARS,
Years.
! s
Per Cent.
6
Per Cent.
7
Per Cent.
8
Per Cent.
9
Per Cent.
10
Per Cent.
Years.
1
.95238095
.94339623
.93457944
.92592593
.91743119
.90909091
1
2
.90702948
.88999644
.87343873
.85733882
.84167999
.82644628
2
3
.86383700
.83961928
.81629788
.79383224
.77218348
.75131480
3
4
.82270247
.79209366
.76289521
! .73502985
.70842521
.68301346
4
5
.78352617
.74725817
.71298618
.68058320
.64993139
.62092132
5
6
.74621540
.70496054
.66634222
| .63016963
.59626733
.56447393
6
7
.71068133
.66505711
.62274974
.58349040
.54703424
.51315812
7
8
.67683930
.62741237
.58200910
.54026888
.50186628
.46650738
8
9
.64460892
.59189846
.54393374
.50024897
.46042778
.42409762
9
10
.61391325
.55839478
.50834929
.46319349
.42241081
.38554329
10
11
.58467929
.52678752
.47509280
.42888286
.38753285
.35049390
11
12
.55683742
.49696936
.44401196
.39711376
.35553473
.31863082
12
13
.53032135
.46883902
.41496445
.36769792
.32617865
.28966438
13
14
.50506795
.44230090
.38781724
.34046104
.29924647
.26333125
14
15
.48101710
.41720500
.36244602
.31524171
.27453804
.23939205
15
16
.45811152
.39364028
.33873400
.29189047
.25186976
.21762914
16
17
.43629669
.37136442
.31657439
.27026895
.23107318
.19784467
17
18
.41552065
.35034379
.29586392
.25024903
.21199374
.17985879
18
19
.39573396
.33051301
.27650833
.23171206
.19448967
.16350799
19
20
.37688948
.31180473
.25841900
.21454821
.17843089
.14864363
20
21
.35894230
.29415540
.24151309
.19865575
.16369806
.13513057
21
22
.34184987
.27750510
.22571317
.18394051
.15018171
.12284597
22
23
.32557130
.26179720
.21094688
.17031528
.13778139
.11167816
23
24
.31006791
.24697855
.19714662
.15769934
.12640494
.10152560
24
25
.29530277
.23299863
.18424918
.14601790
.11596784
.09229600
25
26
.28124073
.21981003
.17219549
.13520176
.10639251
.08390545
26
27
.26784832
.20730795
.16093037
.12518682
.09760781
.07627768
27
28
.25509364
.19503014
.15040221
.11591372
.08954845
.06934335
28
29
.24294632
.18455674
.14056282
.10732752
.08215454
.06303941
29
30
.23137745
.17411013
.13136712
.09937733
.07537114
.05730855
30
31
.22035947
.10425484
.12277301
.09201605
.06914783
.05209868
31
32
.20986617
.15495740
.11474113
.08520005
.06343838
.04736244
32
33
.19987254
.14618622
.10723470
.07888893
.05820035
.04305676
33
34
.19035480
.13791153
.10021934
.07304531
.05339481
.03914251
34
35
.18129028
.13010522
.09360294
.06763454
.04898607
.03558410
35
36
.17265741
.12274077 '
.08753546
.06262458
.04494135
.03234918
36
37
.16443563
.11579318
.08180884
.05798572
.04123059
.02940835
37
38
.15660530
.10923885
.07645686
.05369048
.03782623
.02673486
38
39
.14914797
.10305552
.07145501
.04971341
.03470296
.02430442
39
40
.14204568
.09722219
.06678038
.04603093
.03183758
.02209493
40
41
.13528100
.09171904
.06241157
.04262123
.02920879
.02008630
41
42
.12883962
.08652740
.05832857
.03946411
.02679706
.01826027
42
43
.12270440
.08162962
.05451268
.03654084
.02458446
.01660025
43
44
.11686133
.07700908
.05094643
.03383411
.02255455
.01509113
44
45
.11129651
.07265007
.04761349
.03132788
.02069224
.01371921
45
46
.10599667
.00853781
.04449859
.02900730
.01898371
.01247201
46
47
.10094921
.06465831
.04158747
.02685861
.01741625
.01133819
47
48
.09614211
.00099840
.03886679
.02486908
.01597821
.01030745
48
49
.09156391
.05754566
.03032410
.02302693
.01465891
.00937041
49
50
.08720373
.05428836
.03394776
.02132123
.01344854
.00851855
50 214
TABLE XXI.
THE PRESENT VALUE OF $1 DUE AT THE END OF ANY NUMBER OF YEARS.
Years.
2
Per Cent.
24
Per Cent.
3
Per Cent.
31
Per Cent.
4
Per Cent.
4*
Per Cent.
Years.
51
.36424302
.28384606
.22146318
.17299843
.13530059
.10594225
51
52
.35710100
.27692298
.21501280
.16714824
.13009672
.10138014
52
53
.35009902
.27016876
.20875029
.16149589
.12509300
.09701449
53
54
.34323433
.26357928
.20267018
.15603467
.12028173
.09283683
54
55
.33650425
.25715052
.19676717
.15075814
.11565551
.08883907
55
56
.32990613
.25087855
.19103609
.14566004
.11120722
.08501347
56
57
.32343738
.24475956
.18547193
! .14073433
.10693002
.08135260
57
58
.31709547
.23878982
.18006983
.13597520
.10281733
.07784938
58
59
.31087791
.23296568
.17482508
.13137701
.09886282
.07449701
59
60
.30478227
.22728359
.16973309
.12693431
.09506040
.07128901
60
61
.29880614
.22174009
.16478941
.12264184
.09140423
.06821915
61
62
.29294720
.21633179
.15998972
.11849453
.08788868
.06528148
62
63
.28720314
.21105541
.15532982
.11448747
.08450835
.06247032
63
64
.28157170
.20590771
.15080565
.11061591
.08125803
.05978021
64
65
.27605069
.20088557
.14641325
.10687528
.07813272
.05720594
65
66
.27063793
.19598593
.14214879
.10326114
.07512761
.05474253
66
67
.26533130
.19120578
.13800853
.09976922
.07223809
.05238519
67
68
.26012873
.18654223
.13398887
.09639538
.06945970
.05012937
68
69
.25502817
.18199241
.13008628
.09313563
.06678818
.04797069
69
70
.25002761
.17755358
.12629736
.08998612
.06421940
.04590497
70
71
.24512511
.17322300
.12261879
.08694311
.06174942
.04392820
71
72
.24031874
.16899805
.11904737
.08400300
.05937444
.04203655
72
73
.23560661
.16487615
.11557997
.08116232
.05709081
.04022637
73
74
.23098687
.16085478
.11221357
.07841770
.05489501
.03849413
74
75
.22645771
.15693149
.10894521
.07576590
.05278366
.03683649
75
76
.22201737
.15310389
.10577205
.07320376
.05075352
.03525023
76
77
.21766408
.14936965
.10269131
.07072827
.04880146
.03373228
77
78
.21339616
.14572649
.09970030
.06833650
.04692449
.03227969
78
79
.20921192
.14217218
.09679641
.06602560
.04511970
.03088965
79
80
.20510973
.13870457
.09397710
.06379285
.04338432
.02955948
80
81
.20108797
.13532153
.09123990
.06163561
.04171570
.02828658
81
82
.19714507
.13202101
.08858243
.05955131
.04011125
.02706850
82
83
.19327948
.12880098
.08600236
.05753750
.03856851
.02590287
83
84
.18948968
.12565949
.08349743
.05559179
.03708510
.02478744
84
85
.18577420
.12259463
.08106547
.05371187
.03565875
.02372003
85
86
.18213157
.11960452
.07870434
.05189553
.03428726
.02269860
86
87
.17856036
.11668733
.07641198
.05014061
.03296852
.02172115
87
88
.17505918
.11384130
.07418639
.04844503
.03170050
.02078579
88
89
.17162665
.11106468
.07202562
.04680679
.03048125
.01989070
89
90
.16826142
.10835579
.06992778
.04522395
.02930889
.01903417
90
91
.16496217
.10571296
.06789105
.04369464
.02818163
.01821451
91
92
.16172762
.10313460
.06591364
.04221705
.02709772
.01743016
92
93
.15855649
.10061912
.06399383
.04078942
.02605550
.01667958
93
94
.15544754
.09816500
.06212993
.03941006
.02505337
.01596132
94
95
.15239955
.09577073
.06032032
.03807736
.02408977
.01527399
95
96
.14941132
.09343486
.05856342
.03678972
.02316324
.01461626
96
97
.14648169
.09115596
.05685769
.03554562
.02227235
.01398685
97
98
.14360950
.08893264
.05520164
.03434359
.02141572
.01338454
98
99
.14079363
.08676355
.05359382
.03318222
.02059204
.01280817
99
100
.13803297
.08464737
.05203284
.03206011
.01980004
.01225663
100 TABLE XXI.
215
THE PRESENT VALUE OF $1 DUE AT THE END OF ANY NUMBER OF YEARS.
Years.
5
Per Cent.
6
Per Cent.
7
Per Cent.
8
Per Cent.
9
Per Cent.
10
Per Cent.
c6
a
CD
51
.08305117
.05121544
.03172688
.01974188
.01233811
.00774414
51
; 52
.07909635
.04831645
.02965129
.01827952
.01131937
.00704013
52
53
.07532986
.04558156
.02771148
.01692548
.01038474
.00640011
53
54
.07174272
.04300147
.02589858
.01567174
.00952728
.00581829
54
55
.06832640
.04056742
.02420428
.01451087
.00874063
.00528935
55
56
.06507276
.03827115
.02262083
.01343599
.00801892
.00480850
56
57
.06197406
.03610486
.02114096
.01244073
.00735681
.00437136
57
58
.05902291
.03406 LI 9
.01975791
.01151920
.00674937
.00397397
58
59
.05621230
.03213320
.01846533
.01066592
.00619208
.00361270
59
60
.05353552
.03031434
.01725732
.00987585
.00568081
.00328427
60
61
.05098621
.02859843
.01612834
.00914431
.00521175
.00298570
61
62
.04855830
.02697965
.01507321
.00846696
.00478142
.00271427
62
63
.04624600
.02545250
.01408711
.00783977
.00438663
.00246752
63
64
.04404381
.02401180
.01316553
.00725905
.0(t402443
.00224320
64
65
.04194648
.02265264
.01230423
.00672134
.00369214
.00203927
65
66
.03994903
.02137041
.01149928
.00622346
.00338728
.00185388
66
67
.03804670
.02016077
.01074699
.00576247
.00310760
.00168535
67
68
.03623495
.01901959
.01004392
.00533562
.00285101
.00153214
68
69
.03450948
.01794301
.00938684
1 .00494039
.00261560
.00139285
69
70
.03286617
.01692737
.00877275
.00457443
.00239963
.00126623
70
71
.03130111
.01596922
.00819883
.00423558
.00220150
.00115112
71
72
.02981058
.01506530
.00766246
.00392184
.00201972
.00104647
72
73
.02839103
.01421254
.00716117
.00363133
.00185296
.00095134
73
74
.02703908
.01340806
.00669269
.00336234
.00169996
.00086485
74
75
.02575150
.01264911
.00625485
.00311328
.00155960
.00078623
75
76
.02452524
.01193313
.00584565
.00288267
.00143082
.00071475
76
77
.02335737
.01125767
.00546323
.00266914
.00131268
.00064978
77
78
.02224511
.01062044
.00510582
.00247142
.00120430
.00059070
78
79
.02118582
.01001928
.00477179
.00228835
.00110486
.00053700
79
80
.02017697
.00945215
.00445962
.00211885
.00101363
.00048819
80
81
.01921617
.00891713
.00416787
.00196190
.00092994
.00044381
81
82
.01830111
.00841238
.00389520
.00181657
.00085315
.00040346
82
83
.01742963
.00793621
.00364038
.00168201
.00078271
.00036678
83
81
.01659965
.00748699
.00340222 1
.00155742
.00071808
.00033344
84
85
.01580919
.00706320
.00317965
.00144205
.00065879
.00030313
85
86
.01505637
.00666339
.00297163 1
.00133523
.00060440
.00027557
86
87
.01433940
.00628622
.00277723
.00123633
.00055449
.00025052
87
88
.01365657
.00593040
.00259554
.00114475
.00050871
.00022774
88
89
.01300626
.00559471
.00242574
.00105995
.00046670
.00020704
89
90
.01238691
.00527803
.00226704
.00098144
.00042817
.00018822
90
91
.01179706
.00497928
.00211873
.00090874
.00039282
.00017111
91
92
.01123529
.00469743
.00198012
.00084142
.00036038
.00015555
92
93
.01070028
.00443154
.00185058
.00077910
.00033063
.00014141
93
94
.01019074
.00418070
.00172952
.00072138
.00030333
.00012855
94
95
.00970547
.00394405
.00161637
.00066795
.00027828
.00011687
95
96
.00924330
.00372080
.00151063
.00061847
.00025530
.00010624
96
97
.00880315
.00351019
.00141180
.00057266
.00023422
.00009658
97
98
.00838395
.00331150
.00131944
.00053024
.00021488
.00008780
98
99
.00798471
.00312406
.00123312
.00049096
.00019714
.00007982
99
100
.00760449
.00294722
.00115245
.00045459
.00018086
.00007257
100 216
TABLE XXII.
THE AMOUNT OF $1 PER ANNUM AT THE END OF ANY NUMBER OF YEARS.
Years.
2
Per Cent.
21
Per Cent.
3
Per Cent.
8*
Per Cent.
4
Per Cent.
Per Cent.
Years.
1
1.000000
1.000000
1.000000
1.000000
1.000000
1.000000
1
2
2.020000
2.025000
2.030000
2.035000
2.040000
2.045000
2
3
3.060400
3.075625
3.090900
3.106225
3.121600
3.137025
3
4
4.121608
4.152516
4.183627
4.214943
4.246464
4.278191
4
5
5.204040
5.256329
5.309136
5.362466
5.416323
5.470710
5
6
6.308121
6.387737
6.468410
6.550152
6.632975
6.716892
6
7
7.434283
7.547430
7.662462
7.779408
7.898294
8.019152
7
8
8.582969
8.736116
8.892336
9.051687
9.214226
9.380014
8
9
9.754628
9.954519
10.159106
10.368496
10.582795
10.802114
9
10
10.949721
11.203382
11.463879
11.731393
12.006107
12.288209
10
11
12.168715
12.483466
12.807796
13.141992
13.486351
13.841179
11
12
13.412090
13.795553
14.192030
14.601962
15.025805
15.464032
12
13
14.680332
15.140442
15.617790
16.113030
16.626838
17.159913
13
14
15.973938
16.518953
17.086324
17.676986
18.291911
18.932109
14
15
17.293417
17.931927
18.598914
19.295681
20.023588
20.784054
15
16
18.639285
19.380225
20.156881
20.971030
21.824531
22.719337
16
17
20.012071
20.864730
21.761588
22.705016
23.697512
24.741707
17
18
21.412312
22.386349
23.414435
24.499691
25.645413
26.855084
18
19
22.840559
23.946007
25.116868
26.357181
27.671229
29.063562
19
20
24.297370
25.544658
26.870374
28.279682
29.778079
31.371423
20
21
25.783317
27.183274
28.676486
30.269471
31.969202
33.783137
21
22
27.298984
28.862856
30.536780
32.328902
34.247970
36.303378
22
23
28.844963
30.584427
32.452884
34.460414
36.617889
38.937030
23
24
30.421862
32.349038
34.426470
36.666528
39.082604
41.689196
24
25
32.030300
34.157764
36.459264
38.949857
41.645908
44.565210
25
26
33.670906
36.011708
38.553042
41.313102
44.311745
47.570645
26
27
35.344324
37.912001
40.709634
43.759060
47.084214
50.711324
27
28
37.051210
39.859801
42.930923
46.290627
49.967583
53.993333
28
29
38.792235
41.856296
45.218850
48.910799
52.966286
57.423033
29
30
40.568079
43.902703
47.575416
51.622677
56.084938
61.007070
30
31
42.379441
46.000271
50.002678
54.429471
59.328335
64.752388
31
32
44.227030
48.150278
52.502759
57.334502
62.701469
68.666245
32
33
46.111570
50.354034
55.077841
60.341210
66.209527
72.756226
33
34
48.033802
52.612885
57.730177 1
63.453152
69.857909
77.030256
34
35
49.994478
54.928207
60.462082
66.674013
73.652225
81.496618
35
36
51.994367
57.301413
63.275944
70.007603
77.598314
86.163966
36
37
54.034255
59.733948
66.174223
73.457869
81.702246
91.041344
37
38
56.114940
62.227297
69.159449
77.028895
85.970336
96.138205
38
39
58.237238
64.782979
72.234233
80.724906
90.409150
101.464424
39
40
60.401983
67.402554
75.401260
84.550278
95.025516
107.030323
40
41
62.610023
70.087617
78.663298
88.509537
99.826536
112.846688
41
42
64.862223
72.839808
82.023196
92.607371
104.819598
118.924789
42
43
67.159468
75.660803
85.483892
96.848629
110.012382
125.276404
43
44
69.502657
78.552323
89.048409
101.238331
115.412877
131.913842
44
45
71.892710
81.516131
92.719861
105.781673
121.029392
138.849965
«
46
74.330564
84.554034
96.501457
110.484031
126.870568
146.098214
46
47
76.817176
87.667885
100.396501
115.350973
132.945390
153.672633
47
48
79.353519
90.859582
104.408396 1
120.388257
139.263206
161.587902
48
49
81.940590
94.131072
108.540648
125.601846
145.833734
169.859357
49
50
84.579401
97.484349
112.796867
130.997910
152.667084
178.503028
50
Formula, Amount = - \ = 1 + (1 + i) + ... + (1
% TABLE XXII.
217
THE AMOUNT OF $1 PER ANNUM AT THE END OF ANY NUMBER OF YEARS.
Years.
5
Per Cent.
6
Per Cent.
7
Per Cent.
8
Per Cent. 1
9
Per Cent.
10
Per Cent.
Years,
1
1.000000
1.000000
1.000000
1.000000
1.000000
1.000000
1
2
2.050000
2.060000
2.070000
2.080000
2.090000
2.100000
2
3
3.152500
3.183600
3.214900
3.246400
3.278100
3.310000
3
4
4.310125
4.374616
4.439943
4.506112
4.573129
4.641000
4
5
5.525G31
5.637093
5.750739
5.866601
5.984711
6.105100
5
6
G.801913
6.975319
7.153291
7.335929
7.523335
7.715610
6
7
8.142008
8.393838
8.654021
8.922803
9.200435
9.487171
7
8
9.549109
9.897468
10.259803
10.636628
11.028474
11.435888
8
9
11.026564
11.491316
11.977989
12.487558
13.021036
13.579477
9
10
12.577893
13.180795
13.816448
14.486562
15.192930
15.937425
10
11
14.206787
14.971643
15.783599
16.645487
17.560293
18.531167
11
12
15.917127
16.869941
17.888451
18.977126
20.140720
21.384284
12
13
17.712983
18.882138
20.140643
21.495297
22.953385
24.522712
13
14
19.598632
21.015066
22.550488
24.214920
26.019189
27.974983
14
15
21.578564
23.275970
25.129022
27.152114
29.360916
31.772482
15
16
23.657492
25.672528
27.888054
30.324283
33.003399
35.949730
16
17
25.840366
28.212880
30.840217
33.750226
36.973705
40.544703
17
18
28.132385
30.905653
33.999033
37.450244
41.301338
45.599173
18
19
30.539004
33.759992
37.378965
41.446263
46.018458
51.159090
19
20
33.065954
36.785591
40.995492
45.761964
51.160120
57.274999
20
21
35.719252
39.992727
44.865177
50.422921
56.764530
64.002499
21
22
38.505214
43.392290
49.005739
55.456755
62.873338
71.402749
22
23
41.430475
46.995828
53.436141
60.893296
69.531939
79.543024
23
24
44.501999
50.815577
58.176671
66.764759
76.789813
88.497327
24
25
47.727099
54.864512
63.249038
73.105940
84.700896
98.347059
25
26
51.113454
59.156383
68.676470
79.954415
93.323977
109.181765
26
27
54.669126
63.705766
74.483823
87.350768
102.723135
121.099942
27
28
58.402583
68.528112
80.697691
95.338830
112,968217
134.209936
28
29
62.322712
73.639798
87.346529
103.965936
124.135356
148.630930
29
30
66.438848
79.058186
94.460786
113.283211
136.307539
164.494023
30
31
70.760790
84.801677
102.073041
123.345868
149.575217
181.943425
31
32
75.298829
90.889778
110.218154
134.213537
164.036987
201.137767
32
33
80.063771
97.343165
118.933425
145.950620
179.800315
222.251544
33
34
85.066959
104.183755
128.258765
158.626670
196.982344
245.476699
34
35
90.320307
111.434780
138.236878
172.316804
215.710755
271.024368
35
36
95.836323
119.120867
148.913460
187.102148
236.124723
299.126805
36
37
101.628139
127.268119
160.337402
203.070320
258.375948
330.039486
37
38
107.709546
135.904206
172.561020
220.315945
282.629783
364.043434
38
39
114.095023
145.058458
185.640292
238.941221
309.066463
401.447778
39
40
120.799774
154.761966
199.635112
259.056519
337.882445
442.592556
40
41
127.839763
165.047684
214.609570
280.781040
369.291865
487.851811
41
42
135.231751
175.950545
230.632240
304.243523
403.528133
537.636992
42
43
142.993339
187.507577
247.776496
329.583005
440.845665
592.400692
43
44
151.143006
199.758032
266.120851
356.949646
481.521775
652.640761
44
45
159.700156
212.743514
285.749311
386.505617
525.858734
718.904837
45
46
168.685164
226.508125
306.751763
418.426067
574.186021
791.795321
46
47
178.119422
241.098612
329.224386
452.900152
626.862762
871.974853
47
48
188.025393
256.564529
353.270093
490.132164
684.280411
960.172338
48
49
198.426663
272.958401
378.999000
530.342737
746.865648
1057.189572
49
50
209.347996
290.335905
406.528929
573.770156
815.083556
1163.908529
50 218
T A BL E XXII.
THE AMOUNT OF $1 PER ANNUM AT THE END OF ANY NUMBER OF YEARS.
Years.
2
Per Cent.
21
Per Cent.
3
Per Cent.
31
Per Cent.
4
Per Cent.
41
Per Cent.
Years.
51
87.270989
100.921458
117.180773
136.582837
159.773767
187.535665
51
52
90.016409
104.444494
121.696197
142.363236
167.164718
196.974769
52
53
92.816737
108.055606
126.347082
148.345950
174.851306
206.838634
53
54
95.673072
1111.756996
131.137495
154.538058
182.845359
217.146373
54
55
98.586534
115.550921
136.071620
160.946890
191.159173
227.917959
55
56
101.558264
119.439694
141.153768
167.580031
199.805540
239.174268
56
57
104.589430
123.425687
146.388381
174.445332
208.797762
250.937110
57
58
107.681218
127.511329
151.780033
181.550919
218.149672
263.229279
58
59
110.834843
131.699112
157.333434
188.905201
227.875659
276.074591
59
60
114.051539
135.991590
163.053437
196.516883
237.990685
289.497954
60
61
117.332570
140.391380
168.945040
204.394974
248.510313
303.525362
61
62
120.679222
144.901164
175.013391
212.548798
259.450725
318.184003
62
63
124.092806
149.523693
181.263793
220.988006
270.828754
333.502283
63
64
127.574662
154.261786
187.701707
229.722586
282.661904
349.509886
64
65
131.126155
159.118330
194.332758
238.762876
294.968380
366.237831
65
66
134.748679
164.096289
201.162741
248.119577
307.767116
383.718533
66
67
138.443652
169.198696
208.197623
257.803762
321.077800
401.985867
67
68
142.212525
174.428663
215.443551
267.826894
334.920912
421.075231
68
69
146.056776
179.789380
222.906858
278.200835
349.317749
441.023617
69
70
149.977911
185.284114
230.594064
288.937865
364.290459
461.869680
70
71
153.977469
190.916217
238.511886
300.050690
379.862077
483.653815
71
72
158.057019
196.689123
246.667242
311.552464
396.056560
506.418237
72
73
162.218159
202.606351
255.067259
323.456800
412.898823
530.207057
73
74
166.462522
208.671509
263.719277
335.777788
430.414776
555.066375
74
75
170.791773
214.888297
272.630856
348.530011
448.631367
581.044362
75
76
175.207608
221.260504
281.809781
361.728561
467.576621
608.191358
76
| 77
179.711760
227.792017
291.264075
375.389061
487.279686
636.559969
77
| 78
184.305996
234.486818
301.001997
389.527678
507.770873
666.205168
78
79
188.992115
241.348988
311.032057
404.161147
529.081708
697.184400
79
80
193.771958
248.382713
321.363019
419.306787
551.244977
729.557699
80
81
198.647397
255.592280
332.003909
434.982524
574.294776
763.387795
81
82
203.620345
262.982087
342.964026
451.206913
598.266567
798.740246
82
83
208.692752
270.556640
354.252947
467.999155
623.197230
835.683557
83
84
213.866607
278.320556
365.880536
485.379125
649.125119
874.289317
84
85
219.143939
286.278570
377.856952
503.367394
676.090123
914.632336
85
86
224.526818
294.435534
390.192660
521.985253
704.133728
956.790791
86
87
230.017354
302.796422
402.898440
541.254737
733.299078
1000.846377
87
88
235.617701
311.366333
415.985393
561.198653
763.631041
1046.884464
88
89
241.330055
320.150491
429.464955
581.840606
795.176282
1094.994265
89
90
247.156656
329.154253
443.348904
603.205027
827.983334
1145.269007
90
91
253.099789
338.383110
457.649371
625.317203
862.102667
1197.806112
91
92
259.161785
347.842687
472.378852
648.203305
897.586774
1252.707387
92
93
265.345021
357.538755
487.550217
671.890421
934.490244
1310.079219
93
94
271.651921
367.477223
503.176724
696.406585
972.869854
1370.032784
94
95
278.084960
377.664154
519.272026
721.780816
1012.784648'
L432.684259
95
96
284.646659
388.105758
535.850186
748.043144
1054.296034
L498.155051
96
97
291.339592
398.808402
552.925692
775.224654
1097.467876
L566.572028
97
98
298.166384
409.778612
570.513463
803.357517
1142.36659111638.067770
98
99
305.129712
421.023077
588.628867
832.475030
1189.0612541712.780819
99
100
312.232306
432.548654
607.287733
862.611657
1237.623705 1790.855956
100 TABLE XXII.
219
THE AMOUNT OF $1 PER ANNUM AT THE END OF ANY NUMBER OF YEARS.
Years.
5
Per Cent.
6
Per Cent.
7
Per Cent.
8
Per Cent.
10
Per Cent.
Years.
51
220.815396
308.756059
435.985955
620.671769
1281.299382
51
52
232.856165
328.281422
467.504971
671.325510
1410.429320
52
53
245.498974
348.978308
501.230319
726.031551
1552.472252
53
54
258.773922
370.917006
537.316442
785.114075
1708.719477
54
55
272.712618
394.172027
575.928593
848.923201
1880.591425
55
56
287.348249
418.822348
617.243594
1 917.837058
2069.650567
56
57
302.715662
444.951689
661.450646
992.264022
2277.615624
57
58
318.851445
472.648790
708.752191
1072.645144
2506.377186
58 |
59
335.794017
502.007718
759.364844
1159.456755
2758.014905
59 1
60
353.583718
533.128181
813.520383
1253.213296
3034.816395
60
61
372.262904
566.115872
871.466810
1354.470360
3339.298035
61
62
391.876049
601.082824
933.469487
1463.827988
3674.227838
62
63
412.469851
638.147793
999.812351
1581.934227
4042.650622
63
64
434.093344
677.436661
1070.799216
1709.488966
4447.915685
64
65
456.798011
719.082861
1146.755161
1847.248083
4893.707253
65
66
480.637912
763.227832
1228.028022
1996.027929
5384.077978
66
67
505.669807
810.021502
1314.989983
2156.710164
5923.485776
67 j
68
531.953298
859.622792
1408.039282
2330.246977
6516.834354
68
69
559.550963
912.200160
1507.602032
2517.666735
7169.517789
69
70
588.528511
967.932170
1614.134174
2720.080074
7887.469568
70
71
618.954936
.1027.008100
1728.123566
2938.686480
8677.216525
71
72
650.902683
1089.628586
1850.092216
3174.781398
9545.938177
72 i
73
684.447817
1156.006301
1980.598671
3429.763910
10501.531995
73 1
74
719.670208
1226.366679
2120.240578
3705.145023
11552.685195
74
75
756.653719
1300.948680
2269.657419
4002.556624
12708.953714
75
76
795.486404
1380.005601
2429.533438
4323.761154
13980.849085
76 !
77
836.260725
1463.805937
2600.600779
4670.662047
15379.933994
77
78
879.073761
1552.634293
2783.642833
5045.315011
16918.927393
78
79
924.027449
1646.792350
2979.497831
5449.940211
18611.820133
79
80
971.228821
1746.599891
3189.062680
5886.935428
20474.002146
80
81
1020.790262
1852.395885
3413.297067
6358.890263
22522.402360
81
82
1072.829776
1964.539638
3653.227862
6868.601484
24775.642596
82
83
1127.471264
2083.412016
3909.953812
7419.089602
27254.206856
83
84
1184.844828
2209.416737
4184.650579
8013.616770
29980.627542
84
85
1245.087069
2342.981741
4478.576120
8655.706112
32979.690296
85
86
1308.341422
2484.560646
4793.076448
9349.162601
36278.659326
86
87
1374.758493
2634.634285
5129.591799
10098.095609
39907.525258
87
88
1444.496418
2793.712342
5489.663225
10906.943258
43899.277784
88
89
1517.721239
2962.335082
5874.939651
' 11780.498718
48290.205562
89
90
1594.607301
3141.075187
6287.185427
12723.938616
53120.226119
90
91
1675.337666
3330.539698
6728.288407
13742.853705
58433.248730
91
92
1760.104549
3531.372080
7200.268595
14843.282002
64277.573603
92
93
1849.109777
3744 254405
7705.287397
16031.744562
70706.330964
93
94
1942.565266
3969.909669
8245.657515
17315.284127
77777.964060
94
95
2040.693529
4209.104250
8823.853541
18701.506857
85556.760466
95
96
2143.728205
4462.650505
9442.523288
20198.627405
94113.436513
96
97
2251.914616
4731.409535
10104.499919
21815.517598
103525.780164
97
98
2365.510346
5016.294107
10812.814913
23561.759006
113879.358180
98
99
2484.785864
5318.271753
11570.711957
25447.699726
125268.293998
99
100
2610.025157
5638.368059
12381.661794
27484.515704
137796.123398
100 220
TABLE XXIII.
THE PRESENT VALUE OF $1 PER ANNUM FOR ANY NUMBER OF YEARS.
00
H
c3
d>
>H
2
Per Cent.
2i
Per Cent.
3
Per Cent.
31
Per Cent.
4
Per Cent.
41
Per Cent.
Years.
1
.980392
.975610
.970874
.966184
.961538
.956938
1
2
1.941561
1.927424
1.913470
1.899694
1.886095
1.872668
2
3
2.883883
2.856024
2.828611
2.801637
2.775091
2.748964
3
4
3.807729
3.761974
3.717098
3.673079
3.629895
3.587526
4
5
4.713460
4.645828
4.579707
4.515052
4.451822
4.389977
5
6
5.601431
5.508125
5.417191
5.328553
5.242137
5.157872
6
7
6.471991
6.349391
6.230283
6.114544
6.002055
5.892701
7
8
7.325481
7.170137
7.019692
6.873956
6.732745
6.595886
8
9
8.162237
7.970866
7.786109
7.607687
7.435332
7.268790
9
10
8.982585
8.752064
8.530203
8.316605
8.110896
7.912718
10
11
9.786848
9.514209
9.252624
9.001551
8.760477
8.528917
11
12
10.575341
10.257765
9.954004
9.663334
9.385074
9.118581
12
13
11.348374
10.983185
10.634955
10.302738
9.985648
9.682852
13
14
12.106249
11.690912
11.296073
10.920520
10.563123
10.222825
14
15
12.849264
12.381378
11.937935
11.517411
11.118387
10.739546
15
16
13.577709
13.055003
12.561102
12.094117
11.652296
11.234015
16
17
14.291872
13.712198
13.166118
12.651321
12.165669
11.707191
17
18
14.992031
14.353364
13.753513
13.189682
12.659297
12,159992
18
19
15.678462
14.978891
14.323799
13.709837
13.133939
12.593294
19
20
16.351433
15.589162
14.877475
14.212403
13.590326
13.007936
20
21
17.011209
16.184549
15.415024
14.697974
14.029160
13.404724
21
22
17.658048
16.765413
15.936917
15.167125
14.451115
13.784425
22
23
18.292204
17.332111
16.443608
15.620410
14.856842
14.147775
23
24
18.913926
17.884986
16.935542
16.058368
15.246963
14.495478
24
25
19.523456
18.424376
17.413148
16.481515
15.622080
14.828209
25
26
20.121036
18.950611
17.876842
16.890352
15.982769
15.146611
26
27
20.706898
19.464011
18.327031
17.285365
16.329586
15.453303
27
28
21.281272
19.964889
18.764108
17.667019
16.663063
15.742874
28
29
21.844385
20.453550
19.188455
18.035767
16.983715
16.021889
29
30
22.396456
20.930293
19.600441
18.392045
17.292033
16.288889
30
31
22.937702
21.395407
20.000428
18.736276
17.588494
16.544391
31
32
23.468335
21.849178
20.388766
19.068865
17.873552
16.788891
32
33
23.988564
22.291881
20.765792
19.390208
18.147646
17.022862
33
34
24.498592
22.723786
21.131837
19.700684
18.411198
17.246758
34
35
24.998619
23.145157
21.487220
20.000661
18.664613
17.461012
35
36
25.488842
23.556251
21.832252
20.290494
18.908282
17.666041
36
37
25.969453
23.957318
22.167235
20.570525
19.142579
17.862240
37
38
26.440641
24.348603
22.492462
20.841087
19.367864
18.049990
38
39
26.902589
24.730344
22.808215
21.102500
19.584485
18.229656
39
40
27.355479
25.102775
23.114772
21.355072
19.792774
18.401584
40
41
27.799489
25.466122
23.412400
21.599104
19.993052
38.566109
41
42
28.234794
25.820607
23.701359
21.834883
20.185627
18.723550
42
43
28.661562
26.166446
23.981902
22.062689
20.370795
18.874210
43
44
29.079963
26.503849
24.254274
22.282791
20.548841
19.018383
44
45
29.490160
26.833024
24.518713
22.495450
20.720040
19.156347
45
46
29.892314
27.154170
24.775449
22.700918
20.884654
19.288371
46
47
30.286582
27.467483
25.024708
22.899438
21.042936
19.414709
47
48
30.673120
27.773154
25.266707
23.091244
21.195131
19.535607
48
49
31.052078
28.071369
25.501657
23.276564
21.341472
19.651298
49
50
31.423606
28.362312
25.729764
23.455618
21.482185
19.762008
50
u 1 — (l+l)-» 1 — Vn
Formula, Present V alue = ——A—-— = —:—•
i i TA BL, E XXIII.
221
THE PRESENT VALUE OF $1 PER ANNUM FOR ANY NUMBER OF YEARS
Years.
5
Per Cent.
6
Per Cent.
7
Per Cent.
8
Per Cent.
9
Per Cent.
10
Per Cent.
Years.
1
.952381
.943396
.934579
.925926
.917431
.909091
1
2
1.859410
1.833393
1.808018
1.783265
1.759111
1.735537
2
3
2.723248
2.673012
2.624316
2.577097
2.531295
2,486852
3
4
3.545951
3.465106
3.387211
3.312127
3.239720
3.169865
4
5
4.329477
4.212364
4.100197
3.992710
3.889651
3.790787
5
6
5.075G92
4.917324
4.766540
4.622880
4.485919
4.355261
6
7
5.786373
5.582381
5.389289
5.206370
5.032953
4.868419
7
8
6.463213
6.209794
5.971299
5.746639
5.534819
5.334926
8
9
7.107822
6.801692
6.515232
6.246888
5.995247
5.759024
9
10
7.721735
7.360087
7.023582
6.710081
6.417658
6.144567
10
11
8.306414
7.880875
7.498674
7.138964
6.805191
6.495061
11
12
8 863252
8.383844
7.942686
7.536078
7.160725
6.813692
12
13
9.393573
8 852683
8.357651
7.903776
7.486904
7.103356
13
14
9.898641
9.294984
8.745468
8.244237
7.786150
7.366687
14
15
10.379658
9.712249
9.107914
8.559479
8.060688
7.606080
15
16
10.837770
10.105895
9.446649
8.851369
8.312558
7.823709
16
17
11.274066
10.477260
9.763223
9.121638
8.543631
8.021553
17
18
11.689587
10.827603
10.059087
9.371887
8.755625
8.201412
18
19
12.085321
11.158116
10.335595
9.603599
8.950115
8.364920
19
20
12.462210
11.469921
10.594014
9.818147
9.128546
8.513564
20
21
12.821153
11.764077
10.835527
10.016803
9.292244
8.648694
21
22
13.163003
12.041582
11.061241
10.200744
9.442425
8.771540
22
23
13.488574
12.303379
11.272187
10.371059
9.580207
8.883218
23
24
13.798642
12.550358
11.469334
10.528758
9.706612
8.984744
24
25
14.093945
12.783350
11.653583
10.674776
9.822580
9.077040
25
26
14.375185
13.003166
11.825779
10.809978
9.928972
9.160945
26
27
14.643034
13.210534
11.986709
10.935165
10.026580
9.237223
27
28
14.898127
13.406164
12.137111
11.051078
10.116128
9.306567
28
29
15.141074
13.590721
12.277674
11.158406
10.198283
9.369606
29
30
15.372451
13.764831
12.409041
11.257783
10.273654
9.426914
30
31
15.592811
13.929086
12.531814
11.349799
10.342802
9.479013
31
32
15.802677
14.084043
12.646555
11.434999
10.400240
9.52G37G
32
33
16.002549
14.230230
12.753790
11.513888
10.464441
9.569432
33
34
16.192904
14.368141
12.854009
11.586934
10.517835
9.608575
34
35
16.374194
14.498246
12.947672 |
11.654508
10.566821
9.644159
35
36
16.546852
14.620987
13.035208
11.717193
10.611763
9.676508
36
37
16.711287
14.736780
13.117017
11.775179
10.652993
9.705917
37
38
16.867893
14.840019
13.193473
11.828869
10.690820
9.732651
38
39
17.017041
14.949075
13.264928
11.878582
10.725523
9.756956
39
40
17.159086
15.046297
13.331709
11.924613
10.757360
9.779051
40
41
17.294368
15.138016
13.394120
11.967235
10.786569
9.799137
41
42
17.423208
15.224543
13.452449
12.006699
10.813366
9.817397
42
43
17.545912
15.306173
13.506962 !
12.043240
10.837951
9.833998
43
44
17.662773
15.383182
13.557908
12.077074
10.860505
9.849089
44
45
17.774070
15.455832
13.605522
12.108402
10.881197
9.862808
45
46
17.880067
15.524370
13.650020
12.137409
10.900181
9.875280
46
47
17.981016
15.589028
13.691608
12.164267
10.917597
9.886618
47
48
18.077158
15.650027
13.730474
12.189136
10.933575
9.896926
48
49
18.168722
15.707572
13.766799
12.212163
10.948234
9.906296
49
50
18.255925
15.761861
13.800746
12 233485
10.961683
9.914814
50 TABLE XXIII.
222
THE PRESENT VALUE OF $1 PER ANNUM FOR ANY NUMBER OF YEARS.
Years.
2
Per Cent.
21
Per Cent.
3
Per Cent.
31-
Per Cent.
4
Per Cent.
41
Per Cent.
Years.
1
51
31.787849
28.646158
25.951227
23.628616
21.617485
19.867950
51
52
32.144950
28.923081
26.166240
23.795765
21.747582
19.969330
52
53
32.495049
29.193249
26.374990
23.957260
21.872675
20.066345
53
54
32.838283
29.456829
26.577660
24.113295
21.992957
20.159181
54
55
33.174788
29.713979
26.774428
24.264053
22.108612
20.248021
55
56
33.504694
29.964858
26.965464
24.409713
22.219819
20.333034
56
57
33.828131
30.209617
27.150936
24.550448
22.326749
20.414387
57
58
34.145226
30.448407
27.331005
24.686423
22.429567
20.492236
58
59
34.456104
30.681373
27.505831
24.817800
22.528430
20.566733
59
60
34.760887
30.908656
27.675564
24.944734
22.623490
20.638022
60
61
35.059693
31.130397
27.840353
25.067376
22.714894
20.706241
61
62
35.352640
31.346728
28.000343
25.185870
22.802783
20.771523
62
63
35.639843
31.557784
28.155673
25.300358
22.887291
20.833993
63
64
35.921415
31.763691
28.306478
25.410974
22.968549
20.893773
64
65
36.197466
31.964577
28.452891
25.517849
23.046682
20.950979
65
66
36.468103
32.160563
28.595040
25.621110
23.121810
21.005722
66
67
36.733435
32.351769
28.733049
25.720880
23.194048
21.058107
67
68
36.993564
32.538311
28.867038
25.817275
23.263507
21.108236
68
69
37.248592
32.720303
28.997124
25.910411
23.330296
21.156207
69
70
37.498619
32.897857
29.123421
26.000397
23.394515
21.202112
70
71
37.743744
33.071080
29.246040
26.087340
23.456264
21.246040
71
72
37.984063
33.240078
29.365087
26.171343
23.515639
21.288077
72
73
38.219670
33.404954
29.480667
26.252505
23.572730
21.328303
73
74
38.450657
33.565809
29.592881
26.330923
23.627625
21.366797
74
75
38.677114
33.722740
29.701826
26.406689
23.680408
21.403634
75
76
38.899132
33.875844
29.807598
26.479892
23.731162
21.438884
76
77
39.116796
34.025214
29.910290
26.550621
23.779963
21.472616
77
78
39.330192
34.170940
30.009990
26.618957
23.826888
21.504896
78
79
39.539404
34.313113
30.106786
26.684983
23.872008
21.535785
79
80
39.744514
34.451817
30.200763
26.748776
23.915392
21.565345
80
81
39.945602
34.587139
30.292003
26.810411
23.957108
21.593631
81
82
40.142747
34.719160
30.380586
26.869963
23.997219
21.620700
82
83
40.336026
34.847961
30.466588
26.927500
24.035787
21.646603
83
84
40.525516
34.973620
30.550086
26.983092
24.072872
21.671390
84
85
40.711290
35.096215
30.631151
27.036804
24.108531
21.695110
85
86
40.893422
35.215819
30.709855
27.088699
24.142818
21.717809
86
87
41.071982
35.332507
30.786267
27.138840
24.175787
21.739530
87
88
41.247041
35.446348
30.860454
27.187285
24.207487
21.760316
88
89
41.418668
35.557413
30.932479
27.234092
24.237969
21.780207.
89
90
41.586929
35.665768
31.002407
27.279316
24.267278
21.799241
90
91
41.751891
35.771481
31.070298
27.323010
24.295459
21.817455
91
92
41.913619
35.874 16
31.136212
27.365227
24.322557
21.834885
92
93
42.072175
35.975235
31.200206
27.406017
24.348612
21.851565
93
94
42.227623
36.073400
31.262336
27.445427
24.373666
21.867526
94
95
42.380023
36.169171
31.322656
27.483504
24.397756
21.882800
95
96
42.529434
36.262606
31.381219
27.520294
24.420919
21.897417
96
97
42.675916
36.353762
31.438077
27.555839
24.443191
21.911403
97
98
42.819525
36.442694
31.493279
27.590183
24.464607
21.924788
98
99
42.960319
36.529458
31.546872
27.623365
24.485199
21.937596
99
3.00
43.098352
36.614105
31.598905
27.655425
24.504999
21.949853
100
Perp.
50.000000
40.000000
33.333333
28.571429
25.000000
22.222222
Perp. TABLE XXIII.
223
THE PRESENT VALUE OF $1 PER ANNUM FOR ANY NUMBER OF YEARS.
Years.
5
Per Cent.
6
Per Cent.
7
Per Cent.
8
Per Cent.
9
Per Cent.
10
Per Cent,
Years.
51
18.338977
15.813076
13.832473
12.253227
10.974021
9.922559
51
52
18.418073
15.861393
13.862124
12.271506
10.985340
9.929599
52
| 53
18.493403
15.906974
13.889836
12.288432
10.995725
9.935999
53
| 54
18.565146
15.949976
13.915735
12.304103
11.005252
9.941817
54
1 55
18.633472
15.990543
13.939939
12.318614
11.013993
9.947107
55
56
18.698545
16.028814
13.962560
12.332050
11.022012
9.951915
56
57
18.760519
16.064919
13.983701
12.344491
11.029369
9.956286
57
58
18.819542
16.098980
14.003459
12.356010
11.036118
9.960260
58
59
18.875754
16.131113
14.021924
12.366676
11.042310
9.963873
59
60
18.929290
16.161428
14.039181
12.376552
11.047991
9.967157
60
61
18.980276
16.190026
14.055309
12.385696
11.053203
9.970143
61
62
19.028834
16.217006
14.070383
12.394163
11.057984
9.972857
62
63
19.075080
16.242458
14.084470
12.402003
11.062371
9.975325
63
64
19.119124
16.266470
14.097635
12.409262
11.066395
9.977568
64
65
19.161070
16.289123
14.109940
12.415983
11.070087
9.979607
65
66
19.201019
16.310493
14.121439
12.422207
11.073475
9.981461
66
67
19.239066
16.330654
14.132186
12.427969
11.076582
9.983147
67
68
19.275301
16.349673
14.142230
12.433305
11.079433
9.984679
68
69
19.309810
16.367617
14.151617
12.438245
11.082049
9.986071
69
70
19.342677
16.384544
14.160389
12.442820
11.084449
9.987338
70
71
19.373978
16.400513
14.168588
12.447055
11.086650
9.988489
71
72
19.403788
16.415578
14.176251
12.450977
11.088670
9.989535
72
73
19.432179
16.429791
14.183412
12.454608
11.090523
9.990487
73
74
19.459218
16.443199
14.190104
12.457971
11.092223
9.991351
74
75
19.484970
16.455848
14.196359
12.461084
11.093782
9.992138
75
76
19.509495
16.467781
14.202205
12.463967
11.095213
9.992852
76
77
19.532853
16.479039
14.207668
12.466636
11.096526
9.993502
77
78
19.555098
16.489659
14.212774
12.469107
11.097730
9.994093
78
79
19.576284
16.499679
14.217546
12.471396
11.098835
9.994630
79
80
19.596460
16.509131
14.222005
12.473514
11.099849
9.995118
80
81
19.615677
16.518048
14.226173
12.475476
11.100778
9.995562
81
82
19.633978
16.526460
14.230069
12.477293
11.101632
9.995965
82
83
19.651407
16.534396
14.233709
12.478975
11.102414
9.996332
83
84
19.668007
16.541883
14.237111
12.480532
11.103132
9.996666
84
85
19.683816-
16.548947
14.240291
12.481974
11.103791
9.996969
85
86
19.698873
16.555610
14.243262
12,483310
11.104396
9.997244
86
87
19.713212
16.561896
14.246040
12.484546
11.104950
9.997495
87
88
19.726869
16.567827
14.248635
12.485691
11.105459
9.997723
88
89
19.739875
16.573421
14.251061
12.486751
11.105926
9.997930
89
90
19.752262
16.578699
14.253328
12.487732
11.106354
9.998118
90
91
19.764059
16.583679
14.255447
12.488641
11.106746
9.998289
91
92
19.775294
16.588376
14.257427
12.489482
11.107107
9.998444
92
93
19.785994
16.592808
14.259277
12.490261
11.107438
9.998586
93
94
19.796185
16.596988
14.261007
12.490983
11.107741
9.998714
94
95
19.805891
16.600932
14.262623
12.491651
11.108019
9.998831
95
96
19.815134
16.604653
14.264134
12.492269
11.108274
9.998938
96
97
19.823937
16.608163
14.265546
12.492842
11.108509
9.999034
97
98
19.832321
16.611475
14.266865
12.493372
11.108724
9.999122
98
99
19.840306
16.614599
14.268098
12.493863
11.108921
9.999202
99
100
19.847910
16.617546
14.269251
12.494318
11.109102
9.999274
100
Perp.
20.000000
16.666667
14.285714
12.500000
11.111111
10.000000
Perp. 224
T A B L E XXI V.
Years.
2
Per Cent.
21-
Per Cent.
8
Per Cent.
34
Per Cent.
4
Per Cent.
Per Cent.
» !
8
CD
r*
1
1.02000000
1.02500000
1.03000000
1.03500000
1.04000000
1.04500000
1
2
0.51504950
0.51882716
0.52261084
0.52640049
0.53019608
0.53399756
2
3
.34675467
.35013717
.35353036
.35693418
.36034854
.36377336
3
4
.26262375
.26581788
.26902705
.27225114
.27549005
.27874365
4
5
.21215839
.21524686
.21835457
.22148137
.22462711
.22779164
5
6
0.17852581
0.18154997
0.18459750
0.18766821
0.19076190
0.19387839
6
7
.15451195
.15749543
.16050035
.16354449
.16660961
.10970147
7
8
J3650980
.13946735
.14245639
.14547665
.14852783
.15160965
8
9
.12251544
.12545689
.12843386
.13144601
.13449299
.13757447
9
10
.11132653
.11425876
.11723051
.12024137
.12329094
.12637882
10
11
0.10217794
0.10510596
0.10807745
0.11109197
0.11414904
0.11724818
11
12
.09455960
.09748713
•10046209
.10348395
.10655217
.10966619
12
13
.08811835
.09104827
.09402954
.09706157
.10014373
.10327535
13
14
.08260197
.08553653
.08852634
.09157073
.09460897
.09782032
14
15
.07782547
.08076646
.08376658
.08682507
.08994110
.09311381
15
16
0.07365013
0.07659899
0.07901085
0.08268483
0.08582000
0.08901537
16
17
.06996984
.07292777
.07595253
.07904313
.08219852
.08541758
17
18
.06670210
.06967008
.07270870
.07581684
.07899333
.08223690
18
19
.06378177
.06676062
.06981388
.07294033
.07613862
.07940734
19
20
.06115672
.06414713
.06721571
.07036108
.07358175
.07687614
20
21
0.05878477
0.06178733
0.06487178
0.06803659
0.07128011
0.07460057
21
22
.05663140
.05964660
.06274739
.06593207
.06919881
.07254565
22
23
.05466810
.05769638
.06081390
.06401880
.06730906
.07068249
23
24
.05287110
.05591282
.05904742
.06227283
.00558683
.06898703
24
25
.05122044
.05427592
.05742787
.06067404
.06401196
.06743903
25
26
0.04969923
0.05276875
0.05593829
0.05920540
0.06256738
0.06602137
26
27
.04829309
.05137687
.05456421
.05785241
.06123854
.06471949
27
28
.04698967
.05008793
.05329323
.05660265
.06001298
.06352081
28
29
.04577835
.04889127
.05211467
.05544538
.05887993
.06241461
29
30
.04464992
.04777764
.05101926
.05437133
.05783010
.06139154
30
31
0.04359635
0.04673900
0.04999893
0.05337240
0.05085535
0.06044345
31
32
.04261061
.04576831
.04904662
.05244150
.05594859
.05956320
32
33
.04168653
.04485938
.04815612
.05157242
.05510357
.05874453
33
34
.04081867
.04400675
.04732196
.05075966
.05431477
.05798191
34
35
.04000221
.04320558
.04653929
.04999835
.05357732
.05727045
35
36
0.03923285
0.04245158
0.04580379
0.04928416
0.05288688
0.05660578
36
37
.03850678
.04174090
.04511162
.04861325
.05223956
.05598402
37
38
.03782057
.04107012
.04445934
.04798214
.05163192
.05540169
38
39
.03717114
.04043615
.04384385
.04738775
.05106083
.05485567
39
40
.03655575
.03983623
.04326238
.04682728
.05052349
.05434315
40
41
0.03597188
0.03926786
0.04271241
0.04629822
0.05001738
0.05386158
41
42
.03541729
.03872876
.04219168
.04579828
.04954020
.05340868
42
43
.03488993
.03821688
.04169811 1
.04532539
.04908989
.05298235
43
44
.03438794
.03773037
.04122985
.04487768
.04806454
.05258071
44
45
.03390962
.03726751
.04078518
.04445343
.04826246
.05220202
45
46
0.03345342
0.03682676
0.04036254
0.04405108
0.04788205
0.05184471
46
47
.03301792
.03640669
.03996051
.04366919
.04752189
.05150734
47
48
.03260184
.03600599
.03957777
.04330646
.04718065
.05118858
48
49
.03220396
.03502348
.03921314
.04296167
.04685712
.05088722
49
50
.03182321
.03525806
.03886546
.04263371
.04655020
.05060215
50
THE ANNUITY WHICH $1 WILL PURCHASE FOR ANY NUMBER OF YEARS.
Formula. Annuity = 7^——rr— = —-
J 1 —(l + 0_w 1 — vn TABLE XXIV.
225
THE ANNUITY WHICH $1 WILL PURCHASE FOR ANY NUMBER OF YEARS.
tn
M
5
6
7
8
9
10
to
S
075896
.9611509
.7169836
.4750462
61
62
.6862635
.4310364
.1782058
.9277272
.6795571
.4336535
62
63
.6650742
.4057305
.1488220
.8943034
.6421306
.3922608
63
64
.6438849
.3804246
.1194382
.8608796
.6047041
.3508681
64
65
2.6226956
2.3551188
2.0900545
3.8274559
3.5672776
3.3094755
65
66
.6015063
.3298129
.0606707
.7940321
.5298511
.2680828
66
67
.5803170
.3045070
.0312869
.7606084
.4924246
.2266901
67
68
.5591277
.2792012
.0019031
.7271846
.4549981
'.1852974
68
69
.5379384
.2538953
3.9725193
.6937609
.4175716
.1439047
69
70
2.5167491
2.2285894
3.9431356
3.6603371
3.3801451
3.1025120
70
71
.4955598
.2032836
.9137518
.6269134
.3427186
.0611194
71
72
.4743705
.1779777
.8843680
.5934896
.3052921
.0197267
72
73
.4531812
.1526718
.8549842
.5600658
.2678657
4.9783340
73
74
.4319919
.1273660
.8256005
.5266421
.2304392
.9369413
74
75
2.4108026
2.1020601
3.7962167
3.4932183
3.1930127
4.8955486
75
76
.3896133
.0767542
.7668329
,4597946
.1555862
.8541559
76
77
.3684240
.0514484
.7374491
.4263708
.1181597
.8127632
77
78
.3472347
.0261425
.7080653
.3929471
.0807332
.7713706
78
79
.3260454
.0008366
.6786816
.3595233
.0433067
.7299779
79
80
2.3048561
3.9755308
3.6492978
3.3260996
3.0058802
4.6885852
80
81
.2836668
.9502249
.6199140
.2926758
4.9684537
.6471925
81
82
.2624775
.9249190
.5905302
.2592521
.9310272
.6057998
82
83
.2412882
.8996132
.5611465
.2258283
.8936007
.5644071
83
84
.2200989
.8743073
.5317627
.1924045
.8561742
.5230144
84
85
2.1989096
3.8490015
3.5023789
3.1589808
4.8187477
4.4816218
85
86
.1777203
.8236956
.4729951
.1255570
.7813212
.4402291
86
87
.1565310
.7983897
.4436113
.0921333
.7438947
.3988364
87
88
.1353417
.7730839
.4142276
.0587095
.7064682
.3574437
88
89
.1141524
.7477780
.3848438
.0252858
.6690417
.3160510
89
90
2.0929631
3.7224721
3.3554600
4.9918620
4.6316152
4.2746583
90
91
.0717738
.6971663
.3260762
.9584383
.5941887
.2332657
91
92
.0505845
.6718604
•2966925
.9250145
.5567622
.1918730
92
93
.0293952
.6465545
.2673087
.8915907
.5193357
.1504803
93
94
.0082059
.6212487
.2379249
.8581670
.4819092
.1090876
94
95
3.9870166
3.5959428
3.2085411
4.8247432
4.4444827
4.0676949
95
96
.9658273
.5706369
.1791573
.7913195
.4070562
.0263022
96
97
.9446380
.5453311
.1497736
.7578957
.3696297
5.9849095
97
98
.9234487
.5200252
.1203898
.7244720
.3322032
.9435169
98
99
.9022594
.4947193
.0910060
.6910482
.2947767
.9021242
99
100
.8810701
.4694135
.0616222
.6576245
.2573502
.8607315
100
50-59
T2.4518321
14.2083033
17.9858411
T9.7840532
21.6025585
23.4409866
50-59
60-69
14.3329025
17.6777170
19.0474634
22.4416777
25.8599085
27.30173 80 60-69
70-79
16.2139725
19.1471303
22.1090857
25.0993021
i28.1172592
31.1624496
70-79
80-89
18.0950425
22.6165439
25.1707079
29.7569267
32.3746095
35.0231810
80-89
90-99
21.9761125
24.0859573
28.2323301
32.4145511
36.6319595
40.8839126
90-99 PART SECOND.
LIFE ANNUITIES AND INSURANCES,
The science of Life Annuities and Insurances is founded upon the system of
Probabilities, or chiefly that portion of the system, which treats of “ mathematical
expectation.” Numerous plans and projects from other sources, have generally ended
in failure. And while “mathematical expectation” is admitted to govern the deter-
mination of premiums and reserves, the same principles, confirmed by long experience
in connection with “ moral expectation,” are essentially recognized in the management
of current risks and investments.
SECTION I.
First Principles of Probability for Reference.
For future reference, let us briefly state some of the axioms or first principles of
the system of Probabilities. These have been variously developed in the able treatises
of Professor De Morgan, especially the Essay on Probabilities in Lardner’s Cabinet
Cyclopaedia, Galloway’s article, Probability in the Encyclopaedia Britannica, also pub-
lished separately, Demoivre in 1756, Lubbock, Lacroix, Laplace, Poisson and others.
A simple apparatus will serve for the experimental trials and comparisons. And a
record of the results will illustrate the irregularities of small numbers and the
increasing regularity of large numbers of trials, as well as the probable limits of
fluctuation from a mean result.
1. Definition. According to the usual definition, the mathematical 'probability of a
contingent event is measured by a fraction, whose numerator is the number of equal ways
in which it can happen, and whose denominator is the number of all the equal ways
in which it can either happen or fail.
The different chances or ways of happening are supposed to be perfectly equal.
Otherwise they must he multiplied by numbers proportional to their respective facilities
or likelihoods. If one chance is twice as likely as another, it must be regarded as two
chances; and so generally.
An urn contains, for example, nine white balls, and one black ball. The chance
of drawing a white ball by a person blindfold, is consequently fa; and that of the
black ball fa.
Thus the chance of throwing heads in the toss of a symmetrical coin, is £ or 0.5,
called the “even chance,” And the probability of throwing the ace point with an 234
LIFE ANNUITIES AND INSURANCES.
ivory die, having six equal faces, is f Generally, if the chances of occurrence are to
those of failure, as a to b, the probabilities of happening and of failing in the next trial
will be, respectively,
a -. b
f, and —-z.
a+ b a + 6
2. Certainty. The sum of the last two fractions is evidently 1, the symbol of
certainty; since one event or the other must take place. The practical applications
of this principle are sometimes called “ hedging,” or “ compassing the main chance.”
Thus a person having effected a temporary insurance on his life for ten years for $5,000,
next buys a pure endowment of $5,000 for the same period. Or what is equivalent, he
effects an Endowment Insurance for $5,000 for ten years. The opposite chances of
living and dying evidently make up a certainty, and are both secured ; so that he must
receive $5,000 at the end of ten years, or sooner in the event of death.
8. The Opposite Probability. If p denote the probability of happening, and q
that of failing in the next trial; then as just stated, p-\- q = 1, and q = 1 — p. That
is, the probability p subtracted from unity, gives the probability of the opposite event.
Thus the probability of throwing the ace point being -jf; the probability of not
throwing the ace, but some other face of the die, is 1—or f-.
In the life table, let l denote the number of persons living at any given age; d the
number dying in the ensuing twelve months; p the probability of surviving; and q
that of dying, in the same year. As before described,
d , l-d
? = r p = i-v = —
J/.. Joint Occurrence. The probability of an event compounded of any number of
simple and independent events is equal to the product of their separate probabilities.
It will be useful to observe that this product will be the same, whether the simple
events are simultaneous or successive, or to occur in any one order. For example, if p,
p', p" denote the tabular probabilities of surviving from the age of 35 to 36, 36 to 37,
37 to 38; the probability of surviving from 35 to 38 will be equal to the product
p p p". In like manner, the chance of surviving from 35 to 37, and dying in the
next year, will be p p' (l—p ).
Since the product of two or more proper fractions is always less than either factor,
the uncertainty of a compound event increases with every new contingency, on which
it is made to depend. For a common illustration, the evidence of eye-witnesses is
preferred to hearsay reports successively transmitted.
5. Addition of Probabilities. When a simple event may happen in several ways
independently, the probability of the event is the sum of its separate probabilities.
Thus, the chance of throwing the ace point in one toss of an ivory die, being
the chance of throwing just one ace when two dice are thrown, is the sum of
f xi+ixf or the same case, for contrast, the chance of two aces by Prin-
ciple 4, is equal to } x } or ■gV- The wide difference between these two classes of
results, was strikingly illustrated in the military concentrations and combinations of
Napoleon Bonaparte. “Military science,” said he, “consists in calculating all the
chances accurately in the first place, and then in giving accident its place in one’s
calculations.”
6. Repetition of Trials. An urn contains 3 white and 2 black balls or 5 in all.
A ball is drawn successively and replaced in the urn after each drawing. The result LIFE ANNUITIES AND INSURANCES.
235
of two trials,—if W, B, denote the issue of a white or black ball respectively,—will be
represented thus:
Possible Events. WW, WB, BW, BB.
Probabilities. fxf, fxf, |x|, fxf.
(p + qf = p* + pq + qp + q\
And so generally, if p denote the probability of any contingent event E, and q
that of its opposite, so that p-\-q = 1; the possible combinations in m trials, will be
represented by the terms of the development of the binomial,
\
(V + (t)m = Vm + mpm~1q + —k—— 'p™~*q 2 +
/V
m(m — 1) (m—2).... (m —n + 1)
H—*—123 L —T—Lpm^qn+ +q™.
The first term expresses the probability that E will happen in every one of the m
trials. The whole second term expresses the probability that E will happen m — 1
times and fail once, in any order or without regard to order. Or omitting its coefficient
m, the remaining product by Principle 4, expresses the probability that E will occur
m — 1 times and fail once in any particidar or assigned order. And so on for the third
term, to the last.
In like manner, instead of two simple events, had there been three, whose proba-
bilities are p, q, r, so that + + ?• = 1; the probabilities in m trials would be shown
in the development of (p + q + r)™. And so, for any greater number of events.
The terms of the development of (p + q)m can be represented by the ordinates of a
curve, termed the probability curve. And the maximum term is proved to be that
which has mp and mq, or numbers nearest to these, for the exponents of the factors
p and q respectively. So in the case of three or more simple events, whose probabilities
are q, r .... the most probable combination in m trials is the maximum term of
(P + q + r+ )m, or
12 3 4: 7YI
—zr —4-' V k X PmP X Xr™ X • • . .
1.2,.. .mp x 1.2... . mq x 1.2.. .mr x ....
The probability curve, here described, when materially simplified for large numbers,
has been made the foundation of the “theory of errors of observations.” Also its mean
area has been tabulated for reference, by De Morgan and others.
7. Mathematical Expectation. The mathematical expectation of a contingent gain,
is the product of the sum at issue, by the probability of receiving it. Or, the sum
of the expectations of gain, diminished by the sum of the expectations of loss, gives the
total expectation. If the chance of winning $90 is f the expectation is 90 x or $15.
In the original application to games of hazard, the conditions of equal play not only
required the mathematical expectations to be equal, but also required the option of
continuing the game through a given series of trials. If p denotes the chance that A
will win the sum a from B, and q denotes the chance that B will win the sum b
from A, at each trial, then the first condition of equality requires that pa = qb; or
a:b::q:p. That is, if the number of trials is not limited, the stakes must be
inversely proportional to the probabilities of winning in each single trial.
In the historic problem proposed to Pascal: Two persons of equal skill (that is,
p = q = j;) sit down to play on condition that the one who first gains three games
shall win the stakes. The first has gained two games, and the second one game, when 236
LIFE ANNUITIES AND INSURANCES.
they agree to separate. What share is each entitled to take ? Not equal shares ; but
the stakes must be divided proportionally to the probabilities of winning the un-
finished games.
By Principle 4, the probability of the second player’s winning the next two games
is y. By Principle 5, the probability of the first player’s winning the next game is \;
that of losing this game and winning the second, total f. So that the share of the
second is and of the first, £; being proportional to their respective chances of finally
winning the stakes.
At the present day, these early problems are still valuable as miniature illustrations
of the correct principles, which govern in questions of Insurance reserves and the dis-
tribution of surplus. The danger in practice, generally lies, as was illustrated by
Demoivre, in ignoring just distinctions and liabilities, such as the effect of unexpired
times and conditions.
8. Moral Expectation. In the solution of problems relating to games of chance,
it was noted at an early period, that in many cases, the numeric expectation was an
insufficient guide, and other circumstances must be taken into account. Among these
various conditions, the amount of capital of the party interested, was generally the
most important. In the great majority of cases, the relation between the capital and
the risk, proposed by Daniel Bernoulli, has led to very satisfactory conclusions. This
principle more plainly stated, is, that when the absolute value of the capital increases
in geometrical progression, its relative advantages, termed the moral value, increase
not so fast, but in arithmetical progression. Or the relation in the first case, is that of
numbers, and in the second case, that of their logarithms.
Formula.—Let a denote the original capital. Let e denote the mathematical, and e' the moral
expectation of the contingent gains h, k, l, of which, the respective probabilities are p, q, r, ....
It is assamed that p + q + r+ .... = 1, so that one or other of the events will certainly happen. By
Principle 7, we have
e = ph + qk + rl+
To this, adding the identical equation a = (p + q-\-r+... .)a, we obtain under another form, the
capital and mathematical expectation
a + e = p{a + h) + q(a + k) + r(a + l) +
Changing from absolute to moral values, or from amounts to their logarithms, as above described,
and from e to e', we thus define the moral expectation :
log(a + e') = p log(a + h) + q log(a + k) + r log(a +1) +
Or passing from logarithms to numbers,
a + e' = {a + h)x.(a + k)\(a + l)r....
Since a + h = a{\ + -j, and p + q + r+ ... = 1, we find by developing either of the last equations
in series,
e' = ph + qk + rl+ .... —$(ph2 + qki+rl*+ ) +
The sum of the first powers in this series is identical with e as stated above ; so that the moral
expectation in general, is less than the mathematical.
When the moral expectation is thus expressed by an Algebraic formula, the chief
advantage appears in the facility with which the parts can be varied to bring up the
value of the expectation to its maximum. Thus from the last formula of the Note
above, it appears that when the risks of an Insurance Office are so numerously dis-
tributed, and each one is so small in comparison with the capital, that the higher powers
A2, A*2, .... may be neglected in comparison with the first, the moral expectation attains
its maximum, and the most favorable disposition of the chances is so far realized. LIFE ANNUITIES AND INSURANCES.
237
The general precept known in commerce under the name of Distribution of Risk,
immediately follows, in connection with that of the selection or rejection of risks.
From the problems relating to games of chance, the moral expectation appears to
be always less than the mathematical. Hence the obvious consequence that betting,
gambling and lotteries are attended with moral disadvantage, even when the chances
of gain or loss are perfectly equal. As Prof. De Morgan has remarked, “it does not
follow, that equal play means prudent play.”
Among other illustrative problems of this theory, is one for determining the
amount of capital which an individual should possess in order to be morally indifferent
whether he insures or not. When the amount of capital is beloAv this limit, its moral
value may be advanced by insurance, even when the premium exceeds the mathematical
value of the risk.
SECTION II.
The Life Table and Law of Mortality.
An important application of the system of Probability to give increased accuracy to
the ratios of mortality or loss, by final series, lias already been described on pages 35-38.
By this method, the annual ratios by Lives and by Amounts, for Male and for Female
Life were severally computed from the experience of the Thirty Offices. According to
Principle 3 of Probability, the ratios of loss or of mortality, denoted by q, when sub-
tracted from unity give 1—q or p, the annual probability of surviving one year. Let
p, p', p" ... denote the tabular probabilities, thus found from the Amounts, of sur-
viving from the age of 10 to 11, 11 to 12, 12 to 13, and so on. Then assuming
100,000 to enter at the precise age of ten years, by Principle 4 of the last Section, the
numbers surviving at the ages 10, 11, 12, 13, .... will be 100,000, 100,000 p, 100,000
p p', 100,000 pp' p", and so on to the oldest age, near 100 years. The results are based
on the fourth columns, pp. 159, 164. The standard Tables XXVI and LXXVII were
thus constructed from the ratios of Amounts Avith slight adjustment in a few instances
toward the extremes, from the ratios of Lives. The graduation of the table for Female
Life, on account of small numbers above the age of seventy, was aided by comparison
with other larger collections, and then adjusted by the method of Woolhouse as
described in the Journal of the Institute of Actuaries, Yol. 21, p. 45.
In the Table XXVI for Male Life, the values from age ten to ninety years, were
calculated by the well-known law of Makeham, taking the ungraduated values of
li0, lS5, /,o, as data. The unknown q is found first, by taking the differences of the
four equations once and again ; and then dividing the last two equations, one by the
other. The formulas and constants are as follows:
lx — k. sx.yQx, Xlx — Xk -)- xXs qx. Xg.
A—=— — Xpx — Xs -\-qx{q — 1) 4£/.
lx
Xh = 5.028244, A(— = 4.475241.
s = 0.9936957, = 0.9993122, 0 = 1.099713375.
Xs = 1.99725340, Xg = —0.000298773, Xq = 0.041279507.
Nap. log. 8 = —0.00632429, Nap. log. g = —0.000687950,
Nap. log. q — 0.095050036. 238
LIFE ANNUITIES AND INSURANCES.
The integer results computed from this formula, terminated with only 2 persons
living at the age of 101 years; and their close agreement with the ungraduated values
secured their entire adoption, with a slight change above the age 90, so that the Table
ends at the age of 99 years.
Here the inquiry arises, why should this Table of Male Life from the experience
of Thirty Offices, spontaneously conform to Makeham’s law of mortality, while previous
Tables showed very considerable deviations? In explanation, the following circum-
stances may be noted ; first, that the observations were far more numerous than any
previous collection of life insurance statistics, and comprised nearly a million of
entrants. Second, that the observations were nearly homogeneous, in respect to the
agencies, medical examinations, climate, nationality, education, and habits of living.
Third, that the observations were reduced with the greatest care by select clerks, and
the results always verified by two independent operations. Fourth, that they contain
the first application of the improvement in construction by final series, already men-
tioned. Fifth, that the observations are essentially free from the mis-statement of
ages adjacent to 30, 35, 40, 45, 50, .... years, a common source of error in population
returns.
The present collection also determines the fluctuating nature of climacterics, or
periods of health and sickness, at the different ages of life. Physicians assert the
existence of periodic alternations, more or less regular, of health and disease, observed
in their own persons, and in their professional practice. It is like the surface of a
body of water, disturbed by the wind or other causes, and taking the form of waves,
having like periods of oscillation. And as the force of gravitation by insensible
degrees, causes the waves gradually to subside, so does the innate vital force tend, from
conditions of disease, to the original conditions of health. Detached portions of the
present statistics, will show these fluctuations; an increase of mortality at one age,
being followed by a decrease at a future age, and conversely. The manifestations prove
to be so different in particular individuals, that in the average of several hundred
thousands, the opposite fluctuations compensate each other, leaving in adult life, only
the mathematical law before stated.
Down to the present time, the statistics of life tables have been regarded in practice
merely as observations, with irregularities to be smoothed down and graduated like any
other set of observations.
For a particular account of the earlier life tables, reference can be made to the
original publications, or to Wolford*s Cyclopcedia, or to the Journal of the Institute of
Actuaries. In respect to the mathematical law of mortality, the celebrated Demoivre
first observed that the portion of the life table between the ages of 22 and 86 years,
Avas approximately represented by the simple formula, l = 8G—x; where l denotes the
number living at the age x. This appeared in the year 1727. Different formulas were
subsequently prepared or published by Lambert in 1776; by Babbage in 1823; by
Thomas Young in 1826; by Littrow in 1832 ; by Moser in 1839; and by Orchard
before 1856. But the chief interest centers, upon the formula of Gompertz, published
in the Philosophical Transactions of 1825; and upon Makeham’s modification of it,
which appeared in 1859 in the Assurance Magazine or Journal, Yol. 8, p. 301.
Gompertz’s formula is lx = Tc\.g\Q\x\ Makeham’s, Ix = Tc.g'F.sx.
Here l denotes the number living at the age x, and Tc, g, q, s are constants to be
derived from the observations. Now with reference to the usual expression of force LIFE ANNUITIES AND INSURANCES.
239
in the science of Mechanics, taking the Napierian logarithms of either formula, and
differentiating twice, we find the relative Force of Mortality
= = log-^)2- 2®*
That is, in the average of a great number of adult males, the relative force of
mortality, increases regularly with advancing age, in geometrical progression.
But the chief Aralue of this kiw, like that of the law of gravitation, is found in its
relations to analysis. Integrating the last formula, we obtain by adding the usual
constant, the Instantaneous Rate of Mortality, computed in Table XLIII:
d log. lx . .
= IfiT = -loS-S-^og.g.log.q.cf.
Vx= - = 1.187752 x A——f-1.114833 x A—.
J5»-l ite
—1 — lx+1) — (lx—2— lx + 2) i
hx = — s nearly.
When the quantities are determined from four fixed ages, as 25, 40, 55, 70 years,
the common Log. q is found to have almost the same value 0.04 in all collections. But
the constant 5 changes from one table or latitude to another, and representing the
uniform influence of exterior causes, may be named the climatic constant. Also in
certain cases, s may be used to represent an auxiliary or relative rate of interest.
Having found the law, par excellence, of adult mortality, above the age of 22 years,
we have extended the same law, in Table XXVI, down to the age of ten, to represent
the very few and exceptional insurances contracted at the youthful ages. With more
complete statistics, like the results of the census, Ave might extend the same law down
to the birthday, and compare with the ungraduated values, to indicate the residual
series or law in infancy and youth.
A numerical example of this kind Avas giv'en by the Avriter in the thirteenth Report
of the Insurance Commissioner of Massachusetts, for the year 18G8, page 100. Aided
by this and some further data, Mr. Makebam happily completed his formula doAvn to
the birthday, as described in the Journal of the Institute for 1871, Vol. 1G, page 344.
Thus, for the entire period of Male Life, the number living l in the life table, at any
age x is
lx = k. g • • •
* - D* ’ Da, •
Again, referring to Section III, and multiplying both numerator and denominator
by vx, the Deferred Life Annuity takes the surprisingly simple form,
„ lx+n • VxJrn Dx+n N®+w+i Na-fw+1
ax ~ 'ax+n ~ “
The results already described, will exemplify the superior advantage of commuta-
tion columns, for determining the value of Life Annuities, whether immediate or
deferred, for the whole life, or a portion of it. So manifest have these advantages
appeared, that all the standard formulas of Life Annuities and Assurances ha.ve been
resolved into similar elements. In all, six columns are required; whereof D, N, S are
termed annuity columns, and C, M, R are termed insurance columns. These columns
are connected with each other, by tbe following simple relations:
Let v = ———, where i denotes the rate of interest;
1
dx = Ix—lx+i, where x denotes the age, cl the deaths, l the living in the life
table; then,
Da, = lx. vx; C® = dx. vx= (/®—Z®+i)'y®+1.
N® = Da, + Da,+1 + Da,+2 + .... M® = 0®-f- Ca,+1 + Ca,t-2+ • •
Sa, = N® + N"®+1 + Na,+2 + • • • • R® = M® + M®+i -f- M® + 2 + • • •
Each series is to be continued to the end of the life table. By writing .r + 1 in
place of x, and then eliminating, we find,
N® = E* + N*+l j Mtc = Ccod-Majfi.
So; — N®-f- S®+i ; R® — -M® d- R®+i.
In the former equations for S and R, substituting in place of N and M the series
of D or C which they represent, and adding, we have
S* — Daj -f~ 2Daj+i + 3Da; f 2 + • • • i R* — Gx -f- 20x+l + 3C = • dx.
For this object, the common logarithms of the factors vx, lx, clx are already given
in Tables XXV and XXVII. The process of adding these logarithms, may be proved
by comparing the sum of every ten results, with the corresponding sum obtained
directly from the Tables. To facilitate this mode of Proof, the sum of each ten
logarithms is added in Table XXV. The numbers answering to the logarithmic
results, have been usually found from Filipowski’s seven place Table of Anti-
logarithms.
A second method devised by Gray, takes advantage of the simple factor (14-i) or
1-r-v for determining D*, after the column Ch has been previously constructed; thus,
hhr — (1 4” ("Dar+l 4" C«).
For example, by the Thirty Offices Table, with 4 per cent, interest, D100 = 0;
and C100 = 0; then,
D99 = 1.04 C99 = .082368 ; C99 = .079200
D98 = 1.04(099 4-098) = .256988 ; etc. x.04= 3168
1.04 C99 = . 082368 COMM VTA TION TA BLES.
247
The operation is easily applied with eight decimal places, and is continuous, so
that the separate computation of D* at every tenth or fifteenth value will verify the
whole series.
A third method proceeds entirely by continuous operations with logarithms and
Gray’s Table of Log. (i+x), according to the formulas,
Da+i = VPxDx. Oa+1 = V^QX.
(lx
N» = D„ AN* = AD* + A (l + j.
Ma, = Qc + mx= AO^+A^l+^j.
In all cases, when the elementary columns D and C have been fully constructed,
the N and M columns can be formed from them, by successive additions; or by the
previous logarithmic formulas, in connection with Gray’s Table; and so with the S
and R columns. The logarithm of vpx is already given in Table XXXIII; and a
column for kdx+t—kdx will be simply the differences of Xdx in Table XXVII, to which
Xv can be added from a separate slip.
Commutation Tables for• Joint Lives. In the older Tables of David Jones and
of David Chisholm, J)xy = vxlxly = Dxly; where x denotes the older life. This dis-
tinction of the older age is not needed in De Morgan’s arrangement, nor in the case
of equal ages. Prof. De Morgan suggested the symmetrical form, adopted in the
English Life Table:
x+y x+y+2
Dxy =z IxlyV 2 • C'xy = (Ixly — lx+1 ?y + l)v 2 = vY)xy—^x+\.y+\-
Njcy — Da?y T" LL+l.y+1 “1“ • • • • Mgjy — Jx+\.yJr 1 4" • • • •
Say Na;y-f- H- .... RX1J Mfty + Maj+l.y-fJ -)- . . . .
For two Equal Ages, L)Xx — lxlxOx — lxJ)x> NAc — A A • • • •
Two joint lives of unequal ages x, y have the same life annuity as two equal lives,
u, u, defined in one way by the equation nxy = auu, and in other ways under Make-
ham’s law of mortality, by the relations,
pxpy = PuVu, Pu = Vpxpy; (f + (pJ = 2qu, u = x+Cor. Table LXY.
Since any two joint lives are thus reducible to the case of two equal ages, the
corresponding D and 1ST columns, annuities and premiums may he found in Tables
LXVI-LXXIV. The radix of these Tables was taken as 100,000 at the age of ten;
or the usual D and X values were divided by the constant factor Dio.
All the preceding Commutation formulas, for single life, become applicable to
joint lives, by merely extending the suffixes to every life. For illustration, multiplying
numerator and denominator of the formula in Section III for joint lives, by v*, or by
x+y
v 2 , we obtain in either way,
IWl.y+l + 11*42.y+2+ • • • • Nz+i.y+l
aXy — TA — T)
xJxy 248
LIFE ANNUITIES AND INSURANCES.
The law is perfectly regular for two, three, or any number of joint lives; thus,
Da+l.y+l.a+l + • • • •
axyz — i) — I)
Uxyz xjxyz
Dxyz—(1 — — Xyz Nx+l.y+l.S+l*
Note 1.—It will be proper to observe, that formulas on the plan of De Morgan, when applied
to the order of survivorship will involve the square root of v. Thus the single premium for an
insurance of $1 “ on x against y,” that is, payable on the death of x provided y be then living, is
®NXy—Nx+l.y+l + + 1 Nx-(-l.^)
The annual premium will be found by simply changing V)X!/ to In each term of the
numerator, the sum of the suffixes added to the exponent of Vv counted as 1, gives the homogeneous
sum x + y + 2.
Note 2.—Repeated additions of Xv. There are two methods of correcting the final decimal.
For example, at 4 per cent. Xv is 1.982,960,6007, which falls between the six-decimal values 1.982,967
and 1.982,966. Firstly then, if no more than ten or fifteen results are to be computed so as to be exact
to six decimal places, the computation can be made with four extra decimals, that is, with ten places,
retaining only six at the end of the process. But if Xv is to be implicitly added a hundred times in
computing so many results exact to six places, the computation can be made with five extra decimals,
in order that the first six places may be accurate in all the results.
Secondly, the computations can be correctly made with six decimals by suitable changes of the
sixth decimal. In using 1.982,967 for Xv, we commit an error of .338 in the sixth place, in excess ;
and in using 1.982,966, we commit an error of .661 in deficiency. A compensation will ensue by
using the endings 7 and 6 in frequency, inversely as those errors; that is, as 661 : 338, or 2 : 1 nearly.
Hence the sixth decimal of Xv at 4 per cent, should be taken as 7, 7, 6, 7, 7, 6,.... in order. At 5 per
cent, the ratio is 7: 3, so that the sixth decimal of Xv should be 9, 9, 8, 9, 9, 8, 9,9, 8,9, when the series
recommences. An adjustment can be made at every tenth value. These corrections have already
been applied to the sixth decimal place in Table XXXIII, and in all the other regular Tables.
SECTION YI.
Single and Annual Premiums of Insurance.
By general custom, the sum insured is assumed in theory, to be payable at the end
of the year of decease. The age at which the insurance is effected is often termed the
age of issue, or the age of entry. The contract between the insuring party, and the
insured is named the policy. The first premium thereon is paid at the time when the
contract is made; and the subsequent payments of premium are due at the commence-
ment of each policy year. The premiums, however, may be single, annual, half-yearly
or quarterly. The single premium pays for the insurance in full at the beginning,
in one payment.
By “deferred premium” is meant, the remaining semi-annual or quarterly pre-
miums to be paid during the current policy year. In the estimate of liabilities, these
deferred premiums, with some reduction for the expense of collection, and for discon-
tinuances, are accounted as Assets; and in case of death, the Company usually deducts
them from the policy. To provide for expenses, fluctuations of mortality and other
contingencies, the net premium of theory is augmented in practice, by a margin or
loading, such as one-fifth, one-fourth or one-third, and is then termed the gross or
Office premium. SINGLE AND ANNUAL PREMIUMS.
249
In order to explain the computation of net premiums, let lx denote the number
living at the age x in the life table, and let i and v represent the given rate of interest.
1. SINGLE PREMIUMS.
Let Ax denote the single premium at the age of entry x, to insure $1 payable at
the end of the year of death. Then by the principles of the system of Probabilities,
* (lx—Zx+i)w-p (lx+t — lx+f)i®-\-lx+2—Zx+3)r3-f-....
■ft-x — 7 •
(X
J J
Here the tabular probability of the life failing in the 1st year is x- a!+1 or ;
I'X f'X
and this multiplied by the sum at risk $1, and again by v, to reduce to present value,
gives the first term. In like manner the remaining terms express the present value of
the mathematical expectation for the 2d year, the 3d year, and so on to the end of the
life table. This sum is therefore the present value of the life insurance.
Writing dx for its equal lx—lx+1, and similarly for other ages, then multiplying
both numerator and denominator by vx, which does not alter the value, the fraction is
changed to the commutation terms:
A _dx. w*+1 + fZx+i. v*+2 + dx+2. v®+3 + ....
— >
lxvx
A 0a?4- Cx + i + Cx + 2 + • • • • Mx
Ax~ D* ~DX’
A* = l-(l-^.
In the first formula, the positive and the negative terms are easily identified with
the series for the life annuity, so that we have also
A.x — v—ax -f- vax = 1 — (1 v) (1 + (if) — ’
To facilitate practical applications, the single premium for life is already computed
in the four different forms of Tables XXX and XXXI, L and LI.
2. ANNUAL PREMIUMS.
Instead of a single premium, the payment for insurance is more usually made by
equal annual premiums at the beginning of each policy year. Let P® denote such
annual payments on $1 insured. The present value of all these premiums being
equal to the single premium, we have for common life policies, on $1,
Ax
Vx(lAax) = A*, whence P* =
p*=rir-d-O =
1+#* -*•
p _ Mj_ 5?_n_ v) = V—^±1. 250
LIFE ANNUITIES AND INSURANCES.
For common life policies, the annual premium P or n is already computed in
Table XXIX, for each 11000 insured.
3. INSURANCE BY A LIMITED NUMBER (n) OF ANNUAL PREMIUMS.
The rule proceeds on the same general principle as before. That is, “divide the
single premium by unity added to the present value of a temporary annuity for one
year less than (n) the number of premiums which are to be paid”:
p A*
“ _ l^W-1 “ N»—
Example. A person at the age of 39 years insures his life for $7000 by ten equal
premiums, to be paid annually at the beginning of the first ten years ; required this
premium when interest is 4 per cent.:
Table LI, - - - - AA39 = 1.552218
“ XLIV, - A(l+qg-i) — 0.908205
...2.644013
“ LXXXII, - - P* - 0.0440568
x 7000
Annual Premium - - - - = $308.3976
Table XXXVI, - - AM39 = 3.796864
X39 = 293,818.6 ) X _ 5.152851
X49 — 151,634.5 ) ...2.644013
142,184.1 P* - 0.0440568
x 7000
Annual Premium - - - - = $308.3976
For five, ten, fifteen or twenty payments, the premiums are already computed in
Table LXI, for each $1000 insured.
Note.—If a special payment is made at the beginning, in order to reduce the future annual
premiums, it can he regarded as a single premium, for one part of the sum insured ; and the annual
premium will be computed on the remaining part.
4. TEMPORARY INSURANCES.
Required the single premium to insure II for 7 or m years. By reference to the
explanation of the first three formulas of this Section, it will be seen that the single
premium will be the sum of the first m terms of the series there described; thus,
\m QjH-pa: +1 + • • • + C x + m—1 _m*-m x+m
IVX — T) — T) *
+’x ux
* rn — Njj+bj) — (Na? + l— + wi + i)
Ax — T) •
The equivalent annual premium is obtained by dividing the single premium by 1
plus the temporary annuity for one year less:
-pm A-x +1—Nx+m + 1 flx
x " 1+af* ~ V _ T+a"~i*
pm Ma;—Ma;+m T)x — T)x+m /-. \
* ~ “ Na^N^- ''* SINGLE AND ANNUAL PREMIUMS.
251
The Premium for Temporary Insurance on each $1 for One Year only, is
T>1 0# + 1
-t® = fy- = v(l —px) = vqx = v T. •
\->x Ux
For example, by Tables LII and LIII, at the age of 35 years, on $1000 insured for
7 years, the single premium is $51.70, or the annual premium is $8,486.
5. PURE ENDOWMENTS.
The single premium or present value of $1 to be received at the end of m years, if
the party now aged x be then living, is
a m x+mrm Dx+m
-n-X 7 V I)
lX ±Jx
The annual premium for the same, payable at the beginning of each year, unless
the risk is voided by death, is found on the principle before described:
Am D
P' =r+-aP = °" * •
Table LXIII, AP™ = 1.—AB”\ on $1 Pure Endowment.
For example, at the age 16, the annual premium is $173.88 on $1000, payable at
the age of 21 if the person be then living. The same result is also found by sub-
tracting the premium of term insurance, Tables LII and LIII, from the premium of
endowment insurance, next to be described.
6. ENDOWMENT INSURANCES.
The sum insured is payable on attaining a specified age, or sooner in the event of
death. By simply adding the single premiums of temporary insurance and of pure
endowment for the term of m years, after the age of entry x, we have at once the
single premium of Endowment Insurance on $1 :
A(m) _ M®-M=
To find the equivalent annual premium, we divide by 1 plus the annuity:
t\>n Ma?—Ma. + m + D(B+Wi 1 n 1
For a limited number (n) of equal annual premiums to be paid at the beginning
of the first (n) years of the whole period m, the divisor is changed as below:
Pn M* — Mcc + ra -j- Dx + m
« = -vr » 0n !t’1*
JNtc — JNaj + w-
The various species of Endowment Premiums are already computed in the seven
Tables LIV to LX. Life annuities and insurances.
252
Computation op Tables,—By taking the difference of single premiums, and observing that
Mxf mfi —M*rm = —Cx-tm — -^»Dx+m + DxHn+i, we obtain the simplified annual difference :
.(»») .(OT+l) (1 *)BX-(-m
Ax x - J)x
This difference can be easily computed by the aid of two moveable slips of paper, the one con-
taining aD, and the other /(I —») + >1). When the period of insurance is only 1 year, the premium is v,
as shown in Table LIV. From v, we subtract the difference (1—»)Dx+i-s-Dx to give the single pre-
mium for the term of 2 years. From this, we subtract (1—r)Dx to give the single premium for
the term of 3 years; and so on.
But in single Examples, the shortest method will be to find the logarithm of 1 plus the tem-
porary annuity for one year less, in Table XLIV-XLVII. This will guide to the required premium,
by mere inspection, in the next or Conversion Tables.
7. DEFERRED INSURANCES.
On the principle of deferred Annuities in Sections III and Y, and by Section YI,
the present values of the temporary and of the complementary deferred insurance,
added together, equal the present value for the whole life. Hence by subtraction, the
present value or single premium to insure a person now aged x, for the period of life
which may remain after the age x + m, will be on each $1 insured,
m\ Mao m Dx+m—(1—w)N*+fR
* ~ ThT ~ d* •
The equivalent annual premium payable m years, will be
mPx = Ma'^rw—, on each $1.
The annual premium payable during the whole life, on $1 insured, is
Mm A
x+m -n-x
N* 1 + Ux
For a Deferred Temporary Insurance, or to be deferred m years, then to continue
in force during r years; if the annual premiums are payable during m + t years, we
have on each $1 insured,
Single Premium =
vx
Annual Premium =
JNa: — -Nx+wi+r
8. RETURN PREMIUMS.
Required the net annual premium P to secure a Pure Endowment of $1 at the end
of m years, if the life now aged x should live so long, or the Return of all Premiums
paid, at the end of the year of death, if it occur before that time.
This plan may he regarded as an endowment of $1, added to a series of m deferred
temporary insurances of P. By making the collective annual premiums equal to P,
then substituting R, and reducing the equation, we find SINGLE AND ANNUAL PREMIUMS.
253
D® + ?ra + + i + .... +Ma,+Wt_i — m^lx+rn) p
M AT —
■L'x — ±*x+m
p + m
■N® -N® + m— R®-!- R®+»i “H mdNi-x+m
Also by changing Dx+m to N®+m, we here determine the annual premium, payable
from age x to age x-f m — 1, to provide a Deferred Life Annuity of $1 per annum,
first payment at age x-\-m. The premiums paid, are to he returned ivithout interest,
six months after death, in the event of death before the age x +-m. Or if the
premiums are not to be returned, cancel R and M in the above formula.
By the same mode of solution, the Single Premium of a Pure Endowment of $1,
with return of the premium six months after death, if it occur before the age x + m,
will be
* (m) P® + m
Required the net premium to insure $1 for the Whole Life, and the return of gross
premiums equal to g times the net, without interest, six months after death.
Let P denote the net annual premium to provide for the ordinary insurance of $1,
together with the series of deferred insurances of
xy SINGLE AND ANNUAL PREMIUMS.
257
A*, = = i_(i_,)(i+aw).
Uxy
Annual Premium, Pxy = (1— v).
±*xy ±*xy
Annual Premium, Pxv — — —— (1—v).
y 1 +axy 1 + axy v 1
For a joint insurance of $1 on three lives, x, y, and z, we have only to substitute
xyz in place of the suffix xy in the last formulas; and so for four or more lives. The
insurance on joint lives continues only while they are all living. Also, the Premiums
may be found from Conversion Tables XLVIII and XLIX.
For a Temporary Insurance of $1 to be paid at the end of the year of the first
death of the two lives x or y, if it occurs within n years:
Single Premium,
n — Mx+n.y + n + — (Xc+iy + i— Nx+re + l.y + w+l)
— ta — V)
*~fxy ajxy
a -it-, • T,n Axy Na;+i.y+i Xx+w + 1.2/ + ra i-l
Annual Premium, Pxv = ~—r = v ~ -,T
1 + a%f $xy-Nx+n.y+n
For a Deferred Insurance of $1 payable on the first death of x or y, if it occur
after the expiration of n years:
Single Premium, nAxy = = P*+<.-^>»-(1-»’)Neh.
i'x/y AJxy
Single Premium, nAxy — x+f'y—.Ax+n,y+n.
Uxy
Annual Premium for n payments, nPxy = —.
xy—™x + n.y+n
Annual Premium during joint life, nPxy — •
For an Insurance of $1 on the Longest of Two Lives, x and y:
Single Premium, Axy = Ax-\- Ay—Axy = 1 — (1—v) (1 +ax-\-ay—axy).
Annual Premium, = i _i_ n~ _T~ (1—v)-
1 + Ux + (ly P axy 1 p nx -f- (ly — axy
For a Temporary Insurance of $1 for n years on the Longest or Survivor of Two
Lives, x and y:
Single Premium, Axy = A” + Ay—Axy.
\ni \n_\n
Annual Premium, Vxy = —Ln-\ ** n-i'
1 p ®x “I- dy — ®xy 258
LIFE ANNUITIES AND INSURANCES.
Tables for Two Joist Lives, Equal Ages.—The agreement of the Thirty
Offices Experience with Makeham’s law of mortality, otters a great advantage, in super-
seding the old and cumbrous system of joint life tables, by the new method of equal
lives or ages, shown in Tables LXV-LXXVII. For, under Makeham's law, if the
joint probability pxpy is equal to pupu, that of two equal lives, at the ages of entry, the
relation of equality will always continue during their joint existence.
Example. When interest is 4 per cent., required the annual and the single
premium to insure 11000, payable at the end of the year of the first death of two lives,
x aged 30 and y aged 40 years. Entering Table LXV, with the difference of ages
y—x, or 10 years, we find the correction 6 Y. 1.8 M., which being added to the younger
age 30, gives the nearest Equal Age 36. That is, 36, 36 may be substituted for the
given ages 30 and 40. Entering Table LXXIII with 36 years, we find the annual
premium $29.74; and in the next Table, the single premium $436.06.
Method for Three Joint Lives.—By Table LXY, change the given Ages x, y,
z, to the two equal Ages u, u, which increase equally in equal times, like the actual
ages, from the date of entry. The current value of u, in any year, increased by the
Annual Correction below, under the proper rate of interest, will be the argument for
entering Tables LXV-LXXVII, for premiums, reserves, etc. The correction can be
adjusted to other rates of interest by the Difference column.
Equal
Ages.
U.
3
Per
Cent.
4
Per
Cent.
5
Per
Cent.
g
fi
Equal
Ages.
U.
3
Per
Cent.
4
Per
Cent.
5
Per
Cent.
g
s
Equal
Ages.
u.
3
Per
Cent.
4
Per
Cent.
5
Per
Cent.
Yr
Y.
M.
Y.
M.
Y.
M.
M.
Yr.
Y
. M.
Y
. M.
Y
. M.
M.
Yr.
Y.
M.
Y. M.
Y. M.
10
13
8.7
14
11.7
16
1.6
13.9
| 40
3
3.2
3
4.9 3
6.6
1.7
70
0
5.5
0 5.5
0 5.5
11
13
2.2
14
4.5
15
5.8
13.3
41
3
0.9
3
2.5 3
4.1
1.6
71
5.1
5.1
5.1
12
12
8.1
13
9.7
14
10.4
12.7
42
2
10.8
3
0.3 3
1.7 1.4
1 72
4.7
4.7
4.7
13
12
2.2
13
3.0
14
3.1
12.1
43
2
8.9
2
10.2
2
11.5 1.3
73
4.3
4.3
4.3
14
11
8.4
12
8.5
13
8.0 11.5
44
2
7.0
2
8.2
2
9.3
1.1
74
4.0
4.0
4.0
15
11
2.7
12
2.1
13
1.1 11.0
45
2
5.2
2
6.3
2
7.3
1.0
75
0
3.7
0 3.7
0 3.7
16
10
9.1
11
7.9
12
6.4110.5
46
2
3.4
2
4.4
2
5.3
.9
76
3.4
3.4
3.4
17
10
3.7
11
1.9
11
11.8
9.9
47
2
1.8
2
2.7
2
3.5
.8
77
3.2
3.2
3.2
18
9
10.5
10
8.1
11
5.5
9.4
48
2
0.3
2
1.1
2
1.9
.8
78
2.9
2.9
2.9
19
9
5.4
10
2.4
10
11.2
8.8
49
1
10.9
1
11.6
2
0.3
.7
79
2.7
2.7
2.7
20
9
0.5
9
8.9
10
5.2
8.3
50
1
9.5
1
10.1
1
10.7
.6
80
0
2.5
0 2.5 0 2.5
21
8
7.7
9
3.6
9
11.4
7.8
51
1
8.1
1
8.7
1
9.2
.5
81
2.3
2.3
2.3
22
8
3.0
8
10.5
9
5.9
7.4
52
1
6.9
1
7.4
l
7.9
.5
82
2.1
2.1
2.1
23
7
10.5
8
5.5
9
0.4
6.9
53
1
5.6
1
6.1
1
6.5
.4
83
1.9
1.9
1.9
24
7
6.2
8
0.8
8
7.3
6.5
54
1
4.6
1
5.0
1
5.4
.4
84
1.8
1.8
1.8
25
7
2.0
7
8.1
8
2.1
6.0
55
1
3.5
1
3.9
1
4.2
.3
85
0
1.6
0 1.6
0 1.6
26
6
9.9
7
3.6
7
9.2
5.0
56
1
2.4
1
2.8
1
3.1
.3
86
1.4
1.4
1.4
27
6
5.9
6
11.3
7
4.4
5.1
57
1
1.6
1
1.9
1
2.1
.2
87
1.2
1.2
1.2
28
6
2.1
6
7.1
6
11.8
4.7
58
1
0.6
1
0.9
1
1.1
.2
88
.9
.9
.9
29
5
10.5
6
3.2
6
7.4
4.2
59
0
11.9
1
0.1
1
0.2
.1
89
.8
.8
.8
30
5
7.1
5
11.4
6
3.2
3.8
60
0
11.1
0
11.3
0
11.4
.1
90
0
.8
0 .8
0 .8
31
5
3.7
5
7.7
5
11.3
3.6
61
0
10.3
0
10.5
0
10.6
.1
91
.8
.8
.8
32
5
0.4
5
4.1
5
7.5
8.4
62
0
9.7
0
9.8
0
9.9
.1
92
.7
.7
.7
33
4
9.5
5
0.8
5
3.9
3.1
63
0
9.0
0
9.1
0
9.2
.1
93
.6
.6
.6
34
4
6.5
4
9.5
5
0.4
2.9
64
0
8.5
0
8.5
0
8.7
.2
94
.5
.5
.5
35
4
3.7
4
6.4
4
9.1
2.7
65
0
7.8
0
7.8
0
8.0
.2
95
0
.4
0 .4
0 .4
36
4
1.0
4
3.5
4
6.0
2.51
66
0
7.3
0
7.30
7.5
.2
96
.4
.4
.4
37
3
10.3
4
0.6
4
2.9
2.3
67
0
6.8
0
6.8 0
7.0
.2
97
.2
.2
.2
38
3
.7.8
3
9.9
4
0.0
2.1
68
0
6.3
0
6.30
6.4
.1
98
.1
.1
.1
39
3
5.4
3
7.3
3
9.2
1.9
69
0
5.9
0
5.90
6.0
.1
THREE LIVES.—ANNUAL CORRECTION (c) TO THE EQUAL AGE u.
Formula, for 3.654, 4.660, 5.666, or the rate of vs, auu = au+CM+C) for 3, 4, 5 per
cent, interest respectively, or the rate of v. SINGLE AND ANNUAL PREMIUMS.
259
Method for Four Joint Lives.—By Table LXV, change the three youngest
Ages to two equal Ages, u, u. With u, u, and the fourth or oldest age, re-enter
Table LXV, and find the equal ages, u', u', for record, at the date of entry. In any
year, the current value of u' augmented by the Annual Correction below, will be the
argument for entering Tables LXV-LXXVII, for the life annuity or other monetary
value.
FOUR LIVES.—ANNUAL CORRECTION (c) TO THE EQUAL AGE n'.
Equal
Ages.
u'.
3
Per
Cent.
4
Per
Cent.
5
Per
Cent.
&
P
Equal
Ages.
u'.
3
Per
Cent.
4
Per
Cent.
5
Per
Cent.
.
Ph
1
P
Equal
Ages.
u'.
3
Per
Cent.
4
Per
Cent.
5
Per
Cent.
Yr.
Y.
M.
Y.
M.
Y.
M.
M.
Yr.
Y
M.
Y.
M.
Y.
M.
M.
Yr.
Y.
M.
Y
M.
Y. M.
10
21
10.4
23
1.6124
4.2
14.6
40
6
.5 6
3.2
6
5.9
2.7
70
0
10.6
0
10.7
0 10.7
11
21
1.8
22
4.5
23
6.7
14.2
41
5
9.1 5
11.6
6
2.1 2.5
71
9.8
9.9
9.9
12
20
5.4
21
7.6 22
9.3
13.7
42
5
4.9
5
7.2
5
9.5 2.8
72
9.0
9.1
9.1
13
19
9.1
20
10.7 22
0.0
13.3
43
5
1.5
5
3.6
5
5.7 2.1
73
8.6
8.6
8.6
14
19
0.9
20
2.0
21
2.8
12.8
44
4
10.1
5
0.0
5
1.9 1.9
74
7.8
7.8
7.8
15
18
4.8
19
5.4
20
5.8
12.4
45
4
7.0 4
8.7
4
10.41.7
75
0
7.3
0
7.3
0 7.3
16
17
9.0
18
9.1
19
9.0
11.9
46
4
3.94
5.5
4
7.1 1.6
76
6.8
6.8
6.8
17
17
1.2
18
0.8
19
0.3
11.5
47
4
1.04
2.4
4
3.81.4
77
6.2
6.2
6.2
18
16
5.6
17
4.7
18
3.7
11.0
48
3
10.2 3
11.5
4
0.8 1.3
78
5.8
5.8
5.8
19
15
10.1 16
8.7
17
7.3
10.6
49
O
7.5 8
8.6
3
9.7 1.1
79
5.3
5.3
5.3
20
15
2.9 16
1.0
16
11.1
10.1
I 50
3
5.0 3
6.0
3
7.0 1.0
80
0
4.9
0
4.9
0 5.0
21
14
7.815
5.4
16
3.0
9.6
51
3
2.5 3
3.4
3
4.3
.9
81
4.4
4.4
4.4
22
14
1.0 14
10.1
15
7.2
9.1
52
3
0.3
3
1.1
3
1.9
.8
82
4.0
4.0
4.0
23
13
6.0il4
2.7
14
11.3
8.6
53
2
10.0
2
10.8
2
11.6
.8
83
3.7
3.7
3.7
24
12
11.5 13
7.7
14
3.8
8.1
54
2
8.0
2
8.7
2
9.4
.7
84
3.3
3.3
3.3
25
12
5.1
13
0.8
13
8.4
7.6
55
2
6.0
2
6.6
2
7.2
.6
85
0
3.1
0
3.1
0 3.0
26
11
10.8|12
6.1
13
1.4
7.3
56
2
4.1
2
4.6
2
5.1
.5
86
2.7
2.7
2.7
27
11
4.6 11
11.5
12
6.4
6.9
57
2
2.3
2
2.8
2
3.2
.4
87
2.4
2.4
2.4
28
10
10.8 11
5.2
11
11.8
6.6
58
2
0.7 2
1.1
2
1.5
.4
88
2.1
2.1
2.1
29
10
5.0 10
11.0
11
5.2
6.2
59
1
11.1
1
11.5
1
11.8
.3
89
1.8
1.8
1.8
30
9
11.410
5.0
10
10.9
5.9
60
1
9.6
1
9.9
1
10.1
.2
90
0
1.6
0
1.6
0 1.8
31
9
6.0
9
11.3
10
4 8
5.5
61
1
8.2
1
8.5
1
8.7
.2
91
1.4
1.4
1.4
' 32
9
0.6
9
5.5
9
10.7
5.2
62
1
6.8
1
7.1
1
7.3
.2
92
1.3
1.3
1.3
33
8
7.6
9
0.2
9
5.0
4.8
63
1
5.7
1
5.9
1
6.0
.1
93
1.2
1.2
1.2
34
8
2.6
8
6.8
8
11.3
4.5
64
1
4.4
1
4.6
1
4.7
.1
94
1.1
1.1
1.1
35
7
9.9
8
1.8
8
5.9
4.1
65
1
3.3
1
3.5
1
3.6
.1
95
0
1.0
0
1.0
0 1.2
36
7
5.2
7
8.9
8
0.7
3.8
66
1
2.2
1
2.4
1
2.5
.1
96
.8
.8
.8
37
7
0.8
7
4.2
7
7.7
3.5
67
1
1.2
1
1.4
1
1.5
.1
97
.7
.7
.7
38
6
8.4
6
11.6
7
2.9
3.3
68
1
0.3
1
0.4
1
0.4
.0
98
.3
.3
.3
39
6
4.4
6
7.3
6
10.3
3.0
69
0
11.4
0
11.5
0
11.5
.0
Formula, for 4.311, 5.324, G.337, or the rate of vs2, aU'u' = au'+c.u'+c> for 3, 4, 5 per
cent, interest respectively, or the rate of v.
Equalization of Annuity Elements.—In explanation of the preceding Tables,
the single life annuity ax, on page 240, is or may be resolved into the sum of products
of elementary factors of the form vpx. In like manner, the annuity axy on two lives,
may be resolved into elementary factors of the form vpxpy; on three lives, into vpxpypz;
and so on. In the logarithmic form of Makeham’s law of mortality, from page 237, we
find these Annuity Elements:
For a Single Life, A (vpx>) = A (vs) + qx'(q—1) Ag.
For Two Joint Lives, A (vpxpy) = k (vs2) + (qx + qy) (q—1) ty.
For Two Equal Lives, A (vpupu) = A (vs2) + (qu + qu) (q—1) Ag.
For Three Joint Lives, A (v'pxpypz) = A (Vs3) -f (qx + qy + qz) (q — 1) Ag, etc.
Now if v = v's, that is, vs2 = v'ss; and qw-\-qu = 2qu = qx-\-qy + qz, the annuity
element of three lives, x, y, z, with the rate v', evidently becomes equal to that of two
equal lives, u, u, with the rate vs. Also the same element with the rate v equals that 260
LIFE ANNUITIES AND INSURANCES.
of the single age x' (defined by qx' = qx -f- qv qe) with the rate vs2. By the last two
Tables giving the correction c, the true annuities first computed for the ages u, u, and
rate vs, or vs2, are more conveniently found through the ages u + c, u + cx and the rate of v.
Method for Simpson’s Rule on Page 242.—For greater convenience, two
Tables are here brought together; the first, from Table LXY. gives the Correction
to change two joint lives to two equal ages, which does not involve the rate of interest.
The second, computed from auu = «M+C", gives the correction c" to change from two
equal ages, u, u, to the equivalent single age, u + c", which last depends on the rate of
interest.
CORRECTION TO BE ADDED TO THE YOUNGER AGE FOR REDUCING TWO
JOINT LIVES TO THE CASE OF TWO EQUAL AGES.
Diff.
Ages.
0
1
2
3
4
5
6
7
8
9
Yr.
Y.
M.
Y.
M.
Y.
M.
Y.
M.
Y.
M.
Y.
M.
Y.
M.
Y.
M.
Y.
M.
Y.
M.
0
0.0
0
6.1
1
0.6
1
7.3
2
2.3
2
9.5
3
5.0
4
0.8
4
8.9
5
5.2
1
6
1.8
6
10.6
7
7.6
8
4.7
9
2.0
9 11.6
10
9.5
11
7.3
12
5.5
13
3.7
2
14
2.0
15
0.6
15
11.2
16 10.0
17
8.8
18
7.7
19
6.7
20
5.9
21
5.0
22
4.3
3
23
3.6
24
3.0
25
2.4
26
1.9
27
1.4
28
1.0
29
0.6
30
0.2
30
11.9
31
11.5
4
82 11.3
33
11.0
34 10.8
35
10.6
36 10.4
37 10.2
38
10.1
39
9.9
40
9.8
41
9.7
5
42
9.6
43
9.5
44
9.4
45
9.3
46
9.2
47
9.2
48
9.1
49
9.0
50
9.0
51
9.0
6
52
8.9
53
8.9
54
8.8
55
8.8
56
8.8
57
8.7
58
8.7
59
8.7
60
8.7
61
8.7
7
62
8.7
63
8.6
64
8.6
65
8.6
66
8.6
67
8.6
68
8.6
69
8.6
70
8.6
71
8.6
CORRECTION (c")
TO
REDUCE TWO
EQUAL
AGES TO
THE
EQUIVALENT
SINGLE
AGE.—INTEREST 4
PER CENT.
Equal
Ages.
0
1
2
3
4
5
6
7
8
9
Yr.
Y.
M.
Y.
M.
Y.
M.
Y.
M.
Y.
M.
Y.
M.
Y.
M.
Y.
M.
Y.
M.
Y.
M.
1
21
5.2
20 10.7
20
4.3
19
10.0
19
3.9
18
10.0
18
4.2
17 10.6
17
5.2
16
11.9
2
16
6.8
16
1.8
15
9.0
15
4.3
14
11.9
14
7.5
14
3.3
13
11.3
13
7.4
13
3.6
3
13
.1
12
8.6
12
5.2
12
2.1
11
11.0
11
8.1
11
5.2
11
2.5
11
.0
10
9.5
4
10
7.2
10
4.9
10
2.8
10
.8
9
10.9
9
9.0
9
7.2
9
5.6
9
4.0
9
2.5
5
9
1.1
8
11.8
8
10.5
8
9.3
8
8.1
8
7.1
8
6.0
8
5.1
8
4.2
8
3.3
6
8
2.5
8
1.8
8
1.0
8
.3
8
11.7
7 11.1
7
10.5
7
10.0
7
9.4
7
8.9
7
7
8.4
7
8.0
7
7.5
7
7.0
7
6.4
7
5.8
7
5.2
7
4.4
7
3.4
7
2.2
8
7
.5
6
10.4
6
7.6
6
4.8
6
2.5
6
0.9
5
11.4
5
10.1
5
9.2
5
8.2
9
5
7.6
5
3.8
4 11.6
4
6.1
3
9.5
3
1.0
2
3.8
Thus for two Equal Ages of 38 years each, the 4 per cent, annual correction addi-
tive, is 11 Y., giving the equivalent single Age 49 Years. Ten years after this, the
equivalent single age for finding the life annuity will be 48 Y. + 9 Y. 4 M., or 57 Y. 4 M.
For a rate (i) of Interest other than I per cent.—First find the 4 per cent, correc-
tion c", and re-enter the Table with the corrected argument w + 10(.04—f)c", which will
guide to the correction sought. Here 10 (.04—i) c" is for 3 per cent. + ; 3| per
cent, -p; 4| per cent. — ; 5 per cent,.—; G per cent. — \c".
Required the single premium to insure $1, to be received at the end of the year in
which x dies, provided another life y be then living. This is sometimes termed the
insurance of “ x against y”
The probability of receiving the $1 at the end of any specified year, as the second,
depends on the probability that both are alive at the beginning of the year,
lx ly
multiplied into the sum of the two probabilities, (1) that x may die in said second year
14. SURVIVORSHIP INSURANCES. SINGLE AND ANNUAL PREMIUMS.
261
and y survive through it, or (1—px+i) py+\. (2) that both may die in the second year,
x dying first, or £(1— px+i) (1—jpy+i). This last expression is multiplied by since it
is considered an even chance for so short a period as one year, whether x or y will die
first, whatever be the difference of their ages.
The sum of (1) and (2) is £(1 —px+1) (1 +py+i), or (l + t-). And the
product of this sum by the previous factor and by v2 gives the second term of the series
or formula for the single premium :
*L lX +1 lyhly+l lx+1 — lx+2 ly + l~tly+2 » .
Axu~~h, —£ + ”"
In this notation for A, the 1 written over the x, expresses the condition that x is
to die before y: otherwise the insurance is void. Performing the multiplications
indicated, we have
A xy = jj , . \_(lXly—lx + lly + 1 + lXly + l — lylX+l)V
"P (Jx+lly+l — lX+%ly+2 "H lX+lly+2— ly+llx+i)^
+ ]•
This expression can be resolved into three simpler series, as shown on page 184 of
Milne’s treatise. Thus, the sum of the terms characterized by the first factor
lxly—lx+\ly+\ is identical with the half joint life premium %Kxy. The factor lxly+1 is a
type of the next series, which being multiplied and divided by lx~\ly, takes the form of
\dx—\.y-^px-\ • And the remaining series is like it; so that
aL 1 / a 1 ax—l y ttx.y—\\
AW_^AW+—)■
A Xy -j- A xy — A xy; Axx = j}Axx.
When x and y are both of the same age, the expression is thus simplified by the
evident cancelling of the last two terms. Also, by changing each term into the Com-
mutation notation, and observing that Vv.px-iDx—\y = T)Xy, we verify the formula
already given on page 248, Note 1, for the single and the annual premium.
Another formula can be obtained by substituting the middle value ly+1 in place of
+ ly+i), and so forth, in the primary formula, which will be sufficiently exact. The
coefficient of v divided by lxly, thus becomes (lx — lx+\)ly+£, or lxly+\—lx+ily+%, the
result for the first year. And the sum of the series for the first and following years,
will be, if is replaced by \ / ——:
ly V Py—1
Ab = , / _L. 'X
V Py—1 \ Px-1 7
Example.—Let x = y = 35 years, with interest 4 per cent. The equal ages to
give the two annuities, are found by Table LXV to be and 34f years, for entering
Table LXXII. And the logarithm of the radical factor is of 0.003500 found in
Table XXVII. Completing the calculation, we find the single premium 0.21416 on
each $1 insured. Verification is obtained by the previous formula, which gives
IA35.35 = .21408; and this appears a trifle more exact than the result above, which
depends substantially on the difference ot annuities tabulated to three or four
decimals. Dividing the single premium by l-\-axy, that is, by l + «35.35 or by 14.8678,
gives the annual premium 0.014399 on each $1 insured.
Changing now the last formula, or rather its previous series, into the notation of
De Morgan and of Chisholm, we may find the values of D, N, in the Tables for Two 262
LIFE ANNUITIES AND INSURANCES.
Joint Lives, or the values of 13, 1ST, S, in the Tables of Single Life, entered with the
equivalent single age from page 259, and thence the value of or Pxy below.
1 iv/rh- 1 x+y+4
Ma7y = (lx — lx+\) (ly+h)v 2 -h(Zx+l — tx+z) (Zy+lj) v 2 +••••
13xy
1_ 1 l 1
ILy — Ma;+I.j, + 1+ Ma;+2.y+2+• ••
11 11
MSjfl-Ma??/ = Mxy'y ILy d” ILy = Rfcy
Mxy — V^~^x.y+\ + l.y+d j —
Insurance of $1 on the life of x, if y be the survivor:
Single Premium, = ; Annual Premium =
t'xy xy
Insurance of $1 on the life x, if y be the survivor, with return of net premium
or premiums paid, if y die first:
Single Premium, — ; Annual Premium = —
D j-y — M xy P.iy
Insurance of $1 on the life of x, provided he die last of the two lives x, y:
Single Premium = A*— ; Annual Premium = ——-
J- ~r CCxy
Insurance of $1 upon the life x, deferred n years, and contingent upon y surviving
that term :
, -r, . M x+n ly+n » it, • + ly + n
Single Premium = ; Annual Premium = =r—,—r~ x •
\Jx 'y Ux iy f J- + (txy)
Deferred amd Temporary Survivorship Insurances.—For the single pre-
mium of the contingent insurance deferred n years, and of the temporary insurance
during n years, we have respectively,
n A — _ + n.y + n A _J_ .
— yy •
Uxy
Am- _ A1- ra A — A — X+n.y + n A _J
xy — **~xy ~±xy — — =Fv • + n*
Uxy
The divisor to give the animal premium will be 1 plus the temporary annuity on
the joint lives for one year less than the whole period of payment.
Attention has been called to this advantageous method of determining the single premium of
deferred or of temporary insurances, by Mr. Peter Gray, who states that it applies to all insurances,
depending on the first death from among any assigned number of lives. The commutation formula for
temporary annuities will furnish another illustration. An equation between two or more benefits is
first presupposed. Then by transposition, or solution, each benefit may be expressed in terms of
the others. SINGLE AND ANNUAL PREMIUMS.
263
General Solution.—For the more general case of an insurance of $1, payable at the end of
the year of first death among the (to) lives x', x", x'", provided all the other (n) lives y', y", y"',
are then surviving, Mr. Makeliam has given a strikingly original formula in the Journal, vol. 9,
page 362, reproduced by Mr. Woolhouse in vol. 15, page 403. First, two auxiliary single ages x, y,
are to he found such that,
.. will denote the ordinary single premium of insurance on all the joint lives.
Multiplying by v in order to change from “ continuous ” to ordinary premiums, we find the formula
equivalent to
A 1 mqv—nqx ( \/v 1 / 1 1 . 1\ . ) m ,
j —( +s+7A* *-
For the simplest case, that of an insurance payable after the death of x, provided y survives him,
we have m — n — 1. And first, if x be the younger life, with interest 4 per cent.
1 ay—x 1
=
qy +i
= a' + b'Axy. A log = 3.801012; A Vv = 1.991483.
Here a' denotes tlie first term, and b' tlie whole coefficient of Axy. To facilitate calculation, their
values are sketched in the following Table. The Italic n is annexed to the logarithm, when the
number b' is negative. When x is greater than y, qx will exceed qV in the general formula ; hence if
x be the older of the two lives, and if the difference of ages is accounted positive,
Lory == — a + (1 — l) ) Aa-y.
VALUE OF a' AND XV ACCORDING TO THE DIFFERENCE OF AGES.
INTEREST 4 PER CENT.
Diff.
Ages.
a'.
XV.
Diff.
Ages.
a'.
XV.
Diff.
'Ages.
a'.
XV.
Diff.
Ages.
a'.
XV.
1
0.00751
1.67080
13
0.08690
1.13536
25
0.13124
2.68871n
37
0.14900
1.09030/2
2
.01499
.64079
14
.09201
.06150
26
.13348
.76515/2
38
.14980
.10202/2
3
.02239
.60891
15
.09684
2.97774
27
.13556
.82546/2
39
.15054
.11246/2
4
.02970
.57494
16
.10140
.88044
28
.13747
.87449/2
40
.15121
.12178/2
5
.03688
.53878
17
.10569
.76339
29
.13923
.91520/2
41
.15182
.13012/2
6
.04390
.50023
18
.10972
.61445
30
.14085
.94956/2
42
.15238
.13760/2
7
0.05074
1.45908
19
0.11349
2.40475
31
0.14234
2.97889/2
43
0.15290
1.14431/2
8
.05738
.41505
20
.11701
.02821
32
.14370
1.00418/2
44
.15336
.15034//,
9
.06379
.36781
21
.12029
3.48477/1
33
.14496
.02615/2
45
.15379
.15577/2
10
.06996
.31690
22
.12334
2.19920//
34
.14610
.04535/2
46
.15418
.16066/2
11
.07587
.26176
23
.12618
,44206/i
35
.14715
.06222n
47
.15453
.16507/2
12
.07974
.22151
24
.12880
.58729/2
36
.14812
.07711/2
48
.15485
.16905/2
Example.—Required the single premium on each $1 of insurance payable after the
death of x aged 17, if y, now aged 57, shall then be living. Interest 4 per cent.
AfiSr = 0.15121-[1.12178] Ai7.57 = 0.07574.
Here the difference of ages, 40 years, gives the Equal Ages 49 Y. 11.3 M.; whence
by Table LXXIV, A17.57 is .57016 on $1, for the remaining operation, with a' and XV at
(x—y) 40 years. The former approximation by annuities verifies the answer 0.07575. 264
LIFE ANNUITIES AND INSURANCES.
15. INSURANCES PAYABLE AT THE INSTANT OF DEATH.
The received system of premiums assumes the sum insured to be paid at the end
of the year of death ; that is, on the average, six months after death. Hence, if the
sum insured (s) were paid at the instant of death, it would be increased at the year’s
end by six months’ interest, that is, by the factor + or the divisor s/v according
to some writers, where i denotes the yearly rate of interest. If the common insurance
sis to be thus increased to the premiums should be increased in the same
ratio, that is, by adding 6 months’ interest, or by the factor 1+-• By similar reason-
ing, if the insurance is to be paid three months after death, that is, on the average,
three months before the expiration of the year of death, then 3 months interest should
be added to the premium, whether the premium be single, annual, or by other install-
ments. The factor will make this last addition, which illustrates the process
for other cases.
It will be proper to add, that a three months increase is more commonly included
in the “ loading” of the office premiums. The subject is more fully analyzed in the
“continuous method,” which assumes all insurances to be payable at the instant
of death.
16. SEMI-ANNUAL AND QUARTERLY PREMIUMS.
According to Mr. Woolhouse in the Assurance Magazine, Vol. XI, page 327; in
official practice, it is a safe and convenient rule to take the yearly premium a year
higher when one-fourth of it is to be paid quarterly, and half a year higher when one-
half of it is to be paid half-yearly. This will rather more than cover the true calcula-
tion, as it should do to provide for the additional cost. Rigorously, if 7i denote the
annual premium on $1 insured for life, and n' the increased value when payable in m
instalments, it is there proved that
n' = ~ — Table XLIII.
SECTION VII.
Reserve or Valuation of Policies.
The annual changes of Reserve or Net Value are exemplified in the common life
policy. By Table LXIII, at the age of entry 46 years, the net Annual Premium is $28
on $1000 insured ; and the Reserve for different Durations, or Policy Years, as the
annual premiums are paid, will be as follows:
Policy Years
1st.
2nd.
3rd.
Etc.
Duration Years....
0
0£
1
1
2
2
3
Premiums paid....
28.00
28.00
28.00
Net Value
28.00
22.82
17.64
45.64
40.66
85.68
63.68
58.89
54.10 VALUATION OF POLICIES.
265
At the beginning of the insurance, the Net Value is equal to the Premium paid:
the risk of death incurred during the following twelve months, continually decreases
this value till the close of the first year. Then the premium paid at the beginning of
the second year, carries up the Reserve per salturn to a much higher point; when the
twelve months decrease again commences; and so on.
In the case of paid-up Policies, the Net Reserve changes continuously; either by
increase, as in Table XXXI, for whole life policies; or by decrease, as in Table LII,
Avhere the Single Premium at the age of 24 is $36.86, and at the age of 25 it is $31.58,
on an Insurance of $1000 terminating at the age of 30.
MONTHS AND DAYS.
The decrease of Net Value from the beginning to the end of each policy year, is
conventionally assumed to be uniform. For example, when a policy has been in force
5 years and 4 months, the decrease in the 4 months is one-third of the whole decrease
between the beginning and the end of the 6th policy year. Or multiply the tabular
difference for the mean half-month dm. cl.) by double the number of months. And
for odd days, take proportional parts of dm. d.) representing 15 days. That is,
account 1 day as fs or -fe, and 2 days as Vo> also 3 days as or r\, and 6 days as T4o,
also 9 days as and 12 days as ; also 5 days as V and 10 days as -§. The correction
will be added or subtracted according to increase or decrease of the tabular Reserve.
Other modes of proportional correction will be indicated in Tables LXIII and
LXXIV.
VALUATION TABLES FOR PAID-UP POLICIES.
At every age, the Net Value evidently coincides with the Single Premium, for the
unexpired term of the same species of insurance. Single Premiums will be found
already computed in Tables XXX, XXXI; by Months in Tables L, LI; for Term and
Endowment Insurances, Tables LII, LIV; and for Joint Lives in Table LXXIV.
ANNUAL PREMIUM POLICIES.
Prospective Forms of Net Valuation.—The general principle is simply,
from the net single premium, of the given species, at the age of valuation, to deduct
the present value of the future net annual premiums receivable.
Let s denote the sum insured, and the annual premium thereon, at the age of
entry x. At the end of the nth policy year, or at the age x-\-n, when the next annual
premium is just due but unpaid, the Net Yalue of a common life policy will be, if
A x+n denote the net single premium on each $1 insured,
Va; + n = S.Ax + n—Px(l + Mx+n)>
If the premium has just been paid, by adding it, and accenting Y,
Y x+n —- S.A-x+n •^x+n•
When the valuation is made by the same table of mortality and rate of interest
with the premium, the former expression will be simplified by substituting for A, P,
their equivalents in terms of v, a; the result is, on each $1 insured,
TT 14~ttx + n &X—Mx+n, -T7-t , ttx+n m
V x+n — 1 1 1 — z ~ -j I „ )
x+n 1 + ax 1 + ax l + «a> 266
LIFE ANNUITIES AND INSURANCES.
T7 1—A-x + n P*4"l—v
\x + n - i — Acc “ P.+. + 1-®’
In the case of two joint lives,
1 1 + Mx+n.y+n
x + n.y+n — 1 ' ’
■*- T" uxy
And on the survivor of two lives,
T7- 1 T ttx+n T" Uy+n — dx + n.y+n
Vx+n.y+n - 1 l + ax + Oy-Oxy
By another mode of reasoning, if ~Px+n denote the annual premium at the
advanced age x + n, then the difference Vx+n—Re measures the annual saving in
premium of the former policy over an assumed new policy. The present value of these
savings will of course be the present value of the former policy; that is, if the premium
is just due, but not paid,
Vjc + n = (Pa;+w—Pa;) (1 + ax+n)-
Rule.—Take the difference between the net annual premium, which would be
required at the advanced age, and the net annual premium charged in the policy,
multiply it by unity added to the value of the annuity at the advanced age of the life
insured; the product will be the Reserve.
The last formula is equal to the first one, since kx+n = Px+n (1 + ax+n)- Again,
since A.x+n = l — (l—v) (1 + ax+n), we have from the first formula on page 265 :
Vas+n = 1 —(1 —v + P*) (!+«*+«), on each $1 insured.
By proper change of the annuity, as noted above, this formula will likewise apply to
two joint lives, or to the survivor of them.
On all Temporary and on Endowment Insurances of $1 insured for the term of m
years, after the age of entry x, the Reserve at the intermediate age x + n, when the next
premium is just due but unpaid, will be
Y— A m p /"i | —n—1 \
x+n — -P+x+n—ix\*--f-Clx+n )•
Here A denotes the single premium of the given species of policy, and a the
temporary annuity at the age x+n for the remaining m—n years. The same Reserve
with the annuity expressed in commutation terms, will be
■\j \m—n r)
V x+n — ±*-x+n —±x- f=v *
iJx+n
The value of an Endowment Insurance for m years, with premiums payable
annually, may be simplified like that of the common life policy, as follows :
1 I_nm-n~1 XT N" d
■\j -I ±^ax + n 1 — slx+m x+n
Valuation by Gross Premiums.—Let denote the net annual premium on
any policy, and P'* the loaded, gross, or office premium. Then will the difference
between the gross and the net valuation be expressed by
(P'z-PzMl+rtSVr1), or (P-»-P,c)y“',~K«+ff
*-*x + n VALUATION OF POLICIES.
267
This is the present value of the loading receivable in the future. Since many
policies will be discontinued, it belongs to the uncertain or “ unrealized assets;” which,
as they accrue, may be the material of future dividends. In the early years of life
insurance in the United States, the Carlisle 4 per cent, mutual premiums were loaded
35 per cent.
RETROSPECTIVE FORMS OF NET VALUATION.
In apparent contrast witli “Prospective Valuation” which has just been shown to
be based on the present single premium, and on future annual premiums, “ Retrospec-
tive Valuation ” is based only on past premiums already received. Yet by a singular
coincidence, in net valuation, the Reserve found by the first method, from premiums
future and unpaid, is precisely the same as the Reserve of the second method from
premiums past and paid. This will be the more evident from the direct transformation
of Prospective formulas into Retrospective. Let us take, for example, the Prospective
formula, giving the Reserve at the end of 6, or more generally, n years, on a ten-pre-
mium life policy for $1 insured:
-y Naj + Ti—Na>+10
Dx + re Na; — Na; + 10 Dx + n
Substituting for M.x+n its equal, (M* — M;c+W),
-y- M# / Na;+» — N*+io\ —Ux + w
*+W“D~n\ / VX + n
Px (Na;— + w) — (Mx — M*+n)
V X -f* Tt — *
Dx + n
M
Here the premium Pa; = == ; and the formula now involves no future age,
JNa: — JNa: + l0
but only the past or retrospective ages x and x-\-n ; hence the name of Retrospective
Formula, proposed in the original memoir by James Meikle, Journal, Vol. XI, p. 245.
If T” denote the annual premium of Temporary Insurance, and E® that of Pure
Endowment, for n years; then dividing both numerator and denominator of the
formula by E®—Naj+w, and substituting equivalents, we have the second form, on
each $1 insured:
•p r\\n
U -LX J-X
V* + n — "
By substituting for Ma; its equal T)x—(1 — and so for M.x+n, we obtain a
third form; and then by substituting —v) N® for its equal J)x, the fourth form
following:
Tr (P® + l-v) (NB-N + » ( 1
V x + n — — tV "r
Ux + n
By an obvious modification, we obtain a fifth and more general formula, which
includes the single premium and others:
it l>x • D* -f- l>x + \ ■ B9'+ i 4- ... . S (M# Ma; + W)
VtC+M Vv *
J h n 268
LIFE ANNUITIES AND INSURANCES.
Here each annual payment bx, bx+1, .... (which may be negative), is treated as a
single premium of pure endowment, due at the age ot valuation x + n. The sum of
these terms, less the cost of temporary insurance of s from age x to the age x + n,
gives the required reserve or net value. All the retrospective formulas, coinciding
entirely with the principle here stated, require no further demonstration.
ELEMENTARY FORMULA OF VALUATION.
Let sx denote the amount of insurance in force during the 1st year; s®+i the
amount during the 2d year: Sx+2 during the 3rd year ; etc. The years are past, but at
the age x, the series may bo resolved into deferred temporary insurances of one year
each; so that the factors to give their present values, would be — _^x+l,
a’ 2, And the common factor to change these present values to equiva-
Ux
Dx
lent endowments or reserve, at the age x-\-n, will be — • So that on cancelling D.c,
-Lts+ra
and writing C®, C®+i, ...., for the differences of M,
(E) Vx+n =
bx • Dx+ bx+l • Dx+1 -)-•••• T* bx+n-l • ha+n-l —Sx • La; — Sai+1 • Cx+1 — .... — ftx+n—1 • La?+»—1
Dx,+»
Return Premiums.—For application, let P* denote the net annual premium to
insure a sum s; and let P'x denote each of the total premiums, whether gross or net,
to be returned without interest, at the end of the year of death, with the sum insured.
Then by the last formula (E), and by Sect. V, the net reserve will be
TT Pte (Nip — N)N*rt].
Assuming , to be premiums on $1000 insured, and adjusting the other
terms accordingly, let /" hereafter denote the former 1000/'. Let the former
f-r-~Dx+n = bx+n; and let b'x+n = bx+nNx+n> Also let the prepared premiums
Pa- — (P*—nx) N* ; P "a; = +1000 (1—v). Then as computed in Tables LXXY
and LXXYI:
Yx+n — f " L P/- bx+n—P a: • b'x+n> on $1000 insured.
Example 1.—An ordinary Policy issued on the Joint Lives aged 37 and 37 has
been in force 5 years 4£ months. Required the 4 per cent. Reserve.
1.84012 = AP"37, Table LXXV.
/" = 975.962 1,09963 = Xb\, Table LXXVI.
-870.46 ....2.93975 Sum, Table LXXXIL
$105.50 = Reserve on $1000 Insured. VALUATION OF POLICIES.
271
Example 2.—A Ten Payment Policy on two Joint Lives was issued at the ages of
entry 43 and 43. Required the 4 per cent. Reserve at the end of 8 years 7£ months.
6.76553 = AP43, 2.01450 = AP"43, Table LXXV.
f" = 985.577 4.01727 = AZ&, 1,01345 = W51, Table LXXYI.
+ 606.46 ....2.78280 ....3.02795 Sums, Table LXXXII.
— 1066.50
$525.54 = Eeserve on $1000 Insured.
Value of a Survivorship Insurance.—At the end of n years, the annual
premium P being just due but unpaid, the Reserve on a Policy of $1 payable on the
death of x, provided y be then living, will be
Reserve = Axln.y+n—P (1 +ax+n.y+n) = («' + £') (l — 1 + a*+-n-y±n\
= {a' + 6') x Reserve on Ordinary Joint x, y, at the end of n years.
This expression is based on the Single Premium a! -\-b'Axy ; and a', b', being con-
stant during the continuance of the same Policy, are tabulated on page 263. The age
x is presupposed to be younger than y. When x is the older age, ci', V, must be
replaced by —a and 1 — b' respectively, in all the expressions. These formulas, first
investigated in the present collection, depend on the comparison of different expres-
sions for equal reserves.
To correct for Months, whether the insurance be for the whole Life, or Temporary,
let V denote the Reserve on 1000, computed from Table LXXVI, as if for ordinary
Joint Lives, or for the same term, if Temporary; then, on the corresponding
Survivorship Policy:
Reserve = {a' + A) V + a' (1000—/").
Insurance of $1 on the life x, if y be the survivor, with return of nV the net annual
premiums paid, if y die first. At the end of n years, when the next premium is just
due, but unpaid:
Reserve = Ax+n.y+n p (l -\-ax+n.y+n)-
1 — (1 + wP) + n y + n
For other rare varieties, reference may be made to the work of Chisholm, who has
tabulated Dxy, ~Nxy, , and for every yearly difference of the Carlisle
Table.
Short Method of Determining the Equivalent Amount Insured by
Paid-up Policy. — A remarkably simple formula was devised by Sprague, and
extended by Macfadyen, showing at once the result of the double process of finding
the reserve, and thence the equivalent amount insured by a paid-up policy. I11 the
common formula of valuation, the term Ais the single premium on $1 of paid-up
insurance, page 266. Dividing the Reserve Vx+n or the whole equation by this term,
will consequently give the equivalent sum insured, which is
1 , m—n p
1 p r + Ux+n rx
~ x~ Am-n — Px+n
■n-x+n , j 272
LIFE ANNUITIES AND INSURANCES.
By dividing the first numerator and denominator by the numerator, the denomi-
nator takes the well-known expression for PxAn, given in the last fraction. Let s
denote the amount insured by the first policy. Multiplying the above result for $1
by s, we have the equivalent Amount
Insured by the Paid-up Policy = s (l — )•
' *-x+rJ
Here P denotes the net premium, or the office premium when loaded with a
uniform percentage.
Example 1.—A temporary insurance of $5000, for ten years, is effected at the age
of 40. After four annual premiums have been paid, at the end of four years, the
equivalent amount of paid-up policy for the remaining six years will be, from
Table LIII, I
= 5000(1 -.83689) =$815.55.
\ 144/ \ 1/C. 59 J/
Example 2.—A person at the age of thirty-five years has an ordinary endowment
insurance of $10,000 on his life, payable at the age sixty-five or at previous death. At
the end of ten years, the sum insured by paid-up policy, equivalent to the ten annual
premiums paid, will be, from Table LV,
s =10’000 (i-iili) = ®4042-28-
This sum is payable at “ Death or 65,” as under the original policy.
Non-forfeiting Plan.—In comparison with this correct result $4042.28, “ the
non-forfeiting plan,” so called, would give a paid-up policy for ten-thirtieths or one-
third of the original $10,000, which is only $3,333.33. This is in the ratio of the
total number of premiums payable, to the number already paid. For shorter periods
of insurance, not much exceeding ten years, the results of this empiric rule will
approach nearer to accuracy. Further discussion of this subject may be found in the
fifteenth and sixteenth volumes of the Journal, and an able article by Mr. McClintock
in the seventeenth volume.
Surrender Value.—The Massachusetts non-forfeiture law provides that on
lapsed policies, four-fifths of the net value at the time of lapsing, less the outstanding
note, if any, shall be applied as a single premium of temporary insurance to continue
the policy. And in case of death during such term of extension, the unpaid premiums
may be deducted from the sum insured. We may observe that the premiums of term
insurance in this collection, are given in Table LII; from which, corresponding results
may be found by proportion.
In ordinary practice, a deduction is made from the Net Value to give the Surren-
der Value, depending on the health of the insured, on the cost of replacing the risk, or
other circumstances. The office system of deductions generally refers to surrenders
made, as they usually are, at the end of the policy year. Todd’s Tables, Edinburgh,
1852, may illustrate this topic. If a surrender is made earlier in the year, as at the
middle of it, a portion of the Premium last paid, will be returnable to the insured, for
risk not incurred in the unexpired part of the year. For illustration, let the tabular
net value during the second year, be separated into two parts thus described: (1) The VALUATION OF POLICIES.
273
net value $17.64 at the end of the first year, increasing uniformly to $35.68 at the end
of the second year, as noted at the beginning of this Section. (2) The net premium
of $28 paid at the beginning of the second year, gradually decreases to 0 during the
ensuing second year. The full reserve at any time in the second year, is made up of
these two continuous parts, (1) and (2) ; and the office percentage is deducted by Todd
from the former part only, while the latter or premium part is directly returnable.
Thus at the middle of a policy year, half the net premium which was paid six months
before, is subtracted from the present tabular reserve; then the office reduction is
made for surrender value, to which the half premium is again restored.
Annual Valuation of an Insurance Office. — The general plan is thus
stated in De Morgan’s Essay on Probabilities, page 275 :—“ The most simple theoretical
way of conducting the process, is to ascertain the value of every policy; that is, to
ascertain how much should be given to the holder of each policy to renounce his
claim, the office also abandoning the future premiums. When this is done, it is
obvious that the office is not solvent, unless the assets arising from the accumulation
of former years be sufficient to pay the values of all the policies, and thus to buy them
all up. Otherwise, calculate the present value of all premiums due to the office, and
also the present value of all claims to which it is liable. To the former add the sum
total of the assets of the office, and to the latter add the present value of a perpetuity
equal to the expenses of management. Thus, let
P = present value of all premiums.
C = present value of all claims.
A := total assets of the office.
M = present value of all expenses of management.
If then P and A together exceed C and M together, the office is solvent. On each
of these items a few remarks may be made.
(P.) All the parties who are of the same office age, may have their several policies
considered as one collective policy, in respect of which the sum of the premiums is paid
as one premium, and the sum of the possible claims is one claim. But as these pre-
miums are payable at all periods of the year, they may be considered as, one with
another, due at six months after the valuation, at which time the present office age of
the parties may be considered to be their real age.
(C.) All bonuses which have actually been added to the policies must be included
in the claims; and the value of each claim must be carefully found, with reference to
the time after death at which it is paid.
(A.) The principal of the assets must be deduced entirely by means of the income
it yields, and must be ascertained from the income by means of the rate of interest
assumed.
(M.) Against the expenses of management may be set, as far as they go, the
incidental profits, when they can be tolerably well ascertained.”
Distribution of Surplus. — The methods employed in upwards of seventy
British Life Offices, are very plainly exhibited by Hewat in the Assurance Magazine,
Vol. XXII, page 286. It appears that nearly all the Companies allow the option of
applying the dividend to reduce the current premium ; and in the general average, the
“ loading ” on the premiums is practically refunded.
The Thirteenth Report, 1868, of the Insurance Commissioner of Massachusetts,
contains a series of communications on the Contribution plan of dividends, originated 274
LIFE ANNUITIES AND INSURANCES.
by Messrs Homans and Packler in 1863. A single illustration and the formula below
are recalled and revised from a former article in the Insurance, Times (New York, 1868),
Vol. I, page 545 :
A party aged thirty-five years, takes out an ordinary life policy of $10,000, at an
annual premium of $273 (Carlisle four per cent, net, with 35 per cent, loading).
Required his annual dividends on the contribution plan, according as the Reserve is
reckoned at three, four, five, or six per cent, interest, by the Carlisle Table; the Com-
pany receiving six per cent on investments, above expenses.
RESERVE AT THREE PER CENT.
RESERVE AT
FOUR PER
CENT.
End of
Year.
Reserve
Y.
Contribution
Dividend.
Premium less
Dividend.
End of
Year.
Reserve
Y.
Contribution
Dividend.
Premium less
Dividend.
1
$128.70
$63.25
$209.75
1
$108.50
$81.26
$191.74
2
259.90
67.19
205.81
2
230.00
83.69
189.31
3
893.20
71.19
201.81
3
334.60
85.78
187.22
4
539.10
76.56
196.44
4
451.20
88.11
184.89
5
663.80
79.30
190.70
5
567.40
90.44
182.56
10
1,322.60
99.07
173.93
10
1,136.06
101.89
117.19
20
3,100.50
152.40
120.60
20
2.782.10
134.73
138.27
30
4,898.00
207.83
65.17
30
4,538.40
169.86
103.14
RESERVE AT
FIVE PER
CENT.
RESERVE AT SIX PER
CENT.
End of
Reserve
Contribution
Premium less
End of
Reserve
Contribution
Premium less
Year.
Y.
Dividend
Dividend.
Year.
V.
Dividend.
Dividend.
1
$92.20
$96.16
$176.84
1
$79.20
$108.33
$164.67
2
188.60
97.12
175.88
2
161.10
108.33
164 67
3
285.90
98.12
174.88
3
245.80
108.33
164.67
4
386.90
99.11
173.89
4
333.60
108.33
164.67
5
486.90
100.11
. 172.88
5
420.80
108.33
164.67
10
977.90
105.02
167.98
10
843.60
108.33
164.67
20
2,498.90
120.23
152.77
20
2,246.60
108.33
164.67
30
4,205.90
137.30
136.70
30
3,898.90
108.33
164.67
In general terms, let x denote the age of entry. Let V denote the Reserve or net
value on each $1 insured at the end of n policy years, when the next premium is jnst
due but not paid. Also let Vi denote the corresponding Reserve twelve months after,
at the date of investigation. Let i denote the rate of interest assumed in computing
the Reserve; and let i' be the rate of interest realized on the Office assets. And let
(P*—e) denote the Office premium less its current expense, and px, the net premium.
The following three-fold expression will then denote the Contribution to Surplus:
(V-j-px) (i i) T (P* — 6—px) (1 ~\~i ) T" {tfx+n — Q aj+w) (1—Nl)*
Here the first term makes restitution for the difference of interest %'—i on the
renewal value of the policy. The next term restores the loading on the premium, with
a year’s interest. And the last term corrects for the excess of the tabular rate of mor-
tality qx+n above the realized rate q'x+n• In the preceding illustration, these two rates
of mortality were assumed to be equal. The original formula differs from the one above
by the subtraction of a zero equation, so that the numeric results will be precisely
the same. LIFE ANNUITIES AND INSURANCES.
275
SECTION VIII.
Life Annuities. Survivorship and Reversionary Annuities.
Successive Lives.
In the received system of life annuities, the annual payment is made at the end of
each year during the life of the annuitant, with no allowance for the proportional part
of the year of death. In the case of a survivorship annuity on the life x to commence
on the death of y, the first yearly payment is made at the end of the year, in which y
dies, and the last payment at the end of the last year before x dies. But special
formulas have been investigated, which give the price or present value of an annuity
payable half-yearly or quarterly, etc., and with a proportional part to the day
of death.
Life Annuity. To recapitulate from Sections III and V, the Present Value of
an annuity of 11 payable at the end of each year during life, is
ax = • On two Joint Lives, axy = 1-y+1.
-L'a? Uxy
When payable at the beginning of each year, the Present Value is
-t , „ Dz + Ntf+l 1 , „ Nry
1 + a*-Wx-—^-x 1 + axv~Bxy-
Temporary Annuity. The Present Value of an annuity of $1 for n years, on a
life aged x, is
n Ha;+i — NaH-w+1 „ . n—1 N# — -Na;+n
itx — tv— = ax—nax. 1 +ax =
Here and in such connections, n denotes the term of years, and is not used as an
Algebraic exponent. On Two Joint Lives aged x and y, the Present Value for
n years, is
n Hjc+l.y+l Nff+Ji+I .y + n+1 | „n—\
axy— t\ * — ~tV
i-’jy J Jxy
Deferred Annuity. The Present Value of a life annuity of $1, deferred for
n years, the first payment being due at the age x + n +1, is
„ D x+n
a,x ~ ~ •ttx+n■
„ -NaM-n+l.y+n+1 D x+n.y+n
naXy = jV = —ti" -a*+n.y+n.
Uxy
The divisor to give the annual premium for n payments, will be 1 + «” 1. And
for a change to the beginning of the year, or first payment of annuity at the age x-\-n:
Present Value =
■L'a;
On Joint Lives, Present Value =
UXy 276
LIFE ANNUITIES AND INSURANCES.
Deferred Temporary Annuity. The Present Value of a temporary annuity
of |1, to be entered upon at the expiration of t years, and then to continue n years, if
the annuitant now aged x shall so long live, is
D x+t-ax+t — • Mx+t+n N*+$4-i—^Na?+<+ra+l
Da; Da;
Increasing or Decreasing Annuity. The Present Value of an annuity on
the life x, for the whole term of life, commencing at la and increasing %b annually, is
readily proved by summations founded on the previous formula, to be
cj. i -r, • -j-
Single Premium — ~
And for a similar life annuity decreasing %b annually, we have
or in • ft-NVfl
Single Premium =
j-'x
The Present Value of a temporary annuity on the life x for n years, commencing
at la and increasing Ib each year, is
a~ (Nff+l — Nff+re+l) ~i~b (Sff+2 -^a:+ra+l)
When the similar annuity decreases Ib annually, simply reversing the sign of b in
the last formula, will give the present value. But in this case b must not exceed
- , that the annuity may not finally become negative.
For the Present Value of an annuity of $1, increasing annually by $1 for n years,
and then continuing constant at In through life, we have
Single Premium = S£±lT^£±!LL1.
-Dx
The Present Value of an Annuity certain for n years, commencing at $a and
increasing $b each year, was not given in Part I. But by taking the derivative D of
the last equation on page 59, with respect to v, we find the required value of this
increasing annuity:
(a_£) v + 6yDV = (a + nb + 6).1—^—
\ X/ X X
The value of the similar annuity decreasing Ib each year, is found by reversing the
sign of 1).
Valuation of Annuity Policies. The Present Value of Single and Joint
Life Annuities at entry, being ax, etc., the Reserve after n years will evidently be
ax+n, a.v+n.y+n > etc. In like manner, the Present Value of a Survivorship Annuity of
II on the life of x after the death of y, being ax—axy, the Reserve n years after entry,
will be cix-\-n—ax+n.y+n•
But if the Survivorship Annuity of $1 on the life of x after the death of y, is con-
ditioned with the return of the single premium, without interest, should x die before y,
this premium and the Reserve n years after entry, are proved to be LIFE ANNUITIES AND INSURANCES.
277
Single Premium = ——a~ ;
1 —A xy
Reserve dx+n dx+n.y+n~\ ~^-Ax+n.y+n'
1 —A xy
When the simple Survivorship Annuity of $1 on the life of x after the death of y,
is paid for by an annual premium during joint life, this Premium and the Reserve
n years after entry, will be
Annual Premium = ;
1 +aXy’
Reserve = Ctx+n Clx+n.y + n— 1 , (1 -f- dX+n.y+n)‘
I "~r CtXy
The premium for one year only, to secure a Survivorship Annuity in case y dies
in the first year, is (1 —py) ax, or qy ax. At some ages, this rate is more, and at other
ages less than the average premium during the joint lives. In the latter case, the
Company would not be fully recompensed in the first years, unless the policy were
continued to maturity. The present divisor to give the annual premium is 1-fif
this were changed to 1 + at-y, the ten payment premium so found, would generally
prove sufficient in the first years; and so of various other scales of premium.
Under an arrangement of the Survivorship Annuity, such that the first yearly
payment is made just one year after the death of y, or more generally m times a year
thereafter, and a proportional part is to be paid up to the date of x’s death, Mr. Wool-
house finds the Present Value
= (1-&n)(ax-a**+if) + mP ■AiS'
This is the formula given in the Journal, Vol. XY, page 113. Mr. Sprague had
previously determined the following approximate values, according as the annuity is
payable
Yearly, Half-Yearly, Quarterly.
(ax—axy) Vv, (ax—axy). 1+^v., (ctx- axy). ——•
Note.—It will sometimes be required that the common life annuity be extended to one more pay-
ment, at the end of the year of death. The present value of such payment being the single premium
Ax on $1 insured, the present value of the extended annuity will be ax-\r Ax.
In case the annuity is to be extended proportionally to the day of death, the present value becomes
ax + \Ax; since death will occur, on the average, at the middle of the last year. This is the method of
Francis Baily. A slight further correction is given by the formula on page 242.
Annuity on the Survivor or Last of any Number of Lives. Let us
take first the case of two lives, of the present ages x and y. Then pXy n = lx+n-t-lx and
py,n = ly+n-i-ly will denote their probabilities of living to the end of any future year
denoted as the nth. By Sect. I, Principles 3, 4, the product (1 —px,n) (1 —Py,n) will
express the probability that both will be deceased ; consequently,
1 — (1 —px, n) (1 —Py, n) = Px, n + Py, n—Px, n Py, n >
expresses the probability that both will not be extinct, or that one at least will be living
to receive the nth annual payment. Therefore multiplying each term by vn, and 278
LIFE ANNUITIES AND INSURANCES.
taking the sum of the products for n = 1, 2, 3, ...., we find the required value on the
longest of two lives to be the sum of the two single annuities, minus the joint annuity; or,
~ "F —®xv'
The same mode of solution, illustrated by the theory of the higher equations of
Algebra, gives the value of an annuity on the longest of three lives, denoted by a line
over the suffixes:
axyz ~ ax “t” (fs (a%y + (,JCZ + ttyz) + Ujcyz .
And generally, the value of an annuity on the longest of any number of lives, is
equal to the sum of the annuities on the single lives, minus the sum of annuities on
the joint lives combined by twos, plus the sum of annuities on the joint lives combined
by threes, and so on; the odd orders being positive, and the even negative.
Possibly this species of annuity might be resolved into binomial or single life
factors, like the elementary probabilities above, aided by assumed constants. In
another expression, if Cm denote the sum of joint life annuities on m lives combined n
in a set, then evidently, extending to n such terms,
= Q>m — Cm-\-....
VALUE OF AN ANNUITY ON THE LAST n SURVIVORS OF m LIVES.
For an annuity to commence immediately and terminate with the last death
but one, of m lives, the present value is demonstrated by Milne, Yol. I, pages 45, 126,
to be
Ct-2.C4+3.Ci-4.c4+....
Thus the present value of an annuity to end with the last death but one of two lives,
is axy\ and similarly of three lives, x, y, z, the present value is
ttj-y -f" dj’z -p (tyz 2 ttxyz •
In every case, the algebraic sum of the coefficients of a is 1, to indicate one
annuity in present possession.
And generally, to commence immediately and terminate with the last death but
n — 1 of m given lives, the present value of the annuity is the series of m—n-\-1
terms of C,
nn a, nw+1 i a, W + 1 nn+2 w + 1 n + 2 nn+s ,
m — n. -p n. —-— . \jm — n. —-— . —-— . \jm -p ...
This general formula is proved by Milne, pages 42, 126, by multiplying out the
probabilities of five lives, and applying the rules of combination. By introducing an
auxiliary t, we obtain the following shorter' demonstration:
Let px> Py, , denote the tabular probabilities that the rn lives x, y, z, ,
will survive h years. And let the continued product of the m probabilities of decease
(1 —px) (1 —Py) (1 —pz) he temporarily denoted by P. The required probability
that “ n or more” out of rn lives will survive h years, will be the sum of elementary
products of the form
factors x (m—n) factors = 2pxpv (1 —ps) (1 —pv) = Ps - -. —Py
1—Px 1 —Py
+ 2 (w + 1) factors x (m—n — 1) factors = etc.,
+ LIFE ANNUITIES AND INSURANCES.
279
The right-hand expressions after P represent the combinations of the fractions
taken n in a set, n -f 1 in a set, and so on to m in a set. By comparison with the
Algebraic theory of the roots of the higher equations, these combinations will be the
coefficients of tn, t,l+1, ... .tm in the development of
pI1+r^),"],) ■ • • = li+[i+(<-!)*]■ • • •
= +{t—1)H C(») + (^—l)w+1 C(re+1)+^—i)«+2.C^+2)+ +(t—l)m.C(m\
Here denotes the sum of the combinations of the m factors px, py, ,
taken n in a set; etc., etc. Developing and omitting all powers of t lower than the
wth; then making t = 1, as above indicated, we find the precise result sought, here
denoted by P':
P' = Q> — + D-p+ Q(n + — ” (”+?) (” + 2) (]{n + 3)_|_
1 • 1 • 2 • 3
Lastly multiplying by vh, and taking the sum for h — 1, 2, 3, ...., we change from
probabilities COh to annuities Cm, and the formula on the preceding page, from Milne,
is verified. The symbolic form is evidently developed as far as Cm.
REVERSIONARY ANNUITIES.
General Problem.—To determine the present value of an annuity dependent
upon the joint existence of the last n survivors out of tn lives x, y, z, ...., after the
failure of the joint existence of the last n' survivors out of m other lives x, y', z!, ....
(Milne, p. 130).
The leading principle of solution is very simple. As just described, let P' denote
the probability that n or more of the in lives will survive h years. And let P" denote
the like probability that n' or more of the m' lives will survive h years; then 1 — P" is
the opposite probability, that they will fail. The joint probability is therefore
P' (1 —P”). Multiplying this by vh, and taking the sum for all values of h, we have the
required value of the annuity in terms of single and joint annuities. Also the required
annuity and its components may all be temporary or all deferred for equal periods.
Example 1.—Required the value of an annuity on the life of x to commence after
the death of y.
Annual Premium = axy-
-L + ttxy
Single Premium = ax—axy •
By aid of the above type P' (1 —P"), this solution and more complex ones yet to
follow, are easily written by inspection. To verify the present result, cancel the
annuity on x, during the joint lives. This latter method in De Morgan’s Essay on
Probabilities, is termed “ the balancing of annuities.”
For future reference, let another form also be given, which is easily proved by substi-
tuting for N below, the equivalent series of D and canceling:
clyT)rax -f- + l + dy+2TI*+2^sc+aT ■ • • •
a*~a*y = lyDx
n n {ly— ly + l) Nff+l T + 1 • • • «
Ctx—U-xy — 7 p*
ly^x 280
LIFE ANNUITIES AND INSURANCES.
2. Annuity on the life of x to commence after the first death of y or z; the
present value or
Single Premium = ax—axyz\ Annual Premium = ——(137/z.
1 d- dxyz
„ {Jy lz — 4:+l) Na + i + {}y+l G + l — ?y + 2 fc + 2) Na> + 2-f- • • • •
ur ax—dxyz — 7 ; 71
(y lz X)X
3. Annuity on the joint lives x, y, to commence after the death of z:
Single Premium = axy—dxyz j Annual Premium = (~-—°^.
f + dxyZ
Jf.. Annuity on the life of x after the death of both y and z:
Single Premium = aT—axy—axz + dxyz •
* tt, • dx — dxy—((xz d- dxyz
Annual Premium = - — •
1 “t“ d'xy “h dxz — dxyz
5. Annuity on the survivor of x and y after the death of z:
Single Premium = ax + dy—axy—axz—dyz + dxyz•
Annual Premium = I^£±£iS.
1 H- dxz H- dyz — dxyz
Verification' of Formulas.—In their present state, the algebraic sum of the
coefficients of a in the single premium is zero, indicating no immediate possession of
the annuity. And the only change from the present state must occur by the first
death of either x, y, or z; by making this supposition in the last five cases, and noting
the results, the solutions given above are again verified. The same methods can be
further applied to the solutions following.
6. Annuity on the life of the survivor of x and y, that is on either one, after the
death of the other. Give an annuity to both, to be restored as long as both are alive:
Single Premium = ax + ay—2dxy, Annual Premium = ax + cly—
1 + dxy
7. Annuity to commence with the first death of x, y, or z, and to end with the
joint life of the other two, that is with the second death: Grant each pair a joint life
annuity to be restored as long as all three are alive:
Single Premium — axy + dxz + ayz—3dxyz >
Annual Premium = ++ 'aS=^2m.
1 -\-dxyz LIFE ANNUITIES AND INSURANCES.
281
PARTITION OF ANNUITIES.
8. An annuity on the last survivor of two lives x, y, is to be equally divided
between them during joint life, and afterwards to go to the survivor. The present
value of
afs interest = ax—\cixy, y’s interest = ay—\axy.
Their sum evidently makes up the whole value of the annuity.
9. An annuity for the longest life is to be divided at each payment equally between
the survivors of three lives, x, y, z:
Value of z’s share = ax—| (axy + axz) + \(ixyz •
Here x may denote either one of the three lives. And generally for equal division
of such annuity between the survivors of m lives successively, if [ ] denote that we
exclude from Cm as before defined, every combination into which x does not enter-
x’s share = [<£,] +*[<&]-*[<&] + .. • ■ ±- [<&].
71
10. Two lives x, y, possess an annuity on the survivor of them. If either dies
before a third person z, the remaining annuity is to be equally divided between z and
the survivor during their joint lives. Required the value of z’s interest-
z’s interest = \axz + \ayz—axyz. (Milne, p. 139.)
11. An annuity on the last two survivors of three lives x, y, z, is to be divided
equally among them while all are living; and after the decease of any one of them is
to be divided equally between the two survivors during the remainder of their joint
lives. The value of
x’s interest = \axy + \axz—\aXyZ ■ (Milne, p. 140.)
12. An annuity, after the decease of x, is to be divided equally between y and z
during their joint lives ; and is then continued to the last survivor for his life.
(Milne, p. 141.)
y’s interest = cty—&xy— \&xyz •
13. An annuity on the last survivor of x, y, z, is to be divided equally between x
and y during their joint existence. After the decease of either of them, it is to be
divided equally between z if then living, and the survivor during their joint existence.
And when there is but one survivor left he shall receive the whole of the remaining
annuity. (Milne, p. 142.)
afs interest = ctx—\u-xy—\u-xz H- \^xyz •
CONDITIONAL PERIODS.
11/.. To find the value of an annuity payable during the joint lives of x, y, and also
during t years after the death of y, provided x shall so long live.
During the first t years, whether y be dead or living, x will he entitled to the
temporary life annuity,
Na+I — Na+c + 1 T)x + t
— Ctx tS -ttx + t'
Ux -L'iC 282
LIFE ANNUITIES AND INSURANCES.
After this, or at the end of any given year denoted by t-\-n, the annual payment
will depend on the combined chance that x is living, and that y was alive t years
previous. This expectation, on both multiplying and dividing by lx+t, also summing
with respect to n, to the end of the life table, gives
t lx + t + n ly+n , lx + t „ lx+t + n,ly + n Dx+t „
2vt + n. —j— . Vn. f = -jr-.ax+t.y
lx 'x 'x + t vy -L'a;
Adding this to the former result, we have the total value,
Dx+t t \
ax yv— [ax+t—flx+t.y).
-L'x
15. An annuity certain for the term of 15 years is to be enjoyed by P and his
heirs, during the joint existence of two lives x, y. If that joint existence fail before the
expiration of 8 years, the annuity is to go to Q and his heirs for the remainder of the
term of 15 years. To find the present value of P’s interest, and that of Q.
The value of P’s interest for the first 8 years is evidently the temporary annuity
on the joint lives x, y\ for the next 7 years, P’s interest is the deferred annuity certain
for 7 years multiplied by the chance that x and y will both live through the first period
of 8 years to secure it. That is,
P’s interest = axy—(ax+8.y+8—~J~)‘
The value of Q’s interest will be obtained by subtracting P’s interest from the
1_ v15
value of the entire annuity —j
16. A reversionary annuity to commence on the death of x and continue for the
remainder of the life of y, is also made payable for t years, whether y is alive or dead :
y’s interest = Ax {ay+t—ax.y+t).
I Uy
17. To find the single and annual premiums for an annuity to commence at the
death of y, and continue payable during the remainder of afs life, but to be payable
only if y dies within t years.
In this problem, which is quoted from the Journal, Yol. 9, page 302, the single
premium will be
ly + / x
ax— Uxy YlT \ax + t — ax+t.y+t)-
And the divisor to give the annual premium, payable through t years, if the joint
life so long continues, will be 1 d-ffixy1- We may observe that the single premium is
evidently the first t terms of the series before stated on page 279.
18. Problem.—To determine the present value of an annuity on the life of x, to
commence with the failure of the joint existence of y and z provided it be z who
dies first.
In case the ages of y and « are equal, the probability that z will die first of the
two is Therefore \ the result of the preceding Problem 2, on p. 280, gives the value:
Annuity on x = (ax—axyy). LIFE ANNUITIES AND INSURANCES.
283
Secondly, if the ages of y and z are unequal, in case x is the oldest life, and
approximately in other cases, the chance that z will die first of the two is usually taken
as in practice. So that of the result of Problem 4 gives the value of the annuity
on x, after the death of y. Problem 3 gives the value until the death of y ; and the
sum of the two will express the total value :
Annuity on x — 4 {ax-\-aXy—aXz—xyz) j
Annuity on * = 2 H = 1, 2, 3, ;
lz My Ux
That is, Annuity on x = 2 c^z+n~1 jjj—.
lz ly D#
Another solution of this Problem will be given in the next Section. For other
combinations of Survivorship, reference may be made to the works of Morgan, Bailey,
Milne, Sang, Farr; and to the admirable papers of Makeham in the Journal, Vols. 9,
10, 12.
ON SUCCESSIVE LIVES.
Problem 1.—To determine the present value of a deferred income of $1 per
annum to continue forever, the first payment to commence at the end of the
death year of x.
The principal l when multiplied by the rate i gives the interest $1 in each year
perpetually. Withdrawing from this principal, the annuity of $1 during the life of x,
we have the required result:
Present Value = \—ax = + Ax.
i \
Here the last member is found by merely substituting for a its equal, from the
common formula
Ax = 1 — (1— v)(l+ax) =
Problem 2.—“ An annuity is to be enjoyed during the existence of a life aged x,
and at the end of the year of his decease, a successor aged y is to be nominated, who is
to enjoy the annuity during his life. Required the present value of the annuity on
the second life y.”
In the system of compound interest, if v" is the factor to discount a sum from
the end, to its present value at the beginning of the third year, v' the factor to
discount through the second year, and v the factor to discount through the first
year; the factor to discount through three years is proved to be the product vv'v".
So, the single premium A is evidently the factor to discount the insurance of $1,
from the end, to its present value at the beginning of the whole term of insurance;
and AA'A" would represent the factor to discount through three successive periods
of life. In the Problem above, the value of $1 annuity during the life y, with first
payment on the day of nomination, 1 ay. The factor to discount this sum
from the future date of nomination, to the present time, being A*, we have the
Present Value of the Annuity 284
LIFE ANNUITIES AND INSURANCES.
During the second life = (1 -\-ay) = Ax + --.j (1—Ay).
During the two successive lives = ax-\-Ax (1 + ny).
For ay, we have above substituted its equal, as in Problem 1. If x, 1, 2, ... .n—1,
n designate the successive lives in order, the present value of $1 annuity to continue
during the wth successive life-time beginning with an immediate payment, will be
Ax Ai A2.... Aw_i (1 -f- an) 5 or -%j Ax (I—An) A\ A2.... An—\ >
Problem 3.—Required the total present value of $1 per annum during n suc-
cessive lives, the first payment to commence at the end of the death year of the
present life x.
The result sought will be found by making n — 1, 2, 3, ... .n, in the last expres-
sion, and taking the sum of all the terms. Thus, omitting the common factor
Ax, and making n = 1, 2, 3, ... ,n, the first result is 1 —Ai; the second is
Ai—Ai A2; the third is Ai A2—Ai A2A3, etc. From these, we see that all the terms
will cancel each other, except the first and the last. Hence on restoring the omitted
factor,
Present Value = A3;(l—AiA2A3. .. . An).
When n is very great or infinite, the right-hand product vanishes, since each A is
less than unity, leaving a result which is independent of the future ages to be
nominated:
Present Value = + A^
When the successive lives, after the life x, are all to be of the same age y at
nomination, the formula for the value of the nth. successive life, given at the end of
Problem 2, becomes
A* (l + %) AJ-1.
In like manner, the formula of Problem 3, for the sum of n lives, will be
(1 + ?) A*(l-A?)-
Problem 4.—“ Instead of an annuity, a fine of $1 is to be paid at the end of the
year in which each life in possession fails, being the time when the next succeeding
life is nominated. Required the present value of each fine.
By Problem 2, when n — 1, Value of the first fine = A^,
n — 2, “ “ second fine = Ax Ai,
n = 3, “ “ third fine = A3; Ai A2.
n = 4, “ “ fourth fine = Ai A2 A3, etc. LIFE ANNUITIES AND INSURANCES
285
The sum of the terms written, will evidently be the present value of the
first four fines. Each term is readily derived from that which precedes, to any
extent.
The present value of the first n fines, if the lives are all of the same age at
nomination, will be the sum of n terms above, as a geometrical series, of which
the ratio is Ay; and the sum is
A® (1—Ay) . . Ax
—~—7 • or Tf n be infinite, the sum is .—
-L—Ay 1—Ay
By subtracting the former expression from the latter, we obtain the present
value of all the fines after the nth., that is, of the (w + l)th and following fines to
perpetuity:
A*Aj
1 Ay
The preceding summary is based chiefly on two original papers by Mr. Peter Gray
in the Assurance Magazine, Vol. 2. In Vol. 21, a memoir and Table for the Enfran-
chisement of Copyholds, is given by Edward Smyth. Further problems relating to
English church livings and copyholds are resolved in De Morgan’s Essay on Proba-
bilities, pages 228-236.
SECTION IX.
Derivative Formulas. The Continuous Method.
Besides the regular Commutation Formulas, several principles of mathematical
analysis have conducted to other formulas, which occasionally offer new advantages.
Especially the common method of Algebraic elimination has been the guide to
valuable improvements. For example, writing x and then x + l in the formula for the
expectation of life, we obtained two equations, from which the series common to both
was easily eliminated, leaving the simple equation between ex and ex+i shown on
page 243. By the same general method, and in other ways, the following Formulas of
Annual Derivation have been obtained:
For the Expectation of Life, ex = px {ex+\ + \).
Annuity on a Single Life, ax = vpx (1 +ax+1).
Annuity on Two Joint Lives, axy = vpxpy (1 +ax+\.y+i)-
Annuity on Three Joint Lives, axyz — vpxpypz (1 + «a;+i.y+i.2+i)-
Temporary Annuity, = vpx{\ +«”+{)•
Temporary Annuity on Joint Lives, afcy = vpxpy (1 +cix+\.y+\)-
Insurance by Single Premium for Life, = vpx j -—l + | •
By Single Premium for Term, A” = vpx j -—1 + A”+{ | • 286
LIFE ANNUITIES AND INSURANCES.
By Single Premium on Joint Lives, Axy = vpxpy ( 1 + y+i)-
\P%Py '
Single Premium of Survivorship Insurance of $1 to be paid at the end of the year
in which x dies, provided y be the survivor,
&l.v = vpxp, | I (A-l) + 1) + A*+<•*<•! | •
The values of Apx and A (vpx) are already given in Tables XXYI and XLIII.
And at the oldest age of the life table, where these computations usually commence, is
0 ; consequently e99 = £; the annuity = 0; the single premium A99 = v; and
= \v (1 +py). From this limit, we can determine the other values in a connected
series, by making x = 98, 97, 96, 95, ...., in successive formulas. The operations
will be greatly aided by the Addition and Subtraction Logarithms given in the Tables
and Formula of Peter Gray, whose able researches first directed general attention to
this method.
For practical illustration, Tables XLIY-XLYII in the present series, were com-
puted by the formula of Temporary Annuities, a” = vpx (1 +a”Li)- First the ages 99,
98, 97, 96, ...., were written at the head of so many columns; under these, the values
of A (vp) from Table XLIII for the ages one year younger, were copied, once for all,
to be repeatedly added each in the same column.
TEMPORARY ANNUITIES. COMPUTATION OF A (1 + aT\
Payments
96
97
98
99
Age x.
n
1.681936
1.627579
1.585026
T.505846
s
1
II
2
0.153574
0.141329
0.120743
A(1 -f-a1).
T.835510
1.768908
T.705769
Sum = a a.
3
0.200677
0.178369
A(l+a*).
T.882613
T.805948
Sum = Xa.
4
0.214754
A(1+a«).
Here at 4 per cent, interest, A = T.505846, A (vpw) = T. 585026, ,
A(l+«98) = 0.120743, A(l-f«(j7) = 0.141329, .... At the first, regarding A (vp) as
Aa, we look out A(l-frt) in Gray’s Table and copy it below, one column to the left.
Every logarithm A (1 + a) so found is next added to X(vp) at the head of the column,
and the sum written underneath. Again regarding each sum as A a, the corresponding
A(l + a) is then found by Gray’s Table, and copied as before, on the next lower line,
one column to the left; and so on. The alternate lines present a complete Table of
A(l + rtx-1) for Temporary Annuities. When n is 1, the tabular result is uniformly 0;
and at the foot of the columns, the last A (1-f a) must coincide with the result of com-
mon life annuities. To prove the correctness of the work, while in progress, every
tenth or fifteenth column can be computed by the independent Commutation formula
or otherwise, for comparison and correction.
It appears scarcely necessary to observe in the use of Gray’s Tables, when the
negative index of the argument is 4, 5, 6, etc., we find Log (l-\-x) as if the index LIFE ANNUITIES AND INSURANCES.
287
were 3, and prefix to the result of Table I, as many decimal cyphers ; or in the use of
Table II, as many extra decimal 9’s, as the excess of the given index above 3, denotes.
Thus when Log a; is 5.500, Log (l-f-z) is 0.0000137, and Log (1—x) is I.99998G2.
There are Tables to seven places by the German author Zech, and by Wittstein;
Gray’s to six places are more conveniently arranged for practical use.
Formula of Valuation.—Three different forms have been published, the first
by Hon. Elizur Wright, under the naitte of “the accumulation formula,” a second by
Mr. D. P. Fackler, and a third by Mr. Emory McClintock. To demonstrate the last,
let i denote the rate of interest; and v the ratio + i)- Pet Vx+n denote the
Reserve at the end of n years on each $1 insured at the age of entry x, with the net
annual Premium Px ; so that Yx+n + will be the Reserve just after the Premium is
paid. Assuming lx+n persons to be thus insured for $1 each, the death losses in the
ensuing twelve months, according to the life table, will be lx+n—lx+n+1- Hence the
Reserve with a year’s interest, after payment of the $1 death losses, evidently gives
lx+n 0 x+nY Pa;) (1 T'O — (Jx+n — fc+n + l) = lx+n + •
Dividing by lx+n+1 and reducing, we have the required formula:
x+n+1 = 1 — ~ [(^—Pa;)—"V x+n\-
vpx + n
At the age x, just before the first premium is paid, the Reserve V is evidently 0,
which is the initial value on the right of the last formula, when n is 0. Also at the
end of the last year, V on the left becomes 0 when a temporary insurance expires, or
1 when an endowment matures, or 1 when a common life policy attains the oldest age
of the life table. Thus the computation of the formula may commence at the begin-
ning, or at the end of the policy. Table XLIII will give the values of A (vp).
By making the annual Premium P to be 0, the formula becomes applicable to
paid-up Policies. Consequently one year before maturity, or just after the last annual
premium is paid, the Reserve on every $1 of endowment, will be v.
In the higher analysis, all the preceding formulas would be classed among
“equations of finite differences of the first order;” which are always integrable,—
Boole’s Finite Differences, Ch. IX, Art. 8. But the integrals are already known in the
Commutation and other forms.
THE CONTINUOUS METHOD.
Maclaurin, Euler, and many others have demonstrated the following standard
formula, which will serve the double purpose of summation, and conversely of quadra-
ture. First, let the values of the ordinate u be taken at the distance h from each
other; and let denote the special sum indicated; then
(A) liJJk = ll + Uh +1(2/1 + • • • • + U(n-i)h + tylnh)
= C+f ....
For the present applications, let nh fall beyond the oldest age of the life table,
where unh and its differential coefficients all vanish. For, by Taylor’s Theorem, when
a continuous function becomes permanently zero beyond a certain limit, its derivatives 288
LIFE ANNUITIES AND INSURANCES.
also vanish. Taking the integral of the right-hand member between the limits nh and
0, and making h to be 1,
tt Cnh 7 1 duo . c/3wo
V'-J0
' Subtracting this from the former equation integrated between the same limits,
and transposing,
(B) U, = * (A*-l) (i‘-l) ....
Application- to Lipe Annuities.—Referring back to the common formula
Dxax = N*+i = Da;+i-fDa,+2-(- , let us here make w0 = Vx, u\ = Da:+i, etc.
Then will Ul — £Da; + Da;+i -f- • • • • = ij-Ibc + Xai + i, and U& =
.... Substituting in the formula (B), dividing by Da;, and reducing,
(C) Ux = jy (Da;+A + + 3A-(- - ■ - )-f (ll — 1)—yV — l)(iua; + d).
The values of D are given in the Commutation Tables, and the values of n and 8,
in Table XLIII. The latter represent the differential coefficient of or lxvx, divided
by that is, of log Da,; since u = -dlx-±-lxdx, and d = —log v = log (1 + ?’)• Ac-
cording to W. S. B. Woolhouse, the pioneer of this mode of calculation, in the Journal,
Vol. XI, page 323, the best intervals to adopt for h, will be 5, 7, or 11 years. In the
case of two Joint Lives, x, y, the derivative (>x will become pxy or nx-\-py: and so on.
Example.—Required the value of the life annuity ax, when x is 60 years, and
interest is 5 per cent.
h — 115 «60 = Ty— (D71 + D82 + D93) -j-5. —9.94 0*60+ d) = 9.1322.
h = 7; f*6o — (D67 + TI74 + D8i + Dss + D95) + 3.—4(f*6o + d) = 9.1336.
The standard value from Table XXVIII is 9.1322. In the above formula for 11 year
intervals, the last coefficient was arbitrarily changed from 10 to 9.94 to compensate for
omitted terms. The annuity so computed for age 40 is 13.7601, which differs very
slightly from the true value 13.7602. With this mode of amendment, devised by the
writer, the intervals li may probably be taken so large that only two or three values of
D will be needed. Thus for thirty-five or other ages adjacent:
20
h — 20 ; dx — tV" (I\: + 20-l-kbE + 40'{~Da: + 6o) "f” 9*5— 32.7023 (px-\~8)'
J-'x
Thus with slight correction of the last coefficient, through a single central value of
a, this method (B) may also be extended to joint lives and survivorships; and further
trials are invited.
Instalments.—Let h now designate a fraction i of 1 year or of II annuity;
and let the equal instalments be payable at the end of each interval h or of the
year. In formula (C), the present value of the 1st instalment is when the LIFE ANNUITIES AND INSURANCES.
289
instalment is 1; or it is 7* x IV+A-f-when the instalment is or — ; and so for
the following terms of H, which evidently make up the required present value a^m\
Therefore, substituting 1-i-m in place of h, and transposing the terms, we find the
present value of the annuity payable in instalments of $- at the end of equal intervals
71Z
— of each year,
m J
(m) , m — 1 m2—1 .
(D)
By adding -, we evidently obtain the present value of the annuity payable m
times a year in advance. This will also be the divisor of the single premium to give
the premium of insurance by m annual instalments. The present addition of — gives
/T,. m~\-1 m2—1.
(E)
Continuous Annuities dx. When the equal instalments are so small as to be
payable momently, the corresponding value of the annuity, denoted by ax, is readily
found from formula (D), by making the number of intervals m very great or infinite:
(F) dx~ax-\-\— + d).
dXy = aXy -(- |—vV T" lly T" d).
On the Survivor of Two Lives, = dx-\-dy—dxy,
, 6
— ax-\-ay—axy-\-\—
Continuous Expectation of Life, ex = ex—^fix.
exy — exy—^
Complete or Apportionable Annuities cax. The characteristic c is prefixed,
when the annuity is to be paid with a proportionate part to the instant of death. A close
and very simple approximation for this case has already been given in the Note of
page 277; that is, Q,ax — ax + = (1 —\iv)ax+%v. Here the proportionate part,
defined by the actual day of death, is to be paid at what would be the time of the next
regular instalment of annuity.
The more exact solution first investigated by Mr. Sprague is verified by Mr. Wool-
house in the Journal, Yol. XV, page 107, substantially as follows : Let x‘ denote the
age at the beginning, and x' + at the end of any regular interval. And let x' +1
denote an intermediate age, increasing uniformly to af + —, so that if death should
occur, $7 will be the proportionate payment. By page 252, ante, and since M is a
decreasing function, the single premium to insure this $7 during the instant dt, is
—t.dM.x Or integrating by parts from t — 0, to t = —, we find the single
premium to insure through this interval to be 290
LIFE ANNUITIES AND INSURANCES.
1 — /»! —
+ / >» M#/+1 dt,
m m J o
both terms to be divided by Dx- From this expression we next find the sum or single
premium for all the intervals by formula (A), which gives the total augmentation of
annuity:
_ j 1 , _1 dMx_ ) _ L_ A x
~ [2m x+12m* dx ' x~2m x 12m2
The mark over M denotes its continuous value; and so of the single premium A.
It will presently be shown that = Ax = 1—d.dx, where d = log (1 + 0 ; also
that d Ma, = —vx.dlx. Therefore, adding this augmentation to formula (D), we obtain
the complete annuity payable with a proportionate part to the day of death:
(G) ca*’"> = W)dx+nri*
= (1-s) (s'+i_ir)+sr>'
Change from Ordinary to Continuous Annuities and Premiums.—By
simply changing the common annuity ax to ax + %, which will rarely require the
slight further correction shown in formula (F), we obtain the continuous annuity dx.
And conversely, ax = dx—%, very nearly.
In respect to Premiums, the operations already explained on page 264, namely,
A = A~Vv, and A = A Vv, prove to be sufficiently exact in common practice. For
illustration, a large variety of single premiums are represented by the well known for-
mula, A = 1 — (1—v)(l + «). Eliminating the annuity a by comparison with the
third type of continuous equation stated below, we have
/tt\ d pd A
-|;)« + (ft-+|)«+ • • • •
This type will be of general application, if A, d, depend on all the lives; while
x, y, , designate only each life, by whose death the insurance would become
payable. Thus,
~r - ddx _ ~j~ / i \ - ddxy dttxy j.
A»; — yxax—-jj— ? Axy — (yx + fiy) axy > e^0. LIFE ANNUITIES AND INSURANCES.
291
,\ L I, ddxy' a i _ - ddxy
Axy — Ctxy ’ Axy ~ dyaxy
T"l_ .7 d(JXyZ' — \ _ ddxyz
'xyz ~ l x <- AXyZ — 1—ddxyZ) etc.
(K) Ax = = i*); etc.
Here a denotes the ordinary tabular annuity ; and in both, d == Nap. log. (1 -f f).
The fourth type is but another form of the first, where x, y, designate only each
particular life, by whose death the insurance would become payable. Thus,
(L) Ax = ; Axy — f ’ e^c>
~Txdx ('lxaxy) ’ Axyz = ~TxJx (lx (lxyz) ’ etC‘
The derivatives with respect to age, as x, which occur in the preceding equations,
(unlike derivatives with respect to v), are here determined with sufficient accuracy by
first differences. Thus,
ddx 1 , \ daxy daxy
— dx — Y —<**+!)> — t [ax—i.y—i—ctx+i.y+\); etc.
ddxy i / \ ddXy I, \
dx — * \ax—i v—Ox+I.y); — i [cix.y—i — ax.y+i); etc.
dlx lx—I lx+1 n dx—\ y lx + tdx + l.y
flx ~ Tx~ 2/7 xy ~ 2/7
As heretofore, the single premium, when divided by l+cix for a single life, or by
1 + axy for two joint lives, will give the annual premium.
Demonstrations.—It is shown in the Differential and Integral Calculus, in relation
to the area of a curve whose ordinate is denoted by y, that fydx represents the sum of
y ax, when the difference of abscissas Ax is.diminished to its limit. Accordingly, let
the commutation Dx = y. In the common sum of annual values, N«-, = +
+ Daj+2+ ...., the value of ax is 1 year or 1. Diminishing ax to its'limit dx or dt,
we have the continuous value N = fJ)x+tdt.
In the similar sum, M* = (/«—/«+i) vx+l + {Ix+i—lx+n) vx+%-\- , the value of
ax is represented by lx—lx+\. Diminishing this a to its limit —dlx, since lx is a
decreasing function, we obtain the continuous value; then integrating by parts, 292
LIFE ANNUITIES AND INSURANCES.
M = — fdl.vx = —lvx + log v J‘lvxdx,
= lvx — d/Ddx — Da, — d.N®.
Dividing by Dx, we verify the second type of equations, before noted, A = 1—6.a.
For another standard example or application:
Ix-dx — Ix+t-vt.dt; lx-A# —fvKdUt-
Integrating by parts, and observing that v = elos *; d = —log v = log (1 +»);
lx Ax = —vt.lx+t+log vflx+t-vWt.
Applying the limits t = oo and t = 0, also substituting lx dx for its equal,
lx Ag, :— lx log V . lx • dx ) Aa, — 1 — d. dx •
In another form, noting the principle of the Calculus that —,
differentiating the first equation with respect to x, then substituting in the second,
7 X d {lx das) m ~r - ddx
*"• * “ ~ dx~’ - , xa = 1; v' = vq; u = log -. qx.
vs g
Demonstration of the Formula of Survivorship Insurance on P. 263. Having
tabulated the coefficients a’ and V, let us here adopt the notation, and the two auxiliary
ages, x, y, stated on page 263. The probability of the m lives all surviving the time t
is —^“ ■ or smt.gix (^-b; and the probability of the n other lives surviving the
time t is entirely similar. Hence by the same reasoning as for two single lives, x, y:
-r- 1 f* d (lx'4-t ■ lx"+t • • • ) Ul'\t • lv"+t,.... 4
\x'x»..)y'y>~. — — / — j—i -• —7 -V\
lx' • lx" • • • • ly' • ly"....
= —f(m log s.dt-\-log g.qx.dql).,dm+n)t.gi'f0+qy)
Here the differential has been more conveniently found by the latter form of the
identity dl—l.d log l — d log l.l. Interchanging the lives m and n,
A(yy'.= —f{n s.dt + log g.qv.dq*) $(»»+«) *.^(2*+?1')(^-i).
Eliminating the last term of the two equations,
qv A(Ja,//..)y'y,>,.—qx A(yy/..)a/*?/“ = — (mqV—nqx) log s.dx>..y>...
At the end, a has been written in place of the equivalent integral on all the lives. 294
LIFE ANNUITIES AND INSURANCES.
Also A(JX"..)y'y".. + A(j,y/= Ax'..y\.. Eliminating the middle term by compar-
ison with the preceding equation,
(S) (qx + qy). A(p^>/. = qx.AX'.y..-{mqy—nqa}) log s.dx'.y'...
Again, eliminating a, by the relation A.v>_y,_ = 1—daX'..ywe obtain the general
formula on page 263, which was to be demonstrated. Multiplying by as there
described,
, _mqy~x—n Vv , 1# v _ m ( 1 | 1 w 1\
a qy~x +1 ' d ' s ’ m-\-n qy~x + 1 \m -}- n d ’ s)
= a Jr b'Axy.
The Table of a' and XV on page 263 is based on the values m = n = 1. But if
a" and b" denote the values in any case where the numbers of lives m and n are equal,
we have a!' = no! ; b" = V — (n — 1) a'-^-Vv.
Probability of Survivorship. For two lives, the probability that y will sur-
vive x is strictly the value of A- when v is 1, and d is 0. The former equation A = 1
— S.d thus becomes A = 1; which is the proper result to be substituted in the pre-
ceding solution (S), also changing dxy to exy. The required value will always be
when the ages are equal.
For a near approximation in other cases, let T denote the “Probable Life”of the
age x—5 years, given in Table XXVI. Then ?y-5+ will approximate to tlie
probability A~ that the younger life y will survive the older x. And 1—A— will be
the opposite probability that the older will survive the younger. At the ages 50, 30,
for example, x—5 or 45 guides to T, 25.8G. Whence /y_5+T-i-V_i5 or A*- is 0.785 ;
and the chance that the older x will outlive the younger y is 1— or 0.215.
Solution for Survivorship Annuities. The preceding expressions for a' and V
have the singular property of remaining constant for the duration of the given lives.
Being easily tabulated, they appear adapted for an important part in the system of
survivorships. For illustration, let the last problem on page 282 be resolved by this
method. Firstly, supposing death to occur on an average at the middle of a policy
year, and the first payment of annuity to the survivor to be made six months after, or
at the end of the year, the present value may be considered equivalent to that of a con-
tinuous annuity commencing at death or survivorship.—Page 290.
Problem.—To determine the present value of an annuity on the life of x, to com-
mence with the failure of the joint existence of y and z, provided it be z who dies first.
Let yz>ax denote the required value. By the continuous method,
- rdlz+t-l'y+t-lx+t-V*' -dx-\-t
v'ax ~ -J TJyOx “ VTPT ’
— J dA-z+t.y+t.x+t ■ (ix+t = & SdA-z+t.y+t.x+t • ®x+t •
Here the differentiation refers to the variable time t; and V enters by the substi-
tution of a' + J'A from the previous investigation. Interchanging y and z, LIFE ANNUITIES AND INSURANCES.
295
y’zdx — b" J dA.z+t y+t x+t .dx+t'
Adding the last two equations, and substituting from page 280,
yz’&'X + = dx = (ax—((xyz.)•
Let 2qu = qx + qv + qz;
V = F(m_2) = + log ~s)•
■p
Annuity, (ax—axyz).
The value of u may be found from Table LXV, page *204. Also the logarithm
of the function F and of 1—F can be easily tabulated by single entry, for the given rate
of interest. And the method of integration is evidently adapted to survivorships involv-
ing four or any number of lives.
ADDITION TO SECTION VII,
Tontines. Dividends of Annual Interest may be made to survivors of all ages,
proportional to their deposits. Dividends from Mortality can be apportioned to the
products of each Deposit x qx+nA- Px+n from Table XXVI; the shares being as if
the Tontines were separate for each age. NOTATION.
i = interest on $1 or any other monetary unit for one year,
v = —= the present value of 1 due a year hence.
d = log (1 + i) = Nap log (1-f i), the nominal yearly rate of interest when
compounded momently.
Log A = AA = notation for the common logarithm of A.
log A = Nap log A = notation for the hyperbolic or Napier logarithm.
e = 2.718 281 828 = the number whose hyperbolic logarithm is 1.
M = Ae = 0.434 294 482.
AM = 1.637 784 311.
lx = number living according to the Life Table, at the precise age x.
dx = lx—lx+1 = the number of persons dying between the ages x and x + \.
lxy =. lx.ly = number living in the Joint Life Table at the combination of ages x, y.
dxy — lXy—lx+i.y+i = annual decrement in the Joint Life Table.
px = — = the probability of a person whose age is x living one year.
I'X lx
qx — y = —= 1 —px = the probability of x dying within one year.
lx lx
pxy = — the probability of the joint existence of two lives x and y, con-
lx • ly
tinuing one year.
q«y — 1—Pxy = the probability that one of the lives x, y, may die within one year.
f*x = log h = limiting ratio of mortality at the age x-
lx—i—lx+1 dx—i -\-dx . , i
Vx = 24—' = —21 > approximately.
pXj n = = the probability that a life aged may survive n years.
lx
1 —w — the probability of z dying within n years.
= the probability that both a and y will survive n years. NOTATION.
297
ex = i- + +1tiy2+ _—: — tjie “ expectation ” or average duration of life.
'x
ex—\ = the “curtate expectation ” of life after the age x.
T* = the “probable life” after the age x, defined by Ix+t = fix-
ax — the present value of |1 Annuity due at the end of each year during the life x.
X + ax = the present value of |1 annuity due at the beginning of each year during
the life x.
a® = the present value of a temporary annuity for the next n years, if x shall so
long live.
1 = the value of the same, when the annuity is due at the beginning of each year.
nax = the present value of a deferred annuity; that is, of a life annuity on x to
commence at the age x-\-n.
axy = the value of an annuity during the joint existence of x and y.
ax- = ax-\-ay—axy — the value of an annuity on the longest life.
a(m) — t[ie yalue of II annuity, when payable by m instalments in each year.
_ fjjQ value of the same, when the instalments are payable in advance,
cafti — complete annuity providing for a proportionate payment to the day of death.
dx = the value of a continuous annuity of |1 per annum, for instalments supposed
payable momently.
Aa; = the single premium to insure II payable at the end of the policy year in
which x shall die.
nx or = annual premium to insure II on the life of x. In some cases it denotes the
premium to insure 11000.
A(w») —- single premium of Endowment Insurance of II for m years.
= single premium of insurance “on x against y,” that is, of II payable on the
death of x, if y be the survivor.
Note 1.—Generally x denotes the age of entry, when the risk commences; x + n the age of
valuation of the policy ; and m or n + m the total period of insurance.
The suffixes x, xy, are often omitted where the meaning is obvious.
Note 2.—The notation relative to other conditions is usually explained on the several pages
where it is introduced and applied, INTERPOLATION BY FINITE DIFFERENCES.
From the series a, 5, c, d, e, etc., whose terms are a unit’s distance from each
other, to find any intermediate term denoted by u:
Let x be the integral or the fractional distance of u from the first term a, and
A, A3, A3, etc., the first terms of the differences. That is,
a —l—a, the first difference;
A2 = c—2b + a, the first of the second differences;
A3 = d—3c + 35—a, the first of the third differences; etc. Then,
x(x—1) 0 x(x—1) (x—2) „ x (x—1) (x—2) (x—3)
“ = «+*A + A +' & + J7374 A + ■ ■''
Note.—The derivative dus-dx at any point as at c, may be found by differentiating with respect
to u and x as the only variables, and at c making x = 2. The well known values of p were thus
found.
The first term a can generally be so located that the required term u shall be near
the middle of the employed portion of the series.
DOUBLE INTERPOLATION.
Let U denote the term to be interpolated at the junction of x, y; where x is the
horizontal, and y the vertical distance from the term a. We might apply the preceding
formula to insert a term u at the distance x from a, then a term u' at the distance x
from a', etc. The same formula applied to u, u', u", .... would evidently give U at
the vertical distance y from u. But the result will be the same, if
A = b—a, A2 = (c—b) — (b—a),
A' = b'—a', etc.
6 = d—a, d2 = (a"—a') — (a'—a),
abed
a' V c' d'
a" b" c" d"
U = a + xA + yd + xy (A'—A) + - A,j -f- — - d2-}-
/i . z
NUMBERS OFTEN USED IN CALCULATIONS.
RATE OF
INTEREST.
NUMBERS.
LOGARITHMS.
V.
1—V.
Xv.
A(l — v).
3 per cent.
.9708 7379
.0291 2621
T.987 1628
2.464 2840
3* “ “
.9661 8357
.0338 1643
1.985 0597
2.529 1277
4 « “
.9615 3846
.0384 6154
1.982 9667
2.585 0267
H “ “
.9569 3780
.0430 6220
1.980 8837
2.634 0962
5 “ “
.9523 8095
.0476 1905
1.978 8107
2.677 7807
6 “ “
.9433 9623
.0566 0377
T.974 6941
2.752 8454
iy u u
.9345 7944
.0654 2056
1.970 6162
2.815 7143
8 “ “
.9259 2593
.0740 7407
1.966 5762
2.869 6662
9 “ “
.9174 3119
.0825 6881
1.962 5735
2.916 8161
10 “ “
.9090 9091
.0909 0909
1.958 6073
2.958 6073 TABLES IN PART SECOND.
THIRTY OFFICES’ EXPERIENCE:
MALE LIFE, Tables XXVI-LXIV. Pages *2-*203
JOINT LIVES, - - - - “ LXV-LXXVI. “ *204-*223
FEMALE LIFE, - - - - “ LXXVII-LXXVTTT, “ *224-*227
MISCELLANEOUS, Tables LXXIX-LXXXII. Pages *228-*253
1 2
TABLE XXYI.
MALE LIFE TABLE. FROM EXPERIENCE OF LOSSES AND AMOUNTS INSURED.
EXPECTATION OF LIFE AND PROBABLE LIFE.
AGE.
X.
Living.
lx-
Decre-
ment.
clx.
Prob. of Dec.
a — dx.
fl% — r
I'X
Prob. of Living.
„ _ +1
]>x — —
I'X
Expectation
of Life.
Years.
Probable
Life.
Years.
Age added
to Ex-
pectation.
Years.
AGE.
X.
10
100,000
648
0.006479
0.993521
49.994
55.40
59.99
10
11
99,352
646
.006502
.993498
49.316
54.56
60.32
11
12
98,706
643
.006516
.993484
48.637
53.73
60.64
12
13
98,063
641
.006536
.993464
47.952
52.89
60.95
13
14
97,422
640
.006568
.993432
47.264
52.05
61.26
14
15
96,782
638
0.006593
0.993407
46.573
51.21
61.57
15
16
96,144
636
.006614
.993386
45.880
50.36
61.88
16
17
95,508
635
.006648
.993352
45.183
49.52
62.18
17
18
94,873
634
.006683
.993317
44.481
48.68
62.48
18
19
94,239
633
.006717
.993283
43.777
47.83
62.78
19
20
93,606
633
0.006763
0.993237
43.069
46.99
63.07
20
21
92,973
633
.006808
.993192
42.359
46.14
63.36
21
22
92,340
633
.006856
.993144
41.646
45.29
63.65
22
23
91,707
634
.006911
.993089
40.930
44.43
63.93
23
24
91,073
635
.006973
.993027
40.211
43.58
64.21
24
25
90,438
636
0.007032
0.992968
39.490
42.73
64.49
25
26
89,802
639
.007115
.992885
38.766
41.88
64.77
26
27
89,163
641
.007190
.992810
38.040
41.03
65.04
27
28
88,522
644
.007275
.992725
37.312
40.18
65.31
28
29
87,878
649
.007385
.992615
36.582
39.33
65.58
29
30
87,229
653
0.007485
0.992515
35.850
38.48
65.85
30
31
86,576
658
.007602
.992398
35.117
37.62
66.12
31
32
85,918
664
.007727
.992273
34.383
36.77
66.38
32
33
85,254
671
.007871
.992129
33.646
35.93
66.65
33
34
84,583
679
.008027
.991973
32.910
35.08
66.91
34
35
83,904
689
0.008212
0.991788
32.172
34.23
67.17
35
36
83,215
698
.008390
.991610
31.434
33.38
67.43
36
37
82,517
709
.008591
.991409
30.696
32.53
67.70
37
38
81,808
722
.008826
.991174
29.957
31.69
67.96
38
39
81,086
736
.009077
.990923
29.219
30.85
68.22
39
40
80,350
752
0.009360
0.990640
28.482
30.01
68.48
40
41
79,598
768
.009647
.990353
27.747
29.17
68.75
41
42
78,830
788
.009998
.990002
27.013
28.34
69.01
42
43
78,042
808
.010351
.989649
26.280
27.51
69.28
43
44
77,234
831
.010761
.989239
25.550
26.68
69.55
44
45
76,403
856
0.011203
0.988797
24.822
25.86
69.82
45
46
75,547
883
.011688
.988312
24.090
25.05
70.09
46
47
74,664
913
.012227
.987773
23.377
*24.23
70.38
47
48
73,751
945
.012814
.987186
22.660
23.42
70.66
48
49
72,806
980
.013459
.986541
21.948
22.62
70.95
49
50
71,826
1,018
0.014175
0.985825
21.241
21.83
71.24
50
51
70,808
1,059
.014955
.985045
20.539
21.04
71o54
51
52
69,749
1,103
.015814
.984186
19.843
20.26
71.84
52
53
68,646
1,150
.016752
.983248
19.154
19.49
72.15
53
54
67,496
1,200
.017779
.982221
18.471
18.73
72.47
54 3
TABLE XXVI.
MALE LIFE TABLE. FROM EXPERIENCE OF LOSSES AND AMOUNTS INSURED.
EXPECTATION OF LIFE AND PROBABLE LIFE.
AGE.
X.
Living.
lx-
Decre-
ment.
dx-
Prob. of Dec.
dx
(/x — y
I'X
Prob. of Living.
ft = 'fi-
f'X
Expectation
of Life.
Years.
Probable
Life.
Years.
Age added
6to Ex-
pectation.
Years.
AGE.
X.
55
66,296
1,255
0.018930
0.981070
17.797
17.97
72.80
55
56
65,041
1,312
.020172
.979828
17.130
17.23
73.13
56
57
63,729
1,374
.021560
.978440
16.473
16.50
73.47
57
58
62,355
1,438
.023060
.976940
15.825
15.78
73.83
58
59
60,917
1,505
.024707
.975293
15.187
15.07
74.19
59
60
59,412
1,576
0.026527
0.973473
14.559
14.37
74.56
60
61
57,836
1,650
.028529
.971471
13.942
13.69
74.94
61
62
56,186
1,725
.030700
.969300
13.336
13.03
75.34
62
63
54,461
1,803
.033108
.966892
12.743
12.38
75.74
63
64
52,658
1,882
.035740
.964260
12.162
11.74
76.16
64
65
50,776
1,962
0.038640
0.961360
11.595
11.12
76.60
65
66
48,814
2,040
.041789
.958211
11.040
10.52
77.04
66
67
46,774
2,118
.045282
.954718
10.500
9.94
77.50
67
68
44,656
2,190
.049043
.950957
9.974
9.37
77.97
68
69
42,466
2,261
.053242
.946758
9.463
8.83
78.46
69
70
40,205
2,323
0.057778
0.942222
8.967
8.30
78.97
70
71
37,882
2,378
.062775
.937225
8.486
7.80
79.49
71
72
35,504
2,422
.068216
.931784
8.021
7.31
80.02
72
73
33,082
2,453
.074149
.925851
7.572
6.84
80.57
73
74
30,629
2,472
.080709
.919291
7.138
6.40
81.14
74
75
28,157
2,472
0.087792
0.912208
6.721
5.97
81.72
75
76
25,685
2,453
.095503
.904497
6.320
5.57
82.32
76
77
23,232
2,416
.103996
.896004
5.934
5.19
82.93
77
78
20,816
2,356
.113181
.886819
5.565
4.82
83.57
78
79
18,460
2,274
.123185
.876815
5.211
4.49
84.21
79
80
16,186
2,170
0.134068
0.865932
4.873
4.16
84.87
80
81
14,016
2,044
.145833
.854167
4.550
3.85
85.55
81
82
11,972
1,900
.158704
.841296
4.242
3.57
86.24
82
83
10,072
1,737
.172458
.827542
3.947
3.30
86.95
83
84
8,335
1,563
.187523
.812477
3.666
3.03
87.67
84
85
6,772
1,379
0.203633
0.796367
3.396
2.81
88.40
85
86
5,393
1,191
.220841
.779159
3.137
2.60
89.14
86
87
4,202
1,008
.239885
.760115
2.885
2.38
89.89
87
88
3,194
829
.259550
.740450
2.637
2.14
90.64
88
89
2,365
692
.292600
.707400
2.386
1.89
91.39
89
90
1,673
549
0.328154
0.671846
2.166
1.71
92.17
90
91
1,124
403
.358541
.641459
1.980
1.57
92.98
91
92
721
281
.389736
.610264
1.808
1.43
93.81
92
93
440
187
.425000
.575000
1.643
1.28
94.64
93
94
253
117
.462451
.537549
1.488
1.14
95.49
94
95
136
68
0.500000
0.500000
1.338
1.00
96.34
95
96
68
38
.558824
.441176
1.176
0.90
97.18
96
97
30
18
.600000
.400000
1.033
0.83
98.03
97
98
12
8
.666666
.333334
0.833
0.75
98.83
98
99
4
4
1.000000
.000000
0.500
0.50
99.50
99 4
TABLE XXVII.
COMMON LOGARITHMS (A) OF THE PRECEDING LIFE TABLE.
AGE.
X.
Xlx-
Ap
lx
•
Mx-
tyx-
1
AGE.
X.
10
5.000000
5.000000
2.811575
3.811575
1.997177
0.002823
10
11
4.997177
.002823
.810233
.813056
7167
2833
11
12
.994344
.005656
.808211
.813867
7161
2839
12
13
.991505
.008495
.806858
.815353
7152
2848
13
14
.988657
.011343
.806180
.817523
7138
2862
14
15
4.985795
5.014205
2.804821
3.819026
1.997127
0.002873
15
16
.982922
.017078
.803457
.820535
7118
2882
16
17
.980040
.019960
.802774
.822734
7103
2897
17
18
.977143
.022857
.802089
.824946
7088
2912
18
19
.974231
.025769
.801404
.827173
7073
2927
19
20
4.971304
5.028696
2.801404
3.830100
1.997053
0.002947
20
21
.968357
.031643
.801404
.833047
7033
2967
21
22
.965390
.034610
.801404
.836014
7012
2988
22
23
.962402
.037598
.802089
.839687
6988
3012
23
24
.959390
.040610
.802774
.843384
6961
3039
24
25
4.956351
5.043649
2.803457
3.847106
1.996935
0.003065
25
26
.953286
.046714
.805501
.852215
6899
3101
26
27
.950185
.049815
.806858
.856673
6866
3134
27
28
.947051
.052949
.808886
.861835
6829
3171
28
29
.943880
.056120
.812245
.868365
6781
3219
29
30
4.940661
5.059339
2.814913
3.874252
1.996737
0.003263
30
31
.937398
.062602
.818226
.880828
6686
3314
31
32
.934084
.065916
.822168
.888084
6631
3369
32
33
.930715
.069285
.826723
.896008
6568
3432
33
34
.927283
.072717
.831870
.904587
6500
3500
34
35
4.923783
5.076217
2.838219
3.914436
1.996419
0.003581
35
36
.920202
.079798
.843855
.923653
6341
3659
36
37
.916543
.083457
.850646
.934103
6253
3747
37
38
.912796
.087204
.858537
.945741
6150
3850
38
39
.908946
.091054
.866878
.957932
6040
3960
39
40
4.904986
5.095014
2.876218
3.971232
1.995916
0.004084
40
41
.900902
.099098
.885361
.984459
5790
4210
41
42
.896692
.103308
.896526
.999834
5636
4364
42
43
.892328
.107672
.907411
2.015083
5481
4519
43
44
.887809
.112191
.919601
.031792
5301
4699
44
45
4.883110
5.116890
2.932474
2.049364
1.995107
0.004893
45
46
.878217
.121783
.945961
.067744
4894
5106
46
47
.873111
.126889
.960471
.087360
4657
5343
47
48
.867768
.132232
.975432
.107664
4399
5601
48
49
.862167
.137833
.991226
.129059
4115
5885
49
50
4.856282
5.143718
3.007748
2.151466
1.993800
0.006200
50
51
.850082
.149918
.024896
.174814
3456
6544
51
52
.843538
.156462
.042576
.199038
3077
6923
52
53
.836615
,163385
.060698
.224083
2663
7337
53
54
.829278
.170722
.079181
.249903
2209
7791
54 5
TABLE XXVII.
COMMON LOGARITHMS (A) OF THE PRECEDING LIFE TABLE.
AGE.
X.
Xlx-
xr
f'X
Xdx.
tyx-
hPx-
A-.
Px
AGE,
X.
55
4.821487
5.178513
3.098644
2.277157
1.991700
0.008300
55
56
.813187
.180813
.117934
.304747
.991150
.008850
56
57
.804337
.195003
.137987
.333050
.990534
.009406
57
58
.794871
.205129
.157759
.302888
.989808
.010132
58
59
.784739
.215201
.177537
.392798
.989135
.010805
59
60
4.773874
5.220120
3.197556
2.423082
1.988324
0.011676
60
61
.702198
.237802
.217484
.455280
.987430
.012570
61
62
.749028
.250372
.230789
.487161
.980458
.013542
62
63
.730080
.203914
.255996
.519910
.985378
.014022
63
64
.721404
.278530
.274020
.553150
.984194
.015800
64
65
4.705058
5.294342
3.292099
2.587041
1.982880
0.017114
65
66
.088544
.311450
.309030
.021086
.981401
.018539
66
67
.070005
.329995
.325920
.055921
.979875
.020125
67
68
.049880
.350120
.340444
.090504
.978101
.021839
68
69
.028041
.371959
.354301
.720200
.970239
.023701
69
70
4.004280
5.395720
3.300049
2.701769
1.974153
0.025847
70
71
.578433
.421507
.370212
.797779
.971844
.028156
71
72
.550277
.449723
.384174
.833897
.909315
.030685
72
73
.519592
.480408
.389698
.870100
.900541
.033459
73
74
.480133
.513807
.393049
.906910
.903453
.030547
74
75
4.449580
5.550414
3.393049
2.943403
1.960094
0.039906
75
76
.409080
.590320
.389098
.980018
.956407
.043593
76
77
.300087
.033913
.383097
1.017010
.95*310
.047690
77
78
.318397
.081003
.372175
.053778
.947835
.052105
78
79
•200232
.733708
.356791
.090559
.942908
.057092
79
80
4.209140
5.790800
3.330400
1.127320
1.937484
0.062516
80
81
.140024
.853370
.310481
.103857
.931543
.068457
81
82
.078107
.921833
.278754
.200587
.924949
.075051
82
83
.003110
.990884
.239800
.230684
.917790
.082210
83
84
3.920900
4.079094
.193959
.273053
.909811
.090189
84
85
3.830717
4.109283
3.139504
1.308847
1,901113
0.098887
85
86
.731830
.208170
.075912
.344082
.891026
.108374
86
87
.023450
.370544
.003401
.380005
.880879
.119121
87
88
.504335
.495005
2,918555
.414220
.809490
.130504
88
89
.373831
.020109
.840100
.400275
.849065
.150335
89
90
3.223490
4.770504
2.739572
1.510076
1.827270
0.172730
90
91
.050700
.949234
.005305
.554539
.807109
.192831
91
92
2.857935
3.142065
.448706
.590771
.785518
.214482
92
93
.043453
.356547
.271842
.628389
.759008
.240332
93
94
.403121
.596879
008180
.005005
.730418
.269582
94
95
2.133539
3.866461
1.832509
1.698970
1.098970
0.301030
95
96
1.832509
2.167491
.579784
.747275
.644012
.355388
96
97
.477121
.522879
.255273
.778152
.602060
.397940
97
98
.079181
.920819
0.903090
.823909
.522879
.477121
98
99
0.602000
1.397940
.602000
0.000000
GO
GO
99 6
TABLE XXYIII.
PRESENT VALUE (a) OF ANNUITY OF $1, PAYABLE AT THE END OF EACH YEAR
DURING LIFE.
AGE,
3
Per Cent.
34
Per Cent.
4
Per Cent.
4?
Per Cent
5
Per Cent.
6
Per Cent.
7
Per Cent.
8
Per Cent.
10
PerCent.
AGE,
10
23.7680
21.6018
19.7316
18.1334
16.7497
14.4869
12.7276
11.3299
9.2650
10
11
23.6406
21.5037
19.6547
18.0729
16.7018
14.4562
12.7074
11.3161
9.2580
11
12
23.5093
21.4020
19.5747
18.0098
16.6517
14.4238
12.6858
11.3014
9.2504
12
13
23.3732
21.2963
19.4912
17.9437
16.5990
14.3896
12.6629
11.2855
9.2422
13
14
23.2329
21.1867
19.4042
17.8745
16.5436
14.3533
12.6385
11.2686
9.2333
14
15
23.0881
21.0732
19.3138
17.8023
16.4856
14.3151
12.6125
11.2505
9.2238
15
16
22.9386
20.9555
19.2196
17.7270
16.4248
14.2747
12.5850
11.2312
9.2135
16
17
22.7840
20.8334
19.1216
17.6480
16.3609
14.2319
12.5556
11.2105
9.2023
17
18
22.6246
20.7069
19.0195
17.5656
16.2938
14.1868
12.5244
11.1884
9.1903
18
19
22.4602
20.5758
18.9133
17.4795
16.2236
14.1392
12.4913
11.1647
9.1774
19
20
22.2904
20.4400
18.8029
17.3896
16.1500
14.0889
12.4560
11.1394
9.1634
20
21
22.1154
20.2994
18.6882
17.2959
16.0730
14.0359
12.4187
11.1125
9.1483
21
22
21.9351
20.1539
18.5689
17.1981
15.9923
13.9800
12.3822
11.0838
9.1322
22
23
21.7491
20.0033
18.4450
17.0961
15.9079
13.9212
12,3373
11.0531
9.1148
23
24
21.5575
19.8475
18.3163
16.9898
15.8195
13.8591
12.2928
11.0204
9.0960
24
25
21.3601
19.6865
18.1827
16.8789
15.7271
13.7939
12.2427
10.9856
9.0758
25
26
21.1568
19.5197
18.0440
16.7634
15.6305
13.7250
12.1954
10.9485
9.0541
26
27
20.9476
19.3477
17.9002
16.6433
15.5296
13.6528
12,1426
10.9091
9.0309
27
28
20.7323
19.1700
17.7510
16.5182
15.4241
13.5768
12.0867
10.8671
9.0059
28
29
20.5107
18.9863
17.5964
16.3880
15.3140
13.4969
12.0276
10.8226
8.9791
29
30
20.2832
18.7970
17.4364
16.2529
15.1993
13.4131
11.9652
10.7753
8.9505
30
31
20.0492
18.6016
17.2706
16.1124
15.0797
13.3251
11.8993
10.7251
8.9198
31
32
19.8089
18.4002
17.0990
15.9664
14.9550
13.2328
11.8298
10.6718
8.8870
32
33
19.5621
18.1924
16.9214
15.8148
14.8250
13.1360
11.7565
10.6153
8.8518
33
34
19.3088
17.9786
16.7379
15.6576
14.6897
13.0346
11.6792
10.5555
8.8142
34
35
19.0490
17.7584
16.5483
15.4946
14.5490
12.9285
11.5979
10.4922
8.7741
35
36
18.7830
17.5321
16.3527
15.3259
14.4030
12.8176
11.5125
10.4254
8.7314
36
37
18.5101
17.2992
16.1507
15.1511
14.2510
12.7017
11.4226
10.3547
8.6858
37
38
18.2306
17.0599
15.9423
14.9702
14.0933
12.5804
11.3281
10.2800
8.6372
38
39
17.9448
16.8142
15.7276
14.7830
13.9297
12.4540
11.2290
10.2012
8.5855
39
40
17.6524
16.5621
15.5065
14.5897
13.7602
12.3222
11.1251
10.1182
8.5306
40
41
17.3538
16.3037
15.2792
14.3903
13.5847
12.1849
11.0163
10.0309
8.4723
41
42
17.0485
16.0387
15.0451
14.1844
13.4028
12.0418
10.9023
9.9389
8.4103
42
43
16.7372
15.7677
14.8050
13.9724
13.2151
11.8932
10.7832
9.8424
8.3447
43
44
16.4197
15.4903
14.5582
13.7538
13.0210
11.7387
10.6587
9.7410
8.2752
44
45
16.0963
15.2069
14.3052
13.5291
12.8208
11.5784
10.5289
9.6348
8.2018
45
46
15.7669
14.9174
14.0460
13.2981
12.6144
11.4121
10.3936
9.5234
8.1242
46
47
15.4321
14.6221
13.7806
13.0609
12.4017
11.2399
10.2527
9.4069
8.0423
47
48
15.0917
14.3212
13.5092
12.8176
12.1829
11.0618
10.1061
9.2853
7.9560
48
49
14.7463
14.0149
13.2320
12.5683
11.9582
10.8777
9.9540
9.1583
7.8652
49
50
14.3959
13.7033
12.9490
12.3129
11.7274
10.6877
9.7960
9.0258
7.7698
50
51
14.0410
13.3869
12.6606
12.0521
11.4908
10.4918
9.6325
8.8881
7.6696
51
52
13.6818
13.0658
12.3669
11.7857
11.2485
10.2902
9.4632
8.7449
7.5647
52
53
13.3187
12.7403
12.0682
11.5139
11.0007
10.0829
9.2883
8.5962
7.4549
53
54
12.9519
12.4109
11.7648
11.2370
10.7475
9.8699
9.1079
8.4421
7.3401
54
Formula, ax =
Ux 7
TABLE XXVIII.
PRESENT VALUE (a) OF ANNUITY OF $1, PAYABLE AT THE END OF EACH YEAR
DURING LIFE.
AGE.
3
Per Cent.
01
Per Cent.
4
Per Cent.
44
Per Cent.
5
Per Cent.
6
Per Cent.
7
Per Cent.
8
Per Cent.
10
Per Cent.
AGE,
55
12.5820
12.0778
11.4569
10.9552
10.4892
9.6515
8.9218
8.2825
7.2203
55
56
12.2095
11.7418
11.1451
10.6691
10.2262
9.4280
8.7305
8.1177
7.0955
56
57
11.8347
11.4029
10.8295
10.3788
9.9586
9.1994
8.5340
7.9476
6.9658
57
58
11.4583
11.0621
10.5108
10.0848
9.6869
8.9663
8.3326
7.7725
6.8312
58
59
11.0806
10.7195
10.1893
9.7874
9.4113
8.7286
8.1263
7.5925
6.6916
59
60
10.7021
10.2696
9.8653
9.4869
9.1322
8.4867
7.9155
7.4076
6.5473
60
61
10.3236
9.9187
9.5395
9.1839
8.8501
8.2410
7.7003
7.2182
6.3983
61
62
9.9456
9.5674
9.2125
8.8791
8.5655
7.9920
7.4813
7.0246
6.2448
62
63
9.5684
9.2158
8.8844
8.5725
8.2787
7.7398
7.2585
6.8268
6.0868
63
64
9.1929
8.8650
8.5562
8.2650
7.9902
7.4851
7.0326
6.6254
5.9248
64
65
8.8197
8.5154
8.2282
7.9571
7.7007
7.2283
6.8037
6.4206
5.7588
65
66
8.4494
8.1676
7.9013
7.6493
7.4107
6.9700
6.5726
6.2130
5.5893
66
67
8.0824
7.8222
7.5757
7.3422
7.1206
6.7104
6.3394
6.0027
5.4164
67
68
7.7198
7.4799
7.2525
7.0365
6.8313
6.4504
6.1049
5.7904
5.2406
68
69
7.3614
7.1410
6.9315
6.7323
6.5428
6.1901
5.8691
5.5762
5.0620
69
70
7.0087
6.8066
6.6142
6.4309
6.2562
5.9305
5.6331
5.3609
4.8813
70
71
6.6616
6.4768
6.3006
6.1324
5.9719
5.6718
5.3970
5.1448
4.6987
71
72
6.3210
6.1525
5.9915
5.8376
5.6905
5.4148
5.1616
4.9286
4.5147
72
73
5.9873
5.8340
5.6873
5.5469
5.4124
5.1599
4.9273
4.7126
4.3298
73
74
5.6608
5.5218
5.3885
5.2607
5.1382
4.9075
4.6944
4.4972
4.1442
74
75
5.3425
5.2168
5.0961
4.9801
4.8687
4.6586
4.4640
4.2834
3.9588
75
76
5.0324
4.9190
4.8100
4.7051
4.6042
4.4134
4.2362
4.0712
3.7738
76
77
4.7307
4.6287
4.5306
4.4360
4.3448
4.1722
4.0113
3.8612
3.5895
77
78
4.4381
4.3468
4.2587
4.1737
4.0916
3.9358
3.7903
3.6541
3.4067
78
79
4.1547
4.0731
3.9943
3.9181
3.8445
3.7044
3.5732
3.4501
3.2256
79
80
3.8806
3.8079
3.7376
3.6696
3.6038
3.4783
3.3605
3.2496
3.0467
80
81
3.6158
3.5514
3.4890
3.4285
3.3699
3.2579
3.1524
3.0529
2.8702
81
82
3.3601
3.3032
3.2480
3.1945
3.1425
3.0429
2.9489
2.8601
2.6963
82
83
3.1138
3.0638
3.0152
2.9679
2.9220
2.8340
2.7506
2.6716
2.5254
83
84
2.8756
2.8318
2.7893
2.7478
2.7075
2.6300
2.5565
2.4866
2.3568
84
85
2.6455
2.6074
2.5704
2.5343
2.4990
2.4313
2.3668
2.3053
2.1909
85
86
2.4216
2.3888
2.3567
2.3255
2.2950
2.2361
2.1800
2.1264
2.0262
86
87
2.2012
2.1731
2.1457
2.1189
2.0927
2.0421
1.9937
1.9474
1.8606
87
88
1.9827
1.9590
1.9357
1.9130
1.8908
1.8478
1.8065
1.7669
1.6925
88
89
1.7581
1.7383
1.7189
1.6999
1.6813
1.6452
1.6105
1.5772
1.5143
89
90
1.5598
1.5432
1.5270
1.5111
1.4955
1.4652
1.4361
1.4080
1.3548
90
91
1.3913
1.3774
1.3638
1.3504
1.3373
1.3117
1.2871
1.2633
1.2182
91
92
1.2340
1.2225
1.2111
1.1999
1.1890
1.1676
1.1470
1.1270
1.0890
92
93
1.0828
1.0733
1.0639
1.0547
1.0457
1.0281
1.0110
.9945
.9629
93
94
.9395
.9319
.9243
.9169
.9096
.8953
.8814
.8679
.8421
94
95
.8002
.7942
.7883
.7825
.7767
.7654
.7544
.7438
.7233
95
96
.6485
.6440
.6397
.6353
.6310
.6227
.6145
.6065
.5912
96
97
.5140
.5109
.5079
.5049
.5019
.4960
.4903
.4847
.4841
97
98
.3236
.3221
.3205
.3190
.3175
.3145
.3115
.3087
.3031
98
99
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
99 8
TABLE XXIX.
ANNUAL PREMIUM (JT) TO INSURE $1000, PAYABLE AT THE END OF THE POLICY
YEAR IN WHICH THE LIFE FAILS.
AGE.
3 Per Cent.
31 Per Cent.
4 Per Cent.
41 Per Cent.
5 Per Cent.
6 Per Cent.
AGE,
10
11.2486
10.4447
9.7739
9.2026
8.7201
7.9672
10
11
11.4571
10.6384
9.9535
9.3682
8.8723
8.0952
11
12
11.6747
10.8410
10.1419
9.5422
9.0327
8.2308
12
13
11.9023
11.0534
10.3400
9.7260
9.2026
8.3754
13
14
12.1400
11.2759
10.5480
9.9194
9.3819
8.5289
14
15
12.3881
11.5085
10.7660
10.1227
9.5709
8.6913
15
16
12.6474
11.7524
10.9953
10.3369
9.7706
8.8640
16
17
12.9188
12.0083
11.2364
10.5629
9.9819
9.0478
17
18
13.2025
12.2763
11.4897
10.8010
10.2050
9.2428
18
19
13.4993
12.5574
11.7560
11.0518
10.4407
9.4499
19
20
13.8099
12.8522
12.0361
11.3163
10.6899
9.6701
20
21
14.1349
13.1614
12.3304
11.5950
10.9531
9.9037
21
22
14.4752
13.4857
12.6399
11.8887
11.2311
10.1517
22
23
14.8317
13.8261
12.9656
12.1985
11.5251
10.4152
23
24
15.2050
14.1835
13.3082
12.5250
11.8357
10.6949
24
25
15.5963
14.5586
13.6687
12.8696
12.1641
10.9919
25
26
16.0068
14.9530
14.0485
13.2333
12.5117
11.3079
26
27
16.4369
15.3669
14.4479
13.6166
12.8786
11.6426
27
28
16.8883
15.8021
14.8689
14.0214
13.2671
11.9986
28
29
17.3623
16.2597
15.3125
14.4487
13.6780
12.3769
29
30
17.8592
16.7403
15.7791
14.8991
14.1119
12.7777
30
31
18.3814
17.2461
16.2712
15.3751
14.5713
13.2038
31
32
18.9302
17.7785
16.7902
15.8779
15.0575
13.6566
32
33
19.5070
18.3389
17.3376
16.4092
15.5722
14.1376
33
34
20.1136
18.9289
17.9149
16.9706
16.1170
14.6487
34
35
20.7516
19.5504
18.5241
17.5638
16.6936
15.1916
35
36
21.4224
20.2045
19.1663
18.1901
17.3035
15.7674
36
37
22.1294
20.8947
19.8452
18.8533
17.9504
16.3801
37
38
22.8743
21.6226
20.5024
19.5549
18.6357
17.0316
38
39
23.6590
22.3903
21.3199
20.2971
19.3616
17.7235
39
40
24.4863
23.2003
22.1206
21.0825
20.1309
18.4590
40
41
25.3587
24.0552
22.9667
21.9137
20.9462
19.2406
41
42
26.2802
24.9588
23.8627
22.7949
21.8117
20.0726
42
43
27.2524
25.9128
24.8099
23.7276
22.7287
20.9564
43
44
28.2801
26.9217
25.8133
24.7167
23.7026
21.8972
44
45
29.3662
27.9887
26.8757
25.7651
24.7359
22.8979
45
46
30.5150
29.1173
28.0014
26.8771
25.8330
23.9627
46
47
31.7306
30.3122
29.1948
28.0572
26.9985
25.0962
47
48
33.0174
31.5772
30.4002
29.3094
28.2364
26.3026
48
49
34.3809
32.9177
31.8029
30.6395
29.5524
27.5876
49
50
35.8261
34.3385
33.2283
32.0524
30.9517
28.9564
50
51
37.3590
35.8450
34.7419
33.5538
32.4399
30.4147
51
52
38.9855
37.4432
36.3501
35.1505
34.0235
31.9688
52
53
40.7128
39.1397
38.0599
36.8490
35.7093
33.6257
53
54
42.5485
40.9412
39.8789
38.6571
37.5051
35.3931
54
Formula, ttx = 1 _ 10oo i —1 (l _ v\ l.
-IN * ( 1 + Ux ) 9
TABLE XXIX.
ANNUAL PREMIUM (n) TO INSURE $1000, PAYABLE AT THE END OF THE POLICY
YEAR IN WHICH THE LIFE FAILS.
AGE,
3 Per Cent.
34 Per Cent.
4 Per Cent.
4i- Per Cent.
5 Per Cent.
6 Per Cent.
AGE.
55
44.5008
42.8559
41.8154
40.5834
39.4191
37.2798
55
56
46.5771
44.8899
43.8764
42.6342
41.4585
39.2920
56
57
48.7878
47.0532
46.0730
44.8211
43.6340
41.4411
57
58
51.1416
49.3527
48.4132
47.1516
45.9537
43.7349
58
59
53.6509
51.7999
50.9095
49.6389
48.4302
46.1862
59
60
56.3282
54.9175
53.5745
52.2951
51.0760
48.8073
60
61
59.1848
57.7696
56.4196
55.1317
53.9026
51.6097
61
62
62.2349
60.8148
59.4582
58.1621
56.9232
54.6062
62
63
65.4953
64.0706
62.7079
61.4038
60.1552
57.8148
63
64
68.9809
67.5521
66.1832
64.8709
63.6129
61.2493
64
65
72.7102
71.2769
69.9017
68.5817
67.3143
64.9276
65
66
76.7006
75.2633
73.8817
72.5538
71.2768
68.8667
66
67
80.9763
79.5344
78.1464
76.8106
75.5240
73.0907
67
68
85.5559
84.1089
82.7146
81.3703
80.0741
77.6169
68
69
90.4709
89.0189
87.6177
86.2649
84.9587
82.4772
69
70
95.7386
94.2812
92.8725
91.5105
90.1937
87.6870
70
71
101.395
99.9310
98.5145
97.1422
95.8145
93.2814
71
72
107.467
105.996
104.570
103.188
101.848
99.2869
72
73
113.991
112.511
111.075
109.682
108.329
105.738
73
74
121.006
119.516
118.069
116.663
115.296
112.673
74
75
128.540
127.039
125.579
124.158
122.776
120.118
75
76
136.645
135.131
133.656
132.219
130.819
128.123
76
77
145.373
143.843
142.352
140.897
139.477
136.739
77
78
154.760
153.212
151.701
150.225
148.784
145.998
78
79
164.871
163.302
161.769
160.269
158.802
155.964
79
80
175.769
174.174
172.614
171.087
169.592
166.694
80
81
187.521
185.898
184.307
182.749
181.221
178.257
81
82
200.225
198.567
196.942
195.348
193.783
190.743
82
83
213.958
212.260
210.594
208.958
207.351
204.224
83
84
228.899
227.155
225.442
223.759
222.103
218.877
84
85
245.187
243.390
241.622
239.883
238.173
234.835
85
86
263.137
261.279
259.448
257.648
255.874
252.409
86
87
283.260
281.333
279.436
277.565
275.722
272.117
87
88
306.139
304.140
302.168
300.223
298.305
294.543
88
89
333.449
331.380
329.342
327.326
325.339
321.442
89
90
361.532
359.385
357.261
355.167
353.099
349.040
90
91
389.061
386.811
384.586
382.396
380.228
375.972
91
92
418.501
416.137
413.801
411.495
409.214
404.733
92
93
451.009
448.516
446.049
443.614
441.206
436.468
93
94
486.462
483.818
481.198
478.615
476.056
471.023
94
95
526.353
523.527
520.729
517.963
515.226
509.839
95
96
577.487
574.439
571.434 I
568.437
565.488
559.670
96
97
631.366
628.022
624.715
621.445
618.210
611.838
97
98
726.375
722.578
718.820
715.099
711.420
704.177
98
99
970.868
966.189
861.537 1
956.947
952.384
943.405
99 10
TABLE XXX.
SINGLE PREMIUM (A) TO INSURE $1,000, PAYABLE AT THE END OF THE POLICY
YEAR IN WHICH THE LIFE FAILS.
AGE.
8 Per Cent.
31 Per Cent.
4 Per Cent.
4t Per Cent.
5 Per Cent.
6 Per Cent.
AGE,
10
278.604
236.069
202.630
176.078
154.779
123.387
10
11
282.310
239.403
205.587
178.678
157.055
125.120
11
12
286.139
242.859
208.665
181.396
159.443
126.951
12
13
290.098
246.450
211.878
184.246
161.956
128.893
13
14
294.189
250.175
215.224
187.224
164.593
130.946
14
15
298.404
254.030
218.699
190.329
167.352
133.108
15
16
302.761
258.031
222.321
193.579
170.250
135.395
16
17
307.260
262.181
226.095
196.977
173.294
137.815
17
18
311.903
266.480
230.019
200.526
176.483
140.369
18
19
316.695
270.935
234.101
204.232
179.827
143.064
19
20
321.639
275.552
238.349
208.103
183.332
145.911
20
21
326.735
280.329
242.763
212.140
187.001
148.911
21
22
331.989
285.275
247.348
216.351
190.842
152.072
22
23
337.407
290.395
252.116
220.745
194.865
155.407
23
24
342.986
295.690
257.065
225.322
199.071
158.917
24
25
348.735
301.166
262.203
230.094
203.470
162.613
25
26
354.658
306.832
267.540
235.069
208.075
166.508
26
27
360.751
312.680
273.069
240.241
212.878
170.597
27
28
367.023
318.727
278.806
245.629
217.899
174.901
28
29
373.475
324.971
284.756
251.234
223.143
179.426
29
30
380.101
331.407
290.909
257.053
228.603
184.166
30
31
386.914
338.051
297.284
263.104
234.301
189.146
31
32
393.916
344.905
303.886
269.391
240.242
194.372
32
33
401.104
351.967
310.714
275.917
246.430
199.850
33
34
408.482
359.244
317.773
282.688
252.871
205.589
34
35
416.049
366.733
325.066
289.707
259.570
211.595
35
36
423.799
374.432
332.588
296.970
266.526
217.868
36
37
431.748
382.357
340.359
304.501
273.761
224.435
37
38
439.886
390.502
348.373
312.294
281.273
231.296
38
39
448.214
398.864
356.630
320.349
289.063
238.453
39
40
456.727
407.446
365.134
328.672
297.136
245.915
40
41
465.427
416.243
373.879
337.259
305.493
253.685
41
42
474.317
425.266
382.879
346.127
314.150
261.783
42
43
483.382
434.498
392.119
355.257
323.091
270.195
43
44
492.630
443.947
401.608
364.666
332.333
278.941
44
45
502.052
453.608
411.338
374.344
341.868
288.018
45
46
511.642
463.473
421.309
384.292
351.699
297.427
46
47
521.399
473.540
431.516
394.509
361.825
307.175
47
48
531.308
483.802
441.953
404.985
372.239
317.257
48
49
541.371
494.255
452.618
415.724
382.945
327.678
49
50
551.574
504.889
463.500
426.712
393.934
338.432
50
51
561.916
515.697
474.594
437.949
405.201
349.520
51
52
572.377
526.667
485.889
449.423
416.738
360.933
52
53
582.952
537.792
497.376
461.124
428.538
372.669
53
54
593.632
549.060
509.046
473.048
440.592
384.721
54
Formula, — = 1000 | 1 — (1 — v) (l + «x) | • 11
TABLE XXX.
1INGLE PREMIUM (A) TO INSURE $1,000, PAYABLE AT THE END OF THE POLICY
YEAR IN WHICH THE LIFE FAILS.
AGE.
3 Per Cent.
8J Per Cent.
4 Per Cent.
41 Per Cent.
5 Per Cent.
6 Per Cent.
AGE.
55
604.410
560.462
520.890
485.184
452.895
397.085
55
56
615.259
571.976
532.882
497.502
465.421
409.736
56
57
626.174
583.596
545.019
510.008
478.165
422.674
57
58
637.137
595.295
557.277
522.666
491.101
435.873
58
59
648.137
607.069
569.642
535.472
504.222
449.325
59
60
659.160
618.901
582.103
548.412
517.514
463.019
60
61
670.187
630.769
594.636
561.458
530.947
476.925
61
62
681.197
642.651
607.214
574.587
544.499
491.020
62
63
692.182
654.535
619.829
587.787
558.159
505.290
63
64
703.118
666.403
632.458
601.030
571.895
519.709
64
65
713.991
678.225
645.069
614.291
585.681
534.246
65
66
724.776
689.986
657.643
627.542
599.490
548.868
66
67
735.462
701.665
670.162
640.767
613.303
563.560
67
68
746.028
713.239
682.598
653.931
627.081
578.278
68
69
756.464
724.701
694.941
667.029
640.823
593.017
69
70
766.738
736.011
707.148
680.010
654.466
607.711
70
71
776.848
747.162
719.211
692.861
668.007
622.352
71
72
786.769
758.131
731.099
705.558
681.408
636.901
72
73
796.488
768.897
742.795
718.075
694.647
651.328
73
74
805.996
779.455
754.288
730.397
707.706
665.613
74
75
815.267
789.771
765.535
742.484
720.536
679.701
75
76
824.297
799.840
776.539
754.324
733.133
693.581
76
77
833.086
809.653
787.285
765.914
745.482
707.235
77
78
841.606
819.189
797.743
777.213
757.544
720.616
78
79
849.863
828.446
807.914
788.216
769.311
733.713
79
80
857.847
837.410
817.783
798.914
780.770
746.509
80
81
865.558
846.089
827.346
809.299
791.910
758.990
81
82
873.004
854.480
836.614
819.377
802.738
771.156
82
83
880.181
862.577
845.569
829.131
813.237
782.984
83
84
887.117
870.420
854.256
838.610
823.450
794.528
84
85
893.822
878.011
862.677
847.807
833.376
805.777
85
86
900.343
885.407
870.893
856.799
843.098
816.823
86
87
906.761
892.697
879.012
865.693
852.727
827.805
87
88
913.125
899.941
887.089
874.559
862.341
838.786
88
89
919.670
907.403
895.433
883.737
872.320
850.272
89
90
925.444
914.000
902.805
891.865
881.165
860.459
90
91
930.356
919.606
909.076
898.785
888.700
869.147
91
92
934.934
924.845
914.958
905.268
895.761
877.308
92
93
939.338
929.891
920.617
911.517
902.585
885.201
93
94
943.509
934.674
925.985
917.453
909.065
892.719
94
95
947.563
939.325
931.219
923.245
915.396
900.072
95
96
951.985
944.407
936.936
929.580
922.331
908.151
96
97
955.903
948.903
942.005
935.197
928.483
915.324
97
98
961.449
955.294
949.212
943.203
937.266
925.612
98
99
970.868
966.189
961.537
956.947
952.384
943.405
99 12
TABLE XXXI.
REVERSION, OR LIFE INSURANCE EQUIVALENT TO SINGLE PREMIUM OF 1.
AGE.
8
Per Ct.
34
Per Ct.
4
Per Ct.
Per Ct.
5
Per Ct.
6
Per Ct.
w
CD
<1
8
Per Ct.
31
Per Ct.
4
Per Ct.
44
Per Ct.
5
Per Ct.
6
Per Ct, ’
10
3.5893
4.2360
4.9351
5.6793
6.4608
8.1046
55
1.6545
1.7842
1.9198
2.0611
2.2080
2.5184
11
.5422
.1771
.8641
.5967
.3672
7.9923
56
.6253
.7483
.8766
.0100
.1486
.4406
12
.4948
.1177
.7924
.5128
.2718
.8771
57
.5970
.7135
.8348
1.9608
.0913
.3659
13
.4471
.0576
.7197
.4275
.1745
.7584
58
.5695
.6798
.7944
.9133
.0362
.2942
14
.3992
3.9972
.6463
.3412
.0756
.6367
59
.5429
.6473
.7555
.8675
1.9833
.2256
15
3.3512
3.9366
4.5725
5.2541
5.9754
7.5127
60
1.5171
1.6158
1.7179
1.8234
1.9323
2.1597
16
.3029
.8755
.4980
.1659
.8737
.3858
61
.4921
.5854
.6817
.7811
.8834
.0968
17
.2545
.8142
.4229
.0767
.7705
.2561
62
.4680
.5561
.6469
.7404
.8366
.0366
18
.2061
.7527
.3475
4.9869
.6663
.1241
63
.4447
.5278
.6133
.7013
.7916
1.9791
19
.1576
.6910
.2717
.8964
.5609
6.9899
64
.4222
.5006
.5811
.6638
.7486
.9242
20
3.1091
3.6291
4.1955
4.8053
5.4546
6.8535
65
1.4006
1.4745
1.5502
1.6279
1.7074
1.8718
21
.0606
.5672
.1192
.7139
.3476
.7154
66
.3797
.4493
.5206
.5935
.6681
.8219
22
.0122
.5054
.0429
.6221
.2399
.5758
67
.3597
.4252
.4922
.5606
.6305
.7744
23
2.9638
.4436
3.9664
.5301
.1318
.4347
68
.3404
.4021
.4650
.5292
.5947
.7293
24
.9155
.3819
.8901
.4381
.0233
.2926
69
.3219
.3799
.4390
.4992
.5605
.6863
25
2.8675
3.3204
3.8138
4.3461
4.9147
6.1496
70
1.3042
1.3587
1.4141
1.4706
1.5280
1.6455
26
.8196
.2591
.7378
.2541
.8060
.0057
71
.2872
.3384
.3904
.4433
.4970
.6068
27
.7720
.1982
.6621
.1625
.6975
5.8618
72
.2710
.3190
.3678
.4173
.4675
■.5701
28
.7246
.1375
.5867
.0712
.5893
.7175
73
.2555
.3006
.3463
.3926
.4396
.5353
29
.6776
.0772
.5118
3.9804
.4814
.5733
74
.2407
.2830
.3258
.3691
.4130
.5024
30
2.6309
3.0174
3.4375
3.8902
4.3744
5.4299
75
1.2266
1.2662
1.3063
1.3468
1.3879
1.4712
31
.5846
2.9581
.3638
.8008
.2680
.2869
76
.2132
.2503
.2878
.3257
.3640
.4418
32
.5386
.8993
.2907
.7121
.1625
.1448
77
.2004
.2351
.2702
.3056
.3414
.4140
33
.4931
.8412
.2184
.6243
.0579
.0038
78
.1882
.2207
.2535
.2866
.3201
.3877
34
.4481
.7836
.1469
.5375
3.9546
4.8641
79
.1767
.2071
.2378
.2687
.2999
.3629
35
2.4036
2.7268
3.0763
3.4518
3.8525
4.7260
80
1.1657
1.1942
1.2228
1.2517
1.2808
1.3396
36
.3596
.6707
.0067
.3673
.7520
.5899
81
.1553
.1819
.2087
.2356
.2628
.3175
37
.3162
.6154
2.9381
.2841
.6528
.4556
82
.1455
.1703
.1953
.2204
.2457
.2968
38
.2733
.5608
.8705
.2021
.5553
.3235
83
.1361
.1593
.1826
.2061
.2297
.2772
39
.2311
.5071
.8040
.1216
.4595
.1937
84
.1272
.1489
.1706
.1924
.2144
.2586
40
2.1895
2.4543
2.7387
3.0425
3.3655
4.0664
85
1.1188
1.1389
1.1592
1.1795
1.1999
1.2410
41
.1486
.4024
.6747
2.9651
.2734
3.9419
86
.1107
.1294
.1482
.1671
.1861
.2243
42
.1083
.3515
.6118
.8891
.1832
.8200
87
.1028
.1202
.1376
.1551
.1727
.2080
43
.0688
.3015
.5502
.8149
.0951
.7010
88
.0951
.1112
.1273
.1434
.1596
.1922
44
.0299
.2525
.4900
.7422
.0090
.5850
89
.0874
.1020
.1168
.1316
.1464
.1761
45
1.9918
2.2046
2.4311
2.6713
2.9251
3.4720
90
1.0806
1.0941
1.1077
1.1212
1.1349
1.1622
46
.9545
.1576
.3736
.6022
.8433
.3622
91
.0749
.0874
.1000
.1126
.1252
.1506
47
.9179
.1117
.3174
.5348
.7638
.2555
92
.0696
.0813
.0929
.1046
.1164
.1399
48
.8821
.0670
.2627
.4692
.6864
.1520
93
.0646
.0754
.0862
.0971
.1079
.1297
49
.8471
.0233
.2094
.4054
.6113
.0518
94
.0599
.0699
.0799
.0900
.1000
.1202
50
1.8130
1.9806
2.1575
2.3435
2.5385
2.9548
95
1.0553
1.0646
1.0739
1.0831
1.0924
1.1110
51
.7796
.9391
.1071
.2834
.4679
.8611
96
.0504
.0589
.0673
.0758
.0842
.1011
52
.7471
.8987
.0581
.2251
.3996
. 7706
97
.0461
.0539
.0616
.0693
.0770
.0925
53
.7154
.8595
.0106
.1686
.3335
.6833
98
.0401
.0468
.0535
.0602
.0669
.0804
54
.6845
.8213
1.9645
.1139
.2697
.5993
99
.0300
.0350
.0400
.0450
.0500
.0600
1000 Da;
Formula, -g- = 13
TABLE XXXII.
ANNUITY WHICH 1,000 WILL PURCHASE, PAYABLE AT THE END OF EACH YEAR
DURING LIFE.
w
c£s
<
3
Per Ct.
3iJ
Per Ct.
4
Per Ct.
44
Per Ct.
5
Per Ct.
6
Per Ct.
H
Ci3
3
Per Ct.
U
Per Ct.
4
Per Ct.
Per &t.
5
Per Ct.
6
Per Ct.
10
42.07
46.29
50.68
55.15
59.70
69.03
55
79.48
82.80
87.28
91.28
95.34
103.6
11
42.30
46.50
50.88
55.33
59.87
69.17
56
81.90
85.17
89.73
93.73
97.79
106.1
12
42.54
46.72
51.09
55.53
60.05
69.33
57
84.50
87.70
92.34
96.35
100.4
108.7
13
42.78
46.96
51.31
55.73
60.25
69.50
58
87.27
90.40
95.14
99.16
103.2
111.5
14
43.04
47.20
51.54
55.95
60.45
69.67
59
90.25
94.17
98.14
102.2
106.3
114.6
15
43.31
47.45
51.78
56.17
60.66
69.86
60
93.44
97.37
101.4
105.4
109.5'
117.8
16
43.59
47.72
52.03
56.41
60.88
70.05
61
96.87
100.8
104.8
108.9
113.0
121.3
17
43.89
48.00
52.30
56.66
61.12
70.27
62
100.6
104.5
108.5
112.6
116.7
125.1
18
44.20
48.29
52.58
56.93
61.37
70.49
63
104.5
108.5
112.6
116.7
120.8
129.2
19
44.52
48.60
52.87
57.21
61.64
70.73
64
108.8
112.8
116.9
121.0
125.2
133.6
20
44.86
48.92
53.18
57.51
61.92
70.98
65
113.4
117.4
121.5
125.7
129.9
138.3
21
45.22
49.26
53.51
57.82
62.22
71.25
66
118.4
122.4
126.6
130.7
134.9
143.5
22
45.59
49.62
53.85
58.15
62.53
71.53
67
123.7
127.8
132.0
136.2
140.4
149.0
23
45.98
49.99
54.22
58.49
62.86
71.83
68
129.5
133.7
137.9
142.1
146.4
155.0
24
46.39
50.38
54.60
58.86
63.21
72,16
69
135.8
140.0
144.3
148.5
152.8
161.5
25
46.82
50.80
55.00
59.25
63.59
72.50
70
142.7
146.9
151.2
155.5
159.8
168.6
26
47.27
51.23
55.42
59.65
63.98
72.86
71
150.1
154.4
158.7
163.1
167.5
176.3
27
47.74
51.69
55.87
60.08
64.39
73.25
72
158.2
162.5
166.9
171.3
175.7
184.7
28
48.23
52.16
56.34
60.54
64.83
73.66
73
167.0
171.4
175.8
180.3
184.8
193.8
29
48.76
52.67
56.83
61.02
65.30
74.09
74
176.6
181.1
185.6
190.1
194.6
203.8
30
49.30
53.20
57.35
61.53
65.79
74.55
75
187.2
191.7
196.2
200.8
205.4
214.7
31
49.88
53.76
57.90
62.06
66.31
75.05
76
198.7
203.3
207.9
212.5
217.2
226.6
32
50.48
54.35
58.48
62.63
66.87
75.57
77
211.4
216.0
220.7
225.4
230.2
239.7
33
51.12
54.97
59.10
63.23
67.45
76.13
78
225.3
230.1
234.8
239.6
244.4
254.1
34
51.79
55.62
59.75
63.87
68.08
76.72
79
240.7
245.5
250.4
255.2
260.1
270.0
35
52.50
56.31
60.43
64.54
68.73
77.35
80
257.7
262.6
267.5
272.5
277.5
287.5
36
53.24
57.04
61.15
65.25
69.43
78.02
81
276.6
281.6
286.6
291.7
296.7
307.0
37
54.02
57.81
61.92
66.00
70.17
78.73
82
297.6
302.7
307.9
313.0
318.2
328.6
38
54.85
58.62
62.73
66.80
70.96
79.49
83
321.2
326.4
331.7
336.9
342.2
352.9
39
55.73
59.47
63.58
67.65
71.79
80.30
84
347.8
353.1
358.5
363.9
369.3
380.2
40
56.65
60.38
64.49
68.54
72.67
81.15
85
378.0
383.5
389.1
394.6
400.2
411.3
41
57.62
61.34
65.45
69.49
73.61
82.07
86
413.0
418.6
424.3
430.0
435.7
447.2
42
58.66
62.35
66.47
70.50
74.61
83.04
87
454.3
460.2
466.1
471.9
477.9
489.7
43
59.75
63.42
67.55
71.57
75.67
84.08
88
504.4
510.5
516.6
522.7
528.9
541.2
44
60.90
64.56
68.69
72.71
76.80
85.19
89
568.8
575.3
581.8
588.3
594.8
607.8
45
62.13
65.76
69.91
73.92
78.00
86.37
90
641.1
648.0
654.9
661.8
668.7
682.5
46
63.42
67.04
71.20
75.20
79.27
87 63
91
718.8
726.0
733.3
740.5
747.8
762.3
47
64.80
68.39
72.57
76.56
80.63
88.97
92
810.4
818.0
825.7
833.4
841.1
856.4
48
66.26
69.83
74.02
78.02
82,08
90.40
93
923.6
931.7
939.9
948.1
956.3
972.7
49
67.81
71.35
75.57
79.57
83.63
91.93
94
1064.
1073.
1082.
1091.
1099.
1117.
50
69.46
72.98
77.23
81.22
85.27
93.57
95
1250.
1259.
1269.
1278.
1288.
1307.
51
71.22
74.70
78.99
82.97
87.03
95.31
96
1542.
1553.
1563.
1574.
1585.
1606.
52
73.09
76.54
80.86
84.85
88.90
97.18
97
1945.
1957.
1969.
1981.
1992.
2016.
53
75.08
78.49
82.86
86.85
90.90
99.18
98
3090.
3105.
3120.
3135.
3150.
3180.
54
77.21
80.57
85.00
88.99
93.05
101.3
99
0000.
0000.
0000.
0000.
0000.
0000.
1000 1000 Dx
Formula, =
ax JNz+i 14
TABLE XXXIII.
COMMUTATION COLUMNS. 3 PER CENT.
AGE,
D,.
N*.
c*.
M*.
E*-
AGE.
10
74,409.3
1,842,967.6
38,012,941.3
468.129
20,730.738
735,794.660
10
11
71,774.1
1,768,558.3
36,169,973.7
453.092
20,262.609
715,063.922
11
12
69,230.4
1,696,784.2
34,401,415.4
437.852
19,809.517
694,801.313
12
13
66,776.2
1,627,553.8
32,704,631.2
423.777
19,371.665
674,991.796
13 |
14
64,407.3
1,560,777.6
31,077,077.4
410.792
18,947.888
655,620.131
14
15
62,120.8
1,496,370.3
29,516,299.8
397.581
18,537.096
636,672.243
15
16
59,913.7
1,434,249.5
28,019,929.5
384.790
18,139.515
618,135.147
16
17
57,784.0
1,374,335.8
26,585,680.0
372.996
17,754.725
599,995.632
17
18
55,727.9
1,316,551.8
25,211,344.2
361.561
17,381.729
582.240.907
18
19
53,743.1
1,260,823.9
23,894,792.4
350.477
17,020.168
564,859.178
19
20
51,827.5
1,207,080.8
22,633,968.5
340.269
16,669.691
547,839.010
20
21
49,977.6
1,155,253.3
21,426,887.7
330.358
16,329.422
531,169.319
21
22
48,191.5
1,105,275.7
20,271,634.4
320.736
15,999.064
514,839.897
22
23
46,467.1
1,057,084.2
19,166,358.7
311.886
15,678.328
498,840.833
23
24
44,801.9
1,010,617.1
18,109,274.5
303.279
15,366.442
483,162.505
24
25
43,193.7
965,815.2
17,098,657.4
294.910
15,063.163
467,796.063
25
26
41,640.7
922,621.5
16,132,842.2
287.671
14,768.253
452,732.900
26
27
40,140.2
880,980.8
15,210,220.7
280.166
14,480.582
437,964.647
27
28
38,690.8
840,840.6
14,329,239.9
273.279
14,200.416
423,484.065
28
29
37,290.7
802,149.8
13,488,399.3
267.380
13,927.137
409,283.649
29
30
35,937.2
764,859.1
12,686,249.5
261.192
13,659.757
395,356.512
30
31
34,629.4
728,921.9
11,921,390.4
255.526
13,398.565
381,696.755
31
32
33,365.1
694,292.5
11,192,468.5
250.345
13,143.039
368,298.190
32
33
32,143.1
660,927.4
10,498,176.0
245.616
12,892.694
355,155.151
33
34
30,961.1
628,784.3
9,837,248.6
241.305
12,647.078
342,262.457
34
35
29,818.2
597,823.2
9,208,464.3
237.727
12,405.773
329,615.379
35
36
28,711.9
568,005.0
8,610,641.1
233.818
12,168.046
317,209.606
36
37
27,641.8
539,293.1
8,042,636.1
230.585
11,934.228
305,041.560
37
38
26,606.1
511,651.3
7,503,343.0
227.974
11,703.643
293,107.332
38
39
25,603.2
485,045.2
6,991,691.7
225.626
11,475.669
281,403.689
39
40
24,631.8
459,442.0
6,506,646.5
223.816
11,250.043
269,928.020
40
41
23,690.6
434,810.2
6,047,204.5
221.921
11,026.227
258,677.977
41
42
22,778.6
411,119.6
5,612,394.3
221.068
10,804.306
247,651.750
42
43
21,894.1
388,341.0
5,201,274.7
220.076
10,583.238
236,847.444
43
44
21,036.4
366,446.9
4,812,933.7
219.749
10,363.162
226,264.206
44
45
20,203.9
345,410.5
4,446,486.8
219.766
10,143.413
215,901.044
45
46
19,395.7
325,206.6
4,101,076.3
220.096
9,923.647
205,757.631
46
47
18,610.6
305,810.9
3,775,869.7
220.945
9,703.551
195,833.984
47
48
17,847.7
287,200.3
3,470,058.8
222.028
9,482.606
186,130.433
48
49
17,105.8
269,352.6
3,182,858.5
223.545
9,260.578
176,647.827
49
50
16,384.1
252,246.8
2,913,505.9
225.449
9,037.033
167,387.249
50
51
15,681.3
235,862.7
2,661,259.1
227.699
8,811.584
158,350.216
51
52
14,997.0
220,181.4
2,425,396.4
230.252
8,583.885
149,538.632
52
53
14,329.9
205,184.4
2,205,215.0
233.071
8,353.633
140,954.747
53
54
13,679.5
190,854.5
2,000,030.6
236.121
8,120.562
132,601.114
54
Formulas, J)x—vxlx ; N*—Da + •••5 S*—Na + Naj+i-j- .... 15
TABLE XXXIII.
COMMUTATION COLUMNS. 3 PER CENT.
AGE.,
D*:
N*.
s*
0*.
Mx.
Rx.
AGE.
55
13,044.8
177,175.0
1,809,176.1
239.750
7,884.441
124,480.552
55
56
12,425.2
164,130.2
1,632,001.1
243.339
7,644.691
116,596.111
56
57
11,820.0
151,705.0
1,467,870.9
247.416
7,401.352
108,951.420
57
58
11,228.3
139,885.0
1,316,165.9
251.398
7,153.936
101,550.068
58
59
10,649.8
128,656.7
1,176,280.9
255.448
6,902.538
94,396.132
59
60
10,084.2
118,006.89
1,047,624.22
259.708
6,647.090
87,493.594
60
61
9,530.75
107,922.69
929,617.33
263.983
6,387.382
80,846.504
61
62
8,989.18
98,391.94
821,694.64
267.944
6,123.399
74,459.122
62
63
8,459.43
89,402.76
723,302.70
271.903
5,855.455
68,335.723
63
64
7,941.12
80,943.33
633,899.94
275.550
5,583.552
62,480.268
64
65
7,434.26
73,002.21
552,956.61
278.896
5,308.002
56,896.716
65
66
6,938.84
65,567.95
479,954.40
281.537
5,029.106
51,588.714
66
67
6,455.22
58,629.11
414,386.45
283.789
4,747.569
46,559.608
67
68
5,983.40
52,173.89
355,757.34
284.889
4,463.780
41,812.039
68
69
5,524.25
46*190.49
303,583.45
285.558
4,178.891
37,348.259
69
70
5,077.78
40,666.24
257,392.96
284.844
3,893.333
33,169.368
70
71
4,645.05
35,588.46
216,726.72
283.095
3,608.489
29,276.035
71
72
4,226.65
30,943.41
181,138.26
279.935
3,325.394
25,667.546
72
73
3,823.61
26,716.76
150,194.85
275.260
3,045.459
22,342.152
73
74
3,436.99
22,893.15
123,478.09
269.313
2,770.199
19,296.693
74
75
3,067.56
19,456.1.6-
100,.584.94
261.468
2,500.886
16,526.494
75
76
2,716.76
16,388.60
81,128.78
251.902
2,239.418
14,025.608
76
77
2,385.73
13,671.84
64,740.18
240.876
1,987.516
11,786.190
77
78
2,075.36
11,286.11
51,068.34
228.052
1,746.640
9,798.674
78
79
1,786.86
9,210.75
39,782.23
213.704
1,518.588
8,052.034
79
80
1,521.12
7,423.887
30,571.484
197.991
1,304.884
6,533.446
80
81
1,278.82
5,902.767
23,147.597
181.063
1,106.893
5,228.562
81
82
1,060.51
4,623.947
17,244.830
163.404
925.830
4,121.669
82
83
866.216
3,563.437
12,620.883
145.035
762.426
3,195.839
83
84
695.951
2,697.221
9,057.446
126.705
617.391
2,433.413
84
85
548.976
2,001.270
6,360.225
108.533
490.686
1,816.022
85
86
424.452
1,452.294
4,358.955
91.0068
382.153
1,325.336
86
87
321.083
1,027.842
2,906.661
74.7799
291.1461
943.1827
87
88
236.951
706.759
1,878.819
59.7092
216.3662
652.0366
88
89
170.341
469.808
1,172.060
48.3900
156.6570
435.6704
89
90
116.9892
299.4671
702.2519
37.2722
108.2670
279.0134
90
91
76.3096
182.4779
402.7850
26.5633
70.9948
170.7464
91
92
47.5237
106,1683
220.3071
17.9823
44.4315
99.7516
92
93
28.1573
58.6446
114.1388
11.6183
26.4492
55.3201
93
94
15.7189
30.4873
55.4942
7.05748
14.83089
28.87092
94
95
8.20357
14.76840
25.00690
3.98231
7.77341
14.04003
95
96
3.98231
6.56483
10.23852
2.16059
3.79110
6.26662
96
97
1.70573
2.58252
3.67369
.993631
1.630512
2.475523
97
98
.662419
.876794
1.091169
.428751
.636881
.845011
98
99
.214375
.214375
.214375
.208130
.208130
.208130
99
0 x—{h—lx+\)vx+1', Mx=Cx-|-Cx+i + • • • S Rc=Mx + Mx+i+ .... 16
TABLE XXXIII.
LOGARITHMS OF COMMUTATION COLUMNS. 3 PER CENT.
AGE.
AD*.
AN*.
Ab*.
ACX.
aM#.
AES.
I
AGE.
10
4.871627
6.265518
7.579931
2.670366
4.316615
5.866757
10
11
4.855968
6.247619
7.558349
2.656186
4.306695
5.854345
n
12
4.840297
6.229627
7.536576
2.641327
4.296874
5.841861
12
13
4.824622
6.211535
7.514609
2.627137
4.287167
5.829298
13
14
4.808935
6.193341
7.492440
2.613622
4.277561
5.816652
14
15
4.793237
6.175039
7.470062
2.599425
4.268042
5.803916
15
16
4.777526
6.156625
7.447467
2 585224
4.258626
5.791084
16
17
4.761808
6.138093
7.424648
2.571704
4.249314
5.778148
17
18
4.746073
6.119438
7.401596
2.558182
4.240093
5.765103
18
19
4.730323
6.100654
7.378303
2.544659
4.230964
5.751940
19
20
4.714559
6.081736
7.354760
2.531822
4.221928
5.738653
20
21
4.698776
6.062677
7.330959
2.518985
4.212971
5.725233
21
22
4.682971
6.043471
7.306889
2.506148
4.204094
5.711672
22
23
4.667146
6.024110
7.282539
2.493996
4.195300
5.697962
23
24
4.651296
6.004587
7.257901
2.481843
4.186573
5.684093
24
25
4.635421
5.984894
7.232962
2.469689
4.177916
5.670056
25
26
4.619518
5.965024
7.207711
2.458896
4.169328
5.655842
26
27
4.603580
5.944966
7.182135
2.447416
4.160787
5.641439
27
28
4.587608
5.924714
7.156223
2.436606
4.152301
5.626837
28
29
4.571601
5.904255
7.129961
2.427128
4.143862
5.612025
29
30
4.555544
5.883581
7.103334
2.416959
4.135443
5.596989
30
31
4.539445
5.862681
7.076327
2.407435
4.127059
5.581718
31
32
4.523292
5.841542
7.048926
2.398540
4.118696
5.566200
32
33
4.507087
5.820154
7.021114
2.390257
4.110344
5.550418
33
34
4.490817
5.798501
6.992874
2.382567
4.101990
5.534359
34
35
4.474481
5.776573
6.964187
2.376079
4.093625
5.518008
35
36
4.458061
5.754352
6.935035
2.368878
4.085221
5.501346
36
37
4.441565
5.731825
6.905398
2.362832
4.076795
5.484359
37
38
4.424981
5.708974
6.875255
2.357885
4.068321
5.467027
38
39
4.408294
5.685783
6.844582
2.353389
4.059779
5.449330
39
40
4.391497
5.662231
6.813357
2.349892
4.051154
5.431248
40
41
4.374575
5.638300
6.781554
2.346198
4.042427
5.412760
41
42
4.357528
5.613969
6.749148
2.344526
4.033597
5.393841
42
48
4.340328
5.589213
6.716110
2.342574
4.024618
5.374468
43
44
4.322971
5.564011
6.682410
2.341926
4.015492
5.354616
44
45
4.305434
5.538336
6.648017
2.341962
4.006183
5.334255
45
46
4.287706
5.512159
6.612898
2.342611
3.996672
5.313356
46
47
4.269761
5.485453
6.577017
2.344284
3.986931
5.291888
47
48
4.251582
5.458185
6.540337
2.346408
3.976928
5.269817
48
49
4.233143
5.430321
6.502816
2.349365
3.966638
5.247109
49
50
4.214422
5.401826
6.464416
2.353049
3.956026
5.223723
50
51
4.195383
5.372659
6.425086
2.357360
3.945054
5.199619
51
52
4.176002
5.342781
6.384782
2.362203
3.933684
5.174753
52
53
4.156242
5.312144
6.343450
2.367488
3.921875
5.149080
53
54
4.136069
5.280702
6.301037
2.373134
3.909586
5.122548
54 17
TABLE XXXIII.
LOGARITHMS OF COMMUTATION COLUMNS. 3 PER CENT.
AGE.
s
AN*.
AS*.
AC*.
AM*.
AR*.
AGE,
55
4.115439
5.248403
6.257481
2.379759
3.896771
5.095102
55
56
4.094302
5.215188
6.212720
2.386212
3.883360
5.066684
56
57
4.072616
5.181000
6.166688
2.393428
3.869311
5.037233
57
58
4.050312
5.145771
6.119311
2.400363
3.854545
5.006680
58
59
4.027342
5.109432
6.070511
2.407303
3.839009
4.974955
59
60
4.003641
5.071906
6.020205
2.414486
3.822632
4.941976
60
61
3.979127
5.033113
5.968304
2.421576
3.805323
4.907662
61
62
3.953720
4.992959
5.914710
2.428044
3.786993
4.871918
62
63
3.927341
4.951351
5.859320
2.434413
3.767561
4.834648
63
64
3.899882
4.908181
5.802021
2.440200
3.746910
4.795743
64
65
3.871238
4.863336
5.742691
2.445442
3.724931
4.755087
65
66
3.841287
4.816692
5.681200
2.449536
3.701491
4.712555
66
67
3.809911
4.768113
5.617405
2.452995
3.676471
4.668009
67
68
3.776948
4.717453
5.551154
2.454676
3.649703
4.621301
68
69
3.742273
4.664552
5.482278
2.455695
3.621061
4.572270
69
70
3.705674
4.609234
5.410597
2.454606
3.590321
4.520737
70
71
3.666990
4.551309
5.335912
2.451932
3.557326
4.466513
71
72
3.625996
4.490568
5.258010
2,447057
3.521843
4.409384
72
73
3.582474
4.426784
5.176655
2.439743
3.483653
4.349125
73
74
3.536178
4.359706
5.091590
2.430257
3.442511
4.285483
74
75
3.486794
4.289057
5.002533
2.417419
3.398094
4.218181
75
76
3.434051
4.214542
4.909175
2.401231
3.350135
4.146922
76
77
3.377621
4.135827
4.811174
2.381793
3.298311
4.071373
77
78
3.317094
4.052544
4.708152
2.358035
3.242203
3.991167
78
79
3.252091
3.964295
4.599690
2.329813
3.181440
3.905906
79
80
3.182162
3.870631
4.485317
2.296645
3.115572
3.815142
80
81
3.106809
3.771056
4.364506
2.257828
3.044105
3.718382
81
82
3.025515
3.665013
4.236659
2.213264
2.966531
3.615073
82
83
2.937626
3.551869
4.101090
2.161473
2.882198
3.504585
83
84
2.842579
3.430916
3.957006
2.102795
2.790560
3.386216
84
85
2.739553
3.301306
3.803473
2.035563
2.690804
3.259121
85
86
2.627829
3.162055
3.639382
1.959073
2.582237
3.122326
86
87
2.506618
3.011926
3.463394
1.873785
2.464111
2.974596
87
88
2.374659
2.849271
3.273885
1.776042
2.335189
2.814272
88
89
2.231318
2.671920
3.068950
1.684756
2.194950
2.639158
89
90
2.068146
2.476349
2.846493
1.571385
2.034496
2.445625
90
91
1.882579
2.261210
2.605074
1.424280
1.851228
2.232352
91
92
1.676910
2.025995
2.343028
1.254844
1.647691
1.998920
92
93
1.449591
1.768228
2.057433
1.065143
1.422413
1.742883
93
94
1.196422
1.484119
1.744248
0.848650
1.171168
1.460461
94
95
0.914003
1.169334
1.398060
0.600135
0.890611
1.147368
95
96
0.600135
0.817223
1.010237
0.334573
0.578765
0.797033
96
1 97
0.231910
0.412043
0.565103
1.997225
0.212324
0.393667
97
98
1.821133
1.942898
0.037892
1.632205
1.804059
1.926863
98
| 99
l__
1.331175
1.331175
1.331175
1.318338
1.318338
1.318338
99 18
TABLE XXXIV.
COMMUTATION COLUMNS. PER CENT.
A&E.
N*.
si.
o*.
M*.
AGE,
10
70,892.0
1,602,284.9
31,098,393.9
443.845
16,735.405
551,200.709
10
11
08,050.8
1,531,392.9
29,496,109.0
427.512
16,291.560
534,465.304
11
12
65,322.1
1,463,342.1
27,964,716.1
411.138
15,864.048
518,173.744
12
13
62,701.9
1,398,020.0
26,501,374.0
395.998
15,452.910
502,309.696
13
14
60,185.5
1,335,318.1
25,103,354.0
382.010
15,056.912
486,856.786
14
15
57,768.4
1,275,132.6
23,768,035.9
367.938
14,674.902
471,799.874
15
16
55,446.7
1,217,364.2
22,492,903.3
354.382
14,306.964
457,124.972
16
17
53,217.4
1,161,917.5
21,275,539.1
341.859
13,952.582
442,818.008
17
18
51,076.0
1,108,700.1
20,113,621.6
329.779
13,610.723
428,865.426
18
19
49,019.0
1,057,624.1
19,004,921.5
318.124
13,280.944
415,254.703
19
20
47,043.2
1,008,605.1
17,947,297.4
307.366
12,962.820
401,973.759
20
21
45,145.0
961,561.9
16,938,692.3
296.972
12,655.454
389,010.939
21
22
43,321.3
916,416.9
15,977,130.4
286.930
12,358.482
376,355.485
22
23
41,569.4
873,095.6
15,060,713.5
277.665
12,071.552
363,997.003
23
24
39,886.1
831,526.2
14,187,617.9
268.698
11,793.887
351,925.451
24
25
38,268.6
791,640.1
13,356,091.7
260.021
11,525.189
340,131.564
25
26
36,714.5
753,371.5
12,564,451.6
252.413
11,265.168
328,606.375
26
27
35,220.5
716,657.0
11,811,080.1
244.640
11,012.755
317,341.207
27
28
33,784.8
681,436.5
11,094,423.1
237.474
30,768.115
306,328.452
28
29
32,404.8
647,651.7
10,412,986.6
231.225
10,530.641
295,560.337
29
30
31,077.8
615,246.9
9,765,334.9
224.782
10,299.416
285,029.696
30
31
29,802.1
584,169.1
9,150,088.0 i
218.844
10,074.634
274,730.280
31
32
28,575.4
554,367.0
8,565,918.9
213.372
9,855.790
264,655.646
32
33
27,395.8
525,791.6
8 011,551.9
208.330
9,642.418
254,799.856
33
34
26,261.0
498,395.8
7,485,760.3
203.684
9,434.088
245,157.438
34
35
25,169.3
472,134.8
6,987,364.5
199.695
9,230.404
235,723.350
35
36
24,118.4
446,965.5
6,515,229.7
195.462
9,030.709
226,492.946
36
37
23,107.3
422,847.1
6,068,264.2
191.828
8,835.247
217,462.237
37
38
22,134.2
399,739.8
5,645,417.1
188.740
8,643.419
208,626.990
38
39
21,196.9
377,605.6
5,245,677.3
185.893
8,454.679
199,983.571
39
40
20,294.2
356,408.7
4,868,071.7
183.512
8,268.786
191,528.892
40
41
19,424.4
336,114.5
4,511,663.0
181.078
8,085.274
183,260.106
41
42
18,586.5
316,690.1
4,175,548.5
179.511
7,904.196
175,174.832
42
43
17,778.4
298,103.6
3,858,858.4
177.843
7,724.685
167,270.636
43
44
16,999.4
280,325.2
3,560,754.8
176.720
7,546.842
159,545.951
44
45
16,247.8
263,325.8
3,280,429.6
175.881
7,370.122
151,999.109
45
46
15,522.5
247,078.0
3,017,103.8
175.293
7,194.241
144,628.987
46
47
14,822,3
231,555.5
2,770,025.8
175.119
7,018.948
137,434.746
47
48
14,145.9
216,733.2
2,538,470.3
175.128
6,843.829
130,415.798
48
49
13,492.4
202,587.3
2,321,737.1
175.473
6,668.701
123,571.969
49
50
12,860.7
189,094.9
2,119,149.8
176.112
6,493.228
116,903.268
50
51
12,249.7
176,234.2
1,930,054.9 |
177.010
6,317.116
110,410.040
51
52
11,658.4
163,984.5
1,753,820.7
178.130
6,140.106
104,092.924
52
53
11,086.1
152,326.1
1,589,836.2
179.440
5,961.976
97,952.818
53
54
10,531.7
141,240.0
1,437,510.1
180.910
5,782.536
91,990.842
54 19
table xxxiv.
COMMUTATION COLUMNS. PER CENT.
AGE,
Djp.
N*.
s*.
c*
M*.
n*.
AGE.
55
9,994.66
130,708.28
1,296,270.07
182.803
5,601.626
86,208.306
55
56
9,473.86
1120,713.62
1,165,561.79
184.644
5,418,823
80,606.680
56
57
8,968.85
1111,239.76
1,044,848.17
186.830
5,234.179
75,187.857
57
58
8,478.73
102,270.91
933,608.41
188.920
5,047.349
69,953.678
58
59
8,803.10
93,792.18
831,337.50
191.036
4,858.429
64,906.329
59
60
7,541.42
84,989.08
737,545.32
193.284
4,667.393
60,047.900
60
61
7,093.12
77,447.66
652,556.24
195.516
4,474.109
55,380.507
61
62
6,657.73
70,354.54
575,108.58
197.491
4,278.593
50,906.398
62
63
6,235.11
63,696.81
504,754.04
199.440
4,081.102
46,627.805
63
64
5,824.80
57,461.70
441,057.23
201.139
3,881.662
42,546.703
64
65
5,426.69
51,636.90
383,595.53
202.598
3,680.523
38,665.041
65
66
5,040.59
46,210.21
331,958.63
203.529
3,477.925
34,984.518
66
67
4,666.62
41,169.62
285,748.42
204.165
3,274.396
31,506.593
67
68
4,304.63
36,503.00
244,578.80
! 203.967
3,070.231
28,232.197
68
69
3,955.10
32,198.37
208,075.80
1 203.459
2,866.264
25,161.966
69
70
3,617.90
28,243.27
175,877.43
! 201.969
2,662.805
22,295.702
70
71
3,293.58
24,625.37
147,634.16
199.759
2,460.836
19,632.897
71
72
2,982.44
21,331.79
123,008.79
196.575
2,261.077
17,172.061
72
73
2,685.02
18,349.35
101,677.00
192.359
2,064.502
14,910.984
73
74
2,401.86
15,664.33
83,327.65
187.293
1,872.143
12,846.482
74
75
2,133.34
13,262.47
67,663.32
180.960
1,684.850
10,974.339
75
76
1,880.24
11,129.13
54,400.85
173.496
1,503.890
9,289.489
76
77
1,643.16
9,248.89
43,271.72
165.101
1,330.394
7,785.599
77
78
1,422.49
7,605.73
34,022.83
j 155.556
1,165.293
6,455.205
78
79
1,218.832
6,183.245
26,417.108
1 145.065
1,009.737
5,289.912
79
80
1,032.552
4,964.413
20,233.863
133.749
864.672
4,280.175
80
81
863.885
3,931.861
15,269.450
121.723
730.923
3,415.503
81
82
712.948
3,067.976
11,337.589
109.321
609.200
2,684.580
82
83
579.518
2,355.028
8,269.613
96.5629
499.8792
2,075.3797
83
84
463.358
1,775.510
5,914.585
83.9517
403.3163
1,575.5005
84
85
363.737
1,312.152
4,139.075
i 71.5639
319.3646
1,172.1842
85
86
279.872
948.415
2,826.923
59.7175
247.8007
852.8196
86
87
210.691
668.543
1,878.508
48.8326
188.0832
605.0189
87
88
154.733
457.852
1,209.965
38.8028
139.2506
416.9357
88
89
110.698
303.119
752.113
31.2950
100.4478
277.6851
89
90
75.6598
192.4206
448.9945
23.9884
69.1528
177.2373
90
91
49.1127
116.7608
256.5739
17.0135
45.1644
108.0845
91
92
30.4385
67.6481
139.8131
11.4618
28.1509
62.9201
92
93
17.9474
37.2096
72.1650
7.36969
16.68907
34.76921
93
94
9.97075
19.26219
34.95535
4.45506
9.31938
18.08014
94
95
5.17852
9.29144
15.69316
2.50170
4.86432
8.76076
95
96
2.50170
4.11292
6.40172
1.35074
2.36262
3.89644
96
97
1.06637
1.61122
2.28880
.618184
1.011883
1.533823
97
98
.412123
.544852
.677581
.265458
.393699
.521940
98
99
.132729
.132729
.132729
.128241
.128241
.128241
99 20
TABLE XXXIV.
LOGARITHMS OF COMMUTATION COLUMNS. 3-1- PER CENT.
AGE.
AD*.
AN*.
AS*.
AC*.
AM*.
AR*.
AGE.
10
4.850597
6.204740
7.492738
2.647231
4.223636
5.741310
10
11
4.832833
6.185087
7.469765
2.630948
4.211963
5.727919
11
12
4.815060
6.165346
7.4466 L0
2.613987
4.200414
5.714475
12
13
4.797281
6.145513
7.423208
2.597693
4.189010
5.700972
13
14
4.779492
6.125585
7.399732
2.582075
4.177736
5.687401
14
15
4.761690
6.105555
7.375993
2.565775
4.166575
5.673758
15
16
4.743876
6.085420
7.352046
2.549471
4.155547
5.660035
16
17
4.726054
6.065175
7.327881
2.533847
4.144655
5.646225
17
18
4.708217
6.044814
7.303490
2.518223
4.133881
5.632321
18
19
4.690364
6.024331
7.278866
2.502597
4.123229
5.618315
19
20
4.672497
6.003721
7.253999
2.487656
4.112700
5.604198
20
21
4.654610
5.982977
7.228880
2.472716
4.102278
5.589962
21
22
4.636702
5.962093
7.203499
2.457776
4.091966
5.575598
22
23
4.618774
5.941062
7.177845
2.443521
4.081763
5.561097
23
24
4.600822
5.919876
7.151909
2.429265
4.071658
5.546451
24
25
4.582842
5.898528
7.125679
2.415008
4.061648
5.531647
25
26
4.564837
5.877009
7.099143
2.402111
4.051738
5.516676
26
27
4.546796
5.855311
7.072290
2.388528
4.041896
5.501527
27
28
4.528721
5.833426
7.045105
2.375616
4.032140
5.486188
28
29
4.510609
5.811341
7.017575
2.364034
4.022454
5.470646
29
30
4.492450
5.789050
0.989687
2.351762
4.012812
5.454891
30
31
4.474247
5.766538
0.961425
2.340135
4.003229
5.438906
31
32
4.455993
5.743798
0.932774
2.329137
3.993692
5.422681
32
33
4.437684
5.720813
6.903717
2.318750
3.984186
5.406199
33
34
4.419311
5.097574
0.874236
2.308958
3.974700
5.389445
34
35
4.400871
5.674060
6.844313
2.300367
3.965221
5.372402
35
36
4.382349
5.050274
6.813930
2.291063
3.955722
5.355055
36
37
4.363750
5.626183
6.783005
2.282913
3.946219
5.337383
37
38
4.345063
5.601778
6.751696
2.275804
3.936686
5.319371
38
39
4.326272
5.577038
6.719802
2.269204
3.927097
5.300994
39
40
4.307372
5.551948
0.687357
2.203064
3.917442
5.282234
40
41
4.288348
5.526487
0.654337
2.257860
3.907695
5.263068
41
42
4.269197
5.500634
6.620714
2.254091
3.897858
5.243472
42
43
4.249893
5.474307
6.580459
2.250036
3.887881
5.223420
43
44
4.230434
5.447602
0.551542
2.247285
3.877765
5.202886
44
45
4.210794
5.420493
6.515931
2.245218
3.867475
5.181841
45
46
4.190961
5.392834
6.479590
2.243764
3.856985
5.160255
46
47
4.170915
5.364055
0.442484
2.243334
3.846272
5.138097
47
48
4.150631
5.335925
6.404572
2.243355
3.835299
5.115330
48
49
4.130090
5.306612
6.365813
2.244209
3.824041
5.091920
49
50
4.109265
5.276680
6.326162
2.245790
3.812461
5.067827
50
51
4.088124
5.246090
6.285570
2.247998
3.800519
5.043008
51
52
4.066640
5.214803
6.243985
2.250737
3.788176
5.017421
52
53
4.044777
5.182774
6.201352
2.253919
3.775391
4.991017
53
54
4.022499
5.149958
6.157008
2.257462
3.762119
4.963744
54 21
TABLE XXXIY.
LOGARITHMS OF COMMUTATION COLUMNS. 3 j PER CENT.
AGE.
XI) x.
AN*.
AS*
AO*.
AM*.
AR*.
AGE,
55
3.9997G8
5.116303
6.112696
2.261984
3.748314
4.935550
55
56
3.970527
5.081756
6.066535
2.266334
3.733905
4.906371
56
57
3.952737
5.046260
6.019053
2.271446
3.718849
4.876148
57
58
3.928331
5.009752
5.970165
2.276278
3.703063
4.844811
58
59
3.903258
4.972167
5.919777
2.281116
3.686496
4.812287
59
60
3.877453
4.929363
5.867789
2.286195
3.669074
4.778498
60
61
3.850837
4.889008
5.814618
2.291182
3.650707
4.743357
61
62
3.823326
4.847292
5.759750
2.295547
3.631301
4.706773
62
63
3.794844
4.804118
5.703080
2.299813
3.610777
4.668645
63
64
3.765281
4.759379
5.644495
2.303497
3.589018
4.628866
64
65
3.734535
4.712960
5.583873
2.306636
3.565909
4.587318
65
66
3.702481
4.664738
5.521084
2.308627
3.541321
4.543876
66
67
3.669002
4.614577
5.455984
2.309982
3.515132
4.498401
67
68
3.633936
4.562329
5.388419
2.309560
3.487171
4.450745
68
69
3.597157
4.507834
5.318222
2.308476
3.457316
4.400744
69
70
3.558456
4.450915
5.245210
2.305284
3.425340
4.348221
70
71
3.517668
4.391383
,5.169187
2.300507
3.391083
4.292985
71
72
3.474572
4.329027
5.089936
2.293529
3.354316
4.234823
72
73
3.428947
4.263620
5.007223
2.284112
3.314815
4.173506
73
74
3.380547
4.194912
4.920789
2.272522
3.272338
4.108784
74
75
3.329060
4.122625
4.830353
2.257582
3.226561
4.040379
75
76
3.274213
4.046461
4.735606
2.239291
3.177216
3.967992
76
77
3.215680
3.966090
4.636204
2.217750
3.123979
3.891292
77
78
3.153050
3.881141
4.531770
2.191888
3.066434
3.809910
78
79
3.085944
3.791216
4.421885
2.161563
3.004208
3.723448
79
80
3.013912
3.695868
4.306079
2.126291
2.936850
3.631462
80
81
2.936456
3.594598
4.183823
2.085372
2.863872
3.533454
81
82
2.853058
3.486852
4.054521
2.038705
2.784760
3.428877
82
83
2.763067
3.371996
3.917485
1.984810
2.698865
3.317098
83
84
2.665917
3.249323
3.771924
1.924029
2.605646
3.197418
84
85
2.560787
3.117984
3.616903
1.854694
2.504287
3.068996
85
86
2.446960
2.976999
3.451314
1.776101
2.394103
2.930857
86
87
2.323646
2.825129
3.273813
1.688710
2.274350
2.781769
87
88
2.189584
2.660725
3.082773
1.588863
2.143798
2.620069
88
89
2.044140
2.481613
2.876283
1.495475
2.001940
2.443553
89
90
1.878865
2.284251
2.652241
1.380000
1.839811
2.248555
90
91
1.691194
2.067297
2.409213
1.230793
1.654796
2.033763
91
92
1.483423
1.830256
2.145548
1.059254
1.449492
1.798790
92
93
1.254001
1.570655
1.858327
0.867449
1.222433
1.541195
93
94
0.998728
1.284706
1.543513
0.648853
0.969388
1.257201
94
95
0.714206
0.968083
1.195710
0.398235
0.687022
0.942542
95
96
0.398235
0.614150
0.806297
0.130570
0.373394
0.590668
96
97
0.027907
0.207154
0.359608
1.791118
0.005129
0.185775
97
98
1.615027
1.736279
1.830961
1.423995
1.595164
1.717621
98
99
.
1.122965
1.122965
1.122965
1.108025
1.108025
1.108025
99 22
TABLE XXXV.
COMMUTATION COLUMNS. 4 PER CENT.
AGE.
D®.
N*.
S*.
o*.
M*.
Ra;.
AGE,
10
67,556.5
1,400,554.9
25,570,207.5
420.929
13,688.944
417,085.094
10
11
64,537.2
1,332,998.4
24,169,652.6
403.490
13,268.015
403,396.150
11
12
61,651.5
1,268,461.2
22,836,654.2
386.169
12,864.525
390,128.135
12
13
58,894.1
1,206,809.7
21,568,193.0
370.161
12,478.356
377,263.610
13
14
56,258.7
1,147,915.6
20,361,383.3
355.369
12,108.195
364,785.254
14
15
53,739.7
1,091,656.9
19,213,467.7
340.634
11,752.826
352,677.059
15
16
51,332.1
1,037,917.2
18,121,810.8
326.505
11,412.192
340.924.233
16
17
49,031.3
986,585.1
17,083,893.6
313.454
11,085.687
329,512.041
17
18
46,832.0
937,553.8
16,097,308.5
300.923
10,772.233
318,426.354
18
19
44,729.9
890,721.8
15,159,754.6
288.893
10,471.310
307,654.121
19
20
42,720.6
845,991.9
14,269,032.8
277.782
10,182.417
297,182.811
20
21
40,799.7
803,271.3
13,423,040.9
267.098
9,904.635
287,000.394
21
22
38,963.4
762,471.6
12,619,769.6
256.825
9,637.537
277,095.759
22
23
37,207.9
723,508.2
11,857,298.0
247.337
9,380.712
267,458.222
23
24
35,529.6
686,300.3
11,133,789.8
238.199
9,133.375
258,077.510
24
25
33,924.9
650,770.7
10,447,489.5
229.398
8,895.176
248,944.135
25
26
32,390.6
616,845.8
9,796,718.8
221.616
8,665.778
240,048.959
26
27
30,923.2
d84j 45o. 2
9,179,873.0
213.759
8,444.162
231,383.181
27
28
29,520.1
553,532.0
8,595,417.8
j 206.499
8,230.403
222,939.019
28
29
28,178.2
524,011.9
8,041,885.8
200.099
8,023.904
214,708.616
29
30
26,894.3
495,833.7
7,517,873.9
193.589
7,823.805
206,684.712
30
31
25,666.4
468,939.4
7,022,040.2
187.568
7,630.216
198,860.907
31
32
24,491.6
443,273.0
6,553,100.8
181.999
7,442.648
191,230.691
32
33
23,367.6
418,781.4
6,109,827.8
176.843
7,260.649
183,788.043
33
34
22,292.1
395,413.8
5,691,046.4
172.069
7,083.806
176,527.394
34
35
21,262.6
373,121.7
5,295,632.6
167.888
6,911.737
169,443.588
35
36
20,276.9
351,859.1
4,922,510.9
163.539
6,743.849
162,531.851
36
37
19,333.4
331,582.2
4,570,651.8
159.727
6,580.310
155,788.002
37
38
18,430.2
312,248.8
4,239,069.6
156.400
6,420.583
149,207.692
38
39
17,564.9
293,818.6
3,926,820.8
153.301
6,264.183
142,787.109
39
40
16,736.0
276,253.7
3,633,002.2
150.609
6.110.882
136,522.926
40
41
15,941.7
259,517.7
3,356,748.5
147.898
5,960.273
130,412.044
41
42
15,180.7
243,576.0
3,097,230.8
145.913
5,812.375
124,451.771
42
43
14,450.9
228,395.3
2,853,654.8
143.862
5,666.462
118,639.396
43
44
13,751.2
213,944.4
2,625,259.5
142.266
5,522.600
112,972.934
44
45
13,080.1
200,193.2
2,411,315.1
140.909
5,380.334
107,450.334
45
46
12,436.1
187,113.1
2,211,121.9
139.764
5,239.425
102,070.000
46
47
11,818.0
174,677.0
2,024,008.8
138.954
5,099.661
96,830.575
47
48
11,224.5
162,859.0
1,849,331.8
138.292
4,960.707
91,730.914
48
49
10,654.5
151,634.5
1,686,472.8
137.898
4,822.415
86,770.207
49
50
10,106.8
140,979.99
1,534,838.27
137.736
4,684.517
81,947.792
50
51
9,580.36
130,873.19
1,393,858.28
137.773
4,546.781
77,263.275
51
52
9,074.11
121,292,83
1,262,985.09
137.978
4,409.008
72,716.494
52
53
8,587.13
112,218.72
1,141,692.26
138.324
4,271.030
68,307.486
53
54
8,118.54
103,631.59
1,029,473.54]
138.787
4,132.706
64,036.456
54 23
TABLE XXXV.
COMMUTATION COLUMNS. 4 PER CENT.
AGE.
D#.
N*.
S*.
0*.
Ms.
R«j.
AGE.
55
7,667.49
95,513.05
925,841.95
139.565
3,993.919
59,903.750
55
56
7,233.03
87,845.56
830,328.90
140.292
3,854.354
55,909.831
56
57
6,814.55
80,612.53
742,483.34
141.271
3,714.062
52,055.477
57
58
6,411.17
73,797.98
661,870.81
142.165
3,572.791
48,341.415
58
59
6,022.43
67,386.81
588,072.83
143.066
3,430.626
44,768.624
59
60
5,647.73
61,364.38
520,686.02
144.053
3,287.560
41,337.998
60
61
5,286.45
55,716.65
459,321.64
145.016
3,143.507
38,050.438
61
62
4,938.11
50,430.20
403,604.99
145.777
2,998.491
34,906.931
62
63
4,602.42
45,492.09
353,174.79
146.508
2,852.714
31,908.440
63
64
4,278.88
40,889.67
307,682.70
147.046
2,706.206
29,055.726
64
65
3,967.26
36,610.79
266,793.03
147.400
2,559.160
26,349.520
65
66
3,667.28
32,643.53
230,182.24
147.366
2,411.760
23,790.360
66
67
3,378.87
28,976.25
197,538.71
147.116
2,264.394
21,378.600
67
68
3.101.79
25,597.38
168,562.46
146.266
2,117.278
19,114.206
68
69
2,836.23
22,495.59
142,965.08
145.200
1,971.012
16,996.928
69
70
2,581.94
19,659.36
120,469.49
143.444
1,825.812
15,025.916
70
71
2,339.19
17,077.42
100,810.13
141.193
1,682.368
13,200.104
71
72
2,108.03
14,738.23
83,732.71
138.274
1,541.175
11,517.736
72
73
1,888.68
12,630.20
68,994.48
134.657
1,402.901
9,976.561
73
74
1,681.38
10,741.52
56,364.28
130.481
1,268.244
8,573.660
74
75
1,486.23
9,060.14
45,622,76
125.463
1,137.763
7,305.416
75
76
1,303.60
7,573.91
36,562.62
119.710
1,012.300
6,167.653
76
77
1,133.76
6,270.31
28,988.71
113.369
892.590
5,155.353
77
78
976.78
5,136.55
22,718.40
106.302
779.221
4,262.763
78
79
832.910
4,159.77
17,581.85
98.6560
672.9185
3,483.5424
79
80
702.220
3,326.861
13,422.083
90.5231
574.2625
2,810.6239
80
81
584.688
2,624.641
10,095.222
81.9874
483.7394
2,236.3614
81
82
480.212
2,039.953
7,470.581
73.2802
401.7520
1,752.6220
82
83
388.462
1,559.741
5,430.628
64.4168
328.4718
1,350.8700
83
84
309.105
1,171.279
3,870.887
55.7346
264.0550
1,022.3982
84
85
241.481
862.174
2,690.608
47.2821
208.3204
758.3432
85
86
184.911
620.693
1,837.434
39.2655
161.0383
550.0228
86
87
138.534
435.782
1,216.741
31.9541
121.7728
388.9845
87
88
101.251
297.248
780.959
25.2689
89.8187
267.2117
88
89
72.088
195.997
483.711
20.2818
64.5498
177.3930
89
90
49.0339
123.9092
287.7138
15.4717
44.2680
112.8432
90
91
31.6761
74.8754
163.8046
10.9204
28.7963
68.5752
91
92
19.5375
43.1993
88.9292
7.32159
17.87594
39.77885
92
93
11.4644
23.6618
45.7299
4.68498
10.55435
21.90291
93
94
6.3385
12.1974
22.0681
2.81850
5.86937
11.34856
94
95
3.27621
5.85884
9.87072
1.57510
3.05087
5.47919
95
96
1.57510
2.58263
4.01188
.846350
1.475769
2.428324
96
97
.668170
1.00753
1.42925
.385483
.629419
.952555
97
98
.256988
.339356
.421724
.164736
.243936
.323136
98
99
.082368
.082368
.082368
.079200
.079200
.079200
99 24
TABLE XXXV.
LOGARITHMS OF COMMUTATION COLUMNS. 4 PER CENT.
AGE.
AD®-
AN*.
AS*.
AC*-
AM®-
AR®
AGE.
10
4.829667
6.146300
7.407734
2.624208
4.136370
5.620225
10
11
4.809810
6.124829
7.383270
2.605832
4.122806
5.605732
11
12
4.789944
6.103277
7.358632
2.586777
4.109394
5.591207
12
13
4.770072
6.081639
7.333814
2.568391
4.096158
5.576645
13
14
4.750190
6.059910
7.308807
2.550679
4.083080
5.562037
14
15
4.730295
6.038086
7.283606
2.532287
4.070142
5.547377
15
16
4.710389
6.016162
7.258202
2.513890
4.057369
5.532658
16
17
4.690473
5.994135
7.232587
2.496174
4.044763
5.517871
17
18
4.670543
5.971996
7.206756
2.478456
4.032306
5.503009
18
19
4.650598
5.949742
7.180692
2.460737
4.020001
5.488063
19
20
4.630637
5.927366
7.154394
2.443704
4.007850
5.473024
20
21
4.610657
5.904862
7.127851
2.426670
3.995839
5.457883
21
22
4.590657
5.882224
7.101051
2.409637
3.983966
5.442630
22
23
4.570635
5.859443
7.073985
2.393289
3.972236
5.427256
23
24
4.550590
5.836514
7.046643
2.376940
3.960632
5.411750
24
25
4.530518
5.813428
7.019012
2.360590
3.949155
5.396102
25
26
4.510419
5.790177
6.991081
2.345601
3.937808
5.380300
26
27
4.490285
5.766751
6.962837
2.329925
3.926557
5.364332
27
28
4.470118
5.743143
6.934267
2.314919
3.915421
5.348186
28
29
4.449913
5.719341
6.905358
2.301245
3.904386
5.331849
29
30
4.429661
5.695336
6.876095
2.286880
3.893418
5.315308
30
31
4.409365
5.671117
6.846463
2.273159
3.882537
5.298549
31
32
4.389017
5.646671
6.816447
2.260068
3.871728
5.281558
32
33
4.368615
5.621987
6.786029
2.247589
3.860976
5.264317
33
34
4.348150
5.597052
6.755192
2.235703
3.850267
5.246812
34
35
4.327616
5.571850
6.723918
2.225019
3.839587
5.229025
35
36
4.307002
5.546369
6.692187
2.213622
3.828908
5.210938
36
37
4.286309
5.520591
6.659978
2-203379
3.818246
5.192534
37
38
4.265529
5.494501
6.627271
2.194237
3.807574
5.173791
38
39
4.244646
5.468079
6.594041
2,185544
3.796864
5.154689
39
40
4.223652
5.441308
6.560266
2.177851
3.786104
5.135206
40
41
4.202535
5.414167
6.525919
2.169961
3.775266
5.115321
41
42
4.181292
5.386634
6.490974
2.164093
3.764354
5.095001
42
43
4.159894
5.358687
6.455401
2.157945
3.753312
5.074229
43
44
4.138342
5.330301
6.419172
2.153101
3.742144
5.052974
44
45
4.116610
5.301449
6.382254
2.148940
3.730809
5.031208
45
46
4.094683
5.272104
6.344613
2.145394
3.719284
5.008898
46
47
4.072544
5.242236
6.306212
2.142871
3.707541
4.986013
47
48
4.050168
5.211812
6.267015
2.140798
3.695544
4.962516
48
49
4.027533
5.180798
6.226979
2.139559
3.683265
4.938371
49
50
4.004615
5.149157
6.186062
2.139048
3.670665
4.913537
50
51
3.981382
5.116851
6.144219
2.139162
3.657704
4.887973
51
52
3.957804
5.083835
6.101398
2.139809
3.644341
4.861633
52
53
3.933848
5.050065
6.057549
2.140898
3.630533
4.834468
53
54
3.909478
5.015492
6.012615
2.142348
3.616235
4.806427
54 25
TABLE XXX Y.
LOGARITHMS OF COMMUTATION COLUMNS. 4 PER CENT.
AGE.
AD*-
AN*-
co
8
A(V
AM*-
AE*-
AGE.
55
3.884653
4.980063
5.966537
2.144777
3.601399
4.777454
55
56
3.859320
4.943720
5.919250
2.147034
3.585951
4.747488
56
57
3.833437
4.906403
5.870687
2.150053
3.569849
4.716466
57
58
3.806937
4.868044
5.820773
2.152792
3.553008
4.684319
58
59
3.779772
4.828575
5.769431
2.155536
3.535374
4.650974
59
60
3.751874
4.787916
5.716576
2.158523
3.516874
4.616349
60
61
3.723164
4.745985
5.662117
2.161417
3.497415
4.580360
61
62
3.693561
4.702691
5.605956
2.163689
3.476903
4.542912
62
63
3.662986
4.657936
5.547990
2.165862
3.455258
4.503906
63
64
3.631330
4.611614
5.488103
2.167453
3.432362
4.463232
64
65
3.598491
4.563609
5.426174
2.168499
3.408097
4.420773
65
66
3.564344
4.513797
5.362072
2.168397
3.382334
4.376401
66
67
3.528771
4.462042
5.295651
2.167659
3.354951
4.329979
67
68
3.491613
4.408196
5.226761
2.165144
3.325778
4.281356
68
69
3.452741
4.352097
5.155230
2.161967
3.294689
4.230371
69
70
3.411946
4.293569
5.080877
2.156682
3.261456
4.176841
70
71
3.369066
4.232422
5.003504
2.149812
3.225922
4.120577
71
72
3.323877
4.168445
4.922895
2.140740
3.187853
4.061367
72
73
3.276158
4.101410
4.838814
2.129230
3.147027
3.998981
73
74
3.225666
4.031066
4.751004
2.115548
3.103203
3.933166
74
75
3.172086
3.957135
4.659182
2.098515
3.056051
3.863645
75
76
3.115146
3.879320
4.563037
2.078130
3.005309
3.790120
76
77
3.054520
3.797289
4.462229
2.054496
2.950652
3.712258
77
78
2.989797
3.710672
4.356378
2.026542
2.891660
3.629691
78
79
2.920598
3.619069
4.245065
1.994123
2.827963
3.542021
79
80
2.846473
3.522035
4.127820
1.956759
2.759111
3.448803
80
81
2.766924
3.419070
4.004116
1.913747
2.684611
3.349542
81
82
2.681433
3.309620
3.873354
1.864986
2.603958
3.243688
82
83
2.589349
3.193052
3.734850
1.808999
2.516498
3.130614
83
84
2.490106
3.068660
3.587811
1.746125
2.421694
3.009620
84
85
2.382883
2.935595
3.429850
1.674697
2.318731
2.879866
85
86
2.266963
2.792877
3.264212
1.594011
2.206928
2.740381
86
87
2.141556
2.639269
3.085198
1.504527
2.085551
2.589932
87
88
2.005401
2.473119
2.892628
1.402587
1.953368
2.426855
88
89
1.857863
2.292249
2.684587
1.307105
1.809896
2.248936
89
90
1.690496
2.093104
2.458961
1.189538
1.646090
2.052475
90
91
1.500732
1.874339
2.214326
1.038238
1.459332
1.836167
91
92
1.290868
1.635477
1.949044
0.864606
1.252269
1.599652
92
93
1,059352
1.374048
1.660200
0.670708
1.023431
1.340502
93
94
0.801987
1.086267
1.343765
0.450019
0.768591
1.054941
94
95
0.515372
0.767812
0.994349
0.197308
0.484424
0.738716
95
96
0.197308
0.412062
0.603348
1.927550
0.169018
0.385307
96
97
1.824887
0.003258
0.155108
1.586005
1.798940
1.978890
97
98
1.409913
1.530656
1.625028
1.216789
1.387276
1.509385
98
99
2,915759
2.915759
2.915759
2.898726
2.898726
2.898726
99 26
TABLE XXXVI.
COMMUTATION COLUMNS. 4|- PER CENT.
AGE.
D®-
S».
c*.
M*.
R®.
AGE,
10
64,392.8
1,232,047.6
21,206,324.6
399.297
11,338.107
318,856.514
10
11
61,220.7
1,167,654.8
19,974.277.0
380.923
10,938.810
307,518.407
11
12
58,203.5
1,106,434.1
18,806,622.2
362.827
10,557.887
296,579.597
12
13
55,334.1
1,048,230.6
17,700,188.1
346.123
10,195.060
286,021.710
13
14
52,605.2
992,896.5
16,651,957.5
330.701
9,848.937
275,826.650
14
15
50,009.3
940,291.3
15,659,061.0
315.471
9,518.236
265,977.713
15
16
47,540.2
890,282.0
14,718,769.7
300.940
9,202.765
256,459.477
16
17
45,192.2
842,741.8
13,828,487.7
287.528
8,901.825
247,256.712
17
18
42,958.6
797,549.6
12,985,745.9
274.714
8,614.297
238,354.887
18
19
40,833.9
754,591.0
12,188,196.3
262.469
8,339.583
229,740.590
19
20
38,813.1
713,757.1
11,433,605.3
251.166
8,077.114
221.401.007
20
21
36,890.5
674,944.0
10,719,848.2
240.351
7,825.948
213,323.893
21
22
35,061.5
638,053.5
10,044,904.2
230.001
7,585.597
205.497.945
22
23
33,321.7
602,992.0
9,406,850.7
220.444
7,355.596
197,912.348
23
24
31,666.4
569,670.3
8,803,858.7
211.284
7,135.152
190,556.752
24
25
30,091.5
538,003.9
8,234,188.4
202.504
6.923.868
183,421.600
25
26
28,593.2
507,912.4
7,696,184.5
194.698
6,721.364
176,497.732
26
27
27,167.2
479,319.2
7,188,272.1
186.897
6.526.666
169.776.368
27
28
25,810.4
452,152.0
6,708,952.9
179.686
6,339.769
163.249.702
28
29
24,519.3
426,341.6
6,256,800.9
173.283
6,160.083
156,909.933
29
30
23,290.1
401,822.3
5,830,459.3
166.843
5,986.800
150,749.850
30
31
22,120.4
378,532.2
5,428,637.0
160.881
5,819.957
144,763.050
31
32
21,007.0
356,411.8
5,050,104.8
155.357
5,659.076
138,943.093
32
33
19,947.0
335,404.8
4,693,693.0
150.234
5,503.719
133,284.017
33
34
18,937.8
315,457.8
4,358,288.2
145.479
5,353.485
127,780.298
34
35
17,976.8
296,520.0
4,042,830.4
141.264
5,208.006
122,426.813
35
36
17,061.5
278,543.2
3,746,310.4
136.947
5,066.742
117.218.807
36
37
16,189.7
261,481.7
3,467.767.2
133.115'
4,929.795
112,152.065
37
38
15,359.5
245,292.0
3,206,285.5
129.719
4,796.680
107,222.270
38
39
14,568.4
229,932.5
2,960,993.5
126.539
4,666.961
102,425.590
39
40
13,814.5
215,364.1
2,731,061.0
123.723
4,540.422
97,758.629
40
41
13,095.9
201,549.6
2,515,696.9
120.914
4,416.699
93,218.207
41
42
12,411.0
188,453.7
2,314,147.3
118.721
4,295.785
88,801.508
42
43
11,757.9
176,042.7
2,125,693.6
116.492
4,177.064
84,505.723
43
44
11,135.0
164,284.8
1,949,650.9
114.648
4,060.572
80,328.659
44
45
10,540.9
153,149.8
1,785,366.1
113.012
3,945.924
76,268.087
45
46
9,973.97
142,608.93
1,632,216.26
111.555
3,832.912
72,322.163
46
47
9,432.90
132,634.96
1,489,607.33
110.380
3,721.357
68,489.251
47
48
8,916.33
123,202.06
1,356,972.37
109.329
3,610.977
64,767.894
48
49
8,423.02
114,285.73
1,233,770.31
108.496
3,501.648
61,156.917
49
50
7,951.85
105,862.71
1,119,484.58
107.849
3,393.152
57,655.269
50
51
7,501.55
97,910.86
1,013,621.87
107.362
3,285.303
54,262.117
51
52
7,071.17
90,409.31
915,711.01
107.007
3,177.941
50,976.814
52
53
6,659.66
83,338.14
825,301.70
106.762
3,070.934
47,798.873
53
54
6,266.11
76,678.48
741,963.56
106.607
2,964.172
44,727.939
54 27
TABLE XXX YI.
COMMUTATION COLUMNS. 4J PER CENT.
AGE.
Dar
N*.
s*.
Qc-
Mar
Rar
AGE.
55
5,889.67
70,412.37
665,285.08
106.692
2,857.565
41,763.767
55
56
5,529.36
64,522.70
594,872.71
106.735
2,750.873
38,906.202
56
57
5,184.51
58,993.34
530,350.01
106.965
2,644.138
36,155.329
57
58
4,854.29
53,808.83
471,356.67
107.127
2,537.173
33,511.191
58
59
4,538.14
48,954.54
417,547.84
107.290
2,430.046
30,974.018
59
60
4,235.43
44,416.40
368,593.30
107.513
2,322.756
28,543.972
60
61
3,945.52
40,180.97
324,176.90
107.715
2,215.243
26,221.216
61
62
3,667.90
36,235.45
283,995.93
107.761
2,107.528
24,005.973
62
63
3,402.20
32,567.55
247,760.48
107.784
1,999.767
21,898.445
63
64
3,147.90
29,165.35
215,192.93
107.662
1,891.983
19,898.678
64
65
2,904.68
26,017.45
186,027.58
107.405
1,784.321
18,006.695
65
66
2,672.20
23,112.77
160,010.13
106.866
1,676.916
16,222.374
66
67
2,450.27
20,440.57
136,897.36
106.174
1,570.050
14,545.458
67
68
2,238.58
17,990.30
116,456.79
105.056
1,463.876
12,975.408
68
69
2,037.12
15,751.72
98,466.49
103.791
1,358.820
11,511.532
69
70
1,845.61
13,714.60
82,714.77
102.045
1,255.029
10,152.712
70
71
1,664.09
11,868.99
69,000.17
99.9629
1,152.9835
8,897.6832
71
72
1,492.47
10,204.90
57,131.18
97.4282
1,053.0206
7,744.6997
72
73
1,330.77
8,712.43
46,926.28
94.4261
955.5924
6,691.6791
73
74
1,179.04
7,381.66
38,213.85
91.0598
861.1663
5,736.0867
74
75
1,037.20
6,202.62
30,832.19
87.1385
770.1065
4,874.9204
75
76
905.403
5.165.420
24,629.570
82.7453
682.9680
4,104.8139
76
77
783.670
4,260.017
19,464.150
77.9877
600.2227
3.421.8459
77
78
671.933
3,476.347
15,204.133
72.7760
522.2350
2.821.6232
78
79
570.223
2,804.414
11,727.786
67.2183
449.4590
2,299.3882
79
80
478.451
2,234.191
8,923.372
61.3819
382.2407
1,849.9292
80
81
396.465
1,755.740
6,689.181
55.3280
320.8588
1,467.6885
81
82
324.064
1,359.275
4,933.441
49.2155
265.5308
1,146.8297
82
83
260.894
1,035.211
3,574.166
43.0558
216.3153
881.2989
83
84
206.604
774.317
2,538.955
37.0744
173.2595
664.9836
84
85
160.632
567.713
1,764.638
31.3014
136.1851
491.7241
85
86
122.413
407.081
1,196.925
25.8699
104.8837
355.5390
86
87
91.2723
284.6678
789.8437
20.9521
79.0138
250.6553
87
88
66.3897
193.3955
505.1759
16.4894
58.0617
171.6415
88
89
47.0415
127.0058
311.7804
13.1716
41.5723
113.5798
89
90
31.8442
79.9643
184.7746
9.99977
28.40072
72.00746
90
91
20.4731
48.1201
104.8103
7.02435
18.40095
43.60674
91
92
12.5671
27.6470
56.6902
4.68696
11.37660
25.20579
92
93
7.33902
15.07986
29.04321
2.98476
6.68964
13.82919
93
94
4.03822
7.74084
13.96335
1.78706
3.70488
7.13955
94
95
2.07726
3.70262
6.22251
.993905
1.917819
3.434666
95
96
.993905
1.62536
2.51989
.531500
.923914
1.516847
96
97
.419605
.631453
.894534
.240922
.392414
.592933
97
98
.160615
.211848
.263081
.102465
.151492
.200519
98
99
.051233
.051233
.051233
.049027
.049027
.049027
99 28
TABLE XXX YI.
LOGARITHMS OF COMMUTATION COLUMNS. PER CENT.
AGE.
AD*.
AN*.
AS*.
AC*.
AM*.
AR*.
AGE.
10
4.808837
6.090628
7.326465
2.601296
4.054541
5.503595
10
11
4.786898
6.067315
7.300471
2.580837
4.038970
5.487871
11
12
4.764949
6.043926
7.274311
2.559699
4.023576
5.472141
12
13
4.742993
6.020457
7.247978
2.539230
4.008390
5.456399
13
14
4.721029
5.996904
7.221465
2.519436
3.993390
5.440636
14
15
4.699051
5.973262
7.194766
2.498960
3.978557
5.424845
15
16
4.677061
5.949528
7.167871
2.478480
3.963919
5.409019
16
17
4.655063
5.925695
7.140775
2.458681
3.949479
5.393148
17
18
4.633050
5.901758
7.113467
2.438880
3.935220
5.377224
18
19
4.611021
5.877712
7.085939
2.419078
3.921144
5.361238
19
20
4.588978
5.853551
7.058183
2.399962
3.907256
5.345180
20
21
4.566915
5.829268
7.030189
2.380845
3.893537
5.329040
21
22
4.544831
5.804857
7.001946
2.361729
3.879990
5.312808
22
23
4.522727
5.780312
6.973444
2.343298
3.866618
5.296473
23
24
4.500599
5.755624
6.944673
2.324866
3.853403
5.280024
24
25
4.478444
5.730785
6.915621
2.306433
3.840349
5.263451
25
26
4.456262
5.705789
6.886275
2.289361
3.827457
5.246739
26
27
4.434045
5.680625
6.856624
2.271602
3.814692
5.229878
27
28
4.411795
5.655284
6.826655
2.254514
3.802074
5.212852
28
29
4.389508
5.629758
6.796352
2.238756
3.789586
5.195651
29
30
4.367172
5.604034
6.765703
2.222308
3.777195
5.178257
30
31
4.344793
5.578103
6.734691
2.206505
3.764920
5.160658
31
32
4.322363
5.551952
6.703300
2.191330
3.752746
5.142837
32
33
4.299877
5.525569
6.671515
2.170769
3.740656
5.124778
33
34
4.277329
5.498941
6.639316
2.162800
3.728637
5.106464
34
35
4.254713
5 472054
6.606686
2.150033
3.716672
5.087876
35
36
4.232016
5.444893
6.573604
2.136553
3.704729
5.068998
36
37
4.209240
5.417441
6.540050
2.124227
3.692829
5.049807
37
38
4.186377
5.389686
6.506002
2.113002
3.680941
5.030285
38
39
4.163411
5.361600
6.471437
2.102226
3.669034
5.010408
39
40
4.140334
5.333173
6.436331
2.092450
3.657096
4.990155
40
41
4.117134
5.304382
6.400658
2.082477
3.645098
4.969501
41
42
4.093808
5.275205
6.364391
2.074526
3.633043
4.948421
42
43
4.070328
5.245618
6.327501
2.066295
3.620871
4.926886
43
44
4.046692
5.215597
6.289957
2.059368
3.608587
4.904870
44
45
4.022877
5.185117
6.251727
2.053124
3.596148
4.882343
45
46
3.998868
5.154147
6.212778
2.047495
3.583529
4.859271
46
47
3.974645
5.122658
6.173072
2.042889
3.570702
4.835623
47
48
3.950186
5.090618
6.132571
2.038734
3.557625
4.811359
48
49
3.925468
5.057992
6.091234
2.035412
3.544273
4.786445
49
50
3.900468
5.024742
0.049018
2.032817
3.530603
4.760839
50
51
3.875151
4.990834
6.005876
2.030849
3.516575
4.734497
51
52
3.849491
4.956214
5.961759
2.029412
3.502146
4.707372
52
53
3.823452
4.920844
5.916613
2.028418
3.487270
4.679418
53
54
3.796998
4.884674
5.870383
2.027785
3.471903
4.650578
54 29
TABLE XXXVI.
LOGARITHMS OF COMMUTATION COLUMNS. 4?, PER CENT.
AGE,
AD*.
AN*.
AS*.
AC*.
AM*
AR*.
AGE,
55
3.770094-
4.847649
5.823008
2.028131
3.455997
4.620800
55
56
3.742675
4.809712
5.774424
2.028305
3.439470
4.590019
56
57
3.714708
4.770803
5.724563
2.029242
3.422285
4.558172
57
58
3.686126
4.730854
5.673350
2.029898
3.404350
4.525189
58
59
3.656878
4.689793
5.620706
2.030559
3.385615
4.490997
59
60
3.626897
4.647543
5.566547
2.031463
3.366004
4.455515
60
61
3.596104
4.604020
5.510782
2.032274
3.345421
4.418653
61
62
3.564418
4.559134
5.453312
2.032463
3.323774
4.380319
62
63
3.531760
4.512785
5.394032
2.032553
3.300980
4.340413
63
64
3.498021
4.464867
5.332828
2.032061
3.276917
4.298824
64
65
3.463099
4.415265
5.269577
2.031024
3.251473
4.255434
65
66
3.426869
4.363852
5.204148
2.028839
3.224512
4.210114
66
67
3.389214
4.310493
5.136395
2.026018
3.195914
4.162728
67
68
3.349972
4.255038
5.066165
2.021420
3.165504
4.113121
68
69
3.309017
4.197328
4.993288
2.016160
3.133162
4.061133
69
70
3.266139
4.137183
4.917583
2.008793
3.098654
4.006582
70
71
3.221176
4.074414
4.838850
1.999839
3.061823
3.949277
71
72
3.173904
4.008809
4.756873
1.988685
3.022437
3.889005
72
73
3.124103
3.940139
4.671416
1.975092
2.980273
3.825535
73
74
3.071528
3.868154
4.582221
1.959327
2.935087
3.758616
74
75
3.015864
3.792575
4.489004
1.940210
2.886551
3.687968
75
76
2.956842
3.713106
4.391457
1.917743
2.834400
3.613293
76
77
2.894133
3.629412
4.289235
1.892026
2.778313
3.534260
77
78
2.827326
3.541123
4.181962
1.861988
2.717866
3.450499
78
79
2.756045
3.447842
4.069216
1.827487
2.652690
3.361612
79
80
2.679837
3.349120
3.950529
1.788040
2.582337
3.267155
80
81
2.598205
3.244460
3.825373
1.742945
2.506314
3.166634
81
82
2.510631
3.133307
3.693150
1.692102
2.424115
3.059499
82
83
2.416464
3.015029
3.553175
1.634031
2.335087
2.945123
83
84
2.315138
2.888918
3.404655
1.569074
2.238697
2.822811
84
85
2.205832
2.754129
3.246656
1.495563
2.134129
2.691722
85
86
2.087829
2.609681
3.078067
1.412794
2.020708
2.550887
86
87
1.960339
2.454338
2.897541
1.321227
1.897703
2.399077
87
88
1.822101
2.286446
2.703443
1.217205
1.763890
2.234623
88
89
1.672481
2.103823
2.493849
1.119640
1.618804
2.055301
89
90
1.503030
1.902896
2.266642
0.999990
1.453329
1.857378
90
91
1.311184
1.682327
2.020404
0.846606
1.264840
1.639553
91
92
1.099236
1.441648
1.753508
0.670891
1.056013
1.401500
92
93
0.865638
1.178398
1.463045
0.474910
0.825403
1.140797
93
94
0.606190
0.888788
1.144990
0.252138
0.568774
0.853671
94
95
0.317491
0.568509
0.793966
1.997345
0.282808
0.535885
95
96
1.997345
0.210950
0.401382
1.725503
1.965632
0.180942
96
97
1.622841
1.800341
1.951597
1.381876
1.593744
1.773005
97
98
1.205785
1.326024
1.420089
1.010577
1.180390
1.302156
98
99
2.709547
2.709547
2.709547
2.690431
2.690431
2.690431
99 30
TABLE XXXVII.
COMMUTATION COLUMNS. 5 PER CENT.
AGE.
D**
N*.
s*.
c«.
Mx.
R®.
AGE,
10
61,391.3
1,089,674.9
17,710,636.6
3*78.872
9,502.066
246,311.154
10
11
58,089.1
1,028,283.6
16.620,961.7
359.717
9,123.194
236,809.088
11
12
54,963.2
970,194.5
15,592,678.1
340.997
8,763.477
227,685.894
12
13
52,004.9
915,231.3
14,622,483.6
323.749
8,422.480
218,922.417
13
14
49,204.8
863,226.4
13,707,252.3
307.851
8,098.731
210,499.937
14
15
46,553.9
814,021.6
12,844,025.9
292.275
7,790.880
202,401.206
15
16
44,044.6
767,467.7
12,030,004.3
277.485
7,498.605
194,610.326
16
17
41,669.8
723,423.1
11,262,536.6
263.856
7,221.120
187,111.721
17
18
39,421.7
681,753.3
10,539,113.5
250.895
6,957.264
179,890.601
18
19
37,293.6
642,331.6
9,857,360.2
238.571
6,706.369
172,933.337
19
20
35,279.2
605,038.0
9,215,028.6
227.210
6,467.798
166,226.968
20
21
33,372.0
569,758.8
8,609,990.6
216.391
6,240.588
159,759.170
21
22
31,566.4
536,386.8
8,040,231.8
206.087
6,024.197
153,518.582
22
23
29,857.1
504,820.4
7,503,845.0
196.583
5,818.110
147,494.385
23
24
28,238.9
474,963.3
6,999,024.6
187.517
5,621.527
141,676.275
24
25
26,706.6
446,724.4
6,524,061.3
178.869
5,434.010
136,054.748
25
26
25,256.0
420.017.8
6,077,336.9
171.155
5,255.141
130,620.738
26
27
23,882.2
394,761.8
5,657,319.1
1 163.515
5,083.986
125,365.597
27
28
22,581.4
370,879.6
5,262,557.3
156.457
4,920.471
120,281.611
28
29
21,349.6
348,298.2
4,891,677.7
150.164
4,764.014
115,361.140
29
30
20,182.8
326,948.6
4,543,379.5
143.895
4,613.850
110,597.126
30
31
19,077.9
306,765.8
4,216,430.9
138.092
4,469.955
105,983.276
31
32
18,031.3
287,687.9
3,909,665.1
132.715
4,331.863
101,513.321
32
33
17,039.9
269,656.6
3,621,977.2
127.728
4,199.148
97,181.458
33
34
16,100.8
252,616.7
3,352,320.6
123.096
4,071.420
92,982.310
34
35
15,211.0
236,515.9
3,099,703.9
118.961
3,948.324
88,910.890
35
36
14,367.7
221,304.9
2,863,188.0
114.776
3,829.363
84,962.566
36
37
13,568.7
206,937.2
2,641,883.1
111.033
3,714.587
81,133.203
37
38
12,811.6
193,368.5
2,434,945.9
107.685
3,603.554
77,418.616
38
39
12,093.8
180,556.9
2,241,577.4
104.546
3,495.869
73,815.062
39
40
11,413.4
168,463.1
2,061,020.5
101.732
3,391.323
70,319.193
40
41
10,768.2
157,049.7
1,892,557.4
98.9489
3,289.591
66,927.870
41
42
10,156.4
146,281.46
1,735,507.74
96.6911
3,190.6419
63,638.2786
42
43
9,576.08
136,125.06
1,589,226.28
94.4239
3,093.9508
60,447.6367
43
44
9,025.68
126,548.98
1,453,101.22
92.4874
2,999.5269
57,353.6859
44
45
8,503.39
117,523.30
1,326,552.24
90.7332
2,907.0395
54,354.1590
45
46
8,007.72
109,019.91
1,209,028.94
89.1382
2,816.3063
51,447.1195
46
47
7,537.27
101,012.19
1,100,009.03
87.7777
2,727.1681
48,630.8132
47
48
7,090.58
93,474.92
998,996.84
86.5279
2,639.3904
45,903.6451
48
49
6,666.40
86,384.34
905,521.92
85.4596
2,552.8625
43,264.2547
49
50
6,263.50
79,717.94
819,137.58
84.5461
2,467.4029
40,711.3922
50
51
5,880.69
73,454.44
739,419.64
83.7630
2,382.8568
38,243.9893
51
52
5,516.89
67,573.75
665,965.20
83.0888
2,299.0938
35,861.1325
52
53
5,171.09
62,056.86
598,391.45
82.5041
2,216.0050
33,562.0387
53
54
4,842.34
56,885.77
536,334.59
81.9917
2,133.5009
31,346.0337
54 TABLE XXXVII.
31
COMMUTATION COLUMNS. 5 PEE CENT.
AGE.
D«.
N*.
s*.
0*.
Mx.
Ea.
AGE.
55
4,529.77
52,043.43
479,448.82
81.6663
2,051.5092
29,212,5328
55
56
4,232.39
47,513.66
427,405.39
81.3100
1,969.8429
27,161.0236
56
57
3,949.55
43,281.27
379,891.73
81.0975
1,888.5329
25,191.1807
57
58
3,680.37
39,331.72
336,610.46
80.8333
1,807.4354
23,302.6478
58
59
3,424.28
35,651.35
297,278.74
80.5710
1,726.6021
21,495.2124
59
60
3,180.65
32,227.07
261,627.39
80.3543
1,646.0311
19,768.6103
60
61
2,948.84
29.046.42
229,400.32
80.1212
1,565.6768
18,122.5792
61
62
2,728.30
26,097.58
200,353.90
79.7744
1,485.5556
16,556.9024
62
63
2,518.61
23,369.28
174,256.32
79.41 JO
1,405.7812
15,071.3468
63
64
2,319.26
20,850.67
150,887.04
78.9433
1,326.3702
13,665.5656
64
65
2,129.88
18,531.41
130,036.37
78.3800
1,247.4269
12,339.1954
65
66
1,950.07
16,401.53
111,504.96
77.6153
1,169.0469
11,091.7685
66
67
1,779.60
14,451.46
95,103.43
76.7456
1,091.4316
9,922.7216
67
68
1,618.11
12,671.86
80,651.97
75.5758
1,014.6860
8,831.2900
68
69
1,465.48
11,053.75
67,980.11
74.3104
939.1102
7,816.6041
69
70
1,321.38
9,588.27
56,926.36
72.7125
864.7998
6,877.4939
70
71
1,185.75
8,266.89
47,338.09
70.8896
792.0873
6,012,6941
71
72
1,058.39
7,081.14
39,071.20
68.7631
721.1977
5,220.6068
72
73
939.232
6,022.748
31,990.065
66.3269
652.4346
4,499.4091
73
74
828.180
5,083.516
25,967.317
63.6577
586.1077
3,846.9745
74
75
725.085
4,255.336
20,883.801
60.6264
522.4500
3,260.8668
75
76
629.931
3,530.251
16,628.465
57.2956
461.8236
2,738.4168
76
77
542.639
2,900.320
13,098.214
53.7442
404.5280
2,276.5932
77
78
463.054
2,357.681
10,197.894
49.9138
350.7838
1.872.0652
78
79
391.090
1,894.627
7,840.213
45.8824
300.8700
1,521.2814
79
80
326.585
1,503.537
5,945.586
41.6991
254.9876
1,220.4114
80
81
269.334
1,176.952
4,442.049
37.4075
213.2885
965.4238
81
82
219.102
907.618
3,265.097
33.1163
175.8810
752.1353
82
83
175.551
688.516
2,357.479
28.8336
142.7647
576.2543
83
84
138.358
512.965
1,668.963
24.7098
113.9311
433.4896
84
85
107.0600
374.6070
1,155.9980
20.7627
89.2213
319.5585
85
86
81.1989
267.5473
781.3912
17.0782
68.4586
230.3372
86
87
60.2541
186.3484
513.8439
13.7658
51.3804
161.8786
87
88
43.6191
126.0943
327.4955
10.7822
37.6146
110.4982
88
89
30.7598
82.4752
201.4012
8.57174
26.8324
72.88355
89
90
20.7233
51.7154
118.9260
6.47659
18.26065
46.05116
90
91
13.2599
30.9921
67.2106
4.52782
11.78406
27.79151
91
92
8.10065
17.73217
36.21852
3.00676
7.25624
16.00745
92
93
4.70813
9.63152
18.48635
1.90567
4.24948
8.75121
93
94
2.57826
4.92339
8.85483
1.13554
2.34381
4.50173
94
95
1.31995
2.34513
3.93144
.628545
1.208272
2.157920
95
96
.628544
1.02518
1.58631
.334520
.579727
.949648
96
97
.264094
.396640
.561125
.150911
.245207
.369921
97
98
.100607
.132546
.164485
.063878
.094296
.124714
98
99
.031939
.031939
.031939 1
1
.030418
.030418
.030418
99 32
TABLE XXXVII.
LOGARITHMS OF COMMUTATION COLUMNS. 5 PER CENT.
AGE.
AD®,
AN®.
AS®.
AC®.
AM®.
AR®.
AGE,
10
4.788107
6.037297
7.248234
2.578493
3.977818
5.391484
10
11
4.764095
6.012113
7.220656
2.555961
3.960147
5.374398
11
12
4.740072
5.986859
7.192921
2.532750
3.942677
5.357336
12
13
4.716044
5.961531
7.165021
2.510208
3.925440
5.340290
13
14
4.692007
5.936125
7.136950
2.488341
3.908417
5.323252
14
15
4.667956
5.910636
7.108701
2.465792
3.891587
5.306214
15
16
4.643893
5.885060
7.080266
2.443239
3.874981
5.289166
16
17
4.619822
5.859393
7.051636
2.421366
3.858605
5.272101
17
18
4.595736
5.833627
7.022804
2.399493
3.842438
5.255008
18
19
4.571634
5.807759
6.993760
2.377618
3.826488
5.237878
19
20
4.547518
5.781783
6.964496
2.356428
3.810757
5.220701
20
21
4.523382
5.755691
6.935003
2.335239
3.795226
5.203466
21
22
4.499225
5.729478
6.905269
2.314050
3.779899
5.186161
22
23
4.475048
5.703137
6.875284
2.293546
3.764782
5.168775
23
24
4.450847
5.676660
6.845037
2.273041
3.749855
5.151297
24
25
4.426619
5.650039
6.814518
2.252535
3.735120
5.133713
25
26
4.402364
5.623268
6.783714
2.233390
3.720584
5.116012
26
27
4.378074
5.596335
6.752611
2.213558
3.706205
5.098178
27
28
4.353751
5.569232
6.721197
2.194396
3.692007
5.080199
28
29
4.329390
5.541951
6.689458
2.176566
3.677973
5.062059
29
30
4.304982
5.514479
6.657379
2.158045
3.664064
5.043744
30
31
4.280530
5.486807
6.624945
2.140168
3.650304
5.025237
31
32
4.256026
5.458922
6.592139
2.122921
3.636674
5.006523
32
33
4.231468
5.430811
6.558945
2.106286
3.623161
4.987583
33
34
4.206847
5.402462
6.525346
2.090244
3.609746
4.968401
34
35
4.182158
5.373861
6.491321
2.075404
3.596412
4.948955
35
36
4.157387
5.344991
6.456850
2.059851
3.583126
4.929228
36
37
4.132539
5.315838
6.421913
2.045453
3.569911
4.909199
37
38
4.107603
5.286386
6.386489
2.032155
3.556731
4.888846
38
39
' 4.082563
5.256614
6.350554
2.019306
3.543555
4.868145
39
40
4.057414
5.226505
6.314082
2.007457
3.530369
4.847074
40
41
4.032141
5.196037
6.277049
1.995411
3.517142
4.825607
41
42
4.006741
5.165189
6.239427
1.985386
3.503878
4.803718
42
43
3.981188
5.133938
6.201186
1.975082
3.490513
4.781379
43
44
3.955480
5.102258
6.162296
1.966083
3.477053
4.758561
44
45
3.929592
5.070124
6.122724
1.957766
3.463451
4.735233
45
46
3.903509
5.037506
6.082436
1.950064
3.449680
4.711361
46
47
3.877214
5.004373
6.041397
1.943384
3.435712
4.686911
47
48
3.850682
4.970694
5.999564
1.937156
3.421504
4.661847
48
49
3.823891
4.936435
5.956899
1.931761
3.407027
4.636130
49
50
3.796817
4.901556
5.913357
1.927093
3.392240
4.609716
50
51
3.769428
4.866018
5.868891
1.923052
3.377098
4.582563
51
52
3.741694
4.829778
5.823452
1.919543
3.361557
4.554624
52
53
3.713582
4.792790
5.776985
1.916476
3.345571
4.525848
53
54
3.685056
4.755003
5.729436
1.913770
3.329093
4.496183
54 33
TABLE XXXVII.
LOGARITHMS OP COMMUTATION COLUMNS. 5 PER CENT.
AGE,
AD*.
AN*.
AS*.
Q
8
AM*.
AR*.
AGE,
55
3.65G076
4.716366
5.680742
1.912043
3.312073
4.465569
55
56
3.626586
4.676818
5.630840
1.910144
3.294432
4.433946
56
57
3.596547
4.636300
5.579660
1.909007
3.276125
4.401248
57
58
3.565892
4.594743
5.527128
1.907590
3.257063
4.367405
58
59
3.534570
4.552076
5.473164
1.906179
3.237192
4.332341
59
60
3.502516
4.508221
5.417684
1.905009
3.216438
4.295976
60
61
3.469651
4.463092
5.360594
1.903747
3.194702
4.258220
61
62
3.435892
4.416600
5.301798
1.901863
3.171889
4.218979
62
63
3.401160
4.368645
5.241189
1.899881
3.147918
4.178152
63
64
3.365349
4.319120
5.178652
1.897315
3.122665
4.135627
64
65
3.328354
4.267908
5.114065
1.894205
3.096015
4.091287
65
66
3.290050
4.214884
5.047294
1.889947
3.067832
4.045001
66
67
3.250322
4.159912
4.978196
1.885054
3.037997
3.996631
67
68
3.209008
4.102840
4.906615
1.878383
3.006332
3.946025
68
69
3.165979
4.043509
4.832382
1.871050
2.972717
3.893018
69
70
3.121029
3.981740
4.755314
1.861609
2.936916
3.837430
70
71
3.073992
3.917342
4.675211
1.850582
2.898773
3.779069
71
72
3.024648
3.850103
4.591857
1.837355
2.858055
3.717721
72
73
2.972773
3.779794
4.505015
1.821689
2.814537
3.653156
73
74
2.918125
3.706164
4.414427
1.803851
2.767978
3.585119
74
75
2.860389
3.628933
4.319810
1.782662
2.718045
3.513332
75
76
2.799293
3.547806
4.220852
1.758122
2.664476
3.437499
76
77
2.734511
3.462446
4.117212
1.730332
2.606948
3.357286
77
78
2.665632
3.372485
4.049136
1.698221
2.545040
3.272321
78
79
2.592277
3.277524
3.894328
1.661647
2.478379
3.182209
79
80
2.513996
3.177114
3.774195
1.620127
2.406519
3.086506
80
81
2.430291
3.070759
3.647583
1.572958
2/328967
2.984718
81
82
2.340645
2.957903
3.513896
1.520042
2.245219
2.876296
82
83
2.244404
2.837914
3.372448
1.459899
2.154621
2.760614
83
84
2.141005
2.710088
3.222446
1.392869
2.056642
2.636979
84
85
2.029627
2.573576
3.062957
1.317285
1.950468
2.504550
85
86
1.909550
2.427401
2.892869
1.232443
1.835428
2.362364
86
87
1.779987
2.270325
2.710831
1.138802
1.710797
2.209189
87
88
1.639677
2.100695
2.515205
1.032707
1.575350
2.043355
88
89
1.487983
1.916323
2.304062
0.933069
1.428659
1.862629
89
90
1.316459
1.713619
2.075277
0.811346
1.261516
1.663240
90
91
1.122540
1.491251
1.827438
0.655890
1.071295
1.443912
91
92
0.908520
1.248762
1.558930
0.478101
0.860712
1.204322
92
93
0.672848
0.983695
1.266851
0.280048
0.628336
0.942069
93
94
0.411327
0.692264
0.947181
0.055202
0.369922
0.653380
94
95
0.120556
0.370167
0.594551
1.798336
0.082165
0.334035
95
96
1.798336
0.010800
0.200388
1.524422
1.763223
1.977563
96
97
1.421759
1.598397
1.749060
1.178721
1.389533
1.568109
97
98
1.002630
1.122367
1.216127
2.805349
2.974493
1.095915
98
99
2.504319
2.504319
2.504319
2.483130
2.483130
2.483130
99 34
TABLE XXXVIII.
COMMUTATION COLUMNS. 6 PER CENT.
AGE,
D*.
sr«.
S*.
o«.
M».
AGE,
10
55,839.4
864,777.4
12,598,316.8
341.358
6,889.834
151,664.904
10
11
52,337.5
808,938.0
11,733,539.4
321.042
6,548.476
144,775.070
11
12
49,054.0
756,600.5
10,924,601.4
301.464
6,227.434
138,226.594
12
13
45,975.8
707,546.5
10,168,000.9
283.515
5,925.970
131,999.160
13
14
43,089.8
661,570.7
9,460,454.4
267.050
5,642.455
126,073.190
14
15
40,383.8
618.480.9
8,798,883.7
251.146
5,375.405
120,430.735
15
16
37,846.7
578,097.1
8,180,402.8
236.188
5,124.259
115,055.330
16
17
35,468.3
540,250.4
7,602,305.7
222.468
4,888.071
109,931.071
17
18
33,238.2
504,782.1
7,062,055.3
209.545
4,665.603
105,043.000
18
19
31,147.3
471,543.9
6,557,273.2
197.372
4,456.058
100,377.397
19
20
29,186.8
440,396.6
6,085,729.3
186.200
4,258.686
95,921.339
20
21
27,348.5
411,209.8
5,645,332.7
175.661
4,072.486
91,662.653
21
22
25,624.8
383,861.3
5,234,122.9
165.718
3,896.825
87,590.167
22
23
24,008.6
358,236.5
4,850,261.6
156.584
3,731.107
83,693.342
23
24
22,493.1
334,227.9
4,492,025.1
147.954
3,574.523
79,962.235
24
25
21,071.9
311,734.8
4,157,797.2
139.799
3,426.569
76,387.712
25
26
19,739.4
290,662.9
3,846,062.4
132.508
3,286.770
72,961.143
26
27
18,489.6
270,923.5
3,555,399.5
125.399
3,154.262
69,674.373
27
28
17,317.6
252,433.9
3,284,476.0
118.855
3,028.863
66,520.111
28
29
16,218.4
235,116.3
3,032,042.1
112.998
2,910.008
63,491.248
29
30
15,187.5
218,897.9
2,796,925.8
107.258
2,797.010
60,581.240
30
31
14,220.5
203,710.4
2,578,027.9
101.962
2,689.752
57,784.230
31
32
13,313.6
189,489.9
2,374,317.5
97.0677
2,587.7904
55,094.4779
32
33
12,463.0
176,176.3
2,184,827.6
92.5386
2,490.7227
52,506.6875
33
34
11,665.0
163,713.3
2,008,651.3
88.3414
2,398.1841
50,015.9648
34
35
10,916.4
152,048.3
1,844,938.0
84.5684
2,309.8427
47,617.7807
35
36
10,213.9
141,131.9
1,692,889.7
80.8236
2,225.2743
45,307.9380
36
37
9,554.90
130,918.01
1,551,757.77
77.4503
2,144.4507
43,082.6637
37
38
8,936.61
121,363.11
1,420,839.76
74.4061
2,067.0004
40,938.2130
38
39
8,356.36
112,426.50
1,299,476.65
71.5555
1,992.5943
38,871.2126
39
40
7,811.80
104,070.14
1,187,050.15
68.9727
1,921.0388
36,878.6183
40
41
7,300.66
96,258.34
1,082,980.01
66.4531
1,852.0661
34,957.5795
41
42
6,820.97
88,957.68
986,721.67
64.3241
1,785.6130
33,105.5134
42
43
6,370.54
82,136.71
897,763.99
62.2233
1,721.2889
31,319.9004
43
44
5,947.73
75,766.17
815,627.28
60.3722
1,659.0656
29,598.6115
44
45
5,550.68
69,818.44
739,861.11
58.6684
1,598.6934
27,939.5459
45
16
5,177.82
64,267.76
670,042.67
57.0933
1,540.0250
26,340.8525
46
47
4,827.64
59,089.94
605,774.91
55.6915
1,482.9317
24,800.8275
47
48
4,498.69
54,262.30
546,684.97
54.3806
1,427.2402
23,317.8958
48
49
4,189.66
49,763.61
492,422.67
53.2026
1,372.8596
21,890.6556
49
50
3,899.32
45,573.95
442,659.06
52.1373
1,319.6570
20,517.7960
50
51
3,626.46
41,674.63
397,085.11
51.1671
1,267.5197
19,198.1390
51
52
3,370.02
38,048.17
355,410.48
50.2765
1,216.3526
17,930.6193
52
53
3,128.99
34,678.15
317,362.31
49.4517
1,166.0761
16,714.2667
53
54
2,902.42
31,549.16
282,684.16
48.6809
1,116.6244
15,548.1906
54 35
TABLE XXXVIII.
COMMUTATION COLUMNS. 6 PER CENT.
AGE.
D#.
N*.
s*.
c*.
M*.
AGE,
55
2,689.45
28,646.74
251,135.00
48.0303
1,067.9435
14,431.5662
55
56
2,489.19
25,957.29
222,488.26
47.3696
1,019.9132
13,363.6227
56
57
2,300.93
23,468.10
196,530.97
46.8001
972.5436
12,343.7095
57
58
2,123.88
21,167.17
173,062.87
46.2075
925.7435
11,371.1659
58
59
1,957.46
19,043.29
151,895.70
45.6231
879.5360
10,445.4224
59
60
1,801.03
17,085.83
132,852.41
45.0711
833.9129
9,565.8864
60
61
1,654.02
15,284.80
115,766.58
44.5164
788.8418
8,731.9735
61
62
1,515.88
13,630.78
100,481.78
43.9056
744.3254
7,943.1317
62
63
1,386.17
12,114.90
86,851.00
43.2933
700.4198
7,198.8063
63
64
1,264.41
10,728.73
74,736.10
42.6323
657.1265
6,498.3865
64
65
1,150.21
9,464.32
64,007.37
41.9288
614.4942
5,841.2600
65
66
1,043.18
8,314.11
54,543.05
41.1279
572.5654
5,226.7658
66
67
943.001
7,270.934
46,228.937
40.2835
531.4375
4,654.2004
67
68
849.339
6,327.933
38,958.003
39.2952
491.1540
4,122.7629
68
69
761.967
5,478.594
32,630.070
38.2728
451.8588
3,631.6089
69
70
680.564
4,716.627
27,151.476
37.0965
413.5860
3,179.7501
70
71
604.947
4,036.063
22,434.849
35.8253
376.4895
2,766.1641
71
72
534.878
3,431.116
18,398.786
34.4228
340.6642
2,389.6746
72
73
470.180
2,896.238
14,967.670
32.8900
306.2414
2,049.0104
73
74
410.676
2,426.058
12,071.432
31.2686
273.3514
1,742.7690
74
75
356.161
2,015.382
9,645.374
29.4987
242.0828
1,469.4176
75
76
306.503
1,659.221
7,629.992
27.6150
212.5841
1,227.3348
76
77
261.538
1,352.718
5,970.771
25.6589
184.9691
1,014.7507
77
78
221.075
1,091.180
4,618.053
23.6054
159.3102
829.7816
78
79
184.956
870.105
3,526.873
21.4942
135.7048
670.4714
79
80
152.993
685.149
2,656.768
19.3502
114.2106
534.7666
80
81
124.982
532.156
1,971.619
17.1949
94.8604
420.5560
81
82
100.713
407.174
1,439.463
15.0788
77.6655
325.6956
82
83
79.9335
306.4614
1,032.2888
13.0049
62.5867
248.0301
83
84
62.4041
226.5279
725.8274
11.0398
49.5818
185.4434
84
85
47.8320
164.1238
499.2995
9.18883
38.54196
135.86156
85
86
35.9357
116.2918
335.1757
7.48689
29.35313
97.31960
86
87
26.4147
80.3561
218.8839
5.97784
21.86624
67.96647
87
88
18.9417
53.9414
138.5278
4.63802
15.88840
46.10023
88
89
13.2315
34.9997
84.5864
3.65240
11.25038
30.21183
89
90
8.83015
21.76824
49.58671
2.73362
7.59798
18.96145
90
91
5.59670
12.93809
27.81847
1.89306
4.86436
11.36347
91
92
3.38684
7.34139
14.88038
1.24526
2.97130
6.49911
92
93
1.94988
3.95455
7.53899
.781790
1.726036
3.527809
93
94
1.05772
2.00467
3.58444
.461455
.944246
1.801773
94
95
.536392
.946947
1.579772
.253015
.482791
.857527
95
96
.253015
.410555
.632825
.133387
.229776
.374736
96
97
.105306
.157540
.222270
.059607
.096389
.144960
97
98
.039738
.052234
.064730
.024993
.036782
.048571
98
99
.012496
.012496
.012496
.011789
.011789
.011789
99 36
TABLE XXXVIII.
LOGARITHMS OF COMMUTATION COLUMNS. 6 PER CENT.
AGE.
AD*.
X•
AS*.
AC*.
AM*.
AR*.
AGE.
10
4.746941
5.936905
7.100313
! 2.533211
3.838209
5.180885
10
11
4.718813
5.907915
7.069429
2.506562
3.816140
5.160694
11
12
4.690674
5.878866
7.038405
2.479235
3.794309
5.140591
12
13
4.662529
5.849755
7.007236
2.452576
3.772759
5.120571
13
14
4.634375
5.820576
6.975912
2.426592
3.751468
5.100623
14
15
4.606207
5.791327
6.944427
2.399927
3.730411
5.080738
15
16
4.578028
5.762000
6.912775
2.373257
3.709631
5.060907
IB
17
4.549840
5.732595
6.880945
2.347268
3.689138
5.041120
17
18
4.521637
5.703104
6.848931
2.321278
3.668908
5.021367
18
19
4.493420
5.673522
6.816723
2.295286
3.648951
5.001636
19
20
4.465187
5.643844
6.784313
2.269981
3.629276
4.981915
20
21
4.436934
5.614064
6.751690
2.244675
3.609860
4.962192
21
22
4.408661
5.584174
6.718844
2.219369
3.590711
4.942456
22
23
4.380367
5.554170
6.685766
2.194749
3.571838
4.922691
23
24
4.352049
5.524042
6.652442
2.170127
3.553218
4.902885
24
25
4.323704
5.493786
6.618864
2.145505
3.534859
4.883023
25
26
4.295334
5.463389
6.585016
2.122242
3.516769
4.863092
26
27
4.266927
5.432847
6.550888
2.098294
3.498898
4.843073
27
28
4.238487
5.402148
6.516466
2.075016
3.481280
4.822953
28
29
4.210009
5.371283
6.481735
2.053069
3.463894
4.802714
29
30
4.181485
5.340242
6.446681
2.030431
3.446694
4.782338
30
31
4.152916
5.309013
6.411288
2.008438
3.429712
4.761810
31
32
4.124296
5.277586
6.375539
1.987075
3.412929
4.741108
32
33
4.095621
5.245948
6.339418
1.966323
3.396325
4.720214
33
34
4.066884
5.214084
6.302904
1.946164
3.379883
4.699109
34
35
4.038078
5.181981
6.265982
1.927208
3.363583
4.677769
35
36
4.009191
5.149625
6.228628
1.907538
3.347384
4.656174
36
37
3.980226
5.116999
6.190824
1.889023
3.331316
4.634303
37
38
3.951173
5.084087
6.152545
1.871609
3.315341
4.612129
38
39
3.922017
5.050869
6.113768
1.854643
3.299419
4.589628
39
40
3.892751
5.017327
6.074469
1.838677
3.283536
4.566774
40
41
3.863362
4.983438
6.034620
1.822515
3.267656
4.543541
41
42
3.833846
4.949183
5.994195
1.808374
3.251787
4.519901
42
43
3.804176
4.914537
5.953162
1.793953
3.235854
4.495821
43
44
3.774351
4.879475
5.911492
1.780837
3.219864
4.471271
44
45
3.744346
4.843970
5.869151
1.768404
3.203765
4.446219
45
46
3.714147
4.807993
5.826103
1.756585
3.187528
4.420630
46
47
3.683735
4.771513
5.782311
1.745789
3.171121
4.394466
47
48
3.653086
4.734498
5.737737
1.735444
3.154497
4.367689
48
49
3.622179
4.696912
5.692338
1.725933
3.137626
4.340259
49
50
3.590989
4.658717
5.646069
1.717149
3.120461
4.312131
50
51
3.559483
4.619872
5.598884
1.708991
3.102955
4.283259
51
52
3.527633
4.580334
5.550731
1.701365
3.085060
4.253595
52
53
3.495404
4.540056
5.501555
1.694181
3.066727
4.223087
53
54
3.462761
4.498988
5.451301
1.687359
3.047907
4.191680
54 37
TABLE XXXVIII.
LOGARITHMS OF COMMUTATION COLUMNS. 6 PER CENT.
AGE.
AD*.
AN*.
AS*.
AC*.
AM*.
AE*.
AGE.
55
3.429664
4.457075
5.399907
1.681515
3.028548
4.159313
55
56
3.396059
4.414259
5.347307
1.675499
3.008563
4.125924
56
57
3.361903
4.370478
5.293431
1.670247
2.987909
4.091446
57
58
3.327131
4.325663
5.238204
1.664713
2.966491
4.055805
58
59
3.291693
4.279742
5.181546
1.659185
2.944254
4.018926
59
60
3.255522
4.232636
5.123370
1.653898
2.921121
3.980725
60
61
3.218540
4.184259
5.063583
1.648520
2.896990
3.941113
61
62
3.180664
4.134521
5.002088
1.642520
2.871763
3.899992
62
63
3.141817
4.083319
4.938775
1.636420
2.845358
3.857260
63
64
3.101889
4.030548
4.873531
1.629738
2.817649
3.812805
64
65
3.060777
3.976089
4.806230
1.622512
2.788518
3.766507
65
66
3.018357
3.919816
4.736739
1.614137
2.757825
3.718233
66
67
2.974512
3.861590
4.664913
1.605127
2.725452
3.667845
67
68
2.929081
3.801262
4.590597
1.594339
2.691218
3.615188
68
69
2.881936
3.738669
4.513618
1.582890
2.655003
3.560099
69
70
2.832869
3.673631
4.433794
1.569333
2.616566
3.502393
70
71
2.781717
3.605958
4.350923
1.554190
2.575753
3.441878
71
72
2.728255
3.535435
4.264789
1.536846
2.532327
3.378338
72
73
2.672264
3.461835
4.175154
1.517064
2.486064
3.311544
73
74
2.613499
3.384901
4.081758
1.495109
2.436721
3.241240
74
75
2.551646
3.304357
3.984319
1.469803
2.383964
3.167145
75
76
2.486434
3.219904
3.882525
1.441146
2.327531
3.088963
76
77
2.417535
3.131207
3.776031
1.409239
2.267099
3.006359
77
78
2.344540
3.037896
3.664459
1.373012
2.202244
2.918964
78
79
2.267069
2.939571
3.547390
1.332321
2.132595
2.826380
79
80
2.184671
2.835785
3.424354
1.286685
2.057706
2.728164
80
81
2.096849
2.726039
3.294823
1.235400
1.977085
2.623824
81
82
2.003086
2.609780
3.158200
1.178367
1.890228
2.512812
82
83
1.902729
2.486376
3.013801
1.114107
1.796482
2.394504
83
84
1.795213
2.355122
2.860833
1.042961
1.695322
2.268211
84
85
1.679719
2.215172
2.698362
0.963260
1.585934
2.133097
85
86
1.555526
2.065549
2.525272
0.874302
1.467654
1.988200
86
87
1.421846
1.905019
2.340214
0.776544
1.339774
1.832295
87
88
1.277419
1.731922
2.141537
0.666332
1.201070
1.663703
88
89
1.121609
1.544064
1.927300
0.562578
1.051167
1.480177
89
90
0.945968
1.337823
1.695365
0.436739
0.880698
1.277871
90
91
0.747932
1.111870
1.444333
0.277165
0:687025
1.055511
91
92
0.529795
0.865778
1.172614
0.095261
0.472947
0.812854
92
93
0.290008
0.597098
0.877313
1.893090
0.237050
0.547505
93
94
0.024370
0.302043
0.554421
1.664129
1.975085
0.255700
94
95
1.729482
T. 976326
0.198594
T. 403146
1.683759
T. 933248
95
96
1.403146
1.613372
1.801283
1.125115
1.361304
1.573725
96
97
1.022452
1.197391
1.346881
2.775298
2.984027
1.161248
97
98
2.599206
2.717953
2.811106
2.397809
2.565635
2.686377
98
99
2.096779
2.096779
2.096779
2.071474
2.071474
2.071474
99 38
TABLE XXXIX.
COMMUTATION COLUMNS. 7 PER CENT.
AGE.
D
N*.
s„.
CL
M».
K*.
AGE,
10
50,834.9
697,841.8
9,179,521.1
307.8601
5,181.6666
97,313.2662
10
11
47,201.4
647,006.9
8.481,679.3
286.8318
4,873.8065
92,131.5996
11
12
43,826.7
599,805.5
7.834,672,4
266.8222
4.586.9747
87,257.7931
12
13
40,692.6
555,978.8
7,234.866.9
248.5908
4,320.1525
82,670.8184
13
14
37,781.9
515,286.2
6,678,888.1
231.9654
4,071.5617
78,350.6659
14
15
35,078.3
477,504.3
6,163,601.9
216.1127
3,839.5963
74,279.1042
15
16
32,567.3
442,426.0
5,686,097.6
201.3413
3,623.4836
70,439.5079
16
17
30,235.4
409,858.7
5,243,671.6
187.8736
3,422.1423
66,816.0243
17
18
28,069.5
379,623.3
4,833,812.9
175.3063
3,234.2687
63,393.8820
18
19
26,057.9
351,553.8
4,454,189.6
163.5792
3,058.9624
60,159.6133
19
20
24,189.6
325,495.9
4.102,635.8
152.8778
2,895.3832
57,100.6509
20
21
22,454.2
301,306.3
3,777,139.9
142.8764
2,742.5054
54,205.2677
21
22
20,837.6
278,852.1
3,475,833.6
133.5294
2,599.6290
51,462.7623
22
23
19,345.3
258,014.5
3,196,981.5
124.9910
2,466.0996
48.863.1333
23
24
17,954.7
238,669.2
2,938,967.0
116.9982
2,341.1086
46,397.0337
24
25
16,666.9
220,714.5
2,700,297.8
109.5163
2,224.1104
44,055.9251
25
26
15,463.5
204,047.6
2,479,583.3
102.8345
2,114.5941
41,831.8147
26
27
14,349.1
188,584.1
2,275,535.7
96.4078
2,011.7596
39,717.2206
27
28
13,313.9
174,235.0
2,086,951.6
90.5224
1,915.3518
37,705.4610
28
29
12,352.4
160,921.1
1,912,716.6
85.2573
1,824.8294
35,790.1092
29
30
11,459.0
148,568.72
1,751,795.48
80.1708
1,739.5721
33,965.2798
30
31
10,629.2
137.109.72
1,603,226.76
75.4997
1,659.4013
32,225.7077
31
32
9,858.32
126,480.52
1,466,117.04
71.2038
1,583.9016
30,566.3064
32
33
9,142.19
116,622,20
1,339,636.52
67.2472
1,512.6978
28,982.4048
33
34
8,476.86
107,480.01
1,223,014.32
63.5971
1,445.4506
27,469.7070
34
35
7,858.71
99,003.15
1,115,534.31
60.3119
1,381.8535
26,024.2564
35
36
7,284.27
91,144.44
1,016,531.16
57.1026
1,321.5416
24,642.4029
36
37
6,750.61
83,860.17
925,386.72
54.2079
1,264.4390
23,320.8613
37
38
6,254.78
77,109.56
841,526.55
51.5905
1,210.2311
22,056.4223
38
39
5,794.01
70,854.78
764,416.99
49.1504
1,158.6406
20,846.1912
39
40
5,365.81
65,060.77
693,562.21
46.9335
1,109.4902
19,687.5506
40
41
4,967.83
59,694.96
628,501.44
44.7964
1,062.5567
18,578.0604
41
42
4,598.04
54,727.13
568,806.48
42.9560
1,017.7603
17,515.5037
42
43
4,254.28
50,129.09
514,079.35
41.1647
974.8043
16,497.7434
43
44
3,934.80
45,874.81
463,950.26
39.5668
933.6396
15,522.9391
44
45
3,637.81
41,940.01
418,075.45
38.0908
894.0728
14,589.2995
45
46
3,361.73
38,302.20
376,135.44
36.7217
855.9820
13,695.2267
46
47
3,105.08
34,940.47
337,833.24
35.4854
819.2603
12,839.2447
47
48
2,866.47
31,835.39
302,892.77
34.3263
783.7749
12,019.9844
48
49
2,644.61
28,968.92
271,057.38
33.2688
749.4486
11,236.2095
49
50
2,438.33
26,324.31
242,088.46
32.2979
716.1798
10,486.7609
50
51
2,246.51
23,885.98
215,764.15
31.4007
683.8819
9,770.5811
51
52
2,068.15
21,639.47
191,878.17
30.5658
652.4812
9,086.6992
52
53
1,902.28
19,571.32
170,238.70
29.7834
621.9154
8,434.2180
53
54
1,748.05
17,669.04
150,667.38
29.0451
592.1320
7,812.3026
54 39
TABLE XXXIX.
COMMUTATION COLUMNS. 7 PER CENT.
AGE.
D*.
sy.
a,
M*.
K(B.
1
AGE.
55
1,604.65
15,920.99
132,998.34
28.3891
563.0869
7,220.1706
55
56
1,471.28
14,316.34
117,077.35
27.7369
534.6978
6,657.0837
56
57
1,347.29
12,845.06
102,761.01
27.1474
506.9609
6,122.3859
57
58
1,232.00
11,497.77
89,915.95
26.5532
479.8135
5,615.4250
58
59
1,124.85
10,265.77
78,418.18
25.9723
453.2603
5,135.6115
59
60
1,025.29
9,140.916
68,152.409
25.4183
427.2880
4,682.3512
60
61
932.797
8,115.626
59,011.493
24.8708
401.8697
4,255.0632
61
62
846.904
7,182.829
50,895.867
24.3003
376.9989
3,853.1935
62
63
767.199
6,335.925
43,713.038
23.7374
352.6986
3,476.1946
63
64
693.269
5,568.726
37,377.113
23.1566
328.9612
3,123.4960
64
65
624.758
4,875.457
31,808.387
22.5616
305.8046
2,794.5348
65
66
561.326
4,250.699
26,932.930
21.9239
283.2430
2,488.7302
66
67
502.680
3,689.373
22,082.231
21.2730
261.3191
2,205.4872
67
68
448.521
3,186.693
18,992.858
20.5572
240.0461
1,944.1681
68
69
398.621
2,738.172
15,806.165
19.8352
219.4889
1,704.1220
69
70
352.709
2,339.551
13,007.993
19.0459
199.6537
1,484.6331
70
71
310.588
1,986.842
10,728.442
18.2213
180.6078
1,284.9794
71
72
272.048
1,676.254
8,741.600
17.3443
162.3865
1,104.3716
72
73
236.906
1,404.206
7,065.346
16.4172
145.0422
941.9851
73
74
204.991
1,167.300
5,661.140
15.4620
128.6250
796.9429
74
75
176.118
962.309
4,493.840
14.4504
113.1630
668.3179
75
76
150.146
786.191
3,531.531
13.4013
98.71266
555.15498
76
77
126.922
636.045
2,745.340
12.3357
85.31136
456.44232
77
78
106.283
509.123
2,109.295
11.2423
72.97566
371.13096
78
79
88.0874
402.8396
1,000.1718
10.1412
61.73336
298.15530
79
80
72.1835
314.7522
1,197.3322
9.04428
51.59216
236.42194
80
81
58.4168
242.5687
882.5800
7.96179
42.54788
184.82978
81
82
46.6334
184.1519
640.0113
6.91672
34.58609
142.28190
82
83
36.6660
137.5185
455.8594
5.90966
27.66937
107.69581
83
84
28.3576
100.8525
318.3409
4.96979
21.75971
80.02644
84
85
21.5326
72.4949
217.4884
4.09788
16.78992
58.26673
85
86
16.0260
50.9623
144.9935
3.30768
12.69204
41.47681
86
87
11.6699
34.9363
94.0312
2.61630
9.38436
28.78477
87
88
8.29016
23.26642
59.09493
2.01094
6.76806
19.40041
88
89
5.73687
14.97626
35.82851
1.56879
4.75712
12.63235
89
90
3.79277
9.23939
20.85225
.16319
3.18833
7.87523
90
91
2.38145
5.44662
11.61286
.797990
2.025136
4.686905
91
92
1.42767
3.06517
6.16624
.520014
1.227146
2.661769
92
93
.814258
1.637496
3.101066
.323420
.707132
1.434623
93
94
.437568
.823238
1.463570
.189116
.383712
.727491
94
95
.219827
.385670
.640332
.102723
.194596
.343779
95
96
.102723
.165843
.254662
.053648
.091873
.149183
96
97
.042354
.063120
.088819
.023750
.038225
.057310
97
98
.015833
.020766
.025699
.009865
.014475
.019085
98
99
.004933
.004933
.004933
.004610
.004610
.004610
99 40
TABLE XXXIX.
LOGARITHMS OF COMMUTATION COLUMNS. 7 PER CENT.
AGE.
AD*.
AN*.
AS*.
AC*.
AM*.
AR*.
AGE.
10
4.706162
5.843757
6.962820
2.488353
3.714470
4.988172
10
11
4.673955
5.810909
6.928482
2.457627
3.687868
4.964409
11
12
4.641739
5.778010
6.894021
2.426222
3.661526
4.940804
12
13
4.609516
5.745058
6.859431
2.395485
3.635499
4.917352
13
14
4.577284
5.712049
6.824704
2.365423
3.609761
4.894043
14
15
4.545038
5.678977
6.789834
2.334680
3.584286
4.870867
15
16
4.512782
5.645841
6.754814
2.303933
3.559126
4.847816
16
17
4.480516
5.612635
6.719636
2.273866
3.534298
4.824881
17
18
4.448235
5.579353
6.684290
2.243798
3.509776
4.802047
18
19
4.415939
5.545992
6.648769
2.213728
3.485574
4.779305
19
20
4.383628
5.512545
6.613063
2.184344
3.461706
4.756641
20
21
4.351298
5.479008
6.577163
2.154961
3.438147
4.734041
21
22
4.318847
5.445374
6.541059
2.125577
3.414911
4.711493
22
23
4.286575
5.411644
6.504740
2.096879
3.392011
4.688981
23
24
4.254179
5.377796
6.468195
2.068179
3.369422
4.666490
24
25
4.221856
5.343831
6.431412
2.039479
3.347156
4.644004
25
26
4.189308
5.309732
6.394379
2.012139
3.325227
4.621507
26
27
4.156823
5.275505
6.357084
1.984112
3.303576
4.598979
27
28
4.124305
5.241135
6.319512
1.956756
3.282249
4.576404
28
29
4.091750
5.206613
6.281651
1.930731
3.261222
4.553763
29
30
4.059148
5.171927
6.243484
1.904016
3.240442
4.531035
30
31
4.026501
5.137068
6.204995
1.877945
3.219951
4.508202
31
32
3.993803
5.102024
6.166169
1.852503
3.199728
4.485243
32
33
3.961050
5.066781
6.126987
1.827674
3.179752
4.462134
33
34
3.928235
5.031328
6.087431
1.803438
3.160004
4.438854
34
35
3.895351
4.995649
6.047483
1.780403
3.140462
4.415378
35
36
3.862386
4.959730
6.007120
1.756656
3.121081
4.391683
36
37
3.829343
4.923556
5.966323
1.734062
3.101898
4.367745
37
38
3.796212
4.887108
5.925068
1.712570
3.082868
4.343535
38
39
3.762979
4.850369
5.883330
1.691527
3.063949
4.319027
39
40
3.729635
4.813319
5.841085
1.671483
3.045123
4.294192
40
41
3.696167
4.775938
5.798306
1.651243
3.026352
4.269000
41
42
3.662573
4.738203
5.754964
1.633024
3.007645
4.243423
42
43
3.628826
4.700090
5.711030
1.614525
2.988917
4.217425
43
44
3.594923
4.661574
5.666471
1.597331
2.970179
4.190974
44
45
3.560840
4.622628
5.621254 ’
1.580820
2.951373
4.164034
45
46
3.526563
4.583224
5.575344
1.564923
2.932465
4.136569
46
47
3.492073
4.543329
5.528702
1.550050
2.913422
4.108539
47
48
3.457347
4.502910
5.481289
1.535627
2.894191
4.079904
48
49
3.422362
4.461933
5.433061
1.522037
2.874742
4.050620
49
50
3.387093
4.420357
5.383974
1.509175
2.855022
4.020641
50
51
3.351509
4.378143
5.333979
1.496940
2.834981
3.989920
51
52
3-315582
4.335247
5.283026
1.485235
2.814568
3.958406
52
53
3-279275
4.291620
5.231059
1.473974
2.793731
3.926045
53
54
3.242554
4.247213
5.178019
1.463073
2.772419
3.892779
54 41
TABLE XXXIX.
LOGARITHMS OF COMMUTATION COLUMNS. 7 PER CENT.
AGE,
AD*.
AN*.
AS*.
AC*.
AM*.
AE*.
AGE.
55
3.205379
4.201970
5.123846
1.453152
2.750575
3.858547
55
56
3.167695
4.155832
5.068473
1.443058
2.728108
3.823284
56
57
3.129462
4.108736
5.011828
1.433728
2.704975
3.786920
57
58
3.090612
4.060614
4.953837
1.424116
2,681073
3.749383
58
59
3.051096
4.011391
4.894417
1.414510
2.656348
3.710592
59
60
3.010847
3.960990
4.833481
1.405146
2.630721
3.670464
60
61
2.969787
3.909322
4.770936
1.395690
2.604085
3.628906
61
62
2.927834
3.856295
4.706683
1.385611
2.576340
3.585821
62
63
2.884908
3.801810
4.640611
1.375434
2.547404
3.541104
63
64
2.840902
3.745756
4.572605
1.364674
2.517145
3.494641
64
65
2.795712
3.688015
4.502541
1.353370
2.485444
3.446310
65
66
2.749215
3.628460
4.430284
1.340917
2.452159
3.395976
66
67
2.701292
3.566953
4.355685
1.327829
2.417171
3.343505
67
68
2.651783
3.503340
4.278591
1.312963
2.380295
3.288734
68
69
2.600560
3.437461
4.198826
1.297436
2.341413
3.231501
69
70
2.547416
3.369133
4.116209
1.279801
2.300277
3.171619
70
71
2.492185
3.298163
4.030537
1.260580
2.250737
3.108896
71
72
2.434645
3.224340
3.941591
1.239158
2.210550
3.043115
72
73
2.374576
3.147431
3.849134
1.215298
2.161494
2.974044
73
74
2.311734
3.067183
3.752904
1.189265
2.109326
2.901427
74
75
2.245803
2.983314
3.652618
1.159881
2.053705
2.824983
75
76
2.176513
2.895528
3.547963
1.127147
1.994373
2.744414
76
77
2.103536
2.803488
3.438596
1.091162
1.931007
2.659386
77
78
2.026462
2.706823
3.324137
1.050857
1.863178
2.569527
78
79
1.944914
2.605132
3.204166
1.006088
1.790520
2.474442
79
80
1.858438
2.497969
3.078215
0.956374
1.712584
2.373688
80
81
1.766538
2.384835
2.945754
0.901011
1.628878
2.266772
81
82
1.668697
2.265176
2.806188
0.839900
1.538901
2.153150
82
83
1.564263
2.138361
2.658830
0.771562
1.441999
2.032199
83
84
1.452669
2.003687
2.502892
0.696338
1.337653
1.903233
84
85
1.333096
1.860307
2.337436
0.612559
1.225049
1.765421
85
86
1.204825
1.707249
2.161349
0.519523
1.103531
1.617805
86
87
1.067067
1.543277
1.973272
0.417688
0.972405
1.459163
87
88
0.918563
1.366730
1.771550
0.303398
0.830464
1.287811
88
89
0.758675
1.175403
1.554229
0.195566
0.677344
1.101484
89
90
0.578956
0.965644
1.319153
0.065649
0.503563
0.896263
90
91
0.376842
0.736127
1.064939
1.901997
0.306454
0.670886
91
92
0.154628
0.486455
0.790021
1.716015
0.088896
0.425170
92
93
1.910762
0.214181
0.491511
1.509767
1.849510
0.156738
93
94
1.641046
1.915525
0.165413
1.276727
1.584005
1.861828
94
95
1.342080
1.586216
1.806405
1.011666
1.289134
1.536279
95
96
1.011666
1.219697
1.405964
2.729557
2.963188
1.173719
96
97
2.626895
2.800167
2.948506
2.375662
2.582347
2.758230
97
98
2.199571
2.317353
2.409916
3.994096
2.160619
2.280692
98
99
3.693066
3.693066
3.693066
3.663682
3.663682
3.663682
99 42
TABLE XL
COMMUTATION COLUMNS. 8 PER CENT.
AGE.
"Dx-
N*.
s*.
0*.
M*.
B*.
•
AGE,
10
46,319.3
571,113.6
6,833,398.9
277.916
4,014.670
64,935.891
10
11
42,610.4
524,794.3
6,262,285.3
256.536
3.736.754
60,921.221
11
12
39,197.6
482,183.9
5,737,491.0
236.430
3,480.218
57,184.467
12
13
36,057.5
442,986.3
5,255,307.1
218.235
3,243.788
53,704.249
13
14
33,168.4
406,928.8
4,812,320.8
201.755
3,025.553
50,460.461
14
15
30,509.8
373,760.4
4,405,392.0
186.226
2,823.798
47,434.908
15
16
28,063.5
343,250.6
4,031,631.6
171.891
2,637.572
44,611.110
16
17
25,812.9
315,187.1
3,688,381.0
158.908
2,465.681
41,973.538
17
18
23,741.9
289,374.2
3,373,193.9
146.906
2,306.773
39,507.857
18
19
21,836.4
265,632.3
3,083,819.7
135.809
2,159.867
37,201.084
19
20
20,083.0
243,795.9
2,818,187.4
125.749
2,024.058
35,041.217
20
21
18,469.6
223,712.9
2,574,391.5
116.434
1,898.309
33,017.159
21
22
16,985.1
205,243.3
2,350,678.6
107.810
1,781.875
31,118.850
22
23
15,619.1
188,258.2
2,145,435.3
99.9814
1,674.0644
29,336.9753
23
24
14,362.2
172,639.1
1,957,177.1
92.7214
1,574.0830
27,662.9109
24
25
13,205.6
158,276.9
1,784,538.0
85.9883
1,481.3616
26,088.8279
25
26
12,141.4
145,071.3
1,626,261.1
79.9944
1,395.3733
24; 607.4663
26
27
11,162.1
132,929.9
1,481,189.8
74.3007
1,315.3789
23,212.0930
27
28
10,260.9
121,767.8
1,348,259.9
69.1189
1,241.0782
21,896.7141
28
29
9,431.72
111,506.91
1,226,492.13
64.4959
1,171.9593
20,655.6359
29
30
8,668.58
102,075.19
1,114,985.22
60.0865
1.107.4634
19,483.6766
30
31
7,966.40
93,406.61
1,012,910.03
56.0616
1,047.3769
18,376.2132
31
32
7,320.22
85,440.21
919,503.42
52.3823
991.3153
17,328.8363
32
33
6,725.60
78,119.99
834,063.21
49.0134
938.9330
16,337.5210
33
34
6,178.39
71,394.39
755,943.22
45.9239
889.9196
15,398.5880
34
35
5,674.82
65,216.00
684,548.83
43.1483
843.9957
14,508.6684
35
36
5,211.31
59,541.18
619,332.83
40.4740
800.8474
13,664.6727
36
37
4,784.80
54,329.87
559,791.65
38.0666
760.3734
12,863.8253
37
38
4,392.31
49,545.07
505,461.78
35.8931
722.3068
12,103.4519
38
39
4,031.07
45,152.76
455,916.71
33.8788
686.4137
11,381.1451
39
40
3,698.59
41,121.69
410,763.95
32.0512
652.5349
10,694.7314
40
41
3,392.56
37,423.10
369,642.26
30.3084
620.4837
10,042.1965
41
42
3,110.96
34,030.54
332,219.16
28.7942
590.1753
9,421.7128
42
43
2,851.72
30,919.58
298,188.62
27.3380
561.3811
8,831.5375
43
44
2,613.15
28,067.86
267,269.04
26.0335
534.0431
8,270.1564
44
45
2,393.54
25,454.71
239,201.18
24.8303
508.0096
7,736.1133
45
46
2,191.41
23,061.17
213,746.47
23.7162
483.1793
7,228.1037
46
47
2,005.37
20,869.76
190,685.30
22.7055
459.4631
6,744.9244
47
48
1,834.12
18,864.39
169,815.54
21.7604
436.7576
6,285.4613
48
49
1,676.50
17,030.27
150,951.15
20.8948
414.9972
5,848.7037
49
50
1,531.42
15,353.77
133,920.88
20.0972
394.1024
5,433.7065
50
51
1,397.88
13,822.35
118,567.11
19.3580
374.0052
5,039.6041
51
52
1,274.98
12,424.47
104,744.76
18.6688
354.6472
4,665.5989
52
53
1,161.87
11,149.49
92,320.29
18.0225
335.9784
4,310.9517
53
54
1,057.78
9,987.62
81,170.80
1
17.4130
317.9559
3,974.9733
54 43
TABLE XL.
COMMUTATION COLUMNS. 8 PER CENT.
AGE.
D*.
s».
cx
Ma,.
R®*
AGE.
55
962.011
8,929.843
71,183.180
16.8622
300.5429
3,657.0174
55
56
873.891
7,967.832
62,253.337
16.3222
283.6807
3,356.4745
56
57
792.835
7,093.941
54,285.505
15.8274
267.3585
3.072.7938
57
58
718.279
6,301.106
47,191.564
15.3376
251.5311
2,805.4353
58
59
649.736
5,582.827
40,890.458
14.8632
236.1935
2,553.9042
59
60
586.745
4,933.091
35,307.631
14.4114
221.3303
2,317.7107
60
61
528.870
4,346.346
30.374.540
13.9705
206.9189
2,096.3804
61
62
475.724
3,817.476
26,028.194
13.5236
192.9484
1,889.4615
62
63
426.962
3,341.752
22,210.718
13.0881
179.4248
1,696.5131
63
64
382.247
2,914.790
18,868.966
12.6496
166.3367
1,517.0883
64
65
341.283
2,532.543
15,954.176
12.2104
153.6871
1,350.7516
65
66
303.792
2,191.260
13,421.633
11.7554
141.4767
1,197.0645
66
67
269.534
1,887.468
11,230.373
11.3008
129.7213
1,055.5878
67
68
238.268
1,617.934
9,342.905
10.8195
118.4205
925.8665
68
69
209.798
1,379.666
7,724.971
10.3428
107.6010
807.4460
69
70
183.916
1,169.868
6,345.305
9.83926
97.25822
699.84498
70
71
160.452
985.952
5,175.437
9.32613
87.41896
602.58676
71
72
139.241
825.500
4,189.485
8.79508
78.09283
515.16780
72
73
120.132
686.259
3,363.985
8.24783
69.29775
437.07497
73
74
102.985
566.127
2,677.726
7.69603
61.04992
367.77722
74
75
87.6605
463.1417
2,111.5992
7.12596
53.35389
306.72730
75
76
74.0413
375.4812
1,648.4575
6.54739
46.22793
253.37341
76
77
62.0095
301.4399
1,272.9763
5.97096
39.68054
207.14548
77
78
51.4451
239.4304
971.5364
5.39136
33.70958
167.46494
78
79
42.2430
187.9853
732.1060
4.81826
28.31822
133.75536
79
80
34.2957
145.7423
544.1207
4.25731
23.49996
105.43714
80
81
27.4979
111.4466
398.3784
3.71307
19.24265
81.93718
81
82
21.7480
83.9487
286.9318
3.19582
15.52958
62.69453
82
83
16.9412
62.2007
202.9831
2.70523
12.33376
47.16495
83
84
12.9811
45.2595
140.7824
2.25393
9.62853
34.83119
84
85
9.76558
32.27839
95.52286
1.84129
7.37460
25.20266
85
86
7.20090
22.51281
63.24447
1.47247
5.53331
17.82806
86
87
5.19504
15.31191
40.73166
1.15391
4.06084
12.29475
87
88
3.65633
10.11687
25.41975
.878699
2.906926
8.233909
88
89
2.50679
6.46054
15.30288
.679154
2.028227
5.326983
89
90
1.64194
3.95375
8.84234
.498897
1.349073
3.298756
90
91
1.02142
2.31181
4.88859
.339094
.850176
1.949683
91
92
.606667
1.290385
2.57678
.218926
.511082
1.099507
92
93
.342803
.683718
1.286398
.134899
.292156
.588425
93
94
.182511
.340915
.602680
.078150
.157257
.296269
94
95
.090841
.158404
.261765
.042056
.079107
.139012
95
96
.042056
.067563
.103361
.021761
.037051
.059905
96
97
.017180
.025507
.035798!
.009544
.015290
.022854
97
98
.006363
.008327
.010291
.003928
.005746
.007564
98
99
.001964
.001964
.001964
.001818
.001818
.001818
99 44
TABLE X LI.
COMMUTATION COLUMNS. 9 PER CENT.
AGE.
Da;
Na,
Sa;-
0».
Ma,
Ra>
AGE.
10
42,241.1
473,006.5
5,184,561.6
251.121
3,185.523
44,923.415
10
11
38,502.2
430,765.4
4,711,555.1
229.675
2,934.402
41,737.892
11
12
35,093.4
392,263.2
4,280,789.7
209.733
2,704.727
38,803.490
12
13
31,986.0
357,169.8
3,888,526.5
191.817
2,494.994
36,098.763
13
14
29,153.2
325,183.8
3,531,356.7
175.704
2,303.177
33,603.769
14
15
26,570.4
296,030.6
3,206,172.9
160.693
2,127.473
31,300.592
15
16
24,215.8
269,460.2
2,910,142.3
1 146.963
1,966.780
29,173.119
16
17
22,069.4
245,244.4
2,640,682.1
134.616
1,819.817
27,206.339
17
18
20,112.5
223,175.0
2,395,437.7
123.306
1,685.201
25,386.522
18
19
18,328.6
203,062.5
2,172,262.7
112.947
1,561.895
23,701.321
19
20
16,702.2
184,733.9
1,969,200.2
103.621
1,448.948
22,139.426
20
21
15,219.5
168,031.7
1,784,466.3
95.0650
1,345.3274
20,690.4780
21
22
13,867.8
152,812.2
1,616,434.6
87.2156
1,250.2624
19,345.1506
22
23
12,635.5
138,944.4
1,463,622.4
80.1407
1,163.0468
18,094.8882
23
24
11,512.1
126,308.9
1,324,678.0
73.6396
1,082.9061
16,931.8414
24
25
10,487.9
114,796.8
1,198,369.1
67.6656
1,009.2665
15,848.9353
25
26
9,554.26
104,308.94
1,083,572.28
62.3714
941.6009
14,839.6688
26
27
8,703.02
94,754.68
979,263.34
57.4006
879.2295
13,898.0679
27
28
7,927.00
86,051.66
884,508.66
52.9075
821.8289
13,018.8384
28
29
7,219.58
78,124.66
798,457.00
48.9159
768.9214
12,197.0095
29
30
6,574.55
70,905.08
720,332.34
45.1535
720.0055
11,428.0881
30
31
5,986.56
64,330.53
649,427.26
41.7425
674.8520
10,708.0826
31
32
5,450.50
58,343.97
585,096.73
38.6450
633.1095
10,033.2306
32
33
4,961.82
52,893.47
526,752.76
35.8279
594.4645
9,400.1211
33
34
4,516.29
47,931.65
473,859.29
33.2615
558.6366
8,805.6566
34
35
4,110.13
43,415.36
425,927.64
30.9646
525.3751
8,247.0200
35
36
3,739.80
39,305.23
382,512.28
28.7789
494.4105
7,721.6449
36
37
3,402.23
35,565.43
343,207.05
26.8188
465.6316
7,227.2344
37
38
3,094.49
32,163.20
307,641.62
25.0555
438.8128
6,761.6028
38
39
2,813.93
29,068.71
275,478.42
23.4325
413.7573
6,322.7900
39
40
2,558.15
26,254.78
246,409.71
21.9650
390.3248
5,909.0327
40
41
2,324.96
23,696.63
220,154.93
20.5801
368.3598
5,518.7079
41
42
2,112.41
21,371.67
196,458.30
19.3726
347.7797
5,150.3481
42
43
1,918.62
19,259.26
175,086.63
18.2241
328.4071
4,802.5684
43
44
1,741.98
17,340.64
155,827.37
17.1952
310.1830
4,474.1613
44
45
1,580.95
15,598.66
138,486.73
16.2501
292.9878
4,163.9783
45
46
1,434.16
14,017.71
122.888.07
15.3786
276.7377
3,870.9905
46
47
1,300.37
12,583.55
108,870.36
14.5881
261.3591
3,594.2528
47
48
1,178.41
11,283.18
96,286.81
13.8527
246.7710
3,332.8937
48
49
1,067.26
10,104.77
85,003.63
13.1796
232.9183
3,086.1227
49
50
965.955
9,037.511
74,898.859
12.5602
219.7387
2,853.2044
50
51
873.636
8,071.556
65,861.348
11.9872
207.1785
2,633.4657
51
52
789.514
7,197.920
57,789.792
11.4544
195.1913
2,426.2872
52
53
712.871
6,408.406
50,591.872
10.9564
183.7369
2,231.0959
53
54
643.053
5,695.535
44,183.466
10.4888
172.7805
2,047.3590
54 45
TABLE X LI.
COMMUTATION COLUMNS. 9 PER CENT.
AGE,
Da,.
s».
C*.
Ma,.
Ka;.
AGE.
55
579.469
5,052.482
38,487.931
10.06385
162.2917
1,874.5785
55
56
521.559
4,473.013
33,435.449
9.65214
152.22788
1,712.28676
56
57
468.843
3,951.454
28,962.436
9.27363
142.57574
1,560.05888
57
58
420.857
3,482.611
25,010.982
8.90421
133.30211
1,417.48314
58
59
377.204
3,061.754
21,528.371
8.54962
124.39790
1,284.18103
59
60
337.508
2,684.550
18,466.617
8.21372
115.84828
1,159.78313
60
61
301.427
2,347.042
15,782.067
7.88935
107.63456
1,043.93485
61
62
268.649
2,045.615
13,435.025
7.56693
99.74521
936.30029
62
63
238.901
1,776.966
11,389.410
7.25604
92.17828
836.55508
63
64
211.918
1,538.065
9,612.444
6.94860
84.92224
744.37680
64
65
187.472
1,326.147
8,074.379
6.64584
77.97364
659.45456
65
66
165.347
1,138.675
6,748.232
6.33950
71.32780
581.48092
66
67
145.355
973.328
5,609.557
6.03843
64.98830
510.15312
67
68
127.315
827.973
4,636.229
5.72817
58.94987
445.16482
68
69
111.074
700.658
3,808.256
5.42557
53.22170
386.21495
69
70
96.4773
589.5838
3,107.5975
5.11408
47.79613
332.99325
70
71
83.3973
493.1065
2,518.0137
4.80291
42.68205
285.19712
71
72
71.7082
409.7092
2,024.9072
4.48787
37.87914
242.51507
72
73
61.2997
338.0010
1,615.1980
4.17001
33.39127
204.63593
73
74
52.0681
276.7013
1,277.1970
3.85533
29.22126
171.24466
74
75
43.9136
224.6332
1,000.4957
3.53700
25.36593
142.02340
75
76
36.7507
180.7196
775.8625
3.22001
21.82893
116.65747
76
77
30.4963
143.9689
595.1429
2.90958
18.60892
94.82854
77
78
25.0686
113.4726
451.1740
2.60305
15.69934
76.21962
78
79
20.3957
88.4040
337.7014
2.30500
13.09629
60.52028
79
80
16.4067
68.0083
249.2974
2.01796
10.79129
47.42399
80
81
13.0340
51.6016
181.2891
1.74385
8.77333
36.63270
81
82
10.2140
38.5676
129.6875
1.48715
7.02948
27.85937
82
83
7.88346
28.35357
91.11993
1.24731
5.54233
20.82989
83
84
5.98522
20.47011
62.76636
1.02969
4.29502
15.28756
84
85
4.46134
14.48489
42.29625
.833461
3.265334
10.992535
85
86
3.25950
10.02355
27.81136
.660399
2.431873
7.727201
86
87
2.32997
6.76405
17.78781
.512777
1.771474
5.295328
87
88
1.62481
4.43408
11.02376
.386898
1.258697
3.523854
88
89
1.10376
2.80927
6.58968
.296293
.871799
2.265157
89
90
.716326
1.705505
3.780411
.215656
.575506
1.393358
90
91
.441525
.989179
2.074906
.145234
.359850
.817852
91
92
.259835
.547654
1.085727
.092906
.214616
.458002
92
93
.145475
.287819
.538073
.056722
.121710
.243386
93
94
.076742
.142344
.250254
.032559
.064988
.121676
94
95
.037846
.065602
.107910
.017361
.032429
.056688
95
96
.017361
.027756
.042308
.008900
.015068
.024259
96
97
.007027
.010395
.014552
.003868
.006168
.009191
97
98
.002579
.003368
.004157
.001577
.002300
.003023
98
99
.000789
.000789
.000789
.000723
.000723
.000723
99 46
TABLE X LI I.
COMMUTATION COLUMNS. 10 PER CENT.
AGE,
Da?.
No,
s*.
c*.
M«.
B*.
AGE,
10
38,554.3
395,760.5
4,000,286.2
227.120
2,576.107
32,098.169
10
11
34,822.3
357,206.2
3,604,525.7
205.836
2,348.987
29,522.062
11
12
31,450.8
322,383.9
3,247,319.5
186.254
2,143.151
27,173.075
12
13
28,405.3
290,933.1
2,924,935.6
168.795
1,956.897
25,029.924
13
14
25,654.2
262,527.8
2,634,002.5
153.211
1,788.102
23,073.027
14
15
23,168.9
236,873.6
2,371,474.7
* 138.847
1,634.891
21,284.925
15
16
20,923.7
213,704.7
2,134,601.1
125.829
1,496.044
19,650.034
16
17
18,895.8
192,781.0
1,920,896.4
114.210
1,370.215
18,153.990
17
18
17,063.8
173,885.2
1,728,115.4
103.664
1,256.005
16,783.775
18
19
15,408.8
156,821.4
1,554,230.2
94.0914
1,152.3412
15,527.7698
19
20
13,913.9
141,412.6
1,397,408.8
85.5376
1,058:2498
14,375.4286
20
21
12,563.5
127,498.7
1,255,996.2
77.7615
972.7122
13,317.1788
21
22
11,343.6
114,935.2
1,128,497.5
70.6923
894.9507
12,344.4666
22
23
10,241.7
103,591.6
1,013,562.3
64.3673
824.2584
11,449.5159
23
24
9,246.26
93,349.9
909,970.7
58.6080
759.8911
10,625.2575
24
25
8,347.07
84,103.61
816,620.82
53.3639
701.2831
9,865.3664
25
26
7,534.87
75,756.54
732,517.21
48.7414
647.9192
9,164.0833
26
27
6,801.16
68,221.67
656,760.67
44.4491
599.1778
8,516.1641
27
28
6,138.41
61,420.51
588,539.00
40.5974
554.7287
7,916.9863
28
29
5,539.77
55,282.10
527,118.49
37.1933
514.1313
7,362.2576
29
30
4,998.96
49,742.33
471,836.39
34.0205
476.9380
6.848.1263
30
31
4,510.50
44,743.37
422,094.06
31.1645
442.9175
6,371.1883
31
32
4,069.28
40,232.87
377,350.69
28.5897
411.7530
5,928.2708
32
33
3,670.76
36,163.59
337,117.82
26.2646
383.1633
5,516.5178
33
34
3,310.79
32,492.83
300,954.23
24.1616
356.8987
5,133.3545
34
35
2,985.64
29,182.04
268,461.40
22.2886
332.7371
4,776.4558
35
36
2,691.94
26,196.40
239,279.36
20.5270
310.4485
4,443.7187
36
37
2,426.69
23,504.46
213,082.96
18.9550
289.9215
4,133.2702
37
38
2,187.13
21,077.77
189,578.50
17.5478
270.9665
3,843.3487
38
39
1,970.75
18,890.64
168,500.73
16.2619
253.4187
3,572.3822
39
40
1,775.33
16,919.89
149,610.09
15.1049
237.1568
3,318.9635
40
41
1,598.83
15,144.56
132,690.20
14.0239
222.0519
3,081.8067
41
42
1,439.46
13,545.73
117,545.64
13.0810
208.0280
2,859.7548
42
43
1,295.52
12,106.27
103,999.91
12.1936
194.9470
2,651.7268
43
44
1,165.55
10,810.75
91,893.64
11.4007
182.7534
2,456.7798
44
45
1,048.19
9,645.20
81,082.89
10.6761
171.3527
2,274.0264
45
46
942.224
8,597.01
71,437.69
10.0116
160.6766
2,102.6737
46
47
846.555
7,654.785
62,840.682
9.41070
150.66503
1,941.99711
47
48
760.185
6,808.230
55,185.897
8.85503
141.25433
1,791.33208
48
49
682.221
6,048.045
48,377.667
8.34818
132.39930
1,650.07775
49
50
611.854
5,365.824
42,329.622
7.88353
124.05112
1,517.67845
50
51
548.346
4,753.970
36,963.798
7.45549
116.16759
1,393.62733
51
52
491.041
4,205.624
32,209.828
7.05933
108.71210
1,277.45974
52
53
439.342
3,714.583
28,004.204
6.69103
101.65277
1,168.74764
53
54
392.711
3,275.241
24,289.621
6.34722
94.96174
1,067.09487
54 47
TABLE XLII.
COMMUTATION COLUMNS. 10 PER CENT.
AGE.
D®.
N*.
s*.
oa.
M*.
Rai-
AGE.
55
350.662
2,882.531
21,014.380
6.03467
88.61452
972.13313
55
56
312.750
2,531.869
18,131.849
5.73523
82.57985
883.51861
56
57
278.583
2.219.119
15,599.980
5.46023
76.84462
800.93876
57
58
247.796
1,940.536
13,380.861
5.19506
71.38439
724.09414
58
59
220.075
1,692.740
11,440.325
4.94283
66.18933
652.70975
59
60
195.125
1,472.664
9,747.585
4.70546
61.24650
586.52042
60
61
172.681
1,277.539
8,274.921
4.47855
56.54104
525.27392
61
62
152.504
1,104.858
6,997.382
4.25647
52.06249
468.73288
62
63
134.384
952.354
5,892.524
4.04449
47.80602
416.67039
63
64
118.122
817.970
4,940.170
3.83791
43.76153
368.86437
64
65
103.546
699.848
4,122.200
3.63732
39.92362
325.10284
65
66
90.4955
596.3028
3,422.3523
3.43811
36.28630
285.17922
66
67
78.8306
505.8073
2,826.0495
3.24507
32.84819
248.89292
67
68
68.4191
426.9757
2,320.2422
3.05034
29.60312
216.04473
68
69
59.1488
358.5566
1,893.2665
2.86294
26.55278
186.44161
69
70
50.9087
299.4078
1,534.7099.
2.67404
23.68984
159.88883
70
71
43.6066
248.4991
1,235.3021
2.48851
21.01580
136.19899
71
72
37.1539
204.8925
986.8030
2.30414
18.52729
115.18319
72
73
31.4721
167.7386
781.9105
2.12148
16.22315
96.65590
73
74
26.4895
136.2665
614.1719
1.94356
14.10167
80.43275
74
75
22.1378
o
ID
o
477.9054
1.76687
12.15811
66.33108
75
76
18.3585
87.6392
368.1284
1.59390
10.39124
54.17297
76
77
15.0956
69.2807
280.4892
1.42714
8.79734
43.78173
77
78
12.2961
54.1851
211.2085
1.26518
7.37020
34.98439
78
79
9.91311
41.88903
157.02340
1.11014
6.10502
27.61419
79
80
7.90178
31.97592
115.13437
.963058
4.994877
21.509173
80
81
6.22038
24.07414
83.15845
.824671
4.031819
16.514296
81
82
4.83022
17.85376
59.08431
.696884
3.207148
12.482477
82
83
3.69422
13.02354
41.23055
.579181
2.510264
9.275329
83
84
2.77920
9.32932
28.20701
.473785
1.931083
6.765065
84
85
2.05276
6.55012
18.87769
.380009
1.457298
4.833982
85
86
1.48614
4.49736
12.32757
.298365
1.077289
3.376684
86
87
1.05267
3.01122
7.83021
.229564
.778924
2.299395
87
88
.727410
1.958552
4.818985
.171635
.549360
1.520471
88
89
.489646
1.231142
2.860433
.130246
.377725
.971111
89
90
.314886
.741496
1.629291
.093937
.247479
.593386
90
91
.192323
.426610
.887795
.062687
.153542
.345907
91
92
.112152
.234287
.461185
.039736
.090855
.192365
92
93
.062221
.122135
.226898
.024040
.051119
.101510
93
94
.032524
.059914
.104763
.013674
.027079
.050391
94
95
.015894
.027390
.044849
.007226
.013405
.023312
95
96
.007225
.011496
.017459
.003670
.006179
.009907
96
97
.002898
.004271
.005963
.001580
.002509
.003728
97
98
.001054
.001373
.001692
.000639
.000929
.001219
98
99
.000319
.000319
.000319
.000290
.000290
.000290
99 48
TABLE XLII1.
COMMON LOGARITHMS FOR SPECIAL FORMULAS.
Value of X of pXt X nXj v, d, and Ad.
AGE.
X.
A (vpx).
A^ar
AGE.
X.
3
Per Cent.
31
Per Cent.
4
Per Cent.
41
Per Cent.
5
Per Cent.
6
Per Cent.
10
1.984341
1.982236
1.980143
1.978061
1.975988
1.971872
0.006493
3.81245
10
11
.984329
.982227
.980134
.978051
.975977
.971861
.006513
.81378
11
12
.984325
.982221
.980128
.978044
.975972
.971855
.006530
.81491
12
13
.984313
.982211
.980118
.978036
.975963
.971846
.006547
.81604
13
14
.984302
.982198
.980105
.978022
.975949
.971832
.006572
.81770
14
15
1.984289
1.982186
T. 980094
1.978010
1.975937
1.971821
0.006600
3.81954
15
16
.984282
.982178
.980084
.978002
.975929
.971812
.006624
.82112
16
17
.984265
.982163
.980070
.977987
.975914
.971797
.006653
.82302
17
18
.984250
.982147
.980055
.977971
.975898
.971783
.006687
.82523
18
19
.984236
.982133
.980039
.977957
.975884
.971767
.006721
.82743
19
20
1.984217
1.982113
1.980020
1.977937
f. 975864
1.971747
0.006762
3.83008
20
21
.984195
.982092
.980000
.977916
.975843
.971727
.006807
.83296
21
22
.984175
.982072
.979978
.977896
.975823
.971706
.006854
.83594
22
23
.984150
.982048
.979955
.977872
.975799
.971682
.006906
.83923
23
24
.984125
.982020
.979928
.977845
.975772
.971655
.006965
.84292
24
25
1.984097
f. 981995
1.979901
1.977818
1.975745
1.971630
0.007027
3.84677
25
26
.984062
.981959
.979866
.977783
' .975710
.971593
.007098
.85114
26
27
.984028
.981925
.979833
.977750
.975677
.971560
.007176
.85588
27
28
.983993
.981888
.979795
.977713
.975639
.971522
.007258
.86082
28
29
.983943
.981841
.979748
.977664
.975592
.971476
.007354
.86652
29
30
1.983901
1.981797
1.979704
1.977621
1.975548
1.971431
0.007460
3.87274
30
31
.983847
.981746
.979652
.977570
.975496
.971380
.007569
.87904
31
32
.983795
.981691
.979598
.977514
.975442
.971325
.007692
.88604
32
33
.983730
.981627
.979535
.977452
.975379
.971263
.007829
.89371
33
34
.983664
.981560
.979466
.977384
.975311
.971194
.007978
.90189
34
35
f. 983580
1.981478
1.979386
1.977303
1.975229
1.971113
0.008148
3.91105
35
36
.983504
.981401
.979307
.977224
.975152
.971035
.008330
.92065
36
37
.983416
.981313
.979220
.977137
.975064
.970947
.008524
.93064
37
38
.983313
.981209
.979117
.977034
.974960
.970844
.008744
.94171
38
39
.983203
.981100
.979006
.976923
.974851
.970734
.008987
.95361
39
40
1.983078
1.980976
1.978883
1.976800
1.974727
1.970611
0.009254
3.96633
40
41
.982953
.980849
.978757
.976674
.974600
.970484
.009544
.97973
41
42
.982800
.980696
.978602
.976520
.974447
.970330
.009866
.99414
42
43
.982643
.980541
.978448
.976364
.974292
.970175
.010221
2.00949
43
44
.982463
.980360
.978268
.976185
.974112
.969995
.010606
.02555
44
45
1.982272
1.980167
1.978073
1.975991
1.973917
1.969801
0.011035
2.04277
45
46
.982055
.979954
.977861
.975777
.973705
.969588
.011504
.06085
46
47
.981821
.979716
.977624
.975541
.973468
.969351
.012021
.07994
47
48
.981561
.979459
.977365
.975282
.973209
.969093
.012589
.09999
48
49
.981279
.979175
.977082
.975000
.972926
.968810
.013213
.12100
49
50
1.980961
1.978859
1.976767
1.974683
1.972611
1.968494
0.013901
2.14305
50
51
.980619
.978516
.976422
.974340
.972266
.968150
.014658
.16607
51
52
.980240
.978137
.976044
.973961
.971888
.967771
.015488
.19000
52
53
.979827
.977722
.975630
.973546
.971474
.967357
.016402
.21490
53
54
.979370
.977269
.975175
.973093
.971020
.966903
.017402
.24060
54
T71 D.r+ 1
Formula, vpx = -j— = •
dl/x
~ lxdx 49
COMMON LOGARITHMS FOR SPECIAL FORMULAS.
TABLE XLIII.
AGE.
A (vpx).
f4r
tyx-
AGE.
X.
3
Per Cent.
3*
Per Cent.
4
Per Cent.
Per Cent.
5
Per Cent.
6
Per Cent.
X.
55
1.978863
1.976759
1.974667
1.972584
f. 970510
1.966395
0.018508
2.26736
55
56
.978314
.976210
.974117
.972033
.969961
.965844
.019725
.29502
56
57
.977696
.975594
.973500
.971418
.969345
.965228
.021064
.32354
57
58
.977030
.974927
.972835
.970752
.968678
.964562
.022541
.35297
58
59
.976299
.974195
.972102
.970019
.967946
.963829
.024151
.38294
59
60
1.975486
1.973384
1.971290
1.969207
1.967135
L963018
0.025921
2.41365
60
61
.974593
.972489
.970397
.968314
.966241
.962124
.027883
.44534
61
62
.973621
.971518
.969425
.967342
.965268
.961153
.030027
.47751
62
63
.972541
.970437
.968344
.966261
.964189
.960072
.032385
.51034
63
64
.971356
.969254
.967161
.965078
.963005
.958888
.034988
.54392
64
65
1.970049
1.967946
f. 965853
f. 963770
1.961696
1.957580
0.037853
2.57810
65
66
.968624
.966521
.964427
.962345
.960272
.956155
.040996
.61274
66
67
.967037
.964Q34
.962842
.960758
.958686
.954569
.044457
.64794
67
68
.965325
.963221
.961128
.959045
.956971
.952855
.048249
.68349
68
69
.963401
.961299
.959205
.957122
.955050
.950933
.052427
.71956
69
70
1.961316
1.959212
1.957120
1.955037
f. 952963
1.948848
0.057039
2.75617
70
71
.959006
.956904
.954811
.952728
.950656
.946538
.062089
.79301
71
72
.956478
.954375
.952281
.950199
.948125
.944009
.067652
.83028
72
73
.953704
.951600
.949508
.947425
.945352
.941235
.073746
.86774
73
74
.950616
.948513
.946420
.944336
.942264
.938147
.080484
.90571
74
75
1.947257
1.945153
T. 943060
1.940978
f. 938904
1.934788
0.087902
2.94400
75
76
.943570
.941467
.939374
.937291
.935218
.931101
.095996
.98225
76
77
.939473
.937370
.935277
.933193
.931121
.927005
.104940
1.02094
77
78
.934997
.932894
.930801
.928719
.926645
.922529
.114801
.05995
78
79
.930071
.927968
.925875
.923792
.921719
.917602
.125606
.09901
79
80
1.924647
1.922544
f. 920451
1.918368
1.916295
f. 912178
0.137506
1.13832
80
81
.918706
.916602
.914509
.912426
.910354
.906237
.150570
.17774
81
82
.912111
.910009
.907916
.905833
.903759
.899643
.164978
.21743
82
83
.904953
.902850
.900757
.898674
.896601
.892484
.180795
.25719
83
84
.896974
.894870
.892777
.890694
.888622
.884506
.198181
.29706
84
85
1.888276
1.886173
1.884080
1.881997
1.879923
1.875807
0.217377
1.33721
85
86
.878789
.876686
.874593
.872510
.870437
.866320
.238266
.37706
86
87
.868041
.865938
.863845
.861762
.859690
.855573
.261501
.41747
87
88
.856659
.854556
.852462
.850380
.848306
.844190
.286671
.45738
88
89
.836828
.834725
.832633
.830549
.828476
.824359
.321451
.50711
89
90
1.814433
f. 812329
1.810236
1.808154
f. 806081
f. 801964
0.371233
1.56965
90
91
.794331
.792229
.790136
.788052
.785980
.781863
.421033
.62432
91
92
.772681
.770578
.768484
.766402
.764328
.760213
.468238
.67047
92
93
.746831
.744727
.742635
.740552
.738479
.734362
.522329
.71794
93
94
.717581
.715478
.713385
.711301
.709229
.705112
.585986
.76789
94
95
1.686132
f. 684029
1.681936
1.679854
1.677780
f. 673664
0.655714
1.81671
95
96
.631775
.629672
.627579
.625496
.623423
.619306
.748812
.87437
96
97
.589223
.587120
.585026
.582944
.580871
.576754
.871770
.94040
97
98
.510042
.507938
.505846
.503762
.501689
.497573
.988400
.99493
98
V
6
Ad
0.970874
.029559
2.470687
0.966184
.034401
2.536577
0.961538
.039221
2.593516
0.956938
.044017
2.643619
0.952381
.048790
2.688332
0.943396
.058269
2.765437
V
6
A 6
Formula, 6 = Nap. log (1 + i) = — Nap. log v. 50
TABLE XLIY.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
The first Value under the Age X is A(1 + (t-x) for the whole Life.
4 PER CENT.
Pay-
ments.
x—10
11
12
13
14
15
16
17 Pay-
ments.
n.
1.316633
1.315019
1.313333
1.311567
1.309720
1.307791
1.305773
1.303662
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.291215
.291210
.291207
.291203
.291196
.291191
.291187
.291179
2
3
.457564
.457556
.457549
.457538
.457527
.457516
.457505
.457491
3
4
.572834
.572822
.572810
.572794
.572777
.572761
.572743
.572722
4
5
0.G60148
0.660132
0.660114
0.660093
0.660071
0.660048
0.660024
0.659995
5
6
.729805
.729785
.729763
.729736
.729707
.729679
.729647
.729610
6
7
.787302
.787276
.787248
.787216
.787181
.787145
.787106
.787060
7
8
.835915
.835884
.835851
.835813
.835771
.835728
.835681
.835626
8
9
.877761
.877725
.877686
.877642
.877593
.877542
.877486
.877423
9
10
0.914284
0.914242
0.914198
0.914146
0.914090
0.914032
0.913967
0.913893
10
11
.946514
.946466
.946415
.946357
.946293
.946226
.946152
.940068
11
12
.975210
.975157
.975099
.975033
.974962
.974886
.974802
.974708
12
13
1.000952
1.000892
1.000827
1.000754
1.000673
1.000588
1.000494
1.000388
13
14
.024186
.024119
.024047
.023966
.023877
.023782
.023677
.023560
14
15
1.045269
1.045196
1.045116
1.045026
1.044928
1.044822
1.044707
1.044576
15
16
.064488
.064407
064319
.064220
.064112
.063996
.063868
.063725
16
17
.082076
.081987
.081891
.081783
.081665
.081537
.081397
.081240
17
18
.098228
.098131
.098026
.097908
.097779
.097640
.097486
.097316
18
19
.113106
.113001
.112886
.112758
.112618
.112466
.112299
.112113
19
20
1.126848
1.126733
1.126610
1.126470
1.126318
1.126153
1.125972
1.125770
20
21
.139570
.139446
.139311
.139161
.138996
.138818
.138622
.138404
21
22
.151373
.151239
.151094
.150932
.150754
.150561
.150349
.150114
22
23
.162344
.162201
.162044
.161869
.161678
.161470
.161241
.160988
23
24
.172559
.172405
.172237
.172049
.171843
.174620
.171374
.171102
24
25
1.182086
1.181921
1.181740
1.181538
1.181317
1.181077
1.180814
1.180521
25
26
.190983
.190805
.190610
.190395
.190158
.189901
.189618
.189305
26
27
.199299
.199108
.198901
.198671
.198418
.198142
.197840
.197505
27
28
.207084
.206879
.206659
.206412
.206142
.205847
.205524
.205166
28
29
.214377
.214160
.213922
.213660
.213371
.213057
.212712
.212330
29
30
1.221216
1.220984
1.220731
1.220451
1.220144
1.219809
1.219441
1.219034
30
31
.227633
.227386
.227116
.226818
.226491
.226135
.225743
.225310
31
32
.233659
.233396
.233109
.232792
.232444
.232065
.231648
.231188
32
33
.239320
.239041
.238736
.238399
.238029
.237626
.237183
.236695
33
34
.244642
.244345
.244022
.243663
.243271
•.242843
.242374
.241854
34
35
1.249645
1.249332
1.248988
1.248607
1.248192
1.247738
1.247239
1.246690
35
36
.254019
.253055
.253252
.252811
.252331
.251802
.251220
36
37
.258780
.258427
.258042
.257616
.257148
.256640
.256081
.255465
37
38
.262948
.262574
.262165
.261715
.261221
.260682
.260091
.259440
38
39
.266868
.266473
.266042
.265565
.265043
.264473
.263848
.263161
39
40
1.270557
1.270140
1.269684
1.269180
1.268627
1.268027
1.267367
1.266642
40
41
.274028
.273587
.273105
.272574
.271992
.271357
.270661
.269897
41
42
.277293
.276827
.276318
.275758
.275144
.274474
.273742
.272936
42
43
.280363
.279872
.279335
.278744
.278098
.277391
.276620
.275772
43
44
.283247
.282730
.282165
.281543
.280862
.280119
.279307
.278415
44
10
11
12
13
14
15
16
17
Continued on Page *60.
Formula, A(l + an_1) = a 6-1.
x Dz vjjx-t TABLE XLIY.
51
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
4 PER CENT.
Pay-
ments.
18
19
20
21
22
23
24
25
Pay-1
ments.
U.
1.301453
1.299144
1.296729
1.294205
1.291567
1.288808
1.285924
1.282910
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.291172
.291165
.291155
.291145
.291134
.291123
.291110
.291096
2
3
.457477
.457460
.457441
.457421
.457400
.457376
.457349
.457320
3
4
.572G99
.572674
.572645
.572615
.572581
.572546
.572504
.572459
4
5
0.659963
0.659929
0.659890
0.659848
0.659803
0.659753
0.659698
0.659636
5
6
.729569
. 729525
.729476
.729422
.729364
.729300
.729229
.729150
6
7
.787011
.786956
.786896
.786830
.786758
.786679
.786591
.786494
7
8
.835567
.835502
.835429
.835351
.835265
.835170
.835065
.834949
8
9
.877353
.877276
.877193
.877100
.876999
.876888
.876765
.876629
9
10
0.913813
0.913725
0.913628
0.913521
0.913404
0.913276
0.913134
0.912977
10
11
.945977
.945876
.945765
.945644
.945510
.945364
.945202
.945024
11
12
.974604
.974491
.974365
.974228
.974078
.973912
.973730
.973528
12
13
1.000273
1.000145
1.000005
.999851
.999683
.999498
.999293
.999068
13
14
.023431
.023289
.023133
1.022963
1.022775
1.022569
1.022342
1.022091
14
15
1.044434
1.044277
1.044105
1.043916
1.043709
1.043481
1.043229
1.042953
15
16
.063569
.063396
.063207
.062999
.062771
.062520
.062243
.061939
16
17
.081069
.080880
.080672
.080444
.080194
.079920
.079617
.079283
17
18
.097128
.096922
.096695
.096447
.096174
.095874
.095543
.095180
18
19
.111909
.111685
.111438
.111168
.110871
.110545
.110185
.109790
19
20
1.125549
1.125306
1.125038
1.124745
1.124423
1.124069
1.123679
1.123251
20
21
.138164
.137901
.137612
.137295
.136946
.136564
.136142
.135678
21
22
.149856
.149572
.149159
.148917
.148541
.148128
.147674
.147173
22
23
.160710
.160404
.160168
.159699
.159294
.158850
.158360 .157822
23
24
.170803
.170474
.170113
.169717
.169282
.168804
.168278
.167700
24
25
1.180200
1.179848
1.179460
1.179035
1.178569
1.178056
1.177492
1.176873
25
26
.188961
.188584
.188169
.187714
.187214
.186665
.186062
.185399
26
27
.197137
.196733
.196289
.195803
.195269
.194682
.194037
.193329
27
28
.204773
.204342
.203868
.203349
.202778
.202152
.201464
.200709
28
29
.211911
.211452
.210946
.210392
.209784
.209117
.208384
.207579
29
30
1.218588
1.218098
1.217559
1.216970
1.216322
1.215612
1.214832
1.213976
30
31
.224835
.224314
.223741
.223114
.222425
.221670
.220841
.219932
31
32
.230683
.230129
.229521
.228854
.228123
.227321
.226441
.225476
32
33
.236159
.235571
.234925
.234218
.233442
.232591
.231659
.230636
33
34
.241286
.240662
.239978
.239228
.238406
.237506
.236518
.235436
34
35
1.246087
1.245427
1.244701
1.243908
GO
CO
O
S3
t-H
1.242085
1.241040
1.239896
35
36
.250583
.249883
.249116
.248276
.247357
.246349
.245246
.244037
36
37
.254790
.254051
.253240
.252352
.251381
.250317
.249153
.247879
37
38
.258727
.257946
.257089
.256153
.255127
.254006
.252778
.251436
38
39
.262408
.261584
.260680
.259692
.258612
.257430
.256138
.254726
39
40
1.265848
1.264979
1.264027
1.262986
1.261848
1.260605
1.259246
1.257762
40
41
.269060
.268144
.267141
.266046
.264850
.263543
.262116
.260559
41
42
.272055
.271091
.270037
.268886
.267629
.266257
.264760
.263128
42
43
.274845
.273832
.272724
.271515
.270196
.268758
.267190
.265481
43
44
.277441
.276377
.275214
.273946
.272564
.271057
.269416
.267631
44
18
19
20
21
22
23
24
25
Continued on Page *59. 52
TABLE XLIV.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF Yl
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
4 PER CENT.
Pay-
ments.
n.
x=26
27
28
29
30
31
32
33
Pay-
ments
n.
1.2797581.276466
1.273025
1.269428
1.265675
1.261752
1.257654
1.253372
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.291079
.291064
.291045
.291022
.291001
.290975
.290948
.290919
2
3
.457288
.457255
.457214
.457170
.457126
.457073
.457019
.456957
3
4
.572410
.572357
.572296
.572229
.572159
.572081
.571995
.571899
4
5
0.659568
0.659496
0.659413
0.659322
0.659227
0.659119
0.659000
0.658868
5
6
.729063
.728970
.728864
.728748
.728626
.728486
.728335
.728167
6
7
.786388
.786273
.786143
.786001
.785849
.785678
.785492
.785285
7
8
.834821
.834684
.834530
.834359
.834177
.833973
.833750
.833503
8
9
.876480
.876318
.876137
.875937
.875724
.875485
.875223
.874934
9
10
0.912805
0.912619
0.912409
0.912180
0.911933
0.911657
0.911355
0.911021
10
i 11
.944827
.944614
.944376
.944115
.943833
.943518
.943174
.942794
11
12
.973307
.973066
.972798
.972503
.972184
.971830
.971441
.971013
12
13
.998820
.998551
.998250
.997921
.997564
.997167
.996732
.996253
13
14
1.021816
1.021516
1.021182
1.020816
1.020419
1.019977
1.019494
1.018962
14
15
1.042648
1.042317
1.041948
1.041543
1.041104
1.040617
1.040083
1.039494
15
16
.061604
.061239
.060834
.060389
.059906
.059370
.058783
.058136
16
17
.078917
.078517
.078073
.077586
.077057
.076471
.075828
.075120
17
18
.094780
.094344
.093861
.093329
.092751
.092112
.091411
.090640
18
19
.109355
.108881
.108355
.107778
.107150
.106455
.105694
.104856
19
20
1.122779
1.122265
1.121695
1.121070
1.120389
1.119637
1.118812
1.117905
20
21
.135169
.134613
.133997
.133322
.132586
.131773
.130882
.129903
21
22
.146624
.146024
.145361
.144633
.143839
.142963
.142004
.140949
22
23
.157231
.156585
.155872
.155089
.154236
.153294
.152263
.151130
23
24
.167066
.166372
.165606
.164766
.163850
.162840
.161734
.160520
24
25
1.176193
1.175449
1.174629
1.173729
1.172748
1.171667
1.170483
1.169184
25
26
.184671
.183876
.182998
.182037
.180988
.179833
.178568
.177181
26
27
.192552
.191702
.190765
.189739
.188620
.187388
.186039
.184561
27
28
.199880
.198974
.197976
.196882
.195689
.194377
.192942
.191370
28
29
.206696
.205731
.204669
.203506
.202237
.200843
.199318
.197648
29
30
1.213037
1.212011
1.210883
1.209647
1.208299
1.206819
1.205202
1.203432
30
31
.218935
.217846
.216648
.215337
.213908
.212340
.210627
.208753
31
32
.224419
.223264
.221995
.220607
.219094
.217434
.215623
.213643
32
33
.229516
.228293
.226950
.225482
.223882
.222129
.220217
.218128
33
34
.234251
.232957
.231538
.229987
.228298
.226449
.224433
.222234
34
35
1.238644
1.237278
1.235780
1.234144
1.232364
1.230416
1.228294
1.225982
35
36
.242716
.241275
.239697
.237974
.236099
.234051
.231822
.229395
36
37
.246486
.244968
.243306
.241494
.239524
.237374
.235035
.232492
37
38
.249970
.248373
.246627
.244723
.242656
.240402
.237953
.235292
38
39
.253185
.251507
.249674
.247678
.245512
.243152
.240592
.237814
39
40
1.256144
1.254384
1.252462
1.250372
1.248106
1.245641
1.242969
1.240074
40
41
.258862
.257017
.255006
.252821
.250455
.247883
.245100
.242089
41
42
.261351
.259421
.257320
.255039
.252571
.249894
.247001
.243875
42
43
.263623
.261608
.259415
.257038
.254470
.251688
.248686
.245447
43
44
.265691
.263588
.261305
.258832
.256164
.253279
.250170
.246822
44
26
27
28
29
30
31
32
33
Continued on Page *58. TABLE XLIV.
53
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
4 PER CENT.
Pay-
ments,
34
35
36
37
38
39
40
41
Pay-
ments,
n.
1.248902
1.244234
1.239367
1.234282
1.228972
1.223433
1.217656
1.211632
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.290884
.290845
.290806
.290765
.290713
.290660
.290600
.290538
2
3
.456885
.456808
.456729
.456641
.456537
.456426
.456306
.456176
3
4
.571790
.571674
.571551
.571413
.571255
.571087
.570901
.570699
4
5
0.658721
0.658564
0.658395
0.658205
0.657991
0.657761
0.657507
0.657231
5
6
.727980
.727778
.727559
.727317
.727043
.726748
.726424
.726069
6
7
.785055
.784808
.784538
.784239
.783904
.783540
.783142
.782706
7
8
.833229
.832932
.832609
.832251
.831851
.831418
.830941
.830419
8
9
.874614
.874265
.873886
.873467
.872999
.872491
.871934
.871322
9
10
0.910652
0.910251
0.909813
0.909329
0.908792
0.908205
0.907561
0.906857
10
n
.942375
.941917
.941418
.94081)7
.940254
.939587
.938854
.938051
11
12
.970540
.970024
.969460
.968838
.968148
.967396
.966570
.965665
12
13
.995724
.995146
.994515
.993820
.993049
.992208
.991285
.990272
13
14
1.018374
1.017733
1.017032
1.016259
1.015403
1.014470
1.013444
1.012321
14
15
1.038846
1.038137
1.037363
1.036510
1.035565
1.034535
1.033404
1.032165
15
16
.057424
.056645
.055794
.054856
.053820
.052689
.051448
.050088
16
17
.074341
.073490
.072558
.071533
.070401
.069164
.067808
.066323
17
18
.089791
.088863
.087848
.086731
.085499
.084154
.082678
.081062
18
19
.103935
.102927
.101825
.100613
.099277
.097817
.096218
.094466
19
20
1.116908
1.115817
1.114624
1.113313
1.111869
1.110291
1.108563
1.106671
20
21
.128827
.127649
.126362
.124948
.123391
.121691
.119830
.117793
21
22
.139791
.138524
.137138
.135617
.133944
.132117
.130117
.127931
22
23
.149886
.148525
.147038
.145406
.143612
.141654
.139512
.137171
23
24
.159186
.157729
.156136
.154389
.152471
.150377
.148088
.145589
24
25
1.167759
1.166201
1.164499
1.162633
1.160585
1.158352
1.155913
1.153250
25
26
.175660
.173998
.172183
.170195
.168014
.165638
.163044
.160214
26
27
.182941
.181172
.179240
.177126
.174809
.172286
.169532
.166533
27
28
.189648
.187767
.185716
.183472
.181016
.178341
.175427
.172254
28
29
.195820
.193825
.191651
.189274
.186675
.183847
.180768
.177420
29
30
1.201495
1.199383
1.197082
1.194569
1.191824
1.188841
1.185595
1.182070
30
31
.206705
.204473
.202043
.199392
.196348
.193357
.189943
.186240
31
32
.211481
.209126
.206564
.203772
.200729
.197428
.193845
.189964
32
33
.215849
.213369
.210674
.207739
.204544
.201083
.197332
.193274
33
34
.219835
.217228
.214397
.211319
.207973
.204351
.200433
.196199
34
35
1.223463
1.220727
1.217760
1.214538
1.211039
1.207259
1.203174
1.198769
35
36
.226753
.223888
.220784
.217417
.213768
.209831
.205583
.201010
36
37
.229727
.226732
.223491
.219981
.216183
.212092
.207685
.202950
37
38
.232404
.229279
.225902
.222252
.218307
.214065
.209505
.204614
38
39
.234802
.231548
.228038
.224248
.220161
.215774
.211067
.206029
39
40
1.236940
1.233559
1.229916
1.225992
1.221767
1.217241
1.212394
1.207218
40
41
.238834
.235328
.231558
.227503
.223146
.218488
.213510
.208205
41
42
.240502
.236874
.232980
.228800
.224319
.219537
.214437
.209014
42
43
.241959
.238214
.234202
.229904
.225305
.220407
.215197
.209668
43
44
.243222
.239366
.235241
.230832
.226124
.221122
.215811
.210187
44
34
35
36
37
38
39
40
41
Continued on Page *57. 54
TABLE XLIY.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
4 PER CENT.
Pay-
ments.
II
£0
43
44
45
46
47
48
49
Pay-
ments.
n.
1.205342
1.198793
1.191959
1.184839
1.177421
1.169692
1.161644
1.153265
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0. 000000
0.000000
1
2
.290463
.290386
.290301
.290205
.290101
.289986
.289861
.289722
2
3
.456026
.455868
.455691
.455496
.455283
.455047
.454790
.454507
3
4
.570471
.570227
.569956
.569656
.569329
.568968
.568573
.568137
4
5
0.656921
0.656587
0.656216
0.655808
0.655361
0.654869
0.654329
0.653735
5
6
.725673
.725245
.724770
.724248
.723676
.723046
.722356
.721596
6
7
.782220
.781693
.781110
.780468
.779765
.778992
.778144
.777212
7
8
.829838
.829208
.828512
.827746
.826906
.825983
.824972
.823860
8
9
.870643
.869904
.869089
.868194
.867212
.866133
.864951
.863650
9
10
0.906075
0.905224
0.904285
0.903253
0.902123
0.900882
0.899521
0.898025
10
11
.937162
.936194
.935126
.933953
.932668
.931257
.929710
.928011
11
12
.964664
.963572
.962370
.961050
.959604
.958016
.956275
.954366
12
13
.989154
.987935
.986593
.985120
.983506
.981735
.979794
.977665
13
14
1.011081
1.009729
1.008241
1.006608
1.004820
1.002858
1.000711
.998356
14
15
1.030798
1.029307
1.027667
1.025869
1.023900
1.021742
1.019379
1.016790
15
16
.048589
.046954
.045157
.043187
.041030
.038668
.036084
.033255
16
17
.064687
.062902
.060942
.058794
.056444
.053871
.051058
.047981
17
18
.079284
.077344
.075214
.072882
.070333
.067543
.064496
.061164
18
19
.092541
.090440
.088136
.085614
.082859
.079847
.076558
.072965
19
20
1.104593
1.102326
1.099842
1.097126
1.094159
1.090918
1.087382
1.083525
20
21
.115561
.113119
.110450
.107533
.104349
.100874
.097087
.092959
21
22
.125532
.122919
.120059
.116936
.113531
.109818
.105775
.101374
22
23
.134605
.131811
.128756
.125423
.121792
.117837
.113536
.108859
23
24
.142852
.139872
.136618
.133071
.129211
.125011
.120449
.115496
24
25
1.150337
1.147169
1.143712
1.139948
1.135856
1.131410
1.126586
1.121356
25
26
.157122
.153761
.150098
.146115
.141790
.137096
.132012
.126509
26
27
.163258
.159702
.155832
.151628
.147069
.142129
.136786
.131012
27
28
.168794
.165041
.160962
.156536
.151745
.146561
.140963
.134925
28
29
.173774
.169822
.165533
.160888
.155865
.150441
.144593
.138298
29
30
1.178236
1.174086
1.169589
1.164725
1.159475
1.1538151.147726
1.141185
30
31
.182218
.177871
.173168
.168089
.162617
.156729
.150408
.143630
31
32
.185755
.181213
.176307
.171018
.165331
.159224
.152681
.145682
32
33
.188880
.184146
.179041
.173550
.167656
.161340
.154589
.147383
33
34
.191624
.186703
.181406
.175719
.169629
.163117
.156170
.148775
34
35
1.194016
1.188913
1.183433
1.177560
1.171285
1.164590
1.157466
1.149897
35
36
.196086
.190809
.185153
.179107
.172659
.165797
.158510
.150789
36
37
.197862
.192420
.186599
.180390
.173785
.166770
.159339
.151484
37
38
.199370
.193773
.187799
.181442
.174693
.167543
.159986
.152015
38
39
.200637
.194896
.188782
.182290
.175414
.168146
.160481
.152413
39
40
1.201690
1.195816
1.189576
1.182964
1.175977
1.168607
1.160852
1.152703
40
41
.202553
.196559
.190206
.183490
.176408
.168952
.161122
.152910
41
42
.203249
.197150
.190698
.183892
.176730
.169204
.161314
.153050
42
43
.203803
.197610
.191074
.184193
.176965
.169384
.161445
.153141
43
44
.204234
.197963
.191356
.184413
.177133
.169505
.161530
.153197
44
42
43
44
45
46
47
48
49
Continued on Page *56. TABLE XLIY.
55
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF 11
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
4 PER CENT.
Pay-
ments
50
51
52
53
54
55
56
57
Pay-
ments.
11.
1.144542
1.135469
1.126031
1.116217
1.106014
1.095410
1.084400
1.072966
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.289571
.289400
.289216
.289015
.288795
.288548
.288281
.287982
2
3
.454190
.453849
.453474
.453064
.452609
.452108
.451558
.450952
3
4
.507059
.567130
.566556
.565925
.565228
.564458
.563617
.562691
4
5
0.053082
0.652363
0.651580
0.650718
0.649767
0.648721
0.647576
0.646315
5
6
.720703
.719845
.718842
.717740
.716526
.715191
.713729
.712123
6
7
.770190
.775063
.773833
.772481
.770993
.769358
.767569
.765604
7
8
.822040
.821297
.819829
.818217
.816445
.814499
.812371
.810033
8
9
.802225
.860656
.858942
.857061
.854994
.852727
.850245
.847525
9
10
0.890385
0.894583
0.892613
0.890452
0.888081
0.885479
0.882637
0.879518
10
11
.920149
.924104
.921869
.919419
.916732
.913789
.910569
.907045
11
12
.952275
.949978
.947469
.944720
.941709
.938409
.934807
.930865
12
13
.975335
.972779
.969987
.966931
.963583
.959922
.955925
.951558
13
14
.995779
.992955
.98987.1
.986498
.982807
.978774
.974375
.969578
14
15
1.013900
1.010859
1.007475
1.003778
0.999736
0.995322
0.990519
0.985279
15
16
.030102
.026777
.023087
.019057
1.014656
1.009861
1.004640
.998959
IB
17
.044619
.040945
.036941
.032573
.027811
.022622
.016986
1.010859
17
18
.057527
.053556
.049232
.044522
.039390
.033811
.027756
.021188
18
19
.069049
.064775
.060127
.055071
.049569
.043596
.037127
.030117
19
20
1.079322
1.074743
1.069768
1.064362
1.058491
1.052128
1.045244
1.037800
20
21
.088468
.083579
.078276
.072521
.066284
.059531
.052239
.044373
21
22
.096590
.091392
.085760
.079657
.073056
.065920
.058234
.049957
22
23
.103782
.098274
.092315
.085870
.078908
.071404
.063332
.054663
23
24
.110126
.104310
.098030
.091247
.083938
.076071
.067634
.058591
24
25
1.115697
1.109578
1.102979
1.095870
1.088222
1.080014
1.071227
1.061835
25
26
.120563
.114144
.107239
.099814
.091845
.083309
.074197
.064480
26
27
.124787
.118078
.110875
.103148
.094873
.086035
.076619
.066608
27
28
.128426
.121439
.113953
.105940
.097381
.088259
.078569
.068295
28
29
.131537
.124284
.116531
.108251
.099428
.090050
.080116
.069610
29
30
1.134173
1.126669
1.118665
1.110140
1.101077
1.091471
1.081322
1.070611
30
31
. 136383
.128644
.120411
.111662
.102385
.092580
.082240
.071362
31
32
.138214
.130260
.121817
.112869
.103407
.093423
.082929
.071909
32
33
.139713
.131563
.122933
.113809
.104183
.094057
.083430
.072299
33
34
.140922
.132596
.123803
.114528
.104767
.094518
.083788
.072563
34
35
1.141880
1.133402
1.124468
1.115067
1.105193
1.094847
1.084031
1.072733
35
36
.142628
.134018
.124966
.115459
.105495
.095070
.084187
.072838
36
37
.143200
.134479
.125329
.115739
.105701
.095214
.084284
.072899
37 “
38
.143628
.134816
.125587
.115928
.105834
.095303
.084339
.072935
38
39
.143940
.135055
.125763
.116051
.105916
.095354
.084373
.072953
39
40
1.144163
1.135218
1.125876
1.116127
1.105963
1.095385
1.084388
1.072961
40
41
. 144314
.135323
.125946
.116171
.105991
.095399
.084396
.072964
41
42
.L44412
. 135388
.125987
.116195
.106004
.095406
.084399
.072965
42
43
. 144472
.135426
.126010
.116208
.106011
.095409
.084399
.072966
43
44
45
46
47
48
49
.144507
1.144526
.144537
.144541
.144541
.144542
.135447
1.135458
.135463
.135466
.135467
.135469
.126022
1.126028
.126029
.126030
.126031
.116214
1.116216
.116217
.116217
.106013
1.106013
.106014
.095409
1.095410
.084400
44
45
46
47
48
49
50
51
52
53
54
55
56
57 56
TABLE XLIY.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF W
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. , 11XTrn
4 PER CENT.
Pay-
8
II
O*
00
59
60
61
62
63
64
65
Pay-
merits.
n.
1.061107
1.048803
1.036042
1.022821
1.009130
0.994950
0.980284
0.965118
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.287660
.287305
.286912
.286481
.286011
.285490
.284921
.284292
2
3
.450294
.449567
.448766
.447888
.446929
.445869
.444707
.443431
3
4
.561680
.560568
.559343
.558003,
.556534
.554915
.553142
.551197
4
5
0.644940
0.643427
0.641765
0.639943
0.637947
0.635751
0.633349
0.630712
5
6
.710369
.708441
.706324
.704004
.701465
.698676
.695623
.692286
6
7
.763457
.761100
.758512
.755680
.752582
.749182
.745471
.741409
7
8
.807481
.804679
.801608
.798249
.794577
.790557
.786166
.781369
8
9
.844552
.841294
.837727
.833828
.829572
.824914
.819834
.814292
9
10
0.876115
0.872389
0.868313
0.863865
0.859010
0.853706
0.847930
0.841641
10
11
.903200
.898995
.894402
.889392
.883930
.877975
.871501
.864458
11
12
.926569
.921878
.916756
.911176
.905105
.898497
.891319
.883533
12
13
.946804
.941615
.935959
.929808
.923125
.915860
.907991
.899465
13
14
.964355
.958664
.952472
.9457481
.938453
.930543
.921985
.912738
14
15
0.979585
0.973391
0.966663
0.9593071
0.951469
0.942918
0.933690
0.923742
15
16
.992791
.986096
.978832
.9709621
.962477
.953304
.943427
.932811
16
17
1.004222
.997024
.989233
.980817!
.971743
.961967
.951472
.940221
17
18
.014080
1.006388
.998078
.9891231
.979490
.969141
.958060
.946210
18
19
.022546
.014370
1.005558
.9960821
.985917
.975027
.963393
.950996
19
20
1.029777
1.021132
1.011835 1.001865
0.991199
0.979798
0.967661 0.954769
20
21
.035912
.026815
.017058
.006623
.995485
.983621
.971029
.957693
21
22
.041075
.031550
.021361
.010490
.998923
.986641
.973642
.959917
22
23
.045381
.035454
.024861
.013593
1.001641
.988985
.975629
.961578
23
24
.048935
.038632
.027672
.016048
.003753
.990769
.977115
.962787
24
25
1.051830
1.041187
1.029897
1.017958
1.005300
0.992103
0.978197
0.963646
25
26
.054158
.043210
.031629
.019410
.006562
.993075
.978966
.964228
26
27
.056003
.044786
.032946
.020498
.007439
.993766
.979487
.964606
27
28
.057441
.045985
.033934
.021291
.008062
.994234
.979825
.964837
28
29
.058535
.046884
.034654
.021855
.008484
.994538
.980032
.964974
29
30
1.059355
1.047539
L. 035166 1.022237
1.008758
0.994723
0.980154
0.965049
30
31
.059953
.048005
.035512
.022484
.008925
.994833
.980222
.965088
31
32
.060379
.048320
.035736
.022636
.009024
.994894
.980257
.965107
32
33
.060667
.048524
.035875
.022725
.009079
.994926
.980274
.965116
33
34
.060853
.048650
.035955
.022775
.009108
.994941
.980282
.965117
34
35
1.060968
1.048723
1.036001
1.022802
1.009121
0.994948
0.980282
0.965118
35
36
.061035
.048766
.036025
.022813
.009128
.994948
.980284
37
.061074
.048788
.036035
.022820
.009128
.994950
43
42
38
.061094
.048797
.036041
.022820
.009130
44
1.205342
58
39
.061102
.048801
.036041
. 0^/33/311
45
1.198793
. 205342
57
40
1.061106
1.048802
1.036042
1.191959
.198792
.205342
56
41
.061107
.048803
47
43
1.184839
.191960
.198792
.205341
55
42
.061107
48
1.177421
.184838
.191959
1.198790
1.205338
54
49
1.169692
.177420!
.184838
.191958
.198787
.205332
53
1.161644
.169691
.177420
.184836
.191955
.198781
.205321
52
51
1.153265
.161644
.169691
.177418
.184833
.191948
.198769
.205300
51
50
.153265
.161644
.169689
.177414'
.184826
.191935
.198747
.205266
50
49
1.153264
1.161642
1.169685
1.177407
1.184812
1.191912
1.198710
1.205209
49
48
.153262
.161638
.169677
.177392
.184787
.191872
.198650
.205122
48
47
.153258
.161629
.169661
.177365
.184745
.191808
.198557
.204993
47
46
.153248
.161612
.169633
.177320
.184676
.191708
.198419
.204812
46
45
.153230
.161582
.169584
.177246
.184570
.191562
.198227
.204565
45
49
48
47
43
45
44
43
42
Continued from Page *54. TABLE XLIY.
57
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
4 PER CENT.
Pay-
ments.
66
67
68
69
70
71
72
73
Pay-
n.
0.949453
0.933271
0.916583
0.899356
0.881623
0.863356
0.844568
0.825252
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.283608
.282848
.282028
.281111
.280119
.279023
.277826
.276518
2
3
.442034
.440493
.438825
.436973
.434958
.432742
.430325
.427673
3
4
.549065
.546733
.544183
.541374
.538313
.534961
.531297
.527292
4
5
0.627836
0.624679
0.621234
0.617447
0.613328
0.608817
0.603900
0.598536
5
6
.688638
.684643
.680284
.675511
.670316
.664644
.658470
.651748
6
7
.736976
.732127
.726850
.721076
.714804
.707970
.700546
.692480
7
8
.776139
.770433
.764225
.757451
.750105
.742118
.733460
.724082
8
9
.808262
.801690
.794553
.786785
.778375
.769254
.759395
.748748
9
10
0.834803
0.827369
0.819311
0.810560
0.801106
0.790885
0.779868
0.768008
10
11
.856820
.848531
.839564
.829851
.819388
.808109
.795989
.782987
11
12
.875100
.865969
.856116
.845477
.834046
.821761
.808606
.794547
12
13
.890254
.880304
.869599
.858073
.845724
.832499
.818391
.803367
13
14
.902769
.892036
.880516
.868150
.854946
.840860
.825882
.810003
14
15
0.913050
0.901568 0.889279
0.876134
0.862149
0.847278
0.831533
0.814911
15
16
.921430
.909241
.896241
.882386
.867691
.852130
.835721
.818460
16
17
.928190
.915351
.901704
.887206
.871888
.855732
.838754
.820971
17
18
.933583
.920154
.905923
.890861
.875008
.858343
.840902
.822670
18
19
.937829
.923868
.909125
.893580
.877272
.860194
.842356
.823765
19
20
0.941117
0.926690 0.911510
0.895556
0.878878
0.861448
0.843294
0.824438
20
21
.943616
.928793
.913245
.896956
.879966
.862256
.843870
.824833
21
22
.945481
.930324
.914476
.897908
.880668
.862754
.844209
.825051
22
23
.946838
.931411
.915310
.898522
.881100
.863046
.844396
.825164
23
24
.947802
.932147
.915849
.898899
.881354
.863208
.844493
.825219
24
25
0.948455
0.932623 0.916181
0.899121
0.881494
0.863291
0.844540
0.825242
25
26
.948878
.932916
.916376
.899243
.881567
.863332
.844560
.825250
26
27
.949138
.933089
.916483
.899307
.881602
.863349
.844567
.825252
27
28
.949291
.933183
.916539
.899338
.881617
.863355
.844568
29
.949375
.933232
.916566
.899351
.881622
.863356
35
34
1.248902
66
30
0.949419
0.93o2ou
0.916578
0.899355
0.881624
31
.949440
.933267
.916581
.899356
37
36
1.244234
.248901
65
32
.949450
.933269
.916583
38
1.239367
1.244234
1.248901
64
33
.949451
.933271
39
1.234282
.239367
.244234
.248900
63
34
.949453
40
1.228972
.234283
.239366
.244233
.248898
62
1.223433
.228971
.234283
.239366
.244231
.248894
61
41
1.217656
.223433
.228970
.234282
.239364
.244228
.248888
60
59
1.211632
1.217656
1.223433
1.228969
1.234280
1.239360
1.244220
1.248875
59
58
.211632
.217656
.223432
.228967
.234276
.239352
.244207
.248853
58
57
.211632
.217655
.223430
.228963
.234267
.239338
.244184
.248819
57
56
.211630
.217652
.223425
.228954
.234252
.239314
.244147
.248765
56
55
.211628
.217647
.223415
.228938
.234227
.239274
.244090
.248686
55
54
1.211622
1.217637
1.223398
1.228911
1.234185
1.239214
1.244006
1.248574
54
53
.211612
.217619
.223369
.228867
.234121
.239125
.243889
.248422
53
52
.211592
.217588
.223323
.228799
.234027
.239000
.243727
.248219
52
51
.211560
.217538
.223250
.228698
.233894
.238829
.243512
.247953
51
50
.211507
.217461
.223144
.228557
.233712
.238600
.243230
.247613
50
49
1.211425
1.217348
1.222994
1.228364
1.233470
1.238301
1.242869
1.247184
49
48
.211304
.217189
.222789
.228111
.233153
.237919
.242415
.246654
48
47
.211134
.216971
.222515
.227770
.232746
.237437
.241854
.246009
47
46
.210902
.216679
.222157
.227339
.232236
.236842
.241169
.245321
46
45
.210592
.216299
.221699
.226796
.231603
.236115
.240345
.244308
45
41
40
39
38
37
36
35
34
Continued from Page *53. 58
TABLE X L I V.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. .
4 PER CENT.
Pay-
ments.
£=74
75
76
77
78
79
80
81
Pay-
ments.
n.
0.805400
0.785049
0.764174
0.742769
0.720875
0.698471
0.675562
0.652146
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.275065
.273492
.271774
.269873
.267807
.265547
.263074
.260385
2
3
.424759
.421598
.418136
.414336
.410203
.405700
.400793
.395450
3
4
.522908
.518143
.512941
.507254
.501088
.494380
.487095
.479197
4
5
0.592676
0.586317
0.579393
0.571857
0.563698
0.554856
0.545289
0.534963
5
6
.644425
.636495
.627890
.618553
.608478
.597600
.585882
.573294
6
7
.683722
.674263
.664030
.652969
.641076
.628292
.614587
.599946
7
8
.713931
.702999
.691214
.678528
.664943
.650410
.634915
.618441
8
9
.737262
.724930
.711690
.697499
.682372
.666274
.649192
.631143
9
10
0.755259
0.741621
0.727040
0.711486
0.694987
0.677512
0.659077
0.639575
10
11
.769063
.754228
.738439
.721678
.703979
.685338
.665675
.644944
11
12
.779550
.763641
.746788
.728977
.710270
.690583
.669890
.648213
12
13
.787413
.770563
.752788
.734101
.714497
.693941
.672462
.650123
13
14
.793214
.775551
.757010
.737552
.717209
.695994
.673966
.651175
14
15
0.797403
0.779068
0.759860
0.739769
0.718869
0.697196
0.674796
0.651718
15
16
.800361
.781446
.761692
.741127
.719842
.697859
.675225
.651979
16
17
.802362
.782975
.762816
.741923
.720379
.698202
.675430
.652090
17
18
.803651
.783914
.763475
.742363
.720656
.698366
.675517
.652133
18
19
.804443
.784464
.763838
.742590
.720789
.698436
.675551
.652146
19
20
0.804907
.805163
0.784768
0.764026
0.742699
.742746
0.720846
.720868
0.698463
.698471
0.675562
26
21
22
.784925
.785001
.764117
.764156
.805295
.805360
.805387
.742764
.742769
.720875
27
28
1.279758
74
23
.785033
.764171
29
1.276466
.279758
73
24
.785046
.764174
30
1.273025
.276467
.279758
72
25
26
0.805397
.805400
0.785049
31
1.269428
.273025
.276467
.279757
.279756
71
1.265675
70
32
.269428
.273025
.276467
33
1.261752
1.265676
1.269428
1.273024
1.276465
1.279753
69
1.257654
.261752
.265676
.269427
.273023
.276463
.279749
68
67
1.253372
.257654
.261752
.265675
.269426
.273020
.276458
.279741
67
66
.253373
.257654
.261751
.265674
.269423
.273015
.276450
.279727
66
65
.253373
.257653
.261749
.265671
.269418
.273006
.276435
.279705
65
64
1.253372
1.257652
1.261746
1.265665
1.269409
1.272991
1.276412
1.279670
64
63
.253371
.257649
.261740
.265655
.269392
.272966
.276375
.279620
63
62
.253367
.257642
.261730
.265638
.269366
.272928
.276322
.279548
62
61
.253361
.257631
.261712
.265610
.269325
.272871
.276246
.279451
61
60
.253349
.257612
.261682
.265568
.269266
.272792
.276143
.279320
60
59
1.253329
1.257581
1.261637
1.265504
1.269182
1.272683
1.276006
1.279150
59
58
.253296
.257533
.261570
.265416
.269067
.272537
.275826
.278932
58
57
.253245
.257462
.261476
.265294
.268913
.272348
.275596
.278658
57
56
.253170
.257363
.261348
.265132
.268713
.272105
.275307
.278319
56
55
.253065
.257227
.261177
.264920,
.268456
.271799
.274949
.277906
55
54
1.252921
1.257046
1.260953
1.264649
1.268133
1.271421
1.274513
1.277409
54
53
.252729
.256809
.260666
.264308
.267734
.270961
.273988
.276819
53
52
.252478
.256505
.260305
.263886
.267247
.270407
.273365
.276126
52
51
.252157
.256124
.259858
.263371
.266662
.269749
.272634
.275321
51
50
.251753
.255651
.259314
.262752
.265967
.268976
.271783
.274392
50
49
1.251253
1.255075
1.258660
1.262017
1.265150
1.268077
1.270803
1.273332
49
48
.250643
.254383
.257882
.261154
.264200
.267041
.269682
.272128
48
47
.249910
.253560
.256968
.260149
.263105
.265858
.268411
.270773
47
46
.249038
.252592
.255906
.258991
.261854
.264514
.266979
.269255
46
45
.248014
.251467
.254680
.257667
.260433
.263000
.265375
.267565
45
33
32
31
30
29
28
27
26
Continued from Page *52. TABLE X LIV.
59
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. „
4 PER CENT.
j Pay-
82
83
84
85
86
87
88
89
Pay-
n.
0.628187
0.603703
0.578554
0.552712
0.525914
0.497713
0.467718
0.434386
n.
1
0.000000
0.000000
0.000000
0.000000
!0.000000
0.000000
0.000000
0.000000
1
2
.257424
.254238
.250719
.246926
.242837
.238267
.233496
.225360
2
3
.389631
.383339
.376475
.369109
.361116
.352422
.341682
.326255
3
4
.470629
.461402
.451421
.440707
.429204
.415840
.399517
.378512
4
5
0.523818
0.511880
0.499024
0.485329
0.470086
0.452376
0.431695
0.406483
5
6
.559791
.545384
.529994
.513258
.494585
.473468
.449522
.421206
6
7
.584318
.567754
.549865
.530408
.509034
.485385
.459073
.428627
7
8
.600972
.582338
.562249
.540626
.517295
.491835
.463930
.432148
8
9
.611946
.591516
.569691
.546524
.521796
.495133
.466244
.433635
9
10
0.618899
0.597065
0.574008
0.549752
0.524104
0.496709
0.467223
0.434205
10
11
.623121
.600294
.576377
.551409
.525209
.497375
.467599
.434386
11
12
.625583
.602070
.577596
.552205
.525677
.497632
.467718
13
.626938
.602984
.578182
.552541
.525857
.497713
18
14:
.627637
.603424
.578428
.552671
.525914
20
19
1.301453
82
15
0.627973
0.603609
0.578525
0.552712
oi
1.299144
.301453
81
16
.628115
.603682
.578554
1.296729
.299144
.301453
80
17
.628170
.603703
23
22
1.294205
1.296729
1.299144
1.301453
79
18
. 62818 /
24
1.291567
.294205
.296729
.299144
.301452
78
25
1.288808
.291567
.294205
.296729
.299143
.301450
77
1.285924
.288808
.291567
.294205
.296728
.299141
.301448
76
75
1.282910
.285924
.288808
.291566
.294204
.296726
.299138
.301442
75
74
1.282909
1.285924
1.288808
1.291565
1.294202
1.296723
1.299133
1.301433
73
73
.282909
.285924
.288807
.291563
.294199
.296717
.299123
.301419
74
72
.282908
.285923
.288805
.291560
.294193
.296707
.299108
.301396
72
71
.282907
.285921
.288801
.291553
.294182
.296691
.299084
.301362
71
70
.282905
.285917
.288794
.291542
.294165
.296666
.299049
.301316
70
69
1.282901
1.285909
1.288783
1.291524
1.294138
1.296629
1.298999
1.301251
69
68
.282893
.285897
.288764
.291496
.294099
.296577
.298932
.301166
68
67
.282880
.285877
.288734
.291455
.294045
.296506
.298841
.301054
67
66
.282859
.285846
.288691
.291397
.293970
.296410
.298724
.300910
66
65
.282826
.285800
.288630
.291319
.293870
.296287
.298573
.300730
65
64
1.282778
1.285736
1.288547
1.291213
1.293739
1.296128
1.298383
1.300507
64
63
.282711
.285648
.288436
.291075
.293572
.295928
.298148
.300236
63
62
.282618
.285531
.288291
.290899
.293361
.295681
.297863
.299909
62
61
.282495
.285378
.288105
.290677
.293101
.295380
.297519
.299522
61
60
.282333
.285182
.287871
.290403
.292784
.295018
.297111
.299067
60
59
1.282127
1.284936
1.287582
1.290069
1.292403
1.294589
1.296632
1.298539
59
58
.281867
.284631
.287230
.289668
.291950
.294084
.296076
.297931
58
57
.281546
.284260
.286807
.289191
.291419
.293498
.295435
.297236
57
56
.281154
.283814
.286304
.288631
.290802
.292824
.294704
.296449
56
55
.280683
.283284
.285714
.287981
.290091
.292053
.293875
.295563
55
54
1.280124
1.282661
1.285028
1.287231
1.289279
1.291180
1.292942
1.294572
54
53
.279468
.281938
.284238
.286375
.288359
.290197
.291898
.293470
53
52
.278704
.281105
.283336
.285405
.287322
.289097
.290737
.292250
52
51
.277825
.280153
.282312
.284312
.286163
.287873
.289451
.290906
51
50
.276820
.279073
.281160
.283090
.284873
.286518
.288035
.289431
50
49
1.275681
1.277857
1.279870
1.281729
1.283444
1.285025
1.286481
1.287819
49
48
.274397
.276496
.278435
.280222
.281869
.283386
.284781
.286063
48
47
.272960
.274981
.276844
.278561
.280141
.281593
.282929
.284154
47
46
.271360
.273302
.275091
.276736
.278250
.279640
.280916
.282087
46
45
.269587
.271450
.273165
.274740
.276188
.277516
.278735
.279852
45
25
24
23
22
21
20
19
18
Continued from Page *51. 60
TABLE XLIY.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
4 PER CENT.
Pay-
ments.
11.
II
CD
O
91
92
93
94
95
98
97
Pay-
ments.
11.
0.402G08
0.373607
0.344609
0.314696
0.284280
0.252440
0.214754
0.178369
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.216431
.208653
.200520
.191140
.180950
.170487
.153574
.141329
2
3
.310577
.357604
.296383
.338221
.281311
.317835
.264501
.295797
.246836
.272024
.226526
.246291
.200677
.214754
.178369
3
4
10
5
0.381571
0.358362
0.334359
0.308421
0.281336
0.252440
11
.393460
.367723
.341183
.313181
.284280
1.316633
90
6
.399061
.401417
.402319
.371634
.373129
.373607
.343779
.344609
.314696
12
7
8
13
1.315019
1.336633
89
14
1.313333
.315020
.316633
88
9
15
1.311567
.313333
.313333
.315020
.316633
87
10
0.402608
16
1.309720
.311567
.315019
.316632
86
1.307791
.309720
1.309720
.309720
.311567
1.311567
.313333
1.313333
.313331
.315019
1.315018
.315016
.316631
1.316629
.316626
85
84
83
17
1.305773
1.307791
83
1.303662
.305774
307791
.311566
82
.303662
.305774
.307791
.309719
.311565
.313329
.315011
.316620
82
81
.303662
.305773
.307790
.309718
.311563
.313325
.315006
.316610
81
80
.303661
.305772
.307789
.309716
.311558
.313318
.314995
.316595
80
79
1.303660
1.305771
1.307786
1.309712
1.311551
1.313308
1.314979
1.316572
79
78
.303659
.305769
.307782
.309705
.311540
.313291
.314956
.316541
78
77
.303656
.305764
.307775
.309692
.311522
.313266
.314923
.316498
77
76
.303651
.305755
.307761
.309674
.311496
.313231
.314878
.316441
76
75
.303643
.305742
.307742
.309646
.311460
.313184
.314817
.316367
75
74
1.303629
1.305722
1.307713
1.309608
1.311410
1.313120
1.314739
1.316271
74
73
.303607
.305692
.307673
.309556
.311343
.313038
.314638
.316150
73
72
.303575
.305649
.307618
.309486
.311257
.312932
.314511
.316000
72
71
.303531
.305591
.307544
.309395
.311145
.312799
.314354
.315819
71
70
.303470
.305514
.307448
.309277
.311005
.312634
.314163
.315601
70
69
1.303388
1.305413
1.307325
1.309130
1.310832
1.312434
1.313934
1.315342
69
68
.303282
.395283
.307170
.308948
.310621
.312193
.313662
.315038
68
67
.303146
.305120
.306979
.308727
.310368
.311906
.313343
.314685
67
66
.302975
.304919
.306746
.308460
.310067
.311571
.312972
.314279
66
65
.302763
.304674
.306466
.308144
.309714
.311181
.312545
.313814
65
64
1.302505
1.304379
1.306133
1.307773
1.309305
1.310732
1.312058
1.313289
64
63
.302195
.304029
.305743
.307342
.308833
.310220
.311506
.312699
63
62
.301827
.303619
.305290
.306846
.308295
.309640
.310885
.312039
62
61
.301395
.303142
.304768
.306279
.307685
.308987
.310191
.311305
61
60
.300893
.302593
.304172
.305638
.306998
.308258
.309420
.310494
60
59
1.300315
1.301966
1.303497
1.304916
1.306231
1.307447
1.308567
1.309601
59
58
.299655
.301255
.302737
.304109
.305378
.306550
.307628
.308622
58
57
.298907
.300456
.301888
.303211
.304435
.305562
.306599
.307554
57
56
.298066
.299563
.300944
.302219
.303395
.304480
.305475
.306391
56
55
.297125
.298568
.299898
.301125
.302256
.303298
.304252
.305130
55
54
1.296078
1.297469
1.298748
1.299926
1.301012
1.302010
1.302926
1.303765
54
53
.294920
.296257
.297486
.298617
.299658
.300614
.301489
.302293
53
52
.293644
.294928
.296107
.297191
.298188
.299103
.299939
.300707
52
51
.292245
.293476
.294606
.295643
.296597
.297472
.298270
.299003
51
50
.290715
.291895
.292976
.293968
.294879
.295714
.296476
.297176
50
49
1.289049
1.290178
1.291211
1.292159
1.293029
1.293825
1.294552
1.295218
49
48
.287239
.288318
.289304
.290209
.291038
.291798
.292490
.293124
48
47
.285278
.286309
.287250
.288112
.288903
.289626
.290285
.290888
47
46
.283160
.284142
.285039
.285861
.286613
.287302
.287928
.288502
46
45
.280875
.281811
.282666
.283446
.284163
.284817
.285413
.285958
45
17
16
15
14
13
12
11
10
Continued from Page *50. 61
TABLE XLV.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
The first Value under the Age X is A(1 -)- ax) for tlie wliole Life.
42 PER CENT.
Pay-
ments.
x= 10
11
12
13
14
15
16
17
Pay-
ments.
n.
11.
1.281791
1.280417
1.278977
1.277464
1.275875
1.274211
1.272467
1.270632
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.290199
.290194
.290191
.290187
.290180
.290174
.290170
.290163
2
3
.455548
.455540
.455533
.455523
.455510
.455500
.455490
.455475
3
4
.509836
.569823
.569811
.569796
.569778
.569762
.569745
.569723
4
5
0.656184
0.656166
0.656149
0.656129
0.656106
0.656083
0.656060
0.656031
5
6
.724891
.724869
.724847
.724822
.724792
.724764
.724734
.724696
6
7
.781454
.781427
.781400
.781369
.781333
.781299
.781260
.781213
7
8
.829151
.829119
.829087
.829049
.829008
.828964
.828918
.828863
8
9
.870097
.870061
.870022
.869979
.869929
.869879
.869825
.869761
9
10
0.905737
0.905094
0.905650
0.905600
0.905544
0.905486
0.905423
0.905350
10
11
.937099
.937051
.937001
.936944
.936880
.936814
.936742
.936657
11
12
.964945
.964891
.964834
.964770
.964699
.964624
.964542
.964448
12
13
.989852
.989792
.989728
.989657
.989577
.989493
.989401
.989296
13
14
1.012269
1.012202
1.012131
1.012052
1.011963
1.011870
1.011767
1.011651
14
15
1.032550
1.032477
1.032398
1.032310
1.032214
1.032109
1.031996
1.031867
15
16
.050984
.050902
.050815
.050721
.050612
.050498
.050374
.050235
16
17
.067801
.067712
.067620
.067513
.067396
.067271
.067137
.066984
17
18
.083199
.083105
.083000
.082885
.082758
.082624
.082476
.082306
18
19
.097341
.097235
.097123
.096998
.096862
.096715
.096551
.096368
19
20
1.110358
1.110246
1.110124
1.109991
1.109842
1.109680
1.109504
1.109307
20
21
.122374
.122252
.122123
.121978
.121815
.121641
.121451
.121239
21
22
.133486
.133357
.133216
.133057
.132884
.132696
.132493
.132263
22
23
.143783
.143643
.143489
.143322
.143134
.142935
.142713
.142466
23
24
.153339
.153186
.153024
.152842
.152644
.152427
.152190
.151925
24
25
1.162219
1.162058
1.161882
1.161690
1.161476
1.161243
1.160989
1.160705
25
26
.170486
.170312
.170127
.169919
.169689
.169441
.169169
.168867
26
27
.178187
.178005
.177804
.177582
.177338
.177072
.176783
.176459
27
28
.185374
.185177
.184962
.184727
.184465
.184183
.183874
.183528
28
29
.192081
.191871
.191644
.191392
.191115
.190814
.190483
.190119
29
30
1.198348
1.198120
1.197884
1.197617
1.197322
1.197000
1.196653
1.19625-9
30
31
.204210
.203974
.203716
.203433
.203118
.202781
.202404
.201993
31
32
.209695
.209443
.209171
.208868
.208539
.208174
.207780
.207340
32
33
.214828
.214504
.214272
.213955
.213600
.213219
.212798
.212332
33
34
.219039
.219355
.219050
.218709
.218339
.217932
.217486
.216993
34
35
1.224142
1.223846
1.223519
1.223163
1.222768
1.222337
1.221866
1.221347
35
36
.228367
.228049
.227708
.227329
.226910
.226455
.225960
.225410
36
37
.232323
.231991
.231629
.231226
.230785
.230308
.229782
.229201
37
38
.236034
.235084
.235298
.234875
.234412
.233905
.233351
.232739
38
39
.239513
.239141
.238735
.238291
.237800
.237267
.236684
.236038
39
40
1.242772
1.242380
1.241955
1.241485
1.240969
1.240408
1.239792
1.239114
40
41
.245827
.245417
.244966
.244473
.243930
.243338
.242691
.241977
41
42
.248692
.248257
.247785
.247265
.246693
.246072
.245391
.244640
42
43
.251372
.250917
.250420
.249872
.249273
.248619
.247903
.247117
43
44
.253883
.253405
.252881
.252308
.251677
.250990
.250240
.249414
44
10
11
12
13
14
15
16
17
Continued on Page *71.
Formula, A(1 = a
Dx vpx-i 62
TABLE XLY.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
4-L PER CENT.
I Pay-
x=18
19
20
21
22
23
24
25
Pay-
ments.
71.
n.
1.268708
1.266691
1.264573
1.262353
1.260026
1.257585
1.255025
1.252341
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.290155
.290148
.290139
.290129
.290119
.290107
.290094
.290081
2
3
.455460
.455445
.455426
.455406
.455385
.455361
.455335
.455306
3
4
.569701
.569677
.569647
.569617
.569584
.569547
.569506
.569462
4
5
0.655999
0.655965
0.655926
0.655884
0.655840
0.655789
0.655733
0.655673
5
6
.724655
.724612
.724562
.724509
.724452
.724387
.724316
.724239
6
7
.781164
.781111
.781050
.780984
.780914
.780835
.780748
.780653
7
8
.828805
.828741
.828668
.828590
.828506
.828411
.828307
.828194
8
9
.869692
.869617
.869532
.869441
.869342
.869231
.869110
.868976
9
10
0.905269
0.905182
0.905086
0.904980
0.904866
0.904739
0.904598
0.904444
10
11
.936567
.936469
.936358
.936238
.936109
.935963
.935802
.935627
11
12
.964346
.964234
.964110
.963976
.963828
.963663
.963483
.963285
12
13
.989182
.989057
.988920
.988768
.988602
.988418
.988217
.987997
13
14
1.011523
1.011387
1.011233
1.011063
1.010880
1.010676
1.010453
1.010207
14
15
1.031729
.031577
1.031404
1.031219
1.031017
1.030792
1.030544
1.030273
15
16
.050082
.049911
.049724
.049521
j .049298
.049051
.048779
.048481
16
17
.066813
.066629
.066425
.066202
.065958
.065687
.065391
.065065
17
18
.082122
.081922
.081699
.081456
.081189
.080896
.080572
.080217
18
19
.096170
.095952
.095710
.095445
.095157
.094836
.094486
.094102
19
20
1.109092
1.108856
1.108593
1.108308
1.107993
1.107649
1.107270
1.106852
20
21
.121006
.120750
.120468
.120157
.119821
.119449
.119037
.118586
21
22
.132011
.131737
.131430
.131099
.130736
.130332
.129891
.129406
22
23
.142196
.141899
.141573
.141217
.140823
.140392
.139917
.139397
23
24
.151634
.151318
.150968
.150583
.150164
.149700
.149192
.148633
24
25
1.160396
1.160056
1.159680
1.159270
1.158819
1.158325
1.157780
1.157181
25
26
.168535
.168171
.167771
.167331
.166852
.166321
.165738
.165099
26
27
.176104
.175718
.175288
.174822
.174308
.173741
.173120
.172441
27
28
.183153
.182737
.182283
.181783
.181235
.180633
.179973
.179247
28
29
.189714
.189277
.188791
.188258
.187676
.187037
.186333
.185563
29
30
1.195835
1.195367
1.194849
1.194284
1.193668
1.192987
1.192240
1.191422
30
31
.201539
.201041
.200493
.199896
.199239
.198517
.197725
.196859
31
32
.206858
.206332
.205753
.205118
.204421
.203656
.202820
.201900
32
33
.211823
.211266
.210652
.209978
.209241
.208434
.207546
.206577
33
34
.216456
.215865
.215214
.214503
.213726
.212870
.211935
.210909
34
35
1.220777
1.220152
1.219465
1.218714
1.217892
1.216991
1.216002
1.214921
35
36
.224807
.224148
.223423
.222631
.221765
.220810
.219771
.218631
36
37
.228566
.227870
.227107
.226272
.225356
.224355
.223258
.222059
37
38
.232069
.231337
.230532
.229651
.228690
.227635
.226482
.225222
38
39
.235334
.234563
.233714
.232789
.231777
.230669
.229458
.228136
39
40
1.238372
1.237559
1.236669
1.235696,
1.234634
1.233471
1.232201
1.230815
40
41
.241196
.240344
.239408
.238388
.237273
.236054
.234723
.233272
41
42
.243822
.242926
.241946
.240875
.239707
.238430
.237037
.235520
42
43
.246257
.245319
.244290
.243169
.241946
.240610
.239156
.237571
43
44
.248515
.247530
.246454
.245280
.244002
.242608
.241088
.239436
44
18
19
SO
21
22
23
24
25
Continued on Page *70. 63
TABLE XLY.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
41, PER CENT.
Pay-
26
27
28
29
30
31
32
33
Pay-
ments.
n.
1.249527
1.246580
1.243489
1.240250
1.236862
1.233310
1.229589
1.225692
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.290064
.290047
.290029
.290005
.289984
.289960
.289933
.289902
2
3
.455272
.455239
.455200
.455154
.455111
.455060
.455003
.454942
3
4
.569412
.569360
.569299
.569232
.569163
.569084
.568999
.568902
4
5
0.655606
0.655533
0.655452
0.655360
0.655265
0.655159
0.655039
0.654910
5
6
.724152
.724060
.723955
.723838
.723718
.723578
.723428
.723261
6
7
.780547
.780431
.780303
.780162
.780009
.779842
.779656
.779452
7
8
.828066
.827929
.827777
.827605
.827427
.827224
.827003
.826758
8
9
.868827
.868667
.868486
.868290
.868077
.867843
.867582
.867296
9
10
0.904273
0.904086
0.903882
0.903653
0.903410
0.903137
0.902838
0.902507
10
11
.935431
.935221
.934987
.934729
.934449
.934140
.933798
.933423
11
12
.963067
.962828
.962565
.962273
.961958
.961609
.961225
.960801
12
13
.987751
.987485
.987190
.986864
.986512
.986122
.985691
.985221
13
14
1.009935
1.009639
1.009312
1.008950
1.008560
1.008125
1.007650
1.007127
14
15
1.029973
1.029647
1.029285
1.028888
1.028454
1.027977
1.027452
1.026874
15
16
.048152
.047793
.047398
.046958
.046485
.045961
.045384
.044750
16
17
.064705
.064314
.063878
.063401
.062883
.062309
.061679
.060985
17
18
.079827
.079397
.078926
.078406
.077841
.077217
.076529
.075775
18
19
.093675
.093211
.092699
.092135
.091522
.090843
.090099
.089281
19
20
1.106391
1.105889
1.105335
1.104725
1.104060
1.103327
1.102522
1.101637
20
21
.118090
.117548
.116950
.1162911
.115574
.114785
.113915
.112964
21
22
.128872
.128288
.127644
.126935
.126165
.125313
.124382
.123356
22
23
.138823
.138195
.137503
.136744
.135915
.135004
.134001
.132903
23
24
.148018
.147344
.146605
.145789
.144904
.143925
.142855
.141679
24
25
1.156522
1.155804
1.155011
1.154142
1.153192
1.152150
1.151004
1.149749
25
26
.164398
.163628
.162784
.161854
.160844
.159730
.158510
.157172
26
27
.171690
.170873
.169970
.168983
.167905
.166720
.165422
.163999
27
28
.178451
.177578
.176621
.175569
.174424
.173165
.171785
.170274
28
29
.184715
.183790
.182772
.181657
.180440
.179104
.177641
.176040
29
30
1.190524
1.189540
1.188463
1.187279
1.185990
1.184574
1.183026
1.181332
30
31
.195905
.194866
.193723
.192471!
.191105
.189609
.187973
.186184
31
32
.200894
.199792
.198584
.197259
.195818
.194238
.192512
.190625
32
33
.205511
.204348
.203071
.201674
.200153
.198489
.196669
.194684
33
34
.209785
.208556
.207211
.205738;
.204437
.202384
.200471
.198385
34
35
1.213735
1.212440
1.211023
1.209475
1.207790
1.205948
1.203940
1.201749
35
36
.217382
.216020
.214532
.212904
.211134
.209202
.207094
.204803
36
37
.220746
.219316
.217752
.216043
.214189
.212162
.209959
.207560
37
38
.223844
.222343
.220702
.218913
.216968
.214851
.212540
.210046
38
39
.226691
.225117
.223399
.221525
.219495
.217275
.214880
.212273
39
40
1.229301
1.227653
1.225856
1.223900
1.221773
1.219474
1.216972
1.214262
40
41
.231688
.229966
.228090
.226043
.223840
.221440
.218840
.216027
41
42
.233865
.232068
.230107
.227987
.225689
.223196
.220500
.217586
42
43
.235845
.233967
.231936
.229727
.227341
.224757
.221965
.218952
43
44
.237634
.235690
.233575
.231282
.228809
.226135
.223249
.220140
44
26
27
28
29
30
31
32
33
Continued on Page *69, 64
TABLE XLY.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
4.', PER CENT.
Pay-
II
CO
35
36
37
38
39
40
41
Pay-
ments.
n.
1.221612
1.217341
1.212877
1.208201
1.203309
1.198189
1.192839
1.187248
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.289869
.289829
.289792
.289749
.289699
.289644
.289585
.289524
2
3
.454871
.454795
.454715
.454627
.454525
.454414
.454295
.454164
3
4
.568797
.568679
.568557
.568419
.568263
.568095
.567910
.567709
4
5
0.654765
0.654607
0.654438
0.654249
0.654038
0.653808
0.653556
0.653281
5
6
.723078
.722875
.722658
.722416
.722146
.721852
.721530
.721178
6
7
.779226
.778977
.778710
.778412
.778081
.777721
.777325
.776892
7
8
.826489
.826192
.825872
.825516
.825123
.824691
.824218
.823701
8
9
.866980
.866634
.866258
.865844
.865382
.864877
.864325
.863717
9
10
0.902145
0.901745
0.901314
0.900835
0.900303
0.899723
0.899084
0.898388
10
11
.933010
.932558
.932065
.931518
.930915
.930253
.929530
.928738
11
12
.960338
.959827
.959270
.958656
.957975
.957233
.956419
.955525
12
13
.984702
.984130
.983509
.982821
.982064
.981235
.980325
.979329
13
14
1.006550
1.005917
1.005226
1.004465
1.003625
1.002706
1.001698
1.000592
14
15
1.026239
1.025539
1.024779
1.023939
1.023014
1.022001
1.020890
1.019673
15
16
.044051
.043286
.042450
.041530
.040516
.039405
.038187
.036854
16
17
.060224
.059387
.058475
.057470
.056362
.055150
.053823
.052366
17
18
.074947
.074037
.073045
.071951
.070747
.069432
.067987
.066407
18
19
.088382
.087397
.086320
.085135,
.083834
.082406
.080844
.079135
19
20
1.100668
1.099602
1.098438
1.097161
1.095753
1.094215
1.092530
1.090686
20
21
.111917
.110769
.109518
.108140
.106626
.104971
.103159
.101175
21
22
.122232
.121000
.119654
.118175
.116551
.114775
.112830
.110708
22
23
.131699
.130377
.128935
.127351
.125612
.123711
.121635
.119364
23
24
.140390
.138977
.137436
.135743
.133886
.131860
.129643
.127224
24
25
1.148374
1.146867
1.145221
1.143417
1.141441
1.139282
1.136925
1.134351
25
26
.155707
.154101
.152351
.150434
.148332
.146040
.143535
.140805
26
27
.162440
.160735
.158877
.156840
.154613
.152181
.149530
.146642
27
28
.168621
.166813
.164842
.162687i
.160327
.157756
.154955
.151907
28
29
.174289
.172375
.170291
.168011
.165519
.162806
.159854
.156641
29
30
1.179481
1.177459
1.175258
1.172852
1.170225
1.167370
1.164262
1.160891
30
31
.184230
.182098
.179777
.177243
.174483
.171480
.168222
.164682
31
32
.188567
.186321
.183879
.181220
.178318
.175174
.171756
.168056
32
33
.192519
.190158
.187597
.184803
.181768
.178472
.174904
.171042
33
34
.196110
.193636
.190948
.188028
.184851
.181412
.177691
.173673; 34
35
1.199368
1.196774
1.193966
1.190912
1.187599
1.184017
1.180148
1.175973
35
36
.202308
.199601
.196666
.193483
.190035
.186313
.182297
.177971
36
37
.204958
.202131
.199075
.195764
.192183
.188323
.184163
.179690
37
38
.207330
.204389
.201212
.197775
.194065
.190068
.185770
.181162
38
39
.209449
.206393
.203096
.199538
.195698
.191571
.187146
.182407
39
40
1.211329
1.208160
1.204749
1.201068
1.197106
1.192859
1.188311
1.183447
40
41
.212988
.209710
.206183
.202387
.198311
.193949
.189283
.184307
41
42
.214443
.211056
.207420
.203517
.199333
.194860
.190088
.185011
42
43
.215707
.212218
.208481
.204475
.200186
.195613
.190746
.185576
43
44
.216799
.213213
.209379
.205275
.200892
.196229
.191275
.186022
44
34
35
36
37
38
39
40
41
Continued on Page *68. 65
TABLE XL V.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
41 PER CENT.
Pay-
ments.
42
43
44
45
46
47
48
49
Pay-
ments.
n.
1.181397
1.175290
1.168905
1.162240
1.155279
1.148013
1.140432
1.132524
11.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.289449
.289373
.289286
.289192
.289088
.288973
.288847
.288710
2
3
.454015
.453858
.453681
.453487
.453275
.453040
.452784
.452501
3
4
.507483
.567240
.566968
.566672
.566344
.565986
.565592
.565158
4
5
0.652973
0.652640
0.652271
0.651865
0.651421
0.650931
0. 650393
0.649804
5
6
.720784
.720358
.719885
.719369
.718799
.718173
.717488
.716733
6
7
.776409
.775885
.775307
.774671
.773972
.773206
.772363
.771439
7
8
.823124
.822501
.821809
.821051
.820219
.819303
.818300
.817198
8
9
.863047
.862316
.861507
.860623
.859649
.858581
.857409
.856121
9
10
0.897616
0.890773
0.895845
0.894825
0.893707
0.892480
0.891131
0.889653
10
11
.927859
.926903
.925846
.924690
.923421
.922025
.920497
.918819
11
12
.954538
.953460
.952274
.950975
.949545
.947980
.946262
.944379
12
13
.978226
.977025
.975703
.974251
.972661
.970916
.969003
.966906
13
14
.999373
.998042
.996575
.994972
.993212
.991281
.989166
.986851
14
15
1.018330
1.016862
1.015252
1.013488
1-011551
1.009429
1.007108
1.004566
15
16
.035381
.033778
.032014
.030082
.027964
.025648
.023112
.020338
16
17
.050764
.049016
.047094
.044990
.042689
.040168
.037413
.034397
17
18
.064668
.062770
.060685
.058408
.055912
.053185
.050201
.046944
18
19
.077254
.075201
.072953
.070491
.067800
.064857
.061645
.058136
19
20
1.088658
1.086451
1.084027
1.081380
1.078486
1.075327
1.071877
1.068119
20
21
.099001
.096628
.094029
.091190
.088092
.084708
.081023
.077005
21
22
.108377
.105838
.103057
.100025
.096714
.093108
.089178
.084904
22
23
.116876
.114165
.111201
.107969
.104448
.100610
.096439
.091903
23
24
.124573
.121688
.118536
.115104
.111365
.107301
.102882
.098085
24
25
1.131535
1.128473
1.125132
1.121495
1.117542
1.113243
1.108579
1.103526
25
26
.137823
.134581
.131047
.127208
.123033
.118504
.113598
.108290
26
27
.143489
.140065
.136340
.132291
.127900
.123143
.117998
.112438
27
28
.148581
.144978
.141054
.136802
.132195
.127213
.121831
.116027
28
29
.153147
.149356
.145240
.140784
.135967
.130761
.125150
.119106
29
30
1.157219
1.153246
1.148938
1.144285
1.139256
1.133836
1.127999
1.121733
30
31
.160839
.156685
.152192
.147339
.142108
.136476
.130431
.123949
31
32
.164042
.159713
.155031
.149988
.144558
.138731
.132484
.125799
32 •
33
.166863
.162356
.157496
.152266
.146652
.140635
.134197
.127326
33
34
.169327
.164652
.159615
.154212
.148421
.142224
.135612
.128572
34
35
1.171468
1.166626
1.161427
1.155857
1.149897
1.143537
1.136767
1.129572
35
36
.173309
.168315
.162959
.157230
.151117
.144609
.137694
.130360
36
37
.174885
.169743
.164237
.158365
| .152114
.145471
.138425
.130975
37
38
.176217
.170935
.165295
.159294
.152915
.146149
.138995
.131442
38
39
.177330
.171922
.166160
.160039
.153545
.146678
.139428
.131789
39
40
1.178251
1.172728
1.166854
1.160625
1.154037
1.147080
1.139750
1.132039
40
41
.179004
.173376
.167401
.161083
.154411
.147380
.139982
.132219
41
42
.179609
.173886
.167827
.161432
.154690
.147595
.140149
.132342
42
43
. 180080
.174284
.168153
.161691
.154890
.147750
.140263
.132419
43
44
.180457
.174588
.168394
.161877
.155034
.147856
.140335
.132467
44
42
43
44
45
46
47
48
49
Continued on Page *67. 66
TABLE XLV.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
Pay-
ments.
II
Or
O
51
52
53
54
55
56
57
Pay-
ments.
n.
1.124274
1.115680
1.106723
1.097392
1.087676
1.077558
1.067037
1.056095
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.288556
.288390
.288206
.288005
.287785
.287538
.287272
.286974
2
3
.452188
.451848
.451473
.451062
.450610
.450110
.449562
.448958
3
4
.564681
.564158
.563584
.562954
.562262
.561495
.560657
.559736
4
5
0.649153
0.648441
0.647660
0.646801
0.645857
0.644815
0.643677
0.642425
5
6
.715902
.714995
.713995
.712899
.711694
.710368
.708916
.707320
6
7
.770422
.769307
.768083
.766740
.765266
.763642
.761866
.759916
7
8
.815985
.814658
.813200
.811601
.809846
.807916
.805805
.803489
8
9
.854707
.853158
.851458
.849594
.847549
.845302
.842846
.840152
9
10
0.888028
0.886250
0.884299
0.882161
0.879816
0.877243
0.874430
0.871348
10
11
.916977
.914961
.912751
.910330
.907677
.904767
.901588
.898107
11
12
.942311
.940051
.937573
.934860
.931889
.928634
.925079
.921195
12
13
.964608
.962094
.959341
.956327
.953031
.949422
.945488
.941187
13
14
.984313
.981539
.978501
.975182
.971551
.967585
.963259
.958538
14
15
1.001782
0.998739
0.995413
0.991777
0.987811
0.983476
0.978755
0.973610
15
16
.017299
1.013984
1.010359
1.006408
1.002094
.997388
.992268
.986694
16
17
.031102
.027505
.023584
.019304
.014642
1.009560
1.004038
.998040
17
18
.043382
.039506
.035274
.030668
.025652
.020193
.014273
1.007849
18
19
.054311
.050142
.045603
.040663
.035293
.029461
.023143
.016300
19
20
1.064019
1.059560
1.054708
1.049435
1.043715
1.037510
1.030799
1.023545
20
21
.072632
.067879
.062713
.057113
.051043
.044470
.037375
.029718
21
22
.080253
.075205
.069732
.063803
.057388
.050456
.042984
.034942
22
23
.086975
.081638
.075856
.069604
.062853
.055568
.047738
.039328
23
24
.092885
.087257
.081173
.074606
.067525
.059906
.051733
.042973
24
25
1.098053
1.092142
1.085762
1.078886
1.071493
1.063554
1.055055
1.04597]
25
26
.102550
.096362
.089692
.082525
.074833
.066590
.057791
.048407
26
27
.106438
.099978
.093036
.085590
.077614
.069092
.060014
.050357
27
28
.109772
.103058
.095855
.088143
.079907
.071126
.061794
.051896
28
29
.112614
.105655
.098204
.090249
.081772
.072756
.063200
.053089
29
30
1.115011
1.107820
1.100142
1.091963
1.083268
1.074042
1.064290
1.053994
30
31
.117011
.109607
.101720
.093338
.084448
.075041
.065117
.054670
31
32
.118661
.111063
.102987
.094423
.085365
.075798
.065735
.055158
32
33
.120007
.112232
.103986
.095266
.086060
.076364
.066181
.055506
33
34
.121087
.113154
.104763
.095906
.086579
.076773
.066499
.055742
34
35
1.121939
1.113871
1.105353
1.096383
1.086955
1.077064
1.066714
1.055891
35
36
.122602
.114416
.105792
.096729
.087222
.077261
.066851
.055985
36
37
.123106
.114821
.106111
.096975
.087403
.077387
.066937
.056038
37
38
.123481
.115115
.106338
.097141
.087519
.077466
.066986
.056069
38
39
.123752
.115325
.106491
.097248
.087591
.077511
.067014
.056082
39
40
1.123946
1.115467
1.106589
1.097314
1.087633
1.077536
1.067026
1.056091
40
41
.124078
.115557
.106650
.097353
.087656
.077547
.067034
.056094
41
42
.124161
.115613
.106686
.097374
.087666
.077555
.067037
.056094
42
43
.124213
.115646
.106706
.097383
.087673
.077557
.067037
.056095
43
44
45
46
47
48
49
.124244
1.124261
.124268
.124273
.124274
.124274
.115665
1.115672
.115678
.115679
.115679
.106714
1.106720
.106721
.106722
.097389
1.097390
.097391
.087675
1.087675
.077557
1.077558
.067037
44
45
46
47
48
49
50
51
52
53
54
55
56
57
4-1 PER CENT. TABLE XLY.
67
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. A x
'ivf PER CENT.
Pay-
ments.
58
59
60
61
62
63
64
65
Pay-
ments.
11.
1.044728
1.032915
1.020646
1.007916
0.994716
0.981025
0.966846
0.952166
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.286652
.286298
.285906
.285476
.285008
.284488
.283919
.283293
2
3
.448302
.447578
.446779
.445905
.444948
.443891
.442734
.441463
3
4
.558730
.557622
.556403
.555067
.553605
.551993
.550229
.548291
4
5
0.641055
0.639550
0.637896
0.636084
0.634099
0.631916
0.629525
0.626906
5
6
.705576
.703660
.701557
.699252
.696730
.693958
.690928
.687611
6
7
.757784
.755444
.752876
. 750067i
.746991
.743619
.739934
.735903
7
8
.800957
.798179
.795136
.7918061
.788167
.784181
.779828
.775073
8
9
.837207
.833982
.830450
.826591
.822376
.817764
.812734
.807248
9
10
0.867981
0.864297
0.860267
0.855867,
0.851067
0.845822
0.840110
0.833891
10
11
.894310
.890158
.885620
.880672
.875279
.869397
.863002
.856049
11
12
.916957
.912327
.907277
.901773!
.895785
.889266
.882188
.874509
12
13
.936501
.931391
.925819
.919761
.913178
.906024
.898272
.889876
13
14
.953401
.947802
.941713
.935098
.927923
.920143
.911726
.902632
14
15
0.968015
0.961932
0.955322
0.948156
0.940400
0.932002
0.922939
0.913173
15
16
.980646
.974076
.966953
.959245
.950914
.941918
.932235
.921823
16
17
.991536
.984488
.976859
.968617
.959731
.950161
.939883
.928862
17
18
1.000898
.993377
.985252
.976495
.967078
.956958
.946119
.934533
18
19
.008908
1.000926
.992321
.983072
.973146
.962510
.951152
.939045
19
20
1.015723
1.007294
0.998233
0.988513
0.978111
0.966998
0.955160
0.942585
20
21
.021481
.012629
1.003131
.992971
.982129
.970575
.958310
.945317
21
22
.026311
.017054
.007148
.996582
.985335
.973388
.960742
.947387
22
23
.030321
.020686
.010405
.999466
.987858
.975563
.962586
.948926
23
24
.033616
.023634
.013009
1.001737
.989810
.977212
.963958
.950040
24
25
1.036292
1.025991
1.015060
1.003495
0.991290
0.978440
0.964950
0.950829
25
26
.038433
.027859
.016649
.004829
.992393
.979328
.965654
.951364
26
27
.040122
.029290
.017854
.005822
.993191
.979957
.966131
.951703
27
28
.041431
.030382
.018752
.006542
.993756
.980385
.966433
.951914
28
29
.042424
.031197
.019403
.007051
.994140
.980655
.966622
.952037
29
30
1.043165
1.031787
1.019863
1.007398
0.994383
0.980825
0.966731
0.952105
30
31
.043701
.032205
.020178
.007616
.994536
.980922
.966792
.952138
31
32
.044082
.032489
.020375
.007754
.994623
.980977
.966821
.952155
32
33
.044340
.032669
.020500
.007833
.994672
.981003
.966836
.952163
33
34
.044504
.032782
.020571
.007877
.994696
.981016
.966843
.952165
34
35
1.044607
1.032847
1.020611
1.007899
0.994707
0.981023
0.966845
0.952166
35
36
.044666
.032883
.020630
.007909
.994714
.981024
.966846
37
.044699
.032900
.020640
.007915
.994714
.981025
43
42
38
.044714
.032909
.020645
.007915
.994716
44
1.181397
58
39
.044723
.032914
.020645
.007916
45
1.175290
.181395
57
40
1.044727
1.032914
1.020646
46
1.168905
.175289
1 7P\‘>8Q
.181395
1813Q5
56
55
41
42
.044/2/
.044728
. JId
47
1.155279
1 U/V/v-TrU
1.162239
1.168905
1.175288
1.181393
54
A Ck
48
1.148013
.155278
.162239
.168904
.175286
.181389
53
1.140432
.148012
.155278
.162238
.168902
.175282
.181380
52
51
1.132524
.140431
.148012
.155277
.162235
.168897
.175272
.181362
51
50
.132523
.140431
.148011
.155273
.162230
.168886
.175253
.181334
50
49
1.132523
1.140430
1.148007
1.155268
1.162218
1.168866
1.175223
1.181287
49
48
.132522
.140426
.148002
.155255
.162197
.168834
.175173
.181211
48
47
.132518
.140420
.147988
.155233
.162162
.168780
.175091
.181102
47
46
.132511
.140405
.147964
.155195
.162104
.168692
.174975
.180951
46
45
.132495
.140379
.147923
.155133
.162011
.168568
.174813
.180741
45
49
48
47
45
44
43
42
Continued from Page *65. 68
TABLE XLV.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. pER
Pay-
CO
CD
II
67
68
69
70
71
72
73
Pay-
ments.
n.
0.936983
0.921279
0.905066
0.888311
0.871044
0.853238
0.834905
0.816036
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.282611
.281852
.281035
.280120
.279130
.278037
.276843
.275538
2
3
.440070
.438541
.436872
.435027
.433019
.430811
.428403
.425760
3
4
.546174
.543847
.541309
.538512
.535465
.532128
.528482
.524492
4
5
0.624044
0.620902
0.617476
0.613710
0.609614
0.605131
0.600238
0.594904
5
6
.683985
.680015
.675686
.670944
.665786
.660148
.654017
.647340
6
7
.731504
.726692
.721455
.715727
.709502
.702722
.695357
.687355
7
8
.769890
.764233
.758081
.751367
.744086
.736172
.727594
.718300
8
9
.801278
.794771
.787707
.780017
.771693
.762667
.752909
.742371
9
10
0.827131
0.819779
0.811813
0.803163
0.793818
0.783714
0.772824
0.761101
10
11
.848506
.840319
.831469
.821880
.811550
.800415
.788450
.775614
11
12
.866193
.857190
.847477
.836986
.825715
.813603
.800634
.786774
12
13
.880805
.871009
.860468
.849118
.836958
.823938
.810047
.795255
13
14
.892832
.882275
.870947
.858787
.845804
.831952
.817225
.801609
14
15
0.902672
0.891393
0.879328
0.866419
0.852685
0.838081
0.822618
0.806288
15
16
.910659
.898708
.885960
.872370
.857958
.842695
.826596
.809660
ro
17
.917083
.904508
.891141
.876940
.861935
.846104
.829467
.812031
17
18
.922185
.909047
.895127
.880391
.864877
.848567
.831487
.813631
18
19
.926183
.912543
.898140
.882946
.867004
.850301
.832852
.814656
19
20
0.929266
0.915188
0.900372
0.884796
0.868504
0.851473
0.833726
0.815284
20
21
.931600
.917150
.901990
.886101
.869518
.852224
.834262
.815650
21
22
.933334
.918572
.903132
.886983
.870167
.852685
.834575
.815852
22
23
.934590
.919577
.903904
.887547
.870566
.852955
.834747
.815954
23
24
.935478
.920257
.904397
.887895
.870799
.853102
.834834
.816005
24
25
0.936080
0.920690
0.904702
0.888097
0.870927
0.853177
0.834877
0.816025
25
26
.936462
.920959
.904879
.888209
.870991
.853213
.834895
.816034
26
27
.936700
.921115
.904977
.888264
.871023
.853229
.834902
.816036
27
28
29
.936838
.936914
.921201
.921243
.905025
.905050
.888293
.888304
.871036
.871042
.853235
.853238
.834905
34
35
30
31
0.936951
0.921265
.921274
0.905060
0.888309
0.871044
36
1.221612
66
.936971
.936979
.905065
.905066
.8883111
1.217341
.221612
65
37
32
.921278
38
1.212877
1.217340
1.221612
64
33
.936982
.936983
.921279
39
1.208201
.212875
.217340
.221611
63
34
40
1.203309
.208199
.212875
.217339
.221610
62
41
1.198189
.203304
.208199
.212875
.217338
.217336
1.217329
.221608
61
1.192839
.198187
1.198187
.203304
.208199
.212874
.221601
60
59
1.187248
1.192837
1.203304
1.208198
1.212872
1.221591
59
58
.187246
.192837
.198187
.203303
.208196
.212864
.217318
.221575
58
57
.187246
.192837
.198186
.203301
.208188
.212852
.217301
.221547
57
56
.187246
.192835
.198184
.203292
.208175
.212834
.217271
.221502
56
55
.187244
.192833
.198175
.203278
.208156
.212802
.217223
.221437
55
54
1.187241
1.192824
1.198160
1.203258
1.208122
1.212751
1.217154
1.221346
54
53
.187232
.192808
.198138
.203222
.208068
.212677
.217058
.221221
53
52
.187215
.192784
.198099
.203164
.207989
.212575
.216925
.221051
52
51
.187189
.192743
.198037
.203080
.207880
.212434
.216744
.220829
51
50
.187145
.192677
.197948
.202964
.207730
.212241
.216508
.220545
50
49
1.187074
1.192581
1.197824
1.202804
1.207524
1.211989
1.216205
1.220184
49
48
.186972
.192449
.197653
.202584
.207256
.211667
.215821
.219736
48
47
.186831
.192266
.197419
.202299
.206913
.211259
.215345
.219187
47
46
.186635
.192017
.197115
.201933
.206478
.210751
.214761
.218526
46
45
.186369
.191692
.196724
.201469
.205937
.210130
.214056
.217733
45
41
40
39
38
! 37
36
35
34
Continued from Page *64. TABLE XL Y.
69
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. „ t
PER CENT.
Pay-
ments.
74
75
76
77
78
79
80
81
Pay-
ments.
n.
n.
0.796626
0.776711
jO. 756264
0.735279
0.713797
0.691797
0.669283
0. 646255
1
0.000000
0.000000
'o. 000000
0.000000
0.000000
0.000000
0.000000
O.OOOOCO
1
2
.274089
.272521
.270807
.268910
.266851
.264595
.262129
.259447
2
3
.422855
.419707
.416256
.412470
.408356
.403865
.398975
.393652
3
4
.520129
.515387
.510208
.504550
.498410
.491733
.484479
.476622
4
5
0.589077
0.582756
0.575872
0.568376
0.560266
0.551472
0.541963
0.531696
5
6
.640087
.632192
.623643
.614371
.604364
.593563
.581926
.569428
6
7
.678666
.669282
.659130
.648157
.636363
.623682
.610091
.595572
7
8
.708241
.697409
.685732
.673164
.659708
.645312
.629962
.613646
8
9
.731003
.718801
.705699
.691659
.676693
.660765
.643866
.626012
9
10
0.748499
0.735023
0.720614
0.705242
0.688938
0.671670
0.653455
0.634186
10
11
.761871
.747230
.731645
.715100
.697634
.679234
.659827
.639362
11
12
.771990
.756308
.739691
.722134
.703690
.684280
.663875
.642509
12
13
.779547
.762954
.745453
.727048
.707743
.687494
.666342
.614336
13
14
.785096
.767728
.749487
.730344
.710329
.689456
.667776
.645339
14
15
0.789090
0.771075
0.752198
0.732450
0.711910
0.690597
0.668563
0.645853
15
16
.791895
.773328
.753933
.733738
.712829
.691223
.668967
.646101
16
17
.793785
.774771
.754994
.734488
.713334
.691545
.669162
.646203
17
18
.794996
.775654
.755612
.734900
.713594
.691701
.669242
.646243
18
19
.795738
.776169
.755952
.735112
.713719
.691764
.669274
.646255
19
20
21
22
23
24
0.796170
.796408
.796529
.796589
.796613
0.776452
.776596
.776668
.776696
.776708
0.756126
.756211
.756245
.756259
0.735215
.735256
.735273
.735279
0.713770
.713791
.713797
0.691790
.691797
0.669283
26
27
28
1.249527
74
29
1.246580
.249527
73
.756264
30
1.243489
.246580
.249527
72
25
0.796623
0.776711
31
1.240250
.243489
.246580
.246580
.249527
.249526
71
26
.796626
1.236862
.240249
.243489
70
32
33
1.233310
1.236861
1.240249
1.243489
1.246579
.246577
.246572
1.249524
69
1.229589
.233309
.236861
.240249
.243488
.249519
.249514
68
67
1.225692
.229589
.233309
.236861
.240248
.243486
67
66
.225692
.229589
.233309
.236860
.240246
.243481
.246566
.249502
66
65
.225692
.229588
.233308
.236858
.240241
.243474
.246554
.249485
65
64
1.225691
1.229587
1.233306
1.236853
1.240233
1.243462
1.246536
1.249459
64
63
.225690
.229585
.233300
.236845
.240221
.243443
.246508
.249418
63
62
.225688
.229579
.233292
.236832
.240201
.243413
.246465
.249363
62
61
.225682
.229570
.233278
.236810
.240169
.243368
.246406
.249285
61
60
.225672
.229556
.233255
.236776
.240121
.243305
.246323
.249178
60
59
1.225657
1.229531
1.233218
1.236725
1.240054
1.243217
1.246211
1.249039
59
58
.225631
.229492
.233164
.236654
.239961
.243098
.246063
.248863
58
57
.225589
.229434
.233089
.236556
.239835
.242942
.245877
.248641
57
56
.225528
.229354
.232985
.236422
.239669
.242745
.245641
.248361
56
55
.225442
.229244
.232843
.236246
.239460
.242494
.245345
.248022
55
54
1.225325
1.229093
1.232656
1.236024
1.239194
1.242181
1.244985
1.247613
54
53
.225165
.228895
.232420
.235741
.238862
.241799
.244552
.247121
53
52
.224955
.228644
.232119
.235389
.238456
.241339
.244030
.246546
52
51
.224688
.228325
.231746
.234958
.237969
.240786
.243420
.245868
51
50
.224349
.227928
.231289
.234441
.237382
.240139
.242702
.245088
50
49
1.223928
1.227443
1.230740
1.233819
1.236695
1.239378
1.241875
1.244192
49
48
.223412
.226859
.230079
.233089
.235888
.238500
.240924
.243168
48
47
.222790
.226157
.229303
.232233
.234956
.237491
.239839
.242011
47
46
.222045
.225332
.228394
.231243
.233886
.236340
.238612
.240712
46
45
.221167
.224366
.227343
.230107
.232664
.235038
.237234
.239258
45
33
| 32
31
30
29
28
27
26
Continued from Page *63. 70
TABLE XLY.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. a* DU15
PER CENT.
Pay-
ments.
(S*
CO
II
5*
83
84
85
86
87
88
89
Pay-
ments.
n.
0.022676
0.598565
0.573780 0.548297
0.521852
0.493999
0.464345
0.431342
n.
1
0.000000
0.000000
0.000000 0.000000
0.000000
0.000000
0.000000
0.000000
2
.256494
.253316
.249806
.246024
.241946
.237388
.232632
.224517
2
3
.387854
.381584
.374746
.367409
.359446
.350786
.340091
.324725
3
4
.468093
.458911
.448980
.438319
.426874
.413580
.397349
.376450
4
5
0.520618
0.508753
0.495976
0.482365
0.467218
0.449629
0.429089
0.404031
5
6
.556023
.541721
.526442
.509831
.491308
.470362
.446604
.418493
6
7
.580075
.563652
.545916
.526616
.505459
.482025
.455950
.425749
7
8
.596346
.577893
.557991
.536606
.513513
.488313
.460680
.429178
8
9
.607023
.586809
.565240
.542334
.517883
.491513
.462925
.430618
9
10
0.613752
0.592192
0.569416
0.545455
0.520115
0.493035
0.463870
0.431167
10
11
.617831
.595305
.571699
.547053
.521178
.493677
.464230
.431342
11
12
.620196
.597009
.572869
.547814
.521627
.493921
.464345
13
.621492
.597883
.573427
.548135
.521798
.493999
19
18
14
.622157
• 598301
.573662
.548258
.521852
20
1.268708
82
15
0.622476
0.598476
0.573752
0.548297
o 1
1.266691
.268709
81
16
.622609
.598544
.573780
1.264573
.266692
.268709
80
17
.622661
.598565
23
22
4.262353
1.264574
1.266692
1.268709
79
18
.622676
24
1.260026
.262354
.264574
.266692
.268708
78
25
1.257585
. .260027
.262354
.264574
.266691
.268706
77
1.255025
.257585
.260027
.262354
.264573
.266689
.268703
76
75
1.252341
.255025
.257585
.260027
.262353
.264571
.266686
.268699
75
74
1.252342
1.255025
1.257585
1.260026
11.262351
1.264567
1.266682
1.268692
74
73
.252342
.255025
.257584
.260024
.262347
.264563
.266674
.268681
73
72
.252342
.255024
.257582
.260020
.262343
.264555
.266663
.268665
72
71
.252341
.255022
.257578
.260016
.262334
.264543
.266646
.268639
71
70
.252339
.255017
.257574
.260006
.262322
.264525
.266618
.268603
70
69
1.252334
1.255013
1.257563
1.259993
1.262302
1.264496
1.266580
1.268554
69
68
.252329
.255002
.257549
.259972
.262271
.264456
.266528
.268487
68
67
.252318
.254987
.257527
.259910
.262229
.264400
.266457
.268398
67
66
.252302
.254964
.257493
.259896
.262170
.264325
.266363
.268285
66
65
.252277
.254928
.257447
.259833
.262091
.264226
.266244
.268144
65
64
1.252239
1.254879
1.257381
1.259749
1.261986
1.264100
1.266095
1.267966
64
63
.252187
.254809
.257291
.259638
1 .261853
.263943
.265907
.267749
63
62
.252113
.254714
.257174
.259498
.261687
.263744
.265677
.267490
62
61
.252012
.254590
.257026
.259322
.261477
.263501
.265404
.267178
61
60
.251881
.254433
.256839
.259100
| .261220
.263212
.265074
.266812
60
59
1.251715
1.254235
1.256604
1.258828
1.260914
1.262863
1.264687
1.266381
59
58
.251505
.253986
.256317
.258504
.260545
.262454
.264231
.265885
58
57
.251241
.253683
.255973
.258113
.260112
.261972
.263707
.265319
57
56
.250921
.253319
.255560
.257655
.259602
.261418
.263108
.264670
56
55
.250535
.252881
.255075
.257116
.259016
.260784
.262422
.263936
55
54
1.250071
1.252368
1.254505
1.256496
1.258345
1.260058
1.261646
1.263114
54
53
.249528
.251764
.253848
.255785
.257577
.259238
.260776
.262195
53
52
.248888
.251068
.253095
.254972
.256709
.258317
.259805
.261172
52
51
.248152
.250271
.252235
.254054
.255736
.257290
.258722
.260045
51
50
.247307
.249360
.251262
.253023
.254648
.256143
.257529
.258799
50
49
1.246341
1.248329
1.250171
1.251871
1.253434
1.254881
1.256211
1.257433
49
48
.245249
.247174
.248950
.250586
.252097
.253486
.254765
.255935
48
47
.244024
.245880
.247589
.249170
.250620
.251955
.253180
.254304
47
46
.242653
.244437
.246089
.247605
.248998
.250276
.251452
.252529
46
45
.241122
.242847
.244430
.245886
.247220
.248445
.249572
.250601
45
25
24
23
22
21
20
19
18
Continued from Page *62. TABLE XL V.
71
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. , pER CENT
1 Pay-
ments.
n.
90
91
92
93 ,
94
95
96
97
Pay- 1
ments.|
0.39986G
0.371143
0.342412
0.312760
0.282598
0.251018
0.213605
0.177500
11.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.215615
.207859
.199751
.190399
.180240
.169812
.152954
.140751
2
3
.309111
.355648
.294974
.279967
.263229
.294182
.245642
.270547
.225430
.244967
.199694
.213605
.177500
3
4
.336370
.316098
10
5
0.379276
0.356220
0.332378
0.306615
0.279714
0.251018
11
6
.390948
.396423
.365406
.339071
.341606
.311281
.282598
1.281791
90
.312760
12
7
.369226
13
1.280417
1.281791
.281791
89
8
9
.398714
.399587
.370680
.371143
.342412
14
1.278977
.280417
88
15
1.277464
.278977
.280417
.281791
87
10
0.399866
16
1.275875
.277464
.278977
.280417
.281790
86
85
1.274211
.275876
.277464
.278977
.280416
.281789
17
1.272467
1.274213
1.275876
1.277464
1.278976
1.280415
1.281787
84
83
1.270632
.272468
.274213
.275876
.277463
.278975
.280413
.281785
83
82
.270633
.272468
.274213
.275875
.277462
.278973
.280411
.281780
82
81
.270633
.272468
.274212
.275873
.277460
.278971
.280405
.281774
81
80
.270633
.272467
.274210
.275871
.277458
.278965
.280399
.281761
80
79
1.270632
1.272465
1.274208
1.275869
1.277452
1.278958
1.280385
1.281745
79
78
.270630
.272463
.274205
.275863
.277444
.278944
.280369
.281723
78
77
.270628
.272460
.274199
.275854
.277430
.278927
.280345
.281691
77
76
.270624
.272453
.274190
.275840
.277412
.278901
.280312
.281648
76
75
.270617
.272444
.274175
.275820
.277384
.278866
.280267
.281591
75
74
1.270607
1.272428
1.274154
1.275791
1.277348
1.278819
1.280206
1.281518
74
73
.270591
.272406
.274123
.275752
.277297
.278754
.280128
.281426
73
72
.270567
.272373
.274081
.275699
.277229
.278672
.280032
.281310
72
71
.270533
.272329
.274025
.275627
.277142
.278571
.279908
.281170
71
70
.270486
.272270
.273950
.275535
.277036
.278440
.279762
.281003
70
69
1.270423
1.272190
1.273852
1.275423
1.276898
1.278286
1.279585
1.280801
69
68
.270339
.272087
.273734
.275278
.276735
.278099
.279373
.280563
68
67
.270231
.271962
.273581
.275106
.276537
.277875
.279122
.280287
67
66
.270098
.271801
.273399
.274897
.276301
.277610
.278829
.279966
66
65
.269929
.271608
.273179
.274647
.276022
.277301
.278491
.279599
65
64
1.269724
1.271376
1.272915
1.274352
1.275695
1.276945
1.278105
1.279181
64
63
.269479
.271097
.272604
.274007
.275319
.276536
.277662
.278707
63
62
.269184
.270769
.272239
.273610
.274888
.276069
.277162
.278178
62
61
.268838
.270383
.271819
.273156
.274394
.275542
.276604
.277587
61
60
.268430
.269939
.271340
.272634
.273838
.274952
.275980
.276928
60
59
1.267961
1.269433
1.270789
1.272046
1.273215
1.274293
1.275284
1.276203
59
58
.267426
.268851
.270167
.271388
.272519
.273558
.274519
.275403
58
57
.266811
.268194
.269472
.270652
.271742
.272750
.273674
.274526
57
56
.266117
.267459
.268694
.269832
.270889
.271857
.272748
.273566
56
55
.265340
.266637
.267828
.268931
.269945
.270880
.271734
.272521
55
54
1.264471
1.265722
1.266876
1.267934
1.268913
1.269809
1.270629
1.271387
54
53
.263503
.264715
.265822
.266843
.267781
.268641
.269432
.270157
53
52
.262438
.263601
.264668
.265646
.266547
.267376
.268133
.268827
52
51
.261260
.262380
.263403
.264342
.265210
.266003
.266726
.267391
51
50
.259969
.261043
.262024
.262928
.2G3758
.264516
.265208
.265845
50
49
1.258554
1.259585
1.260528
1.261392
1.262186
1.262911
1.263574
1.264181
49
48
.257012
.258002
.258903
.259730
.260489
.261183
.261815
.262395
48
47
.255336
.256282
.257145
.257934
.258660
.259322
.259924
.260481
47
46
.253516
.254422
.255244
.255999
.256691
.257322
.257901
.258429
46
45
.251546
.252409
.253196
.253915
.254575
.255181
.255729
.256233
45
17
16
15
14
13
12
11
10
Continued from Page *61. 72
TABLE XL VI.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
The first Value under the Age X is A(1 (lx) for the whole Life.
5 PER CENT.
Pay-
x= 10
11
12
13
14
15
16
17
Pay-
n.
1.249190
1.248018
1.246787
1.245487
1.244118
1.242680
1.241167
1.239571
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.289190
.289185
.289182
.289178
.289171
.289165
.289161
.289154
2
3
.453548
.453539
.453534
.453523
.453510
.453500
.453490
.453475
3
4
.566862
.566850
.566838
.566823
.566805
.566789
.566773
.566751
4
5
0.652256
0.652239
0. 652223
0.652202
0.652179
0.652157
0.652133
0.652104
5
6
.720028
.720006
.719985
.719959
.719930
.719901
.719871
.719834
6
7
.775673
.775646
.775620
.775588
.775553
.775518
.775480
.775435
7
8
.822470
.822438
.822407
.822370
.822328
.822285
.822240
.822186
8
9
.862534
.862497
.862461
.862417
.862368
.862319
.862265
.862201
9
10
0.897309
0.897268
0.897225
0.897175
0.897120
0.897062
0.896999
0.896928
10
11
.927826
.927779
.927730
.927674
.927611
.927544
.927474
.927390
11
12
.954845
.954791
.954737
.954673
.954601
.954528
.954446
.954353
12
13
.978941
.978883
.978821
.978749
.978671
.978587
.978496
.978394
13
14
1.000565
L.000500
1.000431
1.000353
1.000264
1.000172
1.000073
.999957
14
15
1.020071
1.019999
1.019924
1.019835
1.019740
1.019639
1.019527
1.019401
15
16
.037746
.037668
.037582
.037488
.037384
.037271
.037149
.037010
16
17
.053824
.053737
.053646
.053543
.053427
.053305
.053171
.053022
17
18
.068498
.068405
.068306
.068192
.068067
.067933
.067790
.067625
18
19
.081933
.081831
.081722
.081600
.081463
.081322
.081163
.080983
19
20
1.094262
1.094152
1.094035
1.093901
1.093758
1.093601
1.093428
1.093237
20
21
.105604
.105486
.105358
.105218
.105061
.104890
.104707
.104497
21
22
.116060
.115932
.115798
.115644
.115474
.115293
.115092
.114870
22
23
.125716
.125581
.125434
.125268
.125088
.124891
.124678
.124439
23
24
.134649
.134503
.134344
.134169
.133974
.133765
.133537
.133281
24
25
1.142924
1.142766
1.142598
1.142409
1.142202
1.141980
1.141734
1.141460
25
26
.150597
.150431
.150250
.150049
.149830
.149590
.149329
.149037
26
27
.157725
.157546
.157355
.157141
.156905
.156651
.156367
.156061
27
28
.164349
.164160
.163956
.163728
.163478
.163203
.162911
.162579
28
29
.170513
.170312
.170095
.169853
.169584
.169300
.168983
.168631
29
30
1.176252
1.176039
1.175808
1.175549
1.175270
1.174964
1.174628
1.174257
30
31
.181599
.181373
.181127
.180859
.180559
.180234
.179880
.179483
31
32
.186585
.186344
.186089
.185800
.185483
.185141
.184762
.184343
32
33
.191235
.190985
.190710
.190405
.190072
.189706
.189307
.188867
33
34
.195578
.195310
.195019
.194699
.194344
.193958
.193539
.193069
34
35
1.199628
1.199345
1.199041
1.198700
1.198325
1.197921
1.197473
1.196983
35
36
.203409
.203114
.202790
.202430|
.202038
.201606
.201140
.200619
36
37
.206942
.206629
.206287
.205911
.205493
.205044
.204548
.204001
37
38
.210240
.209909
.209552
.209152
.208717
.208240
.207720
.207147
38
39
.213319
.212974
.212594
.212177
.211716
.211217
.210671
.210067
39
40
1.216197
1.215830
1.215434
1.214992
1.214511
1.213987
1.213412
1.212777
40
41
.218880
.218498
.218078
.217618
.217112
.216561
.215957
.215294
41
42
.221388
.220983
.220546
.220061
.219530
.218951
.218322
.217623
42
43
.223724
.223303
.222842
.222334
.221777
.221174
.220510
.219779
43
44
.225906
.225462
.224979
.224447
.223867
.223231
.222537
.221772
44
10
11
12
13
14
15
16
17
Continued on Page 82.
Formula, A(l + = a a^=1.
Djc y/Ja;—1 TABLE XL VI.
73
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
5 PER CENT.
Pay-
ments.
18
19
20
21
22
23
24
25
Pay-
ments.
71.
1.237891
1.236125
1.234265
1.232309
1.230253
1.228089
1.225813
1.223420
n.
1
0.000000
0.000000
0.000000
0.000000
0. 000000
0.000000
0.000000
0.000000
1
2
.289146
.289139
.289130
.289120
.289110
.289099
.289085
.289072
2
3
.453460
.453445
.453426
.453406
.453386
.453362
.453336
.453307
3
4
.566728
.566704
.566675
.566645
.566613
.566577
.566536
.566492
4
5
0.652073
0.652039
0.652000
0.651959
0.651916
0.651865
0.651810
0.651750
5
6
.719794
.719751
.719702
.719650
.719592
.719528
.719458
.719380
6
7
.775386
.775333
.775274
.775207
.775137
.775059
.774972
.774877
7
8
.822127
.822065
.821991
.821914
.821831
.821736
.821633
.821520
8
9
.862134
.862058
.831975
.861885
.861786
.861677
.861555
.861424
9
10
0.896847
0.896762
0.896667
0.896562
0.896449
0.896322
0.896184
0.896030
10
11
.927301
.927205
.927095
.926976
.926847
.926704
.926545
.926371
11
12
.954254
.954144
.954021
.953887
.953742
.953579
.953401
.953207
12
13
.978281
.978159
.978021
.977873
.977709
.977528
.977330
.977111
13
14
.999833
.999697
.999545
.999379
.999198
.998998
.998777
.998536
14
15
1.019263
1.019114
1.018946
1.018763
1.018564
1.018342
1.018100
1.017833
15
16
.036861
.036695
.036511
.036311
.036091
.035850
.035583
.035291
16
17
.052856
.052675
.052475
.052255
.052017
.051752
.051461
.051143
17
18
.067444
.067249
.057029
.066793
.066532
.066244
.065929
.065581
18
19
.080790
.080575
.080340
.080082
.079800
.079489
.079145
.078770
19
20
1.093024
1.092796
1.092540
1.092262
1.091957
1.091619
1.091250
1.090842
20
21
.L04271
.104023
.103749
.103448
.103119
.102756
.102356
.101916
21
22
.114626
.114360
.114064
.113741
.113388
.112996
.112565
.112097
22
23
.124178
.123891
.123574
.123229
.122847
.122427
.121969
.121463
23
24
.133000
.132693
.132355
.131982
.131575
.131128
.130635
.130094
24
25
1.141161
1.140833
1.140469
1.140072
1.139639
1.139160
1.138632
1.138056
25
26
.148717
.148366
.147978
.147557
.147093
.146581
.146021
.145404
26
27
.155720
.155346
.154936
.154484
.153990
.153447
.152848
.152194
27
28
.162216
.161821
.161382
.160902
.160378
.159798
.159164
.158466
28
29
.168248
.167825
.167359
.166852
.166293
.165680
.165004
.164266
29
30
1.173847
1.173399
1.172908
1.172366
1.171776
1.171123
1.170410
1.169626
30
31
.179049
.178578
.178053
.177483
.176855
.176166
.175412
.174584
31
32
.183888
.183384
.182833
.182226
.181564
.180837
.180039
.179163
32
33
.188380
.187852
.187266
.186628
.185929
.185161
.184317
.183395
33
34
.192561
.191999
.191383
.190710
.189972
.189161
.188273
.187300
34
35
1.196442
1.195852
1.195204
1.194493
1.193715
1.192862
1.191926
1.190902
35
36
.200051
.199431
.198747
.197998
.197180
.196282
.195298
.194222
36
37
.203405
.202751
.202030
.201244
.200384
.199439
.198406
.197278
37
38
.206518
.205829
.205074
.204247
.203342
.202352
.201270
.200085
38
39
.209405
.208685
.207889
.207021
.206073
.205037
.203901
.202663
39
40
1.212086
1.211326
1.210492
1.209583
1.208592
1.207504
1.206318
1.205022
40
41
.214566
.213769
.212897
.211947
.210907
.209772
.208530
.207174
41
42
.216860
.216027
.215116
.214120
.213035
.211847
.210549
.209139
42
43
.218981
.218112
.217157
.216118
.214984
.213743
.212393
.210921
43
44
.220940
.220030
.219034
.217949
.216765
.215474
.214066
.212531
44
18
19
20
21
22
23
24
25
Continued on Page *81. 74
TABLE XLY1.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF U
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
5 PER CENT.
Pay-
x=26
27
28
29
30
31
33
33
Pay-
ments.
n.
1.220904
1.218261
1.215481
1.212561
1.209497
1.206277
1.202896
1.199343
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.289055
.289039
.289020
.288997
.288976
.288951
.288925
.288894
2
3
.453274
.453240
.453200
.453156
.453111
.453061
.453006
.452944
3
4
.560442
.566388
.566328
.566262
.566194
.566115
.566031
.565934
4
5
0.651681
0.651609
0.651528
0.651439
0.651343
0.651237
0.651120
0.650988
5
6
.719293
.719201
.719098
.718982
.718861
.718725
.718574
.718408
6
7
.774771
.774658
.774530
.774390
.774241
.774072
.773889
.773684
7
8
.821394
.821258
.821107
.820940
.820759
.820559
.820340
.820097
8
9
.861276
.861117
.860941
.860744
.860534
.860300
.860044
.859760
9
10
0.895861
0.895679
0.895474
0.895250
0.895007
0.894738
0.894444
0.894116
10
11
.926180
.925971
.925739
.925484
.925209
.924903
.924567
.924195
11
12
.952990
.952755
.952494
.952209
.951897
.951553
.951175
.950756
12
13
.976871
.976607
.976318
.975997
.975650
.975266
.974843
.974377
13
14
.998267
.997977
.997654
.997301
.996916
.996488
.996021
.995504
14
15
1.017539
1.017217
1.016863
1.016473
1.016046
1.015577
1.015062
1.014493
15
16
.034968
.034616
.034228
.033798
.033332
.032818
.032252
.031628
16
17
.050790
.050407
.049980
.049513
.049005
.048442
.047825
.047147
17
18
.065199
.064779
.064317
.063810
.063256
.062644
.061976
.061235
18
19
.078353
.077899
.077400
.076849
.076249
.075589
.074861
.074063
19
20
1.090392
1.089904
1.089363
1.088769
1.088122
1.087405
1.086624
1.085758
20
21
.101435
.100907
.100324
.099686
.098985
.098218
.097372
.096444
21
22
.111577
.111009
.110385
.109695
.108947
.108119
.107214
.106217
22
23
.120906
.120299
.119626
.118891
.118086
.117201
.116231
.115166
23
24
.129499
.128846
.128131
.127342
.126483
.125537
.124501
.123361
24
25
1.137418
1.136724
1.135958
1.135118
1.134201
1.133193
1.132087
1.130873
25
26
.144727
.143985
.143170
.142275
.141299
.140225
.139049
.137758
26
27
.151471
.150684
.149816
.148866
.147827
.146687
.145438
.144066
27
28
.157701
.156862
.155944
.154934
.153833
.152623
.151297
.149846
28
29
.163452
.162567
.161591
.160522
.159355
.158072
.156672
.155136
29
30
1.168768
1.167827
1.166796
1.165665
1.164429
1.163076
1.161596
1.159973
30
31
.173673
.172681
.171591
.170394
.169091
.167664
.16610]
.164394
31
32
.178204
.177155
.176003
.174743
.173370
.171864
.170221
.168423
32
33
.182382
.181275
.180063
.178737
.177290
.175708
.173979
.172090
33
34
.186233
.185069
.183794
.182398
.180879
.179216
.177402
.175422
34
35
1.189780
1.188557
1.187216
1.185752
1.184157
1.182413
1.180513
1.178438
35
36
.193044
.191758
.190354
.188818
.187145
.185320
.183330
.181164
36
37
.196041
.194695
.193222
.191612
.189864
.187954
.185879
.183620
37
38
.198792
.197381
.195837
.194157
.192328
.190338
.188175
.185820
38
39
.201308
.199830
.198221
.196464
.194550
.192487
.190233
.187785
39
40
1.203603
1.202064
1.200382
1.198545
1.196572
1.194413
1.192071
1.189534
40
41
.205698
.204089
.202332
.200439
.198375
.196133
.193708
.191077
41
42
.207597
.205918
.204107
.202129
.199986
.197667
.195152
.192435
42
43
.209312
.207583
.205692
.203639
.201424
.199019
.196424
.193621
43
44
.210875
.209070
.207108
.204987
.202691
.200211
.197534
.194646
44
36
27
28
29
30
31
32
33
Continued on Page *80. TABLE XLVI.
75
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
Pay-
ments.
34
35
36
37
38
39
40
41
Pay-
ments.
n.
1.195615
1.191703
1.187604
1.183299
1.178783
1.174051
1.169091
1.163896
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.288861
.288821
.288783
.288741
.288690
.288637
.288577
.288516
2
3
.452874
.452797
.452719
.452629
.452528
.452418
.452299
.452168
3
4
.565828
.565712
.565590
.565452
.565297
.565131
.564945
.564745
4
5
0.650846 0.650688
0.650521
0.650332
0.650122
0.649894
0.649642
0.649369
5
6
.718225
.718025
.717809
.717569
.717300
.717008
.716688
.716337
6
7
.773461
.773214
.772950
.772653
.772324
.771967
.771573
.771143
7
8
.819830
.819536
.819219
.818865
.818474
.818047
.817578
.817064
8
9
.859447
.859104
.858733
.858320
.857862
.857365
.856815
.856216
9
10
0.893757
0.893362
0.892935
0.892459
0.891934
0.891359
0.890729
0.890039
10
11
.923787
.923339
.922852
.922312
.921715
.921064
.920348
.919565
11
12
.950298
.949793
.949245
.948637
.947967
.947235
.946431
.945549
12
13
.973865
.973302
.972690
.972013
.971266
.970450
.969552
.968571
13
14
.994937
.994313
.993636
.992885
.992059
.991155
.990162
.989074
14
15
1.013868
1.013183
1.012434
1.011610
1.010699
1.009706
1.008612
1.007416
15
16
.030946
.030192
.029375
.028469
.027475
.026385
.025190
.023879
16
17
.046399
.045581
.044686
.043701
.042616
.041430
.040125
.038701
17
18
.060427
.059535
.058565
.057495
.056318
.055028
.053617
.052072
18
19
.073184
.072221
.071171
.070014
.068738
.067349
.065823
.064153
19
20
1.084812
1.083773
1.082642
1.081390
1.080021
1.078522
1.076879
1.075080
20
21
.095425
.094310
.093088
.091749
.090276
.088667
.086900
.084972
21
22
.105126
.103926
.102620
.101183
.099606
.097879
.095991
.093921
22
23
.113995
.112714
.111317
.109780
.108092
.106252
.104229
.102035
23
24
.122113
.120746
.119254
.117614
.115818
.113850
.111711
.109366
24
25
1.129544
1.128087
1.126498
1.124755
1.122839
1.120761
1.118479
1.115994
25
26
. 136344
.134796
.133110
.131254
.129233
.127020
.124606
.121973
26
27
.142567
.140926
.139134
.137178
.135030
.132692
.130139
.127358
27
28
.148258
.146519
. 144630
.142554
.140289
.137821
.135127
.132199
28
29
.153456
.151623
.149623
.147437
.145049
.142448
.139616
.136537
29
30
1.158204
1.156267
1.154162
1.151860
1.149346
1.146614
1.143641
1.140412
30
31
.162527
.160491
.158276
.155855
.153218
.150354
.147238
.143859
31
32
.166461
.164322
.161995
.159458
.156696
.153697
.150441
.146913
32
33
.170032
.167786
.165350
.162696
.159807
.156676
.153279
.149605
33
34
.173263
.170915
.168368
.165594
.162580
.159317
.155783
.151967
34
35
1.176182
1.173729
1.171070
1.168178
1.165041
1.161648
1.157980
1.154022
35
36
.178810
.176251
.173480
.170473
| .167212
.163694
.159893
.155800
36
37
.181165
.178500
.175622
.172497
.169119
.165476
.161548
.157326
37
38
.183266
.180501
.177511
.174276
.170782
.167018
.162969
.158626
38
39
.185136
.182266
.179172
.175828
.172219
.168342
.164179
.159718
39
40
1.186786
1.183818
1.180621
1.177169
1.173454
1.169470
1.165197
1.160629
40
41
.188237
.185172
.181874
.178322
.174507
.170419
.166046
.161380
41
42
.189504
.186343
.182952
.179306
.175392
.171211
.166746
.161989
42
43
.190599
.187351
. 183871
.180132
.176131
.171864
.167313
.162474
43
44
.191542
.188210
.184643
.180822
.176741
.172393
.167765
.162861
44
34
35
36
37
CO
00
39
40
41
5 PER CENT.
Continued on Page *79. 76
TABLE XLVI.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
5 PER CENT
Pay-
ments.
II
43
44
45
46
47
48
49
Pay-
ments.
n.
1.158448
1.152750
1.146778
1.140532
1.133997
1.127159
1.120012
1.112544
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0. oooooo
1
2
.288442
.288366
.288279
.288184
.288081
.287967
.287841
.287704
2
3
.452020
.451864
.451686
.451494
.451283
.451048
.450792
.450511
3
4
.504520
.564278
.564007
.563710
.563388
.563029
.562637
.562206
4
5
0.649063
0.648732
0.648362
0.647962
0.647519
0.647032
0.646498
0.645909
5
6
.715947
.715523
.715054
.714539
.713975
.713353
.712670
.711921
6
7
.770664
.770146
.769569
.768939
.768247
.767485
.766649
.765730
7
8
.816495
.815875
.815189
.814438
.813614
.812705
.811709
.810616
8
9
.855551
.854828
.854026
.853150
.852187
.851127
.849966
.848693
9
10
0.889276
0.888444
0.887524
0.886516
0.885410
0.884194
0.882862
0.881400
10
11
.918698
.917753
.916709
.915566
.914311
.912933
.911424
.909766
11
12
.944576
.943514
.942341
.941057
.939649
.938105
.936409
.934551
12
13
.967486
.966301
.964996
.963567
.962002
.960281
.958396
.956332
13
14
.987874
.986564
.985121
.983544
.981812
.979912
.977833
.975555
14
15
1.006096
1.004656
1.003073
1.001338
0.999437
0.997354
0.995073
0.992575
15
16
.022438
.020865
.019133
.017238
1.015164
1.012890
1.010401
1.007676
16
17
.037133
.035420
.033537
.031478
.029223
.026753
.024051
.021100
17
18
.050372
.048515
.046478
.044248
.041809
.039138
.036222
.033036
18
19
.062316
.060315
.058107
.055713
.053084
.050211
.047074
.043650
19
20
1.073107
1.070944
1.068590
1.066007
1.063188
1.060107
1.056746
1.053080
20
21
.082848
.080546
.078013
.075253
.072237
.068945
.065357
.061448
21
22
.091665
.089195
.086495
.083548
.080334
.076828
.073011
.068857
22
23
.099620
.096992
.094116
.090982
.087567
.083846
.079799
.075400
23
24
.106802
.104008
.100956
.097633
.094016
.090079
.085800
.081156
24
25
1.113273
1.110314
1.107083
1.103570
1.099750
1.095595
1.091087
1.086204
25
26
.119096
.115969
.112559
.108856
.104830
.100460
.095728
.090606
26
27
.124324
.121028
.117440
.113542
.109315
.104734
.099779
.094424
27
28
.129005
.125541
.121770
.117684
.113258
.108468
.103294
.097711
28
29
.133185
.129549
.125600
.121328
.116706
.111711
.106323
.100525
29
30
1.136899
1.133096
1.128973
1.124515
1.119702
1.114507
1.108917
1.102914
30
31
.140188
.136221
.131924
.127287
.122286
.116901
.111121
.104919
31
32
.143087
.138957
.134492
.129679
.124500
.118938
.112971
.106586
32
33
.145628
.141339
.136709
.131730
.126384
.120647
.114510
.107957
33
34
.147839
.143397
.138610
.133475
.127966
.122070
.115776
.109070
34
35
1.149751
1.145161
1.140229
1.134940
1.129283
1.123240
1.116804
1.109957
35
36
.151390
.146665
.141588
.136161!
.130366
.124191
.117623
.110657
36
37
.152788
.147926
.142721
.137165
.131247
.124949
.118270
.111196
37
38
.153961
.148979
.143653
.137982
.131949
.125547
.118768
.111605
38
39
.154940
.149845
.144411
.138632
.132503
126007
.119145
.111910
39
40
1.155746
1.150550
1.145014
1.139147
1.132929
1.126356
1.119427
1.112128
40
41
.156401
.151110
.145493
.139542
.133252
.126617
.119628
.112283
41
42
.156922
.151555
.145859
.139841
.133494
.126803
.119772
.112388
42
43
.157337
.151895
.146137
.140065
.133666
.126937
.119869
.112455
43
44
.157653
.152154
.146345
.140225
.133790
.127026
.119931
.112496
44
42
43
44
45
46
47
48
49
Continued on Page *78. TABLE XLTI
77
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. _
0 PER CENT.
Pay-
ments.
50
51
52
53
54
55
56
57
Pay-
n.
1.104739
1.096590
1.088084
1.079208
1.069947
1.060290
1.050232
1.039753
n.
1
0.000000
0.000000 0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.287551
.287384
.287201
.287001
.286782
.286535
.286270
.285973
2
3
.450200
.449859
.449486
.449078
.448627
.448128
.447584
.446980
3
4
.561729
.561209
.560639
.560012
.559322
.558559
.557725
.556805
4
5
0.645263
0.644555
0.643778
0.642925
0.641986
0. 640948
0.639816
0.638570
5
6
.711096
.710193
.709202
.708114
.706914
.705595
.704155
.702568
6
7
.764721
.763615
.762403
.761068
.759603
.757993
.756230
.754294
7
8
.809416
.808101
.806655
.805069
.803329
.801415
.799323
.797025
8
9
.847294
.845758
.844074
.842230
.840203
.837978
.835545
.832878
9
10
0.879793
0.878033
0.876106
0.873990
0.871670
0.869124
0.866344
0.863295
10
11
.907947
.905956
.903773
.901381
.898758
.895885
.892746
.889308
11
12
.932514
.930282
.927838
.925160
.922230
.919019
.915515
.911680
12
13
.954068
.951590
.948876
.945910
.942661
.939107
.935229
.930992
13
14
.973058
.970325
.967340
.964074
.960503
.956598
.952343
.947700
14
15
0.989837
0.986849
0.983583
0.980013
0.976113
0.971854
0.967218
0.962164
15
16
1.004699
1.001446
.997893
.994014
.989782
.985165
.980146
.974678
16
17
.0L7873
.014351
1.010508
1.006317
1.001749
.996773
.991366
.985490
17
18
.029556
.025761
.021625
.017120
.012215
1.006876
1.001088
.994807
18
19
.039913
.035843
.031411
.026590
.021345
.015651
.009484
1.002802
19
20
1.049085
1.044737
1.040009
1.034869
1.029293
1.023244
1.016704
1.009629
20
21
.057192
.052566
.047540
.042090
.036181
.029785
.022878
.015427
21
22
.064340
.059435
.054118
.048357
.042125
.035386
.028129
.020313
22
23
.070621
.065443
.059836
.053773
.047220
.040155
.032560
.024399
23
24
.076122
.070673
.064783
.058420
.051563
.044185
.036269
.027781
24
25
1.080915
1.075201
1.069031
1.062385
1.055238
1.047560
1.039340
1.030548
25
26
.085069
.079093
.072659
.065743
.058316
.050357
.041855
.032789
26
27
.088643
.082420
.075734
.068558
.060869
.052648
.043894
.034575
27
28
.091700
.085242
.078313
.070893
.062962
.054507
.045519
.035978
28
29
.094294
.087610
.080454
.072809
.064660
.055988
.046795
.037059
29
30
1.096471
1.089575
1.082211
1.074363
1.066013
1.057153
1.047779
1.037879
30
31
.098279
.091190
.083637
.075602
.067078
.058051
.048525
.038487
31
32
.099766
.092501
.084774
.076578
.067900
.058731
.049080
.038924
32
33
.100973
.093546
.085670
.077331
.068522
.059238
.049477
.039232
33
34
.101935
.094370
.086360
.077901
.068985
.059600
.049758
.039441
34
35
1.102694
1.095004
1.086884
1.078325
1.069316
1.059857
1.049948
1.039576
35
36
.103278
.095486
.087273
.078629
.069552
.060030
.050072
.039657
36
37
.103722
.095844
.087553
.078845
.069710
.060144
.050145
.039703
37
38
.104052
.096101
.087751
.078990
.069814
.060210
.050187
.039730
38
39
.104288
.096284
.087884
.079085
.069875
.060248
.050212
.039744
39
40
1.104457
1.096406
1.087971
1.079141
1.069910
1.060271
1.050225
1.039750
40
41
.104570
.096486
.088023
.079173
.069931
.060283
.050230
.039753
41
42
.104643
.096534
.088052
.079193
.069942
.060287
.050233
.039753
42
43
44
45
46
47
48
49
.104687
.104712
1.104728
.104735
.104737
.104739
.104739
.096561
.096578
1.096586
.096588
.096590
.096590
.088071
.088080
1.088082
.088083
.088084
.079203
.079205
1.079208
.079208
.069945
.069948
1.069948
.060290
.060290
1.060290
.050233
.050230
.039753
43
50
51
52
53
54
55
56
57 78
TABLE XLVI.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. pER ce^t
Pay-
£=58
59
60
61
62
63
64
65
Pay-
ments.
n.
1.028851
1.017506
1.005705
0.993441
!o. 980708
0.967485
0.953771
0.939554
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
o.oooooolo. 000000
1
2
.285651
.285299
.284908
.284479
.284011
.283494
.282926
.282300
2
3
.446325
.445605
.444810
.443937
.442984
.441932
.440776
.439509
3
4
.555804
.554702
.553489
.552159
.550702
.549098
.547340
.545411
4
5
0.637208
0.635712
0.634061
). 632263
0.630289
0.628118
0.625740
0.623135
5
6
.700835
.698932
.696841
.694552
.692045
.689292
.686282
.682984
6
7
.752177
.749855
.747308
.744519
.741465
.738121
.734461
.730461
7
8
.794515
.791763
.788747
.785443
.781837
.777885
.773570
.768856
8
9
.829963
.826771
.823271
.819450
.815276
.810711
.805730
.800300
9
10
0.859966
0.856322
0.852337
0.847985
0.843237
0.838051
0.832404
0.826254
10
11
.885556
.881455
.876974
.872086
.866758
.860953
.854637
.847771
11
12
.907499
.902934
.897952
.892522
.886618
.880190
.873209
.865635
12
13
.926377
.921344
.915855
.909888
.903405
.896360
.888724
.880456
13
14
.942646
.937140
.931150
.924644
.917588
.909935
.901658
.892715
14
15
0.956668
0.950694
0.944203
0.937105
0.929544
0.921299
0.912399
0.902806
15
16
.968745
.962305
.955318
.947756
.939587
.930766
.921268
.911058
16
17
.979124
.972221
.964748
.956677
.947976
.938605
.928538
.917744
17
18
.988009
.980655
.972711
.904147
.954939
.945043
.934441
.923111
18
19
.995584
.987792
.979391
.970359
.960667
.950280
.939187
.927363
19
20
1.002005
0.993790
0.984956
0.975478
0.965334
0.954497
0.942953
0.930685
20
21
.007410
.998794
.989548
.979653
.909096
.957846
.945897
.933238
21
22
.011925
1.002928
.993297
.983022
.972087
.960467
.948161
.935165
22
23
.015658
.006306
.996325
.985703
.974429
.962484
.949872
.936589
23
24
.018711
.009036
.998736
.987804
.976232
.964008
.951137
.937618
24
25
1.021181
1.011212
1.000627
0.989422
0.977596
0.965136
0.952050
0.938335
25
26
.023150
.012919
.002083
.990046
.978606
.965950
.952688
.938827
26
27
.024695
.014233
.003186
.991553
.979334
.966519
.953124
.939139
27
28
.025885
.015230
.004003
.992207
.979843
.966908
.953402
.939328
28
29
.026788
.015968
.004592
.992605
.980191
.967156
.953569
.939438
29
30
1.027457
1.016500
1.005005
0.992977
0.980413
0.967305
0.953667
0.939499
30
31
.027939
.016873
.005286
.993177
.980547
.967392
.953722
.939531
31
32
.028277
.017128
.005467
.993297
.980624
.967442
.953750
.939544
32
33
.028508
.017291
.005575
.993366
.980069
.967467
.953762
.939551
33
34
.028656
'.017389
.005637
.993406
.980091
.967477
.953768
.939553
34
35
1.028745
1.017445
1.005673
0.993426
0.980701
0.967483
0.953770
0.939554
35
38
.028796
.017477
.005691
.993435
.980706
.967485
.953771
37
.028825
.017494
.005699
.993439
.980708
.967485
43
42
38
.028841
.017501
.005703
.993441
.980708
44
1.158448
58
39
.028847
.017505
.005704
. 99344l|
45
1.152750
.158448
57
40
1.028851
1.017506
1.005705
46
1.146778
.152750
.158448
56
41
42
• U/COoOl
.028851
.UL iOUb
47
1.133997
-L. JL4:Uc)0/i
1.140532
.146ii0
1.146779
.lo2750
1.152748
.158446
1.158445
55
54
49
48
1.127159
.133997
.140532
.146777
.152747
.158443
53
1.120012
.127160
.133997
.140530
.146776
.152744
.158432
52
51
1.112544
.120013
.127160
.133995
.140529
.146772
.152733
.158439
51
50
.112544
.120013
.127158
.133994
.140525
.146761
.152719
.158395
50
49
1.112544
1.120011
1.127157
1.133989
1.140513
1.146746
1.152693
1.158356
49
48
.112542
.120010
.127151
.133977
.140497
.146718
.152651
.158294
48
47
.112540
.120004
.127138
.133959
.140467
.146672
.152584
.158201
47
46
.112534
.119990
.127119
.133927
.140417
.146600
.152484
.158073
46
45
.112519
.119969
.127084
.133873
.140340
.146493
.152347
.157894
45
49
48
47
46 1
45
44
43
42
Continued from Page *76. TABLE XLYI.
79
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF ft
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
5 PER CENT.
Pay-
ments.
66
67
68
69
70
71
72
73
Pay-
ments.
n.
0.924834
0.909590
0.893832
0.877530
0.860711
0.843350
0.825455
0.807021
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.281620
.280864
.280048
.279136
.278148
.277058
.275867
.274565
2
3
.438123
.436599
.434934
.433097
.431094
.428896
.426495
.423862
3
4
.543304
.540986
.538460
.535675
.532643
.529321
.525690
.521720
4
5
0.620287
0.617164
0.613755
0.610013
0. 605938
0.601478
0.596614
0.591309
5
6
.679383
.675438
.671139
.666429
.661302
.655705
.649613
.642980
6
7
.726095
.721322
.716125
.710441
.704264
.697538
.690228
.682291
7
8
.763720
.758113
.752015
.745362
.738146
.730302
.721801
.712593
8
9
.794390
.787949
.780955
.773345
.765103
.756171
.746513
.736083
9
10
0.819572
0.812303
0.804427
0.795875
0. 786638
0.776651
0.765885
0.754299
10
11
.840323
.832240
.823499
.814035
.803836
.792842
.781031
.768362
11
12
.857436
.848558
.838981
.828638
.817523
.805584
.792797
.779133
12
13
.871525
.861879
.851498
.840321
.828349
.815530
.801851
.787287
13
14
.883078
.872697
.861555
.849600
.836833
.823211
.808728
.793372
14
15
0.892496
0.881419
1.869570
0.856894
,0.843404
0.829062
0.813873
0.797832
15
16
.900110
.888390
.875886
.862558
.848421
.833448
.817651
.801033
16
17
.906210
.893894
.880800
.866890
.852187
.836674
.820367
.803255
17
18
.911035
.898184
.884565
.870146
.854962
.838996
.822254
.804781
18
19
.914800
.901474
.887398
.872547
.856960
.840610
.823550
.805741
19
20
0.917691
0.903953
0.889489 0.874279
0.858350
0.841718
0.824366
0.806326
20
21
.919872
.905783
.890998
.875483
.859305
.842417
.824863
.806666
21
22
.921482
.907105
.892047
.876311
.859907
.842843
.825153
.806852
22
23
.922646
.908025
.892768
.876833
.860274
.843091
.825311
.806948
23
24
.923456
.908657
.893224
.877151
.860488
.843227
.825393
.806993
24
25
0.924013
0.909057
0.893501
0.877337
0.860605
0.843297
0.825431
0.807012
25
26
.924365
.909300
.893663
.877438
.860666
.843329
.825447
.807019
26
27
.924579
.909441
.893752
.877491
.860693
.843343
.825453
.807021
27
28
29
30
31
32
33
34
59
.924703
.924772
0.924808
.924823
.924831
.909519
.909560
0.909577
.909586
.909588
.893798
.893818
0.893828
.893830
.893832
.877514
.877525
0.877528
.877530
38
1.178783
.178781
.178781
1.178780
.860705
• .860709
0.860711
37
.843348
.843350
36
1.187604
.825455
35
1.191703
1.191701
34
1.195615
.195614
1.195614
66
65
64
.924833
.924834
41
1.163896
.909590
40
1.169091
L.169090
39
1.174051
.174050
1.174050
1.183299
.183298
.183298
.183297
1.183297
.187603
.187603
.187612
.187602
1.187602
.191701
.191701
.191701
.191700
1.191692
.195614
.195614
.195613
.195606
1.195598
63
62
61
60
59
58
.163896
.169090
.174049
.178780
.183295
.187592
.191684
.195583
58
57
.163896
.169089
.174048
.178778
.183287
.187584
.191668
.195562
57
56
.163894
.169088
.174046
.178769
.183278
.187568
.191645
.195526
56
55
.163893
.169086
.174037
.178760
.183261
.187548
.191607
.195472
55
54
L.163891
1.169076
1.174027
1.178741
1.183234
1.187502
1.191550
1.195398
54
53
.163881
.169065
.174007
.178712
.183191
.187441
.191471
.195294
53
52
.163869
.169044
.173976
.178666
.183125
.187357
.191359
.195154
52
51
.163846
.169010
.173926
.178596
.183035
.187237
.191210
.194971
51
50
.163810
.168956
.173851
.178499
.182907
.187077
.191015
.194732
50
49
1.163752
1.168876
1.173747
1.178363
1.182736
1.186869
1.190759
1.194437
49
48
.163666
.168765
.173602
.178180
.182514
.186596
.190437
.194052
48
47
.163546
.168609
.173406
.177942
.182221
.186252
.190033
.193587
47
46
.163379
.168400
.173151
.177628
.181853
.185820
.189536
.193023
46
45
.163155
.168126
.172814
.177235
.181392
.185288
.188933
.192346
45
41
40
39
38
37
36
35
34
Continued from Page *75 80
TABLE XL VI.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
5 PER CENT.
Pay-
ments.
a
II
75
76
77
78
79
80
81
Pay-
ments.
n-
0.788039
0.768544
0.748513
0.727935
0.700853 0.685247
0.663118
0.640468
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.273121
.271556
.269845
.267955
.265900
.263651
.261191
.258516
2
3
.420968
.417828
.414391
.410619
.406519
.402044
.397173
.391869
3
4
.517374
.512654
.507501
.501868
• 4:00 i 00
.489111
.481890
.474070
4
5
0.585515
0.579229
0.572385
1.564931
0.556808
0.548124
0.538671
0.528466
5
6
.635757
.627935
.619444
.610236
.600291
.589573
.578019
.565609
6
7
.673671
.664359
.654291
.643406
.631708
.619132
.605652
.591253
7
8
.702624
.691893
.680323
.667874
.654542
.640282
.625077
.608915
8
9
.724834
.712758
.699796
.685902
.671094
.655336
.638617
.620953
9
10
0.741844
0.728525
0.714284
0.699094
0.682982
0.665917
0.647917
0.628878
10
11
.754796
.740342
.724961
.708631
.691389
.673227
.654072
.633875
11
12
.764558
.740097
.732717
.715406
.697167
.678084
.657966
.636899
12
13
.771820
.755481
.738246
.720077
.701106
.681163
.660327
.638648
13
14
.777130
.760045
.742067
.723269
.703574
.683032
.661695
.639602
14
15
0.780935
0.763294
0.744683
0.725271
0.70507410.684118
0.662441
0.640089
15
16
.783573
.765370
.746325
.726489
.705946
.684709
.662822
.640322
16
17
.785383
.766731
.747325
.727198
.706421
.685012
.663004
.640418
17
18
.786522
.767560
.747907
.727584
.706665
.685156
.663079
.640456
18
19
.787215
.768042
.748225
.727782
.706780
.685215
.663109
.640468
19
20
21
0.787618
.787839
0.768306
.768441
0.748387
.748464
0.727876
.727915
0.706828
.706847
0.685239
.685247
0.663118
26
22
23
24
25
26
.787952
.788005
.788028
0.788036
.788039
33
.768504
.768531
.768541
0.768544
.748496
.748508
.748513
31
.727930
.727935
30
1.209497
1.209498
.209498!
.706853
29
1.212561
.212562
1.212562
.212562
28
1.215481
.215483
.215483
1.215483
.215483
27
1.218261
.218262
.218262
.218262
1.218262
.218261
1.220904
.220905
.220905
.220905
.220905
1.220904
.220899
74
73
72
71
70
69
68
32
1.202896
1.206277
.206277
67
1.199343
.202895
.206277
.209498!
.212502
.215482
.218256
.220893
67
66
.199342
.202895
.206277
.209498
.212501
.215476
.218250
.220885
66
65
.199342
.202895
.206277
.209497
.212555
.215470
.218241
‘ .220873
65
64
1.199342
1.202895
1.206276
1.209491
1.212548
1.215460
1.218228
1.220851
64
63
.199342
.202894
.206270
.2094841
.212538
.215446
.218205
.220817
63
62
.199341
.202888
.206263
.209473
.212523
.215422
.218169
.220772
62
61
.199335
.202881
.206251
.209457
.212497
.215384
.218121
.220711
61
60
.199327
.202868
.206234
.209429
.212457
.215333
.218056
.220627
60
59
1.199313
1.202850
1.206204
1.209387
1.212403
1.215263
1.217967
1.220516
59
58
.199294
.202818
.206159
.209329
.212328
.215169
.217849
.220374
58
57
.199260
.202770
.206098
.209249
.212228
.215043
.217697
.220191
57
56
.199209
.202705
.206012
.209142
.212094
.214881
.217503
.219965
56
55
.199140
.202614
.205898
.208999
.211922
.214675
.217262
.219685
55
54
1.199043
1.202491
1.205746
1.208816
1.211702
1.214419
1.216964
1.219347
54
53
.198911
.202330
.205551
.208583
.211429
.214101
.216605
.218940
53
52
.198740
.202121
.205302
.208291!
.211091
.213719
.216172
.218461
52
51
.198516
.201856
.204991
.207931'
.210683
.213258
.215661
.217895
51
50
.198234
.201524
.204607
.207496
.210193
.212713
.215059
.217238
50
49
1.197879
1.201115
1.204143
1.206974
1.209612
1.212073
1.214360
1.216481
49
48
.197443
.200620
.203587
.206355
.208930
.211328
.213554
.215615
48
47
.196915
.200027
.202927
.205628
.208137
.210470
.212631
.214632
47
46
.196282
.199322
.202152
.204783
.207223
.209487
.211585
.213518
46
45
.195529
.198495
.201251
.203808
.206175
.208372
.210398
.212271
45
33
32
31
30
29
28
27
26
Continued from Page *74. TABLE XL VI.
81
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
5 PER CENT.
Pay-
82
83
84
85
86
87
88
89
Pay-
ments.
n.
0.617258
0.593510
0.569083
0.543949
0.517851
0.490338
0.461018
0.428340
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.255571
.252400
.248902
.245128
.241062
.236517
.231773
.223681
2
3
.386091
.379847
.373034
.365722
.357791
.349165
.338513
.323210
3
4
.465583
.456446
.446561
.435952
.424566
.411342
.395201
.374409
4
5
0.517453
0.505659
0.492958
0.479431
0.464382
0.446913
0.426510
0.401609
5
6
.552297
.538098
.522932
.506445
.488069
.467291
.443722
.415812
6
7
.575884
.559600
.542018
.522889
.501928
.478711
.452865
.422907
7
8
.591780
.573508
.553805
.532637
.509785
.484838
.457472
.426267
8
9
.602169
.582179
.560851
.538204
.514027
.487942
.459663
.427642
9
10
0.608689
0.587392
0.564895
0.541222
0.516183
0.489422
0.460562
0.428172
10
11
.612625
.590394
.567093
.542759
.517212
.490030
.460908
.428340
11
12
.614896
.592029
.568214
.543493
.517636
.490264
.461018
13
.616135
.592863
.568750
.543796
.517799
.490338
19
18
14
.616767
.593262
.568971
.543913
.517851
20
1 0Q/19£K
1.237891
82
15
0.617070
0.593427
0.569057
0.543949
21
1.236125
93P.1 97
.237893
937ftQ‘3
81
80
lb
17
. 01 {1 \j~±
.617243
.593510
. ouyuoo
22
1.232309
1.234266
• /Vt_) U _L Aj (
1.236127
. /Vfj 1 Oc/O
1.237893
79
18
.617258
24
fCO
1.230253
.232310
.234266
.236127
.237893
78
OR
1.228089
.230254
.232310
.234266
.236127
.237892
77
1.225813
.228090
.230254
.232310
.234266
.236126
.237889
76
75
1.223420
.225814
.228090
.230254
.232310
.234265
.236123
.237886
75
74
1.223422
1.225814
1.228090
1.230254
1.232309
1.234262
1.236120
1.237879
74
73
.223422
.225814
.228090
.230253
.232306
.234259
.236112
.237874
73
72
.223422
.225814
.228089
.230250
.232303
.234251
.236107
.237859
72
71
.223422
.225813
.228086
.230246
.232295
.234245
.236091
.237837
71
70
.223421
.225809
.228081
.230238
.232288
.234229
.236068
.237810
70
69
1.223416
1.225804
1.228073
1.230230
1.232271
1.234204
1.236039
1.237772
69
68
.223411
.225796
.228064
.230212
.232245
.234173
.235999
.237721
68
67
.223403
.225786
.228045
.230185
.232212
.234131
.235945
.237652
67
66
.223392
.225766
.228016
.230150
.232167
.234073
.235871
.237564
66
65
.223371
.225736
.227979
.230102
.232105
.233995
.235778
.237453
65
64
1.223339
1.225697
1.227928
1.230037
1.232023
1.233896
1.235660
1.237312
64
63
.223297
.225643
.227859
.229950
.231918
.233771
.235510
.237140
63
62
.223240
.225569
.227766
.229838
.231785
.233612
.235328
.236933
62
61
.223161
.225470
.227648
.229696
.231616
.233418
.235108
.236683
61
60
.223056
.225345
.227497
.229517
.231410
.233184
.234842
.236388
60
59
1.222923
1.225184
1.227307
1.229298
1.231161
1.232902
1.234530
1.236040
59
58
.222751
.224982
.227074
.229034
.230862
.232571
.234160
.235636
58
57
.222538
.224735
.226794
.228715
.230510
.232178
.233732
.235172
57
56
.222275
.224436
.226455
.228342
.230093
.231723
.233238
.234641
56
55
.221957
.224077
.226058
.227899
.229609
.231199
.232674
.234037
55
54
1.221575
1.223654
1.225588
1.227385
1.229053
1.230600
1.232034
1.233357
54
53
.221125
.223154
.225041
.226794
.228417
.229920
.231311
.232593
53
52
.220593
.222573
.224412
.226118
.227694
.229152
.230500
.231740
52
51
.219975
.221904
.223694
.225350
.226877
.228290
.229593
.230787
51
50
.219263
.221140
.222877
.224480
.225962
.227327
.228581
.229741
50
49
1.218450
1.220271
1.221953
1.223508
1.224938
1.226251
1.227470
1.228586
49
48
.217526
.219288
.220919
.222420
.223795
.225070
.226242
.227315
48
47
.216480
.218188
.219761
.221204
.222538
.223765
.224891
.225923
47
46
.215309
.216956
.218467
.219867
.221151
.222329
.223412
.224401
46
45
.213998
.215579
.217045
.218391
.219623
.220755
.221794
.222743
45
25
24
23
22
21
20
19
18
Continued from Page *73. 82
TABLE XL YI.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF W
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
5 PER CENT.
Pay-
ments.
n.
li
CD
O
91
92
93
94
95
96
97
Pay-
ments.
0.397160
0.368711 0.340242
0.310847
0.280937
0.249611
0.212464
0.176638
n.
1
0.000000
0.000000 0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.214806
.207072
.198987
.189664
.179538
.169142
.152340
.140179
2
3
.307656
.293579
.278635
.261970
.244460
.224344
.198721
.176638
3
4
.353712
.334538
.314378
.292583
.269086
.243655
.212464
10
5
0.377006
0.354101
0.330417
0.304829
0.278110
.280937
0.249611
11
.309403
1.249190
90
6
.388464
.363173
.366849
.336984
.339458
.310847
12
7
.393850
13
1.248018
1.249191
89
8
.396046
.368262
.368711
. 34:024/3
14
1.246787
.248019
.249191
88
9
.396892
15
1.245487
.246788
.248019
.249191
87
10
0.397160
16
1.244118
.245488
.246788
.248019
.249190
86
1.242680
1.242682
.244119
1.244119
.245488
1.245488
.246788
1.246788
.248019
1.248018
.249190
1.249188
85
84
17
1.241169
83
1.239571
.241169
.242682
.244119
.245488
.246787
.248016
.249186
83
82
.239572
.241169
.242682
.244119
.245487
.246785
.248014
.249182
82
81
.239572
.241169
.242682
.244118
.245485
.246783
.248010
.249179
81
80
.239572
.241169
.242681
.244116
.245482
.246778
.248007
.249170
80
79
1.239572
1.241168
1.242678
1.244113
1.245477
1.246775
1.247997
1.249156
79
78
.239571
.241165
.242675
.244107
.245474
.246764
.247982
.249139
78
77
.239568
.241162
.242669
.244104
.245463
.246749
.247964
.249115
77
76
.239565
.241156
.242666
.244092
.245447
.246729
.247939
.249083
76
75
.239558
.241152
.242653
.244075
.245426
.246703
.247905
.249041
75
74
1.239554
1.241139
1.242635
1.244053
1.245398
1.246667
1.247860
1.248986
74
73
.239540
.241119
.242612
.244023
.245360
.246620
.247802
.248917
73
72
.239519
.241095
.242580
.243983
.245310
.246558
.247730
.248827
72
71
.239494
.241061
.242537
.243930
.245244
.246481
.247633
.248721
71
70
.239458
.241016
.242481
.243860
.245162
.246379
.247521
.248591
70
69
1.239410
1.240955
1.242406
1.243773
1.245055
1.246260
1.247383
1.248433
69
68
.239345
.240877
.242314
.243660
.244928
.246113
.247216
.248250
68
67
.239262
.240779
.242195
.243525
.244773
.245937
.247023
.248032
67
66
.239158
.240652
.242051
.243360
.244586
.245732
.246791
.247781
66
65
.239024
.240500
.241877
.243163
.244368
.245486
.246525
.247491
65
64
1.238863
1.240315
1.241668
1.242932
1.244108
1.245204
1.246217
1.247160
64
63
.238667
.240094
.241422
.242656
.243809
.244878
.245866
.246782
63
62
.238432
.239832
.241130
.242338
.243463
.244506
.245466
.246359
62
61
.238154
.239523
.240792
.241972
.243068
.244082
.245017
.245882
61
60
.237826
.239164
.240404
.241553
.242619
.243606
.244512
.245352
60
59
1.237445
1.238752 1.239959
1.241076
1.242114
1.243070
1.243950
1.244761
59
58
.237008
.238281
.239454
.24054l|
.241546
.242474
.243322
.244109
58
57
.236508
.237745
.238886
.239938
.240914
.241808
.242632
.243391
57
56
.235939
.237142
.238247
.239268
.240207
.241076
.241871
.242604
56
55
.235299
.236464
.237535
.238518
.239431
.240269
.241035
.241741
55
54
1.234580
1.235708
1.236739
1.2376941
1.238575
1.239383
1.240120
1.240799
54
53
.233776
.234863
.235865
.236786
.237604
.238411
.239120
.239775
53
52
.232879
.233935
.234901
.235787
.236603
.237350
.238034
.238662
52
51
.231894
.232912
.233840
.234693
.235477
.236198
.236853
.237455
51
50
.230807
.231785
.232679
.233498
.234254
.234944
.235572
.236152
50
49
1.229610
1.230552
1.231410
1.232199
1.232923
1.233585
1.234189
1.234744
49
48
.228301
.229205
.230030
.230785
.231479
.232116
.232694
.233225
48
47
.226869
.227738
.228528
.229252
.229920
.230528
.231081
.231588
47
46
.225310
.226142
.226899
.227595
.228232
.228815
.229342
.229827
46
45
.223614
.224410
.225138
.225801
.226412
.226967
.227472
.227936
45
_17
16
15
14
13
12
11
10
Continued from Patre *72. 83
TABLE XLVII.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR,
The first Value under the Age & is A(1 + ax) for the whole Life.
6 PER CENT.
Pay-
ments
x—10
11
12
13
14
15
16
17
Pay-
ments.
n.
1.189964
1.189102
1.18819$
1.187226
1.186201
1.185120
1.183972
1.182755
n.
1 1
0.000000
0.000000
0.00000(
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.287194
.287188
.287185
.287181
.287174
.287169
.287165
.287158
2
3
.449596
.449587
.449581
.449571
.449559
.449549
,449538
.449524
3
4
.560996
.560984
.560972
.560957
.560939
.560923
.560907
.560886
4
5
0.644517
0.644499
0.644483
0.644462
0. 644439
0.644419
0.644395
0.644366
5
6
.710457
.710435
.710413
.710387
.710360
.710332
.710301
.710264
6
7
.764310
.764283
.764256
.764227
.764192
.764157
.764119
.764074
7
8
.809356
.809323
.809291
.809257
.809216
.809174
.809128
.809075
8
9
.847709
.847674
.847637
.847594
.847546
.847497
.847444
.847383
9
10
0.880815
0.880773
0.880731
0.880682
0.880627
0.880572
0.880511
0.880439
10
11
.909701
.909654
.909606
.909550
.909490
.909426
.909355
.909277
11
12
.935128
.935076
.935021
.934960
.934891
.934819
.934742
.934650
12
13
.957674
.957615
.957555
.957486
.957409
.957331
.957241
.957142
13
14
.977784
.977721
.977654
.977577
.977495
.977404
.977308
.977197
14
15
0.995818
0.995747
0.995673
0.995591
0.995496
0.995400
0.995293
0.995171
15
16
1.012057
1.011981
1.011901
1.011807
1.011708
1.011602
1.011483
1.011349
16
17
.026739
.026656
.026565
.026468
.026359
.026242
.026112
.025972
17
18
.040054
.039962
.039869
.039762
.039642
.039514
.039378
.039218
18
19
.052165
.052069
.051966
.051849
.051719
.051585
.051431
.051263
19
20
1.063211
0.063107
1.062995
1.062867
1.062732
1.062581
1.062420
1.062235
20
21
.073305
.073193
.073070
.072939
.072788
.072630
.072452
.072256
21
22
.082549
.082426
.082300
.082153
.081995
.081822
.081635
.081421
22
23
.091026
.090900
.090759
.090606
.090434
.090252
.090047
.089827
23
24
.098818
.098678
.098532
.098366
.098185
.097987
.097776
.097535
24
25
1.105982
1.105838
1.105679
1.105504
1.105308
1.105104
1.104873
1.104621
25
26
.112585
.112429
.112261
.112073
.111871
.111649
.111407
.111134
26
27
.118671
.118506
.118327
.118132
.117914
.117680
.117420
.117129
27
28
.124289
.124113
.123927
.123716
.123488
.123238
.122960
.122656
28
29
.129478
.129295
.129095
.128875
.128630
.128363
.128073
.127751
29
30
1.134277
1.134081
1.133872
1.133637
1.133376
1.133097
1.132791
1.132449
30
31
.138712
.138509
.138286
.138035
.137764
.137470
,137145
.136787
31
32
.142821
.142604
.142366
.142105
.141820
.141508
.141168
.140788
32
33
.146622
.146391
.146144
.145870
.145568
.145242
.144881
.144484
33
34
.150141
.149901
.149640
.149352
.149036
.148690
.148313
.147891
34
35
1.153403
1.153150
1.152877
1.152574
1.152240
1.151880
1.151479
1.151038
35
36
.156424
.156160
.155873
.155554
.155206
.154823
.154405
.153943
36
37
.159226
.158948
.158645
.158313
.157944
.157545
.157106
.156621
37
38
.161821
.161529
.161213
.1608621
.160478
.160058
.159599
.159093
38
39
.164224
.163920
.163587
.163221
.162817
.162380
.161900
.161371
39
40
1.166452
1.166132
1.165785
1.165400
1.164980
1.164523
1.164021
1.163468
•40
41
.168514
.168180
.167815
.167415
.166977
.166498
.165975
.165396
41
42
.170423
.170072
.169694
.169277
.168818
.168319
.167771
.167170
42
43
.172188
.171825
.171430
.170994
.170515
.169994
.169425
.168798
43
44
.173823
.173444
.173032
.172576
.172077
.171537
.170943
.170290
44
10
11
12
13
14
15
16
17
Continued on Page 93.
Formula, \(l + aT~U - a x~Nx+n=X?x=±.
v Da; yjOa^l 84
TABLE XLVII.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
Pay-
*=18
19
20
21
22
23
24
25
Pay-
ments.
11.
1,181467
1.180102
1.178657
1.177130
1.175513
1.173803
1.171993
1.170082
11.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.287151
.287143
.287133
.287123
.287113
.287101
.287088
.287076
2
3
.449510
.449493
.449474
.449455
.449433
.449410
.449385
.449358
3
4
.560863
.560838
.560809
.560779
.560748
.560712
.560672
.560629
4
5
0.644335
0.644301
0.644262
0.644223
0.644179
0.644129
0.644075
0.644017
5
6
.710225
.710181
.710134
.710082
.710025
.709962
.709894
.709818
6
7
.764026
.763974
.763914
.763851
.763781
.763705
.763620
.763526
7
8
.809019
.808955
.808885
.808809
.808727
.808635
.808533
.808423
8
9
.847316
.847243
.847160
.847072
.846975
.846868
.846750
.846621
9
10
0.880364
0.880277
0.880184
0.880083
0.879971
0.879848
0.879713
0.879565
10
11
.909188
.909092
.908986
.908870
.908745
.908605
.908452
.908282
11
12
.934553
.934445
.934325
.934196
.934055
.933898
.933724
.933534
12
13
.957033
.956912
.956779
.956636
.956478
.956301
.956109
.955898
13
14
.977076
.976943
.976797
.976638
.976462
.976267
.976055
.975816
14
15
0.995038
0.994897
0.994732
0.994556
0.994362
0.994149
0.993910
0.993659
15
16
1.011210
1.011046
1.010870
1.010676
1.010465
1.010228
1.009977
1.009696
16
17
.025811
.025637
.025443
.025234
.025001
.024752
.024472
.024171
17
18
.039048
.038857
.038649
.038419
.038174
.037898
.037600
.037264
18
19
.051076
.050872
.050644
.050403
.050133
.049839
.049508
.049149
19
20
1.062036
1.061813
1.061575
1.061310
1.061023
1.060699
1.060346
1.059960
20
21
.072039
.071807
.071546
.071266
.070950
.070605
.070227
.069813
21
22
.081196
.080942
.080667
.080359
.080023
.079654
.079251
.078806
22
23
.089580
.089313
.089012
.088685
.088327
.087934
.087501
.087030
23
24
.097276
.096985
.096665
.096318
.095937
.095516
.095058
.094549
24
25
1.104340
1.104030
1.103692
1.103323
1.102916
1.102472
1.101979
1.101446
25
26
.110835
.110508
.110149
.109756
.109327
.108850
.108334
.107758
26
27
.116815
.116469
.116087
.115672
.115213
.114714
.114157
.113549
27
28
.122323
.121955
.121553
.121110
.120629
.120093
.119506
.118864
28
29
.127397
.127011
.126582
.126119
.125603
.125038
.124419
.123741
29
30
1.132078
1.131667
1.131219
1.130723
1.130180
1.129585
1.128932
1.128216
30
31
.136393
.135963
.135484
.134963
.134392
.133764
.133076
.132325
31
32
.140376
.139917
.139415
.138867
.138265
.137605
.136884
.136091
32
33
.144045
.143564
.143037
.142461
.141828
.141137
.140376
.139542
33
34
.147432
.146928
.146373
.145767
.145106
.144378
.143578
.142706
34
35
1.150557
1.150026
1.149445
1.148812
1.148116
1.147351
1.146516
1.145601
35
36
.153436
.152881
.152275
.151609
.150878
.150080
.149205
.148247
36
37
.156092
.155513
.154875
.154177
.153415
.152579
.151664
.150663
37
38
.158540
.157932
.157264
.156536
.155739
.154866
.153910
.152866
38
39
.160792
.160155
.159459
.158698
157867
.156955
.155959
.154871
39
40
1.162862
1.162199
1.161472
1.160679
1.159811
1.158861
1.157824
1.156690
40
41
.164765
.164073
.163316
.162489
.161585
.160598
.159517
.158343
41
42
.166512
.165791
.165002
.164141
.163203
.162174
.161055
.159825
42
43
.168113
.167362
.166542
.165649
.164670
.163606
.162435
.161174
43
44
.169577
.168798
.167947
.167016
.166005
.164892
.163692
.162381
44
18
19
20
21
22
23
24
25
6 PER CENT.
Continued on Page *92. TABLE XLVII.
85
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
Pay-
ments.
26
27
28
29
30
31
32
33
Pay-
71.
1.168055
1.165920
1.163661
1.161274
1.158757
1.156097
1.153290
1.150327
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.287059
.287043
.287025
.287002
.286980
.286955
.286929
.286899
2
3
.449323
.449290
.449251
.449207
.449162
.449113
.449058
.448997
3
4
.560579
.560526
.560467
.560401
.560333
.560256
.560172
.560076
4
5
0.643948
0.643876
0.643796
0.643708
0.643614
0.643509
0.643393
0.643263
5
6
.709730
.709639
.709538
.709425
.709304
.709170
.709021
.708858
6
7
.763421
.763309
.763185
.763046
.762898
.762733
.762552
.762350
7
8
.808298
.808165
.808015
.807851
.807674
.807478
.807261
.807023
8
9
.846476
.846318
.846146
.845954
.845748
.845518
.845267
.844989
9
10
0.879396
0.879218
0.879018
0.878800
0.878561
0.878299
0.878010
0.877689
10
11
.908093
.907889
.907664
.907415
.907145
.906855
.906518
.906156
11
12
.933322
.933094
.932841
.932561
.932266
.931923
.931554
.931148
12
13
.955656
.955408
.955125
.954822
.954477
.954104
.953694
.953242
13
14
.975561
.975279
.974973
.974624
.974250
.973838
.973385
.972884
14
15
0.993372
0.993068
0.992720
0.992342
0.991932
0.991480
0.990980
0.990433
15
16
1.009388
1.009044
1.008669
1.008257
1.007809
1.007313
1.006770
1.006170
16
17
.023826
.023456
.023049
.022602
.022113
.021575
.020982
.020330
17
18
.036894
.036494
.036054
.035568
.035039
.034454
.033813
.033107
18
19
.048751
.048320
.047843
.047324
.046746
.046117
.045424
.044661
19
20
1.059533
1.059067
1.058559
1.057989
1.057375
1.056696
1.055949
1.055129
20
21
.069353
.068858
.068302
.067698
.067037
.066307
.065507
.064628
21
22
.078318
.077778
.077190
.076542
.075832
.075053
.074198
.073257
22
23
.086500
.085936
.085301
.084606
.083850
.083020
.082105
.081104
23
24
.093996
.093381
.092708
.091970
.091166
.090278
.089309
.088243
24
25
1.100846
1.100196
1.099482
1.098699
1.097840
1.096902
1.095871
1.094736
25
26
.107124
.106436
.105679
.104845
.103939
.102943
.101846
.100645
26
27
.112880
.112151
.111346
.110468
.109508
.108449
.107291
.106024
27
28
.118156
.117383
.116537
.115607
.114589
.113472
.112252
.110912
28
29
.122993
.122180
.121285
.120301
.119228
.118053
.116765
.115353
29
30
1.127430
1.126571
1.125625
1.124591
1.123462
1.122224
1.120868
1.119385
30
31
.131495
.130588
.129596
.128508
.127321
.126019
.124596
.123040
31
32
.135216
.134267
.133224
.132081
.130834
.129469
.127978
.126346
32
33
.138627
.137629
.136535
.135336
.134029
.132601
.131038
.129332
33
34
.141745
.140700
.139553
.138298
.136932
.135436
.133804
.132022
34
35
1.144595
1.143500
1.142301
1.140990
1.139561
1.138000
1.136297
1.134440
35
36
.147195
.146052
.144799
.143430
.141940
.140313
.138539
.136607
36
37
.149566
.148372
.147064
.145639
.144086
.142393
.140549
.138540
37
38
.151722
.150476
.149117
.147622
.146017
.144258
.142342
.140261
38
39
.153678
.152385
.150959
.149425
.147749
.145922
.143939
.141784
39
40
1.155454
1.154097
1.152635
1.151034
1.149295
1.147406
1.145353
1.143127
40
41
.157046
.155655
.154131
.152471
.150674
.148719
.146600
.144305
41
42
.158496
.157047
.155467
.153754
.151895
.149877
.147695
.145328
42
43
.159791
.158290
.156661
.154889
.152970
.150895
.148644
.146216
43
44
.160948
.159402
.157717
.155889
.153917
.151777
.149470
.146977
44
26
27
28
29
30
31
32
33
6 PER CENT.
Continued on Page *91. 86
TABLE XLYII.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. „
Pay-
II
CO
35
36
37
38
39
40
41
Pay-
ments.
71.
1.147200
1.143903
1.140434
1.136773
1.132914
1.128852
1.124576
1.120076
n.
1
0.000000
0.000000
0.000000
0.000000 -
0.000000
0.000000
0.000000
0.000000
1
2
.286866
.286826
.286789
.286747
.286697
.286643
.286584
.286522
2
3
.448926
.448850
.448773
.448684
.448583
.448475
.448355
.448225
3
4
.559971
.559856
.559735
.559598
.559446
.559279
.559096
.558898
4
5
0.643122
0.642965
0.642799
0.642614'
0.642406
0.642179
0.641931
0.641661
5
6
.708676
.708478
.708266
.708028
.707762
.707475
.707159
.706811
6
7
.762130
.761888
.761626
.761333
.761011
.760659
.760269
.759846
7
8
.806761
.806471
.806158
.805812
.805429
.805007
.804546
.804042
8
9
.844681
.844344
.843980
.843577
.843127
.842638
.842102
.841512
9
10
0.877337
0.876951
0.876533
0.876067
0.875554
0.874994
0.874376
0.873702
10
11
.905757
.905320
.904844
.904318;
.903737
.903101
.902403
.901637
11
12
.930701
.930209
.929677
.929086
.928434
.927722
.926937
.926080
12
13
.952744
.952199
.951605
.950947,
.950224
.949429
.948559
.947603
13
14
.972336
.971733
.971076
.970351
.969550
.968675
.967712
.966658
14
15
0.989829
0.989167
0.988446
0.987648
0.986771
0.985809
0.984754
0.983601
15
16
1.005511
1.004788
1.0039991.003129
1.002170
1.001121
.999973
.998712
16
17
.019614
.018827
.017970
.017023
.015983
.014846
1.013596
1.012227
17
18
.032330
.031480
.030550
.029527;
.028403
.027171
.025819
.024338
18
19
.043824
.042905
.041903
.0408021
.039589
.038260
.036804
.035217
19
20
1.054229
1.053242
1.052167
1.050982
1.049677
1.048251
1.046694
1.044982
20
21
.063663
.062607
.061453
.060182
.058786
.057265
.055591
.053763
21
22
.072227
.071097
.069863
.068505,
.067020
.065390
.063606
.061657
22
23
.080005
.078799
.077483
.076042
.074455
.072721
.070825
.068755
23
24
.087072
.085789
.084395
.082858
.081173
.079334
.077324
.075128
24
25
1.093493
1.092136
1.090654
1.089024
1.087241
1.085295
1.083167
1.080844
25
26
.099333
.097893
.096322
.094602
.092718
.090661
.088414
.085965
26
27
.104634
.103112
.101456
.0996411
.097653
.095484
.093119
.090545
27
28
.109446
.107843
.106098
.104185
.102093
.099814
.097332
.094629
28
29
.113810
.112125
.110288
.108278
.106081
.103693
.101091
.098256
29
30
1.117764
1.115993
1.114065
1.111956
1.109658
1.107157
1.104432
1.101474
30
31
.121339
.119482
.117462
.115258
.112854
.110238
.107397
.104311
31
32
.124565
.122622
.120512
.118210
.115698
.112974
.110013
.106807
32
33
.127470
.125442
.123241
.120837
.118226
.115389
.112316
.108990
33
34
.130080
.127967
.125671
.123175
.120457
.117516
.114331
.110885
34
35
1.132419
1.130217
1.127834
1.125238
1.122423
1.119377
1.116080
1.112524
35
36
.134503
.132220
.129744
.127057
.124144
.120994
.117594
.113929
36
37
.136359
.133990
.131429
.128650
.125639
.122393
.118892
.115126
37
38
.138000
.135551
.132904
.130034
.126934
.123593
.119998
.116131
38
39
.139448
.136919
.134186
.131233
.128044
.124616
.120927
.116975
39
40
1.140717
1.138107
1.135297
1.132261
1.128991
1.125476
1.121707
1.117669
40
41
.141819
.139137
.136250
.133138
.129788
.126197
.122349
.118236
41
42
.142775
.140021
.137063
.133877
.130455
.126791
.122873
.118690
42
43
.143595
.140775
.137748
.134495
.131005
.127276
.123293
.119054
43
44
.144295
.141411
.138321
.135004
.131454
.127665
.123629
.119335
44
34
35
36
37
38
39
40
41
6 PER CENT.
Continued on Page *90, table xlvii.
87
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
6 PER CENT.
Pay-
ments.
42
43
44
45
46
47
48
49
Pay-
ments.
n.
1.115337
1.110361
1.105124
1.099624
1.093846
1.087778
1.081412
1.074733
71.
\ 1
0.000000
0.000000
o.oooooo'o. 000000
0.000000
0.000000
0.000000
0.000000
1
2
.286418
.286374
.286287
.286192
.286090
.285975
.285852
.285715
2
3
.448079
.447923
.447747
.447556
.447345
.447113
.446859
.446580
3
4
.558614
.558434
.558166
.557873
.557551
.557196
.556808
.556380
4
5
0.641356
0.641030
0.640665
0.640267
0.639830
0.639347
0.638819
0.638237
5
6
.706436
.706008
.705544
.705037
.704479
.703865
.703191
.702450
6
7
.759374
.758863
.758296
.757675
.756992
.756241
.755417
.754513
7
8
.803481
.802873
.802198
.801459
.800647
.799755
.798777
.797703
8
9
.840861
.840150
.839364
.838504
.837558
.836521
.835384
.834132
9
10
0.872953
0.872139
0.871238
0.870252
0.869171
0.867982
0.866678
0.865245
10
11
.900791
.899867
.898848
.897735
.896510
.895165
.893689
.892073
11
12
.925129
.924095
.922956
.921708
.920337
.918829
.917182
.915374
12
13
.946549
.945402
.944136
.942749
.941224
.939566
.937729
.935723
13
14
.965498
.964231
.962834
.961303
.959637
.957791
.955777
.953571
14
15
0.983327
0.980937
0.979405
0.977741
0.975899
0.973886
9.971685
0.969273
15
16
.991323
.995806
.994148
.992319
.990321
.988132
.985737
.983119
16
17
1.010118
1.009083
1.007271
1.005295
1.003131
1.000761
.998173
.995338
17
18
.032118
.020938
.018988
.016856
.014523
.011972
1.009180
1.006134
18
19
.033462
.031552
.029455
.027166
.024662
.021920
.018931
.015668
19
20
1.043104
1.041056
1.038813
1.036362
1.033679
1.030753
1.027562
1.024079
20
21
.051756
.049571
.047176
.044558!
.041702
.038587
.035190
.031487
21
22
.059522
.057194
.054643
.051863
.048831
.045523
.041920
.037999
22
23
.066485
.064012
.061308
.058363
.055151
.051651
.047845
.043706
23
24
.072723
.070106
.067248
.064135
.060743
.057053
.053044
.048690
24
25
1.078304
1.075544
1.072529
1.069247
1.065678
1.061798
1.057589
1.053028
25
26
.083291
.080384
.077210
.073764
.070017
.065951
.061548
.056778
26
27
.087734
.084677
.081351
.077738
.073818
.069572
.064975
.060005
27
28
.091677
.088479
.084996
.081222
.077135
.072708
.067924
.062764
28
29
.095172
.091838
.088196
.084265
.080009
.075408
.070448
.065105
29
30
1.098253
1.094709
1.090991
1.086903
1.082485
1.077721
1.072591
1.067073
30
31
.100961
.097341
.093415
.089177
.084607
.079685
.074393
.068713
31
32
.103339
.099572
.095506
.091126'
.086410
.081337
.075895
.070066
32
33
.105383
.101497
.097299
.092784*
.087927
.082715
.077134
.071165
33
34
.107159
.103148
.098825
.094179
.089193
.083851
.078141
.072049
34
35
1.108682
1.104554
1.100109
1.095342
1.090236
1.084776
1.078952
1.072752
35
36
.109978
.105733
.101180
.096302
.091086
.085520
.079596
.073298
36
37
.111066
.106724
.102064
.097084
.091770
.086111
.080097
.073718
37
38
.111980
.107539
.102784
.097713
.092313
.086571
.080481
.074032
38
39
.112732
.108203
.103363
.098213
.092736
.086923
.080770
.074263
39
40
1.113345
1.108737
1.103824
1.098602
1.093059
1.087189
1.080982
1.074429
40
41
.113838
.109162
.104182
.098899
.093304
.087383
.081133
.074544
41
42
.114231
.109492
.104456
.099125
.093482
.087522
.081239
.074622
42
43
.114535
.109745
.104664
.099289
.093610
.087620
.081311
.074669
43
44
.114769
.109937
1 .104815
.099407
.093700
.087686
.081353
.074698
44
42
1 43
44
45
48
47
00
49
i
Continued on Page *89 88
TABLE XL VII.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
6 PER CENT.
Pay-
ments,
II
Or
O
51
52
53
54
55
56
57
Pay-
ments.
n.
1.067728
1.060389
1.052701
1.044652
1.036227
1.027411
1.018200
1.008575
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.285563
.285397
.285214
.285015
.284796
.284553
.284288
.283991
2
3
.446270
.445932
.445561
.445155
.444709
.444213
.443670
.443073
3
4
.555907
.555391
.554825
.554205
.553521
.552763
.551937
.551026
4
5
0.637596
0.636896
0.636128
0.635284
0.634353
0.633329
0.632207
0.630974
5
6
.701636
.700745
.699766
.698688
.697509
.696206
.694781
.693215
6
7
.753519
.752429
.751230
.749920
.748476
.746889
.745152
.743244
7
8
.796521
.795225
.793807
.792249
.790537
.788657
.786599
.784343
8
9
.832757
.831254
.829604
.827793
.825807
.823625
.821241
.818624
9
10
0.863674
0.861954
0.860066
0.857998
0.855728
0.853240
0.850518
0.847537
10
11
.890299
.888355
.886225
.883890
.881334
.878529
.875466
.872112
11
12
.913389
.911218
.908838
.906235
.903382
.900259
.896841
.893117
12
13
.933525
.931119
.928488
.925607
.922457
.919007
.915244
.911132
13
14
.951153
.948512
.945620
.942461
.939005
.935228
.931109
.926615
14
15
0.966634
0.963748
0.960596
0.957152
0.953390
0.949281
0. 944806
0.939930
15
16
.980250
.977121
.973703
.969973
.965901
.961460
.956629
.951371
16
17
.992243
.988866
.985180
.981160
.976779
.972006
.966820
.961186
17
18
1.002807
.999179
.995223
.990914
.986224
.981121
.975587
.969579
18
19
.012106
1.008226
1.004001
.999405
.994408
. .988981
.983100
.976731
19
20
1.020281
1.016150
1.011657
1.006776
1.001478
0.995729
0.989513
0.982792
20
21
.027455
.023074
.018316
.013155
.007558
1.001499
.994957
.987898
21
22
.033734
.029106
.024088
.018649
.012764
.006404
.999550
.992168
22
23
.039211
.034342
.029065
.023359
.017195
.010546
1.003394
.995707
23
24
.043972
.038862
.033336
.027372
.020941
.014017
.006583
.998615
24
25
1.048085
1.042744
1.036979
1.030767
1.024082
1.016898
1.009206
1.000974
25
26
.051621
.046058
.040064
.033617
.026692
.019269
.011334
.002867
26
27
.054642
.048867
.042655
.035986
.028841
.021194
.013043
.004363
27
28
.057204
.051227
.044810
.037938
.030585
.032741
.014394
.005528
28
29
.059357
.053191
.046586
.039522
.031988
.023964
.015447
.006418
29
30
1.061150
1.054810
1.048028
1.040797
1.033098
1.024917
1.016251
1.007085
30
31
.062629
.056125
.049189
.041806
.033963
.025645
.016853
.007576
31
32
.063830
.057183
.050108
.042593
.034624
.026191
.017297
.007928
32
33
.064797
.058022
.050824
.043194
.035119
.026593
.017616
.008173
33
34
.065564
.058675
.051372
.043644
.035484
.026882
.017837
.008336
34
35
1.066161
1.059176
1.051783
1.043976
1.035746
1.027082
1.017985
1.008440
35
36
.066619
.059551
.052085
.044215
.035928
.027216
.018079
.008503
36
37
.066962
.059827
.052302
.044380
.036049
.027301
.018135
.008539
37
38
.067214
.060025
.052453
.044490
.036127
.027352
.018169
.008558
38
39
.067395
.060163
.052554
.044561
.036173
.027383
.018185
.008567
39
40
1.067521
1.060255
1.052618
1.044603
1.036201
1.027397
1.018193
1.008572
40
41
.067606
.060313
.052656
.044629
.036214
.027404
.018198
.008575
41
42
.067658
.060348
.052680
.044641
.036220
.027409
.018200
.008575
42
43
.067690
.060370
.052691
.044646
.036225
.027410
.018200
.008575
43
44
.067710
.060380
.052695
•044651|
.036226
.027410
.018200
44
45
1.067719
1.060383
1.052700
1.044652
1.036226
1.027411
45
46
.067722
.060388
.052701
.044652
.036227
46
47
.067727
.060389
.052701
47
48
.067728
.060389
48
49
.067728
49
50
51
52
53
54
55
56
57 TABLE XLVII.
89
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF T1IE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
6 PER CENT.
' Pay-
ments .
58
59
60
61
62
63
64
65
Pay-
n.
0.998532
0.988049
0.977114
0.965719
0.953857
0.941502
0,928659
0.915312
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.283673
.283321
.282933
.282505
.282040
.281525
.280960
.280338
2
3
.442422
.441706
.440916
.440049
.439104
.438057
.436911
.435652
3
4
.550033
.548939
.547736
.546419
.544975
.543385
.541642
.539730
4
5
0.629627
0.628145
0.626519
0. 624735
0. 622783
0.620634
0.618282
0.615705
5
6
.691504
.689626
.687562
.685303
.682828
.680110
.677137
.673883
6
7
.741161
.738873
.736365
.733616
.730611
.727313
.723711
.719772
7
8
.781875
.779171
.776205
.772962
.769416
.765533
.761293
.756662
8
9
.815767
.812635
.809208
.805460
.801368
.796893
.792012
.786688
9
10
0.844280
0.840718
0.836820
0.832563
0.827923
0.822851
0.817329
0.811318
10
11
.868454
.864452
.860080
.855314
.850119
.844455
.838297
.831601
11
12
.889049
.884607
.879762
.874480
.868737
.862484
.855695
.848329
12
13
.906654
.901771
.896447
.890658
.884370
.877535
.870129
.862111
13
14
.921727
.916400
.910605
.904312
.897487
.890083
.882077
.873430
14
15
0.934630
0.928866
0.922604
0.915815
0.908466
0.900511
0.891928
0.882677
15
16
.945665
.939468
.932748
.925476
.917619
.909135
.900000
.890181
16
17
.955078
.948458
.941293
.933552
.925210
.916220
.906567
.896212
17
18
.963081
.956050
.948452
.940266
.931459
.921996
.911854
.901017
18
19
.969852
.962422
.954415
.945802
.936563
.926654
.916073
.904792
19
20
0.975546
0.967739
0.959340
0.950331
0.940684
0.930376
0.919392
0.907716
20
21
.980303
.972136
.963374
.953992
.943982
.933306
.921966
.909944
21
22
.984242
.975742
.966638
.956924
.946579
.935580
.923927
.911611
22
23
.987476
.978662
.969254
.959236
.948596
.937315
.925396
.912831
23
24
.990096
.981004
.971319
.961032
.950137
.938615
.926471
.913705
24
25
0.992199
0.982853
0.972923
0.962404
0.951291
0.939565
0.927242
0.914315
25
26
.993860
.984291
.974149
.963432
.952135
.940248
.927780
.914720
26
27
.995153
.985389
.975068
.964185
.952742
.940724
.928137
.914978
27
28
.996140
.986213
.975742
.964726
.953165
.941040
.928364
.915133
28
29
.996880
.986818
.976225
.965103
.953445
.941241
.928501
.915222
29
30
0.997425
0.987252
0.976563
0.965352
0. 953624
0.941471
0.928580
0.915271
30
31
.997815
.987555
.976785
.965512
.953828
.941433
.928623
.915292
31
32
.998087
.987754
.976929
.965694
.953795
.941454
.928641
.915307
32
33
.998266
.987883
.977091
.965665
.953813
.941487
.928655
.915313
33
34
.998383
.988028
.977065
.965681
.953843
.941499
.928660
.915313
34
35
0.998513
0.988005
0.977079
0.965708
0.953853
0.941503
0.928660
0.915312
35
36
.998492
.988017
.977103
.965717
.953857
.941503
.928659
37
.998503
.988039
.977111
.965720
.953857
.941502
43
43
38
.998523
.988046
.977114
.965720
.953857
44
1.115337
58
39
.998529
.988049
.977114
.965719
45
1.110361
.115337
57
40
0.998532
0.988049
0.977114
A a
1.105124
.110361
.115337
56
41
.998532
.988049
*±o
1.099624
.105124
.110361
.115337
55
42
.998532
48
47
1.093846
1.099624
1.105124
1.110361
1.115333
54
4 q
1.087778
.093846
.099624
.105124
.110357
.115333
53
1.081412
.087778
.093846
.099623
.105120
.110357
.115328
52
51
1.074733
.081412
.087778
.093845
.099619
.105120
.110351
.115317
51
50
.074733
.081412
.087777
.093841
.099619
.105113
.110340
.115302
50
49
1.074733
1.081411
1.087773
1.093840
1.099612
1.105101
1.110323
1.115277
49
48
.074732
.081406
.087772
.093833
.099599
.105083
.110296
.115233
48
47
.074727
.081405
.087764
.093819
.099579
.105053
.110248
.115167
47
46
.074725
.081396
.087749
.093797
.099546
.105001
.110176
.115074
46
45
.074716
.081380
.087725
.093761
.099490
.104924
.110076
.114946
45
49
48
47
46
45
44
43
43
Continued from Page *87. 90
TABLE XLYII.
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
6 PER CENT.
Pay-
ments.
11.
x=66
67
68
69
70
71 i 72
73
Pay- 1
ments.
0.901459
0.887078
0.872181
0.856733
0.840762
0.824241
0.807180
0.789571
11.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
0.000000
.279660
.434275
.537641
0.612888
.670330
.715472
.751614
.780896
0.804784
.824339
.840356
.853450
.864109
0.872733
.879649
.885148
.889463
.892801
0.895343
.897243
.898633
.899628
.900323
0,900785
.901078
.901254
.901356
.901412
0.901436
.901453
.901460
.901460
.901459
0.000000
.278908
.432761
.535345
0.609799
.666436
.710770
.746106
.774584
0.797679
.816459
.831725
.844096
.854071
0.862047
.868376
.873334
.877164
.880078
0.882254
.883846
.884984
.885779
.886307
0.886643
.886844
.886961
.887024
.887052
0.887071
.887079
.887079
.887078
0.000000
.278097
.431109
.532840
0.606427
.662192
.705653
.740116
.767731
0.789981
.807939
.822413
.834033
.843293
0.850622
.856350
.860770
.864127
.866633
0.868466
.869774
.870687
.871294
.871680
0.871912
.872046
.872118
.872150
.872172
0.872181
.872182
.872181
0.000000
.277190
.429283
.530080
0.602724
.657542
.700056
.733580
.760272
0.781623
.798712
.812359
.823190
.831734
0.838395
.843526
.847417
.850318
.852439
0.853952
.855007
.855708
.856154
.856423
0.856577
.856660
.856697
.856723
.856733
0.856734
.856733
38
0.000000
.276207
.427295
.527074
0.598694
.652485
.693975
.726492
.752199
0.772595
.788774
.801549
.811586
.819388
0.825383
.829923
.833302
.835770
.837531
0.838757
.839572
.840090
.840402
.840581
0.840678
.840720
.840751
.840762
.840764
0.840762
37
0.000000
.275121
.425110
.523780
0.594284
.646959
.687351
.718788
.743446
0.762839
.778052
.789946
.799157
.806213
0.811546
.815510
.818401
.820462
.821896
0.822850
.823454
.823819
.824029
.824143
0.824191
.824228
.824240
.824243
.824241
36
1.140434
.140433
.140433
.140433
.140432
1.140432
.140428
.140421
.140412
.140397
1.140368
.140327
.140268
.140189
.140075
1.139929
.139737
.139489
.139179
.138793
0.000000
.273936
.422726
.520183
0.589474
.640949
.680156
.710440
.733991
0.752312
.766548
.777520
.785895
.792208
0.796892
.800302
.802731
.804421
.805544
0.806255
.806684
.806931
.807065
.807122
0.807165
.807179
.807183
.807180
35
1.143903
1.143902
.143902
.143902
.143901
.143901
1.143898
.143891
.143883
.143868
.143842
1.143804
.143749
.143675
.143570
.143435
1.143257
.143026
.142739
.142381
.141943
0.000000
.272640
.420111
.516248
0.584227
.634404
.672342
.701403
.723764
0.741000
.754203
.764238
.771776
.777355
0.781410
.784295
.786299
.787631
.788473
0.788982
.789275
.789433
.789500
.789552
0.789568
.789573
.789571
34
1.147200
.147198
1.147198
.147198
.147197
.147197
.147195
1.147188
.147181
.147167
.147142
.147107
1.147056
.146987
.146891
.146765
.146599
1.146385
.146119
.145786
.145379
.144886
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
66
65
64
63
62
61
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
39
1.136773
.136773
.136773
.136773
1.136771
.136771
.136767
.136760
.136750
1.136733
.136702
.136658
.136594
.136509
1.136386
.136228
.136022
.135754
.135420
40
1.124576
1.124575
.124575
.124575
.124572
.124572
1.124567
.124558
.124545
.124524
.124486
1.124430
.124350
.124241
.124089
.123889
1.132914
.132913
.132913
1.132913
.132911
.132911
.132907
.132899
1.132888
.132870
.132837
.132790
.132720
1.132628
.132496
.132325
.132103
.131814
41
1.120076
120076
.120076
.120076
.120073
1.120073
.120068
.120058
.120044
.120021
1.119980
.119919
.119833
.119715
.119551
1.128852
.128851
1.128851
.128851
.128849
.128849
.128844
1.128836
.128824
.128804
.128769
.128718
1.128643
.128543
.128401
.128216
.127976
41
40
39
38
37
36
35
34
Continued from Page *86. TABLE X L V II.
91
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
6 PER CENT.
Pay-
ments.
74
' 75
76
77
78
79
80
81
Pay-
ments.
n.
n.
0.771402
0.752711
0.733470
0.713672
0.693356
0.672502
0.651114
0.629190
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.271204
.269646
,267945
.266064
.264019
.261782
.259335
.256690
2
3
.417238
.414120
.410707
.406961
.402891
.398447
.393621
.388367
3
4
.511942
.507265
.502157
.496577
.490520
.483943
.476798
.469061
4
5
0.578498
0.572282
0.565525
0.558147
0.550179
0.541542
0.532205
0.522129
5
6
.627277
.619569
.611186
.602107
.592309
.581737
.570351
.558124
6
7
.663864
.654696
.644789
.634083
.622576
.610209
.596956
.582800
7
8
.691612
.681078
.669724
.657506
.644426
.630436
.615521
.599672
8
9
.712756
.700933
.688242
.674642
.660148
.644724
.628364
..611079
9
10
0.728834
0.715826
0.701919
0. 687086
0.671351
0.654689
0.637113
0.618529
10
11
.740985
.726906
.711919
.696009
.679211
.661516
.642856
.623185
11
12
.750074
.735048
.719124
.702298
.684619
.666013
.646457
.625980
12
13
.756780
.740935
.724219
.706638
.688190
.668840
.648623
.627578
13
14
.761642
.745L10
.727744
.709510
.690438
.670543
.649863
.628445
14
15
0.765097
0.748002
0.730080
0.711320
0.691795
0.671518
0.650536
0.628809
15
16
-.767494
.749922
.731553
.712413
.692572
.672048
.650818
.629089
16
17
.769086
.751134
.732443
.713040
.692994
.672270
.651035
.629177
17
18
.770092
.751866
.732955
.713380
.693171
.672440
.651104
.629208
18
19
.770700
.752287
.733232
.713523
.693306
.672495
.651128
.629190
19
20
21
22
0.771049
.771239
.771319
0.752515
.752611
.752684
0.733348
.733437
.733466
0.713632
.713667
.713679
0.693350
.693365
.693356
0.672514
.672502
28
1.163661
.163659
.163659
1.163659
0.651114
26
27
1.168055
74
23
24
25
26
.771380
.771400
0.771406
.771402
33
1.150327
.752708
.752716
0.752711
32
.733476
.733470
31
1.156097
.713672
30
1.158757
1.158755
29
1,161274
.161272
1.161272
1.165920
.165918
.165918
.165918
1.165917
.168054
.168054
.168054
.168053
1.168053
73
72
71
70
69
67
1.153290
.153289
.156096
.156096
.158755
.158755
.161272
.161271
.163658
.163658
.165917
.165916
.168052
.168049
68
67
66
.150325
.153289
.156096
.158754
.161271
.163657
.165912
.168045
66
65
.150325
.153289
.156095
.158754
.161269
.163653
.165908
.168037
65
64
1.150325
1.153288
1.156095
1.158752
1.161265
1.163648
1.165900
1.168024
64
63
.150324
.153288
.156093
.158747
.161260
.163640
.165885
.168002
63
62
.150324
.153286
.156087
.158742
.161251
.163624
.165862
.167973
62
61
.150322
.153279
.156082
.158732
.161234
.163599
.165831
.167935
61
60
.150315
.153274
.156071
.158714
.161208
.163566
.165790
.167881
60
59
1.150309
1.153262
1.156051
1.158686
1.161173
1.163522
1.165732
1.167810
59
58
.150296
.153240
.156022
.158649
.161125
.163459
.165656
.167718
58
57
.150273
.153209
.155982
.158597'
.161058
.163378
.165557
.167600
57
56
.150240
.153166
.155926
.158525
.160971
.163271
.165431
.167448
56
55
.150193
.153106
.155849
.158431
.160855
.163131
.165268
.167260
55
54
1.150129
1.153023
1.155747
1.158307
1.160705
1.162961
1.165066
1.167031
54
53
.150039
.152914
.155614
.158146!
.160522
.162744
.164820
.166754
53
52
.149922
.152770
.155441
.157949
.160290
.162480
.164523
.166422
52
51
.149769
.152585
.155229
.157700
.160006
.162162
.164167
.166029
51
50
.149569
.152357
.154962
.157395
.159664
.161780
.163745
.165570
50
49
1.149322
1.152069
1.154634
1.157027
1.159254
1.161326
1.163252
1.165031
49
48
.149013
.151717
.154239
.156586
.158766
.160797
.162674
.164416
48
47
.148634
.151292
.153765
.156063
.158198
.160177
.162013
.163705
47
46
.148177 .150782
.153202
.155452
.157533
.159467
.161250
.162899
46
45
.147628
.150177
.152545
.154737
.156769
.158647
.160385
.161983
45
33
32
1 31
30 |
29
28
27
26
Continued from Page *85. 92
TABLE XLYI1.
LIFE AND TEMPORARY ANNUITIES LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
6 PER CENT.
Pay-
a;=82
83
84
85
86
87
88
89
Pay-
ments.
71.
0.606694
0.583647
0.559909
0.535453
0.510023
0.483173
0.454503
0.422455
71.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.253744
.250591
.247111
.243358
.239313
.234793
.230075
.222029
2
3
.382611
.376412
.369650
.362393
.354522
.345962
.335400
.320218
3
4
.460632
.451581
.441793
.431288
.420014
.406931
.390968
.370386
4
5
0.511220
0.499566
0.487020
0.473656
0.458801
0.441565
0.421437
0.396806
5
6
.544974
.530980
.516032
.499792
.481703
.461257
.438034
.410543
6
7
.567662
.551649
.534368
.515579
.494998
.472187
.446811
.417331
7
8
.582833
.564914
.545601
.524861
.502462
.478024
.451185
.420496
8
9
.592671
.573117
.552261
.530109
.506472
.480948
.453233
.421807
9
10
0.598793
0.578007
0.556044
0.532939
0.508142
0.482321
0.454083
0.422301
10
11
12
13
.602457
.580794
.558089
.534121
.509434
.482891
.454403
.454503
.422455
11
.604549
.605683
• 5S230/i
.558944
.535035
.509829
.483106
.483173
18
.582934
.559606
.535316
.509977
19
14
.606159
.583423
.559810
.535420
.510023
20
1.181467
.181466
82
15
0.606526
0.583574
.583629
.583647
0.559885
0.535453
21
1.180102
81
16
17
18
.606640
.606681
.559909
1.178657
.180101
.181466
80
22
23
1.177130
1.178656
1.180101
1.181466
79
.606694
24
1.175513
.177128
.178656
.180101
.181466
78
25
1.173803
.175512
.177128
.178656
.180101
.181466
77
1.171993
.173802
.175512
.177128
.178656
.180101
.181466
76
75
1.170082
.171993
.173802
.175512
.177128
.178656
.180101
.181464
75
74
1.170080
1.171993
1.173802
1.175512
1.177128
1.178656
1.180099
1.181461
74
73
.170080
.171993
.173802
.175512
.177128
.178653
.180096
.181457
73
72
.170080
.171992
.173802
.175512
.177125
.178650
.180091
.181450
72
71
.170079
.171992
.173801
.175509
.177122
.178645
.180084
.181436
71
70
.170079
.171991
.173798
.175505
.177116
.178637
.180069
.181420
70
69
1.170078
1.171988
1.173794
1.175499
1.177108
1.178621
1.180052
1.181396
69
68
.170075
.171984
.173788
.175490
.177091
.178603
.180027
.181366
68
67
.170071
.171978
.173778
.175472
.177071
.178577
.179994
.181325
67
66
.170064
.171967
.173759
.175451
.177043
.178542
.179950
.181273
66
65
.170052
.171947
.173736
.175421
.177006
.178494
179894
.181205
65
64
1.170031
1.171922
1.173704
1.175381
1.176955
1.178434
1.179821
1.181118
64
63
.170004
.171888
.173661
.175327
.176891
.178356
.179729
.181009
63
62
.169968
.171842
.173604
.175258
.176807
.178257
.179612
.180876
62
61
.169918
.171780
.173529
.175168
.176701
.178132
.179470
.180719
61
60
.169852
.171700
.173433
.175055
.176567
.177980
.179301
.180526
60
59
1.169767
1.171598
1.173311
1.174911
1.176405
1.177799
1.179095
1.180302
59
58
.169657
.171466
.173157
.174737
.176210
.177579
.178855
.180037
58
57
.169515
.171302
.172971
.174528
.175975
.177321
.178571
.179729
57
56
.169340
.171102
.172747
.174276
.175698
.177017
.178241
.179376
56
55
.169126
.170862
.172477
.173979
.175373
.176664
.177863
.178967
55
54
1.168868
1.170573
1.172159
1.173632
1.174995
1.176259
1.177426
1.178504
54
53
.168558
.170232
.171787
.173226
.174561
.175791
.176930
.177980
53
52
.168192
.169833
.171351
.172761
.174059
.175260
.176369
.177389
52
51
.167764
.169365
.170853
.172223
.173491
.174659
.175735
.176728
51
50
.167262
.168831
.170276
.171615
.172847
.173979
.175028
.175988
50
49
1.166689
1.168213
1.169624
1.170924
1.172117
1.173221
1.174234
1.175163
49
48
.166027
.167513
.168883
.170142
.171305
.172371
.173350
.174256
48
47
.165275
.166718
.168045
.169271
.170394
.171423
.172378
.173246
47
46
.164422
.165819
.167110
.168294
.169377
.170381
.171296
.172138
46
45
.163458
.164816
.166063
.167203
.168260
.169221
.170108
.170916
45
25
24
23
22
21
20
19
18
Continued from Page *84. TABLE X L Y II.
93
LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n
ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR.
6 PER CENT
Pay-
ments
1 90
91
92
93
94
95
96
97
Pay-
ments.
n.
0.391855
0.363938
0.335983
0.307090
0.277673
0.246844
0.210226
0.174939
n.
1
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1
2
.213204
.205516
.197480
.188212
.178148
.167818
.151126
.139047
2
3
.304780
.290824
.270008
.259484
.242126
.222201
.196801
.174939
3
4
.349841
.330929
.310990
.289433
.206205
.241074
.210226
5
0.372540
0.349933
0.326562
0.301314
0.274956
0.246844
11
10
6
.383580
.358617
.332882
.305714
.277673
1.189964
90
7
8
.388701
.390812
.362181
.363519
.335244
.335983
.307090
13
12
1.189102
1.189962
89
9
.363938
14
1.188192
.189101
.189962
88
15
1.187226
.188191
. 189101
.189962
87
10
0.391855
1 6
1.180201
.187225
.188191
.189101
.189962
86
17
1.185120
.186200
.187225
.188191
.189101
.189962
85
1.183972
1.185118
1.186200
1.187225
1.188191
1.189101
1.189962
84
83
1.182755
.183972
.185118
.186200
.187225
.188191
.189101
.189962
83
82
.182754
.183972
.185118
.186200
.187225
.188191
.189101
.189960
82
81
.182754
.183972
.185118
.186200
.187225
.188191
.189099
.189957
81
80
.182754
.183972
.185118
.186200
.187225
.188189
.189096
.189952
80
79
1.182754
1.183972
1.185118
1.186200
1.187222
1.188186
1.189091
1.189945
79
78
.182754
.183972
.185118
.186197
.187219
.188181
.189082
.189936
78
77
.182754
.183971
.185115
.186194
.187214
.188171
.189073
.189922
77
76
.182753
.183968
.185112
.186188
.187203
.188161
.189057
.189905
76
75
.182750
.183905
.185105
.186177
.187192
.188144
.189040
.189881
75
74
1.182740
1.183958
1.185094
1.186105
1.187174
1.188126
1.189014
1.189850
74
73
.182739
.183946
.185081
.186146
.187155
.188098
.188981
.189809
73
72
.182720
.183932
.185061
.186125
.187125
.188063
.188938
.189760
72
71
.182711
.183910
.185038
.186093
.187088
.188017
.188885
.189697
71
70
.182088
.183885
.185004
.180054
.187039
.187960
.188817
.189618
70
69
1.182001
1.183849
1.184962
1.180001
1.186977
1.187887
1.188733
1.189527
69
68
.182622
.183804
.184905
.185935
.186899
.187798
.188635
.189414
68
67
.182574
.183744
.184835
.185852
.186804
.187693
.188515
.189284
67
66
.182510
.183668
.184746
.185750
.186692
.187565
.188376
.189129
66
65
.182429
.183574
.184037
.185630
.186555
.187416
.188210
.188950
65
64
1.182328
1.183457
1.184508
1.185484
1.186395
1.187239
1.188018
1.188744
64
63
.182203
.183319
.184352
.185313
.186206
.187034
.187799
.188506
63
62
.182050
.183152
.184169
.185110
.185987
.186799
.187544
.188236
62
61
.181876
.182950
.183952
.184876
.185735
.186526
.187255
.187931
61
60
.181667
.182724
.183701
.184607
.185444
.180218
.186929
.187588
60
59
1.181419
1.182455
1.183414
1.184295
1.185114
1.185869
1.186562
1.187204
59
58
.181131
.182148
.183080
.183942
.184741
.185470
.186152
.186773
58
57
.180802
.181791
.182702
.183543
.184320
.185038
.185691
.186295
57
56
.180420
.181386
.182275
.183093
.183851
.184544
.185179
.185769
56
55
.179987
.180929
.181794
.182591
.183324
.183996
.184617
.185184
55
54
1.179498
1.180414
1.181216
1.182027
1.182736
1.183395
1.183991
1.184542
54
53
.178946
. 179838
.180653
.181398
.182094
.182725
.183304
.183837
53
52
.178329
.179193
.179980
.180710
.181377
.181991
.182549
.183063
52
51
.177638
.178473
.179243
.179943
.180591
.181183
.181722
.182218
51
50
.176868
.177683
.178422
.179101
.179726
.180297
.180817
.181297
50
49
1.176021
1.176804
1.177520
1.178175
1.178777
1.179328
1.179831
1.180293
49
48
.175079
.175838
.176528
.177159
.177740
.178272
.178756
.179200
48
47
.174045
.174775
.175440
.176049
.176010
.177122
.177585
.173014
47
46
.172905
.173609
.174251
.174838
.175377
.175867
.176316
.176727
46
45
.171656
.172335
.172953
.173517
.174033
.174508
.174938
.175333
45
17
16
15
14 |
13
12
11
10
Continued from Page *83. 94
TABLE XLVI1 I.
CONVERSION TABLE FROM ANNUITY OR A(l + «) TO ANNUAL PREMIUM ON 1,000.
INTEREST 4 PER CENT
Applicable to ordinary Life and Endowment Insurances, to Joint Lives and Survivorships, and all
cases of the annexed Formula, where a denotes the Present Value of the Annuity. At 4 per cent. 1—v
is .03840154. P is theAnnual Premium.
For $1,000 Insured, P = 1,000 j — (1—®) j •
For a Rate of Interest other than 4 per cent., apply the following Correction, 1,000(»—the Tabular
result *
For the rate of.. 3 4 41 5 6 7 ® percent.
Subtract .... .... .... 4.001 9.158 18.142 20.959 35013
Add 9.335 4.645 0.000
Example. Interest 5 percent. A(l + a) = 0.130820; the Tabular result found to be 691.306 ; this,
less the above Correction 9.158, leaves the Annual Premium required $082.15.
PROPORTION All PARTS, SURT.
A(l+«).
0,5
1,6
2,7
3,8
4,9
1
2
3
4
5
6
7
8
9
0.00
0
961.54
1
959.24
21956.94
3
954.65
4
952.37
23
461
69
92
115
137]
LOO
L83
200
5
950.09
6
947.82
7
945.55
8
943.29
9
941.03
23
45
68
90
113
136
L58
L81
204
0.01
0
938.78
1
936.53
2
934.29
3
932.05
4
929.82
22
45
67
90
112
134
157
179
202
5
927.59
6
925.37
7
923.15
8
920.94
9
918.73
22
44
36
88
111
133
155
177
199'
0.02
0
916.53
1
914.33
2
912.14'
3
909.96
4
907.78
22
44
66
88
110
131
153
175
197
5
905.60
6
903.43
7 901.26
8
899.10
9
896.94
22
43
65
86
108
130
151
173
195
0.03
0
894.79
1
892.65
2*890.50
3
888.37
4
886.24
21
43
64
86
107
128
150
171
192
5 884.11
6
881.99
7
879.87
8
877.76
9
875.65
21
42
63
85
106
127
148
169
190
0.04
0 873.55
1
871.45
2
869.36
3
867.27
4
865.19
21
42
63
84
105
125
146
167
188
5l863.ll
6
861.04
7
858.97
8
856.90
9 854.84
21
41
62
83
103
124
144
165
186
0.05
0 852.79
1
850.74
2
848.69
3
846.65
4 844.62
20
41
61
82
102
122
143
163
184
5842.59
6
840.56
7838.54
8
836.52
9
834.51
20
40
61
81
101
121
141
161
182
0.06
0 832.50
1
830.50
2828.50
3
826.51
4
824.52
20
40
60
80
100
120
140
160
179
5 822.53
6
820.55
7;818.58
8
816.61
9
814.64
20 39
59
79
99
118
138
158
177
0.07
0812.68
1
810.72
2
808.77
3
806.82
4
804.87
20
39
59
78
98
117
137
156
176
5
802.93
6
801.00
7
799.07
8
797.14
9795.22
19
39
58
77
96
116
135
154
173
0.08
0
793.30
1
791.39
2
789.48
3
787.58
4 785.68
19
38
57
76
95
114
133
152
171
5
783.78
6
781.89
7
780.00
8
778.12
9
776.24
19
38
56
75
94
113
132
151
169
0.09
0
774.37
1
772.50
2
770.63
3
768.77
4
766.92
19
,37
56
74
93
112
130
149
168
5
765.06
6
763.22
7
761.37
8
759.53
9
757.70
18
j37
55
74
92
110
129
147
165
0.10
0
755.87
1
754.04
2
752.22
3
750.40
4
748.58
18
36
55
73
91
109
127
146
164;
5
746.77
6
744.97
7
743.17
8
741.37
9
739.58
18
36
54
72
90
108
126
144
162
0.11
0
737.79
1
736.00
2
734.22
3
732.44
4
730.67
18
36
53
71
89
107
124
142
160:
5
728.90
6
727.14
7
725.37
8
723.62
9
721'. 86
18
35
53
70
88
105
123
140
158!
0.12
0
720.12
1
718.37
2,716.63
3
714.89
4
713.16
17
35
52
70
87
104
122
139
156
5
711.43
6
709.71
71707.99
8
706.27
9
704.56
17
34
51
69
8G
103
120
137
154
0.13
0
702.85
11701.14
2
699.44
3
697.75
4
696.05
17
34
51
68
85
102
119
136
153
5
694.36
61692.68
7
691.00
8
689.32
9
687.64
17
34
50
67
84
101
117
134
151
0.14
0
685.97
1
684.31
2
682.65
3
680.99
4
679.33
17
33
50
66
83
99
116
132
149
5
677.68
6
676.03
7
674.39
8
672.75
9
671.12
16
33
49
66
82
98
115
131
148
1 0.15
0
669.48
1
667.86
2
666.23
3
664.61
4
662.99
16
32
49
65
81
97
113
13C
146
5
661.38
6
659.77
7
658.17
8
656.56
9
654.96
16
32
48
64
80
9f
112
128
144
0.16
0
653.37
1
651.78
2
650.19
3
648.61
4
647.03
16
32
48
63
79
95
111
12r
“143
5
645.45
e
>643.88
7
642.31
8
640.74
9
639.18
16
31
47
63
78
94
11C
121
>141
0.17
C
637.62
]
.636.07
2
634.52
3
632.97
4
631.42
15
31
46
62
77
93
108
124
139
51629.88
(
>628.35
7
'626.81
8
625.28
9
623.76
15
31
46
61
77
92
107
12i
138
0,5
1,6
2,7
3,8
4,9
1
2
3
4
5
6
7
8
9 TABLE XLVTIT.
95
CONVERSION TABLE FROM ANNUITY OR A(l + «) TO ANNUAL PREMIUM ON 1,000.
INTEREST 4 PER CENT.
PROPORTIONAL PARTS, SUBT.
A(l-)-rt).
0,5
1,6
2,7
3,8
4,9
1
2
3
4
5
6
7
8
9
0.18
0
622.23
1
620.71
2
619.20
3
617.68
4
616.17
1 15
30
45
60
76
91
106
121
136
5
614.67
6
613.17
7
611.67
8
610.17
9
608.68
15
30
45
60
75
90
105
120
135
0.19
0
607.19
1
605.71
2
604.23
3
602.75
4
601.27
15
30
44
59
74
89
103
118
133
5
599.80
6
598.33
7
596.87
8
595.41
9
593.95
15
29
44
58
13
88
102
117
131
0.20
0
592.50
1
591.04
2
589.60
3
588.15
4
586.71
14
29
43
58
72
87
101
116
130
5
585.27
6
583.84
7
582.41
8
580.98
9
579.55
14
29
43
57
71
86
100
114
129
0.21
0
578.13
1
576.72
2
575.30
3
573.90
4
572.48
14
28
42
56
71
85
99
113
127
5
571.08
6
569.67
7
568.27
8
566.89
9
565.49
14
28
42
56
70
84
98
112
126
0.22
0
564.10
1
562.71
2
561.33
3
559.95
4
558.57
14
28
41
55
69
83
97
110
124
5
557.20
6
555.83
7
554.46
8
553.10
9
551.74
14
27
41
55
68
82
95
109
123
0.23
0
550.38
1
549.03
2
547.68
3
546.33
4
544.98
13
27
40
54
67
81
94
108
121
5
543.64
6
542.30
7
540.97
8
539.63
9
538.30
13
27
40
53
67
80
93
107
120
0.24
0
536.98
1
535.66
2
534.33
3
533.02
4
531.70
13
26
40
53
66
79
92
105
119
5
530.39
6
529.08
7
527.78
8
526.48
9
525.18
13
26
39
52
65
78
91
104
117
0.25
9
523.89
1
522.59
2
521.30
3
520.02
4
518.72 I
13
26
39
52
65
77
90
103
116
51517.44
6
516.16
7
514.90
8
513.62
9
512.35
13
25
38
51
64
76
89
102
114
0.26
0511.09
1
509.82
2
508.55
3
507.30
4
506.04
13
25
38
50
63
75
88
101
113
5504.80
6
503.54
7
502.29
8
501.05
9
499.81
12
25
37
50
62
75
87
100
112
0.27
0498.57
1
497.34
2
496.10
3'494.87
4
493.65
12
25
37
49
62
74
86
98
111
5
492.42
6
491.20
7
489.98
8 488.77
9
487.56
12
24
36
49
61
73
85
97
109
0.28
0
486.35
1
485.14
2
483.93
3 482.73
4
481.53
12
24
36
48
60
72
84
96
108
5
480.34
6
479.15
7
477.95
8
476.77
9
475.58
12
24
36
48
59
71
83
95
107
0.29
0
474.40
1
473.22
2
472.04
3
470.88
4
469.70
12
23
35
47
59
70
82
94
106
5
468.53
6
467.36
7
466.20
8
465.04
9
463.88
12
23
35
46
58
70
81
93
104
0.30
0
462.73
1
461.57
2
460.42
3
459.28
4
458.13
11
23
34
46
57
69
80
92
103
5
456.99
6
455.85
7
454.71
8
453.58
9
452.45
11
23
34
45
57
68
79
91
102
0.31
0
451.32
1
450.19
2
449.07
3
447.95
4
446.83
11
22
34
45
56
67
79
90
101
5
445.71
6
444.60
7
443.49
8
442.38
9
441.27
11
22
33
44
55
66
78
89
100
0.32
0
440.17
1
439.07
2
437.97
3
436.87
4
435.78
11
22
33
44
55
66
77
88
99
5
434.69
6
433.60
7
432.52
8
431.43
9
430.35
11
22
33
43
54
65
76
87
98
0.33
0
429.27
1
428.20
2
427.12
3
426.05
4
424.99
11
21
32
43
54
64
75
86
96
5
423.92
6
422.86
7
421.80
8
420.74
9
419.68
11
21
32
42
53
63
74
85
95
0.34
0
418.63
1
417.58
2
416.53
3
415.48
4
414.44
10
21
31
42
52
63
73
84
94
5
413.39
6
412.36
7
411.32
8
410.28
9
409.25
10
21
31
41
52
62
72
83
93
0.35
0
408.22
1
407.19
2
406.18
3
405.15
4
404.13
10
20
31
41
51
61
71
82
92
5
403.12
6
402.09
7
401.08
8
400.08
9
399.06
10
20
30
41
51
61
71
81
91
0.36
0
398.05
1
397.05
2
396.06
3
395.05
4
394.05
10
20
30
40
50
60
70
80
90
5
393.06
6
392.07
7
391.07
8
390.09
9
389.10
10
20
30
40
49
59
69
79
89
0.37
0
388.12
1
387.14
2
386.16
3
385.18
4
384.21
10
20
29
39
49
59
68
78
88
5
383.24
6
382.27
7
381.30
8
380.33
9
379.38
10
19
29
39
48
58
68
77
87
0.38
0
378.41
1
377.46
2
376.49
3
375.54
4
374.59
' 10
19
29
38
48
57
67
76
86
5
373.64
6
372.69
7
371.74
8
370.81
9
369.86
9
19
28
38
47
57
66
75
85
0.39
0
368.93
1
367.98
2
367.05
3
366.11
4
365.18
9
19
28
37
47
56
65
75
84
5
364.26
6
363.34
7
362.41
8
361.48
9
360.56
9
18
28
37
46
55
65
74
83
0.40
0
359.65
1
358.73
2
357.82
3'356.91
4
356.00
9
18
27
36
46
55
64
73
82
5
355.10
6
354.18
7
353.28
8 352.39
9
351.48
9
18
27
36
45
54
63
72
81
0.41
0
350.58
1
349.69
2
348.80
31347.91
4
347.02
9
18
27
36
45
53
62
71
80
5
346.13
6
345.25
7
344.36
8 343.48
9
342.60
9
18
26
35
44
53
62
70
79
0,5
1,6
2,7
3,8
4,9
1
2
3
4
5
6
7
8
9 96
TABLE XL Y III.
CONVERSION TABLE FROM ANNUITY OR A(l +a) TO ANNUAL PREMIUM ON 1,0C0.
INTEREST 4 PER CENT.
PROPORTIONAL PARTS, STJBT.
A(l-f-a).
0,5
1,6
2,7
3,8
4,9
1
2
3
4
5
6
7
8
9
0.42
0
341.73
1
340.85
2
339.98
3
339.11
4
338.24
9
17
26
35
44
52
61
70
78
5
337.38
6
336.51
7
335.66
8
334.79
9
333.93
9
17
26
34
43
52
60
69
78
0.43
0
333.07
1
332.22
2
331.37
3
330.52
4
329.67
9
17
26
34
43
51
60
68
77
5
328.82
6
327.98
7
327.13
8
326.29
9
325.45
8
17
25
34
42
50
59
67
76
0.44
0
324.62
1
323.78
2
322.95
3
322.12
4
321.29
8
17
25
33
42
50
58
67
75
5
320.46
6
319.63
7
318.81
8317.99
9
317.17
8
16
25
33
41
49
58
66
74
0.45
0
316.35
1
315.54
2
314.72
3313.91
4
313.10
8
16
24
32
41
49
57
65
73
5
312.29
6
311.48
7
310.68
8:309.88
9
309.07
8
16
24
32
40
48
56
64
72
0.46
0
308.28
1
307.48
2
306.68
3305.89
4
305.10
8
16
24
32
40
48
56
64
71
5
304.31
6
303.52
7
302.73
8301.95
9
301.16
8
16
24
31
39
47
55
63
71
0.47
0
300.38
1
299.60
2
298.83
3 298.05
4
297.28
8
16
23
31
39
47
54
62
70
5
296.50
6
295.73
7
294.96
8,294.20
9
293.43
8
15
23
31
38
46
54
61
69
0.48
0
292.67
1
291.91
2
291.15
3 290.39
4
289.63
8
15
23
30
38
45
53
61
68
5
288.88
6
288.13
7
287.38
8 286.63
9
285.88
8
15
23
30
38
45
53
60
68
0.49
0
285.13
1
284.39
2
283.65
3 282.90
4
282.17
7
15
22
30
37
44
52
59
67
5
281.43
6
280.69
7
279.96
81279.23
9
278.50
7
15
22
29
37
44
51
59
66
0.50
0
277.77
1
277.04
2
276.31
3275.59
4
274.87
7
14
22
29
36
43
51
58
65
5
274.15
6
273.43
7
272.71
8 271.99
9
271.28
7
14
21
29
36
43
50
57
64
0.51
0
270.57
1
269.86
2
269.15
3
268.44
4
267.73
7
14
21
28
35
42
50
57
64
5
267.03
6
266.33
7
265.63
8
264.93
9
264.23
7
14
21
28
35
42
49
56
63
0.52
0
263.53
1
262.84
2
262.15
3
261.45
4
260.77
7
14
21
28
35
41
48
55
62
5
260.08
6
259.39
7
258.71
8
258.02
9
257.34
7
14
21
27
34
41
48
55
62
0.53
0
256.66
1
255.98
2
255.30
3
254.63
4
253.95
7
14
20
27
34
41
47
54
61
5
253.28
6
252.61
7
251.94
8:251.27
9
250.61
7
13
20
27
33
40
47
53
60
0.54
0
249.94
1
249.28
2
248.62
3
247.96
4
247.30
7
13
20
26
33
40
46
53
59
5
246.64
6
245.98
7
245.33
8
244.68
9
244.03
7
13
20
26
33
39
46
52
59
0.55
0
243.38
1
242.73
2
242.08
3
241.44
4
240.79
6
13
19
26
32
39
45
52
58
5
240.15
6
239.51
7
238.87
8)238.23
9
237.60
6
13
19
26
32
38
45
51
57
0.56
0
236.96
1
236.33
2
235.70
3
235.07
4
234.44
6
13
19
25
32
38
44
50
57
5
233.81
6
233.18
7
232.56
8
231.93
9
231.31
6
12
19
25
31
37
44
50
56
0.57
0
230.69
1
230.07
2
229.46
3
228.84
4
228.22
6
12
18
25
31
37
43
49
55
5
227.61
6
227.00
7
226.39
8
225.78
9
225.17
6
12
18
24
30
36
43
49
55
0.58
0
224.57
1
223.96
2
223.36
3
222.75
4
222.15
6
12
18
24
30
36
42
48
54
5
221.55
6
220.96
7
220.36
8
219.76
9
219.17
6
12
18
24
30
36
42
48
53
0.59
0
218.58
1
217.99
2
217.40
3
216.81
4
216.22
6
12
18
24
29
35
41
47
53
5
215.64
6
215.05
7
214.47
8
213.89
9
213.31
6
12
17
23
29
35
41
47
52
0.60
0
212.73
1
212.15
2
211.57
3
211.00
4
210.42
6
12
17
23
29
35
40
46
52
5
209.85
6
209.28
7
208.71
8
208.14
9
207.58
6
11
17
23
28
34
40
45
51
0.61
0
207.01
1
206.44
2
205.88
3
205.32
4
204.76
6
11
17
22
28
34
39
45
51
5
204.20
6
203.64
7
203.08
8 202.53
9
201.97
6
11
17
22
28
33
39
44
50
0.62
0
201.42
1
200.87
2
200.32
3199.77
4
199.22
5
11
16
22
27
33
38
44
49
5
198.68
6
198.13
7
197.59
8 197.04
9
196.50
5
11
16
22
27
33
38
44
49
0.63
0
195.96
1
195.42
2
194.88
3194.35
4
193.81
5
11
16
21
27
32
38
43
48
5
193.28
6
192.75
7192.21
8191.68
9
191.15
5
11
16
21
27
32
37
42
48
0.64
0
190.63
1
190.10
2189.57
3189.05
4
188.53
5
11
16
21
26
32
37
42
47
5
188.00
6
187.48
7186.96
81186.44
9
185.93
5
10
16
21
26
31
36
41
47
0.65
0
185.41
1
184.90
2184.38
3183.87
4
183.36
5
10
15
20
26
31
36
41
46
5
182.85
6
182.34
7181.83
8481.32
.9
180.82
5
10
15
20
25
30
36
41
46
0,5
1,6
2,7
3,8
4,9
1
2
3
4
5
6
7
8
9 T A B L TC X L V III.
97
CONVERSION TABLE FROM ANNUITY OR A(1 +a) TO ANNUAL PREMIUM ON 1,000.
INTEREST 4 PER CENT.
PROPORTIONAL PARTS, STTBT.
A(1+a).
0,5
1,6
2,7
3,8
4,9
1
2
3
4
5
6
7
8
9
0.66
0
180.31
1
179.81
2
179.31
3
178.81
4178.31
5
10
15
20
25
30
35
40
45
5
177.81
6
177.31
7
176.82
8
176.32
9
175.83
5
10
15
20
25
30
35
40
45
0.67
0
175.33
1
174.84
2
174.35
3
173.86
4
173.37
5
10
15
20
24
29
34
39
44
5
172.89
6
172.40
7
171.92
8
171.43
9
170.95
5
10
15
19
24
29
34
39
44
0.83
0
170.47
1
169.99
2
169.51
o
O
169.03
4
168.55
5
10
14
19
24
29
33
38
43
5
LG8.08
8
167.60
7
167.13
8
166.65
9
166.18
5
9
14
19
24
28
33
38
43
0.69
0
165.71
1
165.24
2
164.77
3
164.31
4
163.84
5
9
14
19
23
28
33
37
42
5
163.38
6
162.91
7
162.45
0
161.99
9
161.52
5
9
14
19
23
28
32
37
42
0.70
0
161.06
1
160.61
2
160.15
3
159.69
4
159.24
5
9
14
18
23
27
32
36
41
5
158.78
6
158.33
7
157.87
8
157.42
9
156.97
5
9
14
18
23
27
32l 36
41
0.71
0
156.52
1
156.07
2
155.63
3
155.18
4
154.74
4
9
13
18
22
27
31
36
40
5
154.29
6
153.85
7
153.41
8
152.96
9
152.52
4
9
13
18
22
27
31
35
40
0.72
0152.08
1
151.65
2
151.21
3|150.77
4
150.34
4
9
13
17
22
26
31
35
39
5149.90
6
149.47
7
149.04
8:148.61
9
148.18
4
9
13
17
22
26
30
34
39
0.73
0147.75
1
147.32
2
146.89
3146.47
4
146.04
4
9
13
17
21
26
30
34
38
5145.62
8
145.19
7
144.77
8144.35
9
143.93
4
8
13
17
21
25
30
34
38
0.74
0143.51
1
143.09
2
142.67
3142.26
4
141.84
4
8
12
17
21
25
29
33
37
5141.43
6
141.0L
7;140.60
8 140.19
9
139.78
4
8
12
16
21
25
29
33
37
0.75
0139.37
1
138.96
2 138.55
3138.14
4
137.74
4
812
16
20
24
29
33
37
513 7.33
3
136.93
7136.52
8436.12
9
135.72
4
8
12
1(
20
24
28
32
36
0.76
0135.32
1
134.92
2134.52
3 134.12
4
133.73
4
8
12
16
20
24
28
32
36
5
133.33
3
132.93
7132.54
8132.15
9
131.75
4
8
12
If
20
24
28
32
35
0.77
0
131.33
1
139.97
2130.58
3130.19
4
129.81
4
8
12
If
19
23
27
31
35
5
129.42
6
129.03
7j 128.65
8 128.26
9
127.88
4
8
12
15
19
23
27
31
35
0.78
0
127.50
1
127.12
2126.73
3126.35
4
125.98
4
8
11
15
19
23
27
30
34 !
5
125.60
6
125.22
7
124.84
8 124.47
9
124.09
4
8
11
15
19
23
26
30
34
0.79
0
123.721
1
123.35
2
122.97
3122.60
4
122.23
4
7
11
15
19
22
26
30
33
5
121.861
6
121.49
7
121.13
8120.76
9
120.39
4
7
11
15
18
22
26
29
33
0.80
0
120.03
1
119.66
2
119.30
3118.94
4
118.57
4
7
11
15
18
22
25
29
33
5
118.21
6
117.85
7
117.49
8117.14
9
116.78
4
7
11
14
18
21
25
29
32
0.81
0
116.42
1
116.06
2
115.71
3
115.35
4
115.00
4
7
11
14
18
21
25
28
32
5
114.65
6
114.39
7
113.94
8113.59
0
113.24
4
7
11
14
18
21
25
28
32
0.82
0
112.89
1
112.55
2
112.20
3111.85
4
111.51
3
7
10
14
17
21
24
28
31
5
111.16
6
110.82
7
110.47
8
110.13
£
109.79
3
7
10
14
17
21
24
27
31
0.83
0
109.45
1
109.11
2
108.77
3
108.43
4
108.09
3
7
10
14
17
20
24
27
30
5
107.76
8
107.42
7
107.08
8
106.75
G
K/
106.42
3
7
10
13
17
20
24
27
30
0.84
0
106.08
1
105.75
2
105.42
3
105.08
4
104.76
3
7
10
13
17
20
23
26
30
5
104.43
8
104.10
7
103.77
8
103.44
9
103.12
3
7
10
13
16
20
23
26
30
0.85
0
102.79
1
102.47
2
102.14
3
101.82
4
101.50
3
6
10
13
16
19
2r
26
29
5
101.18
6
100.85
7
100.53
8
100.21
9
99.80
3
6
10
13
16
19
22
26
29
0.86
0
99.58
1
99.26
2
98.94
3
98.63
4
98.31
3
6
9
13
16
19
22
25
28
5
98.00
6
97.68
7
97.37
8
97.06
9
96.75
3
6
9
13
16
19
22
25
28
0.87
0
96.43
1
96.12
2
95.82
3
95.51
4
95.20
3
6
9
12
15
18
22
25
28
5
94.89
8
94.58
7
94.28
8
93.97
0
93.67
3
6
9
12
15
18
21
24
28
0.88
0
93.36
1
93. OG
2
92.76
3
92.46
4
92.16
3
6
9
12
15
18
21
24
27
5
91.86
6
91.56
7
91.26
8
90.96
9
90.66
3
C
9
12
15
18
21
24
27
0.89
0
90.36
1
90.07
2
89.77
3
89.48
4
89.18
3
6
9
1?
15
18
21
24
26
5
88.89
6
I 88.60
7
88.30
8
88.01
9
87.72
3
6
9
12
15
18
20
23
26
1
0,5
|
1,6
1 2,7
3,8
4,9
1
2
3
4
5
6
7
8
9 98
TABLE XLVII1.
CONVERSION TABLE FROM ANNUITY OR A(l-fa) TO ANNUAL PREMIUM ON 1,000.
INTEREST 4 PER CENT.
A(l+«)
0
1
2
3
4
5
6
7
8
9
1
23
4
5
6
7
8
9
0.90
8743
87.14
86.85
86.56
86.28
85.99
85.70
85.42
85.13
84.85
3
6
9
11
14
17|20 23
26
0.91
84-57
84.28
84.00
83.72
83.44
83.16
82.88
82.60
82.32
82.04
3
6
8
11
14
17
20
22
25
0.92
81.77
81.49
81.21
80.94
80.66
80.39
80.12
79.84
79.57
79.30
3
5
8
11
14
16
1922
25
0.93
79.03
78.76
78.49
78.22
77.95
77.68
77.42
77.15
76.88
70.62
3
5
8
11
13
16
1921
24
0.94
76.35
76.09
75.83
75.56
75.30
75.04
74.78
74.52
74.26
74.00
3
5
8
10
13
16
1821
23
0.95
73.74
73.48
73.22
72.97
72.71
72.46
72.20
71.95
71.69
71.44
3
5
8
10
13
15
1820
23
0.96
71.18
70.93
70.68
70.43
70.18
69.93
69.68
69.43
69.19
68.84
2
5
7
10
12
15
1720
22
0.97
68.69
68.44
68.20
67.95
67-71
67.46
67.22
66.98
66.73 66.49
2
5
7
10
12
15
1720
22
0.98
66.25
66.01 65.77
65.53
65.29
65.05
64.81
64.58
64.34 64.10
2
5
7
10
12
14
1719
21
0.99
63.87
63.63 63.40
63.16
62.93
62.69
62.46
62.23
62.00
61.77
2
5
7
9
12
14
16 19
21
1.00
61.54
61.31 61.08
60.85
60-62
60.39
60.17
59.94
59.71
59.49
2
5
7
9
11
14
1618
21
1.01
59.26
59.0458.81
58.59
58-37
58.14
57.92
57.70
57.48
57.25
2
4
7
9
11
13
1618
20
1.02
57.04
56.82 56.60
56.38
56-16
55.95
55.73
55.51
55.30
55.08
2
4
7
9
11
13
1517
20
1.03
54.8654.65
54.44
54.22
54-01
53.80
53.58
53.37
53.16
52,95
2
4
6
8
11
13
15 17
19
1.04
52.74l52.53
52.32
52.11
51.90
51.70
51.49
51.28
51.08
50.87
12
4
6
8
10
12
15 17
19
1.05
50.6650.46
50.25
50.05
49.85
49.64
49.44
49.24
49.04
48.84
2
4
6
8
10
12
1416
18
1.06
48.6448.44
48.24
48.04
47.84
47.64
47.44
47.24
47.05
46.85
2
4
6
8
10
12
1416
18
1.07
46.6546.46
46.26
46.07
45.87
45.68
45.49
45.29
45.10
44.91
2
4
6
8
10
12
1415
17
1.08
44.7244.52
44.33
44.14
43.95
43.76
43.57
43.39
43.20
43.01
2
4
6
8
10
11
13 15
17
1.09
42.82 42.64 42.45
42.26
42.08
41.89
41.71
41.52
41.34
41.15
2
4
6
7
9
11
1315
17
1.10
40.97 40.79 40.61
40.43
40.24
40.06
39.88
39.70
39.52
39.34
2
4
5
7
9
11
13 14
16
1.11
39.16 38.99 38.81
38.63
38.45
38.28
38-10
37.92
37.75
37.57
2
4
5
7
9
11
1214
16
1.12
37.40 37.22 37.05
36.87
36.70
36.53
36.36
36.18
30.01
35.84
2
3
5
7
9
10
12 14
16
1.13
35.67 35.50 35.33
35.16
34.99
34.82
34.65
34.48
34.32
34.15
2
3
5
7
8
10
1214
15
1.14
33.98 33.82 33.65
33.48
33.32
33.15
32.99
32.82
32.66
32.50
2
3
5
7
8
10
1213
15
1.15
32.33 32.17
32.01
31.85
31.68
31.52
31.36
31.20
31.04
30.88
2
3
5
6
8
10
1113
14
1.16
30.7230.56
30.40
30.25
30.09
29.93
29.77
29.62
29.40
29.30
2
3
5
6
8
9
11
13
14
1.17
29.15
28.99
28.84
28.68
28.53
28.37
28.22
28.07
27.91
27.76
3
3
5
0
8
9
11
12
14
1.18
27.61
27.46
27.30
27.15
27.00
26.85
26.70
2 G
26.40
26.25
2
3
5
6
8
9
11
12
14
1.19
26.10
25.96
25.81
25.66
25.51
25.37
25.22
25.07
24.93
24.78
1
3
4
6
7
9
10
12
13
1.20
24.63
24.49
24.34
24.20
24.06
23.91
23.77
23.63
23.48
23.34
1
3
4
6
7
9
10
11
13
1.21
23.2023.06
22.91
22.77
22.63
22.49
22.35
22.21
22.07
21.93
1
3
4
6
7
8
10
11
13
1.22
21.79
21.66
21.52
21.38
21.24
21.10
20.97
20.83
20.69
20.50
i
3
4
5
7
8
10
11
12
1.23
20.42
20.29
20.15
20.02
19.88
19.75
19.61
19.48
19.35
19.22
1
3
4
5
7
8
9
11
12
1.24
19.08
L8.95 18.82
18.6918.55
18.42
18.29
18.16
18.03
17.90
1
3
4
5
7
8
9
10
12
1.25
17.77
17.6417.51
17.39 17.26
17.13
17.00
16.87
10.75
10.62
1
3
4
5
0
8
9
10
12
1.26
16.49
16.37
16.24
16.1115.99
15.86
15.74
15.61
15.49
15.37
1
3
4
5
6
8
9
10
11
1.27
15.24
15.12
14.99
14.87
14.75
14.63
14.50
14.38
14.26
14.14
1
2
4
5
6
7
9
10
11
1.28
14.02
13.90
13.78
13.66
13.54
13.42
13.30
13.18
13.06
12.94
1
2
4
5
6
7
8
10
11
1.29
12.82
12.71
12.59
12.47
12.35
12.24
12.12
12.00
11.89
11.77
1
2
3
5
6
7
8
9
10
1.30
11.66
11.54
11.43
11.31
11.20
11.08
10.97
10.86
10.74
10.63
1
2
3
5
6
7
8
9
10
1.31
10.52
10.40
10.29
10.18 10.07
9.96
9.84
9.73
9.62
9.51
1
2
3
4
6
tv
i
8
9
10
1.32
9.40
9.29
9.18
9.07
8.96
8.85
8.74
8.64
8.53
8.42
1
2
3
4
5
7
8
9
10
L. 33
8.31
8.20
8.10
7.99
7.88
7.78
7.67
7.56
7.46
7.35
1
2
3
4
5
6
7
8
10
1.34
7.25
7.14
7.04
6.93
6.83
6.72
6.62
6.52
6.41
6.31
1
2
3
4
5
6
7
8
0
! 1.35
6.21
6.10
6.00
5.90
5.80
5.70
5.59
5.49
5.39
5.29
1
2
3
4
5
6
7
8
9
1.36
5.19
5.09
4.99
4.89
4.79
4.69
4.59
4.49
4.39
4.29
1
2
3
4
5
6
7
8
9
1.37
4.20
4.10
4.00
3.90
3.81
3.71
3.61
3.51
3.42
3.32
1
2
3
4
5
6
7
8
9
1.38
3.23
3.13
3.03
2.94
2.84
2.75
2.65
2.56
2.46
2.37
'I
2 3
4
5
6
7
8
9
1.39
2.28
2.18
2.09
2.00
1.90
1.81
1.72
1.63
1.53
1.44
1
2
3
4
5
6
7
7
8
0
1
2
3
4
5
6
7
8
9
i
23
4
5
6
7
¥
9
PROPORTIONAL, PARTS, SUBT. TA B L E X LIX.
99
CONVERSION TABLE FROM ANNUITY OR A(l + «) TO SINGLE PREMIUM ON 1,000.
INTEREST 4 PER CENT.
Applicable to ordinary Life and Endowment Insurance, to Joint Lives and Survivorships, and all
cases under the annexed Formula:
For $1,000 Insured, A = 1,000 )l — (1—®) (l + a)j.
For other Rates of Interest: Before entering the Table, correct 7.(1 + a) by 7(l—v') — A(l—v), thus:
For the rate of. 3 31 4 41 5 6 7 8 percent.
Subtract 0.120743 .055899 .000000 .A
Add..., 049070 .092754 .167819 .230688 .284640
Example. Interest 3 per cent. il(l + a) = 0.368943. Subtracting 0.120743. leaves 0.248200. With
this, the Table gives the required Single Premium 931-89.
A(l-t-a).
0
1
2
3
4
5
6
7
8
9
*Ref.
0.00
961.54
961.45
961.36
961.27
961.18
961.09
961.00
960.91
960.82
960.73
1.39
0.01
960.64
960.55
960.46
960.37
960.28
960.19
960.09
960.00
959.91
959.82
1.39
0.02
959.73
959.63
959.54
959.45
959.35
959.26
959.17
959.07
958.98
958.88
1.38
0.03
958.79
958.69
958.60
958.50
958.41
958.31
958.21
958.12
958.02
957.92
1.37
0.04
957.83
957.73
957.63
957.54
957.44
957.34
957.24
957.14
957.04
556.94
1.36
0.05
956.85
956.75
956.65
956.55
956.45
956.35
956.25
956.14
956.04
955.94
1.35
0.06
955.84
955.74
955.64
955.53
955.43
955.33
955.23
955.12
955.02
954.92
1.34
0.07
954.81
954.71
954.60
954.50
954.39
954.29
954.18
954.08
953.97
953.87
1.33
0.08
953.76
953.65
953.55
953.44
953.33
953.22
953.12
953.01
952.90
952.79
1.32
0.09
952.68
952.57
952.46
952.35
952.24
952.13
952.02
951.91
951.80
951.69
1.31
0,10
951.58
951.47
951.36
951.24
951.13
951.02
950.91
950.79
950.68
950.57
1.30
0.11
950.45
950.34
950.22
950.11
949.99
949.88
949.76
949.65
949.53
949.41
1.29
0.12
949.30
949.18
949.06
948.95
948.83
948.71
948.59
948.47
948.36
948.24
1.28
0.13
948.12
948.00
947.88
947.76
947.64
947.52
947.39
947.27
947.15
947.03
1.27
0.14
946.91
946.79
946.66
946.54
946.42
946.29
946.17
946.05
945.92
945.80
1.26
0.15
945.67
945.55
945.42
945.29
945.17
945.04
944.92
944.79
944.66
944.53
1.25
0,16
944.41
944.28
944.15
944.02
943.89
943.76
943.63
943.50
943.37
943.24
1.24
0.17
943.11
942.98
942.85
942.72
942.58
942.45
942.32
942.19
942.05
941.92
1.23
0.18
941.79
941.65
941.52
941.38 941.25
941.11
940.98
940.84
940.70
840.57
1.22
0.19
940.43
940.29
940.16
940.02 939.88
939.74
939.60
939.46
939.32
939.18
1.21
0.20
939.04
938.90
938.76
938.62 938.48
938.34
938.19
938.05
937.91
937.77
1.20
0.21
937.62
937.48
937.33
937.19
937.05
936.90
936.75
936.61
936.46
936.32
1.19
0.22
936.17
936.02
935.88
935.73
935.58
935.43
935.28
935.13
934.98
934.83
1.18
0.23
934.68
934.53
934.38
934.23
934.08
933.93
933.77
933.62
933.47
933.32
1.17
0.24
933.16
933.01
932.85
932.70
932.54
932.39
932.23
932.08
931.92
931.76
1.16
0.25
931.60
931.45
931.29
931.13
930.97
930.81
930.65
930.49
930.33
930.17
1.15
0.26
930.01
929.85
929.69
929.53
929.36
929.20
929.04
928.87
928.71
928.55
1.14
0.27
928.38
928.22
928.05
927.88
927.72
927.55
927.38
927.22
927.05
926.88
1.13
0.28
926.71
926.54
926.37
926.21
926.03
925.86
925.69
925.52
925.35
925.18
1.12
0.29
925.01
924.83
924.66
924.49
924.31
924.14
923.96
923.79
923.61
923.44
1.11
0.30
923.26
923.08
922.90
922.73
922.55
922.37
922.19
922.01
921.83
921.65
1.10
0.31
921.47
921.29
921.11
920.93
920.74
920.56
920.38
920.20
920.01
919.83
1.09
0.32
919.64
919.46
919.27
919.09
918.90
918.71
918.52
918.34
918.15
917.96
1.08
0.33
917.77
917.58
9 L 7.39
917.20
917.01
916.82
916.63
916.43
916.24
916.05
1.07
0 34
915.86
915.66
915.47
915.27
915.08
914.88
914.68
914.49
914.29
914.09
1.06
0.35
913.90
913.70
913.50
913.30
913.10
912.90
912.70
912.50
912.29
912.09
1.05
0 36
911.89
911.69
911.48
911.28
911.07
910.87
910.66
910.46
910.25
910.04
1.04
0.37
909.84
909.63
909.42
909.21
909.00
908.79
908.58
908.37
908.16
907.95
1.03
0.38
907.74
907.52
907.31
907.10
906.88
906.67
906.45
906.24
906.02
905.81
1.02
0.39
905.59
905.37
905.15
904.93
904.71
904.49
904.27
904.05
903.83
903.61
1.01
0.40
903.39
903.17
902.94
902.72
902.50
902.27
902.04
901.82
901.59
901.37
1.00
0.41
901.14
900.91
900.68
900.45
900.22
899.99
899.76
899.53
899.30
899.07
1.00
0.42
898.84
898.60
898.37
898.13
897.90
897.66
897.43
897.19
896.96
896.72
0.99
0
1
2
3
4
5
6
7
8
9
* Ref. This refers to Proportional Parts, common to both, on the preceding Page. 100
TABLE XLIX.
CONVERSION TABLE FROM ANNUITY OR A(l + a) TO SINGLE PREMIUM ON 1,000.
INTEREST 4 PER CENT.
PROPORTIONAL PARTS, STTBT.
A(l-f- a).
0,5
1,6
2,7
3,8
4,9
1
2
3
4
5
6
7
8
9
0.43
0
896.48
1
896.24
2
896.00
3,895.76
4
895.52
2
5
7
10
12
14
17
19
22
5
895.28
6
895.04
7
894.80
8
894.55
9
894.31
2
5
i
10
12
] 5
17
19
22
0.44
0
894.07
1
893.82
2
893.58
3
893.33
4
893.09
2
5
7
10
12
15
17
20
22
5
892.84
6
892.59
7
892.35
8
892.10
9
891.85
2
5
r-
(
10
12
15
17
20
22
0.45
0
891.60
1
891.35
2
891.10
3
890.85
4
890.60
3
5
8
10
13
15
18
20
23
5
890.35
6
890.09
7
889.84
8
889.59
9
889.33
3
5
8
10
13
15
18
20
23
0.46
0
889.08
1
888.82
2
888.56
3
888.31
4
888.05
3
5
8
10
13
15
18
21
23
5
887.79
6
887.53
7
887.27
8
887.01
9
886.75
3
5
8
10
13
16
18
21
23
0.47
0
886.49
1
886.23
2
885.97
3
885.71
4
885.44
3
5
8
10
13
16
18
21
24
5
885.18
6 884.91
7
884.65
8
884.38
9
884.12
3
5
8
11
13
16
19
21
24
0.48
0
883.85
1
883.58
2
883.31
3
883.04
4
882.77
3
5
8
11
14
16
19
22
24
5|882.50
6
882.23
7
881.96
8
881.69
9
881.42
3
5
8
11
14
16
19
22
24
0.49
0881.14
1
880.87
2
880.59
3
880.32
4
880.04
3
5
8
1L
14
16
19
22
25
5 879.77
6
879.49
7
879.21
8
878.93
9
878.65
3
6
811
14
17
20
22
25
0.50
0878.37
1
878.09
2
877.81
3
877.53
4
877.25
3
6
811
14
17
20
22
25
5 876.97
6
876.68
7
876.40
8
876.11
9
875.83
3
6
911
14
17
20
23
26
0.51
0875.54
1
875.25
2
874.97
3
874.68
4
874.39
3
6
912
14
17
20
23
26
5 874.10
6
873.81
7
873.52
8
873.23
9
872.93
3
6
9
12
15
18
20
23
26
0.52
0872.64
1
872.35
2
872.05
3
871.76
4
871.46
3
6
9
12
15
18
21
24
26
5
871.17
6
870.87
7
870.57
3
870.27
9
869.98
3
6
9
12
15
18
21
24
27
0.53
0
869.68
1
869.37
2
869.07
3
868.77
4
868.47
3
6
9
12
15
18
21
24
27
5
868.17
6
867.86
7
867.56
8
867.25
9
866.95
3
6
9
12
15
18
21
24
28
0.54
0
856.64
1
866.33
2
866.02
3
865.72
4
865.41
3
6
9
12
15
18
22
25
28
5
855.10
8
864.78
7
864.47
3
864.16
1
863.85
3
6
9
13
16
19
22
25
28
0.55
0
863.53
1
863.22
2
862.90
3
862.59
4
862.27
3
6
9
13
16
19
22
25
28
5
861.95
6
861.63
7
861.32
8
861.00
9
860.68
3
6
10
13
16
19
22
26
29
0.56
0
860.35
1
860.03
2
859.71
3
859.39
4
859.06
3
6
10
13
16
19
23
26
29
5
858.74
3
858.41
7
858.09
3
857.76
9
857.43
3
7
10
13
16
20
23
26
30
0.57
0
857.10
1
856.77
2
856.44
3
856.11
4
855.78
3
7
10
13
17
20
23
26
30
5
855.45
8
855.11
7
854.78
8
854.45
9
854.11
3
7
10
13
17
20
24
27
30
0.58
0
853.77
1
853.44
2
853.10
3
852.76
4
852.42
3
7
10
14
17
20
24
27
30
5 852.08
6
85 J.74
7
851.40
8
851.05
9
850.71
3
7
10
14
17
21
24
27
31
0.59
0850.37
1
850.02
2
849.68
3
849.33
4
848.98
3
t
1C
14
17
21
24
28
31
5848.63
6
848.29
7
847.94
8
847.59
9
847.23
4
7
11
14
18
21
25
28
32
0.60
0846.88
1
846.53
2
846.18
3
845.82
4
845.47
4
7
11
14
18
21
25
28
32
5 845.11
6
844.75
7
844.39
8
844.04
9
843.68
4
7
11
14
\ 8
21
25
29
32
0.61
0
843.32
1
842.95
2
842.59
3
842.23
4
841.87
4
7
11
15
18
22
25
29
33
5
841.50
6
841.14
7
840.77
8
840.40
9
840.03
4
7
11
15
18
22
26
29
33
0.62
0
839.67
1
839.30
2
838.93
3
838.55
4
838.18
4
7
11
15
19
22
26
30
33
5
837.81
6
837.44
7
837.06
8
836.68
9
836.31
4
8
11
15
19
23
26
30
34
0.63
0
835.93
1
835.55
2
835.17
3
834.79
4
834.41 |
4
8
11
15
19
23
27
30
34
5
834.03
6
833.65
7
833.27
8
832.88
9
832.50
4
8
12
15
19
23
27
3] 35
0.64
0
832.11
1
831.72
2
831.33
3
830.95
4
830.56
4
8
12
16
19
23
27
31 35 i
5
830.17
6
829.77
71829.38
8
828.99
9
828.59
4
8
12
16
20
24 28
32
35
0.65
0
828.20
1
827.80
2827.41
3
827.01
4
826.61
4
8
12
16
20
24
28
32
36
5
826.21
6
825.81
7825.41
8
825.00
9 824.60 |
4
8
12
16
20
24
28
32
36
0.66
0
824.20
1
823.79
2 823.39
3822.98
4822.57
4
8
12
16
20
24
29
33
37
5
822.16
6
821.75
71821.34
8 820.93
9 820.52
4
8
12
16
21
25
29
33 37
0,5
1,6
2,7
3,8
4,9
1
2
3
4
5
6
7
8
9 TABLE X LI X.
101
CONVERSION TABLE FROM ANNUITY OR A(l-)-a) TO SINGLE PREMIUM ON 1,000.
INTEREST 4 PER CENT.
A(1 + a).
0,5
1,6
2,7
3,8
4,9
1
2
3
4
5
6
7
8
9
0.67
0(820.10
1
819.69
2
819.27
3
818.85
4
818.44
4
8
12
17
2-1
25
29
33
37
5^818.02
6
817.60
7
817.18
8
816.76
9
816.33
4
8
13
17
21
25
30
34
38
0.68
0815.91
1
815.49
2
815.06
3
814.64
4
814.21
4
9
13
17
21
26
30
34
38
5 813.78
6
813.35
7
812.92
8
812.49
9
812.06
4
9
13
17
22
26
30
35
39
0.69
0811.62
1
811.19
2
810.75
3
810.32
4
809.88
4
9
13
17
22
26
31
35
39
5 809.44
6
809.00
7
808.56
8 808.12
9
807.68
4
9
13
18
22
26
31
35
40
0.70
0807.24
1
806.79
2
806.35
3
805.90
4
805.45
4
9
13
18
22
27
31
36
40
5805.00
6
804.55
7
804.10
8
803.65
9
803.20
5
9
14
18
23
27
32
36
41
0.71
0!802.75
1
802.29
2
801.84
3
801.38
4
800.92
5
9
14
18
23
27
32
37
41
5
800.46
6
800.00
7
799.54
8799.08
9
798.62
5
9
14
18
23
28
32
37
42
0.72
0
798.15
1
797.69
2
797.22
3 796.75
4
796.28
5
9
14
19
23
28
33
37
42
5
795.81
6
795.34
7
794.87
8794.40
9
793.92
5
9
14
19
24
28
33
38
42
0.73
0
793.45
1
792.97
2
792.50
3 792.02
4
791.54
5
10
14
19
24
29
33
38
43
5
791.06
6
790.58
7
790.09
8 789.61
9
789.12
5
10
15
19
24
29
34
39
44
0.74
0
788.64
1
788.15
2
787.66
3787.17
4
786.68
5
10
15
20
25
29
34
39
44
5
786.19
6
785.70
7
785.20
8 784.71
9
784.21
5
10
15
20
25
30
35
40
45
0.75
0
783.71
1
783.22
2
782.72
3 782.22
4
781.71
5
10
15
20
25
30
35
40
45
5
781.21
6
780.71
7
780.20
8 779.69
9
779.19
5
10
15
20
25
30
35
40
46
0.76
0
778.68
1
778.17
2
777.66
3
777.14
4
776.63
5
10
15
21
26
31
36
41
46
5
776.11
6
775.60
7
775.08
8
774.56
9
774.04
5
10
16
21
26
31
36
41
47
0.77
0
773.52
1
773.00
2
772.48
3
771.95
4
771.43
5
10
16
21
26
31
37
42
47
5
770.90
6
770.37
7
769.84
8
769.31
9
768.78
5
11
16
21
27
32
37
42
48
0.78
0
768.25
1
767.71
2
767.18
3
766.64
4
766.10
5
11
16
22
27
32
38
43
48
5
765.56
6
765.02
7
764.48
8
763.94
9
763.39
5
11
16
22
27
33
38
43
49
0.79
0
762.85
1
762.30
2
761.75
3
761.20
4
760.65
6
11
17
22
28
33
39
44
50
5
760.10
6
759.55
7
758.99
8
758.44
9
757.88
6
11
17
22
28
33
39
44
50
0.80
0
757.32
1
756.76
2
756.20
3
755.64
4
755.08
6
11
17
22
28
34
39
45
51
5
754.51
6
753.95
7
753.38
8
752.81
9
752.24
6
11
17
23
28
34
40
45
51
0.81
0
751.67
1
751.10
2
750.53
3
749.95
4
749.37
6
11
17
23
29
34
40
46
52
5
748.80
6
748.22
7
747.64
8
747.05
9
746.47
6
12
17
23
29
35
41
47
52
0.82
0
745.89
1
745.30
2
744.71
3
744.13
4
743.54
6
12
18
24
30
35
41
47
53
5
742.94
6
742.35
7
741.76
8
741.16
9
740.57
6
12
18
24
30
36
42
48
53
0.83
0
739.97
1
739.37
2
738.77
3
738.17
4
737.56
6
12
18
24
30
36
42
48
54
5
736.96
6
736.35
7
735.74
8
735.13
9
734.52
6
12
18
24
31
37
43
49
55
CO
o
0
733.91
1
733.30
2
732.68
3
732.07
4
731.45
6
12
18
25
31
37
43
49
55
5
730.83
6
730.21
7
729.59
8
728.96
9
728.34
6
12
19
25
31
37
44
50
56
0.85
0
727.71
1
727.09
2
726.46
3725.83
4
725.19
6
13
19
25
32
38
44
50
57
5
724.56
6
723.93
7
723.29
8 722.65
9
722.01
6
13
19
26
32
38
45
51
57
0.86
0
721.37
1
720.73
2
720.08
3719.44
4
718.79
6
13
19
26
32
39
45
52
58
5
718.14
6
717.49
7
716.84
8 716.19
9
715.54
7
13
20
26
33
39
46
52
59
0.87
0
714.88
1
714.22
2
713.56
3712.90
4
712.24
7
13
20
26
33
40
46
53
59
5
711.58
6
710.91
7
710.25
8
709.58
9
708.91
7
13
20
27
33
40
47
53
60
0.88
0
708.24
1
707.57
2
706.89
3
706.22
4
705.54
7
14
20
27
34
41
47
54
61
5
704.86
6
704.18
7
703.50
8
702.82
9
702.13
7
14
21
27
34
41
48
55
62
0.89
0
701.44
1
700.76
2
700.07
3699.37
4
698.68
7
14
21
28
35
41
48
55
62
5
697.99
6
697.29
7
696.59
8
695.89
9
695.19
7
14
21
28
35
42
49
56
63
0.90
0
694.49
1
693.78
2
693.08
3 692.37
4
691.66
7
14
21
28
35
42
50
57
64
5
690.95
6
690.24
7
689.52
8 688.81
9
688.09
7
14
21
29
36
43
50
57
64
0,5
CO
rH
2,7
3,8
4,9
1
2
3
4
5
6
7
8
9
PROPORTIONAL, PARTS, STTBT. 102
TABLE XLIX.
CONVERSION TABLE FROM ANNUITY OR A(l+«) TO SINGLE PREMIUM ON 1,000
INTEREST 4 PER CENT.
PROPORTIONAL PARTS, BUBT.
A(1+«).
0,5
1,6
2,7
3,8
4,9
1
2
3
4
5
6
7
8
9
0.91
0
687.37
1
686.65
2
685.93
3
685.21
4 684.48
7
14
22
29
36
4351
58
65
5
683.75
6
683.02
7
682.29
8
681.56
9 680.83
7
15
22
29
37
44
51
59
66
0.92
0
680.09
1
679.35
2
678.61
3
677.87
4 677.13
7
15
22
30
37
44
52
59
67
5
676.39
6
675.64
7
674.89
8
674.14
9.673.39
8
15
23
30
38
45
53
60
68
0.93
0
672.64
1
671.88
2
671.13
3
670.37
4669.61
8
15
23
30
38
45
53
61
68
5
668.85
6
668.09
7
667.32
8
666.55
9
665.78
8
15
23
31
38
46
54
61
69
0.94
0
665.01
1
664.24
2
663.47
3
662.69
4
661.91
8
16
23
31
39
47
54
62
70
5
661.13
6
660.35
7
659.57
8
658.79
9
658.00
8
10
24
31
39
47
55
63
71
0.95
0
657.21
1
656.42
2
655.63
3
654.83
4
654.04
8
16
24
32
40
48
56
64
71
5
653.24
6
652.44
7
651.64
8
650.84
9
650.03
8
16
2432
40
48
56
64
72
0.96
0
649.23
1
648.42
2
647.61
3
646.80
4
645.98
8
16
2433
41
49
57
65
73
5
645.16
6
644.35
7
643.53
8
642.71
S
641.88
8
16
25 33
41
49
57
66
74
0.97
0
641.06
1
640.23
2
639.40
3
638.57
4
637.73
8
17
25 33
42
50
58
67
75
5
636.90
6
636.06
7
635.22
0
634.38
6
633.54
8
17
25j34
42
50
59
67
76
0.98
0
632.70
1
631.85
2
631.00
3
630.15
4
629.30
9
17
26 34
43
51
60
68
77
5
628.44
G
627.59
7
626.73
8 625.87
9
625.00
9
17
26 34
43
52
60
69
77
0.99
0
624.14
1
623.27
2
622.40
3621.53
4
620.66
9
17
26 35
44
52
61
70
78
5
619.79
6
618.91
7 618.03
8617.15
9
G16.27
9
18
2635
44
53
62
71
79
1.00
0
615.38
1
614.50
2 613.61
3 612.72
4
611.83
9
18
2736
45
53
62
71
80
5
610.93
6
610.03
7,609.14
8 608.23
9
607.33
9
18
27 36
45
54
63
72
81
1.01
0
606.43
1
605.52
2 604.61
3*603.70
4
602.78
9
18
2736
46
55
64
73
82
5
601.87
6
600.95
7600.03
8599.11
9
598.18
9
18
28 37
46
55
65
74
83
1.02
0
597.26
1
596.33
2595.40
3 594.47
4
593.53
9
19
28 37
47
56
65
75
84
5
592.59
6
591.66
7j590.71
8t589.77
9
588.82
9
19
28 38
47
57 66
75
85
1.03
0
587.88
1
586.93
2585.97
3,585.02
4
584.06
10
19
29
38
48
57 67
76
86
5
583.10
G
582.14
7
581.18
8
580.22
9
579.25
10
19
29
39
48
58
67
77
87
1.04
0
578.28
1
577.31
2
576.33
3
575.35
4
574.38
10
20
29
39
49
59
68
78
88
5
573.39
6
572.41
7
571.43
8 570.44
9
569.45
10
20
30
40
49
59
69
79
89
1.05
0
568.45
1
567.48
2
566.46
3 565.46
4
564.46
10
20
30
40
50
60
70
80
90
5
563.46
6
562.45
7
561.44
8
560.43
9
559.42
10
20
30
40
5L
61
71
81
91
1.06
0
558.40
1
557.38
2
556.36
3
555.34
4
554.32
10
2031
41
51
61
72
82
92
5
553.29
6
552.26
7
551.23
8
550.19
9
549.16
10
21 31
41
52
62
72
83
93
1.07
0
548.12
1
547.07
2
546.03
3
544.98
4
543.94
10
2131
42
52
63
73
84
94
5
542.88
6
541.83
7
540.77
0
539.72
9
538.65
11
2132
42
53
63
74
85
95
1.08
0
537.59
1
536.52
2
535.46
3
534.39
4
533.31
11
21 32
43
54
64
75
86
96
5
532.24
6
531.16
7
530.08
8
528.99
9
527.91
11
2233
43
54
65
76
87
98
1.09
0
526.82
1
525.73
2
524.64
3
523.54
4
522.44
11
2233
44
55
66
77
88
99
5
521.34
6
520.24
7
519.13
8
518.02
9
516.91
11
2233
44
55
66
78
89
100
1.10
0
515.80
1
514.68
2
513.56
3
512.44
4
511.32
11
2234
45
56
67
79
90
101
5
510.19
6
509.06
7
507.93
8
506.80
9
505.66
11
23 34
45
57
68
79
91
102
1.11
0
504.52
1
503.38
2
502.23
3
501.08
4
499.93
11
23 34
46
57
69
80
92
103
5
498.78
6
497.63
7 496.47
8
495.31
9
494.14
12
23
35
46
58
70
81
93
104
1.12
0
492.98
1
491.81
2 490.64
3
489.46
4
488.29
12
23
35
47
59
70
82
94
106
5
487.11
6
485.92
7
484.74
8
483.55
9
482.36
12
24
36
48
59
71
83
95
107
1.13
0
481.17
1
479.97
2
478.77
3
477.57
4
476.37
12
24
36
48
60
72
84
96
108
5
475.16
G
473.95
7
472.74
8
471.52
9
470.30
12
24
36
49
61
73
85
97
109
1.14
0
469.08
1
467.86
2
466.63
3
465.40
4
464.17
12
25
37
49
61
74
86
98
111
5
462.94
G
461.70
7
460.46
8
459.21
9
457.97
12
25
37
50
62
75
87
100
112
0,5
1 1,6
2,7
3,8
4,9 |
1
2
3
4
5
6
7
8
0 TABLE XLIX.
103
CONVERSION TABLE FROM ANNUITY OR A(l+ff) TO SINGLE PREMIUM ON 1,000.
INTEREST 4 PER CENT.
PROPORTIONAL PARTS, SUBT.
A(l-f a).
0,5
1,6
2,7
3,8
4,9
1
2
3
4
5
6
7
8
9
1.15
0
456.72
1
455.46
2
454.21
3
452.95
4
451.69
13
25
38
50
63
75
88
101
113
5
450.43
6
449.16
7
447.89
8
446.62
9
445.34
13
25
38
51
64
76
89
102
115
1.16
0
444.06
1
442.78
2
441.50
3
440.21
4
438.92
13
26
39
52
64
77
90
103
116
5
437.62
6
436.33
7
435.03
8
433.73
9
432.42
13
26
39
52
65
78
91
104
117
1.17
0
431.11
1
429.80
2
428.49
3
427.17
4
425.85
13
26
40
53
66
79
92
105
119
5
424.52
6
423.20
7
421.87
8
420.54
9
419.20
13
27
40
53
67
80
93
107
120
1.18
0
417.86
1
416.52
2
415.17
3
413.83
4
412.47
13
27
40
54
67
81
94
108
121
5
411.12
6
409.76
7
408.40
8
407.04
9
405.67
14
27
41
55
68
82
95
109
123
1.19
0
404.30
1
402.93
2
401.55
3
400.17
4
398.79
14
28
41
55
69
83
97
110
124
5
397.40
6
396.01
7
394.62
8
393.23
9
391.83
14
28
42
56
70
84
98
112
125
1,20
0
390.43
1
389.02
2
387.61
3
386.20
4
384.79
14
28
42
56
71
85
99
113
127
5
383.37
6
381.95
7
380.52
8
379.09
9
377.66
14
29
43
57
71
86
100
114
129
1.21
0
376.23
1
374.79
2
373.35
3
371.90
4
370.45
14
29
43
58
72
87
101
116
130
5
369.00
6
367.55
7
366.09
8
364.63
9
363.16
15
29
44
58
73
88
102
117
131
1.22
0
361.70
1
360.23
2
358.75
3
357.27
4
355.79
15
30
44
59
74
89
103
118
133
5
354.31
6
352.82
7
351.33
8
349.83
9
348.33
15
30
45
60
75
90
105
120
135
1.23
0
346.83
1
345.32
2
343.81
3
342.30
4
340.79
15
30
45
60
76
91
106
121
136
5
339.27
6
337.74
7
336.22
8
334.69
9
333.15
15
31
46
61
77
92
107
123
138
1.24
0
331.61
1
330.07
2
328.53
3
326.98
4
325.43
15
31
46
62
77
93
108
124
139
5
323.87
6
322.32
7
320.75
8
319.19
9
317.62
16
31
47
63
78
94
109
125
141
1.25
0
316.05
1
314.47
2
312.89
3
311.30
4
309.72
16
32
48
63
79
95
111
127
143
5
308.13
6
306.53
7
304.93
8
303.33
9
301.72
16
32
48
64
80
96
112
128
144
1.26
0
300.11
1
298.50
2
296.88
3
295.26
4
293.64
16
32
49
65
81
97
113
130
146
5
292.01
6
290.38
7
288.74
8
287.10
9
•185.46
16
33
49
66
82
98
115
131
148
1.27
0
283.81
1
282.16
2
280.51
3
278.85
4
277.19
17
33
50
66
83
99
116
133
149
5
275.52
6
273.85
7
272.17
8
270.50
9
268.82
17
34
50
67
84
101
117
134
151
1.28
0
267.13
1
265.44
2
263.75
3
262.05
4
260.35
17
34
51
68
85
102
119
136
153
5
258.64
6
256.93
7
255.22
8
253.50
9
251.78
17
34
51
69
86
103
120
137
154
1.29
0
250.06
1
248.33
2
246.60
3
244.86
4
243.12
17
35
52
69
87
104
122
139
156
5
241.38
6
239.63
7
237.87
8
236.12
9
234.36
18
35
53
70
88
105
123
141
158
1.30
0
232.59
1
230.82
2
229.05
3
227.27
4
225.49
18
36
53
71
89
107
124
142
160
5
223.71
6
221.92
7
220.12
8
218.32
9
216.52
18
36
54
72
90
108
126
144
162
1.31
0
214.72
1
212.91
2
211.09
3
209.27
4
207.45
18
36
55
73
91
109
127
146
164
5
205.62
6
203.79
7
201.96
8
200.12
9
198.27
18
37
55
74
92
110
129
147
166
1.32
0
196.42
1
194.57
2
192.71
3
190.85
4
188.99
19
37
56
74
93
112
130
149
167
5
187.12
6
185.25
7
183.37
8
181.48
9
179.60
19
38
56
75
94
113
132
151
169
1.33
0
177.71
1
175.81
2
173.91
3
172.01
4
170.10
19
38
57
76
95
114
133
152
172
5
168.18
6
166.27
7
164.35
8
162.42
9
160.49
19
39
58
77
96
116
135
154
173
1.34
0
158.55
1
156.61
2
154.67
3
152.72
4
150.77
19
39
58
78
97
117
136
156
175
5
148.81
6
146.85
7
144.88
8
142.91
9
140.93
20
39
59
79
99
118
138
158
177
1.35
0
138.95
1
136.97
2
134.98
3
132.98
4
130.99
20
40
60
80
100
120
140
160
179
5
128.98
6
126.97
7
124.96
8
122.94
9
120.92
20
40
60
81
101
121
141
161
181
1.36
0
118.90
1
116.87
2
114.83
3
112.79
4
110.74
20
41
61
82
102
123
143
163
184
5
108.69
6
106.64
7
104.58
8
102.52
9
100.45
21
41
62
83
103
124
144
165
186
1.37
0
98.37
1
96.29
2
94.21
3
92.12
4
90.03
21
42
63
84
104
125
146
167
188
5
87.93
6
85.83
7
83.72
8
81.61
9
79.49
21
42
63
85
106
127
148
169
190
1.38
0
77.37
1
75.24
2
73.11
3
70.98
4
68.83
21
43
64
85
107
128
150
171
192
5
| 66.69
6
64.54
7
62.38
8
60.22
9
58.05
22
43
65
86
108
130
151
173
195
1 0,5
1,6
2,7
3,8
4,9
1
2
3
4
5
6
7
8
9 104
TABLE L.
SINGLE PREMIUM OR RESERVE BY MONTHS, ON PAID-UP INSURANCE OF 1,000
FOR THE WHOLE LIFE.
4 PER CENT.
AGE.
0m.
4m.
l£m.
2£m.
3£m.
4£m.
51m.
AGE.
x+n.
* Dec.
Nov.
Oct.
Sept.
Aug.
July.
z-\-n.
10
202.630
202.753
203.000
203.246
203.493
203.739
203.986
10
11
205.587
205.716
205.972
206.228
206.485
206.741
206.997
11
12
208.665
208.799
209.067
209.334
209.602
209.869
210.137
12
13
211.878
212.017
212.296
212.575
212.854
213.133
213.412
13
14
215.224
215.369
215.658
215.948
216.238
216.528
216.817
14
15
218.699
218.850
219.152
219.454
219.755
220.057
220.359
15
16
222.321
222.478
222.793
223.107
223.422
223.736
224.051
16
17
226.095
226.258
226.586
226.913
227.239
227.566
227.894
17
18
230.019
230.189
230.529
230.869
231.210
231.550
231.890
18
19
234.101
234.278
234.632
234.986
235.340
235.694
236.048
19
20
238.349
238.533
238.901
239.269
239.636
240.004
240.372
20
21
242.763
242.954
243.336
243.718
244.100
244.482
244.864
21
22
247.348
247.547
247.944
248.341
248.739
249.136
249.533
22
23
252.116
252.322
252.735
253.147
253.560
253.972
254.385
23
24
257.065
257.279
257.707
258.135
258.564
258.992
259.420
24
25
262.203
262.425
262.870
263.315
263.759
264.204
264.649
25
26
267.540
267.771
268.231
268.692
269.153
269.614
270.074
26
27
273.069
273.308
273.786
274.264
274.742
275.220
275.698
27
28
278.806
279.054
279.550
280.046
280.541
281.037
281.533
28
29
284.756
285.012
285.525
286.038
286.551
287.064
287.577
29
30
290.909
291.175
291.706
292.237
292.768
293.299
293.830
30
31
297.284
297.559
298.109
298.659
299.210
299.760
300.310
31
32
303.886
304.171
304.739
305.308
305.878
306.447
307.015
32
33
310.714
311.008
311.596
312.185
312.772
313.361
313.949
33
34
317.773
318.077
318.685
319.292
319.901
320.508
321.116
34
35
325.066
325.379
326.006
326.633
327.260
327.887
328.514
35
36
332.588
332.912
333.559
334.207
334.854
335.502
336.149
36
37
340.359
340.693
341.361
342.029
342.696
343.364
344.032
37
38
348.373
348.717
349.405
350.093
350.782
351.470
352.158
38
39
356.630
356.984
357.693
358.402
359.110
359.819
360.528
39
40
365.134
365.498
366.227
366.956
367.684
368.413
369.142
40
41
373.879
374.254
375.004
375.754
376.504
377.254
378.004
41
42
382.879
383.264
384.034
384.804
385.574
386.344
387.114
42
43
392.119
392.514
393.305
394.096
394.887
395.678
396.469
43
44
401.608
402.013
402.824
403.635
404.446
405.257
406.068
44
45
411.338
411.753
412.584
413.415
414.246
415.077
415.908
45
46
421.309
421.734
422.585
423.435
424.287
425.137
425.988
46
47
431.516
431.951
432.821
433.690
434.560
435.429
436.299
47
48
441.953
442.397
443.286
444.175
445.064
445.953
446.842
48
49
452.618
453.071
453.978
454.885
455.792
456.699
457.606
49
50
463.500
463.962
464.887
465.811
466.736
467.660
468.585
50
51
474.594
475.065
476.006
476.947
477.888
478.829
479.770
51
52
485.889
486.368
487.325
488.282
489.240
490.197
491.154
52
53
497.376
497.862
498.835
499.807
500.780
501.752
502.725
53
54
509.046
509.539
510.526
511.514
512.500
513.488
514.475
54
* Months of Entry for Annual Valuation Dec. 31. 105
TABLE L.
SINGLE PREMIUM OR RESERVE BY MONTHS, ON PAID-UP INSURANCE OF 1,000
FOR THE WHOLE LIFE.
4 PEE CENT.
AGE.
6 m.
6|m.
7-&m.
84m.
9im.
10^m.
lll-m.
AGE.
X-\-7l.
June.
May.
April.
March.
Feb.
Jan.
x-\-n.
10
204.109
204.232
204.479
204.725
204.971
205.217
205.464
10
11
207.126
207.255
207.511
207.767
208.024
208.280
208.536
11
12
210.271
210.405
210.673
210.940
211.209
211.476
211.744
12
13
213.551
213.690
213.969
214.248
214.527
214.806
215.085
13
14
216.962
217.107
217.396
217.686
217.975
218.265
218.554
14
15
220.510
220.661
220.963
221.265
221.566
221.868
222.170
15
16
224.208
224.365
224.680
224.994
225.309
225.623
225.938
16
17
228.057
228.220
228.548
228.875
229.201
229.528
229.856
17
18
232.060
232.230
232.570
232.910
233.251
233.591
233.931
18
19
236.225
236.402
236.756
237.110
237.464
237.818
238.172
19
20
240.556
240.740
241.108
241.476
241.843
242.211
242,579
20
21
245.055
245.246
245.628
246.010
246.393
246.775
247.157
21
22
249.732
249.931
250.328
250.725
251.123
251.520
251.917
22
23
254.591
254.797
255.210
255.622
256.034
256.446
256.859
23
24
259.634
259.848
260.276
260.704
261.133
261.561
261.989
24
25
264.871
265.093
265.538
265.983
266.428
266.873
267.318
25
26
270.305
270.536
270.996
271.457
271.917
272.378
272.838
26
27
275.937
276.176
276.654
277.132
277.611
278.089
278.567
27
28
281.781
282.029
282.525
283.021
283.516
284.012
284.508
28
29
287.833
288.089
288.602
289.115
289.627
290.140
290.653
29
30
294.096
294.362
294.893
295.424
295.956
296.487
297.018
30
31
300.585
300.860
301.410
301.960
302.511
303.061
303.611
31
32
307.300
307.585
308.153
308.722
309.292
309.861
310.429
32
33
314.243
314.537
315.125
315.714
316.302
316.891
317.479
33
34
321.420
321.724
322.332
322.939
323.547
324.154
324.762
34
35
328.827
329.140
329.767
330.394
331.021
331.648
332.275
35
36
336.473
336.797
337.444
338.092
338.740
339.388
340.035
36
37
344.366
344.700
345.368
346.036
346.703
347.371
348.039
37
38
352.502
352.846
353.534
354.222
354.910
355.598
356.286
38
39
360.882
361.236
361.945
302.654
363.362
364.071
364.780
39
40
369.506
369.870
370.599
371.328
372.057
372.786
373.515
40
41
378.379
378.754
379.504
380.254
381.004
381.754
382.504
41
42
387.499
387.884
388.654
389.424
390.194
390.964
391.734
42
43
396.864
397.259
398.050
398.841
399.631
400.422
401.213
43
44
406.473
406.878
407.689
408.500
409.311
410.122
410.933
44
45
416.323
416.738
417.569
418.400
419.232
420.063
420.894
45
46
426.413
426.838
427.689
428.539
429.390
430.240
431.091
46
47
436.734
437.169
438.039
438.908
439.779
440.648
441.518
47
48
447.286
447.730
448.619
449.508
450.396
451.285
452.174
48
49
458.059
458.512
459.419
460.326
461.233
462.140
.463.047
49
50
469.047
469.509
470.434
471.358
472.283
473.207
474.132
50
51
480.241
480.712
481.653
482.594
483.536
484.477
485.418
51
52
491.633
492.112
493.069
494.026
494.983
495.940
496.897
52
53
503.211
503.697
504.670
505.642
506.615
507.587
508.560
53
54
514.968
515.461
516.448
517.436
518.422
519.410
520.397
54 106
TABLE L.
SINGLE PREMIUM OR RESERVE BY MONTHS, ON PAID-UP INSURANCE OF 1,000
FOR THE WHOLE LIFE.
4 PER CENT.
AGE,
x + n.
0m.
£m.
2£m.
3|m.
3
5|m.
AGE.
x + n.
* Dec.
Nov.
Oct.
Sept.
Aug.
July.
55
520.890
521.390
522.389
523.388
524.388
525.387
526.386
55
56
532.882
533.388
534.399
535.411
536.421
537.433
538.444
56
57
545.019
545.530
546.551
547.573
548.594
549.616
550.637
57
58
557.277
557.792
558.823
559.853
560.884
561.914
562.945
58
59
569.642
570.161
571.200
572.238
573.276
574.314
575.353
59
60
582.103
582.625
583.670
584.714
585.759
586.803
587.848
60
61
594.636
595.160
596.208
597.256
598.305
599.353
600.401
61
62
607.214
607.740
608.791
609.842
610.893
611.944
612.995
62
63
619.829
620.355
621.408
622.460
623.513
624.565
625.618
63
64:
632.458
632.983
634.034
635.085
636.136
637.187
638.238
64
65
645.069
645.593
646.641
647.689
648.736
649.784
650.832
65
66
657.643
658.165
659.208
660.251
661.295
662.338
663.381
66
67
670.162
670.680
671.716
672.753
673.789
674.826
675.862
67
68
682.598
683.112
684.141
685.169
686.198
687.226
688.255
68
69
694.941
695.450
696.467
697.484
698.502
699.519
700.536
69
70
707.148
707.651
708.656
709.661
710.666
711.671
712.676
70
71
719.211
719.706
720.697
721.688
722.678
723.669
724.660
71
72
731.099
731.586
732.561
733.536
734.510
735.485
736.460
72
73
742.795
743.274
744.232
745.189
746.148
747.105
748.063
73
74
754.288
754.757
755.694
756.631
757.568
758.505
759.442
74
75
765.535
765.994
766.910
767.828
768.744
769.662
770.578
75
76
776.539
776.987
777.882
778.778
779.673
780.569
781.464
76
77
787.285
787.721
788.592
789.464
790.335
791.207
792.078
77
78
797.743
798.167
799.014
799.862
800.710
801.558
802.405
78
79
807.914
808.325
809.148
809.970
810.792
811.614
812.437
79
80
817.783
818.181
818.978
819.775
820.573
821.370
822.167
80
81
827.346
827.732
828.505
829.277
830.049
830.821
831.594
81
82
836.614
836.987
837.733
838.480
839.225
839.972
840.718
82
83
845.569
845.931
846.655
847.379
848.103
848.827
849.551
83
84
854.256
854.607
855.309
856.010
856.712
857.413
858.115
84
85
862.677
863.019
863.704
864.389
865.073
865.758
866.443
85
86
870.893
871.231
871.908
872.584
873.262
873.938
874.615
86
87
879.012
879.349
880.022
880.695
881.367
882.040
882.713
87
88
887.089
887.437
888.132
888.827
889.523
890.218
890.913
88
89
895.433
895.740
896.354
896.969
897.583
898.198
898.812
89
90
902.805
903.066
903.589
904.111
904.635
905.157
905.680
90
91
909.076
909.321
909.811
910.301
910.792
911.282
911.772
91
92
914.958
915.194
915.665
916.137
916.608
917.080
917.551
92
93
920.617
920.841
921.288
921.735
922.183
922.630
923.077
93
94
925.985
926.203
926.639
927.075
927.512
927.948
928.384
94
95
931.219
931.457
931.934
932.410
932.887
933.363
933.840
95
96
936.936
937.147
937.570
937.992
938.414
938.836
939.259
96
97
942.005
942.305
942.906
943.506
944.108
944.708
945.309
97
98
949.212
949.726
950.753
951.780
952.806
953.833
954.860
98
99
961.537
963.140
966.345
969.550
972.756
975.961
979.166
99
* Months of Entry for Annual Valuation Dec. 31. 107
TABLE L.
SINGLE PREMIUM OR RESERVE BY MONTHS, ON PAID-UP INSURANCE OF 1,000
FOR THE WHOLE LIFE.
4 PER CENT.
AGE,
6m.
61m.
7im.
8£m.
9£m.
10|m.
H^m.
AGE,
x-\-n.
June.
May.
April.
March.
Eeb.
Jan.
x-\-n.
55
526.886
527.386
528.385
529.384
530.384
531.383
532.382
55
56
538.950
539.456
540.467
541.479
542.490
543.502
544.513
56
57
551.148
551.659
552.680
553.702
554.723
555.745
556.766
57
58
563.460
563.975
565.006
566.036
567.066
568.096
569.127
58
59
575.872
576.391
577.430
578.468
579.507
580.545
581.584
59
60
588.370
588.892
589.937
590.981
592.025
593.069
594.114
60
61
600.925
601.449
602.497
603.545
604.594
605.642
606.690
61
62
613.521
614.047
615.098
616.149
617.201
618.252
619.303
62
63
626.144
626.670
627.723
628.775
629.827
630.879
631.932
63
64
638.763
639.288
640.339
641.390
642.442
643.493
644.544
64
65
651.356
651.880
652.928
653.976
655.023
656.071
657.119
65
66
663.903
664.425
665.468
666.511
667.554
668.597
669.640
66
67
676.380
676.898
677.934
678.971
680.007
681.044
682.080
67
68
688.769
689.283
690.312
691.340
692.370
693.398
694.427
68
69
701.045
701.554
702.571
703.588
704.605
705.622
706.639
69
70
713.179
713.682
714.687
715.692
716.698
717.703
718.708
70
71
725.155
725.650
726.641
727.632
728.622
729.613
730.604
71
72
736.947
737.434
738.409
739.384
740.358
741.333
742.308
72
73
748.542
749.021
749.979
750.936
751.894
752.851
753.809
73
74
759.911
760.380
761.317
762.254
763.192
764.129
765.066
74
75
771.037
771.496
772.412
773.330
774.246
775.164
776.080
75
76
781.912
782.360
783.255
784.151
785.046
785.942
786.837
76
77
792.514
792.950
793.821
794.693
795.564
796.436
797.307
77
78
802.829
803.253
804.100
804.948
805.795
806.643
807.490
78
79
812.848
813.259
814.082
814.904
815.727
816.549
817.372
79
80
822.565
822.963
823.760
824.557
825.354
826.151
826.948
80
81
831.980
832.366
833.139
833.911
834.683
835.455
836.228
81
82
841.091
841.464
842.210
842.957
843.703
844.450
845.196
82
83
849.913
850.275
850.999
851.723
852.446
853.170
853.894
83
84
858.466
858.817
859.519
860.220
860.923
861.624
862.326
84
85
866.785
867.127
867.812
868.497
869.181
869.866
870.551
85
86
874.953
875.291
875.968
876.644
877.321
877.997
878.674
86
87
883.050
883.387
884.060
884.733
885.406
886.079
886.752
87
88
891.261
891.609
892.304
892.999
893.695
894.390
895.085
88
89
899.119
899.426
900.040
900.655
901.269
901.884
902.498
89
90
905.941
906.202
906.725
907.247
907.770
908.292
908.815
90
91
912.017
912.262
912.752
913.242
913.733
914.223
914.713
91
92
917.787
918.023
918.494
918.966
919.438
919.910
920.381
92
93
923.301
923.525
923.972
924.419
924.867
925.314
925.761
93
94
928.602
928.820
929.256
929.692
930.129
930.565
931.001
94
95
934.078
934.316
934.793
935.269
935.745
936.221
936.698
95
96
939.470
939.681
940.104
940.526
940.949
941.371
941.794
96
97
945.609
945.909
946.510
947.110
947.711
948.311
948.912
97
98
955.374
955.888
956.915
957.942
958.969
959.996
961.023
98
99
980.769
982.372
985.577
988.782
991.987
995.192
998.397
99 108
TABLE LI.
LOGARITHM OF THE SINGLE PREMIUM BY MONTHS, ON 1 INSURED FOR THE
WHOLE LIFE.
4 PER CENT.
AGE.
Orn.
|m.
l|m.
2£m.
3£m.
4£m.
5|m.
AGE.
x-\-n.
* Dec.
Nov.
Oct.
Sept.
Aug.
July.
x-\-n.
10
1.306703
1.306967
1.307496
1.308022
1.308549
1.309074
1.309601
10
11
.312996
.313267
.313808
.314348
.314889
.315427
.315964
11
12
.319450
.319728
.320286
.320839
.321395
.321948
.322503
12
13
.326086
.326371
.326942
.327512
.328082
.328651
.329219
13
14
.332890
.333183
.333766
.334349
.334932
.335514
.336093
14
15
1.339847
1.340147
1.340745
1.341344
1.341939
1.342535
1.343131
15
16
.346980
.347287
.347902
.348513
.349126
.349736
.350347
16
17
.354290
.354604
.355233
.355860
.356483
.357107
.357733
17
18
.361763
.362084
.362725
.363366
.364007
.364645
.365282
18
19
.369403
.369731
.370388
.371042
.371696
.372349
.373000
19
20
1.377213
1.377548
1.378218
1.378886
1.379552
1.380218
1.380884
20
21
.385182
.385524
.386207
.386887
.387568
.388247
.388925
21
22
.393309
.393658
.394354
.395049
.395744
.396436
.397128
22
23
.401601
.401955
.402666
.403373
.404081
.404786
.405492
23
24
.410042
.410404
.411126
.411847
.412568
.413286
.414003
24
25
1.418637
1.419005
1.419741
1.420476
1.421207
1.421939
1.422670
25
26
.427389
.427764
.428509
.429255
.429999
.430742
.431482
26
27
.436272
.436653
.437412
.438168
.438925
.439680
.440434
27
28
.445303
.445688
.446459
.447230
.447997
.448764
.449530
28
29
.454473
.454863
.455645
.456424
.457202
.457979
.458754
29
30
1.463757
1.464155
1.464945
1.465736
1.466524
1.467311
1.468096
30
31
.473172
.473573
.474376
.475176
.475976
.476774
.477570
31
32
.482711
.483]18
.483928
.484738
.485548
.486356
.487160
32
33
.492361
.492771
.493592
.494412
.495228
.496045
.496859
33
34
.502117
.502532
.503362
.504188
.505015
.505839
.506662
34
35
1.511971
1.512390
1.513226
1.514060
1.514893
1.515724
1.516554
35
36
.521906
.522330
.523173
.524015
.524855
.525695
.526532
36
37
.531937
.532363
.533214
.534063
.534909
.535755
.536599
37
38
.542045
.542473
.543329
.544184
.545037
.545888
.546738
38
39
.552218
.552649
.553511
.554370
.555227
.556084
.556939
39
40
T. 562452
1.562885
1.563750
1.564614
1.565475
1.566335
1.567193
40
41
.572731
.573166
.574036
.574904
.575770
.576634
.577496
41
42
.583062
.583498
.584369
.585239
.586107
.586974
.587839
42
43
.593418
.593855
.594730
.595602
.596473
.597342
.598209
43
44
.603802
.604240
.605115
.605989
.606861
.607731
.608599
44
45
1.614199
1.614637
1.615512
1.616386
1.617259
1.618129
1.618997
45
46
.624601
.625038
.625914
.626787
.627660
.628529
.629397
46
47
.634997
.635434
.636308
.637179
.638050
.638917
.639784
47
48
.645376
.645812
.646684
.647554
.648423
.649289
.650154
48
49
.655732
.656166
.657035
.657901
.658767
.659630
.660492
49
50
1.666050
1.666483
1.667347
1.668210
1.669071
1.669930
1.670789
50
51
.676322
.676753
.677612
.678470
.679326
.680180
.681033
51
52
.686537
.686965
.687819
.688671
.689522
.690370
.691218
52
53
.696685
.697109
.697957
.698802
.699647
.700489
.701331
53
54
.706757
.707177
.708018
.708858
.709694
.710530
.711364
54
Months of Entry for Annual Valuation Dec. 31. 109
TABLE LI.
LOGARITHM OF THE SINGLE PREMIUM BY MONTHS, ON 1 INSURED FOR THE
WHOLE LIFE.
4 PER CENT.
AGE.
6m.
6|m.
7*m.
8im.
9-im.
10|m.
Him.
AGE.
x-\-n.
June.
May.
April.
March.
Feb.
Jan.
x+n.
10
1.309862
1.310124
1.310649
1.311171
1.311692
1.312214
1.312735
10
11
.316235
.316505
.317041
.317577
.318113
.318648
.319181
11
12
.322779
.323056
.323609
.324159
.324712
.325261
.325811
12
13
.329502
.329784
.330351
.330917
.331482
.332046
.332610
13
14
.336384
.336674
.337252
.337830
.338407
.338984
.339559
14
15
T. 343428
1.343726
1.344320
1.344913
1.345503
1.346095
1.346685
15
16
.350651
.350955
.351564
.352171
.352778
.353384
.353989
16
17
.358043
.358354
.358977
.359599
.360217
.360835
.361456
17
18
.365600
.365918
.366544
.367188
.367824
.368456
.369088
18
19
.373326
.373651
.374301
.374950
.375598
.376244
.376891
19
20
1.381216
1.381548
1.382211
1.382874
1.383533
1.384194
1.384853
20
21
.389264
.389602
.390278
.390953
.391628
.392301
.392973
21
22
.397474
.397820
.398510
.399198
.399886
.400573
.401258
22
23
.405843
.406194
.406898
.407598
.408298
.408996
.409694
23
24
.414362
.414720
.415434
.416148
.416862
.417573
.418283
24
25
T. 423035
1.423398
1.424127
1.424854
1.425580
1.426305
1.427028
25
26
.431854
.432225
.432963
.433701
.434436
.435172
.435905
26
27
.440810
.441186
.441937
.442687
.443437
.444184
.444930
27
28
.449912
.450294
.451057
.451819
.452578
.453337
.454095
28
29
.459141
.459527
.460299
.461071
.461840
.462608
.463375
29
30
1.468489
1.468882
1.469665
1.470446
1.471227
1.472006
1.472783
30
31
.477967
.478364
.479158
.479949
.480741
.481530
.482317
31
32
.487563
.487965
.488766
.489568
.490369
.491167
.491963
32
33
.497265
.497672
.498483
.499294
.500102
.500910
.501715
33
34
.507073
.507484
.508304
.509121
.509937
.510751
.511566
34
35
T. 516967
1.517381
1.518207
1.519032
1.519855
1.520677
1.521498
35
36
.526950
.527368
.528202
.529035
.529866
.530696
.533524
36
37
.537020
.537441
.538282
.539121
.539958
.540793
.541628
37
38
.547161
.547585
.548431
.549275
.550118
.550960
.551799
38
39
.557365
.557791
.558643
.559493
.560339
.561186
.562031
39
40
T.567621
1.568049
1.568904
1.569758
1.570609
1.571460
1.572308
40
41
.577927
.578357
.579216
.580074
.580929
.581783
.582636
41
42
.588271
.588702
.589563
.590422
.591280
.592136
.592991
42
43
.598641
.599074
.599938
.600800
.601659
.602518
.603375
43
44
.609031
.609464
.610329
.611192
.612053
.612913
.613771
44
45
1.619430
1.619863
1.620728
1.621592
1.622454
1.623314
1.624173
45
46
.629830
.630263
.631128
.631990
.632852
.633711
.634569
46
47
.640217
.640649
.641513
.642373
.643234
.644092
.644948
47
48
.650585
.651016
.651878
.652737
.653594
.654451
.655306
48
49
.660921
.661351
.662209
.663065
.663921
.664774
.665625
49
50
1.671216
1.671644
1.672499
1.673351
1.674203
1.675051
1.675899
50
51
.681459
.681885
.682735
.683582
.684428
.685273
.686116
51
52
.691641
.692064
.692908
.693750
.694590
.695429
.696266
52
53
.701750
.702169
.703007
.703843
.704678
.705511
.706342
53
54
.711780
.712196
.713026
.713857
.714684
.715510
.716335
54 110
TABLE LI.
LOGARITHM OF THE SINGLE PREMIUM BY MONTHS, ON 1 INSURED FOR THE
WHOLE LIFE.
4 PER CENT.
AGE.
x-\-n.
0m.
Hm.
2£m.
3im-
4im.
5|m.
AGE.
x-\-n.
* Dec.
Nov.
Oct.
Sept.
Aug.
July.
55
T. 716746
1.717163
1.717994
1.718823 I
1.719652 1
T. 720480
1.721304 1
55
56
.726631
.727043
.727865
.728687
.729506
.730324
.731140
56
57
.736412
.736819
.737631
.738442
.739251
.740060
.740866
57
58
.746071
.746473
.747274
.748074
.748873
.749670
.750466
58
59
.755602
.755998
.756788
.757577
.758364
.759149
.759934
58
60
T. 765000
T. 765389
1.766167
T. 766944
1.767719
1.768492
1.769265
60
61
.774251
.774634
.775398
.776160
.776923
.777683
.778442
61
62
.783342
.783718
.784469
.785217
.785965
.786712
.787457
62
63
.792272
.792641
.793377
.794111
.794845
.795578
.796310
63
61
.801032
.801392
.802113
.802832
.803550
.804267
.804983
64
65
T. 809606
1.809959
1.810664
1.811366
T.812068
1.812769
1.813469
65
66
.817990
.818335
.819023
.819709
.820396
.821080
.821763
66
67
.826180
.826515
.827186
.827856
.828524
.829192
.829858
67
68
.834165
.834492
.835146
.835797
.836450
.837100
.837750
68
69
.841948
.842266
.842900
.843534
.844167
.844799
.845431
69
70
T.849510
T. 849820
1.850436
1.851051
T.851666
1.852280
T. 852892
70
71
.856856
.857156
.857753
.858350
.858945
.859540
.860134
71
72
.863976
.864266
.864844
.865422
.865998
.866574
.867149
72
73
.870869
.871149
.871708
.872266
.872825
.873382
.873938
73
74
.877537
.877807
.878346
.878885
.879422
.879959
.880494
74
75
T. 883965
T. 884225
T. 884744
1.885264
1.885781
1.886300
1.886817
75
76
.890163
.890414
.890914
.891414
.891913
.892411
.892909
76
77
.896132.
.896373
.896852
.897332
.897811
.898290
.898768
77
78
.901863
.902094
.902554
.903015
.903475
.903935
.904394
78
79
.907365
.907586
.908028
.908469
.908909
.909349
.909790
79
80
1.912638
1.912850
1.913272
1.913695
1.914118
1.914539
1.914960
80
81
.917687
.917890
.918296
.918700
.919104
.919508
.919911
81
82
.922525
.922719
.923106
.923493
.923879
.924265
.924650
82
83
.927149
.927335
.927707
.928078
.928449
.928820
.929190
83
84
.931588
.931767
.932123
.932479
.932835
.933191
.933546
84
85
1.935848
1.936021
1.936365
1.936710
1.937053
1.937397
1.937740
85
86
.939965
.940134
.940471
.940807
.941145
.941481
.941817
86
87
.943995
.944162
.944494
.944826
.945157
.945488
.945820
87
88
.947967
.948138
.948478
.948818
.949158
.949496
.949836
88
89
.952033
.952182
.952480
.952778
.953075
.953372
.953669
89
90
1.955594
1.955719
1.955971
T. 956222
T. 956474
1.956724
1.956975
90
91
.958600
.958718
.958951
.959185
.959419
.959653
.959886
91
92
.961401
.961513
.961737
.961961
.962184
.962407
.962630
92
93
.964079
.964185
.964395
.964606
.964817
.965028
.965238
93
94
.966604
.966706
.966911
.967115
.967320
.967524
.967728
94
95
1.969052
T.969163
1.969385
1.969607
1.969829
T. 970051
T. 970272
95
96
.971710
.971808
.972004
.972119
.972395
.972590
.972786
96
97
.974053
.974191
.974468
.974745
.975022
.975298
.975574
97
98
.977363
.977598
.978068
.978537
.979004
.979473
.979940
98
99
.982966
.983689
.985132
.986570
.988004
.989432
.990856
99
* Months of Entry for Annual Valuation Dec. 31. TABLE LI.
111
LOGARITHM OF THE SINGLE PREMIUM BY MONTHS, ON 1 INSURED FOR THE
WHOLE LIFE.
4 PER CENT.
AGE.
6 m.
6|m.
7|m.
8£m.
9^m.
10|m.
Him.
AGE.
x+n.
June.
May.
April.
March.
Feb.
Jan.
x-\-n.
55
1.721717
T. 722129
T. 722951
1.723771
1.724590
1.725407
1.726224
55
56
.731548
.731956
.732770
.733581
.734392
.735202
.736008
56
57
.741268
.741670
.742474
.743276
.744076
.744876
.745673
57
58
.750863
.751260
.752053
.752844
.753634
.754422
.755209
58
59
.760326
.760717
.761499
.762279
.763059
.763836
.764612
59
60
1.769651
T. 770036
T. 770806
1.771574
T. 772340
1.773105
1.773870
60
61
.778821
.779199
.779955
.780710
.781464
.782216
.782967
61
62
.787830
.788202
.788945
.789685
.790427
.791165
.791903
62
63
.796674
.797039
.797768
.798496
.799221
.799946
.800670
63
64
.805340
.805697
.806410
.807122
.807834
.808544
.809253
64
65
T.813818
T. 814168
T. 814866
T.815562
1.816257
1.816951
T. 817644
65
66
.822105
.822446
.823127
.823808
.824486
.825164
.825841
66
67
.830191
.830524
.831187
.831852
.832513
.833175
.833835
67
68
.838073
.838398
.839045
.839692
.840338
.840983
.841626
68
69
.845746
.846061
.846691
.847319
.847946
.848572
.849197
69
70
T.853198
1.853505
T. 854116
1.854726
1.855336
T. 855945
T.856553
70
71
.860431
.860727
.861320
.861912
.862502
.863093
.863682
71
72
.867436
.867723
.868297
.868870
.869442
.870014
.870584
72
73
.874216
.874494
.875049
.875603
.876156
.876709
.877261
73
74
.880763
.881031
.881566
.882099
.882634
.883167
.883699
74
75
T. 887075
T. 887334
1.887849
T. 888365
1.888879
1.889393
T.889906
75
76
.893158
.893407
.893903
.894400
.894895
.895390
.895885
76
77
.899007
.899246
.899723
.900200
.900675
.901151
.901626
77
78
.904623
.904853
.905310
.905768
.906225
.906682
.907137
78
79
.910009
.910229
.910668
.911106
.911545
.911982
.912420
79
80
1.915171
T. 915381
T.915801
T. 916221
1.916640
T. 917060
T. 917478
80
81
.920113
.920314
.920718
.921120
.921522
.921923
.922325
81
82
.924843
.925035
.925420
.925806
.926190
.926574
.926957
82
83
.929375
.929560
.929929
.930299
.930667
.931036
.931404
83
84
.933723
.933901
.934256
.934610
.934965
.935318
.935671
84
85
T. 937912
1.938083
T. 938426
T. 938769
1.939111
T. 939452
1.939795
85
86
.941985
.942153
.942488
.942823
.943159
.943494
.943828
86
87
.945985
.946151
.946482
.946813
.947142
.947473
.947802
87
88
.950005
.950175
.950513
.950851
.951190
.951527
.951865
88
89
.953818
.953965
.954262
.954559
.954855
.955151
.955446
89
90
1.957100
1.957225
T. 957476
T. 957726
1.957976
1.958226
1.958476
90
91
.960003
.960120
.960353
.960586
.960820
.961052
.961285
91
92
.962742
.962854
.963076
.963299
.963522
.963745
.963968
92
93
.965344
.965449
.965659
.965869
.966080
.966289
.966499
93
94
.967830
.967932
.968135
.968339
.968544
.968747
.968950
94
95
1.970383
T. 970494
T.970716
T. 970937
1.971158
T. 971379
T. 971600
95
96
.972883
.972981
.973176
.973371
.973567
.973761
.973956
96
97
.975712
.975850
.976125
.976400
.976676
.976951
.977226
97
98
.980174
.980407
.980874
.981339
.981805
.982269
.982734
98
99
.991567
.992276
.993691
.995101
.996506
.997907
.999304
99 112
TABLE LII.
SINGLE PREMIUM OR RESERVE (A), FOR TEMPORARY INSURANCE OF 1,000 CON-
TINUING m YEARS FROM THE PRESENT AGE.
In the order of time, a; denotes the Age of Entry; x + n the Present Age; m the unexpired Policy
Years; and n + m the whole Period of Insurance.
4 PER CENT.
PRESENT
AGE.
m — 1.
il
im. a.
2.
Si.
|m. a.
3.
3|.
Jm. a.
PRESENT
AGE.
15
6.339
9.376
.253
12.414
15.330
.243
18.247
21.047
.233
15
16
6.361
9.414
.254
12.467
15.398
.244
18.329
21.143
.235
16
17
6.393
9.462
.256
12.530
15.476
.246
18.422
21.255
.236
17
18
6.426
9.510
.257
12.594
15.560
.247
18.526
21.377
.238
18
19
6.459
9.564
.259'
12.669
15.655
.249
18.640
21.511
.239
19
20
6.502
9.628
.261
12.755
15.760
.250
18.766
21.661
.241
20
21
6.547
9.694
.262
12.841
15.873
.253
18.904
21.823
.243
21
22
6.591
9.765
.265
12.939
15.996
.255
19.053
21.997
.245
22
23
6.647
9.848
.267
13.049
16.132
.257
19.215
22.193
.248
23
24
6.704
9.933
.269-
13.161
16.279
.260
19.398
22.406
.251
24
25
6.762
10.028
.272
13.295
16.445
.263
19.595
22.639
.254
25
26
6.842
10.142
.275
13.441
16.629
.266
19.817
22.905
.257
26
27
6.913
10.251
.278
13.590
16.826
.270
20.061
23.192
.261
27
28
6.995
10.385
.283
13.774
17.052
.273
20.331
23.508
.265
28
29
7.101
10.536
.286
13.971
17.300
.277
20.628
23.857
.269
29
30
7.198
10.685
.291
14.172
17.556
.282
20.940
24.228
.274
30
31
7.308
10.853
.295
14.399
17.844
.287
21.289
24.641
.279
31
32
7.431
11.042
.301
14.652
18.164
.293
21.677
25.104
.286
32
33
7.568
11.249
.307
14.931
18.524
.299
22.116
25.616
.292
33
34
7.719
11.485
.314
15.250
18.918
.306
22.586
26.169
.299
34
35
7.896
11.741
.320
15.587
19.343
.313
23.099
26.777
.307
35
36
8.065
12.004
.328
15.943
19.799
.321
23.656
27.436
.315
36
37
8.262
12.307
.337
16.351
20.316
.330
24.281
28.176
.325
37
38
8.486
12.645
.347
116.804
20.890
.341
24.976
28.988
.334
38
39
8.728
13.015
.357
117.302
21.512
.351
1 25.722
29.876
.346
39
40
8.999
13.417
.368
117.836
22.196
.363
j 26.555
30.853
.358
40
41
9.277
13.854
.381
18.430
22.942
.376
27.455
31.917
.372
41
42
9.612
14.350
.395
19.088
23.774
.391
1 28.460
33.101
.387
42
43
9.955
14.877
.410
19.800
24.676
.406
29.551
34.387
.403
43
44
10.346
15.470
.427
20.593
25.674
.423
30.756
35.808
.421
44
45
10.773
16.115
.445
21.458
26.770
.443
32.081
37.368
.441
45
46
11.239
16.826
.466
22.412
27.972
.463
33.532
39.076
.462
46
47
11.758
17.609
.488
23.460
29.294
.486
35.128
40.956
.486
47
48
12.321
18.463
.512
24.606
30.741
.511
36.877
43.014
.511
48
49
12.943
19.407
.539
25.870
32.336
.539
38.801
45.276
.540
49
50
13.628
20.444
.568
27.260
34.086
.569
40.912
47.755
.570
50
51
14.381
21.582
.600
28.783
36.002
.602
43.221
50.464
.604
51
52
15.206
22.828
.635
30.450
38.097
.637
45.744
53.435
.641
52
53
16.108
24.189
.673
32.271
40.397
.677
48.523
56.692
.681
53
54
17.095
25.691
.716
34.286
42.926
.720
51.566
60.266
.725
54
55
18.202
27.350
.762
36.499
45.711
.768
54.924
64.195
.773
55
56
19.396
29.162
.814
38.928
48.756
.819
58.583
68.472
.824
56
57
20.731
31.162
.869 ,
41.593
52.090
.875
62.587
73.157
.881
57
58
22.175
33.333
.930
44.490
55.724
936
66.959
78.268
.942
58
59
23.756
35.715
.997
47.675
59.715
1.003
71.754
83.857
1.009
59
60
25.506
38.345
1.070
51.183
64.089
1.076
76.995
89.966
1.081
60
61
27.432
41.219
1.149
55.007
68.864
1.155
82.721
96.629
1.159
61
62
29.521
44.356
1.236
59.190
74.079
1.241
88.968
103.892
1.244
62
63
31.833
47.808
1.331
63.783
79.796
1.334
95.809
111.819
1.334
63
64
34.366
51.590
1.435
68.814
86.034
1.435
103.254
120.445
1.433
64
65* I
37.154
55.727
1.548
74.300
92.841
1.545
111.382
129.816
1.536
65
* Ages, G6, 67; .... continued on Page *117.
Formula, A%+n = L0QQMa;+w~Ma;+w+,w.
Vx+n TABLE LIT.
113
SINGLE PREMIUM OR RESERVE (A), FOR TEMPORARY INSURANCE OF 1,000 CONTINU-
ING m, YEARS FROM THE PRESENT AGE.
PRESENT
AGE.
4.
4i
5.
51.
im.a.
6.
6i.
im.d.
PRESENT
AGE.
15
23.847
26.535
.224
29.223
31.808
.215
34.392
36.877
.207
15
16
23.957
26.663
.226
29.369
31.970
.217
34.572
37.073
.208
16
17
24.088
26.811
.227
29.535
32.154
.218
34.773
37.296
.210
17
18
24.229
26.971
.229
29.713
32.354
.220
34.994
37.537
.212
18
19
24.382
27.147
.230
29.911
32.574
.222
35.237
37.801
.214
19
20
24.55G
27.344
.232
30.132
32.816
.224
35.501
38.095
.216
20
21
24.742
27.553
.234
30.364
33.080
.226
35.796
38.416
.218
21
22
24.940
27.784
.237
30.628
33.371
.229
36.114
38.764
.221
22
23
25.171
28.043
.239
30.916
33.691
.231
36.466
39.155
.224
23
24
25.415
28.321
.242
31.227
34.043
.235
36.859
39.583
.227
24
25
25.082
28.632
.246
31.581
34.434
.238
37.287
40.051
.230
25
26
25.994
28.982
.249
31.971
34.867
.241
37.762
40.572
.234
26
27
26.322
29.355
.253
32.387
35.330
.245
38.273
41.132
.238
27
28
26.685
29.768
.257
32.851
35.846
.250
38.841
41.755
.243
28
29
27.087
30.225
.262
33.363
36.416
.254
39.469
42.448
.248
29
30
27.515
30.714
.267
33.913
37.034
.260
40.156
43.196
.253
30
31
27.993
31.263
.273
34.534
37.720
.266
40.906
44.018
.259
31
32
28.532
31.871
.278
35.210
38.471
.272
41.731
44.924
.266
32
33
29.115
32.533
.285
35.950
39.296
.279
42.643
45.923
.273
33
34
29.752
33.260
.292
36.768
40.206
.287
43.644
47.022
.282
34
35
30.455
34.060
.300
37.665
41.207
.295
44.748
48.226
.290
35
36
31.216
34.930
.310 1
38.644
42.291
.304
45.938
49.536
.300
36
37
32.071
35.896
.319
39.721
43.494
.314
47.268
50.989
.310
37
38
33.001
36.959
.330
40.918
44.821
.325
48.724
52.583
.322
38
39
34.029
38.124
.341
42.219
46.269
.338
50.319
54.330
.334
39
40
35.151
39.401
.354
43.651
47.861
.351
52.071
56.247
.348
40
41
36.379
40.799
.368
45.218
49.602
.365
53.985
58.343
>63
41
42
37.742
42.345
.384
46.949
51.525
.381
56.102
60.657
,'-.380
42
43
39.223
44.031
.401
48.838
53.623
.399
58.408
63.179
.398
43
44
40.861
45.889
.419
50.918
55.932
.418
60.946
65.954
.417
44
45
42.654
47.926
.439
53.197
58.462
.439
63.727
68.994
.439
45
46
44.621
50.159
.462
55.697
61.236
.462
66.775
72.322
.462
46
47
46.783
52.612
.486
58.441
64.279
.487
70.116
75.968
.488
47
48
49.151
55.297
.512
61.444
67.605
.513
73.767
79.950
.515
48
49
51.751
58.243
.541
64.734
71.247
.543
77.700
84.309
546
49
50
54.598
61.464
.572
68.330
75.235
.575
82.139
89.080
.578
50
51
57.708
64.992
.607
72.276
79.597
.610
86.919
94.292
.614
51
52
61.125
68.855
.644
76.586
84.370
.G49
92.154
99.987
.653
52
53
64.861
73.087
.686
81.312
89.590
.690
97.868
106.198
.694
53
54
68.907
77.723
.730
86.479
95.290
.734
104.101
112.973
.739
54
55
73.465
82.794
.777
92.124
101.518
.783
110.912
120.368
.788
55
56
78.362
88.320
.830
98.278
108.303
.835
118.327
128.405
.840
56
57
83.726
94.367
.887
105.007
115.703
.891
126.399
137.148
.896
57
58
89.578
100.947
.947
112.316
123.742
.952
135.168
146.630
.956
58
59
95.960
108.1^3
1.014
120.287
132.495
1.017
144.704
156.942
1.020
59
60
102.936
115.954
1.085
128.972
142.022
1.088
155.071
168.117
1.087
60
61
110.537
124.479
1.162
138.420
152,358
1.162
166.296
180.211
1.160
61
62
118.817
133.738
1.243
148.660
163.556
1.241
178.452
193.262
1.234
62
63
127.829
143.812
1.332
159.794
175.684
1.324
191.574
207.348
1.315 1
63
64
137.636
154.727
1.424
171.819
188.786
1.414
205.754
222.516
1.397
64
65
148.250
166.550
1.525
184.850
202.929
1.507
221.007
238.801
1.483
65
4 PER CENT.
* Note that \ m. d. is of the difference of the integer or m columns, 5 — 4, etc. 114
TABLE LII.
SINGLE PREMIUM OR RESERVE (A), FOR TEMPORARY INSURANCE OF 1,000 CON-
TINUING m YEARS FROM THE PRESENT AGE.
4 PER CENT.
PRESENT
AGE.
m = 7.
n*
im.d.
8.
H-
im.a.
9.
Jm. d.
PRESENT
AGE.
15
39.362
41.751
.199
44.141
46.442
.192
48.743
50.959
.185
15
16
39.575
41.985
.201
44.394
46.714
.193
49.034
51.269
.186
16
17
39.818
42.247
.202
44.676
47.015
.195
49.354
51.614
.188
17
18
40.081
42.530
.204
44.979
47.345
.197
49.711
51.993
.190
18
19
40.365
42.842
.206
45.320
47.709
.199
50.099
52.407
.192
19
20
40.689
43.191
.209
45.693
48.110
.201
50.526
52.868
.195
20
21
41.035
43.566
.211
46.097
48.549
.204
51.001
53.373
.198
21
22
41.414
43.982
.214
46.550
49.034
.207
51.518
53.925
.201
22
23
41.844
44.445
.217
47.046
49.566
.210
52.087
54.533
.204
23
24
42.307
44.947
.220-
47.587
50.148
.213
52.709
55.198
.207
24
25
42.816
45.499
.224
48.181
50.788
.217
53.394
55.930
.211
25
26
43.381
46.110
.227
48.840
51.496
.221
54.153
56.744
.216
26
27
43.991
46.774
.232
49.556
52.271
.226
54.985
57.630
.220
27
28
44.670
47.513
.237
50.357
53.127
.231
55.897
58.602
.225
28
29
45.427
48.329
.242
51.231
54.065
.236
56.899
59.675
.231
29
30
46.236
49.206
.248
52.175
55.083
.242
57.991
60.841
.238
30
31
47.129
50.176
.254
1 53.223
56.209
.249
59.195
62.129
.245
31
32
48.117
51.246
.261
54.376
57.451
.256
! 60.526
63.545
.252
32
33
49.203
52.426
.269
55.649
58.813
.264
61.978
65.100
.260
33
34
50.401
53.718
.276
57.035
60.308
.273
j 63.581
66.808
.269
34
35
51.704
55.135
.286
58.566
61.949
.282
' 65.332
68.677
.279
35
36
53.134
56.681
.296
60.228
63.737
.292
67.245
70.720
.290
36
37
54.709
58.388
.307
62.067
65.711
.304
69.356
72.970
.301
37
38
56.443
60.266
.319
64.088
67.880
.316
: 71.672
75.442
.314
38
39
58.341
62.319
.332
66.298
70.254
.330
74.209
78.145
.328
39
40
60.422
64.573
.346
68.724
72.856
.344
76.988
81.108
.343
40
41
62.701
67.039
.362
71.376
75.701
.360
80.026
84.346
.360
41
42
65.212
69.753
.378
74.295
78.832
.378
83.369
87.906
.378
42
43
67.951
72.717
.397
77.482
82.249
.397
87.016
91.790
.398
43
44
70.962
75.971
.417
80.981
85.998
.418
91.015
96.045
.419
44
45
74.260
79.535
.440
84.809
90.096
.441
95.384
100.689
.442
45
46
77.870
83.431
.463
88.993
94.573
.465
100.153
105.764
.468
46
47
81.820
87.692
.489
93.564
99.469
.492
105.374
111.309
.495
47
48
86.132
92.349
.518
98.566
104.816
.521
111.065
117.358
.524
48
49
90.859
97.443
.549
104.027
110.656
.552
117.286
123.957
.556
49
50
96.020
103.009
.582
109.998
117.031
.586
124.064
131.142
.590
50
51
101.665
109.085
.618
116.504
123.971
. 622
131.438
138.956
.627
51
52
107.821
115.704
.657
123.588
131.526
.662
139.463
147.453
.666
52
53
114.528
122.916
.699
131.304
139.748
.704
148.192
156.680
.707
53
54
121.844
130.776
.744
139.707
148.685
.748
157.663
166.686
.752
54
55
129.825
139.331
.792
148.837
158.391
.796
167.945
177.534
.799
55
56
138.482
148.609
.844
158.737
168.902
.847
179.067
189.257
.849
56
57
147.898
158.687
.899
169.476
180.291
.901
191.106
201.919
.901
57
58
158.104
169.600
.958
181.095
192.588
.958
204.081
215.554
.956
58
59
169.179
181.413
1.020
| 193.648
205.862
1.018
218.076
230.220
1.012
59
60
181.164
194.189
1.085
207.213
220.162
1.079
233.111
245.966
1.071
60
61
194.125
207.959
1.153
221.793
235.526
1.144
249.259
262.826
1.131
61
62
208.072
222.774
1.225
237.476
252.000
1.210
266.524
280.820
1.191
62
63
223.123
238.706
1.299
254.290
269.629
1.278
284.968
299.989
1.252
63
64
239.277
255.776
1.375
272.275
288.432
1.346
304.590
320.325
1.311
64
65*
256.596
274.023
1.452
291.450
308.421
1.414
325.392
341.837
1.370
65J
* Ages 66, 67, .... continued on Page *117. Table lii.
115
SINGLE PREMIUM OR RESERVE (A), FOR TEMPORARY INSURANCE OF 1,000 CON-
TINUING in YEARS FROM THE PRESENT AGE.
4 PER CENT.
PRESENT
AGE,
10.
io£.
£m. a.
11.
11^.
im.a.
12.
12tt.
im.d.
PRESENT
AGE.
15
53.176
55.310
.178
57.444
59.506
.172
61.568
63.557
.166
15
16
53.503
55.661
.180
57.820
59.902
.174
61.984
63.995
.168
16
17
53.874
56.054
.182
58.234
60.340
.176
62.446
64.487
.170
17
18
54.275
56.480
.184
58.685
60.821
.178
62.958
65.024
.172
18
19
54.715
56.952
.186
59.189
61.353
.180
63.517
65.614
.175
19
20
55.210
57.476
.189
59.742
61.937
.183
64.132
66.262
.178
20
21
55.746
58.045
.192
60.343
62.574
.186
64.804
66.971
.181
21
22
56.332
58.667
.195
61.003
63.272
.189
65.542
67.750
.184
22
23
56.979
59.356
.198
61.732
64.044
.193
66.356
68.612
.188
23
24
57.686
60.107
.202
62.529
64.892
.197
67.255
69.557
.192
24
25
58.466
60.941
.206
63.415
65.825
.201
68.235
70.589
.196
25
26
59.336
61.860
.210
64.385
66.851
.206
69.316
71.730
.201
26
27
60.274
62.857
.215
65.439
67.968
.211
70.497
72.976
.207
27
28
61.308
63.957
.221
66.606
69.202
.216
71.799
74.350
.213
28
29
62.450
65.170
.227
67.890
70.563
.223
73.235
75.859
.219
29
30
63.691
66.491
.233
69.291
72.040
.229
74.790
77.503
.226
30
31
65.063
67.944
.240
70.826
73.669
.237
76.511
79.313
.234
31
32
66.565
69.544
.248
72.522
75.459
.245
78.396
81.301
.242
32
33
68.222
71.300
.257
74.378
77.422
.254
80.467
83.482
.251
33
34
70.034
73.225
.266
76.416
79.577
.263
82.737
85.872
.261
34
35
72.023
75.336
.276
78.650
81.937
.274
85.224
88.491
.272
35
36
74.194
77.641
.287
81.087
84.513
.286
87.939
91.350
.284
36
37
76.585
80.178
.299
83.772
87.348
.298
90.925
94.491
.297
37
38
79.211
82.963
.313
86.715
90.456
.312
94.197
97.934
.311
38
39
82.082
86.008
.327
89.933
93.854
.327
97.775
101.696
.327
39
40
85.227
89.342
.343
93.457
97.573
.343
101.689
105.812
.344
40
41
88.666
92.987
.360
97.308
101.636
.361
105.964
110.302
.362
41
42
92.444
96.988
.379
101.533
106.089
.380
110.645
115.216
.381
42
43
96.564
101.350
.399
106.136
110.938
.400
115.740
120.569
.402
43
44
101.074
106.121
.421
111.167
116.242
.423
121.316
126.417
.425
44
45
105.994
111.329
.445
116.665
122.027
.447
127.390
132.791
.450
45
46
111.375
117.016
.470
122.657
128.337
.473
134.016
139.732
.476
46
47
117.245
123.222
.498
129.199
135.213
.501
141.228
147.281
.504
47
48
123.650
129.983
.528
136.316
142.689
.531
149.062
155.479
.535
48
49
130.629
137.343
.560
144.057
150.817
.563
157.577
164.382
.567
49
50
138.219
145.346
.594
152.472
159.647
.598
166.821
174.033
.601
50
51
146.474
154.042
.631
161.611
169.219
.634
176.827
184.473
.637
51
52
155.444
163.477
.669
171.509
179.582
.673
187.655
195.757
.675
52
53
165.168
173.698
.711
182.229
190.791
.714
199.353
207.936
.715
53
54
175.709
184.765
.755
193.821
202.899
.757
211.977
221.053
.756
54
55
187.123
196.735
.801
206.347
215.956
.801
225.566
235.159
.799
55
56
199.446
209.633
.849
219.820
229.990
.848
240.159
250.270
.843
56
57
212.732
223.526
.900
234.320
245.052
.894
255.784
266.438
.888
57
58
227.028
238.435
.951
249.842
261.166!
.944
272.490
283.677
.932
58
59
242.363
254.418
1.005
266.473
278.382!
.992
290.291
302.013
.977
59
60
258.821
271.520
1.058
284.219
296.719
1.042
309.219
321.461
1.020
60
61
276.393
289.748
1.113
303.102
316.1801.090
329.258
341.994
1.061
61
62
295.116
309.117
1.167
323.118
336.752
1.136
350.387
363.598
1.101
62
63
315.011
329.640
1.219
344.269
358.445
1.181
372.620
386.250
1.136
63
64
336.060
351.307
1.271
366.555
381.215
1.222
395.876
409.865
1.166
64
65
358.282
374.094
1.318
389.906
404.994
1.257
420.081
434.369
1.191
65 116
TABLE LII.
SINGLE PREMIUM OR RESERVE (A), FOR TEMPORARY INSURANCE OF 1,000 CON-
TINUING m YEARS FROM THE PRESENT AGE.
PRESENT
AGE.
m— 13.
13i
£m. a.
14.
14i
£m.d.
15.
15i
]m.d.
PRESENT
AGE.
15
65.546
67.467
.160
69.389
71.250
.155
73.112
74.913
.150
15
16
66.007
67.956
.162
69.905
71.791
.157
73.677
75.504
.152
16
17
66.527
68.501
.165
70.475
72.388
.159
74.300
76.156
.155
17
18
67.091
69.094
.167
71.096
73.039
.162
74.983
76.871
.157
18
19
67.710
69.744
.170
71.779
73.756
.165
75.732
77.656
.160
19
20
68.393
70.463
.173
72.532
74.546
.168
76.560
78.525
.164
20
21
69.138
71.247
.176
73.356
75.413
.171
77.471
79.475
.167
21
22
69.958
72.112
.180
74.267
76.366
.175
78.464
80.514
.171
22
23
70.868
73.066
.183
75.264
77.410
.179
79.557
81.658
.175
23
24
71.858
74.106
.187
76.353
78.554
.183
80.755
82.912
.180
24
25
72.943
75.248
.192
77.554
79.813
.188
82.072
84.292
.185
25
26
74.145
76.512
.197
78.878
81.203
.194
83.527
85.811
.190
26
27
75.454
77.889
.203
80.324
82.715
.199
85.107
87.466
.197
27
28
76.901
79.406
.209
81.911
84.383
.206
86.854
89.291
.203
28
29
78.484
81.073
.216
83.662
86.214
.213
88.767
91.292
.210
29
30
80.215
82.890
.223
85.565
88.210
.220
90.854
93:474
.218
30
31
82.116
84.887
.231
87.659
90.404
.229
93.149
95.871
.227
31
32
84.205
87.082
.240
89.958
92.811
.238
95.665
98.502
.236
32
33
86.497
89.487
.249
92.478
95.451
.248
98.424
101.383
.247
33
34
89.007
92.124
.260
95.240
98.342
.259
101.444
104.537
.258
34
35
91.759
95.011
.271
98.263
101.506
.270
104.748
107.987
.270
35
36
94.760
98.160
.283
101.560
104.956
.283
108.353
111.750
.283
36
37
98.058
101.620
.297
105.182
108.745
.297
112.308
115.876
.297
37
38
101.670
105.408
.312
109.146
112.889
.312
116.632
120.385
.313
38
39
105.618
109.545
.327
113.473
117.411
.328
121.348
125.299
.329
39
40
109.934
114.067
.344
118.199
122.345
.346
126.491
130.661
.348
40
41
114.641
118.993
.363
123.346
127.723
.365
132.101
136.501
.367
41
42
119.787
124.384
.383
128.981
133.602
.385
138.222
142.875
.388
42
43
125.398
130.252
.405
135.106
139.994
.407
144.882
149.801
.410
43
44
131.518
136.655
.428
141.792
146.961
.431
152.130
157.332
.433
44
45
138.191
143.626
.453
149.060
154.528
.456
159.997
165.504
.459
45
46
145.448
151.200
.479
156.952
162.744
.483
168.536
174.366
.486
46
47
153.334
159.428
.508
165.523
171.659
.511
177.794
183.962
.514
47
48
161.896
168.356
.538
174.815
181.309
.541
187.803
194.329
.544
48
49
171.188
178.029
.570
184.870
191.745
.573
198.621
205.522
.575
49
50
181.244
188.492
.604
195.740
203.015
.606
210.290
217.582
.608
50
51
192.119
199.793
.640
207.468
215.161
.641
222.854
230.545
.641
51
52
203.860
211.982
.677
220.104
228.225
.677
236.345
244.451
.676
52
53
216.518
225.099
.715
233.680
242.246
.714
250.812
259.328
.710
53
54
230.129
239.190
.755
248.250
257.258
.751
266.266
275.209
.745
54
55
244.753
254.291
.795
263.830
273.298
.789
282.767
292.121
.779
55
56
260.381
270.419
.837
280.456
290.372
.826
300.288
310.048
.813
56
57
277.091
287.618
.877
298.141
308.501
.863
318.860
329.006
.846
57
58
294.864
305.875
.918
316.887
327.671
.899
338.455
348.957
.875
58
59
313.736
325.216
.957
336.696
347.875
.932
359.055
369.888
.903
59
60
333.702
345.624
.994
357.545
369.097
.963
380.648
391.755
.926
60
61
354.730
367.071
1.028
379.413
391.279
.989
403.145
414.468
.944
61
62
376.810
389.514
1.059
402.217
414.338
1.010
426.459
437.938
.957
62
63
399.880
412.885
1.084
425.890
438.206
1.026
450.523
462.071
.962
63
64
423.853
437.100
1.104
450.348
462.770
1.035
475.192
486.720
.961
64
65*
448.657
462.055
1.117
I 475.452
487.886
1.036
500.319
511.728
.951
65
4 PER CENT.
* Ages 68, 67, .... continued on Page *117, 117
TABLE LIT.
SINGLE PREMIUM OR RESERVE (A), FOR TEMPORARY INSURANCE OF 1,000 CON-
TINUING m YEARS FROM THE PRESENT AGE.
4 PER CENT.
PRESENT
AGE.
m= 1.
2.
3.
4.
5.
6.
7.
PRESENT
AGE.
66
40.184
80.300
120.184
159.777
198.892
237.392
275.097
66
67
43.540
86.829
129.802
172.255
214.042
254.965
294.818
67
68
47.155
93.967
140.213
185.732
230.311
273.724
315.790
68
69
51.195
101.770
151.552
200.305
247.783
293.788
338.024
69
70
55.557
110.241
163.796
215.949
266.485
315.078
361.442
70
71
G0.3G0
119.471
177.037
232.818
286.453
337.628
386.094
71
72
65.594
129.472
191.369
250.886
307.673
361.453
411.880
72
73
71.297
140.383
206.812
270.195
330.221
386.505
438.740
73
74
77.604
152.223
223.420
290.846
354.069
412,745
466.583
74
75
84.417
164.963
241.243
312.767
379.147
440.055
495.220
75
76
91.830
178.796
260.341
336.020
405.461
468.353
524.567
76
77
99.995
193.756
280.772
360.616
432.931
497.565
554.382
77
78
108.829
209.830
302.505
386.441
461.463
527.411
584.471
78
79
118.447
227.130
325.565
413.546
490.885
557.801
614.568
79
80
128.910
245.665
350.020
441.753
521.122
588.455
644.371
80
81
140.224
265.556
375.719
471.043
551.910
619.066
673.718
81
82
152. GOO
286.742
402.805
501.266
583.033
649.575
702.195
82
83
1G5.825
309.300
431.016
532.095
614.353
679.402
731.612
83
84
180.310
333.275
460.305
563.681
645.430
711.044
761.097
84
85
195.801
358.404
490.730
595.372
679.361
743.431
788.654
85
PRESENT
AGE.
>n = 8.
9.
10.
11.
12.
13.
14.
PRESENT
AGE.
66
311.816
347.395
381.607
414.250
445.163
474.150
501.052
66
67
333.435
370.566
405.995
439.548
471.009
500.207
526.998
67
68
356.239
394.832
431.382
465.653
497.459
526.643
553.076
68
69
380.231
420.203
457.683
492.467
524.384
553.291
579.128
69
70
405.351
446.522
484.732
519.792
551.547
579.928
604.877
70
71
431.538
473.713
512.411
547.460
578.788
606.326
630.152
71
72
458.680
501.622
540.515
575.278
605.835
632.275
654.704
72
73
486.669
530.079
568.879
602.986
632.496
657.530
678.320
73
74
515.345
558.929
597.240
630.389
658.509
681.863
700.867
74
75
544.526
587.868
625.369
657.183
683.602
705.102
722.104
75
76
573.981
616.735
653.006
683.126
707.638
727.022
742.581
76
77
603.542
645.245
679.879
708.063
730.351
748.240
761.886
77
78
632.877
673.076
705.800
731.669
752.433
768.263
779.443
78
1 79
661.711
700.075
730.413
754.764
773.339
786.450
795.241
79
80
689.875
725.860
754.742
776.775
792.326
802.752
809.424
80
81
716.936
751.624
778.085
796.763
809.285
817.298
822.118
81
82
744.430
776.649
799.389
814.636
824.392
830.261
833.541
82
83
771.440
799.552
818.399
830.460
837.715
841.770
843.949
83
84
796.426
820.113
835.270
844.388
849.484
852.222
853.469
84
85
818.973
838.374
850.046
856.569
860.073
861.670
862.352
85
j 118
TABLE LIII.
ANNUAL PREMIUM (P) OF TEMPORARY INSURANCE OF 1000 FOR m YEARS.
ALSO a(tt — P)* .
v ’ 4 PER CENT.
AGE,
X.
= 1.
2.
3.
4.
I
AGE.
X.
P.
A(tt-P).
P.
A(tt-P).
P.
A(tt-P).
P.
3
1
3
15
6.339
0.646159
6.349
0.645088
6.363
0.643740
6.378 0.642295
15
16
6.361
.666022
6.377
.664539
6.392
.663059
6.408
.661562
16
17
6.393
.685150
6.409
.683844
6.425
.682346
6.443
.680708
17
18
6.426
.704502
6.442
.7030691
6.461
.701461
6.481
.699713
18
19
6.459
.724063
6.480
.722280
6.501
.720520
6.522
.718763
19
20
6.502
0.743023
6.524
0.741335
6.545
0.739587
6.569
0.737749
20
21
6.547
.762213
6.568
.760500
6.594
.758675
6.619
.756706
21 j
22
6.591
.781648
6.619
.779692
6.646
.777720
6.673
.775721
22
23
6.647
.800593
6.675
.798740
6.703
.796760
6.735
.794541
23
24
6.704
.819800
6.733
.817846
6.767
.815632
6.801
.813358
24
25
6.762
0.839277
6.801
0.836814
6.836
0.834539
6.873
0.832188
25
26
6.842
.857724
6.876
.855689
6.914
.853366
6.958
.850718
26
27
6.913
.877101
6.953
.874716
7.000
.872014
7.046
.869338
27
28
6.995
.896179
7.047
.893342
7.095
.890663
7.145
.887858
28
29
7.101
.914412
7.148
.911939
7.199
.909193
7.253
.906320
29
30
7.198
0.933538
7.252
0.930807
7.309
0.927915
7.369
0.924798
30
31
7.308
.952468
7.368
.949558
7.432
.946425
7.498
.943141
31
32
7.431
.971234
7.498
.968128
7.568
.964853
7.644
.961235
32
33
7.568
.989881
7.641
.986585
7.722
.982942
7.802
.979305
33
34
7.719
1.008430
7.805
1.004747
7.888
1.001140
7.975
.997379
34
35
7.896
1.026460
7.979
1.023012
8.068
1.019327
8.166
1.015296
35
36
8.065
.045362
8.161
.041571
8.264
.037507
8.372
.033181
36
37
8.262
.063840
8.371
.059760
8.485
.055417
8.604
.050823
37
38
8.486
.081934
8.604
.077663
8.730
.073089
8.857
.068422
38
39
8.728
.100102
8.860
.095477
8.993
.090880
9.136
.085754
39
40
8.999
1.117984
9.135
1.113515
9.286
1.108366
9.442
1.103102
40
41
9.277
.136381
9.440
.131161
9.604
.125917
9.776
.120288
41
42
9.612
.153845
9.779
.148674
9.959
.143115
10.147
.137178
42
43
9.955
.171864
10.146
.166291
10.344
.160342
10.551
.154077
43
44
10.346
.189423
10.554
.183514
10.771
.177300
10.999
.170673
44
45
10.773
1.206904
11.000
1.200718
11.240
1.194123
11.489
1.187120
45
46
11.239
.224347
11.492
.217759
11.754
.210787
12.028
.203401
46
47
11.758
.241475
12.032
.234588
12.320
.227262
12.622
.219402
47
48
12.321
.258627
12.624
.251332
I 12.941
.243507
13.273
.235204
48
49
12.943
.275547
13.276
.267802
13.625
.259532
13.989
.250767
49
50
13.628
1.292263
13.995
1.284037
14.377
1.275337
14.775
1.266078
50
51
14.381
.308803
14.782
.300163
15.200
.290977
15.635
.281179
51
52
15.206
.325195
15.645
.316105
16.101
.306388
16.583
.295944
52
53
16.108
.341466
16.588
.331851
17.095
.321472
17.622
.310432
53
54
17.095
.357628
17.633
.347236
18.187
.336294
18.768
.324502
54
55
18.202
1.373155
18.781
1.362382
19.393
1.350687
20.027
1.338234
55
56
19.396
.388816
20.044
.377181
20.711
.364847
21.404
.351666
56
57
20.731
.403844
21.431
.391695
22.158
.378680
22.918
.364650
57
58
22.175
.418941
22.940
.406085
23.742
.392188
24.577
.377242
58
59
23.756
.433834
24.603
.420046
25.485 j
.405246
26.395
.389423
59
60
25.506
1.448213
26.437
1.433548
27.397
1.417933
28.394
1.401053
60
61
27.432
.462217
28.441
.446845
29.494
.430164
30.584
.412207
61
62
29.521
.476214
30.637
.459719
31.791
.441965
32.988
.422773
62
63
31.833
.489607
33.054
.472085
34.319
.453137
35.622
.432525
63
64
34.366
.502669
35.708
.483974
37.085
.463577
38.511
.442041
64
65
37.154
.515179
38.610
.495002
40.122
.473918
41.668
.450779
65
Formula, P“ = nx= 10°°M*-
■JM®—JNX TABLE LIU.
119
ANNUAL PREMIUM (P) OF TEMPORARY INSURANCE OF 1,000 FOR m YEARS.
ALSO A (tt-P).*
4 PER CENT.
AGE,
5.
6.
7.
8.
AGE,
X.
P.
A(tt-P).
P.
A(tt-P).
P.
3
1
3
P.
pH
1
X.
15
6.393
0.640832
6.409
0.639220
6.426
0.637504
6.443
0.635731
15
16
6.425
.659952
6.443
.658221
6.461
.656452
6.481
.654585
16
17
6.462
.678969
6.481
.677201
6.502
.675334
6.523
.673364
17
18
6.501
.697950
6.523
.696094
6.545
.694120
6.568
.692079
18
19
6.545
.716909
6.569
.714943
6.593
.712909
6.619
.710715
19
20
6.594
0.735785
6.619
0.733765
6.646
0.731583
6.675
0.729294
20
21
6.645
.754701
6.674
.752516
6.704
.750229
6.735
.747838
21
22
6.704
.773521
6.735
.771232
6.767
.768846
6.802
.766238
22
23
6.768
.792264
6.801
.789893
6.838
.787280
6.876
.784583
23
24
6.836
.810996
6.876
.808378
6.915
.805678
6.957
.802847
24
25
6.915
0.829545
6.957
0.826847
7.000
0.824014
7.046
0.821056
25
26
7.002
.848043
7.047
.845218
7.094
.842270
7.144
.839126
26
27
7.094
.866511
7.144
.863582
7.196
.860440
7.251
.857136
27
28
7.197
.884938
7.252
.881811
7.309
.878511
7.371
.874936
28
29
7.310
.903209
7.371
.899934
7.436
.896365
7.502
.892676
29
30
7.433
0.921515
7.501
0.917931
7.571
0.914245
7.644
0.910386
30
31
7.571
.939550
7.644
.935869
7.720
.932026
7.801
.927918
31
32
7.721
.957574
7.801
.953739
7.885
.949646
7.974
.945294
32
33
7.886
.975510
7.974
.971437
8.037
.967094
8.165
.962484
33
34
8.068
.993329
8.165
.988999
8.268
.984402
8.374
.979615
34
35
8.267
1.010988
8.375
1.006399
8.486
1.001641
8.604
0.996488
35
36
8.486
.028590
8.602
.023843
8.727
.018690
8.855
1.013311
36
37
8.726
.046103
8.856
.040956
8.991
.035596
9.133
.029884
37
38
8.993
.063290
9.135
.057964
9.284
.052274
9.439
.046233
38
39
9.284
.080462
9.440
.074798
9.604
.068780
9.774
.062407
39
40
9.605
1.097463
9.776
1.091462
9.955
1.085120
10.143
1.078367
40
41
9.956
.114305
10.144
.107983
10.341
.101252
10.547
.094105
41
42
10.345
.130901
10.551
.124215
10.767
.117103
10.993
.109558
42
43
10.769
.147405
10.996
.140321
11.233
.132806
11.481
.124785
43
44
11.237
.163630
11.486
.156159
11.747
.148180
12.019
.139689
44
45
11.751
1.179693
12.025
1.171750
12.311
1.163306
12.610
1.154311
45
46
12.316
.195498
12.617
.187096
12.930
.178153
13.257
.168630
46
47
12.937
.211053
13.267
.202162
13.610
.192696
13.968
.182614
47
48
13.619
.226367
13.980
.216961
14.356
.206942
14.749
.196212
48
49
14.368
.241427
14.763
.231474
15.176
.220813
15.606
.209439
49
50
15.189
1.256204
15.624
1.245613
16.076
1.234321
16.548
1.222199
50
51
16.093
.270661
16.568
.259449
17.065
.247406
17.581
.234542
51
52
17.083
.284815
17.606
.272856
18.150
.260083
18.713
.246429
52
53
18.174
.298553
18.746
.285880
19.339
.272329
19.955
.257783
53
54
19.370
.311934
19.995
.298493
20.645
.284063
21.319
.268563
54
55
20.685
1.324922
21.369
1.310618
22.080
1.295250
22.814
1.278781
55
56
22.125
.337489
22.875
.322259
23.649
.305935
24.452
.288353
56
57
23.708
.349562
24.526
.333402
25.372
.315984
26.247
.297240
57
58
25.439
.361242
26.334
.343990
27.258
.325424 I
28.212
.305386
58
59
27.339
.372347
28.317
.353971
29.326
.334127
30.362
.312640
59
60
29.426
1.382882
30.494
1.363248
31.591
1.341971
32.720
1.319205
60
61
31.715
.392792
32.876
.371730
34.072
.349229
35.294
.324817
61
62
34.218
.401931
35.486
.379712
36.782
.355582
38.110
.329362
62
63
36.967
.410631
38.341
.386804
39.752
.360895
41.188
.332839
63
64
39.969
.418541
41.470
.392957
42.996
.365247
44.549
.335136
64
65
43.262
.425541
44.887
.398203
46.541
.368478
48.217
.336167
65
* In valuing Term Policies by Table LXIII, (■ —P) Bf will be subtractive. 120
TABLE L111.
ANNUAL PREMIUM (P) OF TEMPORARY INSURANCE OF 1000 FOR m YEARS.
ALSO A(77—P).*
4 PER CENT.
AGE.
m-.
=9.
10.
11.
IS.
AGE,
X.
. P.
A(tt-P).
P.
A(tt-P)-
P.
A(tt-P).
P.
3
1
3
X,
15
6.462
0.633870
6.482
0.631890
6.502
0.629841
6.523
0.627645
15
16
6.501
.652608
6.522
.650560
6.545
.648358
6.569
.646055
16
17
6.545
.671315
6.569
.669115
6.593
.666813
6.619
.664401
17
18
6.593
.689886
6.619
.687584
6.646
.685183
6.675
.682581
18
19
6.646
.708419
6.674
.706021
6.705
.703421
6.736
.700709
19
20
6.704
0.726900
6.736
0.724295
6.769
0.721592
6.803
0.718741
20
21
6.768
.745231
6.803
.742532
6.839
.739681
6.877
.736702
21
22
6.838
.763537
6.876
.760694
6.916
.757714
6.957
.754542
22
23
6.916
.781738
6.957
.778765
7.001
.775606
7.046
.772270
23
24
7.000
.799881
7.046
.796722
7.094
.793392
7.145
.789811
24
25
7.094
0.817907
7.144
0.814585
7.197
0.811003
7.252
0.807277
25
26
7.197
.835813
7.253
.832240
7.311
.828519
7.371
.824619
26
27
7.310
.853562
7.371
.849848
7.434
.845953
7.501
.841813
27
28
7.435
.871237
7.501
.867354
7.571
.863218
7.644
.858826
28
29
7.571
.888805
7.645
.884685
7.721
.880305
7.802
.875655
29
30
7.720
0.906271
7.801
0.901898
7.886
0.897254
7.974
0.892401
30
31
7.885
.923554
7.974
.918920
8.066
.914081
8.164
.908872
31
32
8.067
.940668
8.164
.935843
8.266
.930648
8.372
.925198
32
33
8.266
.957678
8.373
.952498
8.485
.947064
8.602
.941282
33
34
8.486
.974448
8.603
.969034
8.726
.963270
8.854
.957146
34
35
8.727
0.991097
8.856
0.985349
8.990
0.979248
9.132
0.972783
35
36
8.990
1.007578
9.132
1.001491
9.280
.995046
9.435
.988189
36
37
9.282
.023816
9.437
.017393
9.599
1.010559
9.769
1.003301
37
38
9.602
.039835
9.772
.033028
9.950
.025796
10.137
.018117
38
39
9.953
.055630
10.140
.048425
10.335
.040775
10.540
.032621
39
40
10.339
1.071190
10.544
1.063574
10.759
1.055448
10.982
1.046804
40
41
10.762
.086519
10.988
.078425
11.223
.069814
11.468
.060645
41
42
11.229
.101504
11.476
.092935
11.734
.083808
12.002
.074092
42
43
11.741
.116252
12.011
.107164
12.293
.097490
12.587
.087184
43
44
12.303
.130650
12.599
.121025
12.908
.110770
13.230
.099802
44
45
12.921
1.144737
13.244
1.134535
13.583
1.123621
13.934
1.111980
45
46
13.597
158485
13.953
.147629
14.323
.136048
14.708
.123637
46
47
14.342
.171819
14.730
.160304
15.136
.147960
15.556
.134772
47
48
15.158
.184764
15.584
.172490
16.027
.159378
16.485
.145355
48
49
16.055
.197237
16.520
.184203
17.003
.170264
17.504
.155314
49
50
17.038
1.209248
17.546
1.195400
18.073
1.180543
18.620
1.164597
50
51
18.116
.220788
18.672
.206028
19.247
.190187
19.841
.173207
51
52
19.298
.231771
19.905
.216038
20.531
.199176
21.178
.181035
52
53
20.595
.242162
21.255
.225422
21.938
.207409
22.641
.188040
53
54
22.016
.251953
22.736
.234076
23.478
.214850
24.243
.194125
54
55
23.574
1.261047
24.358
1.241974
25.166
1.221407
25.993
1.199187
55
56
25.280
.269440
26.133
.249038
27.008
.226989
27.906
.203328
56
57
27.149
.277017
28.075
.255146
29.024
.231688
29.992
.206313
57
58
29.191
.283700
30.197
.260457
31.223
.235299
32.269
.208025
58
59
31.428
.289625
32.515
.264700
33.625
.237667
34.750
.208422
59
60
33.872
1.294528
35.050
1.267751
36.245
1.238777
37.455
1.207347
60
61
36.544
.298311
37.815
.269622
39.102
.238491
40.398
.204706
61
62
39.460
.300979
40.830
.270165
42.211
.236713
43.595
.200380
62
63
42.646
.302366
44.118
.269269
45.595
.233311
47.073
.194101
63
64
46.119
.302418
47.697
.266860
49.277
.228062
50.844
.185804
64
65
49.902
.301039
51.592
.262680
53.272
.220883
54.929
.175306
65
* In valuing Term Policies by Table LXIII, (tt—P) B/ will be subtractive. TABLE LIII.
121
ANNUAL PREMIUM (P) OF TEMPORARY INSURANCE OF 1000 FOR m YEARS.
ALSO A (tt—P).*
4 PER CENT.
AGE,
= 13.
14.
15.
AGE,
X.
P.
X (tt—P).
P.
1
P.
A(tt-P).
X.
15
6.546
0.625336
6.569
0.622924
6.594
0.620319
15
16
6.593
.643643
6.620
.641039
6.647
.638319
16
17
6.647
.661795
6.675
.659077
6.705
.656216
17
18
6.705
.679867
6.736
.677006
6.769
.674007
18
19
6.769
.697854
6.803
.694858
6.839
.691677
19
20
6.839
0.715751
6.877
0.712573
6.917
0.709226
20
21
6.916
.733532
6.958
.730182
7.002
.726594
21
22
7.001
.751204
7.047
.747616
7.095
.743870
22
23
7.095
.768685
7.145
.764946
7.198
.761015
23
24
7.198
.786078
7.252
.782156
7.310
.777990
24
25
7.310
0.803363
7.371
0.799202
7.434
0.794785
25
26
7.435
.820466
7.501
.816058
7.572
.811378
26
27
7.571
.837412
7.644
.832740
7.721
.827846
27
28
7.721
.854168
7.801
.849285
7.886
.844048
28
29
7.886
.870786
7.975
.865563
8.067
.860072
29
30
8.067
0.887182
8.164
0.881705
8.265
0.875876
30
31
8.265
.903407
8.372
.897595
8.483
.891421
31
32
8.484
.919399
8.601
.913241
8.723
.906715
32
33
8.725
.935143
8.853
.928636
8.987
.921720
33
34
8.989
.950662
9.130
.943764
9.276
.936437
34
35
9.279
0.965909
9.433
0.958602
9.594
0.950845
35
36
9.596
.980904
9.765
.973169
9.942
.964925
36
37
9.946
.995592
10.132
.987375
10.325
.978631
37
38
10.331
1.009931
10.534
1.001221
10.746
.991945
38
39
10.753
.023943
10.975
.014702
11.207
1.004864
39
40
11.216
1.037598
11.460
1.027799
11.713
1.017357
40
41
11.724
.050885
11.989
.040486
12.267
.029368
41
42
12.282
.063739
12.573
.052671
12.876
.040863
42
43
12.893
.076163
13.211
.064406
13.543
.051815
43
44
13.564
.088101
13.913
.075567
14.274
.062175
44
45
14.301
1.099505
14.681
1.086178
15.074
1.071927
45
46
15.108
.110376
15.522
.096198
15.951
.080996
46
47
15.992
.120667
16.444
.105546
16.911
.089324
47
48
16.961
.130316
17.453
.114180
17.961
.096891
48
49
18.022
.139272
18.557
.122080
19.109
.103598
49
50
19.184
1.147512
19.765
1.129137
20.364
1.109387
50
51
20.455
.154948
21.086
.135317
21.735
.114174
51
52
21.845
.161532
22.530
.140519
23.231
.117843
52
53
23.365
.167167
24.106
.144633
24.864
.120438
53
54
25.026
.171744
25.827
.147716
26.643
.121755
54
55
26.842
1.175338
27.704
1.149560
28.583
1.121651
55
56
28.820
.177745
29.750
.150034
30.692
.120077
56
57
30.979
.178815
31.977
.149081
32.985
.116863
57
58
33.329
.178529
34.399
.146556
35.475
.111893
58
59
35.888
.176713-
37.031
.142323
38.174
.105012
59
60
38.672
1.173251
39.890
1. 136245
41.102
1.095957
60
61
41.696
.168027
42,989
.128075
44.269
.084614
61
62
44.978
.160787
46.346
.117699
47.688
.070781
62
63
48.537
.151417
49.975
.104913
51.381
.054124
63
64
52.387
.139747
53.896
.089416
55.358
.034441
64
65
56.552
.125466
58.126
.070998
59.636
.011425
65 122
TABLE LIV.
SINGLE PREMIUM OR RESERVE (A), FOR ENDOWMENT INSURANCE OF l,00u CON-
TINUING m YEARS FROM THE PRESENT AGE.
4 PER CENT.
PRESENT
AGE,
m = 1.
|-m. a.
2.
2}.
im.a.
3.
3-g-.
ijrm.d.
PRESENT
AGE.
15
961.538
943.169
1.531
924.800
907.254
1.462
889.708
872.949
1.397
15
16
961.538
943.169
1.531
924.801
907.256
1.462 |
889.711
872.953
1.397
16
17
961.538
943.170
1.531
924.802
907.258
1.462
889.715
872.959
1.396
17
18
961.538
943.171
1.531
924.803
907.261
1.462!
889.718
872.964
1.396
18
19
961.538
943.171
1.531
924.805
907.264
1.462
889.723
872.971
1.396
19
20
961.538
943.172
1.531
924.806
907.267
1.462
889.727
872.978
1.396
20
21
961.538
943.173
1.530
924.808
907.270
1.462
889.733
872.986
1.396
21
22
961.538
943.174
1.530
924.810
907.274
1.461
889.738
872.994
1.395
22
23
961.538
943.175
1.530
924.812
907.278
1.461
889.744
873.003
1.395
23
24
961.538
943.176
1.530
924.814
907.282
1.461
889.751
873.013
1.395
24
25
961.538
943.177
1.530
924.816
907.287
1.461
889.758
873.024
1.395
25
26
961.538
943.178
1.530
924.819
907.292
1.461
889.766
873.037
1.394
26
27
961.538
943.180
1.530
924.822
907.299
1.460
889.775
873.049
1.394
27
28
961.538
943.181
1.530
924.825
907.305
1.460
889.785
873.064
1.393
28
29
961.538
943.184
1.530
924.829
907.312
1.460
889.796
873.082
1.393
29
30
961.538
943.185
1.530
924.833
907.320
1.459
889.808
873.099
1.392
30
31
961.538
943.188
1.529
924.838
907.329
1.459
889.821
873.118
1.392
31
32
961.538
943.190
1.529
924.842
907.339
1.459
889.835
873.139
1.391
32
33
961.538
943.192
1.529
924.847
907.349
1.458
889.851
873.163
1.391
33
34
961.538
943.196
1.529
924.853
907.361
1.458
889.868
873.190
1.390
34
35
961.538
943.199
1.528
924.860
907.374
1.457
889.888
873.219
1.389
35
36
961.538
943.202
1.528
924.867
907.388
1.457
889.908
873.249
1.388
36
37
961.538
943.206
1.528
924.874
907.402
1.456
889.931
873.283
1.387
37
38
961.538
943.210
1.527
924.883
907.420
1.455
889.957
873.322
1.386
38
39
961.538
943.215
1.527
924.892
907.438
1.455
889.985
873.364
1.385
39
40
961.538
943.220
1.527
924.902
907.458
1.454
890.015
873.410
1.384
40
41
961.538
943.225
1.526
924.913
907.480
1.453
890.048
873.460
1.382
41
42
961.538
943.232
1.526
924.926
907.506
1.452
890.086
873.516
1.381
42
43
961.538
943.239
1.525
924.939
907.532
1.451
890.126
873.576
1.379
43
44
961.538
943.246
1.524
924.954
907.562
1.449
890.171
873.644
1.377
44
45
961.538
943.254
1.524
924.971
907.595
1.448
890.220
873.717
1.375
45
46
961.538
943.263
1.523
924.989
907.631
1.447
890.274
873.798
1.373
46
47
961.538
943.273
1.522
925.008
907.670
1.445
890.333
873.887
1.371
47
48
961.538
943.284
1.521
925.030
907.714
1.443
890.399
873.985
1.368
48
49
961.538
943.296
1.520
925.054
907.762
1.441
890.470
874.091
1.365
49
50
961.538
943.309
1.519
925.080
907.814
1.439
890.549
874.210
1.362
50
51
961.538
943.323
1.518
925.109
907.872
1.436
890.635
874.338
1.358
51
52
961.538
943.339
1.517
925.141
907.935
1.434
890.730
874.480
1.354
52
53
961.538
943.357
1.515
925.176
908.004
1.431
890.833
874.635
1.350
53
54
961.538
943.376
1.514
925.214
908.080
1.428
890.947
874.805
1.345
54
55
961.538
943.397
1.512
925.256
908.164
1.424
891.073
874.994
1.340
55
56
961.538
943.420
1.511
925.302
908.256
1.421
891.211
875.199
1.334
56
57
961.538
943.446
1.508
925.354
908.359
1.416
891.363
875.425
1.328
57
58
961.538
943.473
1.505
925.409
908.469
1.412
891.528
875.670
1.322
58
59
961.538
943.504
1.503
925.470
908.589
1.407
891.709
875.940
1.314
59
60
961.538
943.537
1.500
925.537
908.722
1.401
891.908
876.237
1.306
60
61
961.538
943.575
1.497
925.611
908.868
1.395
892.126
876.560
1.297
61
62
961.538
943.615
1.494
925.692
909.028
1.389
892.365
876.915
1.288
62
63
961.538
943.659
1.490
925.781
909.204
1.381
892.627
877.303
1.277
63
64
961.538
943.708
1.486
925.878
909.396
1.374
892.914
877.728
1.266
64
65
961.538
943.761
1.481
925.985
909.606
1.365
893.228
878.192
1.253
65
Formula, A-m.d.
35.
35i
}m.d.
36.
36i.
|m.d.
PRESENT
AGE,
15
327.224
323.411
.318
319.599
315.983
.301
312.366
308.937
.286
15
16
327.953
324.167
.316
320.381
316.791
.299
313.202
309.803
.283
16
17
328.755
324.997
.313
321.240
317.681
.297
314.122
310.754
.281
17
18
329.633
325.906
.311
322.180
318.654
.294
315.128
311.794
.278
18
19
330.594
326.902
.308
323.210
319.719
.291
316.229
312.932
.275
19
20
331.650
327.995
.305
324.340
320.888
.288
317.437
314.181
.271
20
21
332.803
329.189
.301
325.574
322.165
.284
318.756
315.544
.268
21
22
334.061
330.491
.298
326.922
323.559
.280
320.195
317.031
.264
22
23
335.445
331.923
.294
328.401
325.087
.276
321.774
318.661
.259
23
24
336.952
333.482
.289
330.012
326.753
.272
323.493
320.436
.255
24
25
338.600
335.186
.285
331.772
328.570
.267
325.369
322.372
.250
25
26
340.405
337.052
.279
333.699
330.561
.262
327.422
324.490
.244
26
27
342.368
339.081
.274
335.793
332.722
.256
329.651
326.789
.239
27
28
344.511
341.294
.268
338.077
335.078
.250
332.080
329.293
.232
28
29
346.851
343.710
.262
340.569
337.649
.243
334.728
332.021
.226
29
30
349.385
346.325
.255
343.266
340.429
.236
337.592
334.969
.219
30
31
352.146
349.174
.248
346.201
343.453
.229
340.705
338.174
.211
31
32
355.149
352.269
.240
349.390
346.737
.221
344.084
341.648
.203
32
33
358.407
355.626
.232
352.845
350.292
.213
347.740
345.406
.195
33
34
361.937
359.261
.223
356.585
354.138
.204
351.691
349.464
.186
34
35
365.758
363.192
.214
360.627
358.292
.195
355.957
353.842
.176
35
36
369.878
367.429
.204
364.980
362.762
.185
360.543
358.544
.167
36
37
374.332
372.005
.194
369.678
367.581
.175
365.485
363.606
.157
37
38
379.130
376.931
.183
374.731
372.761
.164
370.790
369.035
.146
38
39
384.287
382.219
,.172
380.152
378.311
.153
376.470
374.843
.136
39
40
389.818
387.886
.161
385.954
384.246
.142
382.539
381.041
.125
40
41
395.738
393.945
.149
392.152
390.579
.131
389.007
387.639
.114
41
42
402.068
400.417
.138
398.765
397.329
.120
395.893
394.656
.103
42
43
408.807
407.298
.126
405.790
404.490
.108
403.190
402.081
.092
43
44
415.974
414.608
.114
413.242
412.077
.097
410.913
409.931
.082
44
45
423.570
422.346
.102
421.121
420.089
.086
419.056
418.197
.072
45
46
431.597
430.511
.091
429.426
428.521
.075
427.617
426.874
.062
46
47
440.058
439.107
.079
438.155
437.374
.065
436.592
435.960
.053
47
48
448.943
448.120
.069
447.297
446.631
.056
445.966
445.437
.044
48
49
458.248
457.547
.058
456.845
456.288
.046
455.730
455.294
.036
49
50
467.957
467.369
.049
466.781
466.321
.038
465.862
465.510
.029
50
51
478.055
477.570
.040
477.085
476.714
.031
476.343
476.065
.023
51
52
488.520
488.128
.033
487.736
487.443
.024
487.149
486.934
.018
52
53
499.328
499.017
.026
498.707
498.480
.019
498.254
498.092
.014
53
54
510.454
510.214
.020
509.974
509.803
.014
509.632
509.516
.010
54
55
521.873
521.693
.015
521.512
521.389
.010
521.266
521.187
.007
55
56
533.542
533.411
.011
533.281
533.197
.007
533.113
533.061
.004
56
57
545.442
545.353
.007
545.263
545.208
.005
545.153
545.120
.003
57
58
557.526
557.479
.004
557.433
557.399
.003
557.364
557.345
.002
58
59
569.792
569.756
.003
569.719
569.699
.002
569.679
569.669
.001
59
60
582.186
582.164
.002
582.143
582.132
.001
582.121
582.115
.001
60
61
594.678
594.666
.001
594.654
594.648
.001
594.643
594.641
.000
61
62
607.234
607.228
.001
607.222
607.219
.000
607.217
607.216
.000
62
63
619.838
619.836
.000
619.833
619.831
.000
619.830!
619.830
.000
63
64
632.460
632.459
.000
632.458
632.458
.000
64
65
645.070
645.069
.000
65
4 PER CENT. 134
TABLE LI V.
SINGLE PREMIUM OR RESERVE (A), FOR ENDOWMENT INSURANCE OF 1,000 CON-
TINUING m YEARS FROM THE PRESENT AGE.
PRESENT
AGE.
II
CO
37 J.
38.
H®*
00
00
|m.d.
39.
CO
CD
toll-*
im.a.
PRESENT
AGE.
15
305.509
302.261
.271
299.014
295.941
.256
292.869
289.963
.242
15
16
306.403
303.186
.268
299.969
296.928
.253
293.886
291.014
.239
16
17
307.386
304.202
.265
301.017
298.010
.251
295.003
292.166
.236
17
18
308.460
305.311
.262
302.163
299.193
.248
296.223
293.425
.233
18
19
309.636
306.527
.259
303.417
300.487
.244
297.557
294.800
.230
19
20
310.925
307.857
.256
304.790
301.904
.241
299.018
296.307
.226
20
21
312.332
309.310
.252
306.288
303.449
.237
300.611
297.949
.222
21
22
313.866
310.894
.248
307.921
305.134
.232
302.346
299.737
.217
22
23
315.548
312. 629
.243
309.710
306.978
.228
304.246
301.694
.213
23
24
317.379
314.517
.239
311.656
308.983
.223
306.311
303.820
.208
24
25
319.376
316.577
.233
313.777
311.168
.217
308.559
306.133
.202
25
26
321.558
318.825
.228
316.093
313.553
.212
311.012
308.657
.196
26
27
323.927
321.266
.222
318.605
316.137
.206
313.670
311.389
.190
27
28
326.506
323.922
.215
321.337
318.948
.199
316.559
314.358
.183
28
29
329.313
326.810
.209
324.307
322.001
.192
319.695
317.578
.176
29
30
332.347
329.931
.201
327.515
325.297
.185
323.079
321.051
.169
30
31
335.642
333.318
.194
330.994
328.869
.177
326.744
324.809
.161
31
32
339.213
336.986
.186
334.759
332.732
.169
330.704
328.867
.153
32
33
343.072
340.947
.177
338.822
336.897
.160
334.972
333.237
.145
33
34
347.237
345.219
.168
343.201
341.382
.152
339.564
337.935
.136
34
35
351.726
349.819
.159
347.913
346.205
.142
344.496
342.975
.127
35
36
356.545
354.754
.149
352.962
351.367
.133
349.773
348.364
.117
36
37
361.727
360.055
.139
358.383
356.905
.123
355.426
354.129
.108
37
38
367.281
365.730
.129
364.179
362.819
.113
361.459
360.276
.099
38
39
373.215
371.788
.119
370.361
369.119
.104
367.878
366.809
.089
39
40
379.543
378.240
.109
376.937
375.815
.094
374.693
373.736
.080
40
41
386.272
385.093
.098
383.915
382.911
.084
381.906
381.058
.071
41
42
393.418
392.363
.088
391.308
390.418
.074
389.529
388.788
.062
42
43
400.973
400.039
.078
399.104
398.326
.065
397.548
396.909
.053
43
44
408.949
408.131
.068
407.313
406.642
.056
405.970
405.427
.045
44
45
417.337
416.631
.059
415.925
415.353
.048
414.782
414.328
.038
45
46
426.132
425.531
.050
424.931
424.453
.040
423.975
423.601
.031
46
47
435.328
434.825
.042
434.322
433.929
.033 1
433.536
433.235
.025
47
48
444.907
444.493
.035
444.079
443.763
.026
443.446
443.209
.020
48
49
454.858
454.524
.028
454.190
453.940
.021
453.690
453.507
.015
49
50
465.158
464.895
.022
464.631
464.438
.016
464.245
464.108
.011
50
51
475.787
475.583
.017
475.380
475.235
.012
475.091
474.993
.008
51
52
486.719
486.567
.013
486.414
486.310
.009 i
486.206
486.139
.006
52
53
497.931
497.821
.009
497.711
497.640
.006 |
497.569
497.525
.004
53
54
509.400
509.325
.006
509.250
509.204
.004
509.157
509.130
.002
54
55
521.107
521.058
.004
521.009
520.980
.002
520.951
520.936
.001
55
56
533.009
532.978
.003
532.948
532.931
.001
532.914
532.905
.001
56
57
545.088
545.071
.001
545.053
545.043
.001
545.034
545.029
.000
57
58
557.326
557.316
.001
557.306
557.302
.000
557.297
557.295
.000
58
59
569.658
569.653
.000
569.648
569.645
.000
569.643
569.643
.000
59
60
582 110
582.107
.000
582.105
582.104
.000
582.103
582.103
.000
60
61
594.638
594.637
.000
594.636
594.636
.000
61
62
607.215
607.215
.000
62
4 PER CENT. TABLE LY.
135
CONTINUED ANNUAL PREMIUM (P), ALSO A (P — tt), FOR ENDOWMENT INSURANCE
OF 1,000, PAYABLE AT DEATH OR AGE 30, 35, 75.
D or 30.
D or 40.
D or 45.
AGE.
X.
P.
A (P - 7T).
AGE.
X.
P.
A(P-tt).
AGE.
X.
P.
A (P - 7T).
10
36.209
1.422188
10
21.626
1.073792
27
42.013
1.440353
11
38.628
1.457505
11
22.610
1.102321
28
45.085
1.480236
12
41.333
1.494033
12
23.674
1.131372
29
48.557
1.521718
13
44.374
1.531914
13
24.828
1.160996
30
52.508
1.565008
14
47.814
1.571314
14
26.080
1.191238
31
57.043
1.610358
15
51.732
1.612430
15
27.444
1.222149
32
62.294
1.658046
16
56.232
1.655497
16
28.933
1.253772
33
68.441
1.708448
17
61.449
1.700814
17
30.564
1.286187
34
75.728
1.762024
18
67.560
1.748737
18
32.357
1.319456
35
84.494
1.819347
19
74.811
1.799719
19
34.333
1.353668
36
95.233
1.881195
20
83.542
1.854341
20
36.521
1.388905
37
108.685
1.948607
21
94.247
1.913374
21
38.955
1.425278
38
126.012
2.023045
22
107.667
1.977848
22
41.674
1.462912
39
149.147
2.106622
23
124.964
2.049214
23
44.730
1.501943
40
181.574
2.202634
24
148.078
2.129593
24
48.186
1.542552
41
230.259
2.316583
25
180.498
2.222272
25
52.122
1.584929
42
311.463
2.458789
26
229.202
2.332749
26
56.639
1.629316
43
473.944
2.652376
27
310.474
2.471330
27
61.873
1.676005
28
473.167
2.661148
28
68.002
1.725368
t» kh
29
75.271
1.777851
30
84.019
1.834038
u or do.
31
94.742
1.894707
10
15.173
0.732305
32
108.178
1.960887
11
15.680
0.757859
10
27.291
1.243468
33
125.490
2.034036
12
16.219
0.783718
11
28.773
1.274618
34
148.615
2.116277
13
16.795
0.809903
12
30.397
1.306530
35
181.039
2.210894
14
17.410
0.836438
13
32.181
1.339279
36
229.732
2.323388
15
18.066
0.863341
14
34.150
1.372945
37
310.967
2.464075
16
18.769
0.890617
15
36.329
1.407614
38
473.559
2.656095
17
19.522
0.918329
16
38.753
1.443390
18
20.330
0.946482
17
41.464
1.480401
19
21.199
0.975105
18
44.510
1.518786
20
22.134
1.004235
19
47.956
1.558715
21
23.142
1.033906
20
51.882
1.600380
10
17.819
0.905510
22
24.232
1.064158
21
56.388
1.644025
1 11
18.510
0.932276
23
25.412
1.095030
22
61.611
1.689944
12
19.250
0.959437
24
26.692
1.126579
23
67.730
1.738495
13
20.045
0.987018
25
28.085
1.158863
24
74.987
1.790134
14
20.901
1.015049
26
29.605
1.191915
25
83.725
1.845446
15
21.821
1.043559
27
31.268
1.225823
26
94.437
1.905194
16
22.814
1.072577
28
33.093
1.260654
27
107.863
1.970416
17
23.888
1.102152
29
35.105
1.296492
28
125.166
2.042566
18
25.052
1.132321
30
37.328
1.333431
29
148.285
2.123761
19
26.315
1.163126
31
39.799
1.371589
30
180.704
2.217287
20
27.690
1.194622
32
42.558
1.411076
31
229.405
2.328653
21
29.191
1.226870
33
45.655
1.452050
32
310.663
2.468160
22
30.834
1.259935
34
49.153
1.494684
33
473.316
2,658944 |
23
32.639
1.293878
35
53.132
1.539172
24
34.628
1.328793
36
57.693
1.585758
25
36.830
1.364772
37
62.972
1.634744
26
39.279
1.401919
38
69.148
1.686510
4 PER CENT.
F<™’ P? = 10 = 1000j^-(l-4 136
TABLE LY.
CONTINUED ANNUAL PREMIUM (P), ALSO A (P — tt), FOR ENDOWMENT INSURANCE
OF 1,000, PAYABLE AT DEATH OR AGE 30, 35, 75.
4 PER CENT.
D or 50.
D or 55.
D or 60.
AGE.
X.
P.
A (P — 7r).
AGE.
X.
P.
A (P - tr).,
AGE.
X.
P.
A (P - tr).
39
76.463
1.741492
46
97.304
1.840747
48
72.131
1.619831
40
85.258
1.800288
47
110.824
1.911844
49
79.567
1.679106
41
96.025
1.863669
48
128.208
1.990107
50
88.483
1.742373
42
109.504
1.932686
49
151.386
2.077669
51
99.369
1.810412
43
126.851
2.008777
50
183.827
2.177823
52
112.954
1.884252
44
150.003
2.094087
1 51
232.476
2.296083
53
130.395
1.965369
45
182.437
2.191901
52
313.525
2.442754
54
153.615
2.055898
46
231.108
2.307724
53
475.564
2.640983
55
186.071
2.159133
47
312.252
2.451875
56
234.677
2.280580
48
474.564
2.647485
57
315.575
2.430562
JJ or bU.
58
477.171
2,632212
D or 55.
10
11.984
0.344432
11
12.290
0.368547
u or do.
10
13.304
0.547799
12
12.613
0.392838
11
13.690
0.572500
13
12.954
0.417355
10
11.069
0.112203
12
14.100
0.597443
14
13.316
0.442119
11
11.321
0.135832
13
14.534
0.622659
15
13.698
0.467164
12
11.586
0.159718
14
14.996
0.648155
16
14.103
0.492439
13
11.867
0.183754
15
15.486
0.673960
17
14.533
0.518014
14
12.163
0.208038
16
16.008
0.700054
18
14.988
0.543869
15
12.474
0.232590
17
16.563
0.726491
19
15.472
0.570029
16
12.804
0.257270
18
17.156
0.753284
20
15.985
0.596509
17
13.152
0.282237
19
17.788
0.780440
21
16.532
0.623363
18
13.520
0.307475
20
18.463
0.807995
22
17.113
0.650570
19
13.909
0.332963
21
19.185
0.835969
23
17.732
0.678154
20
14.320
0.358715
22
19.958
0.864398
24
18.392
0.706146
21
14.756
0.384801
23
20.787
0.893296
25
19.096
0.734600
22
15.217
0.411199
24
21.678
0.922710
26
19.849
0.763488
23
1'5.706
0.437893
25
22.636
0.952662
27
20.655
0.792868
24
16.225
0.464966
26
23.669
0.983189
28
21.518
0.822782
25
16.776
0.492453
27
24.783
1.014332
29
22.445
0.853236
26
17.362
0.520248
28
25.990
1.046160
30
23.440
0.884274
27
17.984
0.548488
29
27.299
1.078678
31
24.512
0.915959
28
18.646
0.577170
30
28.720
1.111975
32
25.668
0.948305
29
19.352
0.606274
31
30.271
1.146109
33
26.918
0.981366
30
20.103
0.635896
32
31.965
1.181126
34
28.271
1.015213
31
20.906
0.666059
33
33.824
1.217123
35
29.741
1.049865
32
21.764
0.696715
34
35.870
1.254188
36
31.340
1.085416
33
22.683
0.727972
35
38.130
1.292396
37
33.086
1.121911
34
23.668
0.759872
36
40.638
1.331870
38
34.999
1.159474
35
24.724
0.792378
37
43.436
1.372737
39
37.101
1.198149
36
25.859
0.825595
38
46.574
1.415162
40
39.420
1.238041
37
27.082
0.859529
39
50.113
1.459295
41
41.990
1.279284
38
28.402
0.894311
40
54.135
1.505350
42
44.852
1.322000
39
29.829
0.929894
41
58.741
1.553575
43
48.055
1.366328
40
31.376
0.966381
42
64.067
1.604276
44
51.663
1.412456
41
33.056
1.003848
43
70.288
1.657804
45
55.756
1.460599
42
34.888
1.042375
44
77.650
1.714635
46
60.435
1.510991
43
36.888
1.082002
45
86.492
1.775362
47
65.834
1.563945
44
39.083
1.122855 TABLE LV.
137
CONTINUED ANNUAL PREMIUM (P), ALSO A (P — rr), FOR ENDOWMENT INSURANCE
OF 1,000, PAYABLE AT DEATH OR AGE 30, 35, 75.
4 PER CENT.
D or 65.
D or 70.
D or 75.
AGE.
X.
P.
A (P — 7t).
AGE.
X.
P.
A (P - tr).
AGE.
X.
P.
A(P-tt).
45
41.499
1.165036
37
23.520
0.565210
24
14.001
1.840420
46
44.169
1.208648
38
24.528
0.598353
25
14.405
1.866996
47
47.135
1.253832
39
25.607
0.632133
26
14.831
1.893595
48
50.447
1.300737
40
26.762
0.666658
27
15.281
1.920697
49
54.167
1.349551
41
28.001
0.701982
28
15.756
1.948119
50
58.375
1.400484
1 42
29.335
0.738154
29
16.258
1.975891
51
63.174
1.453807
43
30.769
0.775181
30
16.789
0.004106
52
68.694
1.509788
44
32.317
0.813167
31
17.350
0.032780
53
75.113
1.568822
45
33.991
0.852175
32
17.943
0.061754
54
82.673
1.631384
46
35.804
0.892256
33
18.571
0.091210
55
91.711
1.698067
47
37.775
0.933497
34
19.237
0.121330
56
102.713
1.769645
48
39.922
0.975974
35
19.942
0.151737
57
116.408
1.847174
49
42.270
1.019810
36
20.689
0.182472
58
133.941
1.932106
! 50
44.845
1.065076
37
21.483
0.214208
59
157.224
2.026593
51
47.682
1.111954
38
22.326
0.246474
60
189.696
2.133927
52
50.821
1.160505
39
23.222
0.279233
61
238.231
2.259621
53
54.313
1.210926
40
24.175
0.312622
62
318.870
2.413989
54
58.220
1.263428
41
25.189
0.346764
63
479.753
2.620184
55
62.621
1.318189
42
26.271
0.381638
56
67.616
1.375481
43
27.424
0.417239
ti m* *yn
57
73.338
1.435614
44
28.655
0.453639
U 01 /U.
58
79.960
1.498957
45
29.973
0.490983
59
87.723
1.566005
46
31.383
0.529174
10
10.461
1.836830
60
96.960
1.637343
47
32.896
0.568331
11
10.678
1.860158
61
108.151
1.713757
48
34.520
0.608526
12
10.907
1.883718
62
122.019
1.796305
49
36.268
0.649812
13
11.148
1.907411
63
139.702
1.886456
50
38.152
0.692256
14
11.402
1.931356
64
163.086
1.986335
51
40.187
0.735998
15
11.669
1.955688
65
195.577
2.099251
52
42.389
0.780987
16
11.950
1.979912
66
243.984
2.230711
53
44.781
0.827415
17
12.247
0.004536
67
324.205
2.391049
54
47.384
0.875339
18
12.560
0.029343
68
483.901
2.603346
55
50.228
0.924925
19
12.889
0.054422
—
56
53.345
0.976281
20
13.237
0.079651
57
56.777
1.029538
21
13.605
0.105272
1! 01 / O.
58
60.571
1.084855
22
13.992
0.131137
59
64.792
1.142461
23
14.402
0.157275
10
10.088
1.497206
60
69.517
1.202557
24
14.835
0.183754
11
10.285
1.520221
61
74.844
1.265398
25
15.293
0.210586
12
10.492
1.543571
62
80.903
1.331322
26
15.777
0.237695
13
10.709
1.567026
63
87.867
1.400704
27
16.289
0.265195
14
10.938
1.590842
64
95.969
1.474015
28
16.833
0.293097
15
11.178
1.614897
65
105.537
1.551885
29
17.409
0.321391
16
11.431
1.638988
66
117.041
1.635077
30
18.018
0.350112
17
11.697
1.663324
67
131.194
1.724712
31
18.666
0.379324
18
11.977
1.687975
68
149.103
1.822091
32
19.354
0.408935
19
12.272
1.712734
69
172.639
1.929527
33
20.086
0.439064
20
12.583
1.737670
70
205.135
2,050237
34
20.865
0.469793
21
12.910
1.762978
71
253.307
2.189751
35
21.694
0.500991
22
13.254
1.788593
72
332.796
2.358365
36
22.577
0.532818
23
13.618
1.814248 138
TABLE LVI.
FIVE ANNUAL PREMIUM (P), ALSO A (P — tt), FOR ENDOWMENT INSURANCE OF 1,000,
PAYABLE AT DEATH OR AGE 30, 35, 75.
4 PER CENT
D or 30.
D or 40.
D or 45.
AGE,
X.
P.
A (P — 7r).
AGE,
X.
P.
A(P-tt).
AGE,
X.
P.
A (P - 7r).
10
106.053
1.983531
10
78.712
1.838461
31
130.936
2.059429
11
109.592
1.998426
11
80.971
1.851367
32
135.574
2.074759
12
113.295
2.013483
12
83.332
1.864452
33
140.430
2.090231
13
117.171
2.028696
13
85.805
1.877745
34
145.515
2.105851
14
121.226
2.044062
14
88.386
1.891195
35
150.839
2.121609
15
125.469
2.059574
15
91.092
1.904855
36
156.415
2.137509
16
129.909
2.075234
16
93.919
1.918679
37
162.261
2.153558
17
134.558
2.091037
17
96.874
1.932666
38
168.393
2.169764
18
139.423
2.106984
18
99.967
1.946830
39
174.824
2.186120
19
20
144.514
149.845
2.123062
2.139278
19
20
103.201
106.584
1.961159
1.975654
40
181.574
2.202634
21
155.425
2.155624
21
110.123
1.990308
D or 50.
22
161.267
2.172098
22
113.825
2.005115
23
167.383
2.188697
23
117.697
2.020079
24
173.788
2.205423
24
121.750
2.035195
10
61.869
1.716798
25
180.498
2.222272
25
26
125.989
130.427
2.050457
2.065872
11
12
63.338
64.876
1.727415
1.738256
D or 35.
27
135.068
2.081421
13
66.482
1.749286
28
139.927
2.097112
14
68.161
1.760517
29
145.018
2.112958
15
69.913
1.771933
10
90.774
1.908484
30
150.343
2.128929
16
71.744
1.783539
11
93.597
1.922434
31
155.922
2.145044
17
73.659
1.795342
12
96.551
1.936559
32
161.763
2.161289
18
75.659
1.807330
13
99.642
1.950861
33
167.890
2.177688
19
77.749
1.819498
14
102.876
1.965331
34
] 74.310
2.194224
20
79.934
1.831855
15
16
106.257
109.796
1.979964
1.994759
35
181.039
2.210894
21
22
82.215
84.598
1.844383
1.857078
17
18
113.499
117.373
2.009715
2.024826
D or 45.
23
24
87.089
89.690
1.869954
1.882991
19
121.427
2.040090
25
92.409
1.896195
20
125.669
2.055504
10
69.242
1.774283
26
95.250
1.909563
21
130.108
2.071064
11
71.058
1.786070
27
98.217
1.923083
22
134.754
2.086765
12
72.955
1.798052
28
101.318
1.936761
23
139.616
2.102608
13
74.940
1.810234
29
104.560
1.950596
24
144.707
2.118590
14
77.015
1.822608
30
107.945
1.964569
25
150.036
2.134709
15
79.184
1.835168
31
111.483
1.978692
26
155.616
2.150963
16
81.451
1.847914
32
115.184
1.992966
27
161.458
2.167347
17
83.821
1.860843
33
119.051
2.007377
28
167.578
2.183865
18
86.298
1.873951
34
123.098
2.021947
29
173.989
2.200512
19
88.890
1.887247
35
127.329
2.036649
30
180.704
2.217287
20
21
91.600
94.431
1.900715
1.914349
36
37
131.752
136.383
2.051484
2.066468
22
97.391
1.928147
38
141.234
2.081604
23
100.488
1.942117
39
146.311
2.096879
24
103.723
1.956242
40
151.632
2.112306
25
107.108
1.970529
4i
157.208
2.127887
26
110.648
1.984974
42
463.061
2.143633
27
114.348
1.999566
43
169.200
2.159538
28
118.219
2.014313
44
175.653
2.175626
29
122.269
2.029209
45
182.437
2.191901
-30
126.504
2.044246
Formula, Px = 1000 ---— —
N* - Naj+5 1 + a*x TABLE LVI.
139
FIVE ANNUAL PREMIUM (P), ALSO A (P — 7r), FOR ENDOWMENT INSURANCE OF 1,000,
PAYABLE AT DEATH OR AGE 30, 35, 75.
4 PER CENT.
D or 55.
D or 60.
D or 65.
AGE,
X.
P.
A(P-tt).
AGE.
X.
P.
A(P-TT).
AGE.
X.
P.
A (P - rr).
10
56.208
1.666834
16
58.094
1.678507
17
55.747
1.648466
11
57.413
1.676324
17
59.996
1.688058
18
56.906
1.657208
12
58.672
1.686008
18
61.353
1.697785
19
. 58.113
1.666112
13
59.987
1.695894
19
62.770
1.707688
20
59.371
1.675182
14
61.361
1.705974
20
64.248
1.717770
21
60.682
1.684414
15
62.795
1.716242
1 21
65.790
1.728022
22
62.049
1.693804
16
64.292
1.726698
22
67.396
1.738435
23
63.472
1.703346
17
65.856
1.737349
23
69.074
1.749028
24
64.955
1.713045
18
67.487
1.748169
24
70.822
1.759771
25
66.500
1.722890
19
69.194
1.759203
25
72.645
1.770675
26
68.110
1.732886
20
70.976
1.770409-
26
74.546
1.781737
27
69.781
1.742988
21
72.835
1.781787
27
76.528
1.792950
28
71.528
1.753272
22
74.774
1.793333
28
78.594
1.804308
29
73.345
1.763670
23
76.801
1.805060
29
80.748
1.815814
30
75.232
1.774175
24
78.915
1.816947 1
30
82.991
1.827444
31
77.198
1.784805
25
81.122
1.829002
31
85.328
1.839207
32
79.243
1.795555
26
83.427
1.841224
32
87.768
1.851120
33
81.372
1.806412
27
85.831
1.853593
33
90.309
1.863151
34
83.586
1.817376
28
88.341
1.866119
34
92.957
1.875306
35
85.889
1.828432
29
90.962
1.878804
35
95.717
1.887577
36
88.281
1.839568
30
93.694
1.891620
36
98.591
1.899954
37
90.771
1.850807
31
96.547
1.904586
37
101.589
1.912455
38
93.362
1.862128
32
99.527
1.917698
38
104.715
1.925070
39
96.055
1.873523
33
102.637
1.930947
39
107.974
1.937790
40
98.855
1.884991
34
105.884
1.944330
40
111.372
1.950617
41
101.767
1.896527
35
109.276
1.957855
41
114.910
1.963548
42
104.798
1.908140
36
112.814
1.971497
42
118.610
1.970595
43
107.949
1.919805
37
116.513
1.985281
43
122.476
1.989744
44
111.225
1.931540
38
120.379
1.999202
44
126.509
2.003011
45
114.644
1.943338
39
124.418
2.013249
45
130.723
2.016394
46
118.200
1.955198
40
128.640
2.027427
46
135.128
2.029898
47
121.904
1.967124
41
133.056
2.041744
47
139.738
2.043533
48
125.766
1.979119
42
137.679
2.056206
48
144.560
2.057300
49
129.794
1.991187
43
142.517
2.070803
49
149.628
2.071237
50
134.001
2.003344
44
147.588
2.085556
50
154.940
2.085333
51
138.398
2.015592
45
152.904
2.100467
51
160.524
2.099620
52
142.996
2.027943
46
158.482
2.115545
52
166.399
2.114107
53
147.814
2.040421
47
164.341
2.130805
53
172.594
2.128833
54
152.872
2.053054
48
170.502
2.146258
54
179.140
2.143829
55
158.191
2.065863
49
176.988
2.161921
55
186.071
2.159133
56
163.792
2.078876
50
183.827
2.177823
57
169.708
2.092141
—
58
175.966
2.105692
JJ or bb.
59
182.614
2.119599
U or bU.
60
189.696
2.133927
10
48.874
1.592176
10
51.955
1.625114
11
49.735
1.599683
11
52.962
1.633556
12
50.635
1.607377
12
54.012
1.642170
13
51.573
1.615248
13
55.109
1.650978
14
52.553
1.623298
14
56.254
1.659974
15
53.573
1.631514
15
57.448
1.669149
16
54.637
1.639898 140
TABLE LYI.
FIVE ANNUAL PREMIUM (P), ALSO A(P — n), FOR ENDOWMENT INSURANCE OF 1,000,
PAYABLE AT DEATH OR AGE 30, 35, 75.
D or 70.
D or 70.
D or 75.
AGE.
X.
P.
A (P - 7t).
AGE.
X.
P.
A (P — rr).
AGE.
X.
P.
A (P - 77).
10
46.763
1.568076
51
123.245
1.946960
32
69.756
1.723993
11
47.526
1.574872
52
126.967
1.957208
33
71.425
1.733094
12
48.322
1.581829
53
130.845
1.967477
34
73.156
1.742266
13
49.152
1.588960
54
134.884
1.977748
35
74.950
1.751476
14
50.017
1.596258
55
139.099
1.988038
36
76.805
1.760717
15
50.919
1.603719
56
143.499
1.998360
37
78.730
1.770005
16
51.858
1.611327
57
148.107
2.008743
38
80.725
1.779325
17
52.838
1.619113
58
152.933
2.019199
39
82.788
1.788652
18
53.859
1.627054
59
158.008
2.029783
40
84.923
1.797979
19
54.923
1.635154
60
163.357
2.040535
41
87.132
1.807300
20
56.031
1.643405
61
169.011
2.051507
42
89.418
1.816611
21
57.185
1.651808
62
175,008
2.062769
43
91.780
1.825882
22
58.386
1.660356
63
181.398
2.074414
44
94.224
1.835122
23
59.638
1.669058
64
188.230
2.086527
45
96.749
1.844309
24
60.938
1.677877
65
195.577
2.099251
46
99.358
1.853434
25
62.292
1.686848
47
102.055
1.862490
26
63.702
1.695946
Tl av
48
104.841
1.871459
27
65.165
1.705159
.U 01 (O.
49
107.720
1.880339
28
66.690
1.714506
50
110.695
1.889115
29
68.275
1.723966
10
45.444
1.552301
51
113.771
1.897785
30
69.919
1.733517
11
46.145
1.558602
52
116.948
1.906322
31
71.629
1.743179
12
46.875
1.565061
53
120.235
1.914738
32
73.404
1.752924
13
47.638
1.571681
54
123.637
1.923025
33
75.252
1.762785
14
48.432
1.578460
55
127.161
1.931181
34
77.169
1.772715
15
49.259
1.585385
56
130.811
1.939194
35
79.158
1.782714
16
50.121
1.592460
57
134.600
1.947074
36
81.220
1.792769
17
51.019
1.599695
58
138.529
1.954800
37
83.362
1.802892
18
51.955
1.607081
59
142.623
1.962432
38
85.586
1.813074
19
52.929
1.614612
60
146.889
1.969949
39
87.892
1.823293
20
53.943
1.622288
61
151.343
1.977375
40
90.283
1.833548
21
54.998
1.630102
62
156.006
1.984743
41
92.762
1.843827
22
56.096
1.638054
63
160.907
1.992106
42
95.335
1.854139 1
23
57.238
1.646131
64
166.068
1.999498
43
98.000
1.864456
24
58.426
1.654345
1 65
171.528
2.007005
44
100.766
1.874787
25
59.661
1.662689
66
177.323
2.014694
45
103.634
1.885123
26
60.945
1.671141
67
183.513
2.022728
46
106.607
1.895453
27
62.278
1.679703
68
190.152
2.031155
47
109.692
1.905778
28
63.665
1.688381
69
197.334
2.040271
48
112.891
1.916089
29
65.104
1.697159
70
205.135
2.050237
49
116.212
1.926390
30
66.597
1.706013
50
119.661
1.936679
31
68.147
1.714968
4 PER CENT. TABLE LVII.
141
TEN ANNUAL PREMIUM (P), ALSO A (P — n), FOR ENDOWMENT INSURANCE OF 1,000,
PAYABLE AT DEATH OR AGE 80, 35, 75.
4 PEE CENT.
D or 30.
D or 40.
D or 50.
AGE.
X.
P.
1
Oh,
AGE.
X.
P.
A (P - tr).
AGE.
X.
P.
A (P - 77).
10
59.073
1.692836
24
67.925
1.737328
18
42.171
1.486872
11
61.047
1.708370
25
70.306
1.753103
19
43.341
1.499481
12
63.114
1.724049
26
72.800
1.769018
20
44.565
1.512267
13
65.278
1.739872
27
75.410
1.785063
21
45.844
1.525216
14
67.543
1.755833
28
78.146
1.801246
22
47.180
1.538324
15
69.912
1.771927
29
81.015
1.817579
23
48.578
1.551600
16
72.393
1.788154
| 30
84.019
1.834038
24
50.039
1.565032
17
74.991
1.804513
25
51.567
1.578620
18
77.711
1.821000
26
53.165
1.592361
19
80.559
1.837609
27
54.835
1.606247
20
83.542
1.854341
28
56.584
1.620290
1 ft
38
1 4^0311
29
58 413
1 G34480
11
39.582
1.471713
30
60.325
1.648805
u or oo.
12
40.642
1.484297
31
62.326
1.663279
13
41.750
1.497075
32
64.422
1.677900
10
50.562
1.610534
14
42.910
1.510035
33
66.616
1.692658
11
52.138
1.625153
15
44.122
1.523169
34
68.916
1.707581
12
53.786
1.639928
16
45.389
1.536479
35
71.325
1.722638
13
55.512
1.654872
17
46.715
1.549963
36
73.848
1.737844
14
57.318
1.669970
18
48.101
1.563619
37
76.496
1.753204
15
59.207
1.685216
19
49.552
1.577441
38
79.275
1.768732
16
61.185
1.700611
20
51.069
1.591429
39
82.193
1.784423
17
63.255
1.716155
21
52.655
1.605572
40
85.258
1.800288
18
65.421
1.731841
22
54.315
1.619876
19
67.689
1.747668
23
56.052
1.634337
"Pi nr
20
70.063
1.763631
24
57.868
1.648946
21
72.549
1.779731
25
59.770
1.663709
22
75.152
1.795963
26
61.760
1.678621
10
31.308
1.333133
23
77.878
1.812326
27
63.842
1.693674
11
31.982
1.342979
24
80.733
1.828821
28
66.023
1.708878
12
32.685
1.353008
25
83.725
1.845446
29
68.306
1.724224
13
33.420
1.363236
30
70.696
1.739709
14
34.188
1.373646
31
73.202
1.755344
15
34.990
1.384237
D or
| 32
75.827
1.771120
16
35.827
1.395008
33
78.579
1.787046
17
36.703
1.405966
10
43.844
1.532368
34
81.466
1.803123
18
37.616
1.417076
11
45.105
1.545939
35
84.494
1.819347
19
38.572
1.428400
12
46.423
1.559682
20
39.571
1.439877
13
47.803
1.573607
21
40.613
1.451521
14
49.246
1.587684
U or ou.
22
41.702
1.463319
15
50.757
1.601961
—
23
42.839
1.475290
16
52.337
1.616392
10
34.462
1.392484
24
44.027
1.487409
17
53.990
1.630969
11
35.283
1.403625
25
45.269
1.499686
18
55.719
1.645712
12
36.141
1.414955
26
46.566
1.512117
19
57.529
1.660609
13
37.038
1.426482
27
47.920
1.524686
20
59.423
1.675660
14
37.976
1.438201
28
49.336
1.537405
21
61.405
1.690861
15
38.956
1.450095
29
50.816
1.550270
22
63.480
1.706204
16
39.980
1.462172
30
52.361
1.563263
23
65.651
1.721694
17
41.051
1.474435 |
31
53.976
1.576399
-in t> 1 nnn -M-* — + Dx-ym -^4
Formula, P* = 1000 v? = ri^'
IN a; — IN X+10 T" ttx 142
TABLE LVII.
TEN ANNUAL PREMIUM (P), ALSO A (P — 7r), FOR ENDOWMENT INSURANCE OF 1.000.
PAYABLE AT DEATH OR AGE 30, 35, 75.
4 PEE CENT.
D or 55.
D or 60.
D or 65.
AGE.
X.
P.
A (P — tr).
AGE.
X.
P.
A (P — n).
AGE,
X.
P.
A (P - 7t).
32
55.665
1.589673
43
69.090
1.646207
49
73.955
1.624815
33
57.432
1.603081
44
71.458
1.659392
50
76.525
1.636458
34
59.279
1.616620
45
73.944
1.672733
51
79.233
1.648277
35
61.212
1.630306
46
76.557
1.686238
52
82.090
1.660296
36
63.233
1.644112
47
79.305
1.699929
53
85.110
1.672563
37
65.351
1.658064
48
82.200
1.713828
54
88.311
1.685133
38
67.569
1.672161
49
85.255
1.727968
55
91.711
1.698067
39
69.894
1.686402
50
88.483
1.742373
40
72.331
1.700792
T> nr *70
41
74.887
1.715340
42
77.572
1.730049
XJ Ol oo.
43
80.395
1.744960
10
26.048
1.211486
44
83.364
1.760054
10
27.223
1.241780
11
26.474
1.218028
45
86.492
1.775362
11
27.705
1.249228
12
26.919
1.224712
12
28.207
1.256850
13
27.383
1.231551
13
28.732
1.264638
14
27.868
1.238538
D or 60.
14
29.280
1.272591
15
28.372
1.245673
15
29.851
1.280697
16
28.898
1.252928
10
28.939
1.282518
16
30.447
1.288951
17
29.448
1.260341
11
29.502
1.291120
17
31.069
1.297377
18
30.020
1.267882
12
30.089
1.299878
18
31.718
1.305959
19
30.617
1.275560
13
30.702
1.308825
19
32.395
1.314682
20
31.239
1.283360
14
31.343
1.317950
20
33.101
1.323553
21
31.887
1.291287
15
32.010
1.327245
21
33.837
1.332570
22
32.562
1.299331
16
32.708 .
1.336710
22
34.604
1.341721
23
33.266
1.307502
17
33.437
1.346357
23
35.405
1.351001
24
33.998
1.315752
18
34.197
1.356167
24
36.239
1.360423
25
34.761
1.324126
19
34.991
1.366141
25
37.109
1.369963
26
35.556
1.332590
20
35.820
1.376278
26
38.016
1.379632
27
36.383
1.341134
21
36.685
1.386574
27
38.962
1.389409
28
37.245
1.349779
22
37.587
1.397017
28
39.947
1.399291
29
38.142
1.358495
23
38.529
1.407620
29
40.974
1.409287
30
39.074
1.367263
24
39.512
1.418368
30
42.043
1.419366
31
40.045
1.376104
25
40.538
1.429258
31
43.159
1.429547
32
41.055
1.384977
26
41.609
1.440287
32
44.321
1.439816
33
42.108
1.393931
27
42.726
1.451455
33
45.532
1.450171
34
43.202
1.402908
28
43.893
1.462753
34
46.796
1.460606
35
44.341
1.411908
29
45.110
1.474183
35
48.112
1.471107
36
45.525
1.420917
30
46.379
1.485723
36
49.482
1.481668
37
46.757
1.429943
- 31
47.704
1.497390
37
50.912
1.492299
38
48.040
1.438980
32
49.088
1.509178
38
52.404
1.503000
39
49.375
1.448010
33
50.533
1.521081
39
53.960
1.513757
40
50.764
1.457022
34
52.042
1.533097
40
55.583
1.524562
41
52.209
1.466010
35
53.617
1.545218
41
57.277
1.535424
42
53.716
1.474987
36
55.261
1.557442
42
59.047
1.546353
43
55.283
1.483916
37
56.980
1.569782
43
60.895
1.557327
44
56.917
1.492818
38
58.777
1.582234
44
62.828
1.568371
45
58.621
1.501682
39
60.656
1.594795
45
64.850
1.579485
46
60.398
1.510501
40
62.621
1.607464
46
66.966
1.590670
47
62.253
1.519279
41
64.678
1.620253
47
69.184
1.601943
48
64.190
1.528014
42
66.833
1.633167
48
71.511
1.613318
49
66.216
1.536723 143
TABLE L Y11.
TEN ANNUAL PREMIUM (P), ALSO A (P — n), FOR ENDOWMENT INSURANCE OF 1.000,
PAYABLE AT DEATH OR AGE 30, 35, 75.
4 PER CENT.
D or 70.
D or 75.
D or 75.
AGE.
X.
P.
A (P — tt).
AGE.
X.
P.
A (P - tt).
AGE.
X.
P.
A (P - Tt).
50
68.336
1.545404
19
29.505
1.21*9176
43
51.774
1.430786
51
70.559
1.554086
20
30.074
1.256198
44
53.222
1.437888
52
72.887
1.562734
21
30.667
1.263328
45
54.727
1.444843
53
75.339
1.571470
22
31.285
1.270560
46
56.291
1.451631
54
77.920
1.580248
23
31.927
1.277873
47
57.919
1.458248
55
80.643
1.589137
24
32.596
1.285289
48
59.613
1.464674
56
83.519
1.598159
25
33.293
1.292792
49
61.377
1.470913
57
86.571
1.607430
26
34.017
1.300356
50
63.216
1.476939
58
89.810
1.616970
27
34.771
1.307984
51
65.133
1.482745
59
93.265
1.626906
28
35.555
1.315681
52
67.138
1.488374
60
96.960
1.637343
29
36.371
1.323421
53
69.230
1.493740
30
37.217
1.331190
54
71.422
1.498907
31
38.099
1.339006
1 55
73.722
1.503878
U 01 (o•
32
39.014
1.346826
56
76.135
1.508644
33
39.966
1.354664
57
78.676
1.513255
10
25.313
1.191417
34
40.956
1.362511
58
81.351
1.517700
11
' 25.705
1.197311
35
41.984
1.370324
59
84.183
1.522105
12
26.113
1.203340
36
43.050
1.378103
60
87.185
1.526473
13
26.540
1.209510
37
44.159
1.385851
61
90.374
1.530890
14
26.985
1.215815
38
45.311
1.393554
62
93.773
1.535485
15
27.448
1.222238
39
46.508
1.401190
63
97.414
1.540400
16
27.930
1.228785
40
47.750
1.408741
64
101.322
1.545788
17
28.434
1.235465
41
49.040
1.416198
65
105.537
1.551885
18
28.958
1.242263
42
50.382
1.423557 144
TABLE LVIII.
CONTINUED ANNUAL PREMIUM (P), ALSO A(P—7r), for endowment insurance
OF 1,000, PAYABLE AT DEATH OR AT THE END OF m YEARS. - .
PER CErsT.
AGE.
m-
= 5.
10.
15.
20.
AGE.
X.
P.
A(P—n).
P.
A(P-7t).
P.
A(P-tt).
P.
A (P 7r).
X.
15
180.291
2.229234
83.428
1.861309
51.732
1.612429
36.329
1.407614
15
16
180.303
.228678
83.447
.860046
51.757
.610247
36.361
.404235
16
17
180.317
.228095
83.468
.858725
51.783
.607961
36.395
.400686
17
18
180.333
.227483
83.489
.857333
51.813
.605559
36.433
.396956
18
19
180.350
.226842
83.514
.855874
51.845
.603035
36.475
.393033
19
20
180.370
2.226172
83.543
1.854341
51.881
1.600380
36.522
1.388904
20
21
180.391
.225467
83.572
.852734
51.921
.597591
36.572
.384560
21
22
180.414
.224724
83.605
.851045
51.964
.594660
36.628
.379985
22
23
180.439
.223945
83.641
.849268
52.012
.591575
36.689
.375175
23
24
180.467
.223129
83.682
.847402
52.064
.588336
36.755
.370093
24
25
180.498
2.222271
83.724
1.845446
52.122
1.584928
36.831
1.364770
25
26
180.532
.221370
83.774
.843387
52.186
.581343
36.912
.359150
26
27
180.569
.220425
83.826
.841222
52.254
.577567
37.002
.353220
27
28
180.611
.219432
83.884
.838949
52.331
.573597
37.101
.346973
28
29
180.657
.218388
83.950
.836557
52.416
.569415
37.210
.340389
29
30
180.705
2.217289
84.019
1.834038
52.509
1.565009
37.328
1.333433
30
31
180.759
.216135
84.097
.831394
52.610
.560372
37.459
.326094
31
32
180.819
.214920
84.182
.828607
52.722
.555482
37.605
.318354
32
33
180.886
.213646
84.277
.825679
52.846
.550330
37.763
.310173
33
34
180.960
.212308
84.380
.822599
52.982
.544901
37.938
.301516
34
35
181.039
2.210894
84.495
1.819347
53.131
1.539171
38.130
1.292396
35
36
181.125
.209405
84.619
.815924
53.295
.533121
38.341
.282731
36
37
181.221
.207839
84.756
.812315
53.476
.526732
38.574
.272498
37
38
181.329
.206197
84.908
.808520
53.676
.520002
38.829
.261674
38
39
181.446
.204462
85.075
.804513
53.894
.512879
39.111
.250205
39
40
181.574
2.202633
85.259
1.800288
54.135
1.505348
39.420
1.238041
40
41
181.714
.200706
85.459
.795826
54.400
.497389
39.760
.225149
41
42
181.871
.198679
85.682
.791126
54.693
.488973
40.136
.211460
42
43
182.041
.196538
85.925
.786154
55.014
.480051
40.547
.196922
43
44
182.229
.194281
86.195
.780906
55.367
.470606
41.001
.181469
44
45
182.437
2.191901
86.491
1.775361
55.756
1.460599
41.498
1.165036
45
46
182.665
.189389
86.818
.769493
56.184
.449981
42.047
.147534
46
47
182.915
.186730
87.177
.763284
56.655
.438712
42.650
.128887
47
48
183.190
.183924
87.571
.756711
57.174
.426745
43.313
.108998
48
49
183.494
.180960
88.006
.749750
57.746
.414025
44.042
.087771
49
50
183.828
2.177825
88.483
1.742373
58.375
1.400488
44.845
1.065079
50
51
184.196
.174508
89.010
.734554
59.069
.386092
45.728
.040832
51
52
184.598
.170989
89.590
.726243
59.832
.370731
46.698
.014843
52
53
185.041
.167261
90.230
.717419
60.672
.354350
47.765
0.986987
53
54
185.531
.163316
90.935
.708040
61.600
.336868
48.938
.957080
54
55
186.071
2.159134
91.711
1.698067
62.621
1.318187
50.228
0.924935
55
56
186.664
.154692
92.566
.687439
63.746
.298183
51.645
.890354
56
57
187.318
.149973
93.510
.676122
64.986
.276765
53.203
.853083
57
58
188.034
.144951
94.550
.664036
66.351
.253779
54.911
.812780
58
59
188.825
.139615
95.695
.651135
67.856
.229103
56.790
.769348
59
60
189.696
2.133925
96.959
1.637341
69.517
1.202557
58.850
0.722272
60
61
190.655
.127866
98.354
.622568
71.346
.173961
61.110
.671219
61
62
191.711
.121406
99.892
.606746
73.362
.143121
63.586
.615708
62
63
192.878
.114511
101.592
.589771
75.586
.109825
66.299
.555336
63
64
194.161
.107135
103.467
.571521
78.035
.073755
69.269
.489396
64
65 |195.578
.099252
105.537
.551884
80.733
.034697
72.515
.417223
65
Foemula, P?= \ [•
XNce— -LNcC + Wl VX -j- Ux ) TABLE LYIII.
145
CONTINUED ANNUAL PREMIUM (P), ALSO A (P — tt), FOR ENDOWMENT INSURANCE
OF 1,000, PAYABLE AT DEATH OR AT THE END OF m YEARS.
4 PEE CENT.
AGE.
25.
30.
35.
40.
AGE.
X.
P.
A (P-77).
P.
A (P-7T).
P.
A(P-rr).
P.
A(P-tt).
X.
15
27.444
1.222149
21.821
1.043559
18.066
0.863341 (
15.486
0.673951
15
16
27.484
.217189
21.872
.036501
18.131
.853449
15.568
.660192
16
17
27.528
.211979
21.929
.029067
18.203
.843015
15.658
.645629
17
18
27.577
.206491
21.991
.021231
18.281
.831985
15.758
.630204
18
19
27.631
.200711
22.059
.012960
18.368
.820320
15.866
.613842
19
20
27.690
1.194625
22.134
1.004235
18.463
0.807995
15.985
0.596520
20
21
27.755
.188205
22.216
0.995016
18.567
.794948
16.116
.578146
21
22
27.826
.181440
22.307
.985287
18.681
.781138
16.259
.558613
22
23
27.904
.174316
22.406
.975013
18.807
.766517
16.416
.537882
23
24
27.990
.166791
22.516
.964146
18.945
.751017
16.588
.515847
24
25
28.085
1.158857
22.636
0.952657
19.096
0.734592
16.776
0.492411
25
26
28.190
.150480
22.768
.940507
19.262
.717163
16.982
.467475
26
27
28.304
.141635
22.913
.927642
19.444
.698666
17.208
.440909
27
28
28.430
.132292
23.073
.914015
19.644
.679001
17.455
.412578
28
29
28.569
.122425
23.248
.899585
19.864
.658126
17.725
.382395
29
30
28.720
1.111978
23.440
0.884280
20.103
0.635906
18.019
0.350112
30
■ 31
28.888
.100936
23.651
.868050
20.366
.612265
18.340
.315656
31
32
29.072
.089248
23.883
.850818
20.655
.587082
18.690
.278799
32
33
29.274
.076870
24.138
.832515
20.970
.560218
19.073
.239325
33
34
29.496
.063765
24.417
.813067
21.316
.531581
19.489
.197087
34
35
29.741
1.049865
24.724
0.792378
21.694
0.500991
19.943
0.151737
35
36
30.009
035121
25.059
.770351
22.106
.468288
20.434
.103051
36
37
30.303
.019461
, 25.428
.746875
22.557
.433290
20.969
.050650
37
38
30.628
.002853
25.833
.721868
23.050
.395868
21.550
1.994405
38
39
30.984
0.985184
26.277
.695175
23.588
.355739
22.178
.933740
39
40
31.375
0.966381
26.762
0.666658
24.175
0.312622
22.859
T. 868292
40
41
31.805
.946383
27.294
.636187
24.814
.266373
23.594
.797683
41
42
32.278
.925070
27.877
.603577
25.509
.216641
24.389
.721233
42
43
32.796
.902340
28.513
.568636
26.266
.163072
25.245
.638589
43
44
33.366
.878079
29.211
.531159
27.088
.105272
26.167
.548635
44
45
33.991
0.852175
29.973
0.490969
27.980
0.043087
27.158
T. 451172
45
46
34.677
.824464
30.805
.447732
28.947
1.975707
28.223
.344785
46
47
35.429
.794802
31.714
.401245
29.994
.902818
29.364
.228400
47
48
36.254
.763008
32.705
.351120
31.126
.823670
30.586
.099681
48
49
37.160
728914
33.785
.297082
32.350
.737908
31.894
2.959041
49
50
38.151
0.692256
34.961
0.238648
33.669
T. 644340
33.291
2.797960
50
51
39.238
.652894
36.240
.175599
35.091
.542825
34.784
.624282
51
52
40.429
.610479
37.630
.107108
36.620
.430881
36.377
.426511
52
53
41.730
.564737
39.138
.032820
38.263
.307496
38.076
- .206826
53
54
43.155
.515344
40.775
1.952066
40.028
.171141
39.888
3.963788
54
55
44.713
0.461934
42.547
1.864096
41.920
1.018284
41.821
3.681241
55
56
46.412
.404115
44.462
.767823
43.947
2.846337
43.879
.397940
56
57
48.268
.341355
46.533
.662286
46.118
.656098
46.074
4.954243
57
58
50.289
.272978
48.764
.544564
48.442
.462398
48.414
.477121
58
59
52.490
.198904
51.171
.415808
50.926
.209515
50.910
— 00
59
60
54.886
0.117735
53.760
T.269513
53.583
3.934498
60
61
57.488
.028815
56.547
.105851
56.424
.623249
61
62
60.311
1.931305
59.543
2.924279
59.461
.278754
62
63
63.374
.823148
62.761
.724276
62.709
4.845098
63
64
66.687
.702344
66.214
.495544
66.184
.477121
64
65
70.270
.566084
69.920
.240549
65 146
TABLE LIX.
FIVE ANNUAL PREMIUM (P), ALSO A(P — n)f FOR ENDOWMENT INSURANCE OF 1,000,
PAYABLE AT DEATH OR AT THE END OF m YEARS.
AGE.
= 10.
15.
20.
25.
AGE.
X.
P.
A(P-7t).
P.
A(P-tt).
P.
A(P-tt).
P.
A (P 7r).
X.
15
149.726
2.142890
125.469
2.059574
106.257
1.979964
91.092
1.904855
15
16
149.745
.142233
125.501
.058828
106.310
.979162
91.174
.904060
16
17
149.767
.141547
125.537
.058050
106.370
.978331
91.266
.903252
17
18
149.790
.140822
125.577
.057237
106.434
.977471
91.367
.902423
18
19
149.816
.140068
125.621
.056390
106.506
.976578
91.477
.901575
19
20
149.845
2.139278
125.669
2.055504
106.5841.975654
91.600
1.900715
20
21
149.877
.138451
125.722
.054583
106.670
.974696
91.733
.899833
21
22
149.911
.137579
125.779
.053612
106.764
.973702
91.878
.898937
22
23
149.949
.136667
125.842
.052602
106.869
.972680
92.040
.898035
23
24
149.990
.135711
125.912
.051554
106.980
.971610
92.215
.897117
24
25
150.036
2.134709
125.989
2.050457
107.108
1.970529
92.409
1.896195
25
26
150.087
.133661
126.074
.049315
107.247
.969407
92.622
.895275
26
27
150.142
.132561
126.167
.048127
107.397
.968243
92.853
.894347
27
28
150.203
.131407
126.269
.046885
107.563
.967053
93.109
.893429
28
29
150.271
.130199
126.382
.045593
107.747
.965833
93.390
.892525
29
30
150.343
2.128929
126.504
2.044246
107.945
1.964569
93.694
1.891620
30
31
150.424
.127601
126.639
.042843
108.163
.963278
94.028
.890740
31
32
150.513
.126206
126.788
.041385
108.405
.961964
94.396
.889892
32 *
33
150.612
.124745
126.951
.039862
108.669
.960618
94.798
.889077
33
34
150.721
.123218
127.132
.038290
108.957
.959240
95.237
.888305
34
35
150.839
2.121609
127.329
2.036649
109.276
1.957855
95.717
1.887577
35
36
150.967
.119918
127.543
.034937
109.621
.956430
96.239
.886898
36
37
151.110
.118149
127.780
.033162
110.001
.954995
96.810
.886293
37
38
151.269
.116299
128.042
.031328
110.419
.953548
97.435
.885774
38
39
151.442
.114351
128.327
.029412
110.874
.952086
98.115
.885333
39
40
151.632
2.112306
128.640
2.027427
111.372
1.950617
98.855
1.884991
40
41
151.839
.110159
128.983
.025372
111.916
.949142
99.659
.884754
41
42
152.071
.107915
129.361
.023244
112.512
.947677
100.536
.884646
42
43
152.321
.105547
129.773
.021036
113.161
.946212
101,485
.884657
43
44
152.599
.103071
130.226
.018755
113.871
.944766
102.517
.884815
44
45
152.904
2.100467
130.723
2.016394
114.644
1.943338
103.634
1.885123
45
46
153.238
.097733
131.266
.013953
115.486
.941934
104.841
.885585
46
47
.153.606
.094859
131.861
.011427
116.404
.940561
106.146
.886215
47
48
154.010
.091843
132.511
.008817
117.401
.939223
107.552
.887010
48
49
154.453
.088668
133.223
.006124
118.485
.937929
109.067
.887979
49
50
154.940
2.085333
134.001
2.003344
119.661
1.936679
110.695
1.889115
50
51
155.476
.081829
134.852
.000477
120.937
.935482
112.442
.890421
51
52
156.062
.078137
L35.778
1.997507
122.316
.934328
114.310
.891873
52
53
156.705
*074250
136.789
.994446
123.809
.933228
116.307
.893467
53
54
157.413
.070164
137.894
.991293
125.421
.932178
118.436
.895188
54
55
158.191
2.065863
139.099
1.988038
127.161
1.931181
120.704
1.897011
55
56
159.043
.061328
140.409
.984673
129.032
.930214
123.107
.898894
56
57
159.979
.056547
141.836
.981196
131.045
.929276
125.654
.900809
57
58
161.002
.051496
143.382
.977582
133.198
.928319
128.338
.902681
58
59
162.125
.046164
145.063
.973839
135.510
.927370
131.172
.904511
59
60
163.357
2.040535
146.889
1.969949
137.980
1.926372
134.151
1.906207
60
61
164.707
.034576
148.866
.965889
140.616
.925294
137.276
.907713
61
62
166.186
.028279
151.005
.961643
143.421
.924088
140.545
.908951
62
63
167.809
.021607
153.322
.957195
146.406
.922718
143.966
.909865
63
64
169.583
.014521
155.821
.952491
149.573
.921111
147.533
.910359
64
65
171.528
.007005
158.520
.947522
152.928
.919214
151.253
.910363
65
4 PER CENT.
Formula, Px = 1000 \ M* ;+ D*+™ j. = .
1 N« —Na + 6 j 1+flJ. TABLE LIX.
147
FIVE ANNUAL PREMIUM (P) ALSO A (P — n), FOR ENDOWMENT INSURANCE OF 1,000,
PAYABLE AT DEATH, OR AT THE END OF m YEARS.
4 PER CENT.
1
AGE,
30.
35.
40.
AGE.
X.
P.
X (P 7r).
P.
A(P-tt).
P.
X(P-TT).
X.
15
79.184
1.835168
69.913
1.771933
62.795
1.716242
15
16
79.306
834492
70.088
.771532
63.035
.716335
16
17
79.442
.833820
70.281
.771177
63.299
.716526
17
18
79.591
.833154
70.491
.770865
63.588
.716821
18
19
79.754
.832495
70.722
.770603
63.903
.717227
19
20
79.934
1.831855
70.976
1.770409
64.248
1.717770
20
21
80.130
.831226
71.253
.770280
64.625
.718453
21
22
80.344
.830618
71.555
.770228
65.034
.719284
22
23
80.581
.830045
71.887
.770273
65.482
.720297
23
24
80.839
.829502
72.249
.770416
65.969
.721491
24
25
81.122
1.829002
72.645
1.770675
66.500
1.722890
25
26
81.433
.828559
73.078
.771070
67.078
.724518
26
27
81.771
.828166
73.549
.771595
67.704
.726372
27
28
82.143
.827845
74.063
.772279
68.385
.728482
28
29
82.550
.827609
74.625
.773145
69.123
.730870
29
30
82.991
1.827444
75.232
1.774175
69.919
1.733517
30
31
83.474
.827388
75.894
.775415
70.781
.736471
31
32
84.002
.827446
76.614
.776876
71.711
.739740
32
33
84.578
.827632
77.396
.778571
72.714
.743326
33
34
85.206
.827958
78.242
.780516
73.793
.747239
34
35
85.889
1.828432
79.158
1.782714
74.950
1.751476
35
36
86.628
.829058
80.145
.785175
76,186
.756027
36
37
87.434
.829875
81.212
.787932
77.511
.760918
37
38
88.310
.830894
82.362
.790987
78.925
.766137
38
39
89.258
.832112
83.598
.794336
80.429
.771652
39
40
90.283
1.833548
84.923
1.797979
82.025
1.777457
40
41
91.391
.835208
86.342
.801922
83.713
.783522
41
42
92.588
.837119
87.861
.806168
85.499
.789839
42
43
93.875
.839260
89.478
.810687
87.378
.796356
43
44
95.261
.841661
91.199
.815480
89.354
.803052
44
45
96.749
1.844309
93.025
1.820524
91.424
1.809883
45
46
98.341
.847203
94.957
.825787
93.587
.816806
46
47
100.045
.850342
96.997
.831245
95.841
.823778
47
48
101.861
.853702
99.144
.836855
98.185
.830749
48
49
103.794
.857277
101.399
.842585
100.618
.837683
49
50
105.845
1.861037
103.760
1.848385
103.136
1.844523
50
51
108.018
.864963
106.226
.854213
105.739
.851239
51
52
110.311
.869000
108.794
.860000
108.423
.857772
52
53
1L2.726
.873125
111.462
.865709
111.188
.864088
53
54
115.265
.877293
114.230
.871289
114.035
.870148
54
55
117.928
1.881455
117.097
1.876686
116.963
1.875917
55
56
120.709
.885543
120.055
.881833
119.969
.881341
56
57
123.610
.889507
123.109
.886696
123.056
.886392
57
58
126.622
.893259
126.256
.891219
126.224
.891042
58
59
129.756
.896782
129.489
.895309
129.472
.895215
59
60
133.004
1.899981
132.820
1.898975
60
61
136.366
.902797
136.245
.902142
61
62
139.842
.905166
139.765
.904755
62
63
143.437
.907030
143.392
.906786
63
64
147.149
.908303
147.124
.908167
64
65
150.985
.908931
65 148
TABLE LX.
TEN ANNUAL PREMIUM (P) ALSO A(P — tt), FOR ENDOWMENT INSURANCE OF 1,000,
PAYABLE AT DEATH OR AT THE END OF m YEARS.
4 PER CENT.
= 15.
20.
25.
AGE,
AGE.
P.
A (P — tt).
P.
A (P - 77).
P.
A (P - tt).
15
69.912
1.771927
59.207
1.685216
50.757
1.601961
15
16
69.937
.770420
59.243
.683472
50.808
.600017
16
17
69.964
.768843
59.281
.681649
50.864
.598000
17
18
69.994
.767188
59.324
.679742
50.926
.595895
18
19
70.027
.765453
59.371
.677748
50.994
.593706
19
20
70.063
1.763631
59.423
1.675660
51.069
1.591429
20
21
70.103
.761723
59.480
.673478
51.151
.589058
21
22
70.147
.759720
59.542
.671194
51.241
.586594
22
23
70.195
.757617
59.611
.668810
51.340
.584037
23
24
70.248
.755414
59.685
.666304
51.448
.581376
24
25
70.306
1.753103
59.770
1.663709
51.567
1.578620
25
26
70.370
.750677
59.861
.660987
51.698
.575763
26
27
70.440
.748129
59.961
.658133
51.841
.572794
27
28
70.518
.745459
60.071
.655162
51.999
.569729
28
29
70.604
.742655
60.193
.652058
52.173
.566555
29
30
70.696
1.739709
60.325
1.648805
52.361
1.563263
30
31
70.799
.736621
60.470
.645411
52.568
.559867
31
32
70.912
.733376
60.631
.641875
52.796
.556367
32
33
71.037
.729968
60.807
.638180
53.045
.552757
33
34
71.174
.726396
60.999
.634316
53.318
.549045
34
35
71.325
1.722638
61.212
1.630306
53.617
1.545218
35
36
71.489
.718687
61.443
.626103
53.943
.541282
36
37
71.670
.714537
61.698
.621728
54.300
.537246
37
38
71.870
.710185
61.979
.617169
54.691
.533119
38
39
72.090
.705606
62.285
.612419
55.118
.528888
39
40
72.331
1.700792
62.621
I.'607464
55.583
1.524562
40
41
72.595
.695729
62.989
.602305
56.091
.520146
41
42
72.887
.690414
63.394
.596938
56.646
.515653
42
43
73.206
.684812
63.835
.591346
57.249
.511067
43
44
73.558
.678925
64.320
.585530
57.906
.506410
44
45
73.944
1.672733
64.850
1.579485
58.621
1.501682
45
46
74.369
.666212
65.429
.573190
59.398
.496877
46
47
74.835
.659343
66.062
.566645
60.241
.492005
47
48
75.346
.652110
66.754
.559836
61.154
.487056
48
49
75.908
.644490
67.511
.552765
62,145
.482041
49
50
76.525
1.636458
68.336
1.545404
63.216
1.476939
50
51
77.203
.627994
69.237
.537759
64.374
.471757
51
52
77.946
.619053
70.218
.529794
65.622
.466457
52
53
78.762
.609619
71.288
.521507
66.969
.461029
53
54
79.658
.599658
72.453
.512870
68.418
.455442
54
55
80.643
1.589137
73.722
1.503878
69.978
1.449674
55
56
81.721
.578003
75.099
.494476
71.651
.443648
56
57
82.905
.566227
76.598
.484655
73.447
.437335
57
58
84.201
.553741
78.221
.474329
75.367
.430617
58
59
85.624
.540512
79.985
.463526
77.424
.423490
59
60
87.185
1.526473
81.897
1.452135
79.624
1.415803
60
61
88.894
.511543
83.968
.440093
81.973
.407452
61
62
90.767
.495666
86.209
.427329
84.480
.398315
62
63
92.822
.478764
88.635
.413754
87.157
.388268
63
64
95.070
.460703
91.258
.399238
90.014
.377135
64 J
65
97.534
.441409
94.093
.383658
93.062
.364752
65
Formula, P„=1000 | m i =
( Nx—Nx+io ) 1 + a3x TABLE LX.
149
TEN ANNUAL PREMIUM (P), ALSO X (P — tt), FOR ENDOWMENT INSURANCE OF 1,000
PAYABLE AT DEATH OR AT THE END OF m YEARS.
4 PER CENT.
d
CO
35.
40.
AGE.
p.
A (P — 7r).
P.
A (P - tt).
P.
A(P-rr).
AGE.
15
44.122
1.523169
38.956
1.450095
34.990
1.384237
15
16
44.194
.521124
39.057
.448114
35.127
.382584
16
17
44.274
.519013
39.169
.446104
35.278
.380957
17
18
44.362
.516833
39.290
.444056
35.442
.379355
18
19
44.459
.514582
39.424
.441976
35.622
.377787
19
20
44.565
1.512267
39.571
1.439877
35.820
1.376278
20
21
44.681
.509880
39.731
.437760
36.035
.374833
21
22
44.808
.507425
39.906
.435626
36.269
.373455
22
23
44.948
.504910
40.099
.433496
36.526
.372179
23
24
45.101
.502328
40.308
.431367
36.805
.371008
24
25
45.269
1.499686
40.538
1.429258
37.109
1.369963
25
26
45.453
.496993
40.790
.427181
37.441
.369069
26
27
45.654
.494238
41.063
.425131
37.800
.368328
27
28
45.875
.491442
41.362
.423140
38.191
.367770
28
29
46.117
.488608
41.689
.421222
38.616
.367421
29
30
46.379
1.485723
42.043
1.419366
39.074
1.367263
30
31
46.667
.482818
42.430
.417615
39.571
.367350
31
32
46.982
.479895
42.850
.415977
40.108
.367691
32
33
47.327
.476963
43.307
.414470
40.688
.368294
33
34
47.702
.474033
43.804
.413112
41.313
.369173
34
35
48.112
1.471107
44.341
1.411908
41.984
1.370324
35
36
48.556
.468193
44.922
.410868
42.703
.371747
36
37
49.041
.465316
45.541
.409858
43.475
.373458
37
38
49.569
.462496
46.230
.409390
44.301
.375458
38
39
50.142
.459726
46.963
.408964
45.182
.377712
39
40
50.764
1.457022
47.750
1.408741
46.120
1.380204
40
41
51.437
.454392
48.596
.408730
47.116
.382904
41
42
52.168
.451863
49.504
.408944
48.174
.385801
42
43
52.956
.449417
50.475
.409344
49.291
.388829
43
44
53.808
.447077
51.513
.409933
50.471
.391960
44
45
54.727
1.444843
52.620
1.410688
51.715
1.395134
45
46
55.715
.442699
53.798
.411560
53.021
.398287
46
47
56.778
.440651
55.049
.412525
54.393
.401363
47
48
57.918
.438669
56.374
.413523
55.828
.404288
48
49
59.140
.436753
57.776
.414515
57.330
.407009
49
50
60.446
1.434853
59.255
1.415426
58.899
1.409433
50
51
61.841
.432953
60.815
.416198
60.536
.411521
51
52
63.326
.430981
62.456
.416730
62.243
.413174
52
53
64.907
.428896
64.179
.416959
64.021
.414331
53
54
66.586
.426630
65.988
.416797
65.876
.414920
54
55
68.369
1.424120
67.887
1.416166
67.810
1.414878
55
56
70.255
.421252
69.875
.414943
69.824
.414104
56
57
72.252
.417951
71.959
.413074
71.928
.412545
57
58
74.359
.414073
74.144
.410457
74.126
.410143
58
59
76.589
.409583
76.431
.406912
76.421
.406736
59
60
78.943
1.404301
78.834
1.402431
60
61
81.430
.398116
81.358
.396865
61
62
84.057
.390910
84.011
.390105
62
63
86.837
.382544
86.810
.382053
63
64
89.779
.372844
89.764
.372557
64
65
92.898
.361648
65 150
TABLE LXI.
LIMITED OR m ANNUAL PREMIUMS (P), ALSO A. (P — rr), FOR LIFE POLICIES OF 1,000.
AGE,
m =
= 5.
10.
15.
20.
AGE.
X.
P.
A (P - 7r).
P.
P.
A (P — 77).
P.
A (P - 7t).
X.
15
47.841
1.569080
26.657
19.725
0.952279
16.357
0.747458
15
16
48.636
.575656
27.103
20.057
.957224
16.634
.751210
16
17
49.465
.582385
27.568
20.404
.962251
16.925
.754982
17
18
50.327
.589248
28.051
20.765
.967323
17.227
.758723
18
19
51.224
.596246
28.555
21.141
.972434
17.543
.762431
19
20
52.158
1.603385
29.079
21.533
0.977591
17.872
0.766115
20
21
53.129
.610649
29.625
21.942
.982773
18.215
.769739
21
22
54.139
.618034
30.193
22.367
.987966
18.573
.773296
22
23
55.188
.625547
30.784
22.810
.993176
18.947
.776774
23
24
56.279
.633170
31.398
23.271
.998377
19.336
.780144
24
25
57.412
1.640910
32.038
23.751
1.003564
19.742
0.783410
25
26
58.590
.648762
32.703
24.252
.008732
20.166
.786546
26
27
59.810
.656696
33.393
24.772
.013840
20.607
.789489
27
28
61.079
.664734
34.111
25.314
.018900
21.067
.792273
28
29
62.395
.672861
34.857
25.878
.023891
21.548
.794857
29
30
63.757
1.681044
35.631
26.464
1.028767
22.048
0.797184
30
31
65.171
.689305
36.435
27.074
.033544
22.570
.799272
31
32
66.636
.697631
37.270
27.710
.038195
23.115
.801067
32
33
68.154
.706006
38.136
28.371
.042690
23.684
.802534
33
34
69.726
.714424
39.036
29.058
.047022
24.278
.803655
34
35
71.352
1.722864
39.969
29.774
1.051141
24.897
0.804364
35
36
73.032
.731309
40.935
30.517
.055026
25.544
.804637
36
37
74.771
.739775
41.938
31.292
.058665
26.220
.804432
37
38
76.569
.748241
42.979
32.098
.062044
26.926
.803723
38
39
78.426
.756679
44.057
32.937
.065094
27.665
.802425
39
40
80.342
1.765086
45.174
33.810
1.067800
28.437
0.800490
40
41
82.319
.773437
46.331
34.719
.070119
29.246
.797884
41
42
84.361
.781742
47.532
35.667
.072033
30.093
.794523
42
43
86.463
.789955
48.775
36.653
.073469
30.981
.790341
43
44
88.631
.798083
50.063
37.682
.074407
31.912
.785266
44
45
90.864
1.806100
51.398
38.755
1.074798
32.891
0.779236
45
46
93.162
.813987
52.781
39.875
.074582
33.919
.772138
46
47
95.527
.821727
54.214
41.045
.073707
35.001
.763885
47
48
97.960
.829300
55.700
42.267
.072118
36.141
.754371
48
49
100.461
.836691
57.241
43.545
.069761
37.343
.743502
49
50
103.031
1.843872
58.839
44.884
1.066531
38.613
0.731121
50
51
105.672
.850829
60.498
46.287
.062413
39.956
.717154
51
52
108.382
.857524
62.219
47.760
.057275
41.378
.701387
52
53
111.165
.863946
64.008
49.307
.051033
42.887
.683659
53
54
114.022
.870073
65.868
50.936
.043625
44.490
.663842
54
55
116.957
1.875879
67.806
52.653
1.034937
46.198
0.641682
55
56
119.965
.881320
69.822
54.464
.024810
48.016
.616969
56
57
123.054
.886386
71.927
56.381
.013174
49.959
.589480
57
58
126.221
.891022
74.124
58.410
0.999852
52.035
.558900
58
59
129.472
.895215
76.421
60.564
.984707
54.259
.524954
59
60
132.811
1.898927
78.829
62.855
0.967553
56.646
0.487280
60
61
136.241
.902120
81.355
65.296
.948227
59.209
.445480
61
62
139.764
.904747
84.010
67.901
.926461
61.965
.399033
62
63
143.391
.906781
86.809
70.689
.902085
64.934
.347603
63
64
147.124
.908167
89.764
73.679
.874801
68.135
.290458
64
65
150.971
.908858
92.889
76.889
.844309
71.588
.226806
65
4 PER CENT.
Formula, = 1000^-^%— = -—
N* — Na-fw 1 + fl^-1 TABLE LXII.
151
NET VALUE OR RESERVE ON TEN PAYMENT LIFE POLICIES OF 1,000.
4 PER CENT,
Age of
1st Year.
2d Year.
3d Year.
Age of
Entry.
6m.
12m.
i m. d.
6m.
12m.
-g-m. d.
6m.
12m.
T>m. d.
Entry.
15
23.965
21.272
.224
45.724
43.520
.184
68.478
66.779
.142
15
16
24.409
21.716
.224
46.619
44.418
.183
69.838
68.154
.140
16
17
24.869
22.169
.225
47.541
45.346
.183
71.247
69.580
.139
17
18
25.346
22.642
.225
48.504
46.315
.182
72.712
71.059
.138
18
19
25.845
23.135
.226
49.503
47.316
.182
74.231
72.590
.137
19
20
26.360
23.641
.227
50.535
48.349
.182
75.804
74.180
.135
20
21
26.896
24.166
.228
51.609
49.427
.182
77.439
75.826
.134
21
22
27.454
24.715
.228
52.724
50.541
.182
79.133
77.532
.133
22
23
28.030
25.277
.229
53.877
51.692
.182
80.887
79.299
.132
23
24
28.631
25.863
.231
55.076
52.891
.182
82.706
81.122
.132
24
25
29.255
26.472
.232
56.314
54.119
.183
84.583
83.010
.131
25
26
29.895
27.088
.234
57.591
55.390
.183
86.527
84.961
.131
26
27
30.566
27.739
.236
58.923
56.715
.184
88.539
86.969
.131
27
28
31.259
28.407
.238
60.290
58.062
.186
90.606
89.040
.130
28
29
31.969
29.080
.241
61.696
59.454
.187
92.744
91.177
.131
29
30
32.712
29.793
.243
63.164
60.904
.188
94.962
93.389
.131
30
31
33.479
30.524
.246
64.674
62.390
.190
97.244
95.662
.132
31
32
34.272
31.273
.250
66.231
63.918
.193
99.591
97.994
.133
32
33
35.090
32.044
.254
67.832
65.485
.196
102.000
100.380
.135
33
34
35.934
32.832
.258
69.475
67.082
.199
104.478
102.837
.137
34
35
36.801
33.633
.264
71.168
68.733
.203
107.032
105.363
.139
35
36
37.704
34.473
.269
72.922
70.437
.207
109.663
107.954
.142
36
37
38.633
35.328
.275
74.717
72.167
.213
112.351
110.597
.146
37
38
39.585
36.192
.283
76.551
73.932
.218
115.100
113.288
.151
38
39
40.568
37.079
.291
78.434
75.731
.225
117.919
116.050
.156
39
40
41.576
37.977
.300
80.364
77.577
.232
120.802
118.854
.162
40
41
42.621
38.912
.309
82.347
79.452
.241
123.752
121.720
.169
41
42
43.685
39.837
.321
84.360
81.351
.251
126.751
124.620
.178
42
43
44.784
40.794
.333
86.429
83.288
.262
129.818
127.572-
.187
43
44
45.910
41.756
.346
88.531
85.244
.274
132.931
130.556
.198
44
45
47.063
42.'729
.361
90.675
87.222
.288
136.095
133.570
.210
45
46
48.247
43.713
.378
92.855
89.217
.303
139.307
136.616
.224
46
47
49.458
44.701
.396
95.072
91.229
.320
142.561
139.679
.240
47
48
50.701
45.702
.417
97.326
93.250
.340
145.856
142.762
.258
48
49
51.968
46.696
.439
99.606
95.275
.361
149.179
145.841
.278
49
50
53.269
47.699
.464
101.918
97.297
.385
152.529
148.922
.301
50
51
54.593
48.688
.492
104.247
99.309
.411
155.900
151.994
.326
51
52
55.949
49.678
.523
106.608
101.319
.441
159.298
155.058
.353
52
53
57.336
50.665
.556
108.995
103.317
.473
162.703
158.080
.385
53
54
58.755
51.641
.593
111.391
105.273
.510
166.102
161.064
.420
54
55
60.194
52.583
.634
113.792
107.195
.550
169.489
163.977
.459
55
56
61.673
53.524
.679
116.210
109.074
.595
172.867
166.838
.502
56
57
63.173
54.418
.730
118.620
110.895
.644
176.221
169.620
.550
57
58
64.713
55.302
.784
121.053
112.680
.698
179.563
172.321
.603
58
59
66.289
56.158
.844
123.486
114.392
.758
182.858
174.904
.663
59
60
67.898
56.967
.911
125.902
116.008
.825
186.106
177.374
.728
60
61
69.541
57.726
.985
128.316
117.552
.897
189.308
179.709
.800
61
62
71.236
58.463
1.064
130.741
119.008
.978
192.458
181.898
.880
62
63
72.972
59.134
1.153
133.141
120.340
1.067
195.525
183.901
.969
63
64
74.756
59.748
1.251
135.531
121.549
1.165
198.523
185.734
1.066
64
65
76.592
60.295
1.358
137.914
122.644
1.272
201.446
187.359
1.174
65 152
TABLE LX1I
NET VALUE OR RESERVE ON TEN PAYMENT LIFE POLICIES OF 1,000.
4 PER CENT.
Age of
4 th Year.
5tli Year.
6th Year.
Age of
Entry,
6m.
12m.
im.d.
6m.
12m.
d.
6m.
12m.
|m.d.
Entry.
15
92.268
91.100
.097
117.145
116.533
.051
143.157
143.124
.003
15
16
94.116
92.974
.095
119.500
118.923
.048
146.039
146.052
.001
16
17
96.031
94.914
.093
121.939
121.397
.045
149.028
149.091
.005
17
18
98.017
96.925
.091
124.474
123.971
.042
152.133
152.243
.009
18
19
100.080
99.015
.089
127.102
126.635
.039,
155.350
155.510
.013
19
20
102.218
101.177
.087
129.826
129.396
.036
158.687
158.899
.018
20
21
104.434
103.417
.085
132.652
132.261
.033
162.144
162.402
.022
21
22
106.735
105.744
.083
135.579
135.221
.030
165.724
166.034
.026
22
23
109.112
108.141
.081
138.605
138.286
.027
169.432
169.794
.030
23
24
111.573
110.627
.079
141.743
141.461
.024
173.265
173.671
.034
24
25
114.124
113.199
.077
144.983
144.729
.021
177.224
177.681
.038
25
26
116.752
115.840
.076
148.325
148.107
.018
181.317
181.824
.042
26
27
119.470
118.578
.074
151.787
151.602
.015
185.549
186.104
.046
27
28
122.275
121.400
.073
155.357
155.203
.013
189.915
190.515
.050
28
29
125.171
124.308
.072
159.039
158.913
.010
194.414
195.059
.054
29
30
128.167
127.313
.071
162.842
162,740
.009
199.053
199.734
.057
30
31
131.248
130.399
.071
166.749
166.664
.007
203.824
204.550
.060
31
32
134.412
133.560
.071
170.768
170.706
.005
208.739
209.502
.064
32
33
137.666
136.816
.071
174.906
174.860
.004
213.791
214.585
.066
33
34
141.016
140.160
.071
179.157
179.119
.003
I 218.977
219.800
.069
34
35
144.460
143.587
.073
183.522
183.487
.003
224.299
225.142
.070
35
36
147.996
147.104
.074
187.999
187.960
.003
229.761
230.627
.072
36
37
151.612
150.688
.077
192.583
192.539
.004
235.350
236.223
.073
37
38
155.312
154.358
.079
197.271
197.206
.005
241.064
241.943
.073
38
39
159.102
158.096
.084
202.065
201.977
.007
246.908
247.781
.073
39
40
162.970
161.912
.088
206.961
206.835
.010
252.869
253.730
.072
40
41
166.924
165.798
.094
211.956
211.784
.014
258.952
259.788
.070
41
42
170.945
169.737
.101
217.037
216.804
.019
265.135
265.934
.067
42
43
175.044
173.741
.109
222.209
221.902
.026
271.428
272.179
.063
43
44
179.204
177.789
.118
227.457
227.063
.033
277.812
278.498
.057
44
45
183.426
181.884
.129
232.781
232.279
.042
284.283
284.890
.051
45
46
187.705
186.014
.141
238.169
237.543
.052
290.831
291.338
.042
46
47
192.034
190.174
.155
243.613
242.839
.065
297.442
297.831
.032
47
48
196.406
194.350
.171
249.106
248.162
.079
304.112
304.362
.021
48
49
200.806
198.530
.190
254.634
253.497
.095
310.827
310.916
.007
49
50
205.238
202.715
.210
260.197
258.839
.113
317.571
317.464
.009
50
51
209.691
206.890
.233
265.770
264.153
.135
324.326
324.000
.027
51
52
214.154
211.032
.260
271.348
269.444
.159
331.080
330.497
.049
52
53
218.614
215.139
.290
276.911
274.676
.186
337.812
336.940
.073
53
54
223.054
219.177
.323
282.442
279.839
.217
344.509
343.311
.100
54
55
227.461
223.138
.360
287.929
284.914
.251
351.148
349.577
.131
55
56
231.845
227.030
.401
293.372
289.891
.290
357.717
355.720
.166
56
57
236.176
230.806
.447
298.726
294.720
.334
364.181
361.716
.205
57
58
240.453
234.461
.499
304.002
299.418
.382
370.545
367.548
.250
58
59
244.654
237.982
.556
309.170
303.938
.436
376.764
373.168
.300
59
60
248.770
241.338
.619
314.208
308.248
.497
382.812
378.548
.355
60
61
252.785
244.505
.690
319.091
312.323
.564
388.672
383.666
.417
61
62
256.687
247.467
.768
323.817
316.156
.638
394.325
388.484
.487
62
63
260.456
250.201
.855
328.348
319.687
.722
399.742
392.987
.563
63
64
264.089
252.680
.951
332.690
322.936
.813
404.905
397.110
.650
64
65
267.582
254.917
1.055
336.816
325.825
.916
409.779
400.845
.745
65 TABLE LXII
153
NET VALUE OR RESERVE ON TEN PAYMENT LIFE POLICIES OF 1,000.
4 PER CENT.
Age of
7tli Year.
8 th Year.
9 th Year.*
Age of
Entry.
6m.
12m.
-gin. d.
6m.
12m.
-?rm. d.
6m.
12m.
im.d.
Entry.
15
170.354
170.927
.048
198.794
200.004
.101
228.535
230.408
.156
15
16
173.789
174.422
.053
202.805
204.084
.107
233.143
235.100
.163
16
17
177.351
178.043
.058
206.963
208.315
.113
237.928
239.972
.170
17
18
181.047
181.800
.063
211.280
212.709
.119
242.889
245.018
.177
18
19
184.882
185.700
.068
215.755
217.255
.125
248.030
250.251
.185
19
20
188.855
189.732
.073
220.390
221.969
.132
253.363
255.677
.193
20
21
192.970
193.912
.079
225.196
226.855
.138
258.882
261.284
.200
21
22
197.235
198.244
.084
230.169
231.902
.144
264.593
267.091
.208
22
23
201.646
202.713
.089
235.311
237.125
.151
270.505
273.102
.216
23
24
206.203
207.338
.095
240.633
242.530
.158
276.622
279.316
.225
24
25
210.920
212.121
.100
246.136
248.113
.165
282.943
285.735
.233
25
26
215.792
217.057
.105
251.819
253.878
.172
289.472
292.363
.241
26
27
220.826
222.155
.111
257.688
259.828
.178
296.208
299.195
.249
27
28
226.017
227.409
.116
263.737
265.953
.185
303.156
306.248
.258
28
29
231.365
232.813
.121
269.971
272.273
.192
310.323
313.516
.266
29
30
236.878
238.391
.126
276.404
278.785
.198
317.708
320.999
.274
30
31
242.557
244.129
.131
283.022
285.480
.205
325.307
328.699
.283
31
32
248.398
250.024
.135
289.828
292.363
.211
333.121
336.609
.291
32
33
254.400
256.079
.140
296.821
299.427
.217
341.153
344.743
.299
33
34
260.560
262.285
.144
304.003
306.684
.223
349.401
353.083
.307
34
35
266.885
268.658
.148
311.372
314.117
.229
357.863
361.639
.315
35
36
273.370
275.179
.151
318.925
321.736
.234
366.537
370.403
.322
36
37
280.007
281.852
.154
326.658
329.527
.239
375.418
379.371
.329
37
38
286.793
288.665
.156
334.566
337.488
.244
384.502
388.537
.336
38
39
293.729
295.620
.158
342.646
345.615
.247
393.784
397.896
.343
39
40
300.807
302.709
.159
350.891
353.898
.251
403.258
407.444
.349
40
41
308.022
309.926
.159
359.297
362.337
.253
412.918
417.169
.354
41
42
315.362
317.257
.158
367.850
370.911
.255
422.753
427.062
.359
42
43
322.827
324.700
.156
376.548
379.622
.256
432.755
437.114
.363
43
44
330.401
332.241
.153
385.377
388.450
.256
442.913
447.313
.367
44
45
338.077
339.866
.149
394.325
397.385
.255
453.216
457.648
.369
45
46
345.842
347.566
.144
403.381
406.416
.253
463.653
468.109
.371
46
47
353.687
355.328
.137
412.538
415.533
.250
474.207
478.668
.372
47
48
361.602
363.142
.128
421.773
424.705
.244
484.862
489.319
.371
48
49
369.566
370.975
.117
431.071
433.925
.238
495.601
500.036
.370
49
50
377.563
378.824
.105
440.415
443.167
.229
506.405
510.803
.366
50
51
385.577
386.656
.090
449.782
452.411
.219
517.257
521.605
.362
51
52
393.588
394.459
.073
459.162
461.645
.207
528.140
532.417
.356
52
53
401.582
402.216
.053
468.531
470.838
.192
539.026
543.206
.348
53
54
409.537
409.895
.030
477.859
479.955
.175
549.892
553.961
.339
54
55
417.420
417.458
.003
487.124
488.984
.155
560.721
564.652
.328
55
56
425.224
424.906
.026
496.313
497.898
.132
571.484
575.247
.314
56
57
432.919
432.195
.060
505.387
506.653
.105
582.148
585.716
.297
57
58
440.485
439.297
.099
514.323
515.225
.075
592.693
596.038
.279
58
59
447.881
446.174
.142
523.091
523.587
.041
603.093
606.177
.257
59
60
455.090
452.802
.191
531.660
531.689
.002
613.315
616.112
.233
60
61
462.076
459.132
.245
540.006
539.524
.040
623.336
625.793
.205
61
62
468.830
465.165
.305
548.100
547.026
.090
633.118
635.201
.174
62
63
475.305
470.815
.374
555.898
554.171
.144
642.635
644.290
.138
63
64
481.476
476.078
.450
563.376
560.911
.205
651.853
653.031
.098
64
65
487.316
480.897
.535
570.499
567.212
.274
660.750
661.399
.054
65
* Note. Tlie last Premium at the beginning of the 10th Year changes the Policy to the Paid-up class. 154
TABLE LXIII.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
INTEREST, 4 PER CENT.
I. Common Life Policies.—The Yet Value or Reserve is already computed in the
column (Y, 6m.) for the middle of each Policy Year, and in the column (Y, 12m.) for
the end of each Policy Year, when the next Annual Premium is just due and unpaid.
For any other time of year, a correction derived from the column of half monthly oi-
ls days difference (£m. d.) is to he added or subtracted, according as the Reserve is
increasing or decreasing in the direction of the given interval.
For example, if the Age of Entry, x, of a common Life Policy of $1000 he 35 years,
the 4 per cent. Reserve at the middle of the 24th Policy Year (Vff, 6m.) is given $362.47,
and at the end of the same Policy Year, $362.37. Hence, at the end of 3 months in
the same year, the Reserve is $362.52, and at the end of the 9tli month, $362.42.
Formulas, (V, 12m.) = 1000 -j 1 — - Ux+- j- •
* ( 1 + a*x )
1000M*
n* — N
(V, 6m.) = \ {(Y2 \ 12m.) + nx + (V£, 12m.)}.
Bj =
•L'x+n
II. General Valuation.—For Endowment Insurances and all other Annual
Premium Policies, the Reserve V is first found from the Table, and corrected for the
elapsed fraction of a year, when required, as if for a common Life Policy. The remain-
ing part and process is thus indicated :
General Formula, Reserve = V + — nx]) • B* • fx+n.
Middle of Policy Year, Reserve = (V, 6m.) +' — nx) • (Bf)x.
End “ “ “ Reserve = (V, 12m.) + — nx) • B*.
Pure Endowment, Reserve = • B* • fx+n-
Here n + m denotes tlie expired (n) added to the unexpired (m) years of the whole
period of simple endowment. In all cases,
x = the Age of Entry or of original Insurance.
li = fraction elapsed of the current Policy Year.
/ = h + (1 — h) vpx+n-1, Table LXIY.
At 12 months, or the end of the year, f — 1, since h = 1.
Example 1. End of the Year.—A Ten Premium Endowment Insurance of $1000, payable at
Death or Age 50, is contracted at the Age of 80. Required the Net Value at the end of the 9tli Year.
x — 30 ; n + h = 8 yrs. 12m.
1.648805 = A(P - tt), Table LVII.
V
= 92.685, 1.060738 = aBIo, Table LXIII.
+ 512.322 ... .2.709543 Sum. Table LXXXI1.
Beserve
= $605,007 on $1000 Insured.
Example 2. Intermediate Months and Term Policies.—An Insurance of $1000 is entered at the
age of 38, for the Term of 7 years. Required the Net Value at the end of 4 years and 8| months
(5th year).
X = 38 ; x+ n + 7i = 42 yrs. m.
1.05227 = 7i(ir — P),
Table LIII.
1.99387 = A/43,
Table LXIV.
Adjusted V = 68.993,
0.76363 = ABse,
Table LXIII.
-64.531 ...
.1.80977 Sum,
Table LXXXII.
Net Value = $4,462 on
$1000 Insured.
In this case, (P — n) being negative, since P is less than tt, tlie computed term is subtractive, which
is a general characteristic of Simple Term Policies.
Example 3. Middle of the Year.—A Fifteen Premium Insurance of $1000 for the whole Life is
entered at the Age of 42. the Net Value at the end of 7 years and 6 months, (8th Policy
Year).
V, 6m.
1.07203 = A (P - 7r),
Table LXI.
183.75
0.99521 = AB,
Table LXIII.
+ 116.75 ..
. .2.06724 Sum,
Table LXXXII.
Reserve = $250.50 on $1000 Insured.
Note 1. Five, Ten, or Limited Premium Policies can be valued by this method only till one year
after the last Annual Premium is paid. After the beginning of this year, the Policy being Paid-up,
the Reserve is more easily found from the proper Single Premium Table.
Note 2. Although six decimals are given, yet in practice five, and in many cases, four decimals
will be sufficient for logarithms or anti-logaritlmis. Thus the last of the six decimals can generally be
omitted, as in the last Examples. 155
TABLE LXIII.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
4 PER CENT.
.g”
10.
11.
r—H C3
o
n.
ft.
Y.
6m.
Y.
12m.
£m.d
MB/)•
6m.
AB.
12m.
Y.
6m.
Y.
12m.
im.d.
MB/).
6m.
AB.
12m.
0
7.100
4.058
.254
0.010050
0.019872
7.292
4.244
.254
0.010055
0.019882
0
1 1
11.243
8.285
.247
.321134
.330961
11.621
8.657
.247
.321147
.330980
1
2
15.553
12.680
.239
.507365
.517198
16.125
13.253
.239
.507382
.517221
2
3
20.039
17.257
.232
.642526
.652365
20.815
18.037
.232
.642549
.652393
3
4
24.711
22.022
.224
.749741
.759585
25.698
23.019
.223
.749771
.759622
4
5
29.574
26.984
.216
0.839313
0.849164
30.779
28.200
.215
0.839352
0.849210
5
6
34.634
32.143
.208
.916736
.926594
36.065
33.589
.206
.916786
.926651
6
7
39.898
37.510
.199
.985293
.995158
41.559
39.189
.197
.985353
.995228
7
8
45.370
43.088
.190
1.047098
1.056973
47.268
45.007
.188
1.047172
1.057057
8
9
51.056
48.883
.181
.103600
.113485
53.201
51.055
.179
.103687
.113583
9
10
56.966
54.906
.172
1.155828
1.165724
59.365
57.336
.169
1.155932
1.165839
10
11
63.104
61.161
.162
.204546
.214453
65.766
63.855
.159
.204667
.214587
11
12
69.479
67.654
.152
.250333
.260253
72.410
70.625
.149
.250474
.260407
12
13
76.096
74.396
.142
.293639
.303572
79.304
77.643
.138
.293803
.303753
13
14
82.962
81.386
.131
.334825
.344775
86.452
84.922
.128
.335013
.344980
14
15
90.082
88.636
.121
1.374178
1.384145
93.866
92.470
.116
1.374394
1.384379
15
16
97.465
96.153
.109
.406064
.416049
101.545
100.279
.105
.412186
.422194
16
17
105.113
103.930
.099
.448301
.458309
109.494
108.369
.094
.448586
.458615
17
18
113.030
111.988
.087
.483436
.493465
117.726
116.743
.082
.483758
.493813
18
19
121.229
120.328
.075
.517482
.527537
126.246
125.409
.070
.517846
.527927
19
20
129.714
128.958
.063
1.550559
1.560640
135.057
134.365
.058
1.550971
1.561083
20
21
138.488
137.877
.051
.582776
.592888
144.162
143.619
.045
.583243
.593388
21
22
147.557
147.094
.039
.614227
.624372
153.561
153.163
.0.33
.614755
.624939
22
23
156.917
156.599
.027
.644995
.655179
163.261
163.020
.020
.645589
.655812
23
24
166.579
166.417
.014
.675152
.685375
173.276
173.191
.007
.675816
.686081
24
25
176.553
176.547
.001
1.704760
1.715025
183.600
183.669
.006
1.705505
1.715821
25
26
186.836
186.982
.012
.733883
.744199
194.232
194.456
.019
.734721
.745091
26
27
197.424
197.725
.025
.762581
.772951
205.175
205.553
.032
.763519
.773949
27
28
208.322
208.777
.038
.790901
.801331
216.434
216.976
.045
.791949
.802440
28
29
219.537
220.154
.051
.818892
.829383
228.006
228.695
.057
.820062
.830629
29
30
231.060
231.825
.064
1.846599
1.857166
239.886
240.737
.071
1.847906
1.858548
30
31
242.893
243.819
.077
.874069
.884711
252.080
253.084
.084
.875524
.886254
31
132
255.038
256.115
.090
.901340
.912070
264.579
265.733
.096
.902963
.913788
32
33
267.485
268.713
.102
.928458
.939283
277.379
278.685
.109
.930262
.941191
33
34
280.234
281.612
.115
.955459
.966388
290.477
291.929
.121
.957467
.968511
34
35
293.278
294.802
.127
1.982387
1.993431
303.864
305.459
.133
1.984621
1.995791
35
36
306.610
308.277
.139
3.009283
2.020453
317.534
319.270
.145
2.011765
2.023073
36
37
320.226
322.032
.151
.036186
.047494
331.477
333.344
.156
.038944
.050405
37|
38
334.111
336.049
.162
.063143
.074604
345.680
347.675
.166
.066204
.077833
38
39
348.256
350.322
.172
.090195
.101824
360.133
362.251
.176
.093591
.105404
39
40
362.652
364.839
.182
3.117388
2.129201
374.825
377.059
.186
2.121154
2.133168
40
41
377.284
379.587
.192
.144770
.156784
389.742
392.085
.195
.148942
.161177
41
42
392.140
394.552
.201
.172391
.184626
404.864
407.303
.203
.177014
.189496
42
43
107.201
409.708
.209
.200307
.212789
420.173
422.703
.211
.205430
.218178
43
12.
tt = 10.1419.
CO
rH
tt = 10.3400.
Continued on Page *198, 157
TABLE LXIII.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
4 PER CENT.
I s Policy
Years.
14.
15.
1 § Policy
1 • Years.
Y.
6 m.
Y.
12m.
* (»/).
6m.
AB.
12m.
Y.
6m.
Y.
12m.
ilrm.d.
MB/).
6 m.
AB.
12m.
0
7.490
4.432
.255
0.010062
0.019895
7.701
4.636
.255
0.010067
0.019906
0
1
12.013
9.047
.247
.321158
.330997
12.432
9.462
.248
.321169
.331013
1
2
16.724
13.852
.239
.507400
.517244
17.358
14.488
.239
.507417
.517268
2
3
21.627
18.855
.231
.642573
.652424
22.483
19.713
.231
.642600
.652458
3
4
26.731
24.058
.223
.749805
.759663
27.815
25.150
.222
.749841
.759706
4
5
32.038
29.470
.214
0.839395
0.849260
33.357
30.799
.213
0.839442
0.849317
5
6
37.556
35.094
.205
.916839
.926714
39.117
36.668
.204
.916898
.926783
6
7
43.289
40.937
.196
.985419
.995304
45.101
42.768
.194
.985492
.995388
7
8
49.248
47.011
.186
1.047252
1.057148
51.319
49.104
.185
1.047340
1.057247
8
9
55.439
53.318
.177
.103783
.113690
57.775
55.680
.175
.103889
.113809
9
10
61.865
59.865
.167
1.156045
1.165965
64.477
62.509
.164
1.156169
1.166102
10
11
68.539
66.664
.156
.204800
.214733
71.432
69.589
.154
.204946
.214896
11
12
75.462
73.712
.146
.250628
.260578
78.643
76.931
.143
.250798
.260765
12
13
82.641
81.022
.135
.293982
.303949
86.121
84.545
.131
.294179
.304164
13
14
90.086
88.602
.124
.335220
.345205
93.867
92.422
.120
.335448
.345456
14
15
97.797
96.444
.113
1.374633
1.384641
101.885
100.583
.108
1.374897
1.384926
15
16
105.780
104.569
.101
.412461
.422490
110.190
109.030
.097
.412760
.422815
16
17
114.048
112.979
.089
.448897
.458952
118.784
117.772
.084
.449239
.459320
17
18
122.605
121.682
.077
.484112
.494193
127.671
126.805
.072
.484499
.494611
18
19
131.452
130.675
.065
.518246
.528358
136.856
136.141
.060
.518687
.528832
19
20
140.596
139.969
.052
1.551425
1.561570
146.338
145.768
.048
1.551927
1.562111
20
21
150.035
149.553
.040
.583758
.593942
156.122
155.711
.034
.584324
.594547
21
22
159.777
159.453
.027
.615336
.625559
166.224
165.971
.021
.615971
.626236
22
23
169.835
169.668
.014
.646239
.656504
176.639
176.541
.008
.646953
.657269
23
24
180.203
180.190
.001
.676545
.686861
187.365
187.422
.005
.677350
.687720
24
25
190.881
191.024
.012
1.706326
1.716696
198.401
198.615
.018
1.707231
1.717661
25
26
201.869
202.167
.025
.735643
.746073
209.760
210.138
.031
.736654
.747145
26
27
213.177
213.639
.039
.764549
.775040
221.432
221.960
.044
.765681
.776248
27
28
224.798
225.408
.051
.793100
.803667
233.416
234.107
.058
.794368
.805010
28
29
236.728
237.501
.064
.821350
.831992
245.717
246.561
.070
.822764
.833494
29
30
248.975
249.900
.077
1.849341
1.860071
258.324
259.321
.083
1.850922
1.861747
30
31
261.526
262.604
.090
.877126
.887951
271.237
272.386
.096
.878887
.889816
31
32
274.381
275.611
.102
.904746
.915675
284.448
285.745
.108
.906709
.917753
32
33
287.535
288.911
.115
.932249
.943293
297.953
299.394
.120
.934435
.945605
33
34 300.979
302.499
.127
.959679
.970849
311.742
313.325
.132
.962110
.973418
34
35 314.708
316.369
.138
1.987078
1.998386
325.807
327.522
.143
1.989783
2.001244
35
36 328.710
330.502
.149
2.014495
2.025956
340.133
341.979
.154
2.017502
.029131
36
37
342.972
344.895
.160
.041978
.053607
354.713
356.681
.164
.045316
.057129
37
38
357.488
359.533
.170
.069572
.081385
369.533
371.619
.174
.073276
.085290
38
39
372.243
374.404
.180
.097325
.109339
384.581386.776
.183
.101434
.113669
39
40
387.223
389.494
.189
2.125294
2.137529
399.834 402,127
.191
2.129850
2.142332
40
41
402.409
404.777
.197
.153532
.166014
415.278 417.662
.199
.158584
.171332
41
42
417.784 420.243
.205
.182103
.194851
430.888
433.348
.205
.187701
.200749
42
43
433.326435.860
.211
.211067
.224115
446.645
449.177
.211
.217272
.230642
43
14.
7t = 10.5480.
15.
77 = 10.7660.
Continued on Page *197, 158
TABLE LXIII.
0ENEEAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
Policy
Years.
CD
rH
II
17.
o£
Uh cd
v.
V.
im.d.
MB/).
AB.
V.
y.
|m.cl.
MB/)-
AB.
n.
6m.
12m.
6m.
12m.
6m.
12m.
6m.
12m.
n.
0
7.922
4.849
.256
0.010072
0.019916
8.155
5.073
.257
0.010079
0.019930
0
1
12.871
9.898
.248
.321182
.331033
13.329
10.349
.248
.321196
.331054
1
2
18.021
15.148
.239
.507438
.517296
18.711
15.837
.239
.507462
.517327
2
3
23.376
20.609
.231
.642630
.652495
24.307
21.540
.231
.642663
.652538
3
4
28.944
26.285
.222
.749881
.759756
30.120
27.465
.221
.749926
.759811
4
5
34.731
32.181
.213
0.839494
0.849379
36.163
33.624
.212
0.839552
0.849448
5
6
40.743
38.310
.203
.916964
.926860
42.440
40.020
.202
.917036
.926943
6
7
46.990
44.675
.193
.985573
.995480
48.958
46.660
.192
.985661
.995581
7
8
53.476
51.282
.183
1.047437
1.057357
55.725
53.554
.181
1.047544
1.057477
8
9
60.210
58.143
.172
.104004
.113937
62.745
60.701
.170
.104131
.114081
9
10
67.196
65.255
.162
1.156306
1.166256
70.026
68.114
.159
1.156456
1.166423
10
11
74.441
72.632
.151
.205106
.215073
77.575
75.800
.148
.205283
.215268
11
12
81.954
80.281
.139
.250985
.260970
85.394
83.752
.137
.251192
.261200
12
13
89.736
88.195
.128
.294397
.304405
93.489
91.991
.125
.294639
.304668
13
14
97.792
96.394
.117
.335702
.345731
101.873
100.519
.113
.335977
.346032
14
15
106.134
104.880
.105
1.375185
1.385240
110.550
109.344
.101
1.375502
1.385583
15
16
114.769
113.663
.092
.413090
.423171
119.522
118.464
.088
.413451
.423563
16
17
123.699
122.739
.080
.449613
.459725
128.794
127.888
.075
.450028
.460173
17
18
132.925
132.117
.067
.484927
.495072
138.365
137.607
.063
.485400
.495584
18
19
142.451
141.789
.055
.519175
.529359
148.245
147.646
.050
.519711
.529934
19
20
152.281
151.779
.042
1.552479
1.562702
158.443
158.004
.037
1.553083
1.563348
20
21
162.431
162.087
.029
.584944
.595209
168.957
168.675
.024
.585625
.595941
21
22
172.894
172.706
.016
.616668
.626984
179.786
179.660
.010
.617439
.627809
22
23
183.669
183.638
.003
.647741
.658111
190.928
190.960
.003
.648610
.659040
23
24
194.758
194.883
.010
.678238
.688668
202.394
202.593
.017
.679211
.689702
24
25
206.169
206.460
.024
1.708224
1.718715|
214.178
214.527
.029
1.709317
1.719884
25
26
217.896
218.336
.037
.737768
.748335
226.277
226.791
.043
.738994
.749636
26
27
229.935
230.540
.050
.766929
.777571
238.696
239.364
.056
.768299
.779029
27
28
242.294
243.052
.063
.795761
.806491
251.422
252.245
.069
.797295
.808120
28
29
254.959
255.872
.076
.824322
.835147
264.458
265.435
.081
.826034
.836963
29
30
267.932
268.997
.089
1.852659
1.863588
277.797
278.922
.094
1.854571
1.865615
30
31
281.206
282.419
.101
.880825
.891869
291.429
292.701
.106
.882958
.894128
31
32
294.772
296.131
.113
.908869
.920039
305.352
306.766
.118
.911245
.922553
32
33
308.627
310.127
.125
.936840
.948148
319.550
321.098
.129
.939484
.950945
33
34
322.756
324.390
.136
.964785
.976246
334.013
335.693
.140
.967730
.979359
34
35
337.149
338.914
.147
1.992758
2.004387
348.733
350.536
.150
1.996032
2.007845
35
36
351.797
353.685
.157
2.020809
.032622
363.694
365.616
.160
2.024446
.036460
36
37
366.686
368.692
.167
.048988
.061002
378.885
380.918
.169
.053025
.065260
37
38
381.804
383.920
.176
.077349
.089584
394.285
396.416
.178
.081832
.094314
38
39
397.128
399.342
.185
.105954
.118436
409.875
412.099
.185
.110930
.123678
39
40
412.644
414.950
.192
2.134865
2.147613
425.636
427.936
.192
2.140385
2.153433
40
41
428.327
430.709
.198
.164146
.177194
441.543
443.915
.198
.170267
.183637
41
42
444.158
446.612
.205
.193867
.207237
457.584
460.017
.203
.200646
.214371
42
43
460.121
462.635
.209
.224097
.237822
473.731
476.208
.206
.231606
.245724
43
16.
7r = 10.9953.
17.
tr = 11.2364.
4 PER CENT.
Continued on Page *190. 159
TABLE LXIII.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
od
r-H (3
18.
19.
.3 “
ce
Y.
fim.
Y.
. 12m.
Tfm.d.
MB/).
6m.
AB.
12m.
V.
6 m.
V.
12 m.
im.a.
MB/).
6m.
AB.
12m.
o <13
n.
0
8.396
5.303
.258
0.010087
0.019945
8.651
5.545
.259
0.010096
0.019961
0
1
13.806
10.818
.249
.321213
.331078
14.304
11.308
.250
.321231
.331106
1
2
19.429
16.551
.240
.507488
.517363
20.180
17.295
.240
.507516
.517401
2
3
25.274
22.506
.231
.642700
.652585
26.284
23.518
.231
.642741
.652637
3
4
31.346
28.696
.221
.749975
.759871
32.628
29.981
.221
.750030
.759937
4
5
37.655
35.125
.211
0.839615
0.849522
39.213
36.690
.210
0.839685
0.849605
5
6
44.207
41.798
.201
.917116
.927036
46.051
43.656
.200
.917202
.927135
6
7
51.007
48.727
.190
.985758
.995691
53.145
50.878
.189
.985865
.995815
7
8
58.064
55.911
.179
1.047661
1.057611
60.501
58.368
.178
1.047789
1.057756
8
9
65.381
63.361
.168
.104271
.114238
68.130
66.135
.166
.104425
.114410
9
10
72.969
71.087
.157
1.156622
1.166607
76.031
74.171
.155
1.156805
1.166813
10
11
80.829
79.080
.146
.205478
.215486
84.211
82.496
.143
.205695
.215724
11
12
88.965
87.361
.134
.251422
.261451
92.683
91.113
.131
.251671
.261726
12
13
97.392
95.932
.122
.294902
.304957
101.449
100.030
.118
.295191
.305272
13
14
106.112
104.802
.109
.336281
.346362
110.516
109.245
.106
.336613
.346725
14
15
115.130 113.969
.097
1.375850
1.385962
119.884
118.768
.093
1.376233
1.386378
15
16
124.450 123.441
.084
.413851
.423996
129.557
128.589
.081
.414292
.424476
16
17
134.071 133.210
.072
.450485
.460669
139.539
138.733
.067
.450988
.461211
17
18
144.000143.300
.058
.485920
.496143
149.844
149.199
.054
.486489
.496754
18
19
154.250 153.710
.045
.520298
.530563
160.468
159.981
.041
.520942
.531258
19
20
164.817 164.435
.032
1.553745
1.564061
171.409
171.081
.027
1.554477
1.564847
20
21
175.70lll75.477
.019
.586377
.596747
182.668
182.500
.014
.587205
.597635
21
22
186.901486.834
.006
.618288
.628718
194.255
194.254
.000
.619219
.629710
22
23
198.425 198.527
.009
.649563
.660054
206.162
206.313
.013
.650611
.661178
23
24
210.270 210.522
.021
.680282
.690849
218.387
218.705
.026
.681462
.692104
24
25
222.430 222.848
.035
1.710520
1.721162
230.935
231.409
.040
1.711842
1.722572
25
26
234.911235.485
.048
.740340
.751070
243.795
244.426
.053
.741823
.752648
26
27
247.704 248.432
.061
.769808
.780633
256.968
257.754
.065
.771467
.782396
27
28
260.806i261.690
.074
.798981
.809910
270.446
271.382
.078
.800838
.811882
28
29
274.212 275.245
.086
.827919
.838963
284.222
285.305
.090
.829993
.841163
29
30
287.915 289.094
.098
1.856675
1.867845
298.288
299.516
.102
1.858989
1.870297
30
31
301.907 303.231
.110
.885303
.896611
312.636
313.999
.114
.887884
.899345
31
32
316.179 317.636
.121
.913859
.925320
327.250
328.746
.125
.916736
.928365
32
33
330.715 332.305
.132
.942396
.954025
342.123
343.744
.135
.945599
.957412
33
34
345.510
347.224
.143
.970969
.982782
357.241
358.982
.145
.974533
.986547
34
35
360.547
362.381
.153
1.999634
2.011648
372.591
374.444
.154
2.003593
2.015828
35
36
375.816
377.761
.162
2.028445
.040680
388.152
390.104
.163
.032847
.045329
36
37
391.295 393.338
.170
.057468
.069950
403.906
405.951
.170
.062359
.075107
37
38
406.964
409.101
.178
.086766
.099514
419.830
421.953
.177
.092197
.105245
38
39
422.805
425.018
.184
.116406
.129454
435.904
438.100
.183
.122435
.135805
39
40
438.794
441.080
.191
2.146461
2.159831
452.113
454.370
.188
2.153143
2.166868
40
41
454.916
457.263
.196
.176999
.190724
468.428
470.730
.192
.184407
.198525
41
42
471.145
473.537
.199
.208106
.222224
484.821
487.155
.195
.216320
.230869
42
43
487.451
489.874
.202
.239874
.254423
i
501.270
503.629
.197
.248971
.263989
43
18.
77 = 11.4897.
19.
7t = 11.7560.
4 PER CENT.
Continued on Page *195. 160
TABLE LXIII.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
4 PER CENT.
.«g
a ca
x=20.
21.
r-H C3
V.
V.
im.d.
MB/).
AB.
Y.
Y.
A (B/).
AB.
o 03
ft.
6m.
12m.
6m.
12m.
6m.
12 m.
im.d.
6m. •
12m.
ft.
0
8.915
5.795
.260
0.010105
0.019980
9.193
6.056
.261
0.010115
0.020000
0
1
14.823
11.816
.251
.321250
.331135
15.368
12.350
.252
.321271
.331167
1
2
20.963
18.073
.241
.507547
.517443
21.783
18.887
.241
.507581
.517488
2
3
27.340
24.572
.231
.642785
.652692
28.445
25.672
.231
.642834
.652754
3
4
33.963
31.318
.220
.750089
.760009
35.360
32.718
.220
.750153
.760086
4
5
40.839
38.323
.210
0.839761
0.849694
42.535
40.023
.209
0.839844
0.849794
5
6
47.972
45.586
.199
.917298
.927248
49.976
47.599
.198
.917402
.927369
6
7
55.370
53.118
.188
.985981
.995948
57.692
55.454
.186
.986110
.996095
7
8
63.041
60.928
.176
1.047932
1.057917
65.682
63.581
.175
1.048088
1.058096
8
9
70.986
69.008
.165
.104596
.114604
73.957
72.002
.163
.104784
.114813
9
10
79.212
77.380
.153
1.157008
1.167037
82.524
80.717
.151
1.157229
1.167284
10
11
87.731
86.045
.141
.205930
.215985
91.392
89.737
.138
.206189
.216270
11
12
96.546
95.012
.128
.251946
.262027
100.563
99.058
.125
.252246
.262358
12
13
105.663
104.278
.115
.295508
.305620
110.038
108.689
.112
.295859
.306004
13
14
115.085
113.855
.102
.336981
.347126
119.821
118.622
.100
.337387
.347571
14
15
124.810
123.730
.090
1.376658
1.386842
129.917
128.882
.086
1.377124
1.387347
15
16
134.848
133.930
.077
.414777
.425000
140.340
139.468
.073
.415307
.425572
16
17
145.211
144.455
.063
.451538
.461803
151.086
150.374
.059
.452142
.462458
17
18
155.894
155.297
.050
.487113
.497429
162.152
161.601
.046
.487803
.498173
18
19
166.896
166.459
.036
.521653
.532023
173.540
173.149
.033
.522437
.532867
19
20
178.218
177.941
.023
1.555284
1.565714
185.259
185.039
.018
1.556169
1.566660
20
21
189.869
189.761
.009
.588113
.598604
197.303
197.236
.006
.589113
.599680
21
22
201.842
201.888
.004
..620244
.630811
209.667
209.769
.009
.621372
.632014
22
23
214.136
214.348
.018
.651766
.662408
222.359
222.619
.022
.653034
.663764
23
24
226.754
227.124
.031
.682757
.693487
235.372
230.794
.035
.684184
.695009
24
25
239.687
240.213
.044
1.713298
1.724123
248.694
249.264
.047
1.714898
1.725827
25
26
252.932
253.615
.057
.743453
.754382
262.321
263.048
.061
.745248
.756292
26
27
266.485
267.319
.070
.773293
.784337
276.254
277.130
.073
.775303
.786473
27
28
280.337
281.319
.082
.802880
.814050
290.482
291.505
.085
.805126
.816434
28
29
294.482
295.610
.094
.832273
.843581
304. 994
306.152
.097
.834784
.846245
29
30
308.910
310.173
.105
1.861535
1.872996
319.775
321.068
.108
1.864338
1.875967
30
31)323.606
325.003
.116
.890725
.902354
334.818
336.239
.118
.893850
.905663
31
32 338.562
340.085
.127
.919901
.931714
350.110
351.651
.128
.923383
.935397
32
33
353.764
355.408
.137
.949123
.961137
365.635
367.289
.138
.952997
.965232
33
34
369.200
370.956
.146
.978450
.990685
381.374
383.128
.146
.982763
.995245
34
35
384.848
386.703
.155
2.007951
2.020433
397.307
399.157
.154
2.012748
2.025496
35
36
400.689
402.639
.162
.037692
.050440
413.415
415.342
.161
.043024
.056072
36
37
416.702
418.730
.169
.067741
.080789
429.672
431.673
.167
.073668
.087038
37
38
432.867
434.967
.175
.098175
.111545
446.067
448.130
.172
.104750
.118475
38
39
449.165
451.328
.180
.129065
.142790
462.568
464.677
.176
.136361
.150479
39
40
465.572
467.779
.184
2.160496
2.174614
479.148
481.289
.178
2.168593
2.183142
40
41
482.055
484.295
.187
.192564
.207113
495.786
497.952
.180
.201539
.216557
41
42
498.596
500.861
.189
.225357
.240375
512.452
514.623
.181
.235303
.250842
42
43
515.166
517.436
.189
.258982
.274521
529.117)531.281
.180
.270003
.286112
43
20
7T =
12.0361.
21
it = 12.3304.
Continued on Page *194. 161
TABLE LXIII.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
4 PER CENT.
og
3 <£
22.
23.
.O
V.
y.
im.d.
MB/).
AB.
y.
y.
1 m.rl
MB/).
AB.
O®
n.
6m.
12m.
2
6m.
12m.
6m.
12m.
‘2 •
6m.
12m.
n.
0
9.487
6.333
.263
0.010126
0.020022
9.792
6.619
.264
0.010138
0.020045
0
1
15.941
12.909
.253
.321294
.331201
16.537
13.489
.254
.321320
.331240
1
2
22.642
19.736
.242
.507619
.517539
23.539
20.623
.243
.507659
.517592
2
3
29.601
26.825
.231
.642886
.652819
30.804
28.018
.232
.642946
.652896
3
4
36.819
34.174
.220
.750225
760175
38.336
35.689
.221
.750303
.760270
4
5
44.305
41.796
.209
0.839936
0.849903
46.149
43.643
.209
0.840037
0.850022
5
6
52.068
49.699
.197
.917517
.927502
54.241
51.872
.197
.917645
.927653
6
7
60.107
57.876
.186
.986253
.996261
62.618
60.398
.185
.986411
.996440
7
8
68.432
66.348
.174
1.0482621.058291
71.293
69.222
.173
1.048451
1.058506
8
9
77.053
75.117
.161
.104989
.115044
80.271
78.354
.160
.105215
.115296
9
10
85.974
84.191
.149
1.157471
1.167552
89.556
87.792
.147
1.157737
1.167849
10
11
95.199
93.568
.136
.206473
.216585
99.151
97.544
.134
.206786
.216931
11
12
104.734
103.259
.123
.252579
.262724
109.055
107.601
.121
.252947
.263131
12
13
114.575
113.252
.110
.296246
.306430
119.278
117.989
.107
.296672
.306895
13
14
124.733
123.574
.097
.337834
.348057
129.831
128.707
.094
.338322
.348587
14
15
135.220
134.225
.083
1.377634
1.387899
140.711
139.749
.080
1.378193
1.388509
15
16
146.031
145.197
.070
.415889
.426205
151.916
151.117
.067
.416533
.426903
16
17
157.164
156.492
.056
.452809
.463179
163.447
162.810
.053
.453544
.463974
17
18
168.622
168.112
.042
.488563
.498993
175.312
174.848
.039
.489397
.499888
18
19
180.413
180.073
.028
.523297
.533788
187.505
187.197
.026
.524243
.534810
19
20
192.529
192.345
.015
1.557142
1.567709
200.026
199.888
.012
1.558215
1.568857
20
21
204.969
204.954
.001
.590214
.600856
212.876
212.898
.002
.591423
.602153
21
22
217.738
217.882
.012
.622611
.633341
226.046
226.228
.015
.623977
.634802
22
23
230.825
231.128
.025
.654431
.665256
239.535
239.877
.028
.655966
.666895
23
24
244.229
244.690
.038
.6857.53
.696682
253.338
253.833
.041
.687479
.698523
24
25
257.944
258.558
.051
1.716659
1.727703
267.445
268.091
.054
1.718597
1.729767
25
26
271.962
272.726
.064
.747223
.758393
281.851
282.645
.066
.749394
.760702
26
27
286.277
287.188
.076
.777512
.788820
296.544
297.476
.078
.779944
.791405
27
28
300.877
301.925
.087
.807598
.819059
311.510
312.579
.089
.810319
.821948
28
29
315.748
316.932
.099
.837546
.849175
326.742
327.938
.100
.840586
.852399
29
30
330.883
332.194
.109
1.867421
1.879234
342.223
343.543
.110
1.870813
1.882827
30
31
346.268
347.701
.119
.897288
.909302
357.944
359.378
.120
.901068
.913303
31
32
361.888 363.435
.129
.927211
.939446
373.879
375.414
.128
.931424
.943906
32
33
377.722 379.370
.137
.957261
.969743
390.012
391.644
.136
.961956
.974704
33
34
393.753
395.496
.145
.987510
2.000258
406.320
408.031
.143
.992735
2.005783
34
35
409.958
411.780
.152
2.018029
2.031077
422.782
424.567
.149
2.023842
2.037212
35
36
426.316
428.211
.158
.048896
.062266
439.381
441.229
.154
.055353
.069078
36
37
442.809
444.767
.163
.080185
.093910
456.089
457.983
.158
.087359
.101477
37
38
459.411
461.415
.167
.111987
.126105
472.876
474.803
.161
.119955
.134504
38
39
476.092
478.129
.170
.144395
.158944
489.722
491.674
.163
.153236
.168254
39
40
492.831
494.893
.172
2.177503
2.192521
506.597
508.554
.163
2.187309
2.202848
40
41
509.599
511.666
.172
.211417
.226956
523.469
525.419
.162
.222292
.238401
41
42
526.366
528.425
.172
.246253
.262362
540.309
542.232
.160
.258311
.275049
42
43
543.098
545.131
.169
.282139
.298877
557.069
558.941
.156
.295499
.312921
43
22.
7T =
12.6399
23. tt =
12.9656
Continued on Page *198. 162
TABLE LXIII.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
4 PER CENT.
!
03
o a>
Ph>h
ft.
II
25.
§ Policy
• Years.
Y.
6 m.
Y.
12m.
-g-m.d.
MB/).
6m.
AB.
12m.
Y.
6m.
Y.
12m.
£m.d.
MB/).
6m.
AB.
12m.
0
10.112
6.916
.266
0.010152
0.020072
10.450
7.231
.268
0.010167
0.020100
0
1
17.161
14.097
.255
.321348
.331281
17.814
14.728
.257
.321380
.331330
1
2
24.473
21.542
.244
.507704
.517654
25.450
22.504
.246
.507754
.517721
2
3
32.057
29.264
.233
.643009
.652976
33.370
30.567
.234
.643080
.653065
3
4
39.922
37.271
.221
.750390
.760375
41.572
38.908
.222
.750486
.760494
4
5
48.067
45.555
.209
0.840150
0.850158
50.064
47.550
.209
0.840275
0.850304
5
6
56.500
54.137
.197
.917787
.927816
58.857
56.495
.197
.917941
.927996
6
7
65.233
63.021
.184
.986583
.996638
67.958
65.752
.184
.986772
.996853
7
8
74.271
72.214
.171
1.048659
1.058740
77.370
75.319
.171
1.048886
1.058998
8
9
83.618
81.714
.159
.105462
.115574
87.096
85.205
.158
.105735
.115880
9
10
93.277
91.531
.146
1.158031
1.168176
97.137
95.399
.145
1.158357
1.168541
10
11
103.247
101.655
.133
.207134
.217318
107.498
105.929
.131
.207515
.217738 11
12
113.537
112.112
.119
.253351
.263574
118.196
116.794
.117
.253793
.264058
12
13
124.161
122.902
.105
.297138
.307403
129.225
127.987
.103
.297648
.307964
13
14
135.114
134.018
.091
.338857
.349173
140.583
139.510
.089
.339450
.349820
14
15
146.394
145.461
.078
1.378811
1.389181
152.271
151.363
.076
1.379493
1.389923
15
16
158.000
157.232
.064
.417242
.427672
164.299
163.565
.061
.418019
.428510
16
17
169.945
169.350
.050
.454350
.464841
176.659
176.0841 .048
.455238
.465805
17
18
182.220
181.782
.036
.490314
.500881
189.350
188.947
.034
.491325
.501967
18
19
194.824
194.557
.022
.525285
.535927
202.376
202.136
.020
.526430
.537160
19
20 207.759
207.654
.009
1.559392
1.570122
215.726
215.648
.007
1.560689
1.571514
20
21,221.018
221.073
.005
.592756
.603581
229.400
229.483
.007
.594219
.605148
21
22 234.596
234.812
.018
.625477
.636406
243.391
243.630
.020
.627129
.638173
22
23
248.491
248.861
.031
.657656
.668700
257.691
258.083
.033
.659516
.670686
23
24
262.692
263.215
.044
.689379
.700549
272.295
272.837
.045
.691469
.702777
24
25
277.194
277.866
.056
1.720729
1.732037
287.188
287.870
.057
1.723075
1.734536
25
26
291.985
292.795
.068
.751784
.763245
302.359
303.179
.068
.754415
.766044
26
27
307.051
307.999
.079
.782621
.794250
317.798
318.749
.079
.785567
.797380
27
28
322.384
323.461
.090
.813313
.825126
333.493
334.567
.089
.816606
.828620
28
29
337.969
339.170
.100
.843930
.855944
349.427
350.618
.099
.847607
.859842
29
30
353.794
355.109
.110
1.874543
1.886778
365.580
366.874
.108
1.878649
1.891131
30
31
369.835
371.253
.118
.905229
.917711
381.934
383.325
.116
.909810
.922558
31
32
386.076
387.591
.126
.936064
.948812
398.466
399.937
.123
.941170
.954218
32
33
402.493
404.087
.133
.967123
.980171
415.152
416.699
.129
.972813
.986183
33
34
419.064
420.733
.139
.998488
2.011858
431.979
433.589
.134
2.004816
2.018541
34
35
435.773
437.506
.144
2.030237
2.043962
448.915
450.572
.138
2.037274
2.051392
35
36
452.593
454.372
.148
.062461
.076579
465.931
467.622
.141
.070288
.084837
36
37
469.492
471.304
.151
.095258
.109807
483.008
484.724
.143
.103951
.118969
37
38
486.450 488.287
.153
.128724
.143742
500.113
501.834
.143
.138376
.153915
38
39
503.437
505.279
.153
.162967
.178506
517.217
518.930
.143
.17368]
.189790
39
40
520.422
522.257
.153
2.198106
2.214215
534.286
535.973
.141
2.209996
2.226734
40
41
537.373
539.182
.151
.234268
.251006
551.276
552.910
.136
.247454
.264876
41
42
554.246
556.002
.146
.271587
.289009
568.188
569.797
.134
.286206
.304387
42
43
571.042
572.773
.144
.310212
.328393
584.997
586.528
.128
.326408
.345409
43
24.
TT =
13.3082.
25.
7t = 13.6687.
Continued on Page *192, 163
TABLE LXIII.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
4 PER CENT.
Uh c3
26.
27.
,3 03
o
n.
Y.
6m.
Y.
12m.
im-d-
A (BO-
6m.
AB.
12m.
Y.
6m.
Y.
12m.
im.d.
MB/).
6m.
AB.
12m.
0
13.301
9.811
.291
0.010321
0.020402
13.789
10.240
.296
0.010352
0.020464
0
1
23.275
19.950
.277
.321703
.331815
24.200
20.821
.282
.321772
.331917
1
2
33.584
30.428
.263
.508275
.518420
34.946
31.733
.268
.508385
.518569
2
3
44.226
41.233
.249
.643826
.654010
46.037
43.004
.253
.643981
.654204
3
4
55.208
52.394
.235
.751485
.761708
57.488
54.634
.238
.751688
.761953
4
5
66.547
63.909
.220
0.841558
0.851823
69.294
66.615
.223
0.841819
0.852135
5
6
78.235
75.772
.205
.919547
.929863
81.450
78.948
.209
.919877
,930247
6
7
90.274
87.985
.191
.988745
.999115
93.961
91.636
.194
.989152
.999582
7
8
102.661
100.548
.176
1.051275
1.061705
106.836
104.697
.178
1.051765
1.062256
8
9
115.410
113.481
.161
.108589
.119080
120.066
118.097
.164
.109174
.119741
9
10
128.510
126.749
.147
1.161730
1.172297
133.650
131.865
.149
1.162424
1.173066
10
11
141.961
140.383
.131
.211474
.222116
147.592
145.982
.134
.212287
.223017
11
12
155.767
154.361
.117
.258409
.269139
161.883
160.445
.120
.259359
.270184
12
13
169.916
168.682
.103
.303003
.313828
[176.518
175.254
.105
.304103
.315032
13
14
184.409
183.346
.089
.345627
.356556
191.495
190.397
.091
.346896
.357940
14
15
199.238
198.340
.075
1.386588
1.397632
206.801
205.867
.078
1.388047
1.399217
15
16
214.395
213.659
.061
.426142
.437312
222.432
221.659
.064
.427811
.439119
16
17
229.872
229.295
.048
.464505
.475813
238.374
237.751
.052
.466411
.477872
17
18
245.657
245.229
.036
.501868
.513329
254.613
254.137
.040
.504037
.515666
18
19
261.737
261.455
.024
.538396
.550025
271.139
270.803
.028
.540858
.552671
19
20
278.101
277.957
.012
1.574238
1.586051
287.937
287.734
.017
1.577025
1.589039
20
21
294.734
294.722
.001
.609529
.621543
304.994
304.915
.007
.612675
.624910
21
22
311.623
311.734
.009
.644392
.656627
322.284
322.315
.003
.647942
.660424
22
23
328.744
328.964
.018
.678949
.691431
339.788
339.924
.011
.682949
.695697
23
24
346.077
346.400
.027
.713315
.726063
357.483
357.704
.018
.717813
.730861
24
25
363.598
364.006
.034
1.747600
1.760648
375.344
375.646
.025
1.752653
1.766023
25
26
381.284
381.771
.041
.781914
.795284
393.354
393.724
.031
.787576
.801301
26
27
399.117
399.673
.046
.816360
.830085
411.483
411.903
.035
.822702
.836820
27
28
417.068
417.673
.050
.851053
.865171
429.697
430.153
.038
.858152
.872701
28
29
435.103
435.744
.053
.886109
.900658
447.975
448.459
.040
.894042
.909060
29
30
453.202
453.870
.056
1.921640
1.936658
466.285
466.773
.041
1.930498
1.946037
30
31
471.333
472.005
.056
.957771
.973310
484.592
485.073
.040
.967657
.983766
31
32
489.460
490.125
.055
.994634
2.010743
502.863
503.315
.038
2.005660
2.022398
32
33
507.551
508.188
.053
2.032368
.049106
521.048
521.444
.033
.044655
.062077
33
34
525.559
526.139
.048
.071118
.088540
539.151
539.520
.031
.084802
.102983
34
35
543.483 544.038
.046
2.111045
2.129226
557.143
557.428
.024
2.126267
2.145268
35
36
561.299
561.770
.039
.152310
.171311
574.951
575.135
.015
.169242
.189160
36
37
578.932
579.303
.031
.195106
.215024
592.554
592.635
.007
.213929
.234840
37
38
596.362
596.631
.022
.239632
.260543
609.927
609.882
.004
.260544
.282551
38
39
613.565
613.710
.012
.286102
.308109
627.037
626.853
.015
.309326
.332530
39
40
630.507
630.514
.001
2.334755
2.357959
643.858
643.526
.028
2.360524
2,385037
40
41
647.164
647.023
.012
.385839
.410352
660.355
659.845
.043
.414439
.440403
41
42
663.497
663.182
.026
.439653
.465617
676.495
675.808
.057
.471378
.498915
42
43
679.481
678.989
.041
.496504
.524041
692.273
691.399
.073
.531660
.560916
43
32. n = 16.7902.
33.
n = 17.3376.
4 PER CENT.
Continued on Page *188. TABLE LX III.
167
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000,
4 PER CENT.
.2*
CO
35.
t/j
o 4)
o >03
O M
•rH d
&£
n.
Y.
Cm.
Y.
12m.
MB/).
6 m.
AB.
12m.
V.
6m.
Y.
12m.
im.d.
M»/).
6m.
AB.
12m.
0
17.948
13.775
.348
0.010687
0.021117
18.673
14.379
.358
0.010752
0.021243
0
1
31.926
27.956
.331
.322469
.332960
33.238
29.130
.342
.322612
.333179
1
2
46.291
42.504
.316
.509497
.520064
48.193
44.288
.325
.509727
.520369
2
3
61.039
57.453
.299
.645569
.656211
63.542
59.829
.309
.645894
.656624
3
4
76.177
72.780
.283
.753819
.764549
79.273
75.751
.294
.754258
.765083
4
5
91.691
88.482
.267
0.844568
0.855393
95.386
92.054
.278
0.845131
0.856060
5
6
107.582
104.561
.252
.923321
.934250
111.873
108.724
.262
.924029
.935073
6
7
123.842
121.002
.237
.993381
1.004425
128.723
125.755
.247
.994251
1.005421
7
8
140.461
137.798
.222
1.056883
.068053
145.931
143.140
.233
1.057934
.069242
8
9
157.431
154.943
.207
.115290
.126598
163.481
160.855
.219
.116549
.128010
9
10
174.739
172.414
.194
1.169663
1.181124
181.358
178.894
.205
1.171153
1.182782
10
11
192.370
190.205
.180
.220789
.232418
199.551
197.241
.192
.222539
.234352
11
12
210.312
208.299
.168
.269276
.281089
218.044
215.881
.180
.271315
.283329
12
13
228.551
226.682
.156
.315604
.327618
236.822
234.795
.169
.317968
.330203
13
14
247.069
245.335
.145
.360168
.372403
255.856
253.950
.159
.362898
.375380
14
15
265.842
264.227
.135
1.403298
1.415780
275.126
273.335
.149
1.406438
1.419186
15
16
284.846
283.345
.125
.445275
.458023
294.605
292.909
.141
.448873
.461921
16
17
304.058
302.650
.117
.486345
.499393
314.269
312.661
.134
.490455
.503825
17
18
323.450
322.129
.110
.526728
.540098
334.095
332.563
.128
.531402
.545127
18
19
343.004
341.757
.104
.566616
.580341
354.053 352.576
.123
.571924
.586042
19
20
362.686
361.494
.099
1.606200
1.620318
374.105 372.667
.120
1.612218
1.626767
20
21
382.462
381.309
.096
.645659
.660208
394.227 392.819
.117
.652462
.667480
21
22
402.306
401.183
.094
.685160
.700178
414.383|412.981
.117
.692837
.708376
22
23
422.186
421.067
.093
.724871
.740410
434.5381433.127
.118
.733524
.749633
23
24
442.061
440.935
.094
.764965
.781074
454.651
453.209
.120
.774703
.791441
24
25
461.899
460.741
.096
1.805614
1.822352
474.672
473.167
.125
1.816556
1.833978
25
26
481.643
480.424
.102
.846991
.864413
494.600
493.066
.128
.859274
.877455
26
27
501.297
500.049
.104
.889285
.907466
514.407
512.781
.136
.903047
.922048
27
28
520.831
519.493
.112
.932678
.951679
534.011
532.274
.145
.948091
.968009
28
29
540.166
538.717
.121
.977383
.997301
553.390
551.540
.154
.994628
2.015539
29
30
559.277
557.717
.130
2.023618
2.044529
572.517
570.527
.166
2.042891
2.064898
30
31
578.141
576.443
.141
.071613
.093620
591.352
589.210
.178
.093137
.116341
31
32
596.716
594.868
.154
.121622
.144826
609.871
607.565
.192
.145630
.170143
32
33
614.979
612.970
.167
.173906
.198419
628.031
625.530
.208
.200684
. 226648
33
34
632.890
630.688
.184
.228776
.254740
645.801
643.104
.225
.258621
.286158
34
35
650.414
648.020
.199
2.286552
2.314089
663.169
660.268
.242
2.319769
2.349025
35
36
667.544
664.947
.216
.347561
.376817
680.103
676.970
.261
.384531
.415688
36
37
684.244
681.420
.235
.412203
.443360
696.574
693.211
.280
.453328
.486551
37
38 700.490
697.438
.254
.480898
.514121
712.576
708.975
.300
.526608
.562091
38
39
716.271
712.984
.274
.554090
.589573
728.097
724.251
.321
.604873
.642829
39
40
731.577
728.049
.294
2.632282
2.670238
743.134
739.051
.340
2.688661
2.729307
40
41
746.408
742.646
.314
.716010
.756656
757.687
753.356
.361
.778594
.822200
41
42
760.760
756.753
.334
.805894
.849500
771.778
767.233
.379
.875305
.922097
42
43
774.657
770.439
.351
.902565
.949357
785.440
780.679
.397
.979528
3.029839
43
40.
7t = 22.1206.
41.
77 = 22.9667.
Continued on Page *184. TABLE L XIII.
171
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
;
1
4 PER CENT
2 *-*
13 ce
42.
rf*
CO
o o
y.
y.
1
MB/).
AB.
y.
y.
MB/).
AB.
o 0)
P4kH
6m.
12m.
im.d.
6m.
12m.
6 m.
12m.
im.a.
6m.
12m.
n.
0
19.415
14.966
.371
0.010831
0.021398
20.212
15.613
.383
0.010911
0.021553
0
1
34.587
30.345
.354
.322771
.333413
36.021
31.620
.367
.322941
.333671
1
2
50.160
46.113
.337
.509978
.520708
52.225
48.020
.350
.510255
.521080
2
3
66.122
62.267
.321
.646255
.657080
68.821
64.812
.334
.646649
.657578
3
4
82.469
78.808
.305
.754740
.765669
85.802
81.982
.318
.755270
.766314
4
5
99.196
95.722
.289
0.845753
0.856797
103.158
99.524
.303
0.846437
0.857607
5
6
116.293
113.001
.274
.924809
.935979
120.882
117.430
.288
.925665
.936973
6
7
133.752
130.639
*259
.995207
1.006515
138.959
135.677
.274
.996260
1.007721
7
8
151.557
148.613
.245
1.059092
.070553
157.372
154.258
.259
1.060366
.071995
8
9
169.696
166.916
.232
.117934
.129563
176.112
173.155
.246
.119458
.131271
9
10
188.155
185.530
.219
1.172793
1.184606
195.159
192.354
.234
1.174597
1.186611
10
11
206.917
204.442
.206
.224464
.236478
214.500
211.835
.222
.226579
.238814
11
12
225.968
223.631
.195
.273558
.285793
234.105
231.565
.212
.276028
.288510
12
13
245.280
243.066
.185
.320571
.333053
253.953
251.532
.202
.323439
.336187
13
14
264.832
262.734
.175
.365905
.378653
274.018
271.694
.194
.369217
.382265
14
15
284.595
282.594
.167
1.409896
1.422944
294.271
292.038
.186
1.413707
1.427077
15
16
304.546
302.634
.159
.452837
.466207
314.693
312.537
.180
.457198
.470923
16
17
324.661
322.826
.153
.494977
.508702
335.248
333.150
.175
.499957
.514075
17
18
344.910
343.131
.148
.536551
.550669
355.903
353.845
.172
.542225
.556774
18
19
365.255
363.515
.145
.577775
.592324
376.628
374.601
.169
.584217
.599235
19
20
385.669
383.961
.142
1.618845
1.633863
397.389
395.368
.168
1.626145
1.641684
20
21
406.121
404.417
.142
.659955
.675494
418.148
416.118
.169
.668214
.684323
21
22
426.568
424.857
.143
.701297
.717406
438.866
436.803
.172
.710624
.727362
22
23
446.976
445.232
.145
.743062
.759800
459.486
457.360
.177
.753574
.770996
23
24
467.288
465.481
.151
.785436
.802858
480.013
477.856
.180
.797270
.815451
24
25
487.508
485.671
.153
1.828620
1.846801
500.415
498.163
.188
1.841914
1.860915
25
26
507.603
505.673
.161
.872808
.891809
520.607
518.241
.197
.887733
.907651
26
27
527.494
525.451
.170
.918222
.938140
540.567
538.084
.207
.934959
.955870
27
28
547.155
544.997
.180
.965089
.986000
560.268
557.641
.219
.983833
2.005840
28
29
566.561
564.261
.192
2.013644
2.035651
579.667
576.884
.232
2.034619
.057823
29
30
585.670
583.217
.204
2.064148
2.087352
598.742
595.790
.246
2.087587
2.112100
30
31
604.460
601.840
.218
.116868
.141381
617.447
614.294
.263
.143058
.169022
31
32
622.885
620.067
.235
.172122
.198086
635.750
632.395
.280
.201358
.228895
32
33
640.914
637.897
.251
.230233
.257770
653.639
650.074
.297
.262822
.292078
33
34
658.535
655.311
.269
.291532
.320788
671.081
667.278
.317
.327854
.359011
34
35
675.716
672.257
.288
2.356424
2,387581
688.048
684.007
.337
2.396883
2.430106
35
36
692.428
688.736
.308
.425333
.458556
704.530
700.243
.357
.470359
.505842
36
37
708.664
704.729
.328
.498706
.534189
720.515
715.977
.378
.548788
.586744
37
38
724.410
720.228
.348
.577049
.615005
736.004
731.222
.398
.632712
.673358
38
39
739.667
735.244
.369
.660903
.701549
750.994
745.955
.420
.722756
.766362
39
40
754.432
749.757
.'390
2.750890
2.794496
765.507
760.249
.438
2.819556
2.866348
40
41
768.729
763.837
.408
.847643
.894435
779.578
774.098
.457
.923851
.974162
41
42
782.589
777.479
.426
.951901
3.002212
793.262
787.616
.471
3.036438
3.090542
42
43
796.069
790.795
.440
3.064459
.118563
806.697
800.969
.477
.158109
.216302
43
42.
7T = 23.8627.
43.
7r = 24.8099.
Continued on Page *183. 172
TABLE LXIII.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
4 PER CENT.
§ Policy
Years.
II
a
45.
§ Policy
• Years.
Y.
6m.
Y.
12 m.
im.d.
MB/).
6m.
AB.
12m.
Y.
6m.
Y.
12m.
im.d.
* (B/).
6m.
AB.
12m..
0
21.037
16.261
.398
0.011001
0.021731
21.906
16.936
.414
0.01110*3
0.021928
0
1
37.498
32.921
.381
.323134
.333959
39.044
34.276
.397
.323343
.334272
1
2
54.356
49.979
.365
.510559
.521488
56.580
52.007
.381
.510895
.521939
2
3
71.607
67.422
.349
.647085
.658129
74.502
70.122
.365
.647564
.658734
3
4
89.239
85.242
.333
.755854
.767024
92.806
88.613
.349
.756496
.767804
4
5
107.243
103.432
.318
0.847188
0.858496
111.472
107.455
.335
0.848016
0.859477
5
6
125.607
121.969
.303
.926608
.938069
130.487
126.643
.320
.927646
.939275
6
7
144.313
140.844
.289
.997420
1.009049
149.838
146.157
.307
.998696
1.010509
7
8
163.349
160.041
.276
1.061769
.073582
169.508
165.983
.294
1.063313
.075327
8
9
182.700
179.545
.263
.121134
.133148
189.479
186.100
.282
.122976
.135211
9
10
202.346
199.335
.251
1.176579
1.188814
209.725
206.474
.271
1.178762
1.191244
10
11
222.263
219.378
.240
.228909
.241391
230.222
227.093
.261
.231476
.244224
11
12
242.426
239.661
.230
.278749
.291497
250.941
247.913
.252
.281746
.294794
12
13
262.809
260.143
.222
.326597
.339645
271.855
268.921
.245
.330077
.343447
13
14
283.382
280.809
.214
.372866
.386236
292.943
290.090
.238
.376881
.390606
14
15
304.128
301.634
.208
1.417899
1.431624
314.171
311.376
.233
1.422516
1.436634
15
16
325.011
322.574
.203
.462001
.476119
335.499
332.746
.229
.467295
.481844
16
17
345.991
343.596
.200
.505445
.519994
356.902
354.181
.227
.511489
.526507
17
18
367.046
364.682
.197
.548473
.563491
378.341
375.626
.226
.555356
.570895
18
19
388.136
385.778
.197
.591314
.606853
399.778
397.053
.227
.599137
.615246
19
20
409.225
406.858
.197
1.634191
1.650300
421.171
418.414
.230
1.643062
1.659800
20
21
430.271
427.871
.200
.677318
.694056
442.466
439.641
.235
.687354
.704776
21
22
451.218
448.753
.205
.720904
.738326
463.661
460.806
.238
.732240
.750421
22
23
472.071
469.575
.208
.765165
.783346
484.730
481.777
.246
.777940
.796941
23
24
492.795
490.203
.216
.810311
.829312
505.581
502.510
.256
.824695
.844613
24
25
513.308
510.600
.226
1.856575
1.876493
526.194
523.001
.266
1.872749
1.893660
25
26
533.586
530.758
.236
.904196
.925107
546.537
543.197
.278
.922355
.944362
26
27
553.598
550.625
.248
.953419
.975426
566.571
563.069
.292
.973785
.996989
27
28
573.306
570.174
.261
2.004512
2.027716
586.268
582.592
.306
2.027320
2.051833
28
29
592.683
589.379
.275
.057751
.082264
605.584
601.700
.324
.083286
.109250
29
30
611.684
608.177
.292
2.113459
2.139423
624.484
620.392
.341
2.142017
2.169554
30
31
630.278
626.565
.309
.171965
.199502
642.958
638.648
.359
.203853
.233109
31
32
648.451
644.524
.327
.233606
.262862
660.969
656.414
.380
.269207
.300364
32
33
666.169
662.000
.347
.298793
.329950
678.490
673.689
.400
.338509
.371732
33
34
683.404
678.995
.367
.367953
.401176
695.510
690.456
.421
.412215
.447698
34
35
700.148
695.489
.388
2.441538
2.477021
712.018
706.703
.443
2.490838
2.528794
35
36
716.388
711.473
.410
.520060
.558016
728.012
722.446
.464
.574922
.615568
36
37
732.122
726.959
.430
.604061
.644707
743.491
737.660
.486
.665098
.708704
37
38
747.349
741.926
.452
.694168
.737774
758.478
752.420
.505
.762003
.808795' 38
39
762.093
756.446
.471
.791019
.837811
773.009
766.722
.524
.866381
.916692
39
40
776.387
770.516
.489
2.895353
2.945664
787.140
780.682
.538
2.979034
3.033138
40
41
790.289
784.248
.503
3.007972
3.062076
801.015
794.471
.545
3.100754
.158947
41
42
803.937
797.813
.510
.129666
.187859
814.766
808.186
.548
.232640
.295403
42
43
817.466
811.306
.513
.261533
.324296
828.710
822.357
.529
.375628
.443161
43
44.
tt = 25.8133.
45.
TT= 26.8757.
Continued on Page *182. 173
TABLE LX 111.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
4 PER CENT.
j?E
cd
46.
47.
.3 "
Uh ce
(£(h
Y.
Cm.
V.
12m.
|m.d.
MB/).
6 m.
AB.
12m.
Y.
6m.
Y.
12m.
im.d.
MB/).
6m.
AB.
12m.
0 0)
n.
0
22.821
17.640
.432
0.011211
0.022140
23.778
18.361
.451
0.011332
0.022376
0
1
40.658
35.676
.415
.323573
.334617
42.337
37.118
.435
.323827
.334997
1
2
58.890
54.103
.399
.511264
.522434
61.289
56.265
.419
.511668
.522976
2
3
77.508
72.912
.383
.648090
.659398
80.618
75.776
.404
.648669
.660130
3
4
96.496
92.079
.368
.757202
.768663
100.308
95.645
.389
.757980
.769609
4
5
115.839
111.597
.354
0.848927
0.860556
120.346
115.852
.374
0.849929
0.861742
5
6
135.523
131.448
.340
.928788
.940601
140.714
136.381
.361
.930044
.942058
6
7
l00.532
151.615
.326
1.000098
1.012112
161.395
157.213
.349
1.001639
1.013874
7
8
175.847
172.079
.314
.065008
.077243
182.359
178.310
.337
.066875
.079357
8
9
196.442
192.804
.303
.125005
.137487
203.583
199.661
.327
.127241
.139989
9
10
217.292
213.778
.293
1.181167
1.193915
1225.038
221.220
.318
1.183816
1.196864
10
11
238.368
234.957
.284
.234303
.247351
'246.694
242.974
.310
.237418
.250788
11
12
259.642
256.327
.276
.285048
.298418
268.532
264.894
.303
.288680
.302405
12
13
281.095
277.861
.269
.333905
.347630
290.512
286.935
.298
.338120
.352238
13
14
302.687
299.513
.265
.381302
.395420
312.597
309.064
.294
.386176
.400725
14
15
324.383
321.251
.261
1.427604
1.442153
334.759
331.259
.292
1.433208
1.448226
15
16
346.153
343.055
.258
.473124
.488142
356.959
353.465
.291
.479546
.495085
16
17
367.963
364.869
.258
.518148
.533687
379.157
375.653
.292
•525487
.541596
17
18
389.768
386.666
.258
.562943
.579052
401.310
397.772
.295
.571309
.588047
18
19
411.531
408.395
.261
.607761
.624499
423.360
419.753
.301
.617269
.634691
19
20
433.192
429.988
.267
1.652840
1.670262
445.309
441.670
.303
1.663624
1.681805
20
21
454.753
451.518
.270
.698421
.716602
467.124
463.384
.312
.710620
.729621
21
22
476.184
472.849
.278
.744734
.763735
488.716
484.853
.322
.758517
.778435
22
23
497.395
493.940
.288
.792031
.811949
510.060
506.072
.332
.807578
.828489
23
24
518.363
514.784
.298
.840563
.861474
531.126
526.984
.345
.858070
.880077
24
25
539.056
535.327
.311
1.890590
1.912597
551.870
547.561
.359
1.910278
1.933482
25
26
559.435
555.542
.324
.942391
.965595
572.266
567.777
.374
.964494
.989007
26
27
579.472
575.401
.339
.996250
2.020763
592.268
587.564
.392
2.021055
2.047019
27
28
599.120
594.839
.357
2.052499
.078463,611.839
606.919
.410
.080302
.107839
28
29
618.347
613.853
.375
.111476
.139013
630.969
625.823
.429
.142583
.171839
29
30
637.138
632.423
.393
2.173525
2.202781
649.618
644.219
.450
2.208319
2.239476
30
31
655.459
650.494
.414
.239061
.270218
667.761
662.107
.471
.277947
.311170
31
32
673.281
668.067
.434
.308519
.341742
685.385
679.469
.493
.351928
.387411
32
33
690.596
685.123
.456
.382357
.417840
702.479
696.294
.515
.430781
.468737
33
34
707.387
701.651
.478
.461089
.499045
719.042
712.594
.537
.515055
.555701
34
35
723.658
717.664
.500
2.545264
2.585910
735.069
728.349
.560
2.605386
2.648992
35
36
739.403
733.141
.522
.635514
.679120
750.588
743.633
.580
.702416
.749208
36
37
754.649
748.155
.541
.732479
.779271
765.636
758.443
.599
.806893
.857204
37
38
769.430
762.704
.560
.836904
.887215
780.267
772.897
.614
.919623
.973727
38
39
783.804
776.903
.575
.949594
3.003698
794.634
787.176
.621
3.041402
3.099595
39
40
797.917
790.930
.582
3.071343
3.129536
808.875
801.379
.625
3.173332
3.236095
40
41
811.906
804.882
.585
.203250
.266013
823.313
816.052
.605
.316352
.383885
41
42
826.090
819.297
.566
.346253
.413786
837.140
829.032
.676
.475762
.551432
42
43
839.673
832.047
.636
.505651
.581321
849.151
840.075
.756
.656718
.741317
43
46.
tt = 28.0014.
47.
= 29.1948.
Continued on Page *181. 174
TABLE LXIII.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
4 PER CENT.
1
.sg
Uh cd
O 0)
PM>H
ft.
II
00
49.
Cd
O CD
Ph£h
n.
Y.
6m.
Y.
12m.
im.d.
MB/).
6m.
AB.
12m.
Y.
6m.
Y.
12m.
im.d.
M®/).
6m.
AB.
12m.
0
24.784
19.108
.473
0.011464
0.022634
25.844
19.885
.497
0.011610
0.022918
0
1
44.091
38.614
.456
.324105
.335413
45.919
40.149
.481
.324412
.335873
1
2
63.782
58.490
.441
.512114
.523575
66.366
60.781
.465
.512607
.524236
2
3
83.840
78.730
.426
.649307
.660936
87.177
81.769
.451
.650009
.661822
3
4
104.253
99.315
.412
.758835
.770648
108.331
103.090
.437
.759776
.771790
4
5
125.001
120.228
.398
0.851031
0.863045
129.808
124.724
.424
0.852241
0.864476
5
6
146.069
141.449
.385
.931423
.943658
151.581
146.635
.412
.932943
.945425
6
7
167.425
162.941
.374
1.003337
1.015819
173.624
168.809
.401
1.005208
1.017956
7
8
189.046
184.691
.363
.068933
.081681
195.905
191.198
.392
.071198
.084246
8
9
210.902
206.653
.354
.129703
.142751
218.396
213.791
.384
.132416
.145786
9
10
232.963
228.814
.346
1.186735
1.200105
241.075
236.556
.377
1.189945
1.203670
10
11
255.209
251.144
.339
.240843
.254568
263.903
259.447
.371
.244617
.258735
11
12
277.601
273.598
.334
.292679
.306797
286.839
282.429
.368
.297088
.311637
12
13
300.099
296.140
.330
.342768
.357317
309.856 305.480
.365
.347885
.362903
13
14
322.676
318.751
.327
.391542
.406560
332.913
328.542
.364
.397454
.412993
14
15
345.291
341.372
.327
1.439382
1.454921
355.965
351.585
.365
1.446188
1.462297
15
16
367.904
363.975
.327
.486625
.502734
378.972
374.557
.368
.494432
.511170
16
17
390.471
386.508
.330
.533581
.550319
401.873
397.385
.374
.542504
.559926
17
18
412.934
408.900
.336
.580532
.597954
424.667
420.147
.377
.590704
.608885
18
19
435.294
431.227
.339
.627755
.645936
447.324
442.697
.386
.639316
.658317
19
20
457.517
453.347
.347
1.675512
1.694513
469.747
464.995
.396
1.688628
1.708546
20
21
479.512
475.218
.358
.724078
.743996
491.915
487.031
.407
.738930
.759841
21
22 501.256
496.833
.369
.773726
.794637
513.791
508.749
.420
.790508
.812515
22
23 522.715
518.137
.382
.824732
.846739
535.336
530.120
.435
.843667
.866871
23
24 543.848
539.099
.396
.877391
.900595
556.519
551.115
.450
.898710
.923223
24
25
564.626
559.693
.411
1.932000
1.956513
577.292
571.665
.469
1.955992
1.981956
25
26
585.001
579.849
.429
.988903
2.014867
597.617
591.766
.488
2.015862
2.043399
26
27
604.938
599.567
.448
2.048447
.075984
617.484
611.399
.507
.078682
.107938
27
28
624.425
618.824
.467
.110984
.140240
636.853
630.504
.529
.144877
.176034
28
29
643.424
637.564
.488
.176939
.208096
655.694
649.082
.551
.214897
.248120
29
30
661.906
655.787
.510
2.246754
2.279977
673.999
667.113
.574
2.289207
2.324690
30
31
679.860
673.474
.532
.320892
.356375
691.751
684.586
.597
.368335
.406291
31
32
697.273
690.613
.555
.399876
.437832
708.952
701.515
.620
.452837
.493483
32
33
714.146
707.219
.577
.484258
.524904
725.598
717.877
.643
.543353
.586959
33
34
730.473
723.268
.600
.574678
.618284
741.715
733.750
.664
.640532
.687324
34
35
746.283
738.838
.620
2.671779
2.718571
757.342
749.131
.684
2.745128
2.795439
35
36
761.612
753.925
.641
.776312
.826623
772.538
764.143
.700
.857950
.912054
36
37
776.517
768.650
.656
.889086
.943190
787.459
778.972
.707
.979799
3.037992
37
38
791.153
783.195
.663
3.010899
3.069092
802.249
793.722
.711
3.111782
.174545
38
39
805.660
797.664
.666
.142855
.205618
817.243
808.961
.690
.254840
.322373
39
40
820.368
812.612
.646
3.285893
3.353426
831.602
822.441
.763
3.414277
3.489947
40
41
834.453
825.834
.718
.445315
.520985
844.077
833.910
.847
.595252
.679851
41
42
846.689
837.084
.800
.626281
.710880
855.176
844.638
.878
.797429
.889806
42
43
857.576
847.607
.831
.828452
.920829
865.710
854.979
.894
4.020868
4.121378
43
00
7r =
30.4602.
49.
7r = 31.8029.
Continued on Page *180, 175
TABLE LXIII.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
4 PER CENT.
,o£
03
50.
51.
r-H Cfl
o a>
y.
6m.
V.
12m.
Yin.d.
MB/).
6m.
AB.
12m.
y.
6 m.
y.
12m.
im.d.
MB/).
6m.
AB.
12m.
o a>
Ph>h
n.
0
26.952
20.675
.523
0.011770
0.023231
28.120
21.497
.552
0.011951
0.023580
0
1
47.815
41.727
.507
.324751
.336380
49.800
43.361
.537
.325123
.336936
1
2
69.047
63.139
.492
.513148
.524961
71.838
65.574
.522
.513741
.525755
2
3
90.630
84.893
.478
.650780
.662794
94.215
88.113
.509
.651626
.663861
3
4
112.543
106.966
.465
.760807
.773042
116.897
110.940
.496
.761945
.774427
4
5
134.758
129.321
.453
0.853574
0.866056
139.862
134.042
.485
0.855044
0.867792
5
6
157.247
151.945
.442
.934618
.947366
163.076
157.368
.476
.936462
.949510
6
7
179.981 174.789
.433
1.007268
1.020316
186.508
180.905
.467
1.009539
1.022909
7
8
202.928
197.840
.424
.073696
.087066
210.135
204.623
.459
.076441
.090166
8
9
226.068
221.067
.417
.135399
.149124
233.918
228.471
.454
.138685
.152803
9
10
249.358
244.422
.411
1.193480
1.207598
257.813
252.414
.450
1.197378
1.211927
10
11
272.760
267.870
.408
.248778
.263327
281.793
276.429
.447
.253358
.268376
11
12
296.244
291.389
.405
.301944
.316962
305.813
300.456
.446
.307294
.322833
12
13
319.768
314.919
.404
.353523
.369062
329.831
324.463
.447
.359739
.375848
13
14
343.288
338.430
.405
.403973
.420082
353.801
348.396
.450
.411161
.427899
14
15
366.763
361.867
.408
1.453693
1.470431
377.659
372.179
.457
1.461968
1.479390
15
16
390.127
385.159
.414
.503039
.520461
401.407
395.893
.459
.512535
.530716
16
17
413.384
408.382
.417
.552346
.570527
425.011
419.387
.469
.563198
.582199
17
18
436.501
431.391
.426
.601920
.620921
448.373
442.617
.480
.614295
.634213
18
19
459.379
454.140
.437
.652071
.671989
471.467
465.575
.491
.666150
.687061
19
20
481.996
476.624
.448
1.703104
1.724015
494.259
488.201
.505
1.719079
1.741086
20
21
504.318
498.783
.461
.755319
.777326
516.704
510.466
.520
.773414
.796618
21
22
526.299
520.587
.476
.809033
.832237
538.774
532.339
.536
.829479
.853992
22
23
547.911
542.008
.492
.864560
.889073
560.414
553.748
.556
.887644
.913608
23
24
569.105
562.974
.511
.922260
.948224
581.591
574.691
.575
.948279
.975816
24.
25
589.843
583.484
.530
1.982493
2.010030
602.288
595.144
.595
2.011752
2.041008
25
26
610.114
603.515
.550
2.045624
.074880
622.467
615.048
.618
.078508
.109665
26
27
629.875
623.007
.572
.112085
.143242
642.097
634.403
.641
.149002
.182225
27
28
649.098
641.962
.595
.182329
.215552
661.167
653.189
.665
.223712
.259195
28
29
667.775
660.359
.618
.256830
.292313
679.662
671.393
.689
.303173
.341129
29
30
685.887
678.187
.642
2.336116
2.374072
697.582
689.030
.713
2.387949
2.428595
30
31
703.437
695.459
.665
.420748
.461394
714.925
706.077
.737
.478689
.522295
31
32
720.420
712.153
.689
.511371
.554977
731.716
722.614
.758
.576049
.622841
32
33
736.865
728.349
.710
.608637
.655429
747.997
738.638
.780
.680786
.731097
33
34
752.809
744.041
.731
.713299
.763610
763.829
754.277
.796
.793719
.847823
34
35
768.313
759.358
.746
2.826174
2.880278
779.372
769.726
.804
2.915653
2.973846
35
36
783.537
774.487
.754
.948064
3.006257
794.781
785.094
.807
3.047699
3.110462
36
37
798.626
789.537
.757
3.080077
.142840
810.404
800.971
.786
.190804
.258337
37
38
813.925
805.086
.737
.223157
.290690
825.363
815.014
.862
.350273
.425943
38
39
828.576
818.838
.811
.382610
.458280
838.360
826.963
.950
.531271
.615870
39
40
841.304
830.541
.897
3.563596
3.648195
849.922
838.140
.982
3.733462
3.825839
40
41
852.627
841.486
.928
.765780
.858157
860.898
848.913
.999
.956910
4.057420
41
42
863.376
852.037
.945
.989223
4.089733
871.393
859.132
1.022
4.204932
.314823
42
43
873.655
862.045
.968
4.237242
.347133
881.483
869.091
1.033
.481379
.601459
43
50.
tt = 33.2283.
51.
tt = 34.7419.
Continue ! on Pago "r170. 176
TABLE LXIII.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
4 PER CENT.
3 Policy
Years.
x = 52.
53.
g Policy
• Years.
y.
6m.
y.
12m.
im.d.
MB/).
6m.
AB.
12m.
y.
6m.
y.
12m.
im.d.
* (B/).
6 m.
AB.
12m.
0
29.347
22.344
.584
0.012143
0.023956
30.640
23.219
.618
0.012355
0.024369
0
1
51.870
45.045
.569
.325528
.337542
54.029
46.780
.604
.325974
.338209
1
2
74.737
68.079
.555
.514390
.526625
77.741
70.642
.592
.515109
.527591
2
3
97.918
91.408
.543
.652558
.665040
101.747
94.791
.580
.653587
.666335
3
4
121.388
115.017
.531
.763199
.775947
126.012
119.174
.570
.764580
.777628
4
5
145.111
138.856
.521
0.856661
0.869709
150.507
143.779
.561
0.858445
0.871815
5
6
169.058
162.910
.512
.938495
.951865
175.205
168.571
.553
.940729
.954454
6
7
193.205
187.149
.505
1.012034
1.0257591
200.066
193.501
.547
1.014782
1.028900
7
8
217.510
211.521
.499
.079464
.093582
225.045
218.529
.543
.082798
.097347
8
9
241.930
235.990
.495
.142307
.156856
250.110
243.632
.540
.146295
.161313
9
10
266.437
260.533
.492
1.201669
1.216687
275.220
268.748! .539
1.206397
1.221936
10
11
290.985
285.087
.491
.258404
.273943
300.326
293.843
.540
.263967
.280076
11
12
315.529
309.622
.492
.313191
.329300
325.382
318.861
.543
.319696
.336434
12
13
340.026
334.080
.496
.366593
.383331
350.321
343.722
.550
.374152
.391574
13
14
364.408
358.386
.502
.419086
.436508
375.147
368.511
.553
.427831
.446012
14
15
388.679
382.621
.505
1.471097
1.489278
399.820
393.069
.563
1.481162 1.500163
15
16
412.801
406.631
.514
.523003
.542004
424.240
417.352
.574
.534556
.554474
16
17
436.676
430.371
.525
.575172
.595090,
448.382
441.351
.586
.588392
.609303
17
18
460.277
453.834
.537
.627954
.648865
472.207
465.003
.600
.643034
.665041
18
19
483.571
476.957
.551
.681688
.703695
495.670
488.277
.616
.698847
.722051
19
20
506.509
499.711
.567
1.736718
1.759922
518.739
511.142
.633
1.756189
1.780702
20
21
529.063
522.065
.583
.793385
.817898
541.362
533.521
.653
.815454
.841418
21
22
551.179
543.944
.603
.852069
.878033
563.497
555.413
.674
.877034
.904571
22
23
572.821
565.347
.623
.913151
.940688
585.133
576.794
.695
.941318
.970574
23
24
593.973
586.250
.644
.977007
2.006263
606.227
597.600
.719
2.008763
2.039920
24
25
614.596
606.591
.667
2.044089
2.075246
626.746
617.832
.743
2.079840
2.113063
25
26
634.656
626.371
.690
.114858
.148081
646.681
637.469
.768
.155039
.190522
26
27
654.145
645.570
.715
.189801
.225284
666.013
656.498
.793
.234907
.272863
27
28
673.047
664.173
.739
.269455
.307411
684.747
674.935
.818
.320019
.360665
28
29
691.360
682.198
.764
.354390
.395036
702.874
692.754
.843
.411032
.454638
29
30
709.084
699.619
.789
2.445260
2.488866
720.428
710.041
.866
2.508611
2.555403
30
31
726.244
716.520
.810
.542723
.589515
737.446
726.791
.888
.613522
.663833
31
32
742.883
732.896
.832
.647543
.697854
753.995
743.140
.905
.726589
.780693
32
33
759.063
748.879
.849
.760540
.814644
770.245
759.289
.913
.848626
.906819
33
34
774.948
764.667
.857
.882523
.940716
786.351
775.353
.917
.980750
3.043513
34
35
790.694 780.372
.860
3.014606
3.077369
802.681
791.949
.894
3.123910
3.191443
35
36
806.660
796.598
.838
.157737
.225270
818.319
806.629
.974
.283420
.359090
36
37
821.949
810.950
.917
.317225
.392895
831.904
819.120
1.065
.464444
.549043
37
38
835.231
823.162
1.006
.498236
.582835
843.992
830.803
1.099
.666653
.759030
38
39
847.048
834.584
1.039
.700435
.792812'
855.464
842.065
1.117
.890112
.990622
39
40
858.264
845.594
1.056
3.923888
4.024398
866.436
852.747
1.141
4.138140
4.248031
40
41
868.991
856.038
1.079
4.171913
.281804
876.982
863.157
1.152
.414590
.534670
41
42
879.301
866.214
1.091
.448362
.568442
887.874
874.531
1.112
.722205
.852747
42
43
889.949
877.334
1.051
.755976
.886518
898.602
884.614
1.166
5.077717
5.225174
43
44
900.438
887.192
1.104
5.111488
5.258945
44
52.
tr = 36.3501.
53.
7T = 38.0599. TABLE LXIII.
177
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
4 PER CENT.
1 j?B
3 cS
54.
55.
.2 ”
V.
V.
im.a.
MB/).
AB.
V.
Y.
* (B/).
AB.
o «
QnfH
n.
6m.
12m.
6m.
12m.
6m.
12m.
im.a.
6m.
12m.
n.
0
32.000
24.121
.657
0.0125900.024825
33.424
25.033
.699
0.012851
0.025333
0
1
56.275
48.550
.644
.326471
.338953
58.608
50.367
.687
.327016
.339764
1
2
80.851
73.273
.632
.515902
.528650
84.064
75.947
.676
.516776
.529824
2
3
105.694
98.236
.621
.654721
.667769
109.761
101.759
.667
.655969
.669339
3
4
130.771
123.426
.612
.766103
.779473
135.671
127.768
.659
.767775
.781500
4
5
156.056
148.807
.604
0.860405
0.874130
161.752
153.921
.653
0.862562
0.876680
5
6
181.507
174.329
.598
.943189
.957307
187.956
180.177
.648
.945901
.960450
6
7
207.080
199.952
.594
1.017813
1.032362
214.253
206.513
.645
1.021148
1.036166
7
8
232.742
225.652
.591
.086468
.101486
240.594
232.861
.644
.090511
.106050
8
9
258.448
251.365
.590
.150690
.166229
266.932
259.188
.645
.155532
.171641
9
10
284.150
277.057
.591
1.211610
1.227719
293.218
285.433
.649
1.217360
1.234098
10
11
309.803
302.669
.595
.270105
.286843
319.381
311.514
.656
.276869
.294291
11
12
335.334
328.121
,601
.326868
.344290
345.425
337.520
.659
.334781
.352962
12
13
360.750
353.499
.604
.382491
.400672
371.309
363.284
.669
.391685
.410686
13
14
386.010
378.642
.614
.437472
.456473
396.929
388.758
.681
.448111
.468029
14
15
411.011
403.502
.626
1.492270
1.512188
422.254
413.935
.693
1.504537
1.525448
15
16
435.726
428.071
.638
.547312
.568223
447.249
438.748
.708
.561391
.583398
16
17
460.118
452.286
.653
.602984
.624991
471.863
463.163
.725
.619102
.642306
17
18
484.139
476.112
.669
.659685
.682889
496.064
487.150
.743
.678070
.702583
18
19
507.756
499.521
.686
.717790
.742303
519.796
510.628
.764
.738731
.764695
19
20
530.916
522.432
.707
L.777712
1.803676
543.019
533.594
.785
1.801501
1.829038
20
21
553.577
544.844
.728
.839851
.867388
565.716
556.024
.808
.866797
.896053
21
22
575.729
566.734
.750
.904610
.933866
587.846
577.852
.833
.935103
.966260
22
23
597.323
588.034
.774
.972462 2.003619
609.372
599.077
.858
2.006903
2.040126
23
24
618.331
608.748
.799
2.043879
.077102
630.284
619.677
.884
.082711
.118194
24
25
638.739
628.851
.824
2.119367
2.154850
650.566
639.640
.910
2.163082
2.201038
25
26
658.531
648.333
.850
.199471
.237427
670.219
658.982
.936
.248609
.289255
26
27
677.710
667.208
.875
.284780
.325426
689.236
677.676
.963
.339957
.383563
27
28
696.269
685.450
.902
.375951
.419557
707.651
695.811
.987
.437805
.484597
28
29
714.238
703.148
.924
.473657
.520449
725.505
713.383
1.010
.542930
.593241
29
30
731.662
720.296
.947
2.578669
2.628980
742.866
730.534
1.028
2.656166
2.710270
30
31
748.604
737.034
.964
.691818
.745922
759.912
747.476
1.036
.778327
.836520
31
32
765.240
753.567
.973
.813912
.872105
776.810
764.328
1.040
.910546
.973309
32
33
781.730
770.013
.976
.946081
3.008844
793.941
781.739
1.037
3.053775
3.121308
33
34
798.448
787.004
.954
3.089275
.156808
810.346
797.139
1.101
.213334
.289004
34
35
814.457
802.032
1.035
3.248807
3.324477
824.599
810.243
1.196
3.394392
3.478991
35
36
828.3*6
814.820
1.129
.429848
.514447
837.279
822.500
1.232
.596622
.688999
36
37
840.740
826.781
1.163
.632067
.724444
849.314
834.314
1.250
.820094
.920604
37
38
852.485
838.310
1.181
.855532
.956042
860.825
845.521
1.275
4.068129
4.178020
38
! 39
863.718
849.247
1.206
4.103563
4.213454
871.889
856.441
1.287
.344586
.464666
39
40 874.515
859.904
1.218
4.380017 4.500097
883.314
868.373
1.245
4.652202
4.782744
40
41
885.665
871.548
1.176
.687632
.818174
894.570
878.951
1.302
5.007715
5.155172
41
|42
896.649
881.871
1.231
5.043145
5.190602
907.380
893.993
1.116
.410448
.570149
42
54.
7t = 39.8789.
55.
7t = 41.8154. 178
TABLE LXIII.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
4 PER CENT.
•" §
x — 56.
57.
.2 ”
c6
O
Ph&h
V.
V.
MB/).
AB.
Y.
Y.
im.d.
MB/).
AB.
n.
6m.
12m.
6m.
12m.
6m.
12m.
6m.
12m.
n.
0
34.930
25.984
.745
0.013135
0.025883
36.505
26.937
.797
0.013452
0.026500
0
1
61.041
52.221
.735
.327616
.340664
63.564
54.118
.787
.328277
.341647
1
2
87.396
78.696
.725
.517736
.531106
90.849
81.507
.779
.518790
.532515
2
3
113.973
105.373
.717
.657338
.671063
118.314
109.047
.772
.658846
.672964
3
4
140.723
132.197
.711
.769614
.783732
145.907
136.695
.751
.771642
.786191
4
5
167.600
159.128
.706
0.864939
0.879488
173.598
164.428
.768
0.867556
0.882574
5
6
194.572
186.139
.703
.948885
.963903
201.338
192.174
.764
.952172
.967711
6
7
221.589
213.164
.702
1.024822
1.040361
229.072
219.897
.765
1.028870
1.044979
7
8
248.604
240.167
.703
.094965
.111074
256.752
247.534
.768
.099880
.116618
8
9
275.564
267.086
.706
.160875
.177613
284.302
274.998
.775
.166762
.184184
9
10
302.400
293.837
.714
1.223696
1.241118
311.727
302.383
.779
1.230688
1.248869
10
11
329.111
320.510
.717
.284333
.302514
338.985
329.514
.789
.292560
.311561
11
12
355.661
346.936
.727
.343503
.362504
365.963
356.339
.802
.353131
.373049
12
13
381.938
373.064
.740
.401831
.421749
392.632
382.851
.815
.413038
.433949
13
14
407.914
398.887
.752
.459862
.480773
418.952
408.980
.831
.472832
.494839
14
15
433.550
424.337
.768
1.518076
1.540083
444.872
434.691
.848
1.533034
1.556238
15
16
458.796
449.380
.785
.576944
.600148
470.357
459.950
.867
.594117
.618630
16
17
483.620
473.983
.803
.636897
.661410
495.348
484.673
.889
.656575
.682539
17
18
507.961
498.063
.825
.698397
.724361
519.801
508.857
.912
.720871
.748408
18
19
531.779
521.619
.847
.761881
.789418
543.704
532.477
.936
.787461
.816717
19
20
555.060
544.625
.870
1.827783
1.857039
567.006
555.462
.962
1.856856
1.888013
20
21
577.757
567.013
.895
.896600
.927757
589.674
577.813
.988
.929573
.962796
21
22
599.836
588.783
.921
.968831
2.002054
611.696
599.506
1.016
2.006144
2.041627
ll
23
621.285
609.912
.948
2.044998
.080481
633.053
620.528
1.044
.087148
.125104
23
24
642.088
630.388
.975
.125667
.163623
653.748
640.895
1.071
.173193
.213839
24
25
662.245
650.226
1.002
2.211438
2.252084
673.774
660.580
1.099
2.264962
2.308568
25
26
681.751
669.400
1.029
.302984
.346590
693.165
679.677
1.124
.363147
.409939
26
27
700.638
688.000
1.053
.400991
.447783
711.966
698.181
1.149
.468538
.518849
27
28
718.950
706.024
1.077
.506242
.556553
730.248
716.242
1.167
.581980
.636084
28
29
736.758
723.615
1.095
.619575
.673679
748.198,734.082
1.176
.704299
.762492
29
30
754.241
740.992
1.104
2.741811
2.800004
765.992,751.828
1.180
2.836635
2.899398
30
31
771.573
758.277
1.108
.874085
.936848
784.032 770.163
1.156
.979950
3.047483
31
32
789.144
776.135
1.084
3.017354
3.084887
801.308 786.380
1.244
3.139570
.215240
32
33
805.971
791.931
1.170
.176942
.252612
816.316 800.179
1.345
.320669
.405268
33
34
820.589
805.371
1.268
.358020
.442619
829.668,813.085
1.382
.522925
.615302
34
35
833.594
817.942
1.304
3.560262
3.652639
842.342 825.526
1.401
3.746413
3.846923
35
36
845.939
830.060
1.323
.783742
.884252
854.464
837.328
1.428
.994458
4.104349
36
37 857.745
841.554
1.349
4.031781
4.141672
866.114
848.827
1.441
4.270920
.391000
37
38
869.093
852.755
1.361
.308241
.428321
878.146
861.392
1.396
.578540
.709082
38
39
880.812
864.994
1.318
.615858
.746400
889.998
872.531
1.456
.934054
5.081511
39
40
892.357
875.843
1.376
4.971372
5.118829
903.488
888.371
1.260
5.336787
5.496488
40
41
42
905.496
926.405
891.272
917.662
1.185
0.729
5.374105
.847673
.533806
6.027960
924.955
915.465
0.791
.810356
.990643
41
56.
7T = 43.8764.
57.
77 = 46.0730. 179
TABLE LXIII.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
4 PER CENT.
>>03'
.2 ”
3 o3
58.
77 = 48.4132.
59.
77 = 50.9095.
j?E
O
P4>H
n.
V.
6 m.
Y.
12m.
im.a.
MB/).
6m.
AB.
12m.
Y.
6m.
Y.
12m.
£m.d.
MB/).
6m.
AB.
12m.
o
Y.
Y.
im.d.
MB/).
AB.
V.
Y.
|m.d.
MB/)-
AB.
«.
6m.
12m.
6m.
12m.
6m.
12m.
6m.
12m.
n.
0
41.779
29.984
.983
0.014592
0.028710
43.727
31.033
1.058
0.015054
0.029603
0
1
71.822
60.086
.978
.330676
.345225
74.806
62.159
1.054
.331641
.346659
1
2
101.970
90.279
.974
.522636
.537654
105.940
93.301
1.053
.524183
.539722
2
3
132.170
120.487
.974
.664348
.679887
137.069
124.418
1.054
.666567
.682676
3
4
162.366
150.671
.975
.779039
.795148
168.138
155.437
1.058
.782025
.798763
4
5
192.502
180.760
.979
0.877116
0.893854
199.060
186.263
1.066
0.880975
0.898397
5
6
222.498
210.662
.986
.964193
.981615
229.841
217.000
1.070
.969050
.987231
6
7
252.357
240.477
.990
1.043688
1.061869
260.436
247.451
1.082
1.049671
1.068672
7
8
282.033
270.015
1.001
.117859
.136860
290.715
277.560
1.096
.125128
.145046
8
9
311.405
299.221
1.015
.188323
.208241
320.648
307.316
1.111
.197052
.217963
9
10
340.440
328.085
1.030
1.256299
1.277210
350.190
336.643
1.129
1.266672
1.288679
10
11
369.096
356.533
1.047
.322746
.344753
379.282
365.501
1.148
.334978
.358182
11
12
397.316
384.526
1.066
.388471
.411675
407.886
393.852
1.170
.402793
.427306
12
13
425.063
412.026
1.086
.454167
.478680
435.936
421.600
1.195
.470862
.496826
13
14
452.272
438.943
1.111
.520487
.546451
463.382
448.744
1.220
.539848
.567385
14
15
478.895
465.273
1.135
1.588023
1.615560
490.210
475.255
1.246
1.610360
1.639616
15
16
504.918
490.989
1.161
.657331
.686587
516.364
501.053
1.276
.683027
.714184
16
17
530.288
516.014
1.190
.728998
.760155
541.806
526.140
1.305
.758466
.791689
17
18
554.968
540.348
1.218
.803611
.836834
566.524
550.488
1.336
.837290
.872773
18
19
578.944
563.966
1.248
.881753
.917236
590.496 574.083
1.368
.920149
.958105
19
20
602.197
586.854
1.279
1.964052
2.002008
613.723 596.943
1.398
2.007708
2.048354
20
21
624.729
609.029
1.308
2.051156
.091802
636.200
619.038
1.430
.100699
.144305
21
22
646.531
630.460
1.339
.143780
.187386
657.965
640.472
1.458
.199859
.246651
22
23
667.643
651.252
1.366
.242648
.289440
679.067
661.241
1.486
.306018
.356329
23
24
688.112
671.398
1.393
.348577
.398888,
699.586
681.512
1.506
.420055
.474159
24
25
708.017
691.061
1.413
2.462436
2.516540
719.734
701.536
1.516
2.542825
2.601018
25
26
727.560
710.485
1.423
.585071
.643264
739.705
721.454
1.521
.675498
.738261
26
27
746.932
729.806
1.427
.717644
.780407
759.953
742,032
1.493
.819059
.886592
27
28
766.574
749.767
1.401
.861132
.928665
779.343
760.234
1.592
.978853
3.054523
28
29
785.382
767.423
1.497
3.020874
3.096544
796.188
775.722
1.706
3.160070
.244669
29
30
801.722
782.447
1.606
3.202055
3.286654
811.175
790.208
1.747
3.362403
3.454780
30
31
816.260
796.499
1.647
.404365
.496742
825.400
804.172
3.769
.585938
.686448
31
32
830.058
810.043
1.668
.627887
.728397
839.005
817.417
1.799
.834011
.943902
32
33
843.254
822.892
1.697
.875951
.985842
852.081
830.325
1.813
4.110487
4.230567
33
34
855.939
835.412
1.711
4.152423
4.272503
865.586
844.427
1.763
.418116
.548658
34
35
36
869.039
881.943
849.092
861.219
1.662
1.727
4.460049
.815565
4.590591
.963022
878.888
856.929
1.830
4.773633
4.921090
35
48.
49.
906.763
896.073
.891
5.239465
5.386922
48
47
905.321
894.049
.939
5.208451
5.355908
897.099
886.992
.842
4.883955
.014497
47
46
895.469
884.790
.890
4.852941
4.983483
887.290
876.747
.879
.576344
4.696424
46
45
885.469
874.346
.927
.545329
.665409
877.791
867.372
.868
.299901
.409792
45
44
875.785
864.788
.916
.268885
.378776
867.908
857.750
.847
.051887
.152397
44
49.
77 = 31.8029.
48.
tr =30.4602.
4 PER CENT.
Continued from Page *174. 181
TABLE L XI11.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
o ?h
a *
62.
tt= 59.4582.
63.
tt= 62.7079.
.3
Pm!*
n.
y.
6 m.
Y.
12m.
Jm.d.
'mb/).
6m.
AB.
12m.
V.
6 m.
V.
12m.
im.d.
MB/).
6m.
AB.
12m.
'o 3>
Pm!*
11.
0
45.791
32.123
1.139
0.015557
0.030575
47.957
33.206
1.229
0.016117
0.031656
0
1
77.922
64.263
1.138
.332703
.348242
81.150
66.385
1.230
.333876
.349985
1
2
110.048
96.375
1.139
.525890
.541999
114.276
99.460
1.235
.527773
.544511
2
3
142.110
128.388
1.144
.669013
.685751
147.249
132.329
1.243
.671708
.689130
3
4
174.024
160.202
1.152
.785315
.802737
180.070
165.103
1.247
.788943
.807124
4
5
205.791
191.922
1.156
0.885232
0.903413
212.691
197.572
1.260
0.889920
0.908921
5
6
237.365
223.349
1.168
.974401
.993402
244.979
229.677
1.275
.980304
1.000222
6
7
268.614
254.422
1.183
1.056274
1.076192
276.895
261.406
1.291
1.063566
.084477
7
8
299.506
285.132
1.198
.133156
.154067
308.396
292.677
1.310
.142016
.164023
8
9
329.994
315.398
1.216
.206687
.228694
339.416
323,447
1.331
.217330
.240534
9
10
360.018
345.180
1.236
1.278129
1.301333
369.915
353.676
1.353
1.290782
1.315295
10
11
389.538
374.438
1.258
.348487
.373000
399.824
383.264
1.380
.363433
.389397
11
12
418.486
403.076
1.284
.418636
.444600
429.090
412.208
1.407
.436163
.463700
12
13
446.812
431.089
1.310
.489331
.516868
457.696
440.476
1.435
.509753
.539009
13
14
474.498
458.449
1.337
.561254
.590510
485.584
467.984
1.467
.584950
.616107
14
15
501.490
485.074
1.368
1.635084
1.666241
512.713
494.733
1.498
1.662469
1.695692
15
16
527.748
510.964
1.399
.711483
.744706
539.068
520.695
1.531
.742997
.778480
16
17
553.257
536.092
1.430
.791095
.826578
564.628
545.853
1.565
.827247
.865203
17
18
577.996
560.442
1.463
.874598
.912554
589.395
570.229
1.597
.915934
.956580
18
19
601.968
584.035
1.494
.962681
2.003327
613.362
593.788
1.631
2.009829
2.053435
19
20
625.165
606.837
1.527
2.056091
2.099697
636.569
616.642
1.661
2,109709
2.156501
20
21
647.626
628.957
1.556
.155586
.202378
659.069
638.788
1.690
.216433
.266744
21
22
669.403
650.391
1.584
.262008
.312319
680.950
660.403
1.712
.330904
.385008
22
23
690.580
671.312
1.606
.376247
.430351
702.432
681.754
1.723
.454006
.512199
23
24
711.374
691.977
1.616
.499172
.557365
723.727
702.992
1.728
.586925
.649688
24
25
731.984
712.533
1.621
2.631959
2.694722
745.318
724.935
1.699
2.730665
2.798198
25
26
752.881
733.771
1.593
.775604
.843137
765.993
744.343
1.804
.890586
.966256
26
27
772.892
752.555
1.695
.935457
3.011127
783.954
760.857
1.925
3.071889
3.156488
27
28
790.276
768.539
1.811
3.116714
.201313
799.934 776.304
1.969
.274279
.366656
28
29
805.743
783.489
1.854
.319074
.411451
815.103j791.193
1.993
.497847
.598357
29
30
820.423
797.900
1.877
3.542624
3.643134
829.608
805.316
2.024
3.745941
3.855832
30
31
834.464
811.570
1.908
.790707
.900598
843.552
819.079 2.039
4.022428
4.142508
31
32
847.959
824.890
1.922
4.067188
4.187268
857.951
834.1161.986
.330062
.460604
32
33
34
861.897
875.625
839.445
852.347
1.871
1.940
.374819
.730338
.505361
.877795
872.136
847.447,2.057
.685583
.833040
33
46.
47.
909.402
899.781
00
o
5.299758
5.447215
50
49
908.121
897.981
.845
5.269889
5.417346
900.085
891.023
.755
4.944248
5.074790
49
48
898.636
889.067
.797
4.914379
.044921
890.624
881.145
.790
.636639
4.756719
48
47
889.006
879.010
.833
.606769
4.726849
881.465
872.103
.780
.360198
.470089
47
46
879.682
869.807
.823
.330327
.440218
871.935
862.825
.759
.112187
.212697
46
45
869.981
860.362
.802
.082315
.182825
861.970
853.044
.744
3.888759
3.981136
45
44
859.838
850.405
.786
3.858883
3.951260
851.472
842.896
.715
3.686599 3.771198
44
47.
77 — 29.1948.
46.
7t = 28.0014.
4 PER CENT.
Continued from Page *173. 182
TABLE LXIII.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
.gB
8
II
a>
II
66.1832.
65.
77 = 69.9017.
r-H C6
o a>
W.
y.
6m.
y.
12m.
im.d.
MB /)•
6m.
AB.
12m.
y.
6m.
y.
12m.
|m.d.
MB/).
6m.
AB.
12m.
O h
n.
y.
12m.
£m.d.
AB.
12m.
Y.
6m.
V.
12m.
im.d.
A(B/).
6m.
AB.
12m.
n.
0
67.031
41.189
2.154
0.021969
0.042880
70.427
42.339
2.341
0.023182
0.045189
0
1
107.923
81.784
2.178
.346181
.368188
112.426
83.999
2.369
.348727
.371931
1
2
148.193
121.728
2.205
.547542
.570746
153.720
124.928
2.399
.551629
.576142
2
3
187.786
160.971
2.235
.700080
.724593
194.215
164.987
2.436
.705977
.731941
3
4
226.612
199.381
2.269
.827224
.853188
233.837
204.174
2.472
.835200
.862737
4
5
264.604
236.954
2.304
0.939579
0.967116
272.568
242.447
2.510
0.949934
0.979190
5
6
301.739
273.650
2.341
1.042974
1.072230
310.326
279.691
2.553
1.056082
1.087239
6
7
337.941
309.359
2.382
.141097
.172254
347.056
315.907
2.596
.157363
.190586
7
8
373.158
344.084
2.423
.236500
.269723
382.739
351.058
2.640
.256364
.291847
8
9
407.372
377.787
2.465
.331096
.366579
417.346
385.120
2.686
.355071
.393027
9
10
440.553
410.447
2.509
1.426454
1.464410
450.879
418.123
2.730
1.455096
1.495742
10
11
472.705
442.090
2.551
.523913
.564559
483.328
450.020
2.776
.557872
.601478
11
12
503.818
472.673
2.595
.624715
.668321
514.749
480.964
2.815
.664667
.711459
12
13
533.944
502.342
2.633
.729994
.776786
545.213
510.947
2.855
.776732
.827043
13
14
563.153
531.091
2.672
.840901
.891212
574.836
540.212
2.885
.895277
.949381
14
15
591.557
559.150
2.701
1.958570
2.012674
603.923
569.119
2.900
2.021447
2.079640
15
16
619.445
586.867
2.715
2.084085
.142278
632.753
597.874
2.907
.156634
.219397
16
17
647.089
614.437
2.721
.218790
.281553
661.986
627.583
2,867
.302013
.369546
17
18
675.116
642.922
2.683
.363822
.431355
689.978
653.860
3.010
.463094
.538764
18
19
701.956
668.117
2.820
.524658
.600328
714.297
676.219
3.173
.645183
.729782
19
20
725.272
689.555
2.976
2.706581
2.791180
735.932
697.132
3.233
2.848077
2.940454
20
21
746.018
709.607
3.034
.909369
3.001746
756.469
717.291
3.265
3.071958
3.172468
21
22
765.707
728.935
3.064
3.133184
.233694
776.109
736.413
3.308
.320234
.430125
22
23
784.539
747.270
3.106
.381422
.491313
794.986
755.046
3.328
.596822
.716902
23
24
802.640
765.136
3.125
.657988
.778068
814.483
775.406
3.256
.904507
4.035049
24
25
26
821.333
839.746
784.657
801.962
3.056
3.149
3.965663
4.321204
4.096205
.468661
833.688
793.455
3.353
4.260054
4.407511
25
38.
39.
917.390
910.998
.533
5.522155
5.669612
58
57
916.577
909.856
.560
5.495734
5.643191
909.116
903.221
.491
.166647
.297189
57
56
908.196
901.979
.518
.140226
.270768
900.714
894.448
.522
4.859041
4.979121
56
55
899.687
893.093
.550
4.832619
4.952699
892.579
886.418
.513
.582606
.692497
55
54
891.447
884.960
.541
4.556183
4.666074
884.116
878.178
.495
4.334606
4.435116
54
53
882.876
876.615
.522
.308182
.408692
875.267
869.492
.481
.111196
.203573
53
52
873.914
867.817
.508
.084770
.177147
865.944
860.480
.455
3.909066
3.993665
52
51
864.470
858.690
.482
3.882638
3.967237
855.465
850.846
.385
.728163
.803833
51
50
853.857
848.931
.410
.701730
.777400
843.402
839.522
.323
.568832
.636365
50
49
841.640
837.463
.348
3.542394
3.609927
830.807
826.721
.341
3.425923
3.488686
49
48
828.882
824.497
.374
.399476
.462239
818.383
814.330
.338
.294145
.352338
48
47
816.299
811.947
.363
.267686
.325879
805.849
801.874
.331
.172570
.226674
47
46
803.605
799.331
.356
.146094
.200198
793.084
789.263
.318
.060106
.110417
46
45
790.675
786.558
.343
.033609
.083920
779.957,776.343
.301
2.955971
.002763
45
44
777.379
773.472
.325
2.929447
2.976239
766.418 763.009
.284
2.859371
2.902977
44
39.
77 = 21.3199.
38.
77 = 20.5624.
4 PER CENT.
Continued from Page *169. 186
TABLE LXIII.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
4 PER CENT.
j?s
Uh ce
II
<1
to
II
104.570.
73.
77 = 111.075.
oj
O 0)
Ph>h
n.
y.
6 m.
V.
12m.
im.d.
MB/).
6m.
AB.
12m.
y.
6m.
y.
12m.
im.d.
MB/).
6m.
AB.
12m.
n.
0
74.036
43.502
2.545
0.024515
0.047719
77.879
44.682
2.766
0.025979
0.050492
0
1
117.156
86.240
2.576
.351524
.376037
122.086
88.415
2.806
.354626
.380590
1
2
159.441
128.071
2.614
.556152
.582116
165.343
131.196
2.846
.561148
.588685
2
3
200.815
168.990
2.652
.712491
.740028
207.624
172.978
2.887
.719674
.748930
3
4
241.258
208.955
2.692
.844001
.873257
248.845
213.637
2.934
.853740
.884897
4
5
280.685
247.846
2.737
0.961393
0.992550
288.944
253.175
2.981
0.974085
1.007308
5
6
319.040
285.663
2.781
1.070602
1.103825
327.900
291.549
3.029
1.086682
.122165
6
7
356.300
322.368
2.828
.175381
.210864
365.679
328.735
3.079
.195360
.233316
7
8
392.437
357.936
2.875
.278392
.316348
402.287
364.764
3.127
.302827
.343473
8
9
427.452
392.398
2.921
.381666
.422312
437.713
399.586
3.177
.411211
.454817
9
10
461.337
425.705
2.969
1.486911
1.530517
472.014
433.367
3.221
1.522247
1.569039
10
11
494.146
458.017
3.011
.595585
.642377
505.271
466.100
3.264
.637511
.687822
11
12
525.956
489.326
3.053
.709074
.759385
537.611
498.048
3.297
.758458
.812562
12
13
556.890
519.884
3.084
.828692
.882796
569.365
529.607
3.313
.886412
.944605
13
14
587.262-
550.070
3.099
.955661
2.013854
600.840
560.998
3.320
2.022905
2.085668
14
15
617.368
580.096
3.106
2.091434
2.154197
632.752
593.431
3.277
2.169222
2.236755
15
16
647.892
611.118
3.064
.237235
.304768
663.312
622.118
3.433
.330963
.406633
16
17
677.123
638.557
3.214
.398613
.474283
689.861
646.528
3.611
.513497
.598096
17
18
702.516
661.905
3.384
.580902
.665501
713.481
669.359
3.677
.716678
.809055
18
19
725.108
683.742
3.447
.783926
.876303
735.900
691.366
3.711
.940734
3.041244
19
20
746.552
704.792
3.480
3.007885
3.108395
757.341
712.242
3.758
3.189113
3.299004
20
21
767.061
724.760
3.525
.256208
.366099
777.951
732.584
3.781
.465757
.585837
21
22
786.774
744.217
3.546
.532821
.652901
799.235
754.811
3.702
.773472
.904014
22
23
807.132
765.477
3.471
.840520
.971062
820.200
774.514
3.807
4.129033
4.276490
23
24
827.185 784.323
3.572
4.196073
4.343530
37.
36.
918.890
913.103
.482
5.574024
5.218516
5.721481
5.349058
60
59
59
918.161
912.080
.507
5.548246
5.695703
910.810
905.510
.442
58
909.986
904.397
.466
.192738
.323280
902.607
896.944
.472
4.910910
.030990
58
57
901.687
895.730
.496
4.885132
.005212
894.666
889.105
.463
.634476
4.744367
57
56
893.651
887.799
.488
.608697
4.718588
886.402
881.060
.445
.386478
.486988
56
55
885.291
879.659
.469
.360698
.461208
877.763
872.579
.432
.163071
.255448
55
54
876.550
871.078
.456
4.137290
4.229667
868.659
863.780
.407
3.960945
4.045544
54
53
867.339
862.176
.430
3.935162
.019761
858.429
854.373
.338
.780050
3.855720
53
52
856.988
852.658
.361
.754263
3.829933
846.651
843.318
.278
.620731
.688264
52
51
845.073
841.473
.300
.594939
.662472
834.354
830.819
.295
.477838
.540601
51
50
832.629
828.827
.317
.452038
.514801
822.224
818.722
.292
.346082
.404275
50
49
820.357
816.587
.314
3.320271
3.378464
809.986
806.560
.286
3.224535
3.278639
49
48
807.975
804.281
.308
.198711
.252815
797.523
794.247
.273
.112109
.162420
48
47
795.365
791.824
.295
.086267
.136578
784.707
781.633
.256
.008023
.054815
47
46
782.397 779.061
.280
2.982156
.028948
771.488
768.615
.239
2.911484
2.955090
46
45
769.023 765.889
.261
.885589
2.929195
757.836
755.195
.220
.821765
.862411
45
44
755.211j752.312
.242
2.805832
2.836478
743.728
741.310
.201
2.738237
2.776193
44
37.
77 19.8452.
36.
77 =
19.1663.
Continued from Page *168, 187
TABLE LXIII.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
4 PER CENT.
og
as
74.
tr = 118.069.
75.
7r= 125.579.
og
r—3 C6
o
pLlfH
n.
V.
6m.
Y.
12m.
|-m.d.
MB/).
6 m.
AB.
12 m.
V.
6m.
Y.
12m.
-J-m.d.
MB/).
6m.
AB.
12m.
o
n.
0
100.990
50.279
4.226
0.035976
0.069199
106.576
51.383
4.599
0.038642
0.074125
0
1
150.529
99.078
4.288
.375648
.411131
157.164
101.175
4.666
.381265
.419221
1
2
198.573
146.367
4.351
.595120
.633076
206.181
149.418
4.730
.604219
.644865
2
3
245.127
192.185
4.412
.768806
.809452
253.615
196.044
4.798
.782023
.825629
3
4
290.176
236.467
4.476
.920540
.964146
299.545
241.277
4.856
.938556
.985348
4
5
333.797
279.425
4.531
1.061377
1.108169
344.077
285.107
4.914
1.085004
1.135315
5
6
376.088
321.051
4.586
.197679
.247990
387.381
327.886
4.958
.227823
.281927
6
7
417.216
361.679
4.628
.333673
.387777
429.899
370.143
4.980
.371259
.429452
7
8
457.596
401.812
4.649
.472420
.530613
472.044
412.176
4.989
.518708
.581471
8
9
497.622
441.732
4.657
.616620
.679383
514.774
455.604
4.931
.672714
.740247
9
10
538.205
482.976
4.602
1.768380
1.835913
555.695
494.016
5.140
1.839770
1.915440
10
11
577.066
519.456
4.801
.933901
2.009571
591.243
526.701
5.379
2.025850
2.110449
11
12
610.828
550.498
5.028
2.118963
.203562
622.870
557.271
5.467
.231294
.323671
12
13
640.865
579.531
5.111
.323761
.416138
652.889
586.738
5.513
.456730
.557240
13
14
669.374
607.517
5.155
.548804
.649314
681.599
614.691
5.576
.705916
.815807
14
15
696.641
634.064
5.215
2.797761
2.9076521
709.195
641.929
5.605
2.983005
3.103085
15
16
722.849
659.933
5.243
3.074724
3.194804
737.694
671.691
5.500
3.290950
.421492
16
17
18
749.916
776.577
688.198
713.255
5.143
5.277
.382603
.738242
.513145
.885699
765.767
698.074
5.641
.646620
.794077
17
30.
31.
922.527
914.922
918.211
.360
5.722991
5.870448
66
65
921.998
917.469
.377
5.698772
5.846229
911.064
.322
.367484
.498026
65
64
914.325
910.257
.339
5.343264
5.473806
907.-202
903.002
.350
5.059879
5.179959
64
63
906.534
902.121
.368
.035659
.155739!
899.727
895.623
.342
4.783447
4.893338
63
62
898.991
894.676
.360
4.759227
4.869118
891.949
888.051
.325
.535453
.635963
62
61
891.143
887.035
.342
.511233
.611743
883.817
880.068
.312
.312053
.404430
61
60
882.937
878.980
.330
.287832
.380209;
875.249
871.787
.288
.109939
.194538
60
59
874.292
870.624
.306
4.085716
4.170315!
865.620
862.933
.224
3.929062
4.004732
59
58
864.575
861.689
.241
3.904836
3.980506
854,535
852.528
.167
.769768
3.837301
58
57
853.389
851.189
.183
.745539
.813072
842.960
840.763
.183
.626912
.689675
57
56
841.709
839.318
.199
.602677
.665440
831.543
829.377
.181
.495205
.553398
56
55
830.189
827.829
.197
.470964
.529157
820.025
817.930
.175
.373725
.427829
55
54
818.566
816.278
.191
3.349475
3.403579
808.294
806.341
.163
3.261386
3.311697
54
53
806.729
804.584
.179
.237124
.287435
796.231
794.468
.147
.157411
.204203
53
52
794.557
792.603
.163
.133133
.179925
783.789
782.215
.131
.061013
.104619
52
51 782.001
780.239
.147
.036715
.080321
770.939
769.584
.113
2.971467
.012113
51
50 769.035
767.493
.128
2.947144
2.987790
757.661
756.516
.095
.888151
2.926107
50
49 755.636
754.306
.111
2.863799
2.901755
743.957
743.027
.077
2.810456
2.845939
49
48 741.808
740.695
.093
.786069
.821552
729.827
729.108
.060
.737856
.771079
48
47 727.550
726.650
.075
.713426
.746649
715.283
714.767
.043
.669860
.701017
47
46
712.874
712.179
.058
.645379
.676536
700.330
700.019
.026
.606033
.635289
46
45
697.786
697.297
.041
.581495
.610751
684.994
684.863
.011
.545968
.573505
45
44
682.310
682.004
.026
2.521362
2.548899
669.304
669.346
.004
2.489277
2.515241
44
31
77 = 16.2712.
30.
tt = 15.7791.
4 PER CENT.
Continued from Page *165. 190
TABLE LXIII.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
,3 ci
a
II
00
O
II
172.614.
81.
7T = 184.307.
,3 a
O CD
P4t*
n.
y.
6 m.
V.
12m.
im.d.
MB/)-
6m.
AB.
12m.
V.
6m.
y.
12m.
|m.d.
MB/).
6m.
AB.
12m.
O
FMrH
n.
V.
6m.
V.
12m.
2'in.d.
MB/).
6m.
AB.
12m.
V.
6m.
y.
12m.
im.d.
MB/)-
6m.
AB.
12m.
n.
0
125.880
54.817
5.922
0.048478,0.092084
133.429
56.263
6.431
0.052451
0.099243
0
1
179.877
107.996
5.990
.401959
.448751
188.818
110.780
6.503
.410393
.460704
1
2
232.232
159.525
6.059
.637870
.688181
242.683
163.991
6.558
.651621
.705725
2
3
283.142
209.818
6.110
.830995
.885099
295.568
216.552
6.585
.851002
.909195
3
4
333.129
259.498
6.136
1.005502
1.063695
347.991
268.836
6.596
1.033065
1.095828
4
5
382.678
308.916
6.147
1.173060
1.235823
401.142
322.853
6.524
1.209337
1.276870
5
6
432.915
359.972
6.079
.340355
.407888
452.039
370.631
6.784
.390937
.466607
6
7
481.023
405.131
6.324
.516239
.591909
496.256
411.287
7.081
.586356
.670955
7
8
522.815
443.558
6.605
.708048
.792647
535.596
449.312
7.190
.797620
.£89997
8
9
560.000
479.499
6.708
.917087
2.009464
572.936
485.965
7.248
2.026552
2.127062
9
10
595.291
514.142
6.762
2.144692
2.245202
608.646
520.734
7.326
2.277765
2.387656
10
11
629.045
547.005
6.837
.395138
.505029
642.971
554.614
7.363
.555967
.676047
11
12
661.487
579.028
6.872
.672919
.792999
678.421
591.633
7.232
.864483
.995025
12
13
14
694.994
727.998
614.018
645.036
6.748
6.914
.981220
3.337062
3.111762
.484519
713.338
624.450
7.407
3.220429
3.367886
13
26.
27.
70
924.385
920.820
.297
5.817832
5.965289
69
923.956
920.218
.312
5.794407 5.941864
917.023
913.901
.260
5.462325
5.592867
69
68
916.538
913.246
.274
.438899
.569441
909.549
906.096
.288
.154720
.274800
68
67
909.007
905.382
.302
.131295
.251375
902.313
898.953
.280
4.878290
4.988181
67
66
901.715
898.184
.294
4.854864
4.964755
894.783
891.623
.263
.630298
.730808
66
65
894.129
890.798
.278
.606872
.707382
886.911
883.895
.251
.406901
.499278
65
64
886.196
883.012
.265
4.383474
4.475851
878.617
875.878
.228
4.204793
4.289392
64
63
877.839
874.933
.242
.181365
.265964
869.294
867.306
.166
.023923
.099593
63
62
868.446
866.297
.179
.000493
.076163
858.563
857.233
.111
3.864643
3.932176
62
61
857.633
856.147
.124
3.841210
3.908743
847.357
845.844
.126
.721803
.784566
61
60
846.342
844.671
.139
.698366
.761129
836.304
834.821
.124
.590121
.648314
60
59
835.205
833.564
.137
3.566678
3.624871
825.154
823.739
.118
3.468672
3.522776
59
58
823.970
822.398
.131
.445223
.499327
813.798
812.520
.106
.356375
.406686
58
57
812.527
811.094
.119
.332916
.383227
802.119
801.026
.091
.252453
.299245
57
56
800.760
799.512
.104
.228982
.275774
790.075
789.164
.076
.156122
.199728
56
55
788.623
787.560
.089
.132636
.176242
777.635
776.936
.058
.066659
.107305
55
54
776.089
775.239
.071
3.043154
3.083800
764.780
764.285
.041
2.983445
3.021401
54
53
763.137
762.491
.054
2.959917
2.997873
751.514
751.227
.024
.905872
2.941355
53
52
749.769
749.334
.036
.882316
.917799
737.834
737.751
.007
.833417
.866640
52
51
50
735.985
721.798
735.756
721.767
.019
.003
.809828
.741964
.843051
.773121
723.753
709.279
723.868
709.590
.010
.026
.765591
.701964
.796748
.731220
51
50
49
707.213
707.381
.014
2.678291
2.707547
694.431
694.918
.041
2.642128
2.669665
49
48
692.253
692.597
.029
.618404
.645941
679.242
679.896
.055
.585701
.611665
48
47
676.947
677.460
.043
.561916
.587880
663.719
664.539
.068
.532368
.556881
47
46
661.305
661.986
.057
.508516
.533029
647.888
648.849
.080
.481830
.505034
46
45
645.355
646.177
.068
.457901
.481105
631.787
632.878
.091
.433790
.455797
45
44
629.132
630.085
.079
2.409775
2.431782
615.439
616.648
.101
2.388007
2.408918
44
27.
7T =
14.4479.
26.
rr = 14.0485.
4 PER CENT.
Continued from Page *163. 192
TABLE L XIII.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
j?e
ce
x= 84. 71 = 225.442.
*d
00
n = 241.622.
og
£3 ce
o
n.
Y.
6m.
Y.
12m.
im.d.
MB/).
6m.
AB.
12m.
V.
6m.
Y.
12m.
im.d.
* (B/).
6m.
AB.
12m.,
O H
n.
y.
6m.
V.
12m.
£m.d.
MB/).
6m.
AB.
12m.
Y.
6m.
Y.
12m.
-Jrm.d.
A (B/).
6 m.
AB.
12m.
o -t
n.
0
210.928
64.595
12.19
0.105165
0.189764
224.588
64.590
13.33
0.117487
0.209864
0
1
273.435
125.014
12.37
.523682
.616059
288.013
126.850
13.43
.549523
.650033
1
2
332.764
183.252
12.46
.841211
.941721
348.673
185.909
13.56
.885237
.995128
2
3
389.504
238.496
12.58
1.136222
1.246113
406.976
243.458
13.63
1.203501
1.323581
3
4
444.043
292.328
12.64
.436615
.556695
467.192
306.339
13.40
.531244
.661786
4
5
6
500.368
555.849
351.147
403.290
12.44
12.71
1,756106
2.117213
1.886648
2.264670
526.504
362.083
13.70
1.896111
2.043568
5
18.
19.
927.133
920.130
924.679
918.097
.205
.169
5.999652
6.147109
78
77
77
926.847
924.277
.214
5.977398
6.124855
.644145
5.774687
76
919.806
917.660
.179
.621891
5.752433
913.019
910.672
.196
.336541
.456621
76
75
912.658
910.196
.205
.314287
.434367
906.136
903.877
.188
.060113
.170004
75
74
905.738
903.365
.198
5.037858
5.147749
898.973
896.904
.172
4.812123
4.912633
74
73
898.537
896.354
.182
4.789869
4.890379
891.484
889.552
.161
.588731
.681108
73
72
891.008
888.964
.170
.566476
.658853
883.594
881.926
.139
.386631
.471230
72
71
883.076
881.297
.148
.364375
.448974
874.726
873.772
.079
.205773
.281443
71
70
874.161
873.099
.089
.183516
.259186
864.518
864.190
.027
.046509
.114042
70
69
863.898
863.466
.036
4.024251
4.091784
853.858
853.356
.042
3.903695
3.966458
69
68
853.181
852.574
.051
3.881433
3.944196
843.344
842.870
.040
.772045
.830238
68
67
842.611
842.032
.048
.749781
.807974
832.737
832.328
.034
.650642
.704746
67
66
831.947
831.434
.043
.628372
.682476
821.934
821.656
.023
.538403
.588714
66
65
821.087
820.705
.032
.516128
.566439
810.824
810.722
.008
.434555
.481347
65
64
809.919
809.713
.017
3.412273
3.459065
799.366
799.437
.006
3.338318
3.381924
64
63
798.399
798.368
.003
.316026
.359632
787.533
787.805
.023
.248971
.289617
63
62
786.502
786.674
.014
.226667
.267313
775.304
775.770
.039
.165899
.203855
62
61
774.209
774.575
.030
.143581
.181537
762.685
763.349
.055
.088496
.123979
61
60
761.522
762.087
.047
.066161
.101644
749.671
750.530
.072
.016244
.049467
60
59
748.439
749.200
.063
2.993888
3.027111
736.277
737.323
.087
2.948656
2.979813
59
58
734.974
735.923
.079
.926276
2.957433
722.508
723.741
.103
.885306
.914562
58
57
721.131
722.269
.095
.862898
.892154
708.384
709.784
.117
.825791
.853328
57
56
706.932
708.237
.109
.803350
.830887
693.934
695.494
.130
.769729
.795693
56
55
692.406
693.871
.122
.747252
.773216
679.168
680.885
.143
.716813
.741326
55
54
677.559
679.184
.135
2.694294
2.718807
664.108
665.960
.154
2.666744
2.089948
54
53
662.421
664.179
.147
.644178
.667382
648.793
650.767
.165
.619231
.641238
53
52
647.022
648.906
.157
.596612
.618619
633.240
635.328
.174
.574036
.594947
52
51
631.387
633.383
.166
.551358
.572269
617.481
619.662
.182
.530929
.550847
51
50
615.544
617.634
.174
.508185
.528103
601.539
603.810
.189
.489707
.508708
50
49 599.518
601.698
.182
2.466891
2.485892
585.433
587.779
.195
2.450180
2.468361
49
48
583.326
585.582
.188
.427285
.445466
569.228
571.597
.197
.412169
.429591
48
47
567.034
569.314
.190
.389186
.406608
552.949
555.369
.202
.375524
.392262
47
46
550.667
552.998
.194
.352445
.369183
536.592
539,038
.204
.340097
.356206
46
45
534.225
536.581
.196
.316914
.333023
520.203
522.656
.204
.305761
.321300
45
44
517.748
520.112
.197
2.282464
2.298003
503.813
506.261
.204
2.272391
2.287409
44
19.
77 = 11.7560.
18.
7r = 11.4897.
Continued from Page *159. 196
TABLE LX III.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
.38
r—H 03
x = 92. tt =
413.801.
93.
tt = 446.049.
M
2 "
r—H C6
0 O)
Ph>h
n.
V.
6m.
V.
12m.
im.d.
MB/).
6m.
AB.
12m.
y.
6m.
y.
12m.
MB/).
6m.
AB.
12m.
n.
0
240.180
66.559
14.47
0.131006
0.231516
256.844
67.639
15.77
0.147474
0.257365
0
1
305.028
129.696
14.61
.579510
.689401
323.619
133.549
15.84
.615040
.735120
1
2
367.358
191.219
14.68
.936727
1.056807
392.582
205.566
15.59
.996003
1.126545
2
3
4
431.731
495.139
258.442
318.035
14.44
14.76
1.280853
.652883
.411395
.800340
460.511
269.408
] 5.93
1.382805
.530262
3
17.
16.
927.663
925.424
.187
6.043818
6.191275
80
79
927.405
925.061
.195
6.021790
6.169247
920.729
918.907
.152
5.688311
5.818853
79
78
920.436
918.512
.160
5.666283
5.796825
913.690
911.556
.178
.380708
.500788
78
77
913.363
911.125
.187
.358680
.478760
906.874
904.829
.170
.104280
.214171
77
76
906.514
904.365
.179
.082251
.192142
899.783
897.924
.155
4.856291
4.956801
76
75
899.388
897.427
.163
4.834262
4.934772
892.368
890.646
.144
.632899
.725276
75
74
891.937
890.113
.152
4.610871
4.703248
884.555
883.095
.122
4.430800
4.515399
74
73
884.087
882.525
.130
.408771
.493370
875.775
875.021
.063
.249945
.325615
73
72
875.264
874.413
.071
.227914
.303584
865.668
865.534
.011
.090685
.158218
72
71
865.108
864.879
.019
.068652
.136185
855.114
854.807
.026
3.947874
.010637
71
70
854.501
854.100
.033
3.925840
3.988603
844.703
844.425
.023
.816231
3.874424
70
69
844.041
843.667
.031
3.794194
3.852387
834.202
833.987
.018
3.694836
3.748940
69
68
833.487
833.178
.026
.672794
.726898
823.505
823.421
.007
.582608
.632919
68
67
822.740
822.561
.015
.560561
.610872
812.506
812.595
.007
.478774
.525566
67
66
811.686
811.682
.000
.456721
.503513
801.162
801.422
.022
.382554
.426160
66
65
800.287
800.455
.014
.360493
.404099
789.445
789.906
.038
.293229
.333875
65
64
788.513
788.882
.031
3.271157
3.311803
777.338
777.990
.054
3.210183
3.248139
64
63
776.346
776.908
.047
.188098
.226054
764.843
765.691
.071
.132812
.168295
63
62
763.791
764.549
.063
.110712
.146195
751.958
752.999
.087
.060597
.093820
62
61
750.844
751.796
.079
.038479
.071702
738.697
739.923
.102
2.993054
.024211
61
60
737.517
738.656
.095
2.970914
.002071
725.064
726.476
.118
.929755
2.959011
60
59
723.818
725.143
.110
2.907590
2.936846
711.080
712.657
.131
2.870299
2.897836
59
58
709.766
711.257
.124
.848105
.875642
696.774
698.508
.145
.814305
.840269
58
57
695.390
697.039
.137
.792078
.818042
682.152
684.044
.158
.761465
.785978
57
56
680.697
682.504
.151
.739201
.763714
667.243
669.266
.169
.711483
.734687
56
55
665.714
667.654
.162
.689177
.712381
652.078
654.224
.179
.664068
.686075
55
54
650.477
652.539
.172
2.641714
2.663721
636.679
638.937
.188
2.618980
2.639891
54
53
635.002
637.178
.181
.596574
.617485
621.076
623.426
.196
.575994
.595912
53
52
619.324
621.591
.189
.553529
.573447
605.293
607.732
.203
.534904
.553905
52
51
603.464
605.821
.196
.512375
.531376
589.346
591.859
.209
.495523
.513704
51
50
587.438
589.870
.203
.472924
.491105
573.302
575.838
.211
.457672
.475094
50
49
571.316
573.771
.205
2.434995
2.452417
557.183
559.770
.216
2.421202
2.437940
49
48
555.119
557.625
.209
.398440
.415178
540.988
543.601
.218
.385967
.402076
48
47
538.846
541.377
.211
.363112
.379221
524.762
527.381
.218
.351838
.367377
47
46
522.540
525.078
.211
.328882
.344421
508.534
511.148
.218
.318694
.333712
46
45
506.232
508.766
.211
.295629
.310647
492.333
494.924
.216
.286421
.300970
45
44
489.954
492.463
.209
2.263238
2.277787
476.189
478.748
.213
2.254918
2.269036
44
17.
TT =
11.2364.
•
CD
i-H
tt = 10.9953.
4 PER CENT.
Continued from Page *158 197
TABLE LX III.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
4 PER CENT.
CD
77 =
481.198.
95.
77 = 520.729.
og
P4>H
W.
y.
6m.
V.
]2m.
im.d.
MB/).
6m.
AB.
12m.
y.
6 m.
y.
12m.
im.d.
MB/).
6m.
AB.
12m.
O
n.
0
275.945
70.691
17.11
0.166535
0.286615
301.923
83.117
18.23
0.187522
0.318064
0
1
2
349.910
422.768
147.932
216.406
16.83
17.20
.655087
1.076479
.785629
1.223936
380.323
156.799
18.63
.713515
.860972
1
15.
14.
928 144
926.099
.170
.136
6.087566
5.732059
6.235023
82
81
927.909
925.770
.178
6.065742
6.213199
921.272
919.641
5.862601
81
80
921.008
919.283
.144
5.710235
5.840777
914.297
912.356
.162
.424456
.544536
80
79
914.001
911.966
.170
5.402632
5.522712
907.542
905.689
.154
5.148028
5.257919
79
78
907.216
905.270
.162
.126203
.236094
900.516
898.848
.139
4.900040
.000550
78
77
900.158
898.397
.147
4.878215
4.978725
893.167
891.635
.128
.676650
4.769027
77
76
892.778
891.153
.135
.654825
.747202
885.426
884.152
.106
.474551
.559150
76
75
885.002
883.637
.114
.452725
.537324
876.725
876.152
.048
.293698
.369368
75
74
876.262
875.601
.055
4.271871
4.347541
866.710
866.750
.003
4.134440
4.201973
74
73
866.202
866.157
.004
.112612
.180145
856.250
856.121
.011
3.991634
.054397
73
72
855.696
855.480
.018
3.969804
.032567
845.935
845.832
.009
.859997
3.918190
72
71
845.334
845.146
.016
.838164
3.896357
835.528
835.489
.003
.738609
.792713
71
70
834.882
834.757
.010
.716772
.770876
824.929
825.019
.008
.626391
.676702
70
69
824.235
824.240
.000
3.604549
3.654860
814.029
814.291
.022
3.522569
3.569361
69
68 813.286
813.464
.015
.500722
.547514
'802.786
803.219
.036
.426365
.469971
68
671801.995
802.343
.029
.404510
.448116
'791.176
791.806
.052
.337059
.377705
67
66
790.332
790.880
.040
.315195
.355841
779.179
779.998
.068
.254037
.291993
66
65
778.281
779.019
.062
.232161
.270117
766.796
767.811
.085
.176694
.212177
65
64
765.844
766.777
.078
3.154805
3.190288
754.029
755.234
.100
3.104513
3.137736
64
63
753.019
754.144
.094
.082607
.115830
740.887
742.276
.116
.037009
.068166
63
62
739.820
741.129
.109
.015084
.046241
727.377
728.950
.131
2.973756
.003012
62
61
726.249
727.744
.125
2.951809
2.981065
713.520
715.256
.145
.914353
2.941890
61
60
712.330
713.989
.138
.892380
.919917
699.342
701.235
.158
.858419
.884383
60
59
698.090
699.905
.151
2.836417
2.862381
684.854
686.902
.171
2.805649
2.830162
59
58
683.536
685.508
.164
.783613
.808126
670.079
672.258
.182
.755744
.778948
58
57
668.696
670.799
.175
.733670
.756874
655.052
657.352
.192
.708415
.730422
57
56
653.602
655.827
.185
.686299
.708306
639.793
642,204
.201
.663424
.684335
56
55
638.275
640.611
.195
.641262
.662173
624.331
626.833
.209
.620545
.640463
55
54
622.744
625.172
.202
2.598329
2.618247
608.690
611.281
.216
2.579573
2,598574
54
53
607.033
609.550
.210
.557301
.576302
592,888
595.552
.222
.540322
.558503
53
52
591.161
593.751
.216
.517987
.536168
576.987
579.675
.224
.502614
.520036
52
51
575.190
577.804
.218
.480209
.497631
561.015
563.752
.228
.466299
.483037
51
50
559.146
561.810
.222
.443819
.460557
544.968
547.730
.230
.431233
.447342
50
49
543.027
545.716
.224
2.408671
2.424780
528.888
531.657
.231
2.397289
2.412828
49
48
526.876
529.572
.225
.374637
.390176
512.806
515.571
.230
.364345
.379363
48
47
510.723
513.415
.224
.341595
.356613
496.752
499.493
.228
.332289
.346838
47
46
494.598
497.265
.222
.309435
.323984
480.754
483.463
.226
.301020
.315138
46
45
478.528
481.164
.220
.278052
.292170
464.831
467.497
.222
.270452
.284177
45
44
462.535
465.127
.216
2.247362
2.261087
449.013
451.618
.217
2.240494
2.253864
44
15.
7t = 10.7660.
14.
77 =
10.5480.
Continued from Page *157. 198
TABLE L X111.
GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000.
4 PER CENT.
X = 96. 7r =
571.434.
.2*8
f—h ce
o a>
Ph>h
n.
y.
6m.
V.
12m.
im.cl.
MB/).
6m.
AB.
12m.
y.
6m.
y.
12m.
im.d.
MB/).
6m.
AB.
12m.
O
|1<&M
n.
y.
6m.
Y.
12m.
Tfm.d.
MB/).
6m.
AB.
12m.
Y.
6m.
Y.
12m.
Trm.d.
MB/).
6m.
AB.
12m.
O H
n.
928.975
927.266
.142
6.173956
6.321413
86
85
928.783
926.995
.149
6.152485
6.299942
922.212
920.910
.109
5.818449
5.948991
85
84
921.994
920.616
.115
5.796979
5.927521
915.346
913.740
.134
5.510846
5.630926
84
83
915.103
913.419
.140
.489376
.609456
908.699
907.179
.127
.234418
.344309
83
82
908.431
906.833
.133
.212948
.322839
901.783
900.445
.111
4.986431
.086941
82
81
901.488
900.074
.118
4.964959
.065469
894.551
893.346
.100
.763042
4.855419
81
80
894.230
892.949
.107
.741571
4.833948
886.932
885.982
.079
.560946
.645545
80
79
886.583
885.557
.086
4.539474
4.624073
878.368
878.108
.022
4.380096
4.455766
79
78
877.987
877.654
.028
.358623
.434293
868.511
868.855
.029
.220843
.288376
78
77
868.093
868.366
.023
.199370
.266903
858.217
858.393
.015
.078044
.140807
77
76
857.761
857.866
.009
.056569
.119332
848.064
848.267
.017
3.946416
.004609
76
75
847.570
847.702
.011
3.924939
3.983132
837.822
838.087
.022
.825041
3.879145
75
74
837.289
837.484
.016
3.803560
3.857664
827.389
827.782
.033
3.712840
3.763151
74
73
826.819
827.141
.027
.691355
.741666
816.662
817.223
.047
.609040
.655832
73
72
816.051
816.543
.041
.587550
.634342
805.598
806.327
.061
.512862
.556468
72
71
804.945
805.605
.055
.491366
.534972
794.170
795.094
.077
.423588
.464234
71
70
793.476
794.331
.071
.402085
.442731
782.362
783.472
.093
.340606
.378562
70
69
781.623
782.666
.087
3.319093
3.357049
770.175
771.478
.109
3.263312
3.298795
69
68
769.392
770.627
,103
.241788
.277271
757.610
759.099
.124
.191188
.224411
68
67
756.778
758.202
.119
.169651
.202874
744.675
746.346
.139
.123751
.154908
67
66
743.796
745.401
.134
.102199
.133356
731.379
733.231
.154
.060576
.089832
66
65
730.450
732.237
.149
.039006
.068262
717.740
719.753
.168
.001263
.028800
65
64
716.761
718.709
.162
2.979672
3.007209
703.787
705.953
.180
2.945431
2.971395
64
63
702.756
704.859
.175
.923818
2.949782
689.527
691.846
.193
.892777
.917290
63
62
688.443 690.699
.188
.871137
.895650
674.985
677.434
.204
.843004
.866208
62
61
673.848 676.233
.199
.821333
.844537
660.196
662.763
.214
.795822
.817829
61
60
659.002 661.508
.209
.774117
.796124
645.177
647.854
.223
.750995
.771906
60
59
643.928 646.543
.218
2.729253
2.750164
629.959
632.726
.231
2.708297
2.728215
59
58
628.654 631.359
.225
.686513
.706431
614.566
617.419
.238
.667526
.686527
58
57
613.203 615.995
.233
.645696
.664697
599.013
601.939
.244
.628495
.646676
57
56
597.591 600.456
.239
.606615
.624796
583.365
586.313
.246
.591028
.608450
56
55
581.885,584.773
.241
.569092
.586514
567.644
570.642
.250
.554976
.571714
55
54
566.106 569.043
.245
2.532980
2.549718
551.849
554.872
.252
2.520197
2.536306
54
53
550.252 553.214
.247
.498136
.514245
536.024
539.053
.252
.486563
.502102
53
52
534.369 537.337
.247
.464430
.479969
520.196
523.221
.252
.453956
.468974
52
51
50
518.481521.446
502.623 505.563
.247
.245
.431745
.399970
.446763
.414519
504.396
488.650
507.397
491.620
.250
.248
.422264
.391388
.436813
.405506
51
50
49
486.818 489.728
.243
2.369004
2.383122
472.979
475.906
.244
2.361244
2.374969
49
48
471.089 473.955
.239
.338763
.352488
457.410
460.278
.239
.331743
.345113
48
47
455.462 458.269
.234
.309158
.322528
441.971
444.769
.233
.302806
.315854
47
46
439.965 442.701
.228
.280110
.293158
426.674
429.398
.227
.274370
.287118
46
45
424.61l|427.274
.222
.251553
.264301
411.542
414.176
.219
.246367
.258849
45
44
409.424 411.995
.214
2.223421
2.235903
396.596
399.135
.212
2.218737
2.230972
44
11.
7T =
9.9535.
10.
7r =
9.7739.
Continued from Page *155, 200
TABLE LXIY.
VALUE OF /' AND LOG / FOR EVERY YEAR AND MONTH OF AGE.
4 PER CENT.
Year
h = 0m.
fm.
l£m.
3im.
4im.
5^m‘
Year
of
Age.
* Dec.
Nov.
Oct.
Sept.
Aug.
July.
of
Age.
x + n.
/' =.961538
.963141
.966346
.969551
.972756
.975962
.979167
x + n.
10
Log/.
1.980143
T. 980989
T. 982675
1.984355
T. 986029
T.987696
1.989357
10
11
.980134
.980980
.982668
.984348
.986022
.987691
.989352
11
12
.980128
.980974
.982662
.984344
.986019
.987687
.989349
12
13
.980118
.980965
.982654
.984336
.986011
.987681
.989344
13
14
.980105
.980953
.982642
.984326
.986002
.987673
.989337
14
15
1.980094
T. 980942
1.982633
1.984317
1.985994
T.987666
1.989331
15
16
.980084
.980932
.982624
.984309
.985988
.987660
.989325
16
17
.980070
.980919
.982612
.984298
.985978
.987651
.989318
17
18
.980055
.980904
.982599
.984286
.985968
.987642
.989310
18
19
.980039
.980889
.982585
.984274
.985956
.987633
.989302
19
20
1.980020
1.980871
T. 982568
T. 984259
1.985943
1.987621
1.989292
20
21
.980000
.980852
.982551
.984244
.985929
.987608
.989281
21
22
.979978
.980831
.982532
.984226
.985914
.987595
.989269
22
23
.979955
.980809
.982512
.984208
.985898
.987580
.989257
23
24
.979928
.980783
.982488
.984187
.985879
.987564
.989243
24
25
1.979901
T. 980758
T. 982465
T. 984166
1.985860
T.987548
1.989229
25
26
.979866
.980724
.982434
.984138
.985835
.987526
.989210
26
27
.979833
.980692
.982406
.984112
.985813
.987505
.989193
27
28
.979795
.980656
.982373
.984083
.985786
.987483
989173
28
29
.979748
.980611
.982332
.984045
.985753
.987454
.989148
29
30
1.979704
1.980569
T. 982294
1.984011
1.985722
T.987426
T.989124
30
31
.979652
.980519
.982249
.983970
.985686
.987395
.989097
31
32
.979598
.980468
.982201
.983928
.985648
.987361
.989068
32
33
.979535
.980407
.982147
.983879
.985604
.987323
.989035
33
34
.979466
.980341
.982087
.983825
.985556
.987280
.988998
34
35
1.979386
T. 980265
1.982017
T. 983762
T. 985500
1,987232
T.988956
35
36
.979307
.980189
.981948
.983700
.985445
.987182
.988914
36
37
.979220
.980106
.981873
.983632
.985384
.987130
.988868
37
38
.979117
.980008
.981783
.983551
.985312
.987066
.988813
38
39
.979006
.979901
.981686
.983464
.985235
.986998
.988754
39
40
1.978883
1.979784
1.981580
1.983368
T. 985149
T.986923
T.988689
40
41
.978757
.979663
.981470
.983269
.985061
.986845
.988622
41
42
.978602
.979515
.981335
.983148
.984953
.986750
.988540
42
43
.978448
.979368
.981201
.983027
.984845
.986656
.988459
43
44
.978268
.979196
.981045
.982886
.984720
.986545
.988364
44
45
1.978073
T. 979009
1.980876
1.982733
1.984583
T.986426
T.988260
45
46
.977861
.978807
.980691
.982567
.984435
.986296
.988149
46
47
.977624
.978580
.980485
.982382
.984271
.986151
.988023
47
48
.977365
.978332
.980260
.982179
.984090
.985992
.987886
48
49
.977082
.978062
.980013
.981957
.983892
.985819
.987736
49
50
T. 976767
T. 977760
1.979740
1.981711
T. 983673
T.985626
T.987570
50
51
.976422
.977430
.979440
.981441
.983432
.985414
.987388
51
52
.976044
.977069
.979112
.981145
.983169
.985183
.987188
52
53
.975630
.976673
.978752
.980821
.982881
.984930
.986970
53
* Months of Entry for Annual Valuation Dec. 31.
X-\-n-\-h=. Present Age in Years and Fraction ; 1—K)vj —h) vpx + n—1. 201
TABLE EXIT.
VALUE OF f AND LOG / FOR EVERY YEAR AND MONTH OF AGE.
4 PER CENT.
Year
6m.
6£m.
7im.
8jm.
9im.
lOijrm.
Him.
Year
of
Age.
June.
May.
April.
March.
Feb.
Jan.
of
Age.
x-\-n.
/'=. 980769
.982372
.985577
.988782
.991987
.995192
.998397
x-\-n.
Log/.
10
T.990185
1.991011
1.992660
1.994302
1.995937
1.997567
1.999190
10
11
.990180
.991007
.992656
.994300
.995936
.997566
.999190
11
12
.990178
.991004
.992654
.994298
.995934
.997565
.999190
12
13
.990173
.991000
.992651
.994294
.995932
.997564
.999189
13
14
.990167
.990995
.992646
.994291
.995930
.997562
.999189
14
15
T.990161
1.990990
1.992642
1.994288
1.995928
1.997561
1.999188
15
16
.990156
.990985
.992638
.994285
.995926
.997560
.999188
16
17
.990149
.990979
.992633
.994281
.995923
.997558
.999187
17
18
.990142
.990972
.992628
.994277
.995919
.997556
.999187
18
19
.990135
.990965
.992622
.994272
.995917
.997555
.999186
19
20
T.990125
1.990957
1.992614
1.994267
1.995912
1.997552
1.999185
20
21
.990115
.990948
.992608
.994261
.995908
.997550
.999185
21
22
.990104
.990938
.992600
.994255
.995904
.997547
.999184
22
23
.990093
.990928
.992591
.994249
.995900
.997545
.999183
23
24
.990080
.990916
.992581
.994241
.995894
.997541
.999182
24
25
T.990067
1.990903
1.992571
1.994234
1.995889
1.997538
1.999181
25
26
.990050
.990888
.992559
.994224
.995882
.997534
.999179
26
27
.990033
.990873
.992547
.994214
.995875
.997530
.999178
27
28
.990015
.990856
.992533
.994204
.995868
.997525
.999176
28
29
.989992
.990835
.992516
.994191
.995858
.997519
.999175
29
30
T.989971
1.990815
1.992499
1.994178
1.995849
1.997514
1.999173
30
31
.989945
.990792
.992481
.994163
.995839
.997508
.999171
31
32
.989919
.990768
.992461
.994148
.995828
.997501
.999169
32
33
.989888
.990740
.992439
.994130
.995815
.997494
.999166
33
34
.989855
.990709
.992413
.994110
.995802
.997486
.999163
34
35
T.989816
1.990673
1.992384
1.994088
1.995785
1.997476
1.999160
35
36
.989777
.990638
.992355
.994065
.995769
.997466
.999157
36
37
.989735
.990599
.992324
.994041
.995752
.997456
.999154
37
38
.989684
.990553
.992286
.994012
.995732
.997444
.999150
38
39
.989630
.990504
.992246
.993981
.995709
.997431
.999146
39
40
1.989570
1.990449
1.992201
1.993946
1.995684
1.997416
1.999140
40
41
42
43
.989509
.989433
•989358
.990392
.990323
.990254
.992155
.992099
.992043
.993911
.993867
.993824
.995660
.995628
.995597
.997401
.997382
.997364
.999135
.999129
.999123
41
42
43
44
.989270
.990175
.991977
.993773
.995562
.997343
.999116
44
45
1.989175
1.990088
1.991907
1.993718
1.995522
1.997319
1.999108
45
46
.989071
.989993
.991829
.993659
.995480
.997293
.999099
4G
47
.988956
.989887
.991743
.993592
.995432
.997265
.999090
47
48
.988830
.989772
.991649
.993519
.995381
.997234
.999080
48
49
.988692
.989646
.991547
.993440
.995324
.997200
.999069
49
50
T.988539
1.989506
1.991433
1.993351
1.995261
1.997164
1.999057
50
51
.988371
.989352
.991308
.993254
.995192
.997122
.999042
51
52
.988187
.989184
.991170
.993149
.995117
.997077
.999028
52
53
.987986
.989000
.991021
.993032
.995035
.997028
.999012
53 202
TABLE LXIY.
VALUE OF /' AND LOG / FOR EVERY YEAR AND MONTH OF AGE.
Year
of
Age.
h = 0m.
lm-
i£m-
2|-m.
3im-
4fm.
Year
of
Age.
* Dec.
Nov.
Oct.
Sept.
Aug.
J uly.
x-\-n.
/ =.961538
.963141
.966346
.969551
.972756
.975962
.979167
x + n.
Log/.
54
1.975175
1.976238
T. 978357
1.980465
1.982563
T.984651
1.986729
54
55
1.974667
1.975752
T. 977916
1.980068
1.982210
1.984341
T.986462
55
56
.974117
.975227
.977438
.979638
.981827
.984005
.986172
56
57
.973500
.974637
.976903
.979156
.981397
.983628
.985847
57
58
.972835
.974001
.976325
.978636
.980935
.983222
.985497
58
59
.972102
.973301
.975689
.978064
.980426
.982775
.985111
59
60
1.971290
T. 972525
T. 974984
T. 977430
1.979861
1.982280
1.984685
60
61
.970397
.971672
.974210
.976733
.979242
.981736
.984216
61
62
.969425
.970743
.973367
.975975
.978568
.981144
.983706
62
63
.968344
.969710
.972430
.975132
.977818
.980487
.983140
63
64
.967161
.968580
.971404
.974210
.976998
.979768
.982521
64
65
1.965853
1.967331
T. 970271
1.973192
1.976093
1.978974
1.981838
65
66
.964427
.965969
.969036
.972082
.975107
.978110
.981093
66
67
.962842
.964456
.967664
.970850
.974012
.977152
.980268
67
68
.961128
.962819
.966181
.969518
.972829
.976116
.979377
68
69
.959205
.960983
.964519
.968026
.971504
.974956
.978379
69
70
1.957120
T. 958994
1.962718
1.966409
1.970070
T.973700
T.977300
70
71
.954811
.956791
.960724
.964621
.968484
.972312
.976108
71
72
.952281
.954378
.958541
.962664
.966749
.970796
.974805
72
73
.949508
.951734
.956150
.960522
.964851
.969137
973381
73
74
.946420
.948789
.953490
.958140
.962742
.967295
.971800
74
75
T. 943060
T. 945588
T. 950599
T. 955554
1.960453
T.965297
1.970086
75
76
.939374
.942077
.947432
.952722
.957947
.963111
.968215
76
77
.935277
.938175
.943915
.949579
.955171
.960692
.966143
77
78
.930801
.933915
.940078
.946154
.952148
.958059
.963891
78
79
.925875
.929230
.935863
.942396
.948832
.955174
.961425
79
80
T. 920451
T. 924073
1.931229
T. 938268
T. 945195
T.952013
1.958726
80
81
.914509
.918428
.926162
.933761
.941228
.948569
.955790
81
82
.907916
.912169
.920552
.928777
.936848
.944773
.952554
82
83
.900757
.905378
.914475
.923386
.932117
.940677
.949070
83
84
.892777
.897815
.907720
.917403
.926875
.936145
.945221
84
85
T. 884080
1.889581
T. 900378
T. 910913
T. 921200
T.931247
1.941068
85
86
.874593
.880609
.892397
.903873
.915054
.925954
.936587
86
87
.863845
.870458
.883388
.895945
.908148
.920018
.931573
87
88
.852462
.859722
.873887
.887604
.900902
.913804
.926334
88
89
.832633
.841059
.857435
.873216
.888444
.903156
.917386
89
90
T. 810236
1.820041
T. 839009
T.857185
1.874629
T.891400
T.907548
90
91
.790136
.801236
.822619
.842998
.862464
.881095
.898959
91
92
.768484
.781043
.805122
.827936
.849610
.870255
.889961
92
93
.742635
.757028
.784456
.810256
.834607
.857666
.879562
93
94
.713385
.729978
.761375
.790655
.818085
.843885
.868237
94
95
T. 681936
T. 701052
T. 736932
T. 770072
T. 800862
T.829612
1.856578
95
96
.627579
.651472
.695641
.735731
.772430
.806268
.837660
96
97
.585026
.613055
.664207
.709965
.751358
.789146
.823908
97
98
.505846
.542608
.607936
! .664711
.714914
.759910
.800679
98
4 PER CENT.
* Months of Entry for Annual Valuation Dec. 31. 203
TABLE LXIV.
VALUE OF /' AND LOG / FOR EVERY YEAR AND MONTH OF AGE.
Year
of
Age.
x-\-n.
6m.
6^m.
7-J-m.
8|rm.
9im-
lOim.
ll|m.
Year
of
Age.
x+n.
June.
May.
April.
March.
Feb.
Jan.
/'=. 980769
.982372
.985577
.988782
.991987
.995192
.998397
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
Log/.
1.987765
1.987518
.987252
.986952
.986630
.986275
T.985882
.985451
.984982
.984461
.983891
T.983262
.982578
.981819
.980999
.980082
T.979089
.977993
.976796
.975487
.974036
1.972463
.970744
.968843
.966777
.964517
T.962044
.959354
.956394
.953208
.949689
T.945896
.941807
.937237
•932467
.924330
1.915401
.907623
.899490
.890109
.879920
T.869458
.852543
.840299
.819713
1.988798
T. 988572
.988328
.988055
.987760
.987435
T. 987076
.986682
.986253
.985777
.985256
T. 984682
.984057
.983363
.982614
.981777
T. 980871
.979870
.978777
.977584
.976260
1.974825
.973259
.971526
.969645
.967587
T. 965337
.962891
.960200
.957305
.954111
T. 950671
.946966
.942827
.938513
.931164
1.923115
.916118
.908813
.900407
.891297
T. 881965
.866934
.856093
.837947
1.990856
T.990673
.990474
.990251
.990011
.989747
T. 989455
.989134
.988785
.988397
.987974
T. 987507
.986999
.986436
.985827
.985147
T. 984412
.983600
.982714
.981746
.980674
1.979512
.978245
.976844
.975324
.973663
1.971849
.969878
.967713
.965386
.962824
T.960067
.957103
.953798
.950360
.944518
T. 938145
.932624
.926880
.920297
.913193
T.905951
.894359
.886056
.872269
1.992905
1.992763
.992609
.992437
.992251
.992047
T. 991821
.991573
.991302
.991003
.990676
T.990315
.989922
.989486
.989017
.988492
T. 987924
.987299
.986615
.985870
.985044
1.984149
.983174
.982098
.980930
.979656
T. 978264
.976755
.975098
.973320
.971365
T. 969264
.967008
.964498
.961892
.957475
1.952672
.948525
.944226
.939316
.934038
1.928681
.920155
.914085
.904075
T. 994944
1.994843
.994734
.994611
.994479
.994334
1.994173
.993997
.993805
.993593
.993360
T. 993104
.992825
.992517
.992184
.991812
T. 991409
.990966
.990482
.989954
.989370
T. 988737
.988048
.987288
.986465
.985566
T. 984586
.983524
.982360
.981112
.979741
1.978270
.976693
.974941
.973125
.970056
T. 966729
.963865
.960905
.957536
.953929
T. 950280
.944504
.940414
.933710
1.996973
T.996913
.996848
.996774
.996696
.996609
T.996514
.996408
.996294
.996167
.996028
1.995875
.995710
.995526
.995327
.995105
T. 994866
.994602
.994314
.994000
.993653
1.993277
.992869
.992418
.991929
.991397
1.990817
.990190
.989502
.988766
.987958
T. 987093
.986167
.985139
.984076
.982282
T. 980344
.978682
.976968
.975023
.972948
1.970856
.967560
.965238
.961452
T. 998994
1.998974
.998952
.998928
.998901
.998873
1.998841
.998806
.998768
.998726
.998680
1.998630
.998574
.998514
.998448
.998375
T. 998295
.998208
.998113
.998009
.997894
1.997771
.997636
.997487
.997326
.997151
T. 996961
.996754
.996529
.996287
.996023
T. 995740
.995438
.995103
.994757
.994174
T. 993546
.993009
.992457
.991833
.991168
T. 990501
.989454
.988719
.987527
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
4 PER CENT. 204
TABLE LX V.
JOINT LIVES MALE. CORRECTION TO BE ADDED TO THE YOUNGEST AGE FOR REDUCING
TWO LIVES X, y, AND ALSO THREE LIVES X, y, z TO THE CASE OF TWO EQUAL AGES.
Y. M. denote Years and Months. For Two Lives, the Correction is given in the Second Column only.
For Three Lives, if x be the Youngest Age, enter the Table jointly with y—x, z—x to find the Correc-
tion of the Age x, to give u, that is, one of the two Equal Ages. With u, which increases uniformly
with the age, proceed as before explained on Pages 258, 259.
to’ 2>
to
P
Equal
Ages.
Dxa>
10
100,000.0
2,123,700.7
5.000000
6.327093
55
11,622.5
120,383.7
4.065300
5.080567
11
95,833.6
2,023,700.7
4.981518
.306146
56
10,860.8
108,761.2
.035862
.036474
12
91,836.2
1,927,867.1
.963014
.285077
57
10,123.4
97,900.4
.005325
4.990785
13
88,003.5
1,836,030.9
.944500
.263880
58
9,409.27
87,776.97
3.973556
.943381
14
84,326.7
1,748,027.4
.925965
.242548
59
8,718.77
78,367.70
.940455
.894137
15
80,798.8
1,663,700.7
4.907405
6.221075
60
8,051.71
69,648.93
3.905888
4.842915
16
77,414.3
1,582,901.9
.888821
.199454
61
7,407.95
61,597.22
.869698
.789562
17
74,168.8
1,505,487.6
.870221
.177677
62
6,787.68
54,189.27
.831721
.733913
18
71,054.1
1,431,318.8
.851589
.155737
63
6,191.56
47,401.59
.791800
.675793
19
68,065.5
1,360,264.7
.832927
.133624
64
5,619.78
41,210.03
.749719
.615003
20
65,198.3
1,292,199.2
4.814236
6.111330
65
5,073.05
35,590.25
3.705269
4.551331
21
62,446.2
1,227,000.9
.795506
.088845
66
4,552.02
30,517.20
.658204
.484545
22
59,804.5
1,164,554.7
.776734
.066160
67
4,057.79
25,965.18
.608289
.414391
23
57,269.2
1,104,750.2
.757921
.043264
68
3,590.89
21,907.39
.555202
.340591
24
54,835.1
1,047,481.0
.739059
.020146
69
3,152.73
18,316.50
.498687
.262843
25
52,498.3
992,645.9
4.720145
5.996794
70
2,743.64
15,163.77
3.438327
4.180807
26
50,254.7
940,147.6
.701177
.973196
71
2,364.81
12,420.13
.373796
.094127
27
48,099.2
889,892.9
.682138
.949337
72
2,016.73
10,055.32
.304647
.002396
28
46,029.1
841,793.7
.663032
.925206
73
1,699.97
8,038.59
.230440
3.905180
29
44,040.7
795,764.6
.643854
.900785
74
. 1,414.76
6,338.62
.150684
.801994
30
42,128.7
751,723.9
4.624578
5.876058
75
1,160.79
4,923.86
3.064753
3.692305
31
40,291.7
709,595.2
.605216
.851011
76
937.787
3,763.069
2.972104
.575542
32
38,525.6
669,303.5
.585749
.825623
77
744.871
2,825.282
.872081
.451062
33
36,827.7
630,777.9
.566175
.799877
78
580.583
2,080.411
.763864
.318149
34
35,194.4
593,950.2
.546473
.773750
79
443.298
1,499.828
.646696
.176041
35
33,623.0
558,755.8
4.526637
5.747222
80
330.883
1056.530
2.519675
3.023882
36
32,109.7
525,132.8
.506636
.720269
81
240.883
725.647
.381806
2.860725
37
30,653.6
493,023.1
.486481
.692867
82
170.630
484.764
.232055
.685531
38
29,251.6
462,369.5
.466150
.664990
83
117.251
314.134
.069115
.497115
39
27,900.6
433,117.9
.445613
.636606
84
77.9575
196.8833
1.891858
.294209
40
26,598.4
405,217.3
4.424856
5.607688
85
49.9624
118.9258
1.698643
2.075276
41
25,342.5
378,618.9
.403850
.578203
86
30.7632
68.9634
.488032
1.838618
42
24,132.0
353,276.4
.382593
.548115
87
18.1321
38.2002
.258447
.582065
43
22,963.0
329,144.4
.361029
.517387
88
10.1711
20.0681
.007367
.302506
44
21,835.0
306,181.4
.339153
.485979
89
5.41405
9.89699
0.733522
0.995504
45
20,745.2
284,346.4
4.316917
5.453848
90
2.63036
4.48294
0.420015
0.651563
46
19,692.3
263,601.2
.294296
.420947
91
1.15271
1.85258
.061718
.267777
47
18,674.3
243,908.9
.271245
.387228
92
.460488
.699874
1.663218
1.845020
48
17,689.8
225,234.6
.247723
.352635
93
.166501
.239386
.221417
.379099
49
16,737.2
207,544.8
.223683
.317112
94
.053446
.072885
2.727916
2.862638
50
15,815.3
190,807.6
4.199077
5.280595
95
.0149939
.0194392
2.175915
2.288678
51
14,922.4
174,992.3
.173838
.243019
96
.0036393
.0044453
3.561017
3.647901
52
14,057.7
160,069.9
.147913
.204310
97
.0006877
.0008060
4.837404
4.906335
53
13,220.0
146,012.2
.121230
.164389
98
.0001068
.0001183
.028687
.072985
54
12,408.5
132,792.2
.093720
.123172
99
.0000115
.0000115
5.061608
5.060698
Formula, Dot = -•
l10l 207
TABLE LXYII.
COMMUTATION COLUMNS FOR TWO EQUAL AGES. MALE LIFE.
PER CENT.
Equal
Ages.
D**.
N**.
AD**.
AN**.
Equal
Ages.
D**.
Not*
AD**.
AN**.
10
100,000.0
1,961,020.2
5.000000
6.292482
55
9,346.69
93,597.81
3.970658
4.971266
11
95,370.3
1,861,020.2
4.979413
.269751
56
8,691.95
84,251.12
.939117
.925576
12
90,950.9
1,765,649.9
.958807
.246904
157
8,062.64
75,559.17
.906477
.878287
13
86,733.9
1,674,699.0
.938189
.223937
58
7,457.70
67,496.53
.872605
.829281
14
82,708.9
1,587,965.1
.917552
.200841
59
6,877.02
60,038.83
.837400
.778432
15
78,865.7
1,505,256.2
4.896888
6.177610
60
6,320.19
53,161.81
3.800730
4.725599
16
75,197.1
1,426,390.5
.876201
.154238
61
5,786.79
46,841.62
.762438
.670632
17
71,696.4
1,351,193.4
.855497
.130718
62
5,276.63
41,054.83
.722357
.613364
18
68,353.9
1,279,497.0
.834763
.107039
63
4,789.97
35,778.20
.680333
.553618
19
65,162.5
1,211,143.1
.813998
.083196
64
4,326.61
30,988.23 •
.636148
.491197
20
62,116.1
1,145,980.6
4.793204
6.059178
65
3,886.83
26,661.62
3.589596
4.425887
21
59,206.6
1,083,864.5
.772370
.034975
66
3,470.79
22,774.79
.540428
.357454
22
56,428.0
1,024,657.9
.751495
.010579
67
3,079.00
19,304.00
.488410
.285647
23
53,774.8
968,229.9
.730579
5.985979
68
2,711.56
16,225.00
.433219
.210185
24
51,240.7
914,455.1
.709615
.961163
69
2,369.20
13,513.44
.374601
.130766
25
48,819.8
863,214.4
4.688596
5.936118
70
2,051.82
11,144.24
3.312139
4.047050
26
46,507.8
814,394.6
.667526
.910835
71
1,759.97
9,092.42
.245504
3.958680
27
44,298.0
767,886.8
.646384
.885297
72
1,493.66
7,332.45
.174252
.865249
28
42,186.7
723,588.8
.625175
.859492
73
1,252.97
5,838.79
.097942
.766323
29
40,169.1
681,402.1
.603892
.833403
74
1,037.73
4,585.82
.016083
.661417
30
38,239.7
641,233.0
4.582514
5.807016
75
847.323
3,548.091
2.928049
3.549995
31
36,395.5
602,993.3
.561048
.780312
76
681.234
2,700.768
.833296
.431487
32
34,632.2
566,597.8
.539480
.753275
77
538.481
2,019.534
.731170
.305251
33
32,946.0
531,965.6
.517802
.725883
78
417.686
1,481.053
.620850
.170570
34
31,332.6
499,019.6
.495997
.698118
79
317.380
1,063.367
.501579
.026683
35
29,789.1
467,687.0
4.474057
5.669955
80
235.752
745.987
2.372455
2.872731
36
28,310.9
437,897.9
.451954
.641372
81
170.798
510.235
.232483
.707771
37
26,896.5
409,587.0
.429696
.612346
82
120.400
339.437
.080628
.530759
38
25,542.4
382,690.5
.407262
.582848
83
82.3353
219.0374
1.915586
.340518
39
24,244.9
357,148.1
.384621
.552849
84
54.4786
136.7021
.736226
.135776
40
23,001.8
332,903.2
4.361761
5.522318
85
34.7462
82.2235
1.540907
1.914996
41
21,809.9
309,901.4
.338653
.491224
86
21.2909
47.4773
.328193
.676486
42
20,667.7
288,091.5
.315292
.459530
87
12.4884
26.1864
.096505
.418076
43
19,571.5
267,423.8
.291624
.427200
88
6.97143
13.69801
0.843322
.136657
44
18,520.2
247,852.3
.267646
.394193
89
3.69296
6.72658
.567374
0.827795
45
17,510.8
229,332.1
4.243307
5.360465
90
1.78552
3.03362
0.251764
0.481961
46
16,541.7
211,821.3
.218581
.325970
91
.778687
1.248099
1.891363
.096249
47
15,610.9
195,279.6
.193429
.290657
92
.309572
.469412
.490761
1.671554
48
14,716.4
179,668.7
.167802
.254473
93
.111393
.159840
.046857
.203685
49
13,856.7
164,952.3
.141660
.217358
94
.035584
.048447
2.551252
2.685267
50
13,030.2
151,095.6
4.114950
5.179252
95
.0099346
.0128629
3.997148
2.109339
51
12,235.1
138,065.4
.087609
.140085
96
.0023997
.0029283
.380147
3.466616
52
11,470.5
125,830.3
.059581
.099785
97
.0004513
.0005286
4.654431
4.723127
53
10,734.8
114,359.8
.030795
.058273
98
.0000698
.0000773
5.843611
5.888179
54
10,027.2
103,625.0
.001180
.015465
99
.0000075
.0000075
6.874428
6.875061 208
TABLE LXVIII.
COMMUTATION COLUMNS FOR TWO EQUAL AGES. MALE LIFE.
4 PER CENT.
Equal 1
Ages.
D*a>
Na%»
xx•
1
Equal
Ages.
Da*.
Ns*.
10
100.000
1,819,554.9
5.000000
6.259966
55
7,524.42
72,918.61
3.876473
4.862838
11
94,911.8
1,719,554.9
4.977320
.235416
56 6,963.70
65,394.19
.842840
.815539
12
90,078.5
1,624,643.1
.954621
.210758
57 6,428.46
58,430.49
.808107
.766639
13
85,488.9
1,534,564.6
.931910
.185985
5815,917.54
52,002.03
.772141
.716029
14
81,129.7
1,449,075.7
.909180
.161091
59
5,430.55
46,084.49
.734844
.663555
15
76,988.0
1,367,946.0
4.886423
6.136069
60
4,966.85
40,653.94
3.696081
4.609103
16
73,054.0
1,290,958.0
.863644
.110912
61
4,525.80
35,687.09
.655695
.552511
17
69,318.0
1,217,904.0
.840846
.085613
62 4,106.97
31,161.29
.613522
.493615
18
65,768.7
1,148,586.0
.818019
.060164
63
3,710.27
27,054.32
.569405
.432237
19
62,396.8
1,082,817.3
.795162
.034555
64
3,335.24
23,344.05
.523127
.368176
20
59,193.5
1,020,420.5
4.772274
6.008779
65
2,981.82
20,008.81
3.474482
4.301221
21
56,149.6
961,227.0
.749347
5.982826
66
2,649.85
17,026.99
.423221
.231138
22
53,257.4
905,077.4
.726380
.956686
67
2,339.42
14,377.14
.369109
.157673
23
50,509.1
851,820.0
.703370
.930348
68
2,050.34
12,037.72
.311826
.080544
24
47,897.5
801,310.9
.680313
.903801
69
1,782.85
9,987.38
.251115
3.999451
25
45,415.3
753,413.4
4.657202
5.877033
70
1,536.59
8,204.53
3.186559
3.914054
26
43,056.4
707,998.1
.634038
.850032
71
1,311.69
6,667.94
.117832
.823992
27
40,813.4
664,941.7
.610803
.822783
72
1,107.87
5,356.25
.044487
.728861
28
38,681.4
624,128.3
.587502
.795274
73
924.875
4,248.381
2.966083
.628223
29
36,654.4
585,446.9
.564126
.767488.
74
762.311
3,323.506
.882132
.521597
30
34,726.0
548,792.5
4.540655
5.739408
75
619.448
2,561.195
2.792005
3.408443
31
32,892.4
514,066.5
.517096
.711019
76
495.632
1,941.747
.695159
.288193
32
31,148.3
481,174.1
.493434
.682303
77
389.888
1,446.115
.590940
.160203
33
29,489.2
450,025.8
.469663
.653237
78
300.973
1,056.227
.478527
.023757
34
27,910.4
420,536.6
.445766
.623804
79
227.595
755.254
.357163
2.878093
35
26,407.8
392,626.2
4.421732
5.5939791
80
168.247
527.659
2.225946
2.722353
36
24,976.8
366,218.4
397537
.563740
81
121.306
359.412
.083881
.555593
37
23,614.8
341,241.6
.373185
.533062
82
85.1007
238.1057
1.929933
.376770
38
22,318.1
317,626.8
.348658
.501918
83
57.9159
153.0050
.762798
.184706
39
21,082.6
295,308.7
.323925
.470276
84
38.1369
95.0891
.581345
1.978131
40
19,905.4
274,226.1
4.298971
5.438109
85
24.2066
56.9522
1.383933
1.755511
41
18,783.2
254,320.7
.273770
.405392
86
14.7613
32.7456
.169126
.515153
42
17,714.0
235,537.5
.248317
.372060
87
8.61678
17.98425
0.935345
.254892
43
16,693.8
217,823.5
.222555
.338105
88
4.78706
9.36747
.680069
0.971622
44
15,721.1
201,129.7
.196484
.303476
89
2.52364
4.58041
.402027
.660904
45
14,792.9
185,408.6
4.170053
5.268130
90
1.21430
2.05677
0.084325
0.313186
46
13,907.0
170,615.7
.143233
.232019
91
.52702
.84247
1.721831
1.925554
47
13,061.3
156,708.7
.115988
.195093
92
.20851
.31545
.319136
.498930
48
12,253.8
143,647.4
.088269
.157298
93
.07467
.10694
2.873138
.029140
49
11,482.4
131,393.6
.060033
.118574
94
.023738
.032266
.375441
2.508745
50
10,745.6
119,911.2
4.031230
5.078860
95
.0065954
.008528
3.819244
3.930847
51
10,041.5
109,165.6
.001797
.038086
96
.0015854
.0019326
.200150
.286152
52
9,368.61
99,124.1
3.971675
4.996179
97
.0002967
.0003472
4.472341
4.540637
53
8,725.61
89,755.4
.940796
.953061
98
.0000456
.0000505
5.659427
5.703515
54
8,111.27
81,029.8
.909089
.908645
99
.0000049
.0000049
6.688152
6.688153
Formula,
TOV Ao 209
TABLE LXIX.
COMMUTATION COLUMNS FOR TWO EQUAL AGES. MALE LIFE.
4i PER CENT.
Equal
i Ages.
Dxx-
Equal
Ages.
T)xx-
10
100,000
1,695,824.4
5.000000
6.229381
55
6,063.75
56,920.00
3.782741
4.755265
11
94,457.8
1,595,824.4
4.975238
.202985
56
5,585.02
50,856.25
.747025
.706344
12
89,218.7
1,501,366.6
.950456
.176487
57
5,131.07
45,271.23
.710208
.655822
13
84,267.7
1,412,147.9
.925661
.149880
58
4,700.67
40,140.16
.672160
.603579
14
79,588.3
1,327,880.2
.900849
.123159
59
4,293.19
35,439.49
.632780
.549487
15
75,163.8
1,248,291.9
4.876009
6.096316
60
3,907.82
31,146.30
3.591934
4.493407
16
70,981.6
1,173,128.1
.851146
.069345
61
3,543.76
27,238.48
.549465
.435183
17
67,029.5
1,102,146.5
.826266
.042239
62
3,200.44
23,694.72
.505209
.374652
18
63,293.0
1,035,117.0
.801356
.014989
63
2,877.46
20,494.28
.459009
.311633
19
59,760.6
971,824.0
.776415
5.987588
64
2,574.23
17,616.82
.410648
.245927
20
56,421.6
912,063.4
4.751445
5.960025
65
2,290.45
15,042.59
3.359920
4.177323
21
53,264.1
855,641.8
.726435
.932292
66
2,025.70
12,752.14
.306576
.105583
22
50,278.7
802,377.7
.701384
.904379
67
1,779.84
10,726.44
.250382
.030456
23
47,456.1
752,099.0
.676292
.876275
68
1,552.44
8,946.60
.191015
3.951658
24
44,787.0
704,642.9
.651152
.847969
69
1,343.45
7,394.16
.128221
.868889
25
42,262.8
659,855.9
4.625958
5.819449
70
1,152.34
6,050.71
3.061582
3.781806
26
39,875.9
617,593.1
.000711
.790702
71
978.976
4,898.367
2.990772
.690051
27
37,617.8
577,717.2
.575393
.701715
72
822.894
3,919.391
.915344
.593219
28
35,482.1
540,099.4
.550009
.732474
73
683.688
3,096.497
.834858
.490871
29
33,461.9
504,617.3
.524551
.702962
74
560.821
2,412.809
.748824
.382523
30
31,549.8
471,155.4
4.498996
5.673164
75
453.537
1,851.988
2.656613
3.267638
31
29,740.9
439,605.6
.473354
.643063
76
361.148
1,398.451
.557685
.145647
32
28,029.2
409,864.7
.447610
.612640
77
282.737
1,037.303
.451383
.015906
33
26,409.2
381,835.5
.421755
.581876
78
217.213
754.566
.336886
2.877697
34
24,875.7
355,426.3
.395775
.550750
79
163.471
537.353
.213440
.730260
35
23,423.9
330,550.6
4.369659
5.519238
80
120.265
373.882
2.080140
2.572735
36
22,048.6
307,126.7
.343381
.487317
81
86.2963
253.6169
1.935992
.404178
37
20,746.6
285,078.1
.316946
.454964
82
60.2506
167.3206
.779961
.223550
38
19,513.5
264,331.5
.290336
.422149
83
40.8078
107.0700
.610743
.029668
39
18,345.1
244,818.0
.263520
.388843
84
26.7428
66.2622
.427207
1.821266
40
17,237.9
226,472.9
4.236483
5.355016
85
16.8932
39.5194
1.227712
1.596810
41
16,188.2
209,235.0
.209199
.320634
86
10.2523
22.6262
.010822
.354612
42
15,193.7
193,046.8
.181063
.285663
87
5.95604
12.37393
0.774958
.092508
43
14,250.1
177,853.1
.153819
.250061
88
3.29306
6.41789
.517599
0.807392
44
13,355.6
163,603.0
.125664
.213791
89
1.727726
3.124835
.237475
.494826
45
12,506.9
150,247.4
4.097150
5.176807
90
.8273495
1.3971089
T.917689
0.145230
46
11,701.7
137,740.5
.068248
.139062
91
.3573668
.5697594
.553113
1.755692
47
10,937.5
126,038.8
.038919
.100504
92
.1407129
.2123926
.148334
.327139
48
10,212.1
115,101.3
.009117
.061080
93
.0501480
.0716797
2.700254
2.855396
49
9,523.53
104,889.19
3.978798
.020731
94
.0158662
.0215317
.200474
.333078
50
8,869.78
95,365.06
3.947913
4.979392
95
.0043873
.0056654
3.642193
3.753234
51
8,248.90
86,495.88
.910396
.936995
96
.0010496
.0012782
.021017
.106595
52
7,659.35
78,246.98
.884192
.893468
97
.0001954
.0002286
4.291125
4.359084
53
7,099.54
70,587.63
.851230
.848729
98
.0000299
.0000331
5.476129
5.520012
54
6,568.09
63,488.09
.817439
.802692
99
.0000031
.0000032
6.502770
6.502770 210
TABLE LXX.
COMMUTATION COLUMNS FOR TWO EQUAL AGES. MALE LIFE.
5 PER CENT.
Equal
Ages.
Na;a:«
3-DXX'
3-NaXC.
Equal
Ages.
Dxa;-
Naze-
3-Naxr*
10
100,000
1,587,002.3
5.000000
6.200578
55
4,891.66
44,516.96
3.689456
4.648526
11
94,008.0
1,487,002.3
4.973165
.172312
,56
4,484.00
39,625.30
.651666
.597973
12
88,369.2
1,392,994.3
.946309
.143949
57
4,099.94
35,141.30
.612777
.545818
13
83,069.6
1,304,625.1
.919442
.115486
58
3,738.14
31,041.36
.572656
.491941
14
78,083.1
1,221,555.5
.892557
.086913
59
3,397.83
27,303.22
.531202
.436214
15
73,391.2
1,143,472.4
4.865644
6.058226
60
3,078.10
23,905.39
3.488283
4.378496
16
68,977.6
1,070,081.2
.838708
.029417
61
2,778.06
20,827.29
.443742
.318633
17
64,826.9
1,001,103.6
.811755
.000479
62
2,496.97
18,049.23
.397413
.256459
18
60,921.7
936,276.7
.784772
5.971404
63
2,234.29
15,552.26
.349139
.191792
19
57,247.7
875,355.0
.757758
.942184
64
1,989.33
13,317.97
.298706
.124438
20
53,791.7
818,107.3
4.730715
5.912810
65
1,761.59
11,328.64
3.245905
4.054178
21
50,539.6
764,315.6
.703632
.883273
66
1,550.55
9,567.05
.190487
3.980778
22
47,479.7
713,776.0
.676508
.853562
67
1,355.88
8,016.50
.132220
.903985
23
44,600.8
666,296.3
.649343
.823667
68
1,177.01
6,660.62
.070781
.823515
24
41,891.9
621,695.5
.622130
.793578
69
1,013.71
5,483.61
.005913
.739067
25
39,342.6
579,803.6
4.594863
5.763281
70
865.370
4,469.901
2.937202
3.650298
26
36,943.9
540,461.0
.567543
.732764
71
731.675
3,604.531
.864318
.556849
27
34,685.8
503,517.1
.540152
.702014
72
612.094
2,872.856
.786818
.458314
28
32,560.8
468,831.3
.512695
.6710171
73
506.125
2,260.762
.704258
.354255
29
30,560.7
436,270.5
.485163
.639756
74
413.191
1,754.637
.616151
.244187
30
28,677.1
405,709.8
4.457536
5.608215
75
332.558
1,341.446
2.521868
3.127573
31
26,904.3
377,032.7
.429821
.576379'
76
263.552
1,008.888
.420866
.003843
32
25,235.0
350,128.4
.402003
.544227
77
205.348
745.336
.312491
2.872352
33
23,663.3
324,893.4
.374076
.511741;
78
157.008
539.988
.195922
.732384
34
22,183.1
301,230.1
.346023
.478898i
79
117.599
382.980
.070402
.583176
35
20,789.0
279,047.0
4.317834
5.445677
80
86.1051
265.3809
1.935029
2.423870
36
19,475.2
258,258.0
.289482
.412054
81
61.4905
179.2758
.788808
.253522
37
18,237.9
238,782.8
.260975
.378003
82
42.7273
117.7853
.630705
.071091
38
17,072.3
220,544.9
.232292
.343497
83
28.8014
75.0580
.459413
1.875397
39
15,973.9
203,472.6
.203402
.308506,
84
18.7847
46.2566
.273804
.665174
40
14,938.0
187,498.7
4.174293
5.272998
85
11.8096
27.4719
1.072237
1.438889
41
13,961.6
172,560.7
.144936
.236942
86
7.13301
15.66231
0.853273
.194856
42
13,041.5
158,599.1
.115326
.200301
87
4.12417
8.52930
.615336
0.930913
43
12,173.3
145,557.6
.085409
.163035
88
2.26937
4.40513
.355905
.643959
44
11,354.9
133,384.3
.055182
.125104
89
1.18497
2.13576
.073707
.329552
45
10,582.7
122,029.4
4.024595
5.086464
90
.564739
.950788
T.751848
1.978084
46
9,854.15
111,446.70
3.993619
.047067
91
.242772
.386049
.385199
.586642
47
9,166.80
101,592.55
.962218
.006858
92
.0951367
.1432769
2.978348
.156176
48
8,518.11
92,425.75
.930343
4.965793
93
.0337438
.0481402
.528194
2.682508
49
7,905.89
83,907.64
.897951
.923801
94
.0106253
.0143964
.026341
.158254
50
7,328.11
76,001.75
3.864992
4.880824
95
.0029241
.0037711
3.465988
3.576466
51
6,782.71
68,673.64
.831403
.836790
96
.0006962
.0008470
4.842738
4.927886
52
6,267.95
61,890.93
.797125
.791627
97
.0001291
.0001508
.110773
.178401
53
5,782.16
55,622.98
.762090
.745254
98
.0000197
.0000217
5.293704
5.337379
54
5,323.86
49,840.82
.726227
.697585
99
.0000021
.0000021
6.318272
6.318272 211
TABLE LXXI.
COMMUTATION COLUMNS FOR TWO EQUAL AGES. MALE LIFE.
6 PER CENT.
Equal
Ages.
Naia;.
Equal
Ages.
Not*
10
100,000.0
1,405,298.1
5.000000
6.147768
55
3,193.08
27,382.81
3.504210
4.437478
11
93,121.3
1,305,298.1
4.969049
.115710
56
2,899.38
24,189.73
.462305
.383631
12
86,711.6
1,212,176.8
.938077
.083566
57
2,626.03
21,290.35
.419299
.328183
13
80,740.8
1,125,465.2
.907093
.051332
58
2,371.71
18,664.32
.375061
.271012
14
75,178.0
1,044,724.4
.876091
.019002
59
2,135.46
16,292.61
.329491
.211991
15
69,994.0
969,546.4
4.845061
5.986569
60
1,916.26
14,157.15
3.282455
4.150976
16
65,164.2
899,552.4
.814009
.954027
61
1,713.16
12,240.89
.233797
.087813
17
60,665.1
834,388.2
.782939
.921368
62
1,525.29
10,527.73
.183351
.022335
18
56,472.8
773,723.1
.751839
.888585
63
1,351.95
9,002.44
.130962
3.954360
19
52,566.6
717,250.3
.720710
.855671
64
1,192.37
7,650.49
.076412
.883689
20
48,927.2
664,683.7
4.689550
5.822615
65
1,045.91
6,458.12
3.019494
3.810106
21
45,535.5
615,756.5
.658350
.789409
66
911.927
5,412.212
2.959960
.733375
22
42,375.0
570,221.0
.627110
.756043
67
789.907
4,500.285
.897576
.653240
23
39,430.1
527,846.0
.595828
.722507
68
679.235
3,710.378
.832020
.569418
24
36,685.8
488,415.9
.564498
.688790
69
579.477
3,031.143
.763036
.481606
25
34,128.3
451,730.1
4.533114
5.654879
70
490.013
2,451.666
2.690208
3.389461
26
31,745.3
417,601.8
.501679
.620762
71
410.402
1,961.653
.613209
.292622
27
29,523.7
385,856.5
.470171
.586426
72
340.088
1,551.251
.531591
.190682
28
27,453.5
356,332.8
.438597
.551856
73
278.558
1,211.163
.444915
.083203
29
25,524.0
328,879.3
.406948
.517036
74
225.264
932.605
.352691
2.969698
30
23,724.9
303,355.3
4.375205
5.481952
75
179.594
707.341
2.254291
2.849629
31
22,048.2
279,630.4
.343373
.446584
76
140.985
527.747
.149173
.722426
32
20,485.1
257,582.2
.311439
.410916
77
108.813
386.762
.036681
.587444
33
19,028.1
237,097.1
.279395
.374926
78
82.4131
277.9488
1.915996
.443965
34
17,669.6
218,069.0
.247226
.338594
79
61.1449
195.5357
.786360
.291226
35
16,402.9
200,399.4
4.214920
5.301896
80
44.3476
134.3908
1.646870
2.128370
36
15'221.3
183,996.5
.182452
.264810
81
31.3713
90.0432
.496532
1.954451
37
14,119.8
168,775.2
.149828
.227309
82
21.5930
58.6719
.334312
.768430
38
13,092.7
154,655.4
.117028
.189365
83
14.4180
37.0789
.158904
.569127
39
12; 134.5
141,562.7
.084022
.150949
84
9.314896
22.660866
0.969178
.355277
40
11,240.8
129,428.2
4.050796
5.112029
85
5.800895
13.345970
0.763495
1.125350
41
10,406.9
118,187.4
.017323
.072571
86
3.470683
7.545075
.540415
0.877664
42
9'629.35
107,780.51
3.983597
.032540
87
1.987747
4.074392
.298361
.610063
43
8,903.55
98,151.16
.949563
4.991895
88
1.083460
2.086645
.034813
.319448
44
8,226.57
89,247.61
.915219
.950597
89
.560401
1.003185
1.748499
.001381
45
7,594.78
81,021.04
3.880515
4.908598
90
.264559
.442783
1.422523
T.646191
46
7,005.24
73,426.26
.845423
.865851
91
.112657
ol78224
.051757
.250966
47
6,455.13
66,421.02
.809905
.822306
92
.043731
.065567
2.640789
2.816685
48
5,941.73
59'965.89
.773913
.777904
93
.015364
.021836
.186520
.339173
49
5,462.67
54,024.16
.737405
.732588
94
.004792
.006472
3.680550
3.811038
50
5,015.68
48,561.49
3.700330
4.686292
95
.001306
.001680
3.116080
3.225309
51
4,598.58
43,545.81
.662624
.638946
96
.000308
.000374
4.488714
4.572872
52
4'209.50
38,947.23
.624230
.590477
97
.000057
.000066
5.752632
5.819544
53
3,846.61
34,737.73
.585078
.540801
98
.0000085
.0000094
6.931446
6.974733
54
3,508.31
30,891.12
.545098
.489834
99
.0000009
.0000009
7.951898
7.951898 212
TABLE LXXII.
JOINT LIFE ANNUITIES. PRESENT VALUE (<1XX) OF 1, PAYABLE AT THE END OF EACH
YEAR DURING THE JOINT CONTINUANCE OF TWO MALE LIVES.
Equal
Ages.
3
Per Ct.
Per Ct.
4
Per Ct.
44
Per Ct.
5
Per Ct.
6
Per Ct.
Equal
Ages.
3
Per Ct.
31
Per Ct.
4
Per Ct.
41
Per Ct.
5
Per Ct.
6
PerCt.
10
20.237
18.610
17.1956
15.9583
14.870
13.053
55
9.358
9.014
8.6909
8.3869
8.101
7.576
11
20.117
18.514
17.1174
15.8946
14.818
13.017
56
9.014
8.693
8.3907
8.1058
7.837
7.343
12
19.992
18.413
17.0359
15.8279
14.763
12.979
57
8.671
8.372
8.0893
7.8230
7.571
7.107
13
19.863
18.309
16.9504
15.7579
14.705
12.939
58
8.329
8.051
7.7880
7.5392
7.304
6.870
14
19.729
18.200
16.8612
15.6844
14.644
12.897
59
7.988
7.730
7.4862
7.2548
7.035
6.630
15
19.591
18.086
16.7683
15.6076
14.581
12.852
60
7.650
7.411
7.1851
6.9703
6.766
6.388
16
19.447
17.969
16.6713
15.5272
14.514
12.804
61
7.315
7.095
6.8853
6.6863
6.497
6.145
17
19.298
17.846
16.5698
15.4427
14.443 12.754
62
6.983
6.780
6.5874
6.4036
6.228
5.902
18
19.144
17.719
16.4640
15.3543
14.36912.701
63
6.656
6.469
6.2918
6.1224
5.961
5.659
19
18.985
17.587
16.3537
15.2620
14.291
12.645
64
6.333
6.162
5.9992
5.8435
5.695
5.416
20
18.820
17.449
16.2387
15.1652
14.209
12.585
65
6.016
5.859
5.7103
5.5675
5.431
5.175
21
18.649
17.307
16.1190
15.0641
14.123
12.523
66
5.704
5.562
5.4257
5.2952
5.170
4.935
22
18.473
17.159
15.9944
14 9586
14.033
12.457!
67
5.399
5.270
5.1456
5.0266
4.912
4.697
23
18.291
17.005
15.8647
14.8483
13.939
12.387
68
5.101
4.984
4.8711
4.7629
4.659
4.463
24
18.102
16.846
15.7297
14.7332
13.841
12.314
69
4.810
4.704
4.6019
4.5039
4.409
4.231
25
17.908
16.682
15.5894
14.6132
13.737
12.236
70
4.527
4.431
4.3394
4.2508
4.165
4.003
26
17.708
16.511
15.4435
14.4878
13.629
12.155
71
4.252
4.166
4.0835
4.0036
3.926
3.780
27
17.501
16.335
15.2922
14.3579
13.517
12.069
72
3.986
3.909
3.8348
3.7629
3.693
3.561
28
17.288
16.152
15.1351
14.2218
13.399
11.980
73
3.729
3.660
3.5935
3.5291
3.467
3.348
29
17.069
15.963
14.9721
14.0803
13.276
11.885
74
3.480
3.419
3.3598
3.3023
3.247
3.140
30
16.844
15.769
14.8035
13.9337
13.148
11.786
75
3.242
3.187
3.1346
3.0834
3.034
2.939
31
16.612
15.568
14.6287
13.7812
13.014
11.683
76
3.013
2.965
2.9177
2.8722
2.828
2.743
32
16.373
15.360
14.4479
13.6228
12.875
11.574
77
2.793
2.750
2.7091
2.6688
2.630
2.554
33
16.128
15.147
14.2607
13.4584
12.730
11.460
78
2.583
2.546
2.5094
2.4739
2,439
2.373
34
15.876
14.927
14.0674
13.2881
12.579
11.342
79
2.383
2.350
2.3184
2.2872
2.257
2.198
35
15.618
14.700
13.8678
13.1117
12.423
11.217
80
2.193
2.164
2.1362
2.1088
2.082
2.030
36
15.354
14.467
13.6623
12.9295
12.261
11.088,
81
2.012
1.987
1.9629
1.9389
1.916
1.870
37
15.084
14.228
13.4503
12.7410
12.093
10.953
82
1.841
1.819
1.7979
1.7771
1.757
1.717
38
14.807
13.983
13.2318
12.5461
11.918
10.812
83
1.679
1.660
1.6419
1.6238
1.606
1.572
39
14.524
13.731
13.0072
12.3451
11.738
10.666
84
1.526
1.509
1.4934
1.4778
1.462
1.433
40
14.235
13.473
12.7765
12.1381
11.552
10.514
85
1.380
1.366
1.3528
1.3394
1.326
1.301
41
13.940
13.209
12.5401
11.9251
11.360
10.357
86
1.242
1.230
1.2183
1.2069
1.196
1.174
42
13.639
12.939
12.2967
11.7057
11.161
10.193
87
1.107
1.097
1.0871
1.0775
1.068
1.050
43
13.334
12.664
12.0482
11.4808
10.957
10.024
88
.973
.965
.9568
.9489
.941
.926
44
13.023
12.383
11.7936
11.2497
10.747
9.849
89
.828
.821
.8150
.8086
.802
.790
45
12.707
12.097
11.5336
11.0131
10.531
9.668
90
.704
.699
.6938
.6887
.684
.674
46
12.386
11.805
11.2684
10.7710
10.310
9.482
91
.607
.603
.5985
.5943
.590
.582
47
12.061
11.509
10.9979
10.5235
10.083
9.290
92
.520
.516
.5128
.5094
.506
.499
48
11.733
11.209
10.7227
10.2710
9.851
9.092
93
.438
.435
.4322
.4294
. 427
.421
49
11.400
10.904
10.4430
10.0137
9.613
8.890
94
.364
.361
.3593
.3571
.355
.350
50
11.065
10.596
10.1591
9.7518
9.371
8.682
95
.296
.295
.2930
.2913
.290
.286
51
10.727
10.284
9.8715
9.4857
9.125
8.469
96
.221
.220
.2190
.2178
.211
.214
52
10.387
9.970
9.5805
9.2159
8.874
8.252
97
.172
.171
.1703
.1694
.169
.167
53
10.045
9.653
9.2864
8.9426
8.620
8.031
98
.107
.108
.1069
.1063
.106
.105
54
9.702
9.334
8.9898
8.6661
8.362
7.805
99
.001
.0000
.0000
.000
.000
Na: + l.a:+l
Formula, axx = —• TABLE LXXIII.
213
ANNUAL PREMIUM (tt) FOR THE INSURANCE OF 1,000 ON TWO JOINT LIVES MALE,
PAYABLE AT THE END OF THE YEAR OF THE FIRST DEATH.
Equal
Ages.
3
Per Ct.
91
Per Ct.
4
Per Ct.
Per Ct.
5
Per Ct.
6
Per Ct.
Equal
Ages.
3
Per Ct.
34
Per Ct.
4
Per Ct.
Per Ct.
5
Per Ct.
6
Per Ct.
10
17.97
17.18
16.49
15.91
15.39
14.55
55
67.42
66.04
64.73
63.47
62.26
60.01
11
18.23
17.43
16.74
16.12
15.60
14.73
56
70.73
69.35
68.02
66.75
65.54
63.26
12
18.52
17.69
16.98
16.37
15.82
14.93
57
74.28
72.89
71.56
70.28
69.05
66.74
13
18.80
17.98
17.25
16.61
16.06
15.13
58
77.97
76.67
75.33
74.04
72.80
70.47
14
19.11
18.27
17.53
16.87
16.30
15.36
59
82.14
80.73
79.37
78.08
76.83
74.47
15
19.44
18.58
17.82
17.15
16.57
15.59
60
86.48
85.07
83.71
82.41
81.14
78.75
16
19.78
18.90
18.13
17.44
16.84
15.84
61
91.14
89.72
88.36
87.04
85.76
83.36
17
20.14
19.25
18.45
17.76
17.14
16.10
62
96.13
94.71
93.33
92.01
90.63
88.28
18
20.52
19.61
18.80
18.08
17.45
16.39
63
101.50
100.06
98.68
97.34
96.04
93.58
19
20.91
19.99
19.17
18.43
17.78
16.69
64
107.25
105.81
104.42
103.06
101.75
99.25
20
21.33
20.39
19.55
18.80
18.13
17.00
65
113.42
111.97
110.56
109.20
107.88
105.36
21
21.77
20.81
19.95
19.19
18.51
17.35
66
120.05
118.58
117.16
115.79
114.46
111.88
22
22.23
21.26
20.38
19.60
18.90
17.71
67
127.15
125.69
124.25
122.87
121.52
118.92
23
22.71
21.72
20.84
20.04
19.32
18.10
68
134.80
133.31
131.87
130.46
129.09
126.46
24
23.22
22.22
21.31
20.50
19.77
18.51
69
143.01
141.51
140.05
138.63
137.25
134.57
25
23.76
22.74
21.82
20.99
20.23
18.95
70
151/81
150.30
148.82
147.38
145.98
143.27
26
24.33
23.29
22.36
21.51
20.73
19.41
71
161.28
159.75
158.25
156.79
155.37
152.61
27
24.93
23.87
22.92
22.05
21.27
19.91
72
171.45
169.89
168.38
166.89
165.43
162.63
28
25.56
24.49
23.51
22.63
21.84
20.44
73
182.35
180.78
179.24
177.73
176.25
173.38
29
26.21
25.14
24.15
23.25
22.43
21.01
74
194.08
192.48
190.91
189.38
187.86
184.94
30
26.92
25.82
24.82
23.90
23.06
21.60
75
206.63
205.00
203.40
201.83
200.29
197.31
31
27.66
26.54
25.53
24.59
23.74
22.24
76
220.09
218.42
216.79
215.19
213.61
210.55
32
28.44
27.31
26.27
25.33
24.46
22.93 j
77
234.52
232.82
231.15
229.51
227.89
224.75
33
29.26
28.12
27.07
26.10
25.21
23.65
78
249.94
248.20
246.49
244.80
243.14
239.90
34
30.13
28.98
27.91
26.93
26.02
24.42
79
266.44
264.65
262.89
261.16
259.44
256.10
35
31.05
29.88
28.80
27.80
26.88
25.25
80
284.06
282.21
280.39
278.61
276.84
273.39
36
32.02
30.84
29.74
28.73
27.79
26.12
81
302.84
300.93
299.05
297.20
295.37
291.80
37
33.05
31.85
30.74
29.71
28.76
27.06
82
322.87
320.89
318.94
317.03
315.14
311.42
38
34.14
32.93
31.80
30.76
29.79
28.05
83
344.13
342.08
340.06
338.07
336.11
332.24
39
35.30
34.07
32.93
31.87
30.88
29.12
84
366.84
364.70
362.61
360.52
358.47
354.46
40
36.51
35.28
34.13
33.05
32.05
30.25
85
391.00
388.77
386.57
384.40
382.27
378.05
41
37.81
36.56
35.39
34.31
33.29
31.45
86
416.96
414.62
412.33
410.06
407.81
403.40
42
39.18
37.92
36.75
35.65
34.61
32.74
87
445.53
443.09
440.67
438.28
435.91
431.26
43
40.64
39.37
38.18
37.06
36.01
34.11
88
477.71
475.14
472.57
470.04
467.55
462.62
44
42.19
40.91
39.70
38.57
37.51
35.58
89
517.93
515.21
512.50
509.85
507.20
502.02
45
43.83
42.54
41.32
40.18
39.10
37.14
90
557.64
554.77
551.93
549.12
546.35
540.89
46
45.58
44.28
43.05
41.89
40.80
38.80
91
593.09
590.08
587.11
584.17
581.24
575.51
47
47.44
46.12
44.88
43.72 42.61
40.58
92
628.84
625.68
622.55
619.45
616.39
610.37
48
49.41
48.09
46.84
45.66 44.55
42.48
93
666.42
663.10
659.77
656.55
653.32
647.03
49
51.52
50.19
48.93
47.74 46.61
44.51
94
704.16
700.67
697.23
693.81
690.43
683.88
50
53.76
52.43
51.15
49.95 48.79
46.69
95
742.21
738.53
734.92
731.33
727.78
721.03
51
56.15 54.80
53.52
52.3151.14
49.00
96
789.56
785.66
781.89
778.09
774.34
767.24
52
58.70 57.34
56.06
54.83 53.65
51.48
97
824.12
819.89
816.03
812.10
808.19
800.62
53
61.41 60.05
58.75
57.51 56.33
54.13
98
873.91
868.65
865.01
860.83
856.70
848.53
54
64.32^62.95
61.64
60.40 59.20
56.97
99
961.54
956.94
952.38
943.40
Formula, 7rra = 1000 j -(!-«)}- 1000 { N” (1 v) [ ' 214
TABLE LXXIY.
SINGLE PREMIUM (A) OR RESERVE, FOR THE INSURANCE OF 1,000 ON TWO JOINT
LIVES, PAYABLE AT THE END OF THE YEAR OF THE FIRST DEATH.
cd co
$ O
w«*
3
Per Cent.
H
Per Cent.
4
PerCent.
Per Cent.
5
Per Cent.
Equal
Ages.
3
Per Cent.
Per Cent.
4
Per Cent.
4.1
Per Cent.
5
Per Cent.
10
381.45
336.85
300.17
269.75
244.29
55
698.32
661.36
627.27
595.78
566.64
11
384.95
340.12
303.17
272.48
246.77
56
708.33
672.22
638.82
607.88
579.19
12
388.57
343.51
306.32
275.35
249.38
57
718.33
683.09
650.41
620.06
591.84
13
392.34
347.06
309.60
278.37
252.13
58
728.29
693.94
662.00
632.28
604.57
14
396.24
350.74
313.03
281.53
255.04
59
738.20
704.77
673.61
644.54
617.36
15
400.27
354.57
316.60
284.84
258.08
60
748.05
715.56
685.19
656.78
630.18
16
404.45
358.55
320.33
288.30
261.27
61
757.81
726.27
696.70
669.01
643.00
17
408.79
362.69
324.24
291.94
264.63
62
767.47
736.89
708.18
681.18
655.79
18
413.28
367.00
328.31
295.75
268.17
63
777.01
747.41
719.55
693.29
668.54
19
417.92
371.47
332.54
299.72
271.87
64
786.42
757.80
730.79
705.30
681.21
20
422.73
376.12
336.97
303.88
275.77
65
795.66
768.04
741.91
717.19
693.77
21
427.70
380.94
341.57
308.25
279.85
66
804.73
778.10
752.87
728.91
706.19
22
432.83
385.94
346.37
312.79
284.14
67
813.63
787.99
763.64
740.48
718.45
23
438.14
391.13
351.36
317.54
288.61
68
822.31
797.65
774.19
751.83
730.53
24
443.62
396.50
356.54
322.49
293.31
69
830.78
807.12
784.54
762.99
742.41
25
449.28
402.07
361.94
327.66
298.23
70
839.02
816.33
794.63
773.89
754.04
26
455.12
407.84
367.55
333.07
303.37
71
847.03
825.30
804.48
784.54
765.40
27
461.13
413.81
373.37
338.66
308.74
72
854.78
833.99
814.05
794.89
776.51
28
467.33
419.98
379.43
344.52
314.35
73
862.27
842.42
823.32
804.97
787.29
29
473.72
426.36
385.69
350.61
320.21
74
869.50
850.56
832.31
814.73
797.78
30
480.29
432.94
392.17
356.92
326.32
75
876.45
858.40
840.98
824.16
807.93
31
487.05
439.74
398.89
363.49
332.68
76
883.12
865.93
849.32
833.25
817.72
32
493.99
446.75
405.84
370.31
339.30
77
889.52
873.17
857.34
842.02
827.16
33
501.13
453.98
413.05
377.39
346.20
78
895.63
880.09
865.02
850.40
836.23
34
508.46
461.42
420.48
384.73
353.37
79
901.46
886.70
872.36
858.45
844.92
35
515.97
469.08
428.16
392.32
360.81
80
907.00
892.99
879.38
866.14
853.23
36
523.66
476.95
436.06
400.16
368.53
81
912.26
898.98
886.04
873.45
861.17
37
531.54
485.03
444.22
408.29
376.54
82
917.25
904.66
892.38
880.42
868.73
38
539.61
493.34
452.62
416.68
384.85
83
921.97
910.04
898.39
887.02
875.90
39
547.86
501.85
461.26
425.33
393.42
84
926.44
915.14
904.10
893.31
882.74
40
556.27
510.58
470.13
434.25
402.29
85
930.67
919.98
909.51
899.36
889.22
41
564.85
519.49
479.23
443.42
411.44
86
934.71
924.59
914.68
904.96
895.43
42
573.61
528.63
488.59
452.86
420.89
87
938.64
929.09
919.73
910.54
901.51
43
582.51
537.93
498.14
462.55
430.62
88
942.53
933.55
924.74
916.08
907.56
44
591.58
547.44
507.94
472.50
440.62
89
946.76
938.40
930.19
922,12
914.17
45
600.78
557.12
517.94
482.69
450.89
! 90
950.36
942.55
934.85
927.29
919.83
46
610.12
566.97
528.13
493.12
461.44
91
953.19
945.80
938.52
931.35
924.28
47
619.58
576.98
538.54
503.57
472.25
92
955.73
948.72
941.81
935.00
928.28
48
629.15
587.14
549.12
514.65
483.31
93
958.12
951.48
944.91
938.44
932.06
49
638.83
597.44
559.88
525.73
494.61
94
960.28
953.96
947.72
941.57
935.48
50
648.60
607.87
570.80
537.01
506.13
95
962.24
956.22
950.26
944.40
938.59
51
658.44
618.40
581.86
548.46
517.87
96
964.42
958.73
953.11
947.56
942.06
52
668.35
629.04
593.06
560.07
529.80
97
965.86
960.39
954.99
949.64
944.36
53
678.31
639.75
604.37
571.84
541.91
98
967.75
962.53
957.43
952.36
947.34
54
688.30
650.53
615.77
583.75
554.20
1
99
966.13
961.54
956.94
952.38
Formula, Aw = 1000 j 1 — (1—v) (1 + axx) j = 1000 j 1 — (1 — v) ~ j- • TABLE LXXY.
215
TEN PAYMENT PREMIUM (P) ON 1,000, WITH VALUATION FACTORS FOR THE SAME,
ALSO FOR ORDINARY JOINT POLICIES ON TWO MALE LIVES.
FOR THE LATTER POLICIES P' IS 0.
4 PER CENT.
H w
2 9
Ord’y Joint.
Ten Pay’t Joint Life Policies.
r—4 .
o3 m
2 9
Ord’y Joint.
Ten Pay’t Joint Life Policies.
cr* be
AP".
P.
AP'.
AP".
cr* be
AP".
P.
AP'.
AP".
10
1.740034
37.562
7.583526
1.880948
50
1.952370
77.388
6.497778
2.063894
11
.741904
37.945
.561975
.883130
51
.963711
79.516
.452967
.071799
12
.743863
38.347
.540412
.885409
52
.975496
81.753
.406117
.079957
13
.745925
38.766
.518801
.887772
53
.987735
84.105
.357061
.088372
14
.748089
39.206
.497150
.890240
54
2.000444
86.584
.305615
.097068
15
1.750354
39.663
7.475422
1.892787
55
2.013635
89.206
6.251613
2.106081
16
.752732
40.142
.453635
.895442
56
.027301
91.975
.194806
.115399
17
.755233
40.646
.431800
.898218
57
.041468
94.911
.134988
.125066
18
.757855
41.171
.409867
.901091
58
.056112
98.023
.071871
.135084
19
.760607
41.718
.387814
.904064
59
.071289
101.34
.005232
.145512
20
1.763495
42.293
7.365674
1.907167
60
2.086978
104.88
5.934739
2.156372
21
.766521
42.891
.343382
.910371
61
.103184
108.66
.860049
.167676
22
.769694
43.516
.320956
.913695
62
.119907
112.71
.780822
.179470
23
.773022
44.169
.298355
.917141
63
.137168
117.06
.696641
.191790
24
.776512
44.849
.275557
.920700
64
.154951
121.74
.607041
.204667
25
1.780169
45.560
7.252567
1.924391
65
2.173261
126.79
5.511540
2.218145
26
.784006
46.303
.229347
.928214
66
.192083
132.25
.409582
.232263
27
.788020
47.076
.205861
.932157
67
.211436
138.16
.300568
.247044
28
.792228
47.885
.182120
.936245
68
.231282
144.55
.183770
.262479
29
.796638
48.726
.158035
.940455
69
.251664
151.51
.058521
.278689
30
1.801247
49.600
7.133591
1.944787
70
2.272505
159.05
4.923893
2.295592
31
.806077
50.513
.108792
.949266
71
.293840
167.27
.779012
.313301
32
.811131
51.463
.083556
.953878
72
.315626
176.21
.622795
.331774
33
.816426
52.457
.057905
.958653
73
.337860
185.93
.454064
.351006
34
.821962
53.489
.031737
.963554
74
.360535
196.53
.271559
.371052
35
1.827753
54.565
7.005041
1.968607
75
2.383562
208.03
4.073626
2.391802
36
.833797
55.681
6.977727
.973786
76
.406966
220.51
3.858577
.413252
37
.840123
56.847
.949802
.979132
77
.430737
234.06
.624217
.435400
38
.846740
58.062
.921192
.984633
78
.454770
248.69
.367287
.458111
39
.853649
59.327
.891826
.990288
79
.479070
264.49
.083312
.481373
40
1.860862
60.643
6.861641
1.996094
80
2.503593
281.49
2.762652
2.505084
41
.868378
62.013
.830568
2.002056
81
.528288
299.75
.402315
.529188
42
.876257
63.446
.798584
.008206
82
.553163
319.37
.003219
.553679
43
.884450
64.933
.765527
.014498
83
.578092
340.30
1.561001
.578366
44
.893008
66.490
.731399
.020988
84
.603214
362.73
.068323
.603352
45
46
47
48
49
1.901923
.911214
.920895
.930971
.941459
68.111
69.802
71.573
73.422
75.358
6.696056
.659409
.621388
.581854
.540687
2.027645
.034482
.041529
.048766
.056217
85
86
87
88
89
2.628422
.653973
.680453
.708447
.741123
386.63
412.35
440.67
472.57
512.50
0.518142
1.901420
.184758
2.377984
3.397848
2.628483
.653995
.680454
.708447
.741121
—
—
Formula, Yxx =
LN cccc — i*x+n.x + n
Y'xx= {?xx—Kxx) Nxx- P"xx — ?xx+ 1000(1— v). 216
TABLE LXXVI.
GENERAL VALUATION OF JOINT LIFE POLICIES BY ANNUAL PREMIUM ON 1,000.
Note. Ten or Limited Premium Policies are valued by tliis Table only till tlie end of the year
after Payment of the last Premium, and afterwards by Single Premium Table.
4 PER CENT.
Equal
h = |m
. *Dec.
lfm.
Nov.
Oct.
Equal
Ages.
/" = 963.141
966.346
969.551
972.756
Ages.
x-{- n.
Xb.
Xb'.
Xb.
Xb'.
Xb.
XV.
Xb.
x-\-n.
10
5.000969
1.236385
5.002901
1.238317
5.004824
1.240240
5.006739
10
11
.023650
.234408
.025584
.236342
.027508
.238266
.029424
11
12
.046350
.232335
.048284
.234269
.050209
.236194
.052127
12
13
.069062
.230153
.070998
.232089
.072925
.234016
.074843
13
14
.091793
.227862
.093731
.229800
.095660
.231729
.097582
14
15
5.114550
1.225462
5.116491
1.227403
5.118422
1.229334
5.120346
15
16
.137330
.222943
.139272
.224885
.141205
.226818
.143130
16
17
.160129
.220293
.162074
.222238
.164010
.224174
.165937
17
18
.182957
.217512
.184905
.219460
.186843
.221398
.188773
18
19
.205817
.214596
.207766
.216545
.209707
.218486
.211639
19
20
5.228706
1.211532
5.230659
1.213485
5.232604
1.215430
5.234540
20
21
.251635
.208321
.253591
.210277
.255539
.212225
.257479
21
22
.274604
.204952
.276564
.206912
.278515
.208863
.280459
22
23
.297615
.201416
.299580
.203381
.301536
.205337
.303482
23
24
.320675
.197708
.322644
.199677
.324605
.201638
.326556
24
25
5.343789
1.193821
5.345762
1.195794
5.347727
1.197759
5.349683
25
26
.366956
.189739
.368935
.191718
.370907
.193690
.372868
26
27
.390193
.185467
.392179
.187453
.394156
.189430
.396124
27
28
.413498
.180986
.415490
.182978
.417473
.184961
.419448
28
29
.436878
.176286
.438878
.178286
.440870
.180278
.442852
29
30
5.460353
1.171372
5.462361
1.173380
5.464360
1.175379
5.466350
30
31
.483916
.166219
.485933
.168236
.487941
.170244
.489939
31
32
.507583
.160820
.509610
.162847
.511627
.164864
.513635
32
33
.531360
.155164
.533397
.157201
.535425
.159229
.537444
33
34
.555263
.149242
.557313
.151292
.559352
.153331
.561383
34
35
5.579303
1.143043
5.581367
1.145107
5.583422
1.147162
5.585466
35
36
.603506
.136568
.605583
.138645
.607650
.140712
.609708
36
37
.627865
.129783
.629958
.131876
.632040
.133958
.634113
37
38
.652401
.122677
.654512
.124788
.656613
.126889
.658703
38
39
.677144
.115253
.679274
.117383
.681394
.119503
.683502
39
40
5.702109
1.107501
5.704260
1.109652
5.706401
1.111793
5.708531
40
41
.727321
.099381
.729495
.101555
.731657
.103717
.733809
41
42
.752788
.090893
.754988
.093093
.757178
.095283
.759356
42
43
.778564
.082040
.780791
.084267
.783008
.086484
.785212
43
44
.804650
.072780
.806909
.075039
.809157
.077287
.811393
44
45
5.831099
1.063118
5.833392
1.065411
5.835673
1.067692
5.837942
45
46
.857937
.053030
.860268
.055361
.862586
.057679
.864892
46
47
.885203
.042501
.887575
.044873
.889934
.047232
.892281
47
48
.912945
.031519
.915363
.033937
.917767
.036341
.920158
48
49
.941206
.020066
.943674
.022534
.946127
.024987
.948567
49
50
5.970037
1.008123
5.972560
1.010646
5.975069
1.013155
5.977563
50
51
.999501
0.995680
4.002085
0.998264
4.004653
.000832
4.007207
51
52
4.029657
.982718
.032307
.985368
.034942
0.988003
.037561
52
53
.060573
.969218
.063297
.971942
.066003
.974648
.068693
53
Foemula, V = /" +■ p; • bxtn - P" ■ vm; F = (P, - *x) N*.
The suffix is not here repeated for equal Ages, Page 270. = P» + 1000 (1 — V). 217
TABLE LXXYI.
GENERAL VALUATION OF JOINT LIFE POLICIES BY ANNUAL PREMIUM ON 1,000.
Equal
*Sept.
4m.
Aug.
5|m.
July.
6m.
Equal
Ages.
975.962
979.167
980.769
Ages.
x + n.
XV.
XI.
XV.
Xb.
XV.
Xb.
XV.
x-\-n.
10
1.242155
5.008645
1.244061
5.010542
1.245958
5.011488
1.246904
10
11
.240182
.031332
.242090
.033232
.243990
.034178
.244936
11
12
.238112
.054035
.240020
.055936
.241921
.056883
.242868
12
13
.235934
.076754
.237845
.078656
.239747
.079604
.240695
13
14
.233651
.099494
.235563
.101398
.237467
.102348
.238417
14
15
1.231258
5.122260
1.233172
5.124166
1.235078
5.125116
1.236028
15
16
.228743
.145046
.230659
.146954
.232567
.147905
.233518
16
17
.226101
.167855
.228019
.169766
.229930
.170718
.230882
17
18
.223328
.190694
.225249
.192607
.227162
.193560
.228115
18
19
.220418
.213563
.222342
.215479
.224258
.216433
.225212
19
20
1.217366
5.236467
1.219293
5.238384
1.221210
5.239341
1.222167
20
21
.214165
.259409
.216095
.261330
.218016
.262289
.218975
21
22
.210807
.282393
.212741
.284318
.214666
.285278
.215626
22
23
.207283
.305421
.209222
.307350
.211151
.308312
.212113
23
24
.203589
.328499
.205532
.330433
.207466
.331397
.208430
24
25
1.199715
5.351630
1.201662
5.353569
1.203601
5.354535
1.204567
25
26
.195651
.374822
.197605
.376766
.199549
.377735
.200518
26
27
.191398
.398082
.193356
.400032
.195306
.401004
.196278
27
28
.186936
.421412
.188900
.423369
.190857
.424344
.191832
28
29
.182260
.444825
.184233
.446790
.186198
.447769
.187177
29
30
1.177369
5.468331
1.179350
5.470302
1.181321
5.471285
1.182304
30
31
.172242
.491930
.174233
.493910
.176213
.494897
.177200
31
32
.166872
.515633
.168870
.517623
.170860
.518614
.171851
32
33
.161248
.539453
.163257
.541453
.165257
.542450
.166254
33
34
.155362
.563404
.157383
.565415
.159394
.566418
.160397
34
35
1.149206
5.587500
1.151240
5.589525
1.153265
5.590534
1.154274
35
36
.142770
.611756
.144818
.613795
.146857
.614810
.147872
36
37
.136031
.636176
.138094
.638229
.140147
.639252
.141170
37
38
.128979
.660783
.131059
.662853
.133129
.663885
.134161
38
39
.121611
.685601
.123710
.687690
.125799
.688731
.126840
39
40
1.113923
5.710651
1.116043
5.712762
1.118154
5.713813
1.119205
40
41
.105869
.735951
.108011
.738082
.110142
.739143
.111203
41
42
.097461
.761524
.099629
.763681
.101786
.764755
.102860
42
43
.088688
.787406
.090882
.789589
.093065
.790676
.094152
43
44
.079523
.813617
.081747
.815830
.083960
.816933
.085063
44
45
1.069961
5.840200
1.072219
5.842446
1.074465
5.843564
1.075583
45
46
47
.059985
.049579
.867186
.894615
.062279
.051913
.869467
.896937
.064560
.054235
.870604
.898093
.065697
.055391
46
47
48
.038732
.922536
.041110
.924901
.043475
.926079
.044653
48
49
.027427
.950993
.029853
.953406
.032266
.954607
.033467
49
50
1.015649
5.980043
1.018129
5.982508
1.020594
5.983736
1.021822
50
51
.003386
4.009745
.005924
4.012269
.008448
4.013525
.009704
51
52
0.990622
.040164
0.993225
.042751
0.995812
.044039
0.997100
52
53
.977338
.071367
.980012
.074024
.982669
.075347
.983992
53
4 PER CENT.
* Months of Entry corresponding to Annual Valuation, Dec. 31.
Here x + n + h — Present Age in years and fraction ; f 1000 -r (1 ft) j
h + (1 — h) vpx+n—1 . y _ . 1ST
K+n = f>~ " ’ x+n x+n x+n 218
TABLE LXXYI.
GENERAL VALUATION OF JOINT LIFE POLICIES BY ANNUAL PREMIUM ON 1,000,
4 PER CENT.
Equal
June.
71m.
May.
8|m.
April.
9im-
Equal
Ages.
f" = 982.372.
985.577.
988.782.
991.987.
Ages.
x-\-n.
xb.
XV.
Xb.
Xb’.
Xb.
Xb'.
Xb.
x-\-n.
10
5.012432
1.247848
5.014313
1.249729
5.016187
1.251603
5.018052
10
11
.035122
.245880
.037006
.247764
.038880
.249638
.040747
11
12
.057828
.243813
.059712
.245697
.061588
.247573
.063456
12
13
.080549
.241640
.082435
.243526
.084313
.245404
.086182
13
14
.103294
.239363
.105182
.241251
.107062
.243131
.108933
14
15
5.126063
1.236975
5.127954
1.238866
5.129835
1.240747
5.131708
15
16
.148853
.234466
.150744
.236357
.152628
.238241
.154502
16
17
.171667
.231831
.173561
.233725
.175446
.235610
.177323
17
18
.194511
.229066
.196407
.230962 .
.198295
.232850
.200175
18
19
.217386
.226165
.219284
.228063
.221174
.229953
.223056
19
20
5.240295
1.223121
d.242196
1.225022
5.244090
1.226916
5.245975
20
21
.263244
.219930
.265149
.221835
.267046
.223732
.268935
21
22
.286235
.216583
.288143
.218491
.290044
.220392
.291936
22
23
.309271
.213072
.311183
.214984
.313087
.216888
.314984
23
24
.332358
.209391
.334275
.211308
.336183
.213216
.338084
24
25
5.355499
1.205531
o. 357420
1.207452
5.359333
1.209365
5.361238
25
26
.378702
.201485
.380629
.203412
.382548
.205331
.384458
26
27
.401974
.197248
.403906
.199180
.405830
.201104
.407746
27
28
.425316
.192804
.427255
.194743
.429185
.196673
.431107
28
29
.448745
.188153
.450691
.190099
.452630
.192038
.454559
29
30
5.472265
1.183284
5.474218
1.185237
5.476164
1.187183
5.478101
30
31
.495881
.178184
.497844
.180147
.499797
.182100
.501742
31
32
.519604
.172841
.521575
.174812
.523537
.176774
.525491
32
33
.543444
.167248
.545426
.169230
.547399
.171203
.549363
33
34
.567418
.161397
.569411
.163390
.571394
.165373
.573370
34
35
5.591541
1.155281
5.593548
1.157288
5.595544
1.159284
5.597533
35
36
.615823
.148885
.617843
.150905
.619853
.152915
.621853
36
37
.640273
.142191
.642306
.144224
.644330
.146248
.646346
37
38
.664913
.135189
.666965
.137241
.669006
.139282
.671037
38
39
.689769
.127878
.691839
.129948
.693898
.132007
.695947
39
40
5.714861
1.120253
5.716950
1.122342
5.719030
1.124422
5.721099
40
41
.740202
.112262
.742313
.114373
.744412
.116472
.746502
41
42
.765827
.103932
.767962
.106067
.770088
.108193
.772202
42
43
.791761
.095237
.793922
.097398
.796073
.099549
.798212
43
44
.818032
.086162
.820223
.088353
.822403
.090533
.824572
44
45
o. 844680
1.076699
5.846903
1.078922
5.849114
1.081133
5.851314
45
46
.871736
.066829
.873994
.069087
.876241
.071334
.878476
46
47
.899246
.056544
.901543
.058841
.903828
.061126
.906100
47
48
.927253
.045827
.929593
.048167
.931919
.050493
.934234
48
49
.955805
.034665
.958192
.037052
.960565
.039425
.962925
49
50
5.984960
1.023046
5.987398
1.025484
5.989822
1.027908
5.992234
50
51
4.014778
.010957
4.017273
.013452
4.019753
.015932
4.022220
51
52
.045323
0.998384
.047881
.000942
.050422
.003483
.052950
52
53
.076665
.985310
.079291
0.987936
.081900
0.990545
.084494
53 219
TABLE LXXYI.
GENERAL VALUATION OF JOINT LIFE POLICIES BY ANNUAL PREMIUM ON 1,000,
4 PER CENT,
Equal
March.
10-^m
Feb.
llj-m.
Jan.
12m.
Equal
Ages.
Ages.
995.192.
998.397.
1000.000.
x + n.
A V.
a b.
lb'.
lb.
lb'.
lb.
lb'.
10
1.253408
5.019909
1.255325
5.021759
1.257175
5.022680
1.258096
10
11
.251505
.042606
.253364
.044456
.255214
.045379
.256137
11
12
.249441
.065315
.251300
.067167
.253152
.068090
.254075
12
13
.247273
.088043
.249134
.089896
.250987
.090820
.251911
13
14
.245002
.110797
.246866
.112653
.248722
.113577
.249646
14
15
1.242620
0.133573
1.244485
5.135431
1.246343
5.136356
1.247268
15
16
.240115
.156368
.241981
.158228
.243841
.159154
.244767
16
17
.237487
.179192
.239356
.181054
.241218
.181981
.242145
17
18
.234730
.202046
.236601
.203910
.238465
.204838
.239393
18
19
.231835
.224930
.233709
.226796
.235575
.227726
.236505
19
20
1.228801
5.247853
1.230679
5.249722
1.232548
5.250653
1.233479
20
21
.225021
.270815
.227501
.272687
.229373
.273620
.230306
21
22
.222284
.293819
.224167
.295695
.226043
.296630
.226978
22
23
.218785
.316872
.220673
.318751
.222552
.319687
.223488
23
24
.215117
.339976
.217009
.341860
.218893
.342798
.219831
24
25
1.211270
5.363134
1.213166
O.365022
1.215054
5.365962
1.215994
25
26
.207241
.386360
.209143
.388253
.211036
.389197
.211980
26
27
.203020
.409653
.204927
.411552
.206826
.412498
.207772
27
28
.198595
.433020
.200508
.434925
.202413
.435874
.203362
28
29
.193967
.456479
.195887
.458392
.197800
.459345
.198753
29
30
1.189120
5.480028
1.191047
5.481948
1.192967
5.482904
1.193923
30
31
.184045
.503678
.185981
.505606
.187909
.506566
.188869
31
32
.178728
.527436
.180673
.529372
.182609
.530337
.183574
32
33
.173167
.551318
.175122
.553264
.177068
.554234
.178038
33
34
.167349
.575336
.169315
.577293
.171272
.578268
.172247
34
35
1.161273
5.599512
1.163252
5.601481
1.165221
5.602463
1.166203
35
36
.154915
.623845
.156907
.625828
.158890
.626815
.159877
36
37
.148264
.648351
.150269
.650347
.152265
.651342
.153260
37
38
.141313
.673060
.143336
.675072
.145348
.676075
.146351
38
39
.134056
.697987
.136096
.700018
.138127
.701029
.139138
39
40
1.126491
5.723159
1.128551
5.725209
1.130601
5.726230
1.131622
40
41
.118562
.748582
.120642
.750652
.122712
.751683
.123743
41
42^
P.110307
.774307
.112412
.776401
.114506
.777445
.115550
42
43 ’
.101688
.800342
.103818
.802461
.105937
.803516
.106992
43
44
.092702
.826730
.094860
.828877
.097007
.829947
.098077
44
45
1.083333
5.853504
1.085523
5.855683
1.087702
15.856767
1.088786
45
46
.073569
.880698
.075791
.882910
.078003
.884012
.079105
46
47
.063398
.908362
.065660
.910611
.067909
.911731
.069029
47
48
.052808
.936537
.055111
.938827
.057401
.939967
.058541
48
49
.041785
.965272
.044132
.967608
.046468
.968770
.047630
49
50
1.030320
5.994631
1.032717
5.997016
1.035102
5.998203
1.036289
50
51
.018399
4.024672
.020851
4.027111
.023290
4.028325
.024504
51
52
.006011
.055463
.008524
.057960
.011021
.059204
.012265
52
53
0.993139
.087072
0.995717
.089635
0.998280
.090911
0.999556
53 220
TABLE LXXYI.
GENERAL VALUATION OF JOINT LIFE POLICIES BY ANNUAL PREMIUM ON 1,000.
4 PER CENT.
Equal
Ages.
x-\-n.
h = -g-m. *Dec.
Nov.
2ig-m.
Oct.
Hm.
Equal
Ages.
x-\-n.!
/" =
963.141
966.346
969.551
972.756
lb.
lb'.
lb.
lb'.
lb.
IV.
lb.
54
4.092320
0.955158
4.095124
0.957962
4.097911
0.960749
4.100680
54
55
4.124982
0.940521
4.127877
0.943416
4.130752
0.946291
4.133609
55
56
.158664
.925303
.161656
.928295
.164628
.931267
.167580
56
57
.193453
.909482
.196555
.912584
.199635
.915664
.202694
57
58
.229479
.893034 I
.232700
.896255
.235898
.899453
.239072
58
59
.266842
.875945
.270196
.879299
.273523
.882626
.276824
59
60
4.305679
0.858190
4.309178
0.861689
4.312648
0.865159
4.316091
60
61
.346147
.839762
.349806
.843421
.353435
.847050
.357035
61
62
.388409
.820646
.392245
.824482
.396046
.828283
.399815
62
63
.432626
.800802
.436657
.804833
.440651
.808827
.444610
63
64
.479013
.780234
.483260
.784481
.487467
.788688
.491633
64
65
4.527780
0.758918
4.532266
0.763404
4.536708
0.767846
4.541105
65
66
.579173
.736846
.583924
.741597
.588622
.746295
.593270
66
67
.633435
.713979
.638478
.719022
.643465
.724009
.648393
67
68
.690881
.690332
.696244
.695695
.701541
.700992
.706775
68
69
.751776
.665830
.757499
.671553
.763149
.677203
.768725
69
70
4.816533
0.640525
4.822651
0.646643
4.828684
0.652676
4.834634
70
71
.885485
.614346
.892043
.620904
.898503
.627364
.904869
71
72
.959079
.587302
.966124
.594347
.973056
.601279
.979879
72
73
3.037759
.559356
3.045342
.566939
3.052795
.574392
3.060123
73
74
.122022
.530465
.130211
.538654
.138250
.546693
.146141
74
75
3.212493
0.500686
3.221349
0.509542
3.230029
0.518222
3.238540
75
76
.309721
.469924
.319320
.479523
.328712
.488915
.337904
76
77
.414374
.438131 I
.424809
.448566
.435000
.458757
.444957
77
78
.527269
.405362
.538632
.416725
.549706
.427799
.560504
78
79
.649174
.371527
.661577
.383930
.673635
.395988
.685367
79
80
3.781001
0.336594
3.794569
0.350162
3.807726
0.363319
3.820496
80
81
.923750
.300520
.938622
.315392
.953001
.329771
.966919
81
82
2.078479
.263185
2.094829
.279535
2.110586
.295292
2.125791
82
83
.246487
.224618
.264482
.242613
.281761
.259892
.298379
83
84
.428946
.184457
.448823
.204334
.467831
.223342
.486042
84
85
2.627494
0.142647
2.649484
0.164637
2.670414
0.185567
2.690381
85
86
.843590
.098482
.867956
.122848
.891028
.145920
.912936
86
87
1.078896
.050518
1.106049
.077671
1.131603
.103225
1.155737
87
88
.335865
1.996769
.366080
.026984
.394328
.055232
.420851
88
89
.617059
.930245
.652886
1.966072
.685981
1.999167
.716732
89
90
1.938654
T. 864208
T. 981258
1.906812
0.020053
T. 945607
0.055665
90
91
0.304967
.803897
0.354059
.852989
.398162
.897092
.438196
91
92
.712147
.741287
.768670
.797810
.818675
.847815
.863513
92
93
1.164049
.672794
1.230048
.738793
1.287328
.796073
1.337926
93
94
.669216
.600063
.746739
.677586
.812499
.743346
.869597
94
95
2.234475
T. 520627
2.325331
T. 611483
2.400428
T. 686580
2.464435
95
96
.872089
.412726
.988191
.528828
3.079712
.620349
3.155262
96
97
3.617270
.320785
3.754903
.458418
.859255
.562770
.943328
97
98
4.470370
.158523
4.651266
.339419
4.778587
.466740
4.876920
98
* Months of Entry corresponding to Annual Valuation Dec. 31. 221
TABLE LXXVI.
GENERAL VALUATION OF JOINT LIFE POLICIES BY ANNUAL PREMIUM ON 1,000.
4 PER CENT.
Equal
Sept.
4m.
Aug.
5-^m.
July.
6m.
Equal
Ages.
975.962
979.167
980.769
Ages.
x-\-n.
xv.
n.
XV.
n.
XV.
Xb.
XV.
x-\-n.
54
0.963518
4.103431
0.966269
4.106165
0.969003
4.107525
0.970363
54
55
0.949148
4.136447
0.951986
4.139266
0.954805
4.140669
0.956208
55
56
.934219
.170513
.937152
.173425
.940064
.174874
.941513
56
57
.918723
.205732
.921761
.208748
.924777
.210248
.926277
57
58
.902627
.242223
.905778
.245352
.908907
.246908
.910463
58
59
.885927
.280101
.889204
.283353
.892456
.284970
.894073
59
60
0.868602
4.319507
0.872018
4.322897
0.875408
4.324582
0.877093
60
61
.850650
.360603
.854218
.364144
.857759
.365904
.859519
61
62
.832052
.403551
.835788
.407256
.839493
.409096
.841333
62
63
.812786
.448532
.816708
.436819
.804995
.454351
.822527
63
64
.792854
.495759
.796980
.499847
.801068
.501876
.803097
64
65
0.772243
4.545457
0.776595
4.549766
0.780904
4.551904
0.783042
65
66
.750943
.597869
.755542
.602420
.760093
.604678
.762351
66
67
.728937
.653267
.733811
.658087
.738631
.660477
.741021
67
68
.706226
.711947
.711398
.717057
.716508
.719590
.719041
68
69
.682779
.774231
.688285
.779669
.693723
j .782362
.696416
69
70
0.658626
4.840504
0.664496
4.846296
0.670288
4.849162
0.673154
70
71
.633730
.911143
.640004
.917328
.646189
.920387
.649248
71
72
.608102
.986597
.614820
.993212
.621435
i .996482
.624705
72
73
.581720
3.067329
.588926
3.074417
.596014
3.077918
.599515
73
74
.554504
.153892
.562335
.161508
.569951
.165266
.573709
74
75
0.526733
3.246886
0.535079
3.255074
0.543267
3.259112
0.547305
75
76
.498107
.346906
.507109
.355726
.515929
.360069
.520272
76
77
.468714
.454690
.478447
.464210
.487967
.468894
.492651
77
78
.438597
.571041
.449134
.581328
.459421
.586381
.464474
78
79
.407720
.696791
.419144
.707922
.430275
.713382
.435735
79
80
0.376089
3.832902
0.388495
3.844963
0.400556
3.850870
0.406463
80
81
.343689
.980404
.357174
.993484
.370254
.999879
.376649
81
82
.310497
2.140482
.325188
2.154691
.339397
2.161626
.346332
82
83
.276510
.314384
.292515
.329820
.307951
.337337
.315468
83
84
.241553
.503518
.259029
.520320
.275831
.528483
.283994
84
85
0.205534
2.709470
0.224623
2.727757
0.242910
2.736619
0.251772
85
86
.167828
.933791
.188683
.953691
.208583
.963309
.218201
86
87
.127359
1.178600
.150222
1.200320
.171942
1.210785
.182407
87
88
.081755
.445846
.106750
.469481
.130385
.480833
.141737
88
89
.029918
.745449
.058635
.772384
.085570
.785250
.098436
89
90
1.981219
0.088577
0.014131
0.119169
0.044723
0.133694
0.059248
90
91
.937126
.474848
1.973778
.508647
.007577
.524609
.023539
91
92
.892653
.904153
.933293
.941312
1.970452
.958764
T. 987904
92
93
.846671
1.383239
.891984
1.424268
.933013
1.443415
.952160
93
94
.800444
.920054
.850901
.965254
.896101
.986205
.917052
94
95
T. 750587
2.520208
T. 806360
2.569627
T. 855779
2.592377
1.878529
95
96
.695899
3.219597
.760234
3.275620
.816257
3.301135
.841772
96
97
.646843
4.013736
.717251
4.074305
.777820
4.101691
.805206
97
98
.565073
.957048
.645201
5.024669
.712822
5.054902
.743055
98 222
TABLE LXXVI.
GENERAL VALUATION OF JOINT LIFE POLICIES BY ANNUAL PREMIUM ON 1,000.
4 PER CENT.
Equal
Ages.
x\-n.
h = 6g-m. June.
7£m-
May.
8im.
April.
9im.
Equal
Ages.
x-j- n.
f" = 982.372
985.577
988.782
991.987
xv
XV.
Xb.
XV.
Xb.
XV.
Xb.
54
4.108881
0.971719
4.111582
0.974420
4.114264
0.977102
4.116932
54
55
4.142007
0.957606
4.144851
0.960390
4.147616
0.963155
4.150365
55
56
.176318
.942957
.179191
.945830
.182047
.948686
.184883
56
57
.211743
.927772
.214718
.930747
.217673
.933702
.220608
57
58
.248458
.912013
.251543
.915098
.254605
.918160
.257646
58
59
.286581
.895684
.289786
.898889
.292967
.902070
.296123
59
60
4.326260
0.878771
4.329597
0.882108
4.332909
0.885420
4.336195
60
61
.367656
.861271
.371139
.864754
.374595
.868210
.378023
61
62
.410929
.843166
.414572
.846809
.418184
.850421
.421766
62
63
.456273
.824449
.460092
.828268
.463877
.832053
.467630
63
64
.503897
.805118
.507909
.809130
.511884
.813105
.515824
64
65
4.554033
0.785171
4.558258
0.789396
4.562443
0.793581
4.566588
65
66
.606924
.764597
.611381
.769054
.615793
.773466
.620160
66
67
.662853
.743397
.667568
.748112
.672232
.752776
.676847
67
68
.722108
.721559
.727101
.726552
.732038
.731489
.736918
68
69
.785038
.699092
.790342
.704396
.795583
.709637
.800760
69
70
4.852011
0.676003
4.857652
0.681644
4.863221
0.687213
4.868720
70
71
.923425
.652286
.929438
.658299
.935370
.664231
.941221
71
72
.999728
.627951
3.006148
.634371
3.012474
.640697
3.018709
72
73
3.081391
.602988
.088255
.609852
.095013
.616610
.101667
73
74
.168991
.577434
.176349
.584792
.183583
.592026
.190699
74
75
3.263112
0.551305
3.271004
0.559197 '
3.278755
0.566948
3.286369
75
76
.364369
.524572
.372845
.533048
.381157
.541360
.389315
76
77
.473527
.497284
.482647
.506404
.491580
.515337
.500334
77
78
.591376
.469469
.601198
.479291
.610802
.488895
.620199
78
79
.718776
.441129
.729363
.451716
.739700
.462053
.749795
79
80
3.856698
0.412291
3.868124
0.423717
3.879257
0.434850
3.890112
80
81
2.006181
.382951
2,018518
.395288
2.030514
.407284
2.042187
81
82
.168451
.353157
.181788
.366494
.194728
.379434
.207293
82
83
.344726
.322857
.359138
.337269
.373087
.351218
.386601
83
84
.536495
.292006
.552090
.307601
.567143
.322654
.581693
84
85
2.745303
0.260456
2.762168
0.277321
2.778403
0.293556
2.794053
85
86
.972718
.227610
.990948
.245840
1.008442
.263334
1.025259
86
87
1.221005
.192627
1.240749
.212371
.259635
.231257
.277733
87
88
.491896
.152800
.513211
.174115
j .533528
.194432
.552937
88
89
.797745
.110931
.821707
.134893
.844416
.157602
.865997
89
90
0.147748
0.073302
0.174561
0.100115
0.199815
0.125369
0.223681
90
91
.540005
.038935
.569250
.068180
.596649
.095579
.622422
91
92
.975541
.004681
1.007268
.036408
1.036834
.065974
1.064516
92
93
1.461753
1.970498
.496258
.005003
.528222
.036967
.557995
93
94
2.006190
.937037
2.043598
1.974445
2.078038
.008885
2.109946
94
95
2.613993
T. 900145
2.654244
T. 940396
2.691078
T. 977230
2.725031
95
96
3.325234
.865871
3.369758
.910395
3.410138
.950775
3.447080
96
97
4.127452
.830967
4.174798
.878313
4.217486
.921001
4.256352
97
98
| 5.083166
.771319
5.134712
.822865
5.180783
.868936
5.222432
98 223
TABLE LXXYI.
GENERAL VALUATION OF JOINT LIFE POLICIES BY ANNUAL PREMIUM ON 1,000.
4 PER CENT.
Equal
Ages.
x-\-n.
March.
10^m.
Feb.
Him.
Jan.
12m.
Equal
Ages.
x + n.
995.192
998.397
1000.000
Xb'.
Xb.
XV.
Xb.
Xb'.
Xb.
XV.
54
0.979770
4.119582
0.982420
4.122216
0.985054
4.123527
0.986365
54
55
0.965904
4.153096
0.968635
4.155809
0.971348
4.157160
0.972699
55
56
.951522
.187700
.954339
.190500
.957139
.191893
.958532
56
57
.936637
.223523
.939552
.226418
.942447
.227859
.943888
57
58
.921201
.260666
.924221
.263665
.927220
.265150
.928711
58
59
.905226
.299259
.908362
.302371
.911474
.303919
.913022
59
60
0.888706
4.339457
0.891968
4.342695
0.895206
4.344305
0.896816
60
61
.871638
.381425
.875040
.384800
.878415
.386478
.880093
61
62
.854003
.425319
.857556
.428844
.861081
.430595
.862832
62
63
.835806
.471351
.839527
.475040
.843216
.476873
.845049
63
64
.817045
.519727
.820948
.523596
.824817
.525518
.826739
64
65
0.797726
4.570693
0.801831
4.574760
0.805898
4.576779
0.807917
65
66
.777833
.624484
.782157
.628766
.786439
.630891
.788564
66
67
.757391
.681414
.761958
.685933
.766477
.688174
.768718
67
68
.736369
.741744
.741195
.746518
.745969
.748885
.748336
68
69
.714814
.805878
.719932
.810935
.724989
.813441
.727495
69
70
0.692712
4.874149
0.698141
4.879511
0.703503
4.882168
0.706160
70
71
.670082
.946994
.675855
.952692
.681553
.955513
.684374
71
72
.646932
3.024857
.653080
3.030918
.659141
3.033917
.662140
72
73
.623264
.108221
.629818
.114676
.636273
.117868
.639465
73
74
.599142
.197701
.606144
.204590
.613033
.207995
.616438
74
75
0.574562
3.293853
0.582046
3.301209
0.589402
3.304841
0.593034
75
76
.549518
.397321
.557524
.405183
.565386
.409060
.569263
76
77
.524091
.508913
.532670
.517327
.541084
.521473
.545230
77
78
.498292
.629396
.507489
.638403
.516496
.642837
.520930
78
79
.472148
.759662
.482015
.769310
.491663
.774054
.496407
79
80
0.445705
3.900703
0.456296
3.911041
0.466634
3.916119
0.471712
80
81
.418957
2.053555
.430325
2.064633
.441403
2,070067
.446837
81
82
.391999
.219505
.404211
.231383
.416089
.237202
.421908
82
83
.364732
.399708
.377839
.412431
.390562
.418655
.396786
83
84
.337204
.595771
.351282
.609406
.364917
.616067
.371578
84
85
0.309206
2.809158
0.324311
2.823756
0.338909
2.830874
0.346027
85
86
.280151
1.041449
.296341
1.057056
.311948
1.064655
.319547
86
87
.249355
.295107
.266729
.311813
.283435
.319931
.291553
87
88
.213841
.571515
.232419
.589332
.250236
.597973
.258877
88
89
.179183
.886555
.199741
.906184
.219370
.915675
.228861
89
90
0.149235
0.246303
0.171857
0.267805
0.193359
0.278169
0.203723
90
91
.121352
.646751
.145681
.669788
.168718
.680864
.179794
91
92
.093656
1.090538
.119678
1.115089
.144229
1.126862
.156002
92
93
.066740
.585856
.094601
.612038
.120783
.624559
. 133304
93
94
.040793
2.139669
.070516
2.167488
.098335
2.180756
.111603
94
95
0.011183
2.756522
0.042674
2.785883
0.072035
2.799850
0.086002
95
96
1.987717
3.481126
.021763
3.512695
.053332
3.527659
.068296
96
97
.959867
4.292022
1.995537
4.324985
.028500
4.340573
.044088
97
98
.910585
5.260434
.948587
5.295377
1.983530
5.311848
.000001
98 224
TABLE LXXYII.
FEMALE LIFE TABLE. FROM THIRTY OFFICES’ EXPERIENCE OF LOSSES AND
AMOUNTS INSURED.
AGE.
X.
Living.
k
Decre-
ment.
dg>.
Expectation
of Life.
Prob. of Dec.
dx
qx — -j-
Prob. Living.
p* = -p-
Lx
Xpx-
A/;,..
AGE.
X.
10
100,000
314
48.05
0.003140
0.996860
T.998634
5.000000
10
11
99,686
420
47.21
.004212
.995788
.998167
4.998634
11
12
99,266
510
46.40
.005140
.994860
.997762
4.996801
12
13
98,756
581
45.64
.005882
.994118
.997438
4.994563
13
14
98,175
631
44.91
.006426
.993574
.997200
4.992001
14
15
97,544
644
44.19
0.006603
0.993397
T.997123
4.989201
15
16
96,900
678
43.48
.006998
.993002
.996950
4.986324
16
17
96,222
745
42.79
.007741
.992259
.996625
4.983274
17
18
95,477
792
42.12
.008296
.991704
.996382
4.979899
18
19
94,685
820
41.46
.008659
.991341
.996223
4.976281
19
20
93,865
856
40.82
0.009120
0.990880
T.996021
4.972504
20
21
93,009
847
40.19
.009106
.990894
.996027
4.968525
21
22
92,162
943
39.56
.010233
.989767
.995533
4.964552
22
23
91,219
992
38.96
.010873
.989127
.995252
4.960085
23
24
90,227
990
38.38
.010973
.989027
.995208
4.955337
24
25
89,237
1,006
37.80
0.011274
0.988726
T.995076
4.950545
25
26
88,231
1,024
37.23
.011606
.988394
.994930
4.945621
26
27
87,207
998
36.66
.011442
.988558
.995002
4.940551
27
28
86,209
977
36.08
.011333
.988667
.995050
4.935553
28
29
85,232
975
35.49
.011440
.988560
.995003
4.930603
29
30
84,257
971
34.89
0.011524
0.988476
1.994966
4.925606
30
31
83,286
966
34.29
.011599
.988401
.994933
4.920572
31
32
82,320
916
33.69
.011126
.988874
.995141
4.915505
32
33
81,404
869
33.06
.010675
.989325
.995339
4.910646
33
34
80,535
903
32.42
.011212
.988788
.995103
4.905985
34
35
79,632
882
31.78
0.011076
0.988924
T.995163
4.901088
35
36
78,750
877
31.13
.011137
.988863
.995136
4.896251
36
37
77,873
878
30.47
.011274
.988726
.995076
4.891387
37
38
76,995
887
29.81
.011522
.988478
.994967
4.886463
38
39
76,108
863
29.16
.011338
.988662
.995048
4.881430
39
40
75,245
885
28.48
0.011763
0.988237
T.994861
4.876478
40
41
74,360
893
27.82
.012009
.987991
.994753
4.871339
41
42
73,467
821
27.15
.011174
.988826
.995120
4.866092
42
43
72,646
791
26.45
.010889
.989111
.995245
4.861212
43
44
71,855
805
25.74
.011203
.988797
.995107
4.856457
44
45
71,050
799
25.02
0.011244
0.988756
1.995089
4.851564
45
46
70,251
792
24.30
.011274
.988726
.995076
4.846653
46
47
69,459
757
23.57
.010900
.989100
.995240
4.841729
47
48
68,702
766
22.83
.011149
.988851
.995131
4.836969
48
49
67,936
803
22.08
.011820
.988180
.994836
4.832100
49
50
67,133
821
21.33
0.012230
0.987770
T.994656
4.826936
50
51
66,312
889
20.59
.013407
.986593
.994138
4.821592
51
52
65,423
955
19.87
.014597
.985403
.993614
4.815730
52
53
64,468
976
19.15
.015139
.984861
.993375
4.809344
53
54
63,492
1,009
18.44
.015891
.984109
.993043
4.802719
54 TABLE LXXVII.
225
FEMALE LIFE TABLE. FROM THIRTY OFFICES’ EXPERIENCE OF LOSSES AND
AMOUNTS INSURED.
AGE.
X.
Living.
x•
Decre-
ment.
d®.
Expectation
of Life.
Prob. of Dec.
dx
qx — -y-
tx
Prob. Living.
VX
AGE.
X. '
55
62,483
1,082
17.73
0.017318
0.982682
T. 992413
4.795762
55
56
61,401
1,138
17.03
.018532
.981468
.991876
4.788175
56
57
60,263
1,221
16.35
.020262
.979738
.991110
4.780051
57
58
59,042
1,302
15.67
.022051
.977949
.99D316
4.771161
58
59
57,740
1,360
15.02
.023554
.976446
.989648
4.761477
59
60
56,380
1,429
14.37
0.025345
0.974655
T. 988851
4.751125
60
61
54,951
1,501
13.73
.027315
.972685
.987972
4.739976
61
62
53,450
1,617
13.10
.030254
.969746
.986658
4.727948
62
63
51,833
1,690
12.49
.032605
.967395
.985604
4.714606
63
64
50,143
1,771
11.90
.035318
.964682
.984384
4.700210
64
65
48,372
1,856
11.31
0.038370
0.961630
1.983008
4.684594
65
66
46,516
1,940
10.74
.041705
.958295
.981499
4.667602
66
67
44,576
2,024
10.19
.045405
.954595
.979819
4.649101
67
68
42,552
2,107
9.65
.049515
.950485
.977945
4.628920
68
69
40,445
2,185
9.13
.054024
.945976
.975880
4.606865
69
70
38,260
2,257
8.62
0.058991
0.941009
T. 973594
4.582745
70
71
36,003
2,322
8.13
.064495
.935505
.971046
4.556339
71
72
33,681
2,376
7.65
.070544
.929456
.968229
4.527385
72
73
31,305
2,416
7.20
.077176
.922824
.965119
4.495614
73
74
28,889
2,442
6.76
.084532
.915468
.961643
4.460733
74
75
26,447
2,449
6.34
0.092599
0.907401
1.957799
4.422376
75
76
23,998
2,435
5.93
.101467
.898533
.953534
4.380175
76
77
21,563
2.397
5.55
.111163
.888837
.948822
4.333709
77
78
19,166
2,335
5.18
.121829
.878171
.943579
4.282531
78
79
16,831
2,246
4.82
.133445
.866555
.937796
4.226110
79
80
14.585
2,133
4.49
0.146245
0.853755
1.931333
4.163906
80
81
12,452
1,994
4.17
.160134
.839866
.924210
4.095239
81
82
10.458
1,833
3.88
.175273
.824727
.916310
4.019449
82
83
8,625
1,654
3.59
.191768
.808232
.907536
3.935759
83
84
6,971
1,462
3.33
.209725
.790275
.897778
3.843295
84
35
5,509
1,262
3.08
0.229081
0.770919
1.887009
3.741073
85
86
4,247
1,062
2.84
.250059
.749941
.875027
3.628082
86
87
3,185
868
2.62
.272527
.727473
.861817
3.503109
87
88
2,317
687
2.42
.296503
.703497
.847262
3.364926
88
89
1,630
- 526
2.23
.322700
.677300
.830781
3.212188
89
90
1,104
387
2.05
0.350544
0.649456
T. 812550
3.042969
90
91
717
272
1.89
.379358
.620642
.792841
2.855519
91
92
445
183
1.73
.411236
.588764
.769941
2.648360
92
93
262
116
1.59
.442747
.557253
.746052
2.418301
93
94
146
70
1.46
.479452
.520548
.716461
2.164353
94
95
76
39
1.34
0.513158
0.486842
1.687388
1.880814
95
96
37
20
1.23
.540541
.459459
.662247
1.568202
96
97
17
10
1.09
.588235
.411765
.614649
1.230449
97
98
7
4
.93
.571429
.428571
.632023
0.845098
98
99
3
3
.50
1.000000
.000000
00
0.477121
99 226
TABLE L X X VIII.
FEMALE LIFE. D, N, Ctx, AND CORRECTION OF AGE.
Example. An actual Age, 32 years 3 months, added to its Correction, gives 34 years, the equivalent
Age at which the equal Life Annuity and Premium are found in the preceding tables for Male Life.
4 PER CENT.
AGE.
X.
Da;*
ADa*.
N*.
AN*.
ANa; — ADa>
Life An’y.