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' + 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 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 nn uu 00 00 00 00 00 01 23 45 67 89 02 46 80 24 68 03 69 25 81 47 04 82 60 48 26 uu 1 9 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 i b 00 00 00 00 01 01 23 45 67 9x 02 46 80 24 69 03 69 25 81 48 04 82 60 48 27 1 & 1 Q 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 1 Q iu 00 00 00 00 11 01 23 45 67 9x 02 46 80 24 79 03 69 25 81 58 04 82 60 48 37 1 o 1 ft 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 1 ft X U 00 00 00 01 11 01 23 45 68 9x 02 46 80 25 79 03 69 25 82 58 04 82 60 49 37 10 1 7 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 00 00 00 11 11 01 23 45 78 9x 02 46 80 35 79 03 69 25 92 58 04 82 60 59 37 1 7 20 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 on 00 00 01 11 11 01 23 46 78 90 02 46 81 35 79 03 69 26 92 58 04 82 61 59 37 OQ 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 OQ 00 00 01 11 12 01 23 46 78 91 02 46 81 35 7x 03 69 26 92 59 04 82 61 59 38 <«o oft 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 Oft 00 00 11 11 22 01 23 56 78 xl 02 46 91 35 8x 03 69 36 92 69 04 82 71 59 48 29 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 OQ 00 00 11 12 22 01 23 56 79 xl 02 46 91 36 8x 03 69 36 93 69 04 82 71 5x 48 30 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 00 00 11 12 22 01 23 56 79 01 02 46 91 36 80 03 69 36 93 69 04 82 71 50 48 Qd. 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 Q/1 ! 00 01 11 22 23 01 24 56 89 02 02 47 91 46 81 03 6x 36 x3 6x 04 83 71 60 49 o~t 3ft 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 QQ 00 01 11 22 33 01 24 56 89 12 02 47 91 46 91 03 6x 36 x3 7x 04 83 71 60 59 OO 40 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 AC\ 00 01 12 22 33 01 24 57 89 12 02 47 92 46 91 03 60 37 03 70 04 83 72 60 59 /IQ 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 A O rtO 00 01 12 23 33 01 24 57 8x 12 02 47 92 47 91 03 60 37 04 70 04 83 72 61 59 Aft 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 A ft 00 01 12 23 34 01 24 57 8x 13 02 47 92 47 92 03 60 37 04 71 04 83 72 61 5x ftn 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 ftn 00 11 22 33 44 01 34 67 90 23 02 57 02 57 02 03 70 47 14 81 04 93 82 71 60 OU KC 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 OD 00 11 22 33 45 01 34 67 90 24 02 57 02 57 03 03 70 47 14 82 04 93 82 71 61 ob ftQ 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 ft Q 00 11 22 34 45 01 34 67 91 24 02 57 02 58 03 03 70 47 15 82 04 93 82 72 61 Oo fiO 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 cn 00 11 23 34 45 01 34 68 91 24 02 57 08 58 03 03 70 48 15 82 04 93 83 72 61 DU CQ 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 bo 00 11 23 34 55 01 34 68 91 34 02 57 03 58 13 03 70 48 15 92 04 93 83 72 71 bo art 01 23 45 6/ 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 00 12 23 44 56 01 35 68 xl 35 02 58 03 68 14 03 71 48 25 93 04 94 83 82 72 b 7 r?n 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 (U 00 12 23 45 50 01 35 68 01 35 02 58 03 68 14 03 71 48 25 93 04 94 83 82 72 7U *vn 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 7 <4 00 12 23 45 56 01 35 68 02 35 02 58 03 69 14 03 71 48 26 93 04 94 83 83 72 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 7 0 00 12 33 45 66 01 35 78 02 45 02 58 13 69 24 03 71 58 26 x3 04 94 93 83 82 70 >VQ 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 7 O 00 12 33 45 67 01 35 78 02 40 02 58 13 69 25 03 71 58 26 x4 04 94 93 83 83 70 on 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 Qn oU 00 12 34 45 67 01 35 79 02 46 02 58 14 69 25 03 71 59 26 04 04 94 94 83 83 oU Q/| 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 Q A 04 00 12 34 55 67 01 35 79 12 40 02 58 14 79 op: 6k) 03 71 59 36 04 04 94 94 93 83 84 q n 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 q n ob 00 12 34 56 67 01 35 79 13 46 02 58 14 7x 25 03 71 59 37 04 04 94 94 94 83 ob QQ 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 QQ oo 00 12 34 56 77 01 35 79 13 56 02 58 14 7x 35 03 71 59 37 14 04 94 94 94 93 oo QQ 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 QQ 00 12 34 56 78 01 35 79 13 57 02 58 14 7x 36 03 71 59 37 15 04 94 94 94 94 O KJ 90 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 an 00 12 34 56 78 01 35 79 13 57 02 58 14 70 36 03 71 59 37 15 04 94 94 94 94 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 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 r\r\ 05 05 05 05 05 06 28 40 62 84 07 41 85 29 63 08 64 20 86 42 09 87 65 48 21 UU 1 2 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 1 o 05 05 05 05 06 06 28 40 62 85 07 41 85 29 64 08 64 20 86 43 09 87 65 43 22 i ti i q 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 JLO 05 05 05 05 16 06 28 40 62 95 07 41 85 29 74 08 64 20 86 53 09 87 65 43 32 lo i 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 1 05 05 05 06 16 06 28 40 63 95 07 41 85 2x 74 08 64 20 87 53 09 87 65 44 32 ID 1 7 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 05 05 05 16 16 06 28 40 73 95 07 41 85 3x 74 08 64 20 97 53 09 87 65 54 32 17 20 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 o a 05 05 06 16 16 06 28 41 73 95 07 41 86 30 74 08 64 21 97 53 09 87 66 54 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 -JOi GO 62 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 ©x 4s- c© to Ki 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 Oi 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 62 Cm 02 h-i 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 *1 1G4.0 h-4 k-4 h-4 h-4 M Of Oi 0 Q Cn i—i 62 62 CO l—i CO 4- 62 CO 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 00 esD 00 0 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 •vj GO -Q O 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 =5 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 0q k G) 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 • 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 WS © bfl < £a CD M H o - 8.3 5** 73 O) 5 M _ QQ M o g s53 Deatl Claim & © 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 T3 a> 5 73 . 1 hf) ’H ' 73 o 5 o s 3~ 5 a §‘3 PO 73 . II wS “ cS d £ £d M tub 0 M w A 73 3 § gfl a* 73 o 3 73 . n HS .2 M H A §2 o 2 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 —4 *—4 »—4 H-4 h"4 CD OD -- b W I—1 o< 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 —i 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 ■€£ 1—1 to to h-4 to «» Existing. V* ►—4 © k^ On © 03 jo *0 OS l—i 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 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 « 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* Oi © GO h-1 © -3 -3 if* tO © 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 • 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 -3 00 Discon- tinued. 0 d ft > crq ■ CD to l—i to H to it* h-A 1—1 h-i h-* H w it* Died. H P h* 1—1 OOOMOiO l—i Cu © Oi GO bi bi bi oi 25 s M) O ZD P c+- bd y c+- 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 •1 'O • Is « s jf M H 1 ~~ £ g g S’3 ?•§ so wS d CD he < 03 03 Jh d rj ® OT hi: G *3 H ■ "d o § gn 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 ■ 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 •- 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* © *a 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 -i 62 05 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+- Pd 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 & d 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 *3 *3 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 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 -3 00 O 62 05 H OS H M Oe be Oe Oe a H S 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 -3 O 00 CO l(i *¥e 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 L. CO PS ►H 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 + • • • * - 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.