PRACTICAL LESSONS IN ACTUARIAL SCIENCE AN ELEMENTARY TEXT-BOOK CONTAINING, ALSO, ALL MORTALITY TABLES THAT HAVE EVER BEEN STANDARD ANYWHERE, WITH CORRESPONDING COMMUTATION COLUMNS. BY MILES MENANDER DAWSON CONSULTING ACTUARY AUTHOR OF " ELEMENTS OF LIFE INSURANCE," "ASSESSMENT LIFE INSURANCE," ETC. THE SPECTATOR COMPANY 95 WILLIAM STREET NEW YORK COPYRIGHTED, 1898, BY THE SPECTATOR COMPANY ERRATA AND EXPLANATIONS. IN so large and comprehensive a work as Practical Les- sons in Actuarial Science, it is not to be expected that the first edition will be wholly free from errors. But one error in the text has been reported, viz., on page 94, James J. Barker should be Jesse J. Barker. It is believed that the formulas are free from material errors. The following errors have been discovered in the tables: First, instead of 100,000 in D column, page 454, read 1,000,000. Second, The following columns had not been changed from the Farr notation to the Davies: N column, pages 316 and 317, and pages 318 and 319; S column, pages 408, 409, 410, 411, 414, 415, 416, 417. In using these columns in the formulas given in this book, the figures opposite age 21 should be used as if opposite age 20, those opposite age 22 as if opposite 21, and so on. Third, originally Craig's Modified Actuaries' Table was called in this book " Craig's Industrial Table, " Mr. Daw- son being or the opinion that it was suitable for use in industrial insurance. Mr. Craig protested against this form of putting it, on the ground that it subjected him to the inference of being responsible for that use of the table. Mr. Craig prepared the table for impaired lives only and desired that it should be so known. Accordingly, the name "Craig's Impaired Life Table" is used in this book in order that no misunderstanding may exist as to what Mr. Craig intended it for. A new introduction and suitable headings have been substituted for those which originally appeared. The use of the tables on pages 70 to 72, inclusive, and on pages 382 to 391, inclusive, and on pages 491 to 499, inclusive, has been sanctioned by the Institute of Actuaries of Great Britain. INTRODUCTION. AN elementary text-book in actuarial science has long been needed ; and a consciousness of that need, growing out of my own experience, prompts the preparation of this book. The student has either been compelled to turn to books from which he could only pick out what he needed piece- meal after a great waste of energy; or to books which were wofully incomplete; or to books which were barren of explanations intelligible to him. The best literature he has had, when beginning the study, has been Elizur Wright's books, none of which was written primarily as a text-book. Perhaps the best of these was the Massachusetts Reports, in which the instruction is delightfully clear, but only enough in any case to illustrate the point of Commissioner Wright's arguments, which were directed toward convincing his readers of the desirability of certain proposed laws. Naturally, at best such instruction has its manifest dis- advantages and does not tend toward a rounded education. The text of " The Principles and Practice of Life Insur- ance " is also excellent, but does not cover enough, and besides requires some elucidation before becoming com- prehensible to the beginner. All of these are now not to be recommended as first-books for students because they employ mainly a notation not corresponding to the Universal Standard Notation, adopted in 1895 by the International Congress of Actuaries. To start by studying them is to invite confusion; while, after the universal notation has been mastered, these may be learned without more effort than merely to add certain alternative symbols. The Universal Standard Notation, used in this text-book 4 INTRODUCTION. for the first time in an American book, is the notation also employed in the Institute of Actuaries' Text-Book; the Congress merely confirmed that system of notation. Unquestionably the Institute of Actuaries' Text-Book is the book of books in actuarial science. It is a veritable mine of information; but like other mines, it requires to be mined. And therein, especially when no teacher is at hand to explain, lies its inapplicability to the case of begin- ners, who would find it incomprehensible as a whole and would not know how to select what they needed. In America there are many who desire a working knowl- edge of the science, but whose mathematical attainments are not more than is included in elementary algebra. This book is designed for their easy comprehension ; and yet it is made to cover all the problems that the actuary or man- ager of a small company, issuing only policies of the usual form, will probably be required to deal with. Not much of the work is original and still less is original otherwise than in the mode of putting the argument. Its chief virtue lies in its simplicity and its arrangement. It is written straightforward in chapters of convenient size. Consequently it may be studied consecutively to advantage; though if not immediately interested, the student may skip the chapters on constructing mortality tables, etc., from page 39 to 72, inclusive. My thanks are due Messrs. Weeks, Frankland, Beckley and Robertson, of the New York Life Insurance Com- pany's actuarial force; Emory McClintock and John Tat- lock, of the Mutual Life Insurance Company; Win. A. Marshall and G. L. Plumley, of the Home Life Insurance Company, and W. D. Whiting, consulting actuary, for substantial assistance rendered. Thanks are also due the Connecticut Mutual Life Insur- ance Company for consent to republish tables copyrighted by them. TABLE OF CONTENTS. Introduction 5 Scope of the Science 9 Rudiments of Probabilities, I *3 Rudiments of Probabilities, II 17 Simple Problems in Rate-Making , 21 Probabilities of Mortality 27 Annual Life Contingencies 3 1 Mortality Tables 35 Mortality Tables, Graphic System of Graduation, 1 39 Mortality Tables, Graphic System of Graduation, II 43 Mortality Tables, Graduation by Finite Differences 48 Mortality Tables, Graduation by Formula 52 Mortality Tables, Force of Mortality Employed in Graduation 57 Interpolation and Graduation by Interpolation 62 Modified Graduation by Interpolation, " Woolhouse's Formula " . 66 Testing the Accuracy of Graduated Tables 69 Life Probabilities 73 Pure Endowments and Annuities 77 Commutation Columns N and D 80 Deferred Annuities 83 Temporary Annuities 85 Conversion Formulas, Life, Temporary and Deferred Annuities ... 87 Deferred Temporary and Other Intercepted Annuities 89 Whole Life Insurance 95 Whole Life Insurance, Commutation Columns C and M 97 Deferred, Temporary and Deferred Temporary Insurances 100 Definition of Premium 92 Conversion Formulas Annuities and Insurances 103 Endowment Insurances 107 -Endowments and Insurances with Return of Net Single Premium . i n Endowments and Insurances with Return of Gross Single Premium 1 14 Varying Benefits in Annuities and Insurances 117 Temporarily Varying Benefits in Annuities and Insurances 121 Inter- Relation Between Single Premiums 124 Annual Premiums Life and Limited Payment-Life 125 Annual Premiums Term and Deferred Insurances, Annuities and Endowments. 128 Annual Office or Gross Premiums. The Loading 132 Endowments and Insurances with Return of Net Annual Pre- miums, Continuous Throughout Term 136 Endowments and Insurances with Return of Net Annual Pre- miums, Limited Payments 140 Endowments and Insurances with Return of Gross Annual Pre- mium, Continuous Throughout Term 144 Endowments and Insurances with Return of Gross Annual Pre- miums, Limited Payments 148 Age and Premiums Known to Approximate the Term 1 53 Valuation Gross and Aggregate 160 Net Valuation, Annuities, Insurance and Endowments, Prospect- ive Method.. . 165 6 TABLE OF CONTENTS. PAGE Net Valuation, Annuities, Insurances and Endowments, Retro- spective Method 169 Tontine Accumulation and Retrospective Valuation 174 Valuation Additional Formulas 179 Valuation Return-Premium Insurances 182 Valuation True Reserves 184 Actual Insurance 187 Cost of Insurance 190 Computing Actual Insurances, etc 193 Distribution of Surplus 195 Tontine and Other Long Term Dividends 201 Reversionary Dividends, Applied as Insurance or Annuity 205 Participation of Annuities 209 Surrender Values 212 Insurance Value 217 Individual Accounts 224 Insured Instalment Loans or Advanced Endowments 227 - Expectation of Life 230 Joint Life Annuities. 233 Joint Life Annuities Equal Age Formulas 236 Joint Life Insurances Single Premiums 239 Joint Life Insurances Annual and Limited Premiums 242 Joint Life Endowments and Endowment Insurances 245 Joint Life Net Valuations 247 Joint Life Valuations. Retrospective Continuous Methods. . . . 250 Probabilities of Survivorship 252 Survivorship Insurance 254 Equivalent Equal Ages in Survivorship Insurances ... 258 Reversionary or Survivorship Annuities .265 Reversionary Annuities. The Survivorship Insurance of a Tempo- rary or Deferred Annuity 267 Reversionary Annuities. Combinations 271 Conversion Formulas. Joint Life, Successive Lives and Survivor- ship Annuities 274 Valuations Survivorship Contracts 276 Sickness Insurance 279 Conversion Tables 283 Correspondences and Equivalents 287 Comparative Notation 289 Short Methods of Computation 291 Calculating Machines and Tables 297 Interpolation 299 The Use of Prepared Tables 304 TABLES. AMERICAN EXPERIENCE TABI,E, INTRODUCTION TO 309 American Experience Mortality Table 311 Commutation Columns, 4^2 Per Cent 312 Commutation Columns, 4 Per Cent 314 Commutation Columns, 3^ Per Cent 316 Commutation Columns, 3 Per Cent 318 Valuation Columns, 3 Per Cent, 3^ Per Cent, 4 and 4^ Per Cent 320 TABLE OF CONTENTS. 7 PAGE ACTUARIES, COMBINED EXPERIENCE OR 17 OFFICES' TABLE, INTRODUCTION TO 323 Actuaries' Mortality Table ' 325 Commutation Columns, 4 Per Cent 326- Commutation Columns, $ l / 2 Per Cent 328- Commutation Columns, 3 Per Cent 330 Valuation Columns, 3 Per Cent, 3^ Per Cent and 4 Per Cent. 332. Wright's Table for Single Premiums for Temporary Insurance, 4 Per Cent Insert CRAIG'S MODIFIED ACTUARIES' TABLE (IMPAIRED LIVES), INTRO- DUCTION TO 335 Craig's Modified Actuaries' Table 337 Commutation Columns, 4 Per Cent 338 NORTHAMPTON TABLES FOR INDUSTRIAL LIVES, INTRODUCTION TO 3411 Northampton Mortality Table 343. Commutation Columns, 4 Per Cent 344 Commutation Columns, 3 Per Cent 348 Commutation Columns ( D and N), 5 Per Cent 350^ Annuities at Various Rates 351 Joint Life Annuities, 4 Per Cent, 5 Per Cent, 6 Per Cent 352 INSTITUTE OF ACTUARIES' MORTALITY TABLES, INTRODUCTION TO 363 H m Table ( Woolhouse's Formula) 364 Commutation Columns, 4^ Per Cent 366 Commutation Columns, 4 Per Cent . 368 Commutation Columns, $ l / 2 Per Cent 370 Commutation Columns, 3 Per Cent 372 H m 5 Table (Woolhouse's Formula) 375 Commutation Columns, 4 Per Cent 376 Commutation Columns, 3^ Per Cent 378 Commutation Columns, 3 Per Cent 380 H m Table (Makeham's Formula), King & Hardy's Graduation 382 Commutation Columns, 4^ Per Cent 384 Commutation Columns, 4 Per Cent 386 Commutation Columns, 3^ Per Cent 388 Commutation Columns, 3 Per Cent 390 H F Table (Woolhouse's Formula) , 392 Commutation Columns, 4%, Per Cent 394 Commutation Columns, 3^ Per Cent 396 Commutation Columns, 4 Per Cent 398 Commutation Columns, 3 Per Cent 400 FARR'S ENGLISH MORTALITY TABLES, No. 3, INTRODUCTION TO. . 403 Farr's Mortality Table No. 3, Males 4OS Commutation Columns, 5 Per Cent 406 Commutation Columns, 4 Per Cent 408 Commutation Columns, 3 Per Cent 410 Farr's Mortality Table No. 3, Females 413 Commutation Columns, 4 Per Cent 414 Commutation Columns, 3 Per Cent 416 CARLISLE MORTALITY TABLE, INTRODUCTION TO 419 Carlisle Mortality Table 421 Commutation Columns, 4 Per Cent 422 Commutation Columns, 3^ Per Cent 424 Commutation Columns, 3 Per Cent 426 8 TABLE OF CONTENTS PAGE AMERICAN TROPICAL MORTALITY TABLE, INTRODUCTION TO 423 American Tropical Tables 431 Commutation Columns, 4 Per Cent 432 Commutation Columns, 3^ Per Cent 434 Commutation Columus, 3 Per Cent 436 TWENTY-THREE GERMAN OFFICES' MORTALITY TABLE, INTRO- DUCTION TO 439 Twenty-three German Offices' Table 441 Commutation Columns, 4 Per Cent 442 Commutation Columns, 3^ Per Cent 444 Commutation Columns, 3 Per Cent 446 FRENCH ACTUARIES' MORTALITY TABLES, INTRODUCTION TO .... 449 French Actuaries' Mortality Table (Insurances) A. F 451 Commutation Columns, 4 Per Cent 452 Commutation Columns, 3^ Per Cent 454 Commutation Columns, 3 Per Cent 456 French Actuaries' Mortality Table (Annuities) R. F 459 Commutation Columns, 4 Per Cent 460 Commutation Columns, 3^ Per Cent 462 Commutation Columns, 3 Per Cent 464 DES PARCIEUX MORTALITY TABLE, INTRODUCTION TO 467 Des Parcieux Mortality Table 468 Commutation Columns, 4 Per Cent 469 DUVILLARD MORTALITY TABLE, INTRODUCTION TO 471 Duvillard Mortality Table 472 Commutation Columns (Annuities), 4 Per Cent 474 COMPANY TABLES, INTRODUCTION TO ... 475 Australian Mutual Provident Mortality Table 476 Canada Life Table 477 Canada Life (5) Table 478 POPULATION TABLES, INTRODUCTION TO 479 Milne's Swedish Tables, Males 480 Milne's Swedish Tables, Females 481 New Zealand Table, Males 482 New Zealand Table, Females 483 JOINT LIFE TABLES, INTRODUCTION TO 485 AMERICAN TROPICAL TABLES. Force of Mortality 487 Commutation Columns, 4 Per Cent 488 HM (KING & HARDY) TABLES. Annuities, 4^ Per Cent 491 Annuities, 4 Per Cent 492 Annuities, 3^ Per Cent 493 Annuities, 3 Per Cent 494 CONVERSION TABLES: Single Premiums 496 Single Premiums, Differences 497 Annual Premiums 498 Annual Premiums, Differences 499 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. SCOPE OF THE SCIENCE. To define actuarial science is rendered somewhat difficult by the circumstance that not all who use the term mean by it the same thing. As used in other countries, it is so broad as to cover all kinds of expert knowledge of the use of com- pound interest and discount in solving financial problems, as well as a knowledge of the application of the laws of probability to insurance. Thus the formulation of plans for a building association and the distribution of its profits among its members are there regarded as actuarial func- tions. This comprehensive significance of the term, how- ever, is only beginning to be recognized in the United States. Here we have in the past commonly understood the field of an actuary to be limited to technical life insurance as practiced by the regular companies. When actuaries have on occasion undertaken other financial problems, such as the distribution of building association's profits, they have been rather apologetic about it, as if it were not altogether pro- fessional. And it is true that in these days of prepared tables and ready reckoners the solution of such problems is not attended with difficulties which are insuperable to per- sons who are not actuaries. Eliminating this, then, we have remaining a distinctive field for actuarial science which may be defined, therefore, as the application of the laws of proba- bility to insurance. This definition calls for an accurate conception of insur- ance. Insurance is furnishing to persons, suffering a loss in a specified manner, indemnity through apportioning the io PRACTICAL LESSONS IN ACTUARIAL SCIENCE. loss among a number of persons who were, during a stated time, subject to the risk of a similar loss and who were cov- ered by a similar contract of indemnity. When this appor- tionment is made after the losses actually occur, the opera- tion may be simple enough, especially when the exposures are considered to involve equivalent hazards. In that case, it is only necessary to divide the loss ratably among the insured, each bearing his share. But, in practice, ordinarily, this division needs to be made before the losses occur, though in many cases subject to future revision ; in other words, it is necessary to compute in advance the probability of the loss for each individual. To do this, it is required to deal with large aggregates so as to reach reliable averages, it being the primary law of probabilities that when, in a large number of cases in which an event might either hap- pen or not happen, it does happen a certain number of times, the chance of its happening is ascertained by dividing the number of times it did happen by the total number of cases. Sometimes, these simple probabilities are very diffi- cult to determine ; and, when it is remembered that this primary problem is multiplied and involved with other more intricate problems, it appears that actuarial science has an extended and peculiar field. Insurance is of the nature of a "hedge." We find our- selves, in the nature of things, subject to certain hazards, such as of loss of property or health or life ; we cannot alto- gether escape these hazards, however careful we may be. We can, however, in a degree, guard against the worst con- sequences by "hedging." The office of a "hedge" is to cancel a hazard already incurred. But the "hedge " is nec- essarily of the nature of a bet; and it follows, therefore, that the same technical knowledge of probabilities which fits one for the direction of an insurance company, designed to PRACTICAL LESSONS IN ACTUARIAL SCIENCE. n diminish the inequalities of fortune, also fits one to direct a lottery, designed to increase those inequalities. Such employment of the skill attained by a study of the science is closely analogous to the employment of medical skill to destroy life instead of to preserve it. Speaking absolutely, there is no such thing as chance; all phenomena are the results of causation. But from the view- point of the individual, everything is chance which he does not voluntarily cause or fail to prevent. If we could discern and compute the significance of all causes, there would be no question about probability in anything. We are able to do this to a considerable degree and one of the first tasks of the actuary is to so discriminate that, so far as he is able to determine prima facie, each individual in the aggregate is equally liable to the same hazard. Thus, for instance, it would be foolish to derive the probability of the death within the year of a man aged eighty, from division of the number dying by the lives exposed in a total made up of all ages ; for, prima facie, the hazards on the individuals making up the aggregate are not the same. The advance payment by the assured to cover his indem- nity is called a premium. When the actuary has determined from a consideration of aggregates what the hazard is, a basis for this premium has been reached. Another problem at once presents itself, namely: to provide for expenses, a provision which must also be incorporated in the premium. And, finally, if the premium covers insurance for a consid- erable term, there enters into the calculation the matter of the interest which current balances of premium may earn, which interest is, in a very real sense, a part of the premium itself. The adjustment of the actual costs for insurance and expense to the assumed costs after the term has expired is 12 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. the next problem in order, together with the proper appor- tionment of the net profit or loss among the stockholders or policyholders. This is an operation often requiring the greatest nicety of the actuary's skill. And as the policies of a company are not precisely concurrent, beginning on the same day and expiring on the same day, it follows that in making this adjustment the question of a sufficient reserve to cover the unexpired contracts is directly involved. While the three items of rate-making, surplus or loss apportionment and reserve computation comprise the major part of the mathematical phase of actuarial science, there are other intricate problems though of less moment. And, in particular, it is always assumed that an actuary is versed in all technical questions relating to life insurance, including the legal construction of contracts. The want of such knowl- edge will seriously impede his success. The Institute of Actuaries of Great Britain recognizes this by requiring an examination on these matters as well as insurance mathe- matics. This little text-book will treat insurance mathematics only, and it only so far as it can be comprehended and made use of by persons who are familiar with ordinary algebra. RUDIMENTS OF PROBABILITIES. THE science of probabilities includes so much more than is necessary as a basis for applied actuarial science, that a knowledge of the mere rudiments of probabilities is all that the actuarial student needs. To be sure, he cannot have too much and, if he is prepared for it and has the required mathematical training 1 , he will do well to make a more extended study of the discoveries in the realm of this science. In this elementary treatise, we can only treat of the simpler problems, a knowledge of which is essential. The science of probabilities assumes the individual as a microcosm to reproduce the characteristics of the group as a macrocosm. We have already seen that this assumption is not really true and that if, for instance, we could accu- rately calculate the causes at work, it would be foreseen that of a certain group just the persons who actually die within a year were the only ones who were likely to die and that all others were certain to survive. But we cannot fore- see this ; and we can, by deducing averages from past expe- rience, forecast with some accuracy how many of the whole group will die. Therefore, having selected as carefully as possible a group in which the causes known to be at work apply equally to each individual, we say that the probability of the death of each is represented by the fraction, formed by dividing the number of actual deaths in such a group by the number of lives exposed in the group. Stated as a bald proposition in the science of probabilities, this is : If an event may happen in i o ways and not happen in 990 ways, the probability of its happening is represented by a fraction, formed by dividing the ways it may happen 14 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. (10) by the sum of the ways it may happen and the ways it may not happen (10 + 990), or 10 10 i = = = .01. 10 + 990 1000 ioo For convenience in carrying forward our investigations, let us hereafter designate the probability of the event hap- pening by the letter p, and the probability of its not hap- pening by the letter q. By applying the rule already given, we find that the probability in the case stated of its not hap- pening, is = = 990 = _ 990 99 10 + 990 ~ 1000 ioo If we add together the probabilities of its happening (.01), and of its not happening (.99), we have .01 + .99 = i for the sum. That is, expressed algebraically, p + q= i Or, in words, the sum of the alternative probabilities that an event will happen and that it will not happen exhaust all the probabilities and make a certainty. The fractions are, then, complementary to one another. It follows that the value of either of them may be determined when the value of the other is known. For, by a simple algebraic process, we derive from the foregoing equations : p= i-q and q= i-p These simple deductions cover all the permutations of simple probabilities and they will be found very useful in the practice of actuarial science. But probabilities are often complex, involving three or any number of events which may happen to the individuals of a group. Taken singly, the probability of each event may be computed by reference to the rule given ; but the PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 15 application of the same rules would hardly suffice to compute the value of the complex probabilities. For instance, the probability that both of two events would happen to a single individual would not be the sum of the probabilities that each would happen ; for it is sufficiently plain that, unless one were the cause of the other, one might very well hap- pen without the other happening. It is evident that the probability of all the events happening to one individual becomes smaller and smaller, the more the events are mul- tiplied. To illustrate this mathematically, let us take a group of 1,000,000, in which one event happens to 100,000, another to 10,000 and a third to 1000. Let us designate the proba- bility of the first happening as p 1 , the probability of the sec- ond as p 2 and of the third as p 3 . By the application of the rule for simple probabilities, we find 100,000 i 1,000,000 10 '* 10,000 i p = - - = =.oi 1,000,000 100 3 1,000 i p == === =^.001 1,000,000 1,000 Suppose the first event to have happened to its 100,000 individuals in the group ; each individual in this new and smaller group of 100,000 to which the first event has hap- pened, is on that account no more liable to the happening of the second event. We have found the probability of the second event (p 2 ) to be .01 for each individual, from which we find that in the smaller group of 100,000 only 1,000 would have the second event happen. Taking this number (1000) as a new group, by applying the value of p 3 (.001), we find that of it only i person would have the third event happen. To but one out of the 1 6 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. original 1,000,000, therefore, all three events would happen. The chance, then, of the three events happening to any one individual is not measured by adding the probabilities (. i + .01 + .001 = .in), but by multiplying them (. i x -01 X .001 = .00,000,1). Expressed algebraically, this is p 1 ' 281 " 13 = p 1 x p 2 x p 3 or, in words, the probability that a number of events will happen to one individual is equal to the product of the probabilities that each will happen. We have seen that the probability that an event will not happen is equal to unity less the probability that it will hap- pen (q = i p). It is equally evident that the sum of the probabilities that all the events will happen to one individual and that at least one of the events will fail to happen exhausts all the probabilities and, therefore, also equals i ; from which it appears that the probability that at least one event will fail is equal to the difference between unity and the probability that all the events will happen. Expressed algebraically, this is: qvor. = l _ (pi x p* x p ). RUDIMENTS OF PROBABILITIES. II. WE have considered the probability that all will happen and that at least one will fail to happen to one individual of a group and have found them to be complementary to each other. Let us now consider the probability that all will fail, which on the surface would have appeared to be com- plementary to the probability that all will happen. It is at once evident that the probability that all will fail is decreased instead of increased when the number of different events is multiplied. Reverting to our group of 1,000,000, in which the first event will happen to 100,000, the second event to 10,000 and the third to 1000, we find that, taken severally, the probabilities that each will fail are as follows : 900,000 9 1,000,000 10 ~ 990,000 99 q = - = =.QO 1,000,000 100 999, 999 Q 000 Q t ^ e rst an( j seconc i both failing to 1,000,000 i8 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. be 89I ' and of all three failing 89 ' r 9 ; of these frac- I,OOO,OOO I,OOO,OOO tions, the second is the product of the first by the value of q 2 , the probability of the second failing (. 99) ; and the third is the product of that product by q 3 , the probability of the third failing (.999). Algebraically expressed, this means q i,-i q i X q 2 xq 3 . Just as the probability that all will fail is decreased by the multiplication of the different events, so the probability that at least one will happen is increased. And these two things are found to be complementary probabilities ; for it is evident that to all in the original group of 1,000,000 to which all of the events have not failed to happen, at least one of them must have happened. Therefore the sum of those to which all the events failed to happen and of those to which at least one event happened equals the whole num- ber. Consequently, the probability that all will fail plus the probability that at least one will happen is equal to cer- tainty or unity; and the probability that at least one will happen is equal to unity less the probability that all will fail. Algebraically expressed, this means P 1 ' 2or3 = i (q 1 X q 2 X q 3 ). To ascertain the probability that the first will happen and the other two events fail to happen, let us recur to the origi- nal group of 1,000,000, to 100,000 of which the first event happens, to 10,000 the second and to 1000, the third. Out of the 100,000 to which the first event happens, the second fails to happen to q 2 x 100,000 or 99,000; and out of this 99,000, the third fails to happen to q 3 x 99,000 or 98,901. PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 19 Thus we find the probability that the first will happen and the second fail is - and that the third will also fail 1,000,000 ; s 9 >9 01 /pk e f ormer O f these fractions is equal to the 1,000,000 product of p 1 , the probability that the first will happen, into q 2 , the probability that the second will fail (. i x -99 = .099) ; and the latter of these fractions is the product of this product into q 3 , the probability that the third will fail (.099 X .099= .09,890,1). Expressed algebraically, this means An equivalent result would be reached if we sought the value of the probability that the second would happen and the first and third fail ; hence, it is evident that the formula is of general application. We have found that the probability that two or more events will happen is the product of the several probabilities that each will happen (p 1 x P 2 X . . ) ; that the probability that two or more will fail is the product of the several probabili- ties that each will fail (q x q* X ) ; an d that the probability that one will happen and the others fail is the product of the probability that one will happen into the probability that the others will fail (p 1 x q 2 X q 3 X ) We might in the same way develop the fact that the probability that one will fail and the others happen is the product of the probability that one will fail into the probability that the others will happen (q 1 x p 2 x p 3 X .) And it follows, in order, that the combined proba- bility that two or more events will happen and two or more fail is the product of the probability that the two or more will happen into the probability that the two or more will fail. 20 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. Algebraically expressed, this is p'''q M =(p'xp s )(q 3 xq 4 ). All other compound probabilities may be derived from an elaboration on these formulas. Corollary i. If events be such that in the nature of things they cannot concur, that is, occur to the same individual of the group, the probability that either will happen must be computed separately and the probability that one will occur is equal to the sum of the separate probabilities. The rea- son this does not apply to the probability that at least one will happen when the events can concur, is that in that case there is the additional probability that they may concur. Corollary 2. If two events stand related as cause and effector in such manner that the second cannot occur unless the first has happened, the situation is materially modified. In the latter case, the probability that the second event will happen is exactly the equivalent of the probability that both will happen, viz; p 1 x P 8 - SIMPLE PROBLEMS IN RATE-MAKING. ONE year is the ordinary unit of time in insurance ; pre- miums for a longer or shorter time are commonly deduced from the annual probability by some means, more or less scientific and exact. In practically all kinds of insurance, excepting only against death and illness, the hazard is toler- ably uniform and, at least, does not necessarily increase continuously, resulting in certainty. Consequently, by separating out a group, to each of which, so far as prima facie appears, the event insured against is equally likely to happen, and observing to how many this event has actually happened in the space of one year, we can ascertain the probability of its happening to each of a like group to whom we furnish insurance. Thus, for instance, suppose that we have separated out from statistical data a group of 1,000,000 dwelling houses which have been exposed to the risk of fire for one year each and that we find that of this number 1000 have burned completely, we find that the probability of a dwelling house being completely destroyed by fire within one year is . oo i . This, however, evidently, does not cover 1,000,000 the total probability of loss by fire, since it might easily happen that the damage done by fires which do not totally consume the buildings is yet greater. Therefore, let us transfer our attention from numbers to amounts, assuming the dwellings to be worth $1000 each, or an aggregate of $1,000,000,000 and the damage to be a total of $2,000,000. Applying our formula, we find the risk of damage by fire to be valued for each dwelling at - - = .002. This 1,000,000,000 22 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. would constitute the net premium, excepting that, if one wished to be very exact, he might ascertain at what period in the year, on the average, the losses occurred or were pay- able and discount the net probability for that term. This might be accomplished by reducing the total damage of $2,000,000 to a present value at the beginning of the year before dividing by the aggregate amount at risk to ascertain the probability. The quotient would be the net, discounted probability and constitute a premium, sufficient when im- proved at interest to meet the demands upon it as they accrue. Whether numbers exposed and number of losses or amounts at risk and amount of losses should be employed in computing a probability for rate-making purposes, even when no partial damages enter into the account and all losses are total, has been a warmly disputed question. It is almost, if not quite, unavoidable to use amounts when making calculations of probabilities, involving partial dam- age. Where this is not the case, the result would be the same, whether by number or amounts, if the amount of each individual risk were the same. When the amounts of the individual risks are different, it appears clear that the pre- miums should vary in direct ratio to the amount at risk ; which would indicate that the probability of loss should be ascertained for some unit of amount at risk and this net pre- mium be increased or diminished according as the insurance is for a greater or less amount than this unit. But the probability deduced from numbers will give the basis for the probability of loss for any unit of amount at risk ; and it follows that the correct system, when dealing with cases not involving partial damages, is to compute the probability from the numbers involved. The distinction is, however, ordinarily of no great moment. PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 23 The assumption, in selecting one year as the unit of time, that the risk of loss,, one year with another, is a reasonably uniform one, is scarcely true in more than a few branches of insurance, possibly such as individual accident. In fire, there is the conflagration hazard which actuaries have never had a fair opportunity to compute ; these conflagrations recur in periods commonly much longer than one year, caused by immediate exposures as in cities or by wood or prairie fires. In credit, surety and allied lines of insurance there is the panic hazard, also recurring commonly at longer periods than one year. If an opportunity were offered actuaries to devise from the data, which insurance companies might fur- nish, premiums and methods of reserve to meet these extraor- dinary contingencies, the computations would doubtless involve operations as nice and as difficult as any arising in life insurance to the problems of which actuarial attention has been mainly directed. In making premiums for a shorter term than one year, commonly more than a mere pro rata part of the annual pre- mium is demanded. This addition to the annual rate is made arbitrarily and more as a penalty than to cover any increase in the hazard. In fact, it is generally assumed in actuarial calculations that the hazard is uniform throughout the year. As will be seen later in these studies, actuaries do sometimes treat the risk of death as a regularly increas- ing one throughout each year ; but most of their calculations proceed on the contrary assumption, which is, in fact, quite as near the fact. The fact is that the law of mortality, while obedient to a general tendency to increase, after the era of childhood is past, is not uniform throughout short periods of time. Mortality at all ages is increased at cer- tain seasons of the year by variable weather, excessive heat or cold or humidity. Likewise, fires are more common and 24 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. destructive in the coldest weather; and there are two or more seasons of liquidation of commercial credits every year when the risk of loss in credit insurance is augmented. Not- withstanding which, no attempt has been made to vary the rates in accordance with these fluctuations in the hazard. In fire insurance, the arbitrary method which has been adopted yields rates which are, perhaps, always high enough. The rule is : charge . 2 the annual rate for one month and . i additional for each month up to and including six, making . 7 for six months ; and . 05 additional for each month beyond six, making .95 for eleven months. The system for computing premiums for longer than one year in fire insurance, known as "term premiums," is equally arbitrary but does not sin on the safe side. The common rule is to charge twice the annual premium for three years, two and one-half times for four years and three times for five years. The absurdity of so great a reduction has been so far recognized that on certain classes of risks the companies now charge two and one-half times the annual premium for three years, three times for four and three and one-half for five. No effort has been made to solve the problem on scientific grounds. To do so offers no serious difficulties. Given an aggre- gate insurance of $1,000,000,000 and an aggregate annual loss of $2,000,000 or .002, probability of loss or net pre- mium. The losses on the $1,000,000,000 insured will, for five years, run as follows : First year. Amount at risk $1,000,000,000 Loss @ .002 thereon , 2,000,000 Second year. Amount at risk 998,000,000 Loss @ .002 thereon 1,996,000 Third year. Amount at risk 996,004,000 Loss @ .002 thereon 1,992,008 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 25 Fourth year. Amount at risk i $994,011,992 Loss @ .002 thereon 1,988,024 Fifth year. Amount at risk 992,023,968 Loss @ .002 thereon 1,984,048 Let us assume that the losses of each year are payable at the beginning of that year (annual premiums are thus pay- able), and then proceed to find what would need to be on hand at the beginning of the first year to pay the first year's losses, the second year's, etc. Assuming that the money will earn interest at five per cent per annum until needed, we discount at that rate with the following results : Sum at beginning of first year to meet losses of first year. . .$2,000,000 second " . . 1,900,952 third "... 1,806,809 fourth " . . 1,717,331 fifth "... 1,632,280 Having reduced the data to this form, to find the amount we need to have on hand to pay the losses of the first and second years, the first three years, etc. , we merely add these sums for the respective years included, thus : Sum beginning of istyearto meet losses of i stand 2nd years. $3, 900, 952 first three 5,707,761 four 7,425,092 ' five 9. 57,372 To ascertain the premium, that is the part which each dollar of insurance would need to contribute, divide the whole amount needed by the sum originally insured, $1,000,- 000,000, which gives the following results: Advance premium for two years 0039 + three " 0057 + four 007425 five " 009057 In reaching this result, we have obtained the loss for the second year by the product of p, the probability of loss into the insured property surviving the year; and so with the loss of each year. We have then discounted them to pres- 26 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. ent value and added the discounted values, dividing the sum by the original sum insured. Precisely the same results may be obtained by an analogous process, dealing with the probabilities only, thus : to obtain the part of the combined advance premium covering the hazard of the second year, multiply the probability of surviving the first year (q 1 ) by the probability of loss the second year (p) and multiplying the product by the present values of $i, discounted one year at five per cent (v 1 ) ; etc. , throughout the series of years. The sum of these parts of the advance premium make up the whole and the formula may be stated generally, thus : Net advance prem. = p + q 1 pv 1 + q 2 pv 2 +- q 3 pv 3 + . . . This simple process and formula will hereafter be found to contain the principle which solves many intricate prob- lems of life insurance rate-making. PROBABILITIES OF MORTALITY. WE have seen that to ascertain the probability of an event, it is first requisite to separate out a group, to each of whom the event is prima facie equally likely to happen. Then, by observing this group for any certain space of time, it separates into those to whom the event has not happened and those to whom it has happened. The prob- ability of the event happening to each of the group is then calculated by dividing the number to whom it has happened by the number of the original group. Suppose, now, that, instead of a group to each of which the event was prima fade equally likely to happen, a hete- rogeneous group were taken, to the individuals of which the event was prima facie likely to happen in widely vary- ing degree. Suppose, for instance, that the group con- tained dwellings, warehouses, saw-mills and powder maga- zines and it were attempted to calculate the probability of loss by fire to any one of these by dividing the aggregate loss by the aggregate values exposed. It is at once clear that the quotient would not represent accurately the prob- ability of loss to any one of the classes of property; but would, instead, be less than the probability of loss to the more hazardous and more than the probability of loss to the less hazardous. No fire insurance company would proceed thus to apply the same rate of premium to unequal hazards, and, if it did so, experience would soon demon- strate the error; for the owners of property, by a process of conscious selection, would insure more hazardous prop- erty and leave the less hazardous uninsured. It is prima facie clear that the probability of decease is different at different ages, and that after the period of child- 28 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. hood is passed, the probability of decease increases with age with a certain regularity. In youth and up to middle age, the increase may be slow and might be disregarded ; but there is a very considerable difference between the prob- ability of death at age twenty and the probability at age fifty, other things being equal. It follows that a fraction formed by dividing the number of deaths in a group of per- sons at different ages by the number of the group would not represent the probability of death of persons at any one of those ages. It would also not represent the aggregate probability of death in any other heterogeneous group, unless the group were formed of a proportionate number of persons at each age. Notwithstanding which, there has been a practice in cer- tain life insurance organizations in the United States to charge the same for a year's insurance without regard to age ; this was brought about by raising money to pay death- losses by means of an equal levy upon all the insured, with- out regard to age. The effect was to make premiums as if the probability of decease were the same at all ages and, also, as if the probability of an individual's death were the same in any one year as in any other year. The first of these assumptions has come to be univer- sally conceded to be preposterous and the practice of level charges at all ages has accordingly been generally aband- oned. The absurdity of the second assumption is quite as evident, but it is not yet so universally recognized. In consequence, we find some life insurance organizations violating all the principles of probabilities by proceeding on the basis that the probability of decease remains throughout life identical with the probability of decease at the age of entry into the organization. This is prima facie erroneous, and the consequences can hardly fail to be mis- chievous. PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 29 If we started with a group from a definite point of time, to which none were ever added and which was diminished only by the deaths, the consequence of this system would merely be to apportion the losses by fixed ratios which did not correspond with the actual hazards after the first year, provided the premiums were not fixed. If it were .attempted to furnish insurance for premiums fixed accord- ing to the probabilities of death at age of entry, the conse- quence would be that the premiums of the second year would not cover the aggregate loss, since the probability of death has increased for each insured person, while his premium has remained the same namely, just sufficient to cover the probability of death the year before. And this deficiency would increase every year. The premiums being fixed, the admission of a fresh group each year would not help matters, as the new group would, according to the premises, pay in only enough to cover their own prob- abilities. The premiums not being fixed on the admission of a fresh group each year, the apportionment would place an unfair proportion of the loss-burden on the new group and this would become heavier with each year. From these considerations, it is apparent that if we are to determine accurately the probabilities of decease, for any individual at any age, we must do so by discovering how many persons die out of a group of persons at that age and otherwise with prima fade equal probabilities. It is also apparent that the probability of the death of the same individual when he attains the next higher age must be independently ascertained from the mortality in a group of persons at that age. And so on throughout all ages. Insured lives, though at the outset carefully selected, are not, on the whole, select or healthy lives, because of the circumstance that commonly the renewal of a policy is not 30 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. optional with companies. Consequently, the probability of decease at a certain age, in order to be useful in insur- ance, should be derived from average lives, such as obtain in a company in which the benefit of fresh selection has somewhat worn off. Thus this probabiilty is not that of the death within the year of a man who begins it in good health, but the probability of a man in average condition as to health. ANNUAL LIFE CONTINGENCIES. WE have seen that the proper way to determine the prob- ability of decease or survival is by observing the number dying during a certain period out of a number setting out at the same age and with apparently average chances of sur- vival. This certain period is naturally measured by the unit of one year and the probability of survival or decease in a period longer than one year may be divided into a succession of probabilities during one year, followed by a fraction or not, according as the whole period is an exact number of years or not. For convenience, therefore, actuaries make the unit of measurement the year from birthday to birthday. If a large number of persons could be put under observa- tion who were born on the same day, by following them down through the diminution of their number by death until all were deceased, we could determine, one after another, the probabilities of death and survival for each year of age. For the number starting out from each age, the number dying before attaining the next age and the number surviving to that next age would be ascertained for each age of life; and from these three data all the probabilities of death and survival could be calculated. For instance, let 1 be the number born in the same year and d the number dying in that year, lj the number sur- viving the first year and setting out upon the second, dj the number dying during the second, etc. ; then, first, it is apparent that 1, = 1 d , or, generally, Let us designate the probability of survival as p ; then by 32 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. applying the rule for determining- the value of a simple probability, we find that the probability of surviving the first year of life is measured by the fraction found by divid- ing the number of the survivors by the number who set out, thus: p = = ^p - = i _ T , or, generally, *o *o 1 l*+i 1* d x d x Px = = p = i y l x l x l x Similarly, designating q for the probability of not surviv- ing, that is, of dying, during the year, we have by applying the same rule, a value for the probability of decease meas- ured by the fraction found by dividing the number who died by the number who set out, thus : d ! 1, 1, q = r = : = i T , or, generally, lo i 1 V, It would, however, be very difficult, if not impracticable, to collect such a group and keep them all under observation until the last one died ; and it is by no means necessary. For, since we are dealing with the probability of death during years of age, it is evidently a matter of no import- ance to also have concurrence as to calendar years. It is enough if a large number have been observed from birth to decease, the years of age when the respective deaths occurred being marked, so that it is possible to group the different years of life together. And, by this method, the probabilities for one year of age may be determined abso- lutely independently of any other year, if some large number of persons entering upon that year of age has been observed for that year of age and the deaths noted. It is, in fact, only by computing these probabilities separately that anything PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 33 can be accomplished in actual practice, since the lives under observation are constantly shifting and it is impossible to keep track of any large group throughout a long term of years. This is true concerning insured lives, the insured coming in and going out of the company at all sorts of intervals. Consequently, probabilities which are computed from deaths and survivals among insured lives are separately and individually determined for each year of age. To do this, all the individuals who entered upon that year of age dur- ing the continuance of their insurance are collated, the number who died during that year of age is computed, and from these data the probabilities are readily calcu- lated by the use of formulas already given. But a difficulty is at once met when it is undertaken to collate the number entering upon any given age, namely, that the insured entered at all fractions of the year of age and that, in consequence, it would be very difficult to group them by exact years of age. This difficulty is, however, practically obviated by the custom of the companies to reckon ages from the half-way point between birthdays, which assures that the persons insured as of a certain age are on an average nearly exactly of that age, about as many having been received who were younger as who were older. Life contingencies are also computed from census statis- tics ; but, because no such precaution is taken in collating such statistics as in receiving applicants for insurance, it happens that all the persons who have not passed their next birthday are ranked as of their age at the last birthday. Conse- quently they are, on the average, one-half year older than when at the last birthday. The deaths, on the contrary, are given just as we would desire them, namely, covering all who die while in a certain year of age. On the assumption, 34 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. then, that the deaths are reasonably uniform throughout the year, it appears that the number surviving to one-half year beyond the exact age is smaller than the number set- ting out at that age by just one-half the number who have died. Thus, before applying the formulas to determine the probabilities of survival or decease, we must ascertain the number who set out, thus : MORTALITY TABLES. WE have seen in our last lesson that if we could keep under observation a large number of persons who were born on the same day, until all have died, we would be able to compute successively the probabilities of death and survival for each year from birth to the extreme limit of life. We shall now see that in the course of the observations we should also form a complete table of survivors and decedents for each age, which we call a mortality table. Starting from birth, we obtain successively the number who survive the first year of life, the second year of life, etc., up to the extreme limit of life. The differences between these each year show the respective decedents. Special marks which distinguish this table from a mere col- lection of data from which the probabilities are independ- ently deduced for each age, are that the radix or number setting out from any age less the number of decedents during that year of age gives not merely the survivors of that group, but also the radix or the group setting out from the next age. Thus the data become connected and associ- ated in such manner that the table will show not merely the survivors and decedents for one year but, at a glance, for any integral number of years, out of a group starting from any age. This tabulated form will be found very useful in computing complex probabilities. The practical impossibility of making such a table in this manner has already been pointed out and is apparent. Not only was information such as this not attainable when the earlier tables were made, but information such as we can now obtain from censuses or from the returns as to 36 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. insured lives was also not to be had. For much more than a century after an attempt was made to develop the law of mortality, the only data which could be obtained were tran- scripts of the deaths from parish registers. The first mor- tality table was constructed by Graunt from London regis- ters and the second by the astronomer H alley from the registers of Breslau, Germany. The registers furnished no information as to the number who were exposed at any age. The manner of constructing the mortality tables was to assume that the data fairly represented the mortality which would successively apply to a group equal to the whole number of decedents, born on the same day. For instance, in the case of the table constructed from the statistics of Breslau, the total number of deaths was 1174, which number was accordingly taken as the radix of the table. The deaths before attaining the age of one year were 348, from which the rate of mortality was made T 8 T 4 A, or 29.64 per cent, and the number of survivors 1174 348 = 826. The fact was, however, that these deaths at less than one year of age were among 1238 children, born within the period covered by the observations of mortality instead of among 1174. And it must be evi- dent that, unless the population were precisely stationary, the births balancing the deaths each year and nobody removing from or into the field of observation, the assump- tion that the mere number of deaths during any period could furnish a radix for a mortality table constructed as if from lives observed from the time of birth was sure to be erroneous. As the populations of Breslau and other places, which furnished statistics used to compile these tables, were in each case increasing, it followed that the radix was in each case too small to correctly represent the mortality at the younger ages and this, in turn, made the PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 37 mortality proportionately too small at older ages. Where these tables were used for insurances, this proved an error on the safe side ; where used for annuities, it was an unfor- tunate error. The observation of insured lives and the lives of annui- tants indicated the faults of these tables, and, parentheti- cally, it may be said, that they were found faulty in just the respects which have been mentioned and which might have been suspected from the fault of the system. The making of tables from data which independently demonstrate the probabilities of decease and survivorship for each age, is attended by somewhat greater difficulties than the mere employment of the whole number of deaths as a radix, as in the system just described. For instance, statistics from the census or from observations on insured lives, when reduced to simple forms, set forth the decedents and survivors out of a certain number of years of life exposed, all the figures being different at the various ages. It is not possible to trace directly from these data the annual diminution of a group of actual individuals by death up to the extreme limit of life. In fact, the individuals are wholly lost sight of when they pass from one group into another; they merely appear as integers of years of life exposed. What can be done is this : we may compute from these data the probability of survival and decease for each year of age independently and we will find it possible to con- struct a mortality table by using these ratios. For, starting out from any large group, such as 1,000,000, as a radix, we may, by applying the values of p x and q x (the probabil- ities of survival and decease), successively ascertain the number of decedents and survivors, using the survivors of one year as the radix for computing the decedents and survivors of the next. Thus we may as well express the 38 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. law of mortality as if it were possible to put under observa- tion such a number born on the same day until exhausted by death. To do this from data taken from the census, we may determine the probabilities of death and survival from what is known as the meridian, mean or central death rate, which means the number dying during any year of age divided by the number surviving to one-half year beyond the initial age. This central death rate is desig- nated by the algebraic expression, m x . We may trace its relations to p x and q x as follows : By our former lessons we have found that p x = * + I and q x = We now find that m v = , ?? We may derive a more workable formula, however, by the following ingenious process. Suppose 900 1000 50 50 950 50 950 1000 1000 50-1-50 9504- 50 50 95 IOO IOO q x = and m x = 1000 950. Expressing - - in terms of m x , we have i YZ rn x _ 2 m x = i 4- y 2 m x "" 2 + m x ' 2 m x 2 4- m x 24- m x and q x == i p x = i - 2 -f- m x 2 -(- m x 2 m = , which values may then be used directly in con- 2 -{- m x structing the mortality table. MORTALITY TABLES. GRAPHIC SYSTEMS OF GRADUATION, I. UNLESS the number of lives dealt with is at each age very large, the values of the probabilities of death and survival derived from them will not be as uniform as is desirable. Some things, for instance, are certainly known about the force of mortality. Both from what is prima facie reason- able and from the testimony of a large aggregation of sta- tistics, we know that the rate of mortality tends to fall from infancy up to about ten years of age and then gradually to increase. The increment itself is also at an augmenting ratio, so that in old age the increase is very rapid. The increase during the years of men's prime, say from twenty to fifty, is not so much the direct result of aging as it is of the fact that merely living longer multiplies the chance that the system is already undermined by disease. In other words, the probability of survival for one year may not be much different for either of two lives, freshly selected by medical examination, though one be at twenty years of age and the other at fifty. Bu L the expectation or probable life of the former will be much longer than that of the lat- ter, and the hazard of the former dying during his fiftieth year, if he survives so long, is much greater than of his dying during his twentieth year, for there is the hazard of his not reaching fifty in unimpaired health. This last hazard applies to each of the group, forming the radix of a mortality table. Consequently, independently of the increase of mortality from actual old age, we expect an increase with each year (after childhood has been safely 40 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. passed) because of deteriorated average health conditions. Deviations, such as a decrease of mortality, will be found to be followed at a greater or less interval by a sudden and disproportionate increase, both phenomena being the results of the inadequate number of lives dealt with. In most cases, therefore, it is to be expected that a mortality table, constructed as per the instructions given in the last lesson, will exhibit glaring irregularities and want of uniformity. As has already been indicated, these irregularities coun- terbalance each other and the correct ratios may, therefore, be arrived at by some system of graduation which will spread out the excesses so as to fill up the deficiencies and yield a uniform table, showing a gradual but regular increase in the mortality. No method has yet been invented to accomplish this which is so simple and so self-explanatory as the graphic method. The principle is to lay out the trend of mortality, as devel- oped by the actual investigation, upon a scale, drawing lines so as to connect the values of q r at the different ages, thus representing the force of mortality by a broken line, as in the accompanying illustration. This exhibit is taken from the mortality experience of the Australian Mutual Provident Society and excellently illustrates the graphic method of graduation. The vertical spaces represent a scale of mortality differ- ing by .005 per cent of the number of lives exposed. The variations in the force of the mortality as ex- pressed by the ratios deduced from the society's actual experience are indicated graphically by the broken line which starts at nearly .034 and rapidly descends until at age three it is less than .009, and then after several reac- tions it is at about .001 at age eleven. Then it reascends, at first very slowly and with many variations up and down PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 41 0*0 or I 000 *r f * " 42 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. until age thirty, after which time the upward tendency very strongly asserts itself. It must be at once clear that the trend of this broken line may be closely approximated by a more or less regular curve, which will avoid the violent variations, while at the same time representing on the average faithfully the facts which the unadjusted table presents. This curved line must also correspond with the known characteristics of the force of mortality enumerated in the beginning of this les- son. Thus prepared for the task, we may sketch freehand, but with care, the curved line in the diagram, keeping a middle course. From this curved line, by reference to the scale, the ratios of mortality for the different ages may be taken and a graduated mortality table constructed by suc- cessive application of these ratios, in accordance with the instructions in the last lesson. This diagram is reduced to small proportions, so as to print in the page of an ordinary book. In actual practice, there are advantages in occupying more space and using a larger scale. MORTALITY TABLES. GRAPHIC SYSTEM OF GRADUATION. ANOTHER system of graduation, which calls for more space for its successful development, is to construct a similar curve from the data of an ungraduated mortality table. The unadjusted table is first constructed by taking the values of q and p (the probabilities of death and survival for each year of age), direct from the qriginal data and proceeding as per instructions in lesson 7 of this series to construct a table, taking any large number of lives as a radix. A good specimen of an unadjusted table is found in the published, tables of the experience of the Canada Life Assurance Com- pany and runs as follows : AGE. 1 X d x AGE. 1* d x AGE. i, d x 20 IO,OOO 53 38 Q.OQO 54 56 7,671 IIQ 21 9Q47 37 TO 90*^6 67 C7 7 m2 126 22 Q.QI4 55 4O 8 Q77 71 ^8 7 ^86 146 27 0,850 44 41 . 8 902 58 5o 7 240 176 24. . 0,815 52 42. 8,844 ca 60 7,064 177 25 9,76^ 58 47. . 8,701 65 61 6,887 174 26 9,705 54 44. . 8,726 6? 62 6,717 198 27 0,651 47 45. . 8,650 72 63 6,515 194 28 Q 608 4*2 46 8 t;87 ?8 64 6 721 1 68 2Q . 9c65 48 47 8 ZOQ 88 6* 6 157 204 "3O. . Q CJ7 ej 48 8 421 87 66 5.Q4Q 217 71. . 0,466 45 4Q 8 77.8 7Q 67 5,772 220 32 Q.42I 4C CQ 8 2^0 80 68 5,512 220 33 0,776 54 51 8,170 04 69 5,283 231 74. 97.22 61 C2 8 076 Ql 7O e o^2 221 75. . Q 26l 5Q C7 7 o8c, 117, 71 . . 4,871 ?76 36 Q.2O2 61 54 />y u -> 7,872 127 72 4.555 777 07 9 141 C I c e 7 74^ 114 7-2 4 222 274 44 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. AGE. X d x AGE. 1* d* AGE. 1* d* 74 3.948 293 81 1,694 266 88 308 27 JC 3 655 2QI 82 I 4.28 278 80 QQ 76 3,364 256 83 1,150 24Q QO j J L 283 106 77 o ,108 30-3 84 QOI 142 QI 177 7T 78 2,772; 4IO 85 7CQ 06 Q2 1 06 /* 7q. . 2,^6^ 343 86 663 ICQ Q3 1 06 so :. 2,022 328 87 cod 106 04 106 gc 106 96 97 106 71 35 71 It will be observed that the deaths and survivals under this table for ten year periods are as follows : Inclusive. Radix. Deaths. Survivors. 2O2O IO OOO 483 Q ^17 IQ 3Q Q CI7 C44 8,077 4.O-4.Q . 8.Q73 714 U >V/ J 8,2^Q 8,250 1,105 7.O64 6060 . 7,064 2,012 5,O52 7O 7Q e O52 q, Oqo 2 O22 8080 2 O22 I.73Q 283 QO QQ 283 A > /jy 283 Lay out a line, divided into eight equal spaces to corre- spond with the eight terms of ten years each. From the left hand point of the line, construct a perpendicular of an arbitrary length to represent the radix at age twenty. Complete the parallelogram by drawing the necessary lines. From each point on the base line, 30, 40, 50, 60, 70, 80 and 90, draw lines parallel with the perpendicular and ending in the top line of the parallelogram. On the first perpen- PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 45 46 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. dicular, beginning at the base, lay out a line proportional to the whole perpendicular as the survivors of 20-29 to the radix of 10,000 and draw a line parallel with the base con- necting this point with the second perpendicular. This last line divides the parallelogram contained between the two perpendiculars into two parallelograms, which are pro- portional, the upper to the deaths for the period and the lower to the survivors of the period. Repeat this operation for each of the perpendiculars. Then, describe a curve, touching each of these outer angles, beginning at the upper extremity of the first perpendicular and ending at the right extremity of the base line, as in the diagram given herewith. It is possible to derive a similar curve in a somewhat similar manner from population statistics, and also even without first constructing an unadjusted mortality table. In this case, as was pointed out in an earlier lesson, the number living in each year of age is, approximately, the number who attain one-half year beyond that exact age and so it is neither the radix nor the number of survivors. This brings it about that the principal parallelograms are divided by lines somewhere between the lines drawn parallel to the base in this diagram and the top line. And, accordingly, the curve should be described, starting from the first perpendicu- lar, so as to contain below itself as much space as was con- tained by the original parallelograms, the spaces cut off and the spaces added equalizing one another. To describe such a curve, it is not even necessary for the data to be rearranged so as to conform with a radix ; all that is necessary is that each principal parallelogram be divided into two parallelograms, proportional to the number living and number dying dur- ing the age terms, 20-29, e ^c. This method was first described by J. Milne, author of the Carlisle Tables, and was by him employed in the construction of those tables. PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 47 The values of the probability of survival, p x , and of the probability of dying, q x , may be taken from these tables by use of dividers and a table of proportional parts. If these values are not quite so regular as may be desired usually owing to drawing the diagram on too small a scale they may be again graduated by reference to the first method of deriving those values by the graphic system, described in the last lesson. Sufficient accuracy may be attained by this system to determine the l z and d x columns at ages up to 65 or 70; but the scale used in the diagram would need to be very large to give satisfactory figures for the highest ages. The same difficulty was encountered in the system last described arid is encountered in most of the systems in use. MORTALITY TABLES GRADUATION BY FINITE DIFFERENCES. The first of the graphic methods of graduation described in the preceding papers determines the mortality rate per cent for each age (q x ), while the last determines the value of l x and d x as they appear in the completed mortality table. l x is a constantly diminishing quantity, the differ- ence between l x and l x + j always being d x . d x is itself the product of q x and l x . But as l x is constantly diminishing, while q x is constantly increasing, it follows that d x will not increase so rapidly as q x and in fact that its value for each age is the resultant from two opposing forces, one being the diminishing number of the living and the other the increas- ing proportionate mortality. The consequence is that the values of d x at first tend to increase, then to reach an equi- librium and at last actually to diminish. It is possible to graduate a mortality table by reference to the series of values of d x . For instance, we may take up the unadjusted experience of the Canada Life. Arranged by decades of ages and multiplied by 10 so as to get along without fractions, we have the following short mortality table from which to work: AGES. lx d x 20 to 29 inclusive IOO,OOO 4,830 95,I7O 5,440 89,730 7,140 co to ^o 82 ^QO 11,950 60 to 60 . .... 70, 64O 2O,I2O 7O to 7Q, 5O,52O 30,300 80 to 89, 2O,22O 17,390 2,830 2,830 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 49 From these figures we may at once determine certain tendencies in the values of d x . d x increases continually at the increase itself, increases for many ages ; the maximum value of d x is reached between 70 and 79 at about the median age; after that age (74) the values of d x diminish; for several years before that age (74), the increase in the value of d x must have been diminishing. After that age (74) the decrease in values of d x is at first very rapid, but later the exhaustion of l x is such that the rate of diminu- tion, taking the series as an arithmetical series, is not so great. Having thus determined certain outlines, we may proceed to fill in. Owing to the phenomenon that for each decade up to 60 the values of d x increase another decade follows for which the value of d r also increases, it follows that the average value of d x for each decade is probably reached at an age a little beyond the median or at 25, 35, etc. On this assumption the value of d a5 will be 483, d 36 544, d 45 714, The difference between any two of these values must be equal to the aggregate differences between the successive values of d x included between them. That is, in algebraic form, d 35 d 25 = (d 26 d 25 ) + (d 27 d 26 ) + (dsg d 27 ) +, etc. , up to 35 - d 34- But we may readily determine this aggregate difference by deducting the value of d 25 from d 35 , which is d 35 d 25 = 5 44 48 3 = 6 1 . Knowing as we do already that the series of increments, d 26 d 25 , etc., increases very slowly, we may assume this aggregate difference to be made up thus: 5, 5, 5, 5, 6, 6, 7, 7, 7, 8, total 61; from which, starting with the already ascertained value of d 25 , we may construct : 50 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 25 = 483 31 = 5*5 32=522 33=529 29 =53 The sum of the first five of these values is 2465, which being deducted from 4830, the value of d 20 .29, leaves 2365 to be accounted for by d 20 _ 24 , which we may construct by con- tinuing the series of differences downward thus: 4, 4, 3, 2, i, remembering that the total values must equal 2365. This gives us : 24 = 479 22 =47 2 d 21 = 470 d 20 = 469, the sum of which is 2365. Summing the values of d x from ages 20 to 29, inclusive, we have 4830 for the decade which just accounts for the value of d 20 . 2 j as per the unadjusted table. Proceeding in a similar manner we have : 45 35 = 7 1 4 5 44 = 1 7 - The new ifferences must be a continuation of the series already discovered, namely: i, 2, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 7 7> 7> 8, and the series may be extended thus, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, the sum of which is 171, as near as we can come to 170. Proceeding to construct we have: 36 =553 41 = 623 37 =5 6 3 42 = 643 38 = 575 43 = 665 40 = 605 d 45 =7i5 Going back and summing the values of d x from 30 to 39, PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 51 inclusive, we get a total of 5435, which is certainly very close to the 5440 of the original unadjusted table. In like manner, though with increasing difficulty, as the differences get larger, the values up to d 55 may be sketched out. But if it is undertaken to work to 65 on the basis that d 65 = -^, it will be found that the differences will begin 10 to diminish soon after age 60, instead of near to age 70 as we have a priori decided to be the case. We will, then, be compelled to assume that - - does not represent d x with reasonable accuracy for any intermediate age. The series can, therefore, only be continued by leaping from 55 across to 74 since we have assumed that d 74 corresponds fairly with This course will be found to give reasonably satis- ^70.79 10 factory results. The series may then be extended to 80 by reference to its own tendency and it will be found to change to minus instead of plus quantities before it reaches 80. From 80 to the extreme limit of life, the difficulties are greater and the best results, perhaps, are obtained by deter* mining the series from (first) its tendency at 80, (second) the fact that the aggregate of decrements from 80 on must be just equal to d 80 , and (third) the fact that the value of l x must fall from 1 80 to zero or, in other words, that all the lives must be exhausted. A series which will fulfill these three conditions will answer the purpose. MORTALITY TABLES GRADUATION BY FORMULA. WE have already discovered in our inquiries into methods of graduation of mortality tables that the curve, whether it be the curve in a tabulation of the rate of mor- tality (q x ) or the force of mortality ( ^ ) or of the l x column in the completed mortality table, is governed by a law which may be traced out by the differences. This gives us the clue that there is a law of mortality which may well govern the graduation of all tables. The application of this law will, of course, vary according to the special data, drawn from experience, upon which each table may be based. The first attempt to formulate such a law was by De Moivre in his " Treatise of Annuities on Lives," pub- lished in 1725, before any considerable quantity of mor- tality statistics was available. His theory was that while the rate of mortality increased for each year of life, an arithmetical series with equal decrements for each year would make up a mortality table. Thus, assuming seventy-four people to be living at age twelve, his theory was that one might be expected to die each year, ter- minating the series at age eighty-six. This means that the d x column in a mortality table should remain constant, the same number. Thus in order to graduate a mortality table no information would be requisite other than the radix number of lives and the extreme age attained ; the constant decrement would then be obtained by dividing the radix number by the difference between the radix and the extreme ages, thus assuming the radix age to be ten, the extreme age 100 and the radix number 100,000. PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 53 100,000 100,000 d x = - - = - -=1,1114- 100 10 90 And from this a mortality table would be formed by merely successively subtracting this constant value of d x from 100,000. Just a century later, in 1825, Benjamin Gompertz in- vented a formula which proceeded on the basis that there was with increasing age " an increasing inability to with- stand destruction. ' ' This assumed that the force of mor- tality increased in geometrical progression. That is, in algebraic form, ^ = Be 1 in which formula B represents a basic constant number and c the constant number involved in the geometrical increment, which is raised to a power corresponding to the age attained. That is, concretely, ^ = Bc 30 Gompertz, in the reasoning which led up to this for- mula, supplied the basis for a much better formula, though he did not apply it. For he said : " It is possible that death may be the consequence of two generally co-ex- isting causes; the one, chance, without previous disposi- tion to death or deterioration ; the other, a deterioration or increased inability to withstand destruction. ' ' In his formula he deals with the force of mortality as if the latter of these propositions were alone true ; whereas, in fact, they are in all probability equally true. In 1860 the eminent actuary, Makeham, corrected and improved Gompertz 's formula by adding a quantity to represent this constant element in the force of mortality, which made the formula take this form : /^ = A -f Bc x , or, for a concrete example, // 30 = A + Be 30 54 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. This formula was adopted for use in the graduation of the H m tables by the Institute of Actuaries and has since been employed in the graduation of most important tables. To give thoroughly smooth and satisfactory results it is necessary to apply the formula by means of differential and integral calculus, but in this treatise we will not try to do more than to indicate how the value of the constant quantities may be discovered and how the formula may be roughly and inexactly applied by means of mere algebra and common logarithms. First of all, it is necessary to determine the values of A, B and c, which may be done by the following process. By our fundamental assumption we have : A + Bc* = ^30 and A -f Be 20 = ,%. Subtracting, we get Be 30 Be 50 = /"go i* m or, simplifying B (c 30 c 20 ) = Be 20 (c 10 i) =A /w Again, reverting to the original fundamental assumption we have A+ Bc 40 = /* 40 A 4. Be 30 = /^o whence, by subtracting and simplifying, Be 30 (c 10 l)=A^ 40 -30. Now, for convenience's sake, let us substitute loga- rithms in these two final equations, remembering in our computations that logarithms add when numbers multi- ply, subtract when numbers divide and multiply by the exponent when numbers are raised to powers. We wish in this case to divide the equation: Be 30 (c 10 i) = A /V 80 b y the equation Be 20 (c 10 1)= A /Vso- We may do this directly, which will give us: Be 30 (c 10 i) A /v,o PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 55 c 10 ^^^L an d A /'SO-SO A /"30-20 But these powers and roots, which are very formidable when encountered thus in simple algebraic form, lose all their terrors when represented by logarithms. Therefore we represent the dividend and divisor equations by their logarithmic equivalents and subtract instead of divide, thus : AB + ;icx4 + * (c 10 i) = ;IA ^.30 (c 10 i) = AA ^30.20 A C X 1 = AA /" 40 _30 ^-A ,"30.20 from which we derive A A /"40-30" "^A /"so-20- JO. But the values of u 20 , u 30 and u 40 are known or discover- able from the unadjusted table ; so that c now becomes a known quantity. As a matter of fact its logarithmic expression in almost all tables closely approximates . 04. With this quantity known, we proceed to derive value of A and B. Taking up again the equation : B(c 30 c 20 ) = /" 30 /"so which reduces to B(AC 30 - 20 ) = A ,"30-20, we get B = -S, thus obtaining the value of B in terms of // and c, all of which are known quantities. We may now derive the value of A from the original equation, thus: A -f Be 20 = ^o, whence A = /" 20 Be 20 . As few tables follow this law closely when unadjusted the thirty American offices being a closer approximation than any other it is advisable tpdgte^me these values 56 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. by reference to several groups of ages, and then to aver- age the results. The values of ^ may then be deter- mined for each age, including the highest ages of the table, from this original equation which acts as a formula: The law and consequently the formula fail at ages below twenty, at which ages another adjustment must be made, but they answer better than any known formula to determine from sufficient data at ages below seventy what the normal death rate should be at ages beyond seventy, a problem which, as we saw in recent lessons, is not satis- factorily solved by other methods of graduation. The usual course in constructing a table by this formula is to find values at short, regular intervals and determine inter- mediate values by differencing or integration. THE FORCE OF MORTALITY EMPLOYED IN GRADUATION. WE have already made use of the expression, " force of mortality, ' ' // x , without attempting to assign it a definite meaning. It is not the same as the " rate of mortality " or probability of dying which we designate by q x and which is the quotient of the number living at the beginning of the year into the number dying during the year, y. The lia- i* bility to die must be taken as increasing not in broken steps once a year, but constantly and, consequently, the real force of mortality is constantly increasing. The force of mortality is measured by the same denominator or divisor, l x , as is q x , but the numerator is not the number who die out of the number beginning the year, but the number who, by the same fo'rce of mortality, would have died had the number living been kept full at l x by adding new lives of the same age and conditions exactly whenever lives failed. This numerator would evidently be somewhat larger than d x ; and at some of the higher ages might be considerably larger. The force of mortality is a mathematical expression which accurately measures the curve in the first illustration of the graphic method in these lessons. We express it by the Greek // x , which should not be confused with m x , which is the central death rate. It is not possible to get a simple expression of the value of pz in terms of l x , d x , p x or q x without making use of the calculus. An approximate expression, which is said by the Institute of Actuaries' Text-Book to be " for practical purposes usually sufficiently accurate," may be derived by dividing the number dying for the previous and present 58 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. years by double the number surviving the previous year and beginning the present year, thus : _d f _ 1 +d f U-i I 1+ i, r*X - - - ' 2 1* 2 l x the latter expression being derived from the fact that the deaths for the two years are equal to the number beginning the first year less the number surviving and beginning the third. In considering tabulation from census statistics, we have already found something very similar to this in m x or the central death rate. The population given in such statistics at any age is, approximately, the number surviving to one- half year above that age, while the deaths are all those occurring at that year of age from birthday to birthday ; from which it happens that the quotient from dividing the number dying by this number living is not p x but m x , thus: If we expand this so as to cover two years instead of one, and so double the denominator, we will get just the same as we have already obtained as the value of ^ ; from which it follows that, approximately, m x = >"x + l-2- Having derived a series of values of ^ from the formula given in the last lesson, we may make use of the following simple formula to obtain values of p x . We have already the formulas: p, = -^- and from that p x _i = =-^- . 1* 1* i From the latter we may develop the following as an equivalent of -^ : PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 59 -7- 1 P.. u; We may now insert these values in the equation, 1 --x + l T 1 i_. U, i, which will transform it into 2 This will not reduce to terms of p x alone, as there is no such proportionate mathematical relation between p x and p x _! as will make it possible to express one in terms of the other in a general formula. But it is clear that, when we have secured a starting point we may readily proceed ; that is, if we know the value of p x or of p x _i, we can proceed to find the value of the other, as for instance of p x in terms of p x _j and p. , both known quantities. At age 20, for instance, the values of q and p, are so nearly the same that we may take q and p as being identical. As p^ is a known quantity, having been determined by the formula given in our last lesson, q 20 thus becomes a known quantity also; as likewise does p 20 , which we know to be equal to i q 2 o- We therefore have from the formula, /^= ^ - , applied at age 21, the fol- lowing : P 2 p _ i (fa f- -7- 60 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. Clearing from fractions, we have 2 ^21 = Psi. Deducting from i q 2 <> i q 20 each side of this equation and multiplying by i, we get p2i = 2 an or, reinserting the value 1 q2o pso for its equivalent, i q 20 , we have p2i = 2^si, which maybe made gen- eral, thus: A table of values of ^ having been made according to the formula given in the last lesson, a table of values of p may be derived by using this formula and from that by con- tinued multiplication, starting with a radix number at any age, a table of values of 1 may be derived, remembering that It remains to discover a general formula for 1 in terms of H and known values of 1. This we can evidently do by reference to the original formula : u = I1 1 x * 1 , if we can but predeter- 2l x mine two values of 1; for we can easily get the third value of 1 in terms of the other values and of ^ . In finding a value for p x , we have already assumed that q ao = jUao ; which, being granted, also makes ! 2 i = \ (qJ = \ ( /O- Thus we have two values of 1, 1 20 by assuming any number as a radix and l ai by the above. Applying these in the original form- ula, we have PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 61 1 I p n = 30 M . Whence we get 2 1 91 (/O =1 20 !; and by transposing 1 90 and multiplying both sides by i, we have 1 M - 1--*!,, (/<) Having thus obtained a value for 1 M , we may, by the same process, using // 22 , l gl and 1 M , obtain 1 23 , etc. The formula is therefore general and may be stated : INTERPOLATION AND GRADUATION BY INTERPOLATION. IN several recent lessons there have been references to completing the outlines of a table, values at certain inter- vals being determined by one of the systems described, through the interpolation of the intermediate values. In the description of a system of graduating by finite differ- ences, the work was really a continuous interpolation con- ducted by developing the law governing the series of differ- ences. But this was carefully kept out of the range of the calculus of finite differences and within the comprehension of those who are familiar with ordinary algebra. A merely rudimentary knowledge of this calculus is, how- ever, so useful in every day calculations by an actuary that we will seek to develop a few simple principles and formulas. Suppose a series of values of u, such as u o , u t , u a , u a , and so on. The difference between the first and the second, that is, Uj u o , we call AU O ; the difference between the second and third, or u a u, , is AU S ,etc. Then for the difference between these differences we have A 3 u , etc., thus: Series A A 4 A 6 y etc. A *o AU, AU, AU A a U A 3 U A 4 u AUj A a U, A 3 Uj A 4 u t *'. AU. If we have the value of u o and of all the differences bear- ing the notation u o , we can reconstruct the entire series PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 63 for A 6 u + A 4 UQ = A 4 Ul ; A 4 u o + A 3 u o =A 3 u i and A 9 u, + A 4 i\ l = A 3 u 2 ; and so on. We find the following values of A, A 2 } A 3 ? etc., in terms of u , u ,u t etc. : AU, = It will be observed that this series of values of A follow the well-known law of the binomial theorem, which is also readily proven to hold good through all orders of differ- ences. We find another correspondence with the binomial theorem in the following development of the values of u x in earlier values of u plus differences : U i = U o + AU o Au i ( u o + Au o )+( Au o + A2 u o ) =u o = u +3 A U O + 3 A2 u o + A3 u o This is also readily proven to hold good for all values of u and may be stated generally : n (n i) 2 n(n i)(n 2) 3 A u- ^ -A 8 - An adjustment or graduation of a mortality table may be made by taking the values of 1 at five year intervals such as ten, fifteen, twenty, etc., and interpolating to get the intermediate values. The foregoing formula may be employed to discover these values. Let u n be the desired value of 1, u a value of 1 (to be known as l m ), at the nearest quinqennial period, which is z years younger or older than the age at which the value is desired, z of course is some number of years less than five, usually not more than two, as values further from l m than l, 3 can best be derived from a higher or lower triad of quinquennial values. 64 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. Then we have u u l x ; u, =\ t ; u = l z _ 5 ; ti a = l i+B ; A U O = 1. l z - 5 I ^ =1 1+ . 1. ; and A* u = (U-l )-a -U) =U -^ + W Inserting these values in the general formula and stop- ping at second differences, we have: n(n i)-], r .-i, r n ( n ~li - JW- The interval which in u o , u t , u a , etc., is expressed by unity is here expressed by 5. The interval n is greater or less than 5 by z, that is, n = 5 z, accordingly as l x is below or above l s . Let us treat z as always added to 5 but as being itself a minus or plus quantity according as l x is below or above l s . Now when l x is beyond l z and the interval between u o and u n is consequently greater than 5, it so happens that z will be a minus quantity. Thus if u o be 1 10 , Uj 1 15 and u x 1 20 and u n be 1 16 , then the interval is 6 and the value of z is i so that 16 + z = 16 + ( i) 15. To get an expression for this interval between u o and u then we must deduct z from 5, which in the foregoing case will give 5 2 = 5 ( i) = 5 + i = 6. And this will work equally well in cases where l x is before l g as, for instance, 1 14 , while l a is 1 16 . The value of z is then posi- tive, thus + i, so that 14+ (+i) = 15; and the interval is 5 (.f i) = 4. But the interval in terms of u o , u x , u a , etc. , can be but fractional with 5 for a denominator. Thus the formula becomes : . 5 5 Inserting this value in our equation, it becomes : PRACTICAL LESSONS IN ACTUARIAL SCIENCE. / z \ / z / ZN (I )(l 1 -( T)** ' n ' Z \ 3 Z (?)' -ri.,. 2=; z 2 z 2 trz 5 2 5 5 . z (5+ z )i , 2 5 z 2 , 2(5 z) 1-7, 5 ~T~ I z J-i + 6 5 2 5 5 MODIFIED GRADUATION BY INTERPOLATION. WOOLHOUSE'S FORMULA. THE last lesson described a system of determining inter- mediate values of 1 from quinquennial values taken from an unadjusted table. By this means through continued interpolation a graduated table could be constructed, but it would have many of the faults of the ungraduated. The eminent English actuary, W. S. B. Woolhouse, who was delegated by the Institute of Actuaries and Faculty of Actuaries to adjust the H m table, invented a system by which these inequalities could be avoided; and the first H ra table was accordingly graduated by his system. This table was afterward largely displaced by a new H m table graduated according to Gompertz's law and Makeham's formula. The same original adoption of Woolhouse 's formula and its displacement by Makeham's may be noted in the case of the French tables recently published. The basis of Woolhouse 's system is finding the arithmet- ical mean between values of l x determined by the construc- tion of five different tables by interpolation from five series of quinqennial values of 1 from the unadjusted table, thus: 1 1U L 1 1,., c f l" l", l" I 4 ' r 1 i 1 " l" r l" 18' 33 r. . : When complete tables have been constructed by interpo- lation, we will have five different values of 1 for each age, which we may call I 1 , l a , I 3 , I 4 and I 5 . Woolhouse 's idea is that a very close approximation to the true value may be made by adding these and dividing by 5, thus: I 1 + I 2 + i 3 + I 4 + I 5 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 67 This would, as the author of the plan remarked, be a very laborious and circuitous route ; and a shorter one can be devised, as follows: We have already for our last lesson the general formula for interpolating values of 1 : i z (5 + z) 25 z 3 _z( 5 z) 50 25 50 '+ 6 To get the five values of l x by interpolation it is only necessary to make z 2, i, o, i, and 2, successively. Using these values in the formula, we get : 1, == + 1, + I x = +.i2l x _ 4 +. 9 61 x+1 .o8l x+6 l i = +. 2 81 i _ 3 +.8 4 U 2 -.i2l i+7 5 \ = 1, + 96 1 I+1 +. 84 U a + 28 l z+8 +.12 l x+4 - 8 Ue ' I2 U 7 + . 9 6l i _ i +.8 4 l x _ 8 +.28l x _ 3 +.i2l x _ 4 -.o81 x _ 6 -.i2U For convenience, put Inserting these values in the last equation, it takes the form: 51. =\ +-96 7l +.84 y a -f .28 73 +.12 n -.08 y 6 .12 y T = ! z +(n -047, ) + ( 7l . 16 7a ) + (.0 4 y 3 +.l6y 3 +.o8y 3 )+.I2 y 4 .08 y fl .12 y r = l x + yj + y a .04 [ (y, y 3 ) + 4 ( 7 . 7 8 ) 68 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. For further contraction of the expression, put f /! 7 3 g ^ 7 3 b=7 e 7 a k = 7 T 7 4 Inserting these for their corresponding values in the last equation, we have : Adjusted 5 l x = l x + Yl + 7i .04 (f+4g+ 2 h+ 3 k) Adjusted 1, = ^7 1 + r a -.Q4(f+4g +2 h+ 3 k) Formidable as this appears, it is really merely a little complex, while every one of these values can be derived from the original unadjusted table by the use of addition, subtraction, multiplication and division only, which makes it a very simple and easy formula to operate. It is the perfection of the principles underlying the plans for grad- uation by the graphic system and by differences already described; it has been supplanted by Makeham's because the latter alone expresses a law of mortality in itself. TESTING THE ACCURACY OF GRADUATED TABLES OF MORTALITY. THE graduation of mortality tables is, strictly speaking, not construction, but modification. The trend of mortality is indicated by the statistical data drawn from actual mor- tality experience. From this actual mortality which deals with numbers of lives exposed at certain ages and those of them failing and surviving, respectively, a table of values of p x and q x may be directly derived. But, when the data are not very numerous, the tables will be found to pre- sent many irregularities which may be adjusted. This adjustment is the office of graduation. But the adjusted or graduated experience must be such as to adequately and fairly account for the deaths and sur- vivals according to the original data, taken as a whole, the idea being merely that surplusage in one portion will fill up discrepancies in another portion. But the graduated tables of values of q x , for instance, should not, when applied to the numbers of lives exposed at each of several adjoining ages, result in a widening discrepancy. That would indi- cate that the graduation was something more than a mere adjustment ; that it was, in fact, a complete departure from the data from which it was constructed, and proceeded by some other, quite different law. It follows that the gradu- ated ratios should, when applied to original data, not only account approximately for the aggregate deaths, but also that, within a small number of ages, it should also account for the actual deaths with fair accuracy. For the purpose of testing the success of graduation, therefore, reference must be had back to the original data. A table must be made showing the number of lives exposed PRACTICAL LESSONS IN ACTUARIAL SCIENCE. to risk and the number failing for each age. The number out of those exposed that would have failed according to the graduated table of values of q x must next be computed for each age. This is done by multiplying the number exposed at any given age by the value of q x for that age in the graduated table. The number that should have failed is then set opposite the number which did fail and the error, plus or minus, noted in an adjoining column. The accumulated error is carried forward in another column, so that any wide departure from the original data appears at once and so that also the proper tendency of the graduation to merely smooth out irregularities is manifest if it be suc- cessful. The following table is an example of this system of test- ing the accuracy of graduation. It has historical interest, also, as being the test of the success of the graduation of the famous H m tables according to Makeham's formula by comparing it with the original data. It is taken from the Institute of Actuaries' Text-Book, pages 94 and 95 : AGE. Exposed to Risk. Actual Deaths. Expected Deaths. Error. Accumu- lated Error. 37Q 3 2 I 434 o 2 2 12 401.5 2 2 iq 578 2 2 I<1 731 3 3 O 15 908 2 3 I 2 16 1129 4 4 6 1421 6 6 o 6 18 1810.5 ii 9 2 4 2414 17 13 4 O 20 . . 3203.5 IQ 19 o o 21 4578.5 32 28 4 4 22 6^07 4O 41 i 3 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. AGE. Exposed to Risk. Actual Deaths. Expected Deaths. Error. Accumu- lated Error. 23 8534 66 57 Q 12 24. . IO936 75 76 I II oc 13622. ^ 7O 06 26 T e 26 l6^3Q. ^ 113 118 e 2O 27 10170.^ 124 1 40 16 36 28 21837 171 163 8 28 2Q 24588 181 187 6 34 3O. . 27112.5 224 2OQ 15 IQ 31 . . 29213 215 23O 15 34 32 31232 260 251 q 25 -3-3 22060 274 271 -7, 24. 34535. e 3OO 2QO 10 12 3*5 ISSlS.S 2Q5 3OQ 14 26 36 36840. 5 326 J y 326 o 26 37360 357 34O 17 38 37804.5 389 354 35 26 3Q. . 38112.5 405 360 36 62 40 38195 377 382 5 57 41 . 37838 306" 2Q-7 -3 60 42. . 372^8. 5 3QQ 4O3 A 56 43. . 36^34.5 387 4IO 23 33 44 3^603 421 4l8 36 45 34735.5 429 425 4 4O 46 33660.5 421 431 IO 3O 47 32502 460 437 23 53 48 31228 440 442 2 ci 4Q 3OO55. 5 450 448- II 62 5O. ... ... 28855.5 476 454 ' 22 84 51 27510.5 479 458 21 105 52 26208.5 44.6 462 16 80 53 24785 426 464 38 51 54. . 23426 444 467 23 28 55 22170.5 ^OQ 471 38 66 56 20746 47Q 470 o 75 57 10-777. e 463 460 6 50 58 18116.5 455 47 15 44 59 16890.5 428 469 41 13 60 15672.5 488 468 20 33 61 I43Q2.5 468 461 7 4 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. AGE. Exposed to Risk. Actual Deaths. Expected Deaths. Error. Accumu- lated Error. 62 1*3261 450 458 I 41 6-1 12147.5 454 4^2 2 4^ 64 IIO2I.5 443 442 1 44 65 0084,5 435 412 3 47 66 QOOQ. 5 421 421 o 47 67.. 8081 306 408 12 35 68 7214 qQQ 304 5 40 60 6375.5 380 377 12 52 70. . 5622 315 360 45 7 71 . . 4053 308 344 36 2O 72. . 4378 340 33 iq IO 7-5 2771. 5 2Q7 308 II 21 74 3228 340 286 ^4 7-3 7c 26Q^ 254 250 r 28 76 22^ 24O 2^5 C "\"\ 77. . 1848.5 2O I 2OQ 8 25 78. . 1531 188 188 o 25 70 1257 171 167 4 2o go QQ5 140 144 2 s * 81 WJ 782 125 122 q 28 82 600. ^ 105 IO'? 2 7O 81 4.64. 06 B* I I - 41 84. . 330 61 67 6 35 8e 2^4.^ 55 ce o .. -JC 86 184 4O 4^ q a2 87 128.5 28 72 4 28 88 QI. % 26 25 i 20 80 . 57.5 II 17 6 2 1 } QO. . 43.5 IO 14 4 10 32 IO II i 18 O2 20 QQ 7 2 20 Q-J IO. 5 7 4 -7 2^ Q4 4 o 2 2 21 Q5 . . 3.-S i 2 I 2O 06.. 2 2 I I 21 07. . o O O O 21 Totals II99092.5 20517 20496 422 443 21 LIFE PROBABILITIES. WE have now seen how the probabilities of living can range themselves into an accurate and regular record of the durations of the lives of a number of persons, in the form of a mortality table. In our consideration of these probabilities up to this time we have dealt only with death or survival within one year of age. The probability of surviving one year from the age x we have designated by the expression p x and the probability of dying by q x . In these expressions x has reference only to the age, and there is no character to designate the time covered, one year being understood wherever no time is designated. Whenever any other period of time is to be designated as n years, it appears as a prefix, thus : n p x . We have found that p x = -~ , and we may now give this the gen- U eral form : We also found in our study of probabilities that the probability of two or more events happening is measured by the products of their separate probabilities, thus : p', a , , = p 1 X p a X p 3 , etc. It follows that the probability that two lives, x and y, will survive the term n is expressed by npr X np y = npxy, which latter expression will be hereafter employed. The expression n p xy may be extended in terms of 1 as follows : vx lx + n ^ ly + n lx*n X 1, + n P-' = * x * = TT X TT ~ i xi, ' 74 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. which latter may be condensed into the expression - f and still further by using the expression -y-^. In this and in all future expressions a prefix in the upper position will indicate that the lives are to be increased in age by that quantity. Thus n'pxy* means the probability that three lives aged x + t, y + t and z + t years, respectively, will survive n years. Likewise we have found that - !L k k + i _ From this we may derive the value of q x , covering a longer term than one year. This value we will express by | n q x , a want of uniformity with the expression n p x , which appears in the English usage, and has been adopted into general use by the formal acceptance of the English symbols by the last congress of actuaries. In consequence of this the English symbols will be employed throughout this work. Making our formula general, we have l.q, = -^- = -^- - = i - - -p The probability that two lives, x and y, will fail, will be | n q x X| n q y , which we may express by | n q^, the bar indicating we are dealing with survivorship ot one and not of both. The probability that at least one life will survive is the reverse or complement of the above, namely : The probability that x will survive and y die is the mul- tiple of the respective probabilities, viz. : n p x (i n p y ) n p x -- n p xy . The probability that one will survive and the other die PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 75 is the above probability, increased by the probability that y will survive and x die, thus : ( n^x np ly ) + ( np y npx y ) = ^ + n p y 2 n p xy . The probabilities that one life will fail is the reverse or complement of n p iy , the probability that both will survive, thus: | n q iy = i - - np* y . The probability that x will die in the n th year is the product of the probability that he will survive n i years and the probability that, when he has so survived, he will die in the n th year, thus: n i px X qx+ n t ; or the same prob- ability may be expressed in terms of p alone. For the probability of death the n th year is clearly the probability of surviving n i years less the probability of surviving one year longer or n years, thus : ^-!|q,= n-tpx npx. Or this probability may be directly derived from the mortality table, by dividing the deaths at age x + n i by lx , thus : i . The probability that both x and y will die in the n th year is the product of the probabilities that each will die, thus: .-i|q, X -i|q y = (.-ipx n px ) (.-! p y n p y ) 1, 1, The probability that the first life of x and y will fail in the n th year is the probability that both will survive n i years less the probability that both will survive n years, thus : n-i | q iy = *-i p* y npx y . The probability that the second life of x and y will fail in the n th year is the probability that at least one will survive n i years less the probability that neither will survive n years, thus: 76 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. u-l|qx~7 n-l p,~ -~npr y = n-, PX + .-, p y - n-lpxy - ( npx + npy - npxy) n~l px - n P x + n-l Py np 7 - - (n-l Pxy n p* y ) = .-, | q* + n-i q y n -,|q* y . The probability that one only of the lives, x and y, will fail in the n* year is the sum of the compound probabilities that x will die in the n th year, and y survive that year, n -i |q (i i|q y ), and that y will die in the n* year and x survive, n 1 1 q y ( i n 1 1 q ) , thus : -i|q* (i " --ilqj) +n-i|q y (i .-,|q ) = n-'|q* n-i|qx (n-i q y ) + *-, |q, The probability that neither of the lives will fail in the n th year is the product of the probabilities that each will not fail, thus : The probability that one life at least will fail is the reverse or complement of the last foregoing, thus : i~- (i -- n-i|q* D -i q y +- .-,|qxX .-i|q,) =.-i | q + n-i | q y n-i | qx X n -i q y . The probability that x will survive n years and y, n i years would be n p* X -, p y . We may make this analo- gous to joint life formulas in the following manner : The probability of y surviving n i years is a component part of the probability of y i surviving n years, which latter probability may be stated as first the probability of surviving one year and then of surviving n i years. The probability of y i surviving n years may be thus expressed : n p y -i = p y -, X n-i p y , from which n ,|p y =- . Substituting this value in n p* X n -i p y , we have : nPx X nP y l nPx ' y i - _L ~ PURE ENDOWMENTS AND ANNUITIES. SUPPOSE persons at age x, equal in number to l x , accord- ing to the mortality tables, and that each wishes to secure $i to himself if he survive one year; what equal sum must each pay in advance ? At the end of the year l x+1 persons will survive, requir- ing l x _|_ x dollars. Let the present value of $i due in one year be v ; then the value of 1^ dollars due in one year is vl^. That sum in hand, improved at interest, will amount to 1^ dollars at the end of the year, and is, there- fore, the aggregate sum which the l x persons must pay in advance. As each is to pay the same amount, the quota vl j_ for each will be -^^ dollars. A promise to pay a sum of money upon the con- tingency of survival only, is called a pure endowment. The single premium, paid in advance, is designated by the expression n E x , x standing for the age and n for the term; when the term is one year the prefix is omitted. We may thus write : If a E x were desired, we should find it by dividing the present value of l x + 2 dollars by the number of contribu- tors, l x . But the present value of $i due in two years is v a . So we have 2 E x := p 2 and, generally, E^ *x-l-a a z ~ j In insurance we shall often use the term " present value," not merely in its ordinary arithmetical sense as 78 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. above, but also to express the idea of the present value of a sum certain, modified by the probability of securing it. Thus n E x may be called the present value of a pure endow- ment for n years from age x ; it is the present value of one dollar due in n years, modified by n p* or the probability of surviving n years and may be expressed thus : We shall find this to be equivalent to the former expres- sion ; for IL = Pi , .rb. = ^ and, generally, ^bs. n p x . I X X Writing then our former formula and substituting these values, we have : E. = ^ An annuity is a promise to pay a like sum at the end of i, 2, 3, 4, etc., years; its present value is the present value of the sum due in one year, plus the present value of the sum due in two years, plus etc. The present value of an annuity certain is v + v 2 + v 3 + v 4 + etc. A life annuity is one in which each payment is contingent on surviving the year; its present value is, therefore, the present value of a pure endowment in one year plus the present value of a pure endowment in two years, plus etc. It may be expressed thus: = v p x -f- v 2 2 px + v 3 3 px + v n n pz __ v U, + v 2 l x+2 + v 3 1 T+S + . . . . v n l x+n 1, Where the term of the annuity is the life of the annui- tant, it is called a life annuity ; where it is a term of years, conditioned upon survival, it is called a temporary life annuity. The algebraic expression for the present value of a temporary annuity is n a x , the prefix indicating the PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 79 limited term and the suffix the age. The expression for the present value of a life annuity is a x , the absence of the suffix meaning, not that the term is one year, but that it is for life. To arrive at the present value of any annuity, we must first compute the present value of the simple endowments composing it and must then sum these values. This summation may be somewhat simplified, however, as follows : When one has survived one year, and the first dollar is due, the value of his annuity is one dollar plus the present value of an annuity at an age one year older than his age at the outset, thus : i -f a x+] , in the case of a life annuity. It follows that the value to him at the outset, one year earlier, was the present value of a pure endow- ment of i + a i+1 dollars, due in one year, or a x = E x (i + a x+1 ) = vp x (i + a x+1 ) Starting, therefore, at the extreme age of the mortality table, we may make a complete table of life annuities, thus, for instance : a 99 = v P 99 a ,8 = VPosC 1 + O a 97 = V P 9 r ( T + a 98 )> 6tC - NOTE. This is the English notation throughout. In American notation the symbol for the single premium for an annuity is A x , which also means an immediate annuity or with the first payment due now and not one year from now. It is equal to i + a x , English notation. COMMUTATION COLUMNS N AND D. IN the last paper we derived the following formula for the present value of a life annuity : _ v L+. + v l*+ 2 4- v l x + 3 , etc. "!7~ This formula is general in its application, but only as a formula. We have seen that values of a have a certain relation so that, one value being given, we may derive the next lower. We also know that the numerator in any value of a will consist of values of j_ for ages higher than the pres- ent age of the annuitant, but these values of 1 will not be modified with v raised to the same power. Thus etc. . and not vM . 4- v 3 1 +3 + v 4 l x + 4 , etc. - ' 3 In other words the index of v and the suffix of 1 no longer agree at all; you are constantly dealing with the same values of 1, but modified by changing powers of v. But let us multiply the numerator and denominator of the second term of the equation : a = v U+v'U+v'Wetc. by v* which will not alter the value ; this gives us _ v' +1 U+ v' +a U,+ v'+ 3 U+ etc. v 1 l x Now let us multiply the numerator and denominator of the second term of the equation : _vl, +2 +v a l z+3 +v 3 l x+4 +etc. ~U~ by v x + 1 , which also will not alter the value. We have PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 81 _ v'+'U,+ v'*'U.+ v'*'U.+ etc. V*M. +1 The numerator in this formula for the value of a x+l is the same as the numerator in the formula for the value of a x , except that the first term, v x+1 l x+l , is omitted; the denominator is the value of 1 corresponding to the age of the annuitant (x+i), multiplied by v raised to the same power (x+i). And this will be found to be general, thus: in It follows that when we have "once derived the values v z l x for each age, we may employ them to compute annuity values at any age. Let us designate these values as D x , D x+1 , D x+a , etc. We then have _ v x + 1 l, +i + v x+a l x + 2 + v' +8 U 8 + etc - a * ~ v x l x _D x+1 +D x+2 +D, +3 +etc. D x And, generally, _ D, +n+1 + D r+n+2 + D x+n4 . 3 + etc. ~^T These are the values in the D column of the commutation tables. A still greater saving in time and labor is effected by summing, once for all, the values of D above each age, thus furnishing at sight the numerators for the formula as well as the denominators. The sum of the values above age x is called N x and the formula takes this form _ D, - D x These sums are the values in the N columns of the com- mutation tables. The annuity values with which we have so far dealt, are 82 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. values of annuities with first payment due in one year ; an annuity of one dollar with the first payment due now is worth just one dollar more than this value, thus: i+a x . We may derive its value directly from the commutation tables, thus: N P+P+ D+ etc. etc. D D, D. + D x+1 +D x+2 + D x+8 + etc. D Note. The English notation as agreed upon in the last Congress of Actuaries is used in the foregoing throughout. In American notation, as used in all American publications hitherto, A x instead of a x is the symbol for an annuity and, besides, signifies the value of an immediate annuity. It is therefore equivalent to i+a x in the English notation. N x , in the English notation as here adopted, includes only the values of D at ages above x ; in the American notation it also includes D x . The reason for these differences is that English actuaries first had to do with annuities, first payment due in one year, while American actuaries first had to do with annuity values in connection with insurance premiums in which immediate annuities are employed. The formulas given take the following form in American notation : N N a, = yyS English A x i = -F^-S American and N*_ N i 4- a x = ' f , English = A x = ^-, American. DEFERRED ANNUITIES. WE have so far dealt with annuities, first payment due in one year with a formula as follows : A _ vl.+,+ v 2 !.+,,+ v 3 1.4.,+ 1 l P But let it be required to find the value of this same annuity a year earlier or at age x i, payments not to begin until age x + i as in this annuity. It is evident that the value of such an annuity would be the present value of a pure endowment equal to a x , thus: J E x _ l a x . The value v 1 of jE^, is, by formula, j-*- . Hence v 8 l +3+ v (vl,+,+ v a l I+a + v 3 l x+3 + v a l r+1+ v'U a+ v*U, x _ 1 In other words the fraction, expressing a x , is modified by raising the power of v and substituting !,_, for l x for denominator. The algebraic symbol for a deferred annuity 84 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. is x ; and we may give the formula a general expression : . *I It will be observed that n | a x does not signify an annuity with the first payment in n years, but a life annuity deferred n years, the first payment being due one year later or in n + i years. Thus the first payment on an annuity deferred twenty years is due not in twenty years but in twenty one years. In the foregoing formula we have Ja x = B E X a x+n . We have found the formula: n E x = v n n p x . Substituting, this gives Ja x = n E x a x+n =v nPx (a x+ J. But we have also seen that v n n p x = * +n and that N ' +n . Substituting again, we get the general formula : TEMPORARY ANNUITIES. IN our original consideration of annuities, we have a foundation for formulae for a temporary annuity which may be defined as an annuity limited to a term of years, first payment due in one year. A pure endowment due in one year may be considered a temporary annuity limited to one year. A temporary annuity limited to two years will have a value greater than this by just the value of the additional payment or the pure endowment due in two years. And, generally, adopting the algebraic symbol | n a x , to express a temporary annuity limited to n years, we have : a = E + E + E + ...... It will be observed that this differs from the first two equations of general formula for a life annuity : VI only in the particular that the factors of the numerator stop at n E x and v n l x+n instead of running to the end of the series. As the second two equations in the formula 86 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. are equivalent in every detail to the first two, it follows that This may be verified by multiplying by v 1 the numerator and denominator in the formula : L ^ = vU+v'U+v'U -vW. which does not alter the value and gives the identity of which with I . .-P.+.+ P.+ .+ D.+.+ ..... P.*. La,- - I5 - is evident by inspection. The numerator of this latter fraction is, then, less than N x by all the values in the series beyond the age x + n, that is: byD x+n+1 + D x+n + a + D x+n+3 + ....... But the value of these is summed in N x + n , whence we have ! D, +1 + D.+.+ D z+3 + . . D, +n _N x N, +tt ~ ~ ~~ CONVERSION FORMULAS. LIFE, TEMPORARY AND DEFERRED ANNUITIES. WE have so far considered deferred and temporary annui- ties in comparison with life annuities only; it remains to compare them with each other. Suppose two annuities, one limited to n years from age x and one deferred n years from age x. We find their values to be as follows : | n a x = ,E X + 2 E X + 3 E X + n E x _vU+ v a U,+ v 8 l x4 . 8 4- .. . . v n l x+B l x and Ja x = n E x a x+n n+l E x + n4 . 2 E x - _ v"* 1 l x+n+1 + v n + a l x4 . n+2 + 1, __ D z4 . n+1 + D i+n+a + D In other words, the value of the deferred annuity is expressed by those factors in the series v l x or the series D x beyond the factors expressing the value of the tem- porary annuity. They are thus found to be complementary to each other, their sum being equal to a x , as we may prove by adding them, thus: ( A + ,E + ,E. + - A ) + (.+,E. + .*.E. . .) = A + ,E. + ,E, . . ......... = a, ; 88 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. (D, + ,+ D.+.+ . . D.+.) + (D.+.+, + D.+.+. + . .) D. x and, plainly, - - !^ = -=^- == a x The values of temporary and deferred annuities are thus seen to be parts of a life annuity, which furnishes the fol- lowing formulae for converting the one into the other : DEFERRED TEMPORARY AND OTHER INTER- CEPTED ANNUITIES. AN annuity, the first payment deferred more than one year and payments thereafter limited, is known as an inter- cepted annuity. This, however, is a general term which may also be applied to an interrupted life or temporary annuity where payments are made for a term, discontinued for a term and then permanently or temporarily resumed. A more definite and specific name for the annuity in ques- tion would therefore be a deferred temporary annuity. Suppose, then, it were required to ascertain the value of an annuity on a life aged x to begin in n years and then continue for m years, subject to survival. Let us designate this annuity by the algebraic expression n | m a x . Remem- bering, then, that an annuity is composed of a series of pure endowments, we have .l. i = .*,^.+,,B.H- n+n E _D.+.+. + 0,+,.+, + D.+.+. D, The numerator of this fraction may easily be expressed in terms of N. The sum of the series of D above D x4 . B by definition equals N x+n ; the sum of the series above D x4 . n+m equals N x+m4 . n . Consequently the sum of the values between D x+n and D x4 . n+m4 . , equals N x4 . n N x+n+m . So our formula takes the form The same formula may be derived from the following 90 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. relations: If you deduct from the value of a life annuity deferred n years, the value of a life annuity deferred n + m years, the remainder will be the value of the intercepted annuity, deferred n years and then limited to m years. Thus .L a , = .IV-. + .K _N x+n N,^ m N z+ -N x+n+m " D x D x D x Suppose, now, that it were required to ascertain the value of an annuity running for n years, then interrupted for m years and then resumed for life. We may consider this as, first, a life annuity, less an annuity deferred n years and then limited to m years or, second, as a tempo- rary annuity for n years plus a life annuity deferred n + m years. In the former aspect, the problem takes this form : D. D, In the other aspect, the problem takes the form : + (.++, E < +.*.+, E + ' Suppose, again, that it were required to ascertain the PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 91 value of an annuity running for n years, then interrupted for m years, then limited to m' years more. This would be composed of a temporary annuity plus a deferred temporary annuity. Discarding the intervening steps, we may give the result in commutation symbols, thus: I . , _ (N. - N.Q + (N.+.+. N.+.+.+., ) In a x ~^n + mlm' a x - ~ ) Suppose, again, that it were required to ascertain the value of an annuity deferred n years, then limited to m years, then interrupted for m' years and then again limited to m" years. This would be composed of two deferred temporary annuities and would take the form : ' a + + + / 1 a In a similar manner all complex forms of intercepted annuities may be split up into temporary, deferred tempo- rary and life annuities and their value so ascertained. DEFINITION OF "PREMIUM." IN developing formulas for the present values of life, temporary, deferred and intercepted annuities, we have at the same time discovered formulas, on the same assump- tions, for the lowest price, paid in advance, for which a company could afford to sell such annuity contracts; since no company could afford to sell them at less than cost and, if it made a practice of selling below cost, no company could, from the proceeds of such sales, carry out its con- tracts. As it is commonly the business of actuaries in making these computations to discover this lowest price, they are wont to consider the resultant figures as net pre- miums instead of present values. The Institute of Actuaries' Text-Book defines Premium as follows: " The sum, whether single or periodical, which is payable in consideration of a benefit, is usually called a premium. " This definition does not include the idea that the premium must be sufficient or at least equal to the pres- ent value of the benefit, nor indeed does that idea properly enter into the definition of the word. The distinction is all the more important as the actuary in his ordinary work is engaged in finding present values or sufficient premiums and thus always has the question of the sufficiency of a premium in view. It is beyond question, however, that a sum of money paid and received in consideration for an insurance, endowment or annuity is a premium, whether it be sufficient or not from an actuarial standpoint. A very simple definition would seem to be: "A premium is the price of an insurance, endowment or annuity." Deeming it desirable to have as authoritative a definition PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 93 as possible, I have collated the following, mainly by per- sonal correspondence and interviews : " Premium, Gross. The amount which must be paid at fixed dates named in the policy to keep the insurance in force." Principles and Practice of Life Insurance. " The price or amount paid for insurance." Standard Dictionary. ' ' The amount paid or agreed to be paid in one sum or periodically to insurers as the consideration for a contract of insurance." Century Dictionary. " Premium in life insurance is a sum or sums stipulated to be paid to the company in consideration of future insur- ance benefits." Emory McClintock, LL. D., Actuary Mutual Life Insurance Company, and President Actuarial Society of America. " The price charged for insurance." Asa S. Wing, Actuary Provident Life and Trust Company. ' ' Premium is the sum, fixed in advance, which is paid to an insuring party for insurance." Rufus W. Weeks, Act- uary New York Life Insurance Company. " Premium, in insurance, is an agreed payment, made in advance, to purchase indemnity." D. H. Wells, Actuary Connecticut Mutual Life Insurance Company. "The premium, in life insurance, is the price paid or agreed to be paid as the consideration for the contract." J. A. De Boer, Actuary National Life Insurance Company. ' ' Premium is any sum paid to a company to secure the issue of or to maintain a policy. " Walter C. Wright, Act- uary New England Mutual Life Insurance Company. " A consideration, payable in one sum or instalments, for a contract for a payment contingent upon life or death. " David Parks Fackler, Consulting Actuary. " Premium, as used in life insurance, is the amount paid 94. PRACTICAL LESSONS IN ACTUARIAL SCIENCE. or agreed to be paid by the insured to the company as con- sideration money for the policy." B. J. Miller, Actuary Mutual Benefit Life Insurance Company. " A premium in insurance is the price received by an underwriter for a contract of insurance." James J. Barker, Actuary Penn Mutual Life Insurance Company. " Premium is the consideration required for the benefits guaranteed." /. M. Craig, Actuary Metropolitan Life Insur- ance Company. " Gross Premium. The amount or amounts of money which, if paid at the time or times stipulated in the con- tract, will, plus interest, provide for the payment of the policy's share of the company's claims and expenses." J. G. Van Cise, Assistant Actuary Equitable Life Assurance Society. 1 ' Premium in insurance is the stipulated consideration to be paid, either in one sum or periodically, under a contract securing a benefit or compensation for loss upon the hap- pening of the contingency insured against." L. G. Fouse, President and Actuary Fidelity Mutual Life Association. W. D. Whiting, consulting actuary, endorses the defini- tion in the Standard Dictionary; H. W. St. John, actuary ^Etna Life Insurance Company, endorses the definitions by the Century Dictionary and the Institute of Actuaries' Text- Book, and C. A. Loveland, actuary of the Northwestern Mutual Life Insurance Company, adopts the Century Dic- tionary definition as his own. WHOLE LIFE INSURANCE. ENGLISH and American actuaries in considering insurance have assumed that the sum insured is payable at the close of the policy or insurance year in which the insured dies. In their ordinary computations they assume that the deaths at any age are evenly distributed throughout the year and that, consequently, on the average deaths occur when the year is half over. Taking the sum payable as due at the close of the policy year is therefore assuming that the sum insured is payable on the average six months after the death of the insured. This assumption is plainly erroneous in these days of prompt payment of death claims. French and some other European actuaries, coming later into the field than the English, discarded this fallacy. They for a long time counted the sum insured as payable at the beginning instead of the close of the policy year, considering wisely, it seems to me that money to be devoted to paying claims within the year will earn no interest. Subsequently, how- ever, in the interest of scientific exactitude, they shifted their position, now assuming that the sum insured is payable at the middle of the policy year. Though the English assumption is more at variance with the facts than either of the others, it has persisted and is so embalmed in all the text-books, tables of rates and reserves and computations that it would be rash to attempt to reconstruct the whole science to conform to more accu- rate assumptions, even if the congress of actuaries had not in the interest of uniformity indorsed the English notations as employed in the English text-book. Accordingly the general formulas in this book will be based on the same assumption as the English. 96 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. There is so close a connection between a life annuity and a life insurance that the value of the latter may be readily derived from the former. To define the two is to make the close relationship apparent. A life annuity of one dollar is a promise to pay one dollar at the close of each year that the annuitant survives; a life insurance of one dollar is a promise to pay one dollar at the close of the year that he fails to survive. It is therefore supplementary to a life annuity. But let us suppose that two annuities have been issued on a life aged x, each for one dollar payable at the end of the year, but one for each year that he enters and the other for each year that he survives. The value of the latter is a x ; the value of the former one year from now, after the first payment has been collected, will be a x or before that payment is collected, i + a x and its present value accord- ingly is i + a x discounted by interest for one year or v (i + a x ). But the difterence between the value of a promise to pay one dollar at the end of each year which the annuitant enters and the value of a promise to pay one dollar at the end of each year which he survives is the value of a promise to pay one dollar at the end of the year which he enters, but does not survive, or, in other words, the present value of an insurance on his life of one dollar. Let A x be the symbol of the present value or net single premium of an insurance of $i at age x; then we have A x = v(i + a x ) a x Note. This is according to the English notation ; accord- ing to the American, n, is the symbol for a single pre- mium and the formula becomes n. = v (A. ) - (A, - i) = i + v (A, ) - A, WHOLE LIFE INSURANCE. COMMUTATION COLUMNS C AND M. IN the last lesson a formula was derived to give the value of an insurance of one dollar in terms of an annuity of one dollar at the same age. We may, however, get the value of an insurance from the mortality table directly, as fol- lows: Suppose 1 lives at age x are insured on the same day for one dollar each, payable at the end of the year in which death occurs. The total to be ultimately paid is then l x dollars which will fall due as the deaths occur, that is, d x dollars at the end of one year, d x+i at the end of two years, etc., until all have died. The present values of these sums are vd x , v 2 d x+J , v 3 d i+2 , v 4 d x + 8 , etc. The aggregate present value is v d x + v a d x + ,+ v 3 d x+3 + . . , etc. , and the value of each insurance is that aggregate divided by l x , the number of lives, thus: x Just as we did in the case of the annuities, we may multi- ply numerator and denominator by v x without altering the value. We thus get A - v x l x But v x l x we have expressed by the symbol D x and the numerator now consists of the sum of a series of the form v x+1 d x , which we may derive once for all and call C s , C x + l , C x+a , etc. We then have 98 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. This formula is evidently general ; for let us take any other age as for instance, x + n. We get Multiplying numerator and denominator by v raised to a power equal to the suffix, x + n, we have ' x+ n - v'+ n l x+n But this denominator is equivalent to D x + n and this numerator is equal to the sum of the values of C beginning with C x+n . Conseqiiently, we save labor by summing these values for each age once for all ; which has been done and the sums given the symbol M. Thus our formulas take the very simple form : A M < + - ' +n ~~^7 Except in its aggregate or summated form of M, C is so little used that it is seldom tabulated, the values of M being alone given in the commutation tables. It is noteworthy that D x = v x l x and C x = v x+ * d x ; sub- stituting for d , its value in terms of 1, we have C. = V+'(l.-l. + 1 ) = vD,-D, + 1 Since N x = a series of values of D x+ , or 2 D x+ ,, N x _, = a series of values of D x or 2 D x and M x = a series of values of C x or 2 C x , it follows that: M x ^vN x _ l N x So the value of A x may be derived from the N and D columns directly, thus: v!SL_. N. . The American symbol for a net single premium PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 99 for whole life insurance is n instead of A; but unlike the values of N, the values of M exactly correspond in the English and American notation. In American notation, the formulas take the form *L_ -VN.-N.+ . n ' ' D D DEFERRED, TEMPORARY AND DEFERRED TEMPORARY INSURANCE. IF an insurance on a life now aged x is not to oegin until a later age, it is known as a deferred insurance. Suppose it is to begin in n years; then, after n years has elapsed and the life is aged x + n, the value is Its present value, therefore, would be; adopting n | A x as the symbol for that value : n x ~~ n x D x+n D x < D x+n D x ' If an insurance is to extend only for a limited term, it is called a temporary insurance and its value is designated as | A x . The value of a temporary insurance for one year from age x is the present value of the aggregate number of dollars, d x , payable at the end of one year to the heirs of those of l x insured who do not survive, thus | t A x = -y- - and, multiplying numerator and denominator by v* , I A -V*'d. ._C. ~D ' The value also of one dollar payable at the end of two years to each who survives one year but not the second is v x + 2 d which being similarly modified becomes -^ PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 101 The value, then, of an insurance for both years will be the sum of these and generally A = But the numerator of this fraction is M x or the sum of the values of C beginning with C x lessM x + n or the sum of the values of C beginning with C x+n which gives us the formula I A= M.-M... 1 n * D, An insurance beginning at some time in the future and then running for a limited time is known as a deferred temporary or intercepted insurance, and its value is given the symbol n | m A x , which means an insurance which begins in n years and then continues for in years, or from age x 4- n to age x -f n -f- m. We have just found that the value of a dollar payable at the end of the second year, if one survive the first year and fail to survive the second, is j = Xit., = C. This is th e value of an insurance for one year, deferred one year and the formula may be made general, thus : I A - C ' + - -~' A temporary insurance deferred n years and then con- tinuing for m years would be composed of all these insur- ances for one year from x + n up to x + n 4- m; hence But the sum of these values of C are equal to M x+ n or the sum of the values of C from age x + n less M x+n+m , or the 102 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. sum of the values of C from age x + n + m. From which we derive: .M.+ .-M.+.+. .1.4.= -57- From the interrelations of these various insurances we get the following conversion formulas: B |A, = A, | D A x = n E x A x+n LA = A,-JA. = A, - D E x A x + n Note. The American notation is as follows: Deferred insurance n n m Temporary insurance n xn Deferred temporary insurance n | m n z CONVERSION FORMULAS ANNUITIES AND INSURANCES. EARLY in our consideration of pure endowments and annuities we learned that U E X = v n p x and that n v a The last formula may be stated also as follows: a x = 2V nPl The principle is that the value of a sum, payment being contingent, is the value of a like sum certain, multiplied by the probability of its becoming due. Applying this to insurance, we have for an insurance of one year : ! A x = v q x The value of a second year's insurance becomes: ,l,A, = v ,q. The value of the two years' insurance is, then : 1 1 A x = i v q x + v 2 , | q x or generally : | n A t = vq x + v 2 Jq, + . . . . v^.Jq, This makes the value of a whole life insurance A,= vq, + v a ,|q,+ . . . ^svVJq, But the probability of dying in any year is equal to the probability of surviving to the beginning of that year, less the probability of surviving to its close; thus: .-,
  • a x , from which we get : These equations also yield the following values for a x in terms of A x : From A, = v (i + a r ) a x , A x = v-f- v a x a x = v + (v i) a x = v (i v) a, v = v A From A x = -^ * * -i -i (i + i) A, = i i a x i a x = i ( i + i) A r i-i+i A _ From A x = v (i i a x ), we get also: a * = H ~' This may also be derived from A x = v (i v) a, . From A, = i d (i + a x ) = i d d a x , d a x = i d A x _ i d A x _ i A, a * ~ d^ d~ io6 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. a.. From A. = i ~h a /, J. O \ A M Q IX d / -^~^-x doO t^x Referring again to the fundamental formula, A x = v(i + a x ) a x and the explanation that the value of an insurance of one dollar at the end of the year not survived is the difference between the value of an annuity of one dollar at the end of each year the life enters and of a like annuity at the end of each year the life survives, it is at once clear that the same reasoning applies to temporary insurances and annuities, so that we may substitute as fol- lows: ENDOWMENT INSURANCES.. THE pure endowment or promise to pay only on the con- dition of survival is not a form of contract commonly pur- chased; but a contract promising to pay a sum in event either of death during a term or survival of that term, known as endowment insurance, is common enough. The value of an endowment insurance of one dollar on a life aged x and due in n years or at prior death, we may designate as \ n AL r . It is composed of the successive values of one year's insurance plus the value of a pure endowment due in n years, thus : \ n ^= vq x + v a ,|q x + . . vVjq.-H.E,. =|.A* + .E.. In other words, by its very definition the endowment insurance is composed of a temporary insurance for n years and a pure endowment in n years. We have found the value of | n A x expressed in commuta- tion symbols to be x ~Z ^ and the value of n E * +n nx , Combining these, then, we have : Y 1 A. Referring to the conversion formula, A z = - L and its explanation, which was: A dollar down produces i interest at the end of each year the life survives and amounts to i -Hi, principal and interest, at the end of the year the life does not survive; wherefore an insurance of i -H i is valued at i less the value of the interest during T *- 1 A. survival or i a z and an insurance of i is valued at - io8 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. It is equally clear that the investment at the end of every year as well as the year the life does not survive, amounts, principal and interest, to i + i. If, then, we regard the transaction as limited to n years when the investment is withdrawn, we have : An endowment insurance of i + i is worth the original investment of i, less the value of a temporary annuity of i for n years, which may be stated thus: i + i^=i_ia and But r. v and : = v i = d; from which the i <- i i + i formula takes the form : Referring to the conversion formula, A x = v (i + a x ) a r and its explanation that the value of a promise to pay a dollar at the end of the year a life does not survive is the difference between the values of an annuity at the end of each year the life begins and an annuity at the end of each year the life survives. Suppose now that a man aged x had two annuities, one payable at the end of each year up to n years, inclusive, that he begins and another payable at the end of each year for n i years, inclusive, that he survives. Now the first of these annuities means a certain payment of i at the end of the first year which the life is already entering, and, thereafter, as many payments as are included in the second annuity, except that each payment is deferred one year. The second annuity is evidently a temporary annuity for n i years, | n _ l a x ; and the value of tbe first annuity in terms of the second becomes v (i + n _, a x ). PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 109 During n i years, the difference between these annui- ties each year is precisely equal to the value of that year's insurance; since if death occurs in any year, the second annuity of i is not paid, leaving just i to be paid by the first annuity. At the close of n i years, the second annuity becomes valueless and the first is worth just i, payable at the end of that year, no matter whether the life fails or survives. This covers the insurance of i for the last year and an endowment of i if the life survives. The total difference between the annuities, then, is : the value of an insurance of i for n i years; the value of an insurance of i for the nth year; and the value of an endowment of i in n years. But these three things aggregate to the value of an endowment insurance of i in n years; and give the fol- lowing formula : | B ^ = v(i +L-,a,) |.-,f- The case is sometimes presented of an insurance for one sum and an endowment for a less or greater. The insur- ance and pure endowment elements being so completely separable, these are not difficult to compute. Suppose, for instance, a case of insurance for i for n years and an endowment of J^ in n years, commonly called a semi-endow- ment. We will not try to get one symbol to express the combined value, but will state it separately thus : | ^A,. + *4 n E x , which at once gives the key to the solution. We find the values separately to be : i A _ M T M I+a ~ Combining these we get : _ M, M,+ n D x+n _ 2 (M r no PRACTICAL LESSONS IN ACTUARIAL SCIENCE. In the foregoing, the English notation has been used throughout. The Institute of Actuaries' text-book, how- ever, also adopts the following alternative symbols: A x -, for ! u ^E x and a z - { for | n a x . It introduces these after the others have been made familiar, but expresses a preference for them. With these changes the conversion formula becomes : A x ^-j = v (i -f a x ^zn) a x=T7 The American symbol for a single premium for endow- ment insurance is r n... The same commutation for- mulas are applicable. exn . ENDOWMENTS AND INSURANCE WITH RETURN OF NET SINGLE PREMIUM. SUPPOSE a pure endowment due in n years with return of net single premium if death occurs in n years. The value is composed of the value of an endowment of i in n years and an insurance for n years of the net single premium, which we will designate by E (slanting, instead of verti- cal), thus: M i+n ) E - - Suppose an insurance for n years, with return of net single premium. We have now an insurance of i for n years and an insurance for n years of the net single pre- mium, which we will designate by the symbol A (slanting, not vertical), which symbol is used in the English notation for all special or unusual net single premiums for insurance. Thus we get : A .A. I Or in commutation symbols, we get : A[D (M M + ) ] = M x M, M,-M,+ . H2 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. Suppose a whole life insurance, with return of net single premium ; its value is composed of an insurance of i plus an insurance of A, thus : A = (i + A) A x = A x + (A x )A ^ (i - AJ = A Or in commutation symbols, we get : D A = (i + A) M ^H ..+ M. M.Q D, D, A D i+n + M, M, +n + A (D i+n + M, M + A [D, - (D, t .+ M. - M 1+ .) = D i+n + M. - M. + - Suppose a like case, except that the premium is only returnable if death occurs in n years; the value is com- posed of an endowment of i in n years and an insurance of i + A for n years, thus : PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 113 D x A = D x+n + M x M x+n + A (M x M x+ ) A [D x - (M x - M x + n ) ] = D x + n+ M x - M x D +n D.+.t- M x -M, + n " D x (M x M x+n Again, suppose a like case, except that the premium is returnable only with the endowment upon survival; the value is composed of an endowment in n years of i + A and an insurance of i for 11 years, thus : A = (i + A) A + I . A. = D, A= (i + A~) D I+n + M, M, + n A (D, D, + J = D.+ .+- M, M,+ D >+a+ M. M... D^TD It should be observed that these formulas cover only the cases of policies with return of single premiums. Policies with return of annual premiums will be dealt with after we have considered the general problems of annual premiums. An insurance with return of a single premium is not equiv- alent in value to an insurance with return of the corre- sponding annual premium. Note. A general formula for the single premium of a return-premium insurance, the premium without premium - return being known, may be found thus : The full amount insured is i + A\ but A, the known premium for an insur- ance of i, bears the same ratio to i that A bears to i + A, that is: I = A : : I + A A A=A(i + A) A (i A) = A A= ENDOWMENTS AND INSURANCES WITH RETURN OF GROSS SINGLE PREMIUMS. IT is customary to increase the net premium by a margin or loading to make the gross or office premium, actually charged for the benefit. This loading may take the form of a percentage addition to the net premium or of the arbi- trary addition of a lump sum, or both. That a premium is gross instead of net is indicated by the " prime ", thus: A^, n E^, etc. The relations between gross and net pre- miums in each of the foregoing cases, respectively, indi- cating a percentage addition by k and a lump addition by c, are as follows: A'= (i + k) A A' = A + c A' (i + k) A + c To avoid repetition, we will employ the last of these; because formulas drawn from it will fit each of the other cases by substituting zero for c in the first instance and for k in the second instance. Suppose, then, a pure endowment in n years with the gross premium, E ', returnable if death occurs in n years; the value is composed of an endowment of i in n years and an insurance of E' for n years, thus: - D *+n + [(i + k) E + c] (M M T+B ) D ,JF=D.+ .+ c(M, MH.J+ [O + k) E\ (M, M.+ J [D, (i + k) (M, - M, + m ) ] = D x+n + c (M, - M x + n ) D x+n + c (M x M T+n ) -D x -(i+k)(M x -M,+ n ) Suppose an insurance for n years with return of gross PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 115 premium; this is an insurance of i plus an insurance of A\ thus: c](M. M.Q D,A = (i+c) (M x M.O + [(i +k)^] (M, M I+ .) ^ [D x (i + fc) (M, M X+D ) ] = (i + c) (M, M 1+ .) (i + C)(M. M. M.+.) -D,-(i +k) (M,-M. + J Suppose an insurance for life with return of gross pre- mium; it becomes: A (i + A') A x = [(i + (i + k) A + c] A x ' A[i (i +k) A,] = (i+c) A x (i + c) A x ~ i + (i + k) A x In commutation symbols, we get: _ (i + A') M x _ [i + (i + k) A + c] M x D x "DT" D x ^ = (i + c) M x + [ (i + k) ^] M x A [D x (i + k) M z ] = (i + c) M x (i + c) M x " D x (i + k) M x In the case of an endowment with return of gross pre- mium, either at death or upon survival, we have a tempor- ary insurance of i + A' and a pure endowment of i + A', worth respectively: ( M '" M xO C 1 + O and D + (i + A/) - ; and combined : , _ _ (M, M,+ B + D. t .) [i + (i + k) A + c] n6 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. D x A = (M x M x+u -f D x + n ) [i + (i + k) A 4- c] = (M x M x+n + D x+n ) (i + c) 4- (M x M x+n + D x + n ) [(i + kMl ^[D T (i+k) (M x - M x+n + D x+n ) ] = (i+c) (M x - M x+n + D x+n ) ,_ (i + c) (M x M x+p +D x+n ) -D x -(i+k) (M x -M x+n +D x+n ) In an endowment policy returning gross premiums only in event of death, we have an endowment of i and an insurance of i + A' and combined, in commutation symbols : (M x M x+p ) (i+^) + D x+n A ~~ ~^~ _(M X M xhn ) [i + (i+k)^ + c] + D X+X D X ^ = (M X M x+n ) A [D x (i+k) (M x M x+n ) ] = (i + c)(M x (i 4- c) (M x M x+n ) + D x . n D x -(i+k)(M x M x+n ) In an endowment policy returning the gross premium only in case of survival, the value is composed of an endow- ment of i+A' and an insurance of i ; which, combined, become, in commutation symbols : _ M x - M x+n + [i + (i + k) A + c] D x + n D x D x A = M x - M x+n + (i + c) D x+n + [ (i + k) A} D x+n A [D x (i + k) D x+n ] = M x M x+n + (i 4- c) D x+n _ M x -M x+n + (i + c) D x+n D x (i+k)D x+n It must not be forgotten that these are not formulas for single premiums, corresponding to net annual premiums to return gross annual premiums. VARYING BENEFITS IN ANNUITIES AND INSURANCES. WE have so far developed the following commutation columns : D x = V l x =:V'*. (I, l, + t We have also developed, among others, the following commutation formulas: N Q - - * N N + N -i | n a x or a x -i = - ^-^ -, which we condense to -^- * X A.= M D, M _ M M - n A x = ' D > condensed to ^ I X If A D x or A x -, = We have also discovered the following reciprocal rela- tionships between the values of D and C and N and M : C x = vD x - D x4M M x v N x _, N x Suppose a life annuity beginning at one sum (k) and' annually increasing or decreasing by another sum (h), and designate its value by the symbol (v a) x . The value of n8 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. this annuity will be made up of the values of a number of successive annuities, thus: first, an annuity of k; second, an annuity of h, deferred one year; third, an annuity of h, deferred two years, etc. This combined value may be expressed thus : (v a), = k a x h ( | i a, + 1 2 a x + 3 a x + . . ) k N But the value of k a x is ' ; the value of h | i a x is D x+1 ; of h 1 2 a x , D ' +a , etc. Substituting, we get : (v a) = kN x h(N. + 1 +N x+a + . . . . ) For convenience in solving such problems, we now create a new commutation column, containing values of N summed from each age to the end of the table and call these values S; thus: = (D x+] + D x+2 . . ) + (D I+2 +D x+3 . . . ).+ ; = D X+1 + 2 D x+2 + 3 D x+3 + .......... Substituting in the formula, we get : k N h S + (v&\ = - ^- The case of a decreasing annuity is an infrequent, though not an impossible one in insurance mathematics. The most frequent form of a varying annuity benefit is one increasing annually by an amount equal to the original annuity. The solution of such a case becomes yet simpler ; for assuming k and h in any case to be alike equal to i, and adopting the symbol (I a) x to express its value, we have: (la) _N T + S + ,_ A j ' ~ D. D, Turning now to insurances, suppose an insurance of k to PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 119 increase or decrease by h each year and designate its value by (v A) x . The value of this insurance will be equal to the sum of an insurance of k, of an insurance of h deferred one year, another of h deferred two years, etc., thus: (vA), = k A, 11(1,^4- | 2 A X + | 3 A,+ . . .) But the value of k A x is ' ; of h 1 1 A x , - ' +1 , etc. Substituting, we get : _kM x h(M x + 1+ M x+2 + . . . . ) '" D x For convenience in solving these problems, we now make yet a new commutation column by summing the values of M from each age, to the end of the table which values we call R. thus: = (C, + C, +1 ....->..+ (C x+I +C x+2 + .. = C X + 2C x+J + 3C,+ 2 + Substituting in the formula, we get : The case of a decreasing insurance is also infrequent, though not impossible; the case of an insurance increasing annually by its own amount is also infrequent, except in the form of return of premiums. But, expressing it by the symbol (I A) x , meaning an insurance of i, annually increased by i, we derive the following formula from the last: (I A^> - M ' + R '+- - R * ( X ~DT -DT We have seen that: C, = v D x - D 1+ , M x = v N x _ 1 N x Since R x is the sum of values of M beginning with M x , 120 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. S x _,of values of N beginning with N x _j and S x of the values of N, beginning with N x , we may infer: R, = v S,_ - S, . Substituting, we get: , IA) _ R, _ v s._ .- s. >* ~ D - D But S,_, S r = N._, and v S,_, S, is less than that amount by just d S_,; substituting again: S x But -=j- is the value of (I a) x by formula and -- 1 of (I a) x ; so that we may derive the formula: = v [i + a x + (I a) x ] - (I a) x TEMPORARILY VARYING BENEFITS IN ANNUITIES AND INSURANCES. SUPPOSE an annuity of k at the beginning, increasing or diminishing annually by h for n years and then remaining constant. Let its value be designated by the symbol (v n| a) x . This value will be composed of an annuity of k, plus or minus an annuity of h deferred one year, an annuity of h deferred two years, etc., up to an annuity of h deferred n years, thus. (v 5| a), = k a, .h (Ja x + 2 |a x + . . ,-Ja,) k N But the value of k a x is ' ; the value of h l a x is ' ; of h, |a. , * etc., U ntil h n _, |a, = Substituting, the formula becomes: _kN.h(N. + .+ N. Remembering that S x is the sum of all values of N beginning with N x , we at once see that the above series of values of N are equal to S x+1 , less the values beginning with N x+n or, in other words, less S x+n . Restating: the formula with this substitution, we obtain : ) When k and h are each unity, adopting (I n| a) x as the symbol of this value, the formula becomes in the case of an increasing annuity: Likewise an insurance of k increasing or decreasing by h annually for n years and then constant, represented by 122 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. the symbol (v n| A) x , is composed of an insurance of k, an insurance of h deferred one year, an insurance of h deferred two years, etc. , to an insurance of h deferred n years, mak- ing n i increments, thus: (vHj A) x = kA x h( 1 |A r + ,|A JE +. . . JA X ) Substituting commutation values, we get: x This series of M is plainly equivalent to R x+l , or the values beginning with M x + l , less R x+n or the values begin- ning with Mz+s+j . Substituting, we have: (y -! AX = kM.*h(R.+.-R..) In the case of an insurance of i, increasing annually by i for n years and then constant, adopting the symbol (I n|A) x to represent the value, this formula takes the form : (I n| A), = R x+l R x+n _ R x - R x+ D x D x ' D x Suppose, however, that the annuities and insurances are temporary; that is, suppose an annuity of k, increasing h annually for n years and then expiring. The value of this will be just as much less than (v"na) x as is equal to the value of the annuity (k + (n i ) h) to be paid after n years, which is - ~~ l ) -^*. Giving the value of the temporary annuity the symbol (v a) x - ( , we have : _kN. h (S.+. - S.+.) - [k + (n-- r)h]N > + n V. V /* n| jj _ k (N x - N x4n ) h [S x+ , S x+n (n - i) N x+n ] D x k(N z N T+n )h(S > + 1 S x + n +N x . n -nN x+p ) PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 123 _ k (N r - N x+n ) h (S. +1 S x+n+1 - n N r+n ) D r When an annuity of i, increasing by i each year for n years, and then expiring, is in question, adopting for it the symbol (I a) x -j, we have: _ N, N x hn + S x+1 S, +D+1 n N x + n a). .1 - - The same course of reasoning, identically, in the case of insurances, gives us ( A ^ - - k (M, M Z+B ) h (R g+l R x+n4 . l n M x+n ) v v "-yat n| -Q and (I A); -, = R.-R.-M.,. It will be remembered that the formulas for an annuity and a deferred annuity differed only in the numerators, N N thus : a x = =-^ and n | a x = fr*-- And that a similar differ- ence exists between the formulas for insurance, thus: Since S x is the sum of values of N beginning with N x and R x of M beginning with M x , it is evident that the same relations will exist when S or R is employed, as in the case of varying benefits, as exist when N and M are employed. Thus if the value of a varying annuity or insurance deferred t years be desired, it is only necessary to modify the formulas by using x + t wherever x appears in the numerators only. INTER-RELATIONS BETWEEN SINGLE PRE- MIUMS. SUPPOSE an insurance, either for life or n years, payable as an annuity certain for in years, first payment immediate; the value is equivalent to a single premium for. an insur- ance payable in one sum for an amount equal to the dis- counted or commuted value of the annuity. Suppose an endowment in n years, payable as an annuity certain for m years, first payment immediate ; the value is equivalent to a single premium for an endowment equal to the commuted value of the annuity. Suppose an endowment insurance for n years, payable as an immediate annuity certain for m years; the value is again equivalent to a single premium for an endowment insurance of the commuted value. Combinations may also be made between temporary insurance and endowment, providing either for insurance in a lump sum and endowment in an annuity certain or for insurance in an annuity certain and endowment in a lump sum, by remembering that the value of the element involv- ing an annuity certain is equivalent to a single premium for the commuted value of the annuity. It has been already pointed out that the value of a life or temporary annuity or insurance deferred n years is equiv- alent to the single premium of an endowment for an amount equal to the value of the annuity or insurance n years hence. Even a whole life insurance may be split into a temporary insurance to the age next before the extreme age of the table plus a pure endowment due at the end of the year to all who enter that year of age ; or may be considered an endowment insurance, maturing at the extreme age. ANNUAL PREMIUMS. LIFE AND LIMITED PAYMENT LIFE. THE value of an annual premium of i, first payment immediate and payable for life, is plainly the value of an immediate life annuity of i or i + a x . Consequently there will be as many times i in an annual premium equiv- alent to a single premium for whole life insurance as i + a x is contained in A x . Designating the annual premium by P , we have, then, the formula: To get this formula in terms of A x only, let us substi- tute for a x its equivalent value, found in our conversion formulas, of ^ i; this gives A A I~V., d d To get the formula in terms of a x only, let us substi- tute for A x its equivalent from our conversion formulas: i d ( i + a x ) ; this gives Conversely, we may derive A from P x , thus: 126 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. And also a x from P x , thus : P - x d * rr~a7~ P x (i 4- a x ) = i d (i + a x ) a x (P + d) = i - P - d . . i - (P. + d) _ i PX + d ~ P x + d In commutation symbols, the formula for annual pre- miums for life becomes: A, M x _._ N x _, _ M x = n^ar : : -57 " ~DT 'NT; Or from one alternative formula: Or from the other alternative formula: d A d M M v d M v P = A, " D x D x " D, M x The value of an ann-ual premium of i, first payment immediate, limited to t premiums, is the value of an immediate temporary annuity for t years, thus: i +| t _,a x or i + a x t -zr|. Consequently there will be as many times i in the limited annual premium for t years equal to a single premium as the value of the immediate annuity for t years is contained in the single premium, thus: In commutation symbols, this takes the form : M x ^N_ N + _ M ~ : " D x -N^-N^., Note: TT is the American symbol for an annual pre- mium; in English notation it always signifies a special annual premium. In American notation the formulas are: PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 127 *.&.. dn ' _L a =--*? * ""A, : ~i n x r ~ A, ' : N," D^ dM, - N / "D x -M, ANNUAL PREMIUMS. TERM, DEFERRED INSURANCES, ANNUITIES AND ENDOWMENTS. SUPPOSE an insurance for n years with premiums annu- ally in advance. The value of an annuity due for n years has been found to be i + a z ^zr|. Designating the pre- mium by the symbol, | n P x or P^,, we have: P * In commutation symbols: _M.-M.+._.. "'- It may be desired to have this formula in terms of D or N, instead of M. We have seen that : M x = v N x _, - N x = D x d N x _ r Substituting, then, these values in our formula: ' P . _vN._.-N - (v N, +c _ .- N.Q - Or p , D. D . -- D ,+ . A ~ N,_,- N, + ^, ~ Suppose a pure endowment in n years, annual premiums in advance; we have: P E , = _^- i + a^ri In commutation symbols: D, D x PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 129 Suppose an endowment insurance for n years, annual premiums in advance ; we have : In commutation symbols: _ M. - M.+ .+ D.+ . ^ N x _- N,+._. - _ Of this last formula, ^-^ - x+ " is the insurance "NX- 1 - 1N x + n-l portion and ^= -- ^ - the pure endowment portion. -^x , - 1N x + n -, It may be desired to have this formula in terms of D and Q _ D N only; if so, substituting the value ^^ - ^^ -- d for x I - *+n l the equivalent T z M r+p -, we have: x nl N x __, NI*^, N x _, I = ^_ * d Similarly by substituting, instead, v T ^"xT' +n we obtain: P- - v _ N x _, - N xtn _, Suppose an insurance for n years, annual premiums in advance for t years only ; we have : Ai _ . _ *p' 130 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. In commutation symbols : _ M * M x+n m N x+1 Suppose an endowment insurance in n years, annual pre miums in advance for t years ; we have : p A, a t ^ x ni , __: I ^ a x t-ll In commutation symbols: _ M x M x+n + D g+B ^ N x _ - N x+t , t 1 z ni ~ D z D x Suppose an insurance deferred n years, annual premiums in advance; we have: IP - I A - ' I +".531 In commutation symbols: ,p _M. +n ^N I ..-N l+a _, ' " ' D, D, If paid for by t annual premiums, these formulas become : P |A - .I A - - M ^~ ' ' i+.CT.~N^,-N, + ^, Suppose an insurance for m years deferred n years and they take the forms : Or if paid by t annual premiums : - Ax _ M r+n M x+n+m PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 131 Suppose an annuity deferred n years, annual payments in advance; we have: P la - ' *+n ' * "i + a xn - =ri --N T - 1 -N I+n _ i If paid for by t annual premiums : Suppose an annuity for m years, deferred n years, annual premiums in advance; we have: P n 1 m a * T + * = *T^- ""JSP" i * a* n _i T IM^ JN Z ( . n _ 1 If paid for by t annual premiums : Note: In American notation, the most important for mulas are: _ n. B _ _ M x - M I+n _ N.-- N, +n _, ' XQ ~A '- N x N x - Nx-N z - A x n N x N x+n 7T f e e * L n A ** ANNUAL OFFICE OR GROSS PREMIUMS. THE LOADING. IN considering single premiums we learned that to make the office or gross premium it is customary to add to the net premium either (ist), a percentage of itself, (A + k A) or, (2d), a fixed sum, (A 4- c) or, (sd), in both of these man- ners, thus : A + kA + corA(i + k) + c. And we found that the last formula will answer for all three cases; for when c = o, it becomes identical with the first, and when k = o with the second. The custom in adding kt loading " to annual premiums is similar; such gross premiums taking the form, respectively : P (i +k), P + candP (i +k) + c, the last again comprehend- ing both the others. An analysis of the subject of ** loading "so as to arrive at more exact conceptions of how it should be done may not be out of place in this connection; and to understand the subject at all, it is necessary to consider the purposes for which this extra contribution is levied. The main pur- poses are (ist) to provide a contingent fund to cover unforeseen losses of any sort, and (2d) to meet the expenses. As to the first of these, the better system is now gen- erally acknowledged to be to construct the net premium on such mortality and interest hypotheses as will guarantee its sufficiency and to accumulate a surplus to cover other sorts of losses. The most important thing, then, to be considered is the expense, and usually attention is directed almost exclu- sively to that. Expenses may be divided into several dis- tinct categories, such as: cost of new business, cost of col- PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 133 lection and renewal, cost of general management and cost of care of investments. These costs commonly come in different forms and at different times. The cost of new business comes but once, viz., at the outset of policies. It comprises commissions for new insurances, examiner's fees, agency salaries and expenses, canvassing literature and supplies, etc. The largest item, generally, is for commissions, which are paid mostly as percentages on first premiums. It is, therefore, not possible to escape the conclusions : first, that the expense comes in a lump sum at the outset, and, second, mainly as a percentage charge on premiums. The cost of collection and renewal also comes in the form of a percentage charge on premiums after the first. The general management of a life insurance company is for the end to furnish insurance, and it appears clear that it should be a fixed charge per $1000 insurance carried, or at most should be proportionate, not to the premium, but to the costs of insurance which are the values of the bene- fits currently furnished. The cost of caring for investments is properly chargeable against the income from investments and need not appear in the loading. It should appear in determining the inter- est factor in constructing net premiums and in computing dividends, as only the probable net revenue from invest- ments should be considered. These leave us with three items of widely different nature to incorporate into the loading, viz., cost of new business, coming at the outset and being a lump sum and usually a percentage of the first premium; cost of collection and renewal, a percentage of subsequent premiums; and cost of insurance management, a fixed sum per $1000 insured. Concerning the first of these, it must be premised that 134 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. the first premiums must be large enough to meet this cost and the mortality cost of the year or else moneys contrib- uted by other insurances will be trenched upon. It has been attempted to meet the difficulty in two ways, viz., first, by charging a larger premium the first year to cover the additional expense ; second, by charging the same or a smaller gross premium the first year, but treating it as cov- ering only one year term insurance, the other insurance with its net premium beginning the next year at one year's advance of age. The first plan is difficult to sell and the second has not demonstrated its success. Another plan has been adopted and employed by French companies whose actuaries have devoted a deal of thought to the subject. This is to determine the amount of money required to cover this cost ; to deduct this amount from the net premium for the first year and to add its equivalent value in an annuity to each net premium, including the first year. Designating this cost by F and the correspond- ing annuity by f, we have 'for a life policy: Net premium after ist year 2 P X = P x + f Net premium ist year 1 P X = P x + f F This plan appears to answer all the requirements. The cost of collection and renewal offers no difficulties, being a function, which we will designate as k, of the pre- mium after the first year; but as it will really need to be a function of the gross premium, we defer it fqr a moment. The cost of management may be added to the net pre- mium as a fixed amount (c), each year including the first. The values of P x , F, f and c are, then, determined in advance, and we have the following requirements estab- lished: 'P'l (gross premium ist year) = P' x (gross premium thereafter). PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 135 a P x (net premium after ist year) = P x 4- f 'P, (net premium ist year) = 8 P X F = P x + f F F = f (i + a,J. From these, remembering that the premium when loaded with other costs must be loaded by k per cent of itself for cost of collection and renewal, we get : F T = ( 2 P X 4- c) (i + k) = (P, 4- f + c) (i -f k) 'P' x = ( 'P r 4- F + c) (i + k) =( a P x + c) (i + k) = (P z 4- f + c) (i + k) ENDOWMENTS AND INSURANCES WITH RETURN OF NET ANNUAL PREMIUMS, CONTINUOUS THROUGHOUT TERM. SUPPOSE an insurance for life with return of all net annual premiums. Designate the net annual premium by the symbol TT, used in the English notation to signify any special or unusual annual premium. The value of this insurance is plainly separable into two portions, an insur- ance of i and an insurance of TT , increasing yearly by ?r or, in symbols, A x + ?r (IA) X and the annual premium may be found by dividing this value by i + a x , the value of an annuity due, thus: i + a r JT(I + a, ) = A, + ir(I A), - i + a, - (I A), In commutation symbols this takes the form : a x R N_ M M x Suppose a pure endowment due in n years with return of all net annual premiums if death occurs within n years. The value of this is n E x plus TT(! A)i-| and the premium may be found by dividing by i + a x ^zrj, thus: PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 137 -{n-a,^|)=.E. + *( I A)ia ^+y^IA)^ A In commutation symbols this takes the form: D, -(N._ - N ._,) = D i+o + r (R, - R, + .- n M. + .) ff [N t _- N, +o _- (R, - R i+ - n M, + J ] = D, + . _ _ D.+ . _ " N ,- - N I+n _- (R, - R, + - n M 1+ .) Suppose an insurance for n years with return of net annual premiums. The value is composed of AJ-i + ir (I A)i a and the premium is found by dividing by i + a x ^zn thus : Ai- B| +ir(IA)^ i + a x n -zr i "(i + a x ^i) = A^-, + ir(I A)i a Or in commutation symbols : _ Aj 5 + ,r(I A)!;, - j - i + a x izri _ M, - M, + , + ir(R, - R x - n n M I+n ) ^N x _ t ~ N, + ,_, D, D x _ M, M r+n + TT (R, R. + n n M, + n) 138 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. tEN^- N.^ - (R x - R x+ - n M x+n ) ] = M, - M i+n _ __ M x M, + n _ Z N,_, N ,+(*, R x+n n M x+ J Suppose an endowment insurance, with return of all net annual premiums either at death or maturity. This con- sists of an endowment of i + n ?r and of an insurance for n years of i+ TT, increasing annually by TT, the value being (i + riTr) n E x + Aia + TT(! A)J-, and the premium is found by dividing by i + a x n i, thus: _,(! + ^TT) P x+n + M, M >+n + ff (R, R x+n n M x+n ) ^ N _ - N x+n _. D, _ (i + n TT) D x+B + M, M + + 7r(R R x+ n M + ) R +D -nM x+n ) riN^,- N x+n _- n D x+n - (R x - R x +n - n M x +n ) ] = D x+n + M x M x+n _ __ D x+p + M, -M, + n _ - N,_ - N x+n _- n D x . - (R x -- R x+ - n M, + J Suppose an endowment insurance for n years with all net annual premiums returned only in case of death. The value is then composed of A x -, or n ^E x + K (I A) x -, and the annual premium is found by dividing by i + a x ^=11, thus: _ M x M x+n + D x+n + 7r (R x - R^ B - n M x+n ) ^ D x N _- N x+n _ t D. PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 139 _ M x M x+n + D x + n + * (R, - R x+a n M x+n ) N X _, - N^; irCN^ N x+n _, ) = M x M x+a + D x+n + ,r(R, R x+n - n M X+ J .r [N x _ N x Hn _ i - (R x - R x+n _ n M x+n ) ] = M x - M x+n + Dx +B _ M x -M z+c +D x+n ' N x _, N x+n _ - (R x - R I+n n M x+n ) Suppose an endowment insurance with return of all net annual premiums only upon survival. The value is Aini+C 1 +HTT) n E x and the premium is found, thus : A x -,+ (i +n7r) n E x 7T - - j - i + a x n , M x M x+n + (i + n TT) D x _ n . N x _, N x D x _M X M x+n + (i N _ _ N ^ - N x+n _, )^ M x Ml +n + (i + n ^, - N x+n _ n D x+n ) = M x - M x+n M x M x+n + D x+n N x _ N ENDOWMENTS AND INSURANCES WITH RETURN OF NET ANNUAL PREMIUMS, LIMITED PAYMENTS. SUPPOSE an insurance for life paid for by t annual pre- miums, with return of net annual premiums at death. This is composed of an insurance for life and an insurance of TT, increased by TT annually for t years, and thereafter con- stant. The value is, then, A x + 7r(In A) r and the limited annual premium for t years is found, thus; _ A, + Tr(InA), 7T - - j - 1 + a, , , _ M, + , (R. - R.+.) _._ N,_ - N. +t _, D x D, (N_ N I+t _,) = M, + * (R, R I+t ) tN^,- N in _- (R, - R, +t ) ] = M, _ M, Suppose a like insurance except that return of premiums is only in case death occurs during premium-paying period. The value will then be A z 4- TT (I A)i n and the limited annual premium for t years will be found, thus : x - R z +t t M, _M r + 7r(R r R. +t tM x+t ) N^-N^^ ir (N r _ N x+t _,) = M x + TT (R x R x+t t M x+t ) ir[N,_ N x+t _ (R, - R x+t t M x+t ) ] =_- M x PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 141 M x - N x _ N x+t _ (R x R x+t -t M x+t ) Suppose a pure endowment in n years, paid for by t pre- miums, with return of net premiums in event of death dur- ing the endowment period. The value will then be n E x + 7r(I A) xt - + t TT t |A x ^zn and the limited annual premium for t years may be found, thus: ^ = n E x + *(I A), n + t TT JA,,^ i + a, izn _D x + n + 7r(R T R T+t ^ N x _.- N x+t _, N x _ - N x+t _, TT (N x _- N x+t _,) = D x+n + .(R x R x+t t M x+t ) + ^ [rC- NX ( R * R *+t t M x+t ) t (M x+t - M x+n ) ] = D x+n * = N z _ - l t _- (R, - R x+t - 1 M x+t )- 1 (M x+t - M x+n ) Suppose an endowment insurance for n years paid for by t premiums, with return of premiums in event of death during the endowment period or in event of survival. The valueis(i+t7r) n E x + | n A x + * (I A xFl )+ tirj A x 5-5-1 and the limited annual premium for t years may be found, thus : _ (l + trr) n E r +| ?n A x + 7 r(IA xt -) + t TT t | A x n ~ l TT ~ ; + t^r) D. + .+ M. M.^,+ ^(R x R^ t tM 1+t ) D, " t.(M. +t -M.Q-, _._ N M -N. tt -. D, D, 142 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. (i + tO D x+n + M x M x+n + n (R x R x+t t M x+t ) + t^(M x+t M x+n ) __ N x _ N x+t _, (NU - N x+t _,) = (i + t O D x+n + M x - M x+n + v (R x R x+t t M x+t ) + t TT (M x+t M i+ j ^[N z _ t N x+t _ t t D x+n (R x -- R x+t t. M x+t ) -t (M x+t - M x+n ) ] = M x M x+n + D x + n _ _ M x M x+n +D x+n _ -N^- N I+t _- t D x+n - (R x - R x+t - t M x+t ) t (M x+t Mx +n ) Suppose an endowment insurance for n years paid for by t premiums, with return of net premiums only in event of death during endowment period. The value is A x -, + 7r(I A) x fi 4- t 7r t | A x n -=n and the annual premium may be found, thus: ft= A, a + IT (I A) xt - + 1 7r t |A x ^i i + a, izrr M x M z+n 4-D >+n + 7r (R r R x+t _tM i+t )+t.(M x+t M x+n ) D ,r(N x _- N^.,) = M x - M x+n+ D m+m + r(R. R +t - tM i+t ) + t.(M x+t -M x+n ) 7rt N i _-N x+t _-(R x -R +t -tM x+t )-t(M x+t -M x+ J] = M x -M x+n+ D x+n _ _ M x M x+n + D x+n _ - N j _-N x+t _-(R x _R +t -TM x+t )-t(M i+t - M x+n ) Suppose an endowment insurance for n years, paid for by t annual premiums to be returned only upon survival. The value is (i 4- t TT) n E x + A z -, and the annual premium is found, thus: PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 143 i= (i + tQ.E. + AJa i + a, ;=i| D N,--- N + ,_, ^(N_- N, +t _,) = (i + t O D, + .+ M. M. + . (N._,- N, +t _ - t D.O = D.+.+ M. - M l+n M. M.+ .+ D.+ ,. ENDOWMENTS AND INSURANCES WITH RETURN OF GROSS ANNUAL PREMIUMS, CONTINUOUS THROUGHOUT TERM. TAKING P (i + k) + c as our formula for the value of P', the gross annual premium, suppose an insurance for life with return of gross annual premiums. The value is plainly A x + [7r(i + k) + c] (I A) z and the net annual premium is found thus: _A,+ |>(i + k) + c] (IA) X 7T - - - - i + a, _ M x + |>(i + k) + c] R x _._ N _ " _ M,+ [TT(I + k) + c] R ~^r N^^r M x + [>(l + k) + c] R x [N x _ (i + k) R x ] = M x + c R x M + c R N x _ - (i + k) R x Suppose a pure endowment due in n years with return of all gross annual premiums if death occurs within n years. The value is n E x + [>(i + k) + c] (I A)l nl and the net annual premium is found, thus : n E x + Q(i +k) + c] (I A)j-, c] (R x R x+ n M x+n ) N x _ - N x+n _, r (N B _ - N x+n _ t ) - D I+n + [, (i + k)+ c] (R, - R x+ - n M x+ n ) v [ Nx _ - N X+B _ - (i 4-k) (R. - R x+ - n M x+n ) ] =D x+n+ c(R x -R x+n nM r+n ) PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 145 D, + B +c(R, R, + , nM, + B ) " N x _ -N x+n _- (i +k) (R, - R + - n M.O Suppose an insurance for n years with return of gross annual premiums. The value is 'A ij, + [ir(i + k) + c] (I A)i^ and the net annual premium is found, thus: A i k cIA + a, jzji |>(i +k) + c] (R. R.+ . n D, _ M, - M.+.+ [,(i + k) + c] (R. - R,. n M.+.) N._-N, + ._, (N, _,- N 1+n _.) = M. M, + n + [, (i + k) + c] (R, - R.+. -nM. + J w[Nl _- N, +n _ - (i + k) (R, - R I+ - n M I+ .) = M, - M I+l ,+ c(R,-R +n -nM l+ .) M. M^.+ c (R. R l+n n M..) - N x _,- N i+o _- (i + k) (R. - R k+ - n M i+n ) Suppose an endowment insurance with return of all gross annual premiums either at death or maturity. The value is [i + n (i + k) + n c] n E x + AJ a + [TT (i + k) + c] (I A)^-| and the net annual premium is found, thus: [i+n7r(i +k)+nc],E, + Ajj + |>(i + k)+c] n , [i +n7r(i +k) + nc]D x+n + M x M x+n +|>(i + k) + c] (R r - R T+n nM,Q __ D x [i +n TT (i + k) + n c] D x+ n + M x ~ M x+ n + [> (i + k) + c] 146 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 7r(N x _, N^^Jrzrfi + n ( i + k) + n c] D x+ n + M x M x+ n + [,(i -f- k) + c] (R x - R x+n -n M x+n ) .[N- -N,^- (i + k) (n D x + n )- (i +k) (R x - R X+D - nM,O] = (i + n c) D x+n + M x - M s+n + c (R x R x+n n M x+n ) _ (i + nc) D x + n + M x M,.+ B + c(R x R i+n nM x+p ) - N x _,- N x + n _,- (i + k) ( n D x+n ) - (i + k) (R s - R x+n -nM i+n ) Suppose an endowment insurance for n years with all gross premiums returned only in case of death, the value is A x -, ^ [TT(I + k) + c] (I A)ia, and the net annual pre- mium is found, thus: M, M z+ n + D z+ n + [ ff ( i+ k) + c] (R, R x + D - n M, D. k) + c] (R r - R + - n M, + n N x _ - N x+n _, -(N x - - N^^) = M x - M x+n+ D x+n + KI + k) + c] (R, R x+n -nM I+n ) -[N,., - N,^^ (i + k) (R x - R x+n - n M a O ] = M s M x+n + D x+n + c (R x - R x+n - n M x+n ) M x M x+n + D x+p + c (R, R x+n n M x+n ) - N,_, - N x+n _ (i + k) (R x - R x+n - n M x+n ) Suppose an endowment insurance with return of all gross annual premiums only upon survival. The value is A X ni + [i + n TT (i + k) 4- n c] n E x and the net annual pre- mium is found, thus: _ A! -, 4- [i 4- n TT (i 4- k) + n c] n E x i + a x szr, PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 147 M r M,+ n +[i +n ff (i +k) +nc] D x+n D, _ M x M z+p + [i + n TT (i + k) + n c] D z4 . n N x _ N X+D _, (N^ N^^) = M x M x+n -h [i + n * (i + k) + n c] D x+n [N _ N x+n _ n (i + k) D x+n ] = M x M x+n + (i + n c) D^ B M x M T+n +(i+nc)D, + , - N x _- N x+n _ n (i + k) D x+n ENDOWMENTS AND INSURANCES WITH RETURN OF GROSS ANNUAL PRE- MIUMS, LIMITED PAYMENTS. SUPPOSE an insurance for life paid for by t annual pre- miums, with return of gross annual premiums at death. The value is A x + [TT(I + k) + c] (I t -, A) x and the annual premium is found, thus : ^ A g +|>(i+ k) + c] (I r , A), i + a, jzri _ >I X + |> (i + k) + c] (R r - (R x R x+t ) N x _ N x+t _, *(!*_- N x+t _,) = M x + |>(i + k) H- c] (R, R x+t ) ff [N^- N x+t _, - (i +k) (R x -R x+t ) ] = M x + c (R x -R x+t ) _ _ M x +c(R x R x+t ) - NI __ N x+t _- (i + k) (R x - R x+t ) Suppose a like insurance except that return of gross annual premiums is only in case death occurs during pre- mium-paying period. The value is A x + [TT(I + k) + c] (I A)i n an d the net annual premium is found, thus: 1 + a rai M x + [TT (i + k) + C] (R, R x + t tM x + t _ M, + [ir (i + k) + c] (R, - R x+t - t M x+t ) N x _-N x+t _, PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 149 [!_-]*+,_- (i-k) (R,_R l+t _ tM, +l ) ] = M,+ c(R, R I+t tM I+t ) _ _ M r + c(R t R.+. tM.+ t ) _ - N,_- N i+l _- (i_+ k) (R,_ R I+t - t M, tt ) Suppose a pure endowment in n years, paid for by t pre- miums, with return of gross premiums in event of death during the endowment period. The value is n E x + [TT (i + k) + c] (I A)l r , + t [TT (i + k) + c] J A x jzn and" the net annual premium may be found, thus : t |A x t1 D x+n + [> (i + k) + c] (R x R x+t t M x+t ) + 1 KI + k) + c] N x _ D x+n + [TT (I +k) +C] (R x R x + t M x + t ) + t [)r (i (M x+t M x+n ) N _ ~ N r(N M - N, +r ^) = D^.T[r(i7*k) + c] (R z - R I+t - t M x+t ) + t [ w (i + k) + c] (M x+t M x+n ) ^[N^ N^,, (i+k) (R x _R x+t _tM x+t ) t(i + k) (M x+t M x+n ) = D x+n + c (R x R x+t t M x+t ) + t c (M x+t - M x+n ) D t+n +c(R x R +t tM x+t )+tc(M x+t M x+n ) -N x _-N x+t _ l -(i+k)(R x -R x+t -tM x+t )-t(i+k) (M x+t -M x+n ) Suppose an endowment insurance for n years paid for by t premiums, with return of premiums either in event of death during the endowment period or of survival. The value is [i +t TT(I +k) + t c] A x n - + TT[(I + k) + c] (I A)i r , + [t TT(I 4- k) + t c] J A izi| and the annual premium is found, thus : UNIVERSITY 150 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. [i + ITT (i + k) + t c] A x -, + [r (i + k) + c] (I A) tl + [trr(l+k)+tc] t |A xn | _ _______ (i + tr(i+k) + tc] (M x M x+n + D x+n ) + |>(i (R, '- R, +t - t M, +t ) + [t ff (i -f k) + t c] (M, +t - M, + n ) N 1+ ^, ) = [ . + 1 (i + k) -t c] (M. - M. + i + k) c] (R, - R x+l - t.M, + ,) + [t * (i + k) + t c] (M i+ ,-M l+ J ^[N^-N^^-tCi +k)(M, M, + .+ b. t .) (i+k) (R, - R. t| - t M. +1 ) - t (i -I- k) (M. +t - M.O ] = (i + tc) (M, M x+n + D t+n ) + c (R. R; +t tM, +t ) . +tc(M I+ ,-M, t .)] (i + tc)(M M l+ .+ D l+ .)-i-c (R,- R x+l tM l+ ,) + t c (M... M xt .) ] _ ' - N, tl _-t(i + k) (M.-M. t .+ D,^ (i+k) (R x - R +l _t M, +1 ) - t (i + k) (M, + .- M. + J Suppose an endowment insurance for n years paid for in t years with return of gross annual premiums only in event of survival. The value is [i + t TT (i + k) -f- t c] n E Jl + Ai ni and the net annual premium is found, thusi : [.!.+ t TT (i +k) +t c] D, + n + M x M^-,^ 'N^, N D x D x _ [i +t T(I + k)+tc] D X+P +-M X M x+n N X _ ^ +t _, : (N^ N x+t _, ) = [i + t * d +k) + t c] D x+n +: M x M i+a PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 151 * [N r _,- N x+t _ - t (i + k) D x+n ] = (i + t c) D x+n + M x - M, +B M x M X+D + (i + t c) D x+a - ' N x _-N x+t _-t(i + k)D r+n Suppose an endowment insurance for n years paid for by t premiums with return of gross annual premiums only in event of death during endowment period. The value is A x a + fr (i+ k) + c]~ (I A)J r , + [t * (i + k) + t c] t | A x ^ and the net annual premium is found, thus : A, a + MI + k) + c] (I A) x r, 4- [t ^ (i + k)" +t c] M x -M x+n +D x+n + [.(i+k)+c](R x -R x+t -t^M x+t ) + [t T '(i + k) + -trc] (M x + t M x+n ) --.- N *~~ N x + t^i ~D y M x M x+n + D x+n + J>(i + k) + c] (R x R x+t t M x+t ) + [t ,r (i + k) + t c] (M x+t M x+n ) N x _ N x+t _, ^(N^- N^^) = M x M x+n + D x+n + [ T (i + k) + c] (R x R x+t tM x+t ) + [tn-(i+k) + t c] (M x+t M x+n ) ^[N,. N x+t _-(i+k)(R x R x+t tM x+t ) t(n-k) (M x+t - M x+n ) = M x - M x + n + D x+n + c (R x R, +t t M x+t ) + t c (M x+t - M x+n ) (M x M x+n 4- D x+n + c (R x R x+t t M x+n ) + tc (M x+t - M x+n ) - Nx _ N x+t _- (i + k) R x - R I+t + t M E+ J t(i+k)(M, +t -M x+n ) Suppose an endowment insurance for n years paid for in t years, with return of gross annual premiums only in event of death during premium-paying period. The value is 152 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. A x a 4- [if (i + k) + c] (I A)ir, and the annual premium is found, thus: _A x ^r+7r(i + k)+c] (I A)j r , I+^CT, M. M x + + D >+ n + [ ff (i + k) + C ] (R, - R x+t - 1 M. +t ) D, _ M, - M x4 . n + D I+n + [IT (i + k) + c] (R, - R x+t - t M x+t ) N x _ - N x+t _, = M x - M x+n + D x+a + [ ff (i + k) + c] [N x - - N x+t _ - (i + k) (R, - R +t - t M x+t ) ] = M E M x+t + D x+n + c (R, R x+t - t M x+t ) M, - M x+t + D x+n + c (R, - R I+t t M i+t ) - N x _,- N x+t _^ (i + k) (R, - R x+t - t M. +t ) AGE AND PREMIUM KNOWN, TO APPROXI- MATE THE TERM. GIVEN the age x, the amount of the insurance and the net single premium | n / A x , to find the duration of the insur- ance, n'. We have the formula : In this equation, the first term is a known quantity and both M x and D x are determinable quantities by mere reference to the commutation tables, the age x being known. This leaves but one quantity, M x + n / neither known nor determinable by inspection. We find its value in terms of the known quantities, thus : D,(|. n' will be equal to n + y 2 years; or let M x+n M x+n ' be any fraction of M X+B M x ,. n+i , as w, then n'=-n + w. This may, in order to approximate the excess over n years by days, be stated in the form of a proportion, thus : M x+n M x+ . n+J : M x+n M x+n /" 365 : n' n . ,: ( M x+n - M x+n+] The theory upon which this approximation is based is that of uniform deaths throughout the year, while strict adherence to fact calls for increasing mortality; but in this formula the error is on the safe side, the assumption being that the premium will be expended a little sooner than the mortality calls for. The approximation, however, closely corresponds to the practical system of apportioning mor- tuary cost, commonly employed by American companies. , In the case of annual premiums, the formulas are derived as follows : '.M.-M.+., . N *- NX*,'-, In this equation all terms are known or determinable except M x+n > and N x+n /_ l . We may find the combined value of these thus : I./ P. (N x _ N X+T/ _J = M x M x . +n / , . M x + n > | a ,P x (N x + n ,_J ^ M x -^| nV P, N x _, Now^find the value of - M; +n a /P r -(N^^.), taking for ^ : -^, %any age (near the, age to- which the insurance seems PRACTICAL LESSONS IN: ACTUARIAL SCIENCE. 155 from inspection likely to: run), as a trial value. Continue the trials until two values are found, next under and next over the ascertained value. If an approximation in: years only is desired, take the value most closely approximating the ascertained value; the exponent will be the age (x + n') approximately when the insurance ceases ; and the term may be found by deduct- ing the age at entry, thus :. x + n' x = n'. . If an approximation in days is desired, then having found the values, next over and nextunder the ascertained values, let them be designated by the symbols, X x+n and X x+n4 , 1 and let the ascertained value be designated by the symbol X x+n ,. Then since X x+n X x+n > < X x+a - X X+Q+1 and since if , , we may assume that when X x+n X x+tt / and, generally, when X x + n X x+n / = w (X x , +n X r+n + 1 ), _(w being any fraction) . ^i n'=: n + w Or to approximate the excess over n years by days, we may employ the proportion : X x+n X, +B+1 , : X x+n - X x+n ,: I 365 ! n'- n ' n ,_ n _ 365(X. +n -X. +n Q _ -, v v ~ * x + n X + Q + ! Given the age x, the amount of insurance and the net single premium, A x 71, of an- endowment insurance, to -find the term to maturity, n';. We have the formula: io _M X -^M T ^^D K ^ ... , .7 . 156 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. In this equation, the known or determinable quantities are A x 7,, M x and D x and the unknown are M x+n 'and D x+n >. We may get all the known quantities on one side and the unknown on another, thus : D x A x 7, = M y M S+B / + D lV M x+n D xtB /= M x D x A, 7, Now find the value of M r+n / D x+n >, taking for x + n' any age x + n (near the age at which from inspection the endowment seems likely to mature) as a trial value and repeat until the values next under and next over the ascer- tained value have been found. If an approximation in years only is desired, the age of the value most nearly equally the ascertained value may be taken as the age at which the endowment will mature ; and the term may be found by deducting the age at entry, thus x + n' x = n'. If an approximation in days is desired, then designate the values over and under the ascertained value by the symbols X x+n and X X+B+1 and the ascertained value by X x+n / and proceed as the foregoing calculations where the same symbols were employed. The final formulas will be as before n'= n + w andn'-n= In the case of annual premiums, the formulas may be derived as follows: _ _M X M. +I H- D x+n , P *= N_ - N +n ,_, In this equation P x TJ, M x and N x _j are known as deter- minable quantities and M x + n ', D i+n > and N x+n /_j are unknown quantities. We may get the latter upon one side of the equation, thus: PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 157 M I+n '- D 8+m P x 7, (N,^,) = M x P. 7, N x _, Proceed as before to find values, X I+n and X I+n _ p next under and over the ascertained values, X x+n / by which symbol M x+n / D x+n P x 71 (N,*^) may be designated. The age (x + n or x + n + i) of the value X approxi- mating X x+n / most nearly, will be the approximate age at maturity in even years and the term to maturity is x + n' x n'. The approximations in fractions of a year and in years and days may be found by developing the calculation as before, resulting in the final formulas : n' = n + w Given the age x, the amount of insurance and the limited premium t PJ T, for t years, to find the term, n'. We have the formula : M.-M.+.. In this equation, the known or determinable quantities .are t Pi7i, M r , N x _, and N x+t _ } leaving only M x+n / as the unknown quantity. The equation takes the form : t p i7i ( N X-,- N, + t _,) = M r M x+a , M^^M^P^N.-N^) Find by inspection of the M column, that value of M nearest the ascertained value and the age will approximate in even years the age at which the insurance will cease ; the term is then found by deducting the age at entry, thus: x + n' x = n'. To approximate in years and fractions of a year or in years and days, find by inspection values, M x+n and M x + n+1 , next under and over the ascertained value. Then 158 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. proceed precisely as in finding the approximate term, the net single premium being given, reaching the formulas: n' = n + w 365 (M x+ M + ,) and n' n = -V * + ^^ w. M x+n M, + n + j Given the age x, the limited premium t P x ^, for t years and the amount of endowment insurance, to find the term in which the endowment matures, n'. We have the formula: P - _~ x _ - N x+t _, In this equation, the known or determinable quantities are t P x j, M x , N x _ t and N x+t ._ i , and the unknown quanti- ties M I+D / and D x + n /. Getting the known quantities on one side of the equation, we have : -.Px.M (N M N l+t _,) = M x - M x+n ,+ D. +1/ M, +D >- D x+n , = M x - t P M T, (N x _ - N +l _,) Find the value of M x4 . n D x + n from the commutation tables, taking for x + n any age (near the age at which from inspection the endowment seems likely to mature) as a trial value. Repeat until two values are found, next under and next over the ascertained value. These we may designate X i+n and X x + n+) and the ascertained value X i+n /. The same calculations already repeatedly employed will give the formulas : n' = n + w 365 (X x+n -X t+n ,) n' n = * - = w. ^Z+~~ ^x+ n + ! Given the age x, the amount of the annuity and the net single premium [ n /a r , to find the term for which annuity will continue. We have the formula: PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 159 _r'N.-N.+.. D. In this equation, the known or determinable quantities are n > a x , N z and D x and the unknown quantity N I+n /. Clearing of fractions and separating out the unknown quantity, we get: Find the value N x+n next under the ascertained values, N s+n /; x .+ n will then be the age attained when the last full annuity will be payable and x + n x =n will be the term during which a full annuity is payable. But a fractional annuity will be due one year later, on survival. If the difference N x+n N x+n > were equal to N x+n N x+n+1 , a whole annuity would be then payable. We may therefore approximate the fractional annuity by this proportion, letting w represent the fraction : N x+n -N i+u+I :. N x+n -N x+n , : : i :w ' N x+n - N x+n+l It follows then that the term of the annuity is n i + years, during n years of which a full annuity is payable and a fractional annuity, w, at the end of n + i years. VALUATIONS, GROSS AND AGGREGATE. THE liabilities of a life insurance company, aside from ascertaining immediate claims against it, consist of future death-claims, endowment and annuity payments. The company owes the whole amount insured by it, death being certain, subject to the continuance of the insurance in force ; it must therefore be counted to owe the whole amount insured until some part of the insurance be dis- continued. The folly of diminishing this computed liability by esti- mating discontinunces is at once apparent when the case of insurances for whole life, to be paid for as currently enjoyed, is considered. Counting the company as liable for the whole amount insured and the insured as liable for all future premiums, liabilities and resources precisely balance as they should; discontinuances at any time remove also equivalent resources and liabilities. On the other hand, counting the company as liable for anything less than the whole amount insured because of estimated discontinuances will make the resources apparently greater from the start than the liabilities and will thus destroy the balance which should exist. The company, then, is liable for all insurances for whole life that it assumes and for an ascertain able amount for all insurances for less than whole life. We may sit down with a list of the insurances classified by ages and figure from our mortality table how many deaths will according to that table occur the first year, the second year, the third year every year in fact until all lives have failed. The only things that diminish the company's present liabilities for these death-claims are, first, that they are not now due and, second, that if it had the money in hand PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 161 now to meet them, it could put it at interest until the respective claims were due. For these reasons, the actual present liabilities may be computed as less than the whole amount insured, but only by so much as equals the discount at a reasonable rate on the respective death claims and other payments from the dates when they will accrue to the present date. Thus we discount the claims due in one year by one year' s interest, those due in two years by two years' interest, etc., summing all the discounted amounts to find the immediate liability. The resources consist of two distinct classes, first, the actual cash and other assets and, second, the future pre- miums. These future premiums form a resource which will be realizable at least, as rapidly as is requisite in order to meet death-claims. They may therefore be offset against the death-claims and other payments. This may be accomplished by ascertaining the present or discounted value of theppreiniums, by discounting sums due in one year by one year's interest, etc., and adding up the amounts thus arrived at. It goes without saying that if the present value of the premiums plus the present assets is less than the present value of future death-claims and other payments, the com- pany is insolvent. From this it follows as a necessary corollary that the company in order to be solvent must have in reserve a sum at least equal to the difference between the value of all future death-claims and other payments which it must pay and the value of all future premiums which it will receive. Since the company is insolvent if it does not possess this reserve, it is clear that this reserve possesses the nature and has the effect of a liability instead of a surplus. The company may in its balance-sheet eliminate both the item 162 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. of present value of future death-claims and the item of present value of future premiums and merely charge as a liability against present assets the reserve. Originally balance-sheets exhibited as resources all pres- ent assets plus the present value of all future premiums and as liabilities all immediate claims plus the present value of all future claims. This style of balance-sheet is still generally employed in Great Britain and appears there in advertisements of companies. ^ The shortened balance-sheet exhibits as resources only present assets and as liabilities immediate claims plus the reserve, which is the present value of future death-claims less the present value of future premiums. By converting the first balance-sheet into an equation, we shall plainly see that the second is directly derived from it. We have : Present assets + Present value premiums = Immediate claims + Present value future claims + surplus. Present assets = Immediate claims + (Present value future claims Present value premiums = Reserve) + Surplus. We thus find the aggregate reserve which the company needs to preserve intact in order to meet its liabilities. In computing this reserve originally actuaries inclined to com- pute the present value of future premiums from the gross or office premiums, allowing nothing for expenses. The real assumption was that surplus enough would arise dur- ing the operation to cover expenses, which was a reasonable assumption once, whatever may be thought of it as applied to present conditions. This method is called * ' gross valua- tion," an apt name both because the *' gross" premiums are dealt with and also because the reserve is regarded as one mass and the policies are dealt with in gross and not PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 163 singly. Under this method, the addition of new policies to the insurances reduced instead of increased the net balance of liability, it often being years before the present value of its future premiums fell below the present value of its future costs. The very mention of the influence and effect of the addi- tion of new policies indicates that the aggregate reserve is composed of separate values of individual insurances to be ascertained by finding the differences between the probable future premiums and the probable future costs of each, the sum of which differences equals the aggregate reserves. It was always foolish at best to count the difference a present resource when the value of future premiums exceeded the value of future costs for the reason that no enforceable obligation to pay future premiums exists. At most it is only safe to count future premiums as a resource in an amount equal to future costs. When agency methods and expansion of business made expenses nearly equal to the entire loading which had been added to premiums to meet expenses and other contingent requirements, the gross system of valuation became unavailable. One of the first public acts of Elizur Wright was to make this clear. He proposed instead what is known as net valuation and he procured it to be adopted by the State of Massachusetts as a test of solvency. The net system differs from the gross only in one par- ticular and that is that in figuring resources only net pre- miums are considered, the equations becoming : Present assets + Present value net premiums = Imme- diate claims 4- Present value future claims + Surplus. Present assets = Immediate claims + (Present value future claims Present value net premiums = Reserve) + Surplus. 164 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. The aggregate net reserve is composed of the net indi- vidual reserves, i. e., of the sum of the differences of the discounted probable future costs and the discounted prob- able future premiums of each insurance. Each of these differences will be a net liability, except in the cases of one year term insurance or of whole life insurance to be paid for at current cost, in both of which instances the differences are nil, the liabilities and resources auto- matically balancing. The errors of this system if any, lie not in the principle but in the too stringent or too lax standards employed. NET VALUATIONS, ANNUITIES, INSURANCES AND ENDOWMENTS, PROSPECTIVE METHOD. GIVEN a life annuity, a x , to find its value after n years. There still remain all the payments to be made thereon after age x + n has been passed. But the value of an annuity from age x + n is a x+n . It follows that the value of the annuity a x after n years is a* +n - Given a life insurance for a single premium A x , to find its value after n years. There still remain all the insurance costs after age x + n. The value of an insurance from age x + n, comprising all these costs, is A x + n . It follows that the value of an insur- ance for the single premium A x after n years is A x+n . Given a life insurance for annual premiums, P x , to' find its value after n years. At the outset at age x, the value of the insurance bene- fits and the value of the premiums balance. For by con- struction P x (i +a x ) = A x , the first term representing the value of the premium just received together with all future premiums and the second term the value of future insurance benefits. But at the end of n years, the value of future insurance benefits has become A x + n instead of A x and the value of future premiums P x (i + a x+n ) instead of P x (i + a x ). Moreover A x+n > A x while P x (i + a x+n )

    . N,.,- The problem may also be solved another way, viz. : the insurance now, n years after issue at age x, is worth A x+n or P x+n (i + a x+n ); the future premiums on the contrary are worth but P x (i + a x + D ) and the difference between the value of the insurance and the value of the premiums, n V P x , is wanted. We have the equations : A x+n = P x+n (i -r a x+n ) and A x + n = n V P x + P x (i + a i+n ) n VP x H-P x (i + a x+n ) = P x+n (i +a x+n ) and n VP y :=(P i+n - P,)(* + a .O In language, this may be stated thus : the value or reserve of a life insurance with annual premiums at the end of any policy year is equal to the value of an immediate annuity for the difference between the annual premium at age of entry and the annual premium at age attained. In the case of limited premiums, given the premium t P x , to find the value after n years. At the outset the value of the insurance was A x and the value of the premiums was t P r ( i + 1 1 _, a x ) and the equation stood thus: A,= ,P, (i+Ua.) Now the value of the insurance is A I+n and the value of the premiums t P x (i + | t _ n , a x + n ) only; and the equa- tion becomes: In the case of a temporary insurance with single pre- mium, given the single premium | n A x , to find the value after m years. After m years there will remain an insurance beginning i68 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. at age x + m and running n m years, the value of which is L-,A+ m - In the case of a temporary insurance with annual pre- miums, given the premium j n P x , to find the value after m years. At the outset the values of the insurance and of the pre- miums balanced thus : | m A.= |.P.(i + !.) Now the value of the insurance stands at | n _ m A x ^ m and the value of the premiums at | n P x (i +| n _ m _ ) a x + m ) and the equation stands: I^.A I+ .= .V|.A, + |. P. (i + !__, a, +- ) -VI . A, = |._ m A 1+m - | .P, (i + !_, a^J In the case of an endowment insurance with single pre- mium, given the premium A x - to find the value in m years. After in years there will remain n m years insurance and an endowment due in n m years ; in other words, the value will be that of an endowment insurance f or n m years entered into at age x + m, thus A T+mll |. In the case of an endowment insurance with annual pre- mium, given the premium P x -,, to find the value in m years The value of the remaining endowment and insurance has already been found to be A x + m ^i; the value of the premiums is P r ^ (i + L-^-A-t-m)- Tne equation becomes A T+mn | = m V A x -, + P x ^ (i + |,- m -, a s+m ) m V A, ^ = A x+mtl ^i P x -, (i + (_, a x+n ) NET VALUATION, ANNUITIES, INSURANCES AND ENDOWMENTS. RETROSPECTIVE METHOD. THE prospective method of valuation is the measure of what is required, together with future net premiums, to meet future costs of insurance, endowments or annuities. But there must have been accumulated from past pre- miums an amount sufficient to meet this requirement. If there has not been accumulated enough, it means that premiums were not adequate, in consequence of which, first, not enough has been accumulated from past premi- ums and, second, too much is required to make good the deficiencies of future premiums. If the premiums have been precisely correct, according to the assumed standards, it follows that, according to the same standards, just enough will be left of premiums already paid, with interest realized thereon, to equal the required reserve. We may, therefore, ascertain the amount of the net reserve by considering only premiums paid. To ascertain the aggregate reserve in a company by this method, we would improve the premiums by interest at the rate employed in computing rates and would deduct at the proper intervals for costs of insurance, endowments or annuities, according to the tables. The result would be the aggregate reserve. The principal value, however, of the retrospective sys- tem lies in its convenient application to the computation of individual reserves, as follows: To the net premium of the first year of insurance add interest for one year and deduct the tabular insurance cost 170 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. of the actual net insurance for one year at that age; the remainder is the terminal reserve. Add to this reserve the net premium for the second year and repeat the process. The "actual net insurance" is the difference between the face of the policy and the accumulation at the end of the year, the hypothesis being that death-claims are paid at the close of the insurance year. A frequent and convenient though by no means neces- sary form of this computation is to deduct the tabular cost, discounted one year, at the beginning of the year and then improve only the remainder at interest. This is equivalent to the other form and offers no especial advantages ; still it is frequently employed. What makes the retrospective system of so peculiar value is that through it the values of all sorts of policies may be derived by one formula. We have already seen that a separate and distinct formula was required for each form of annuity, insurance or endowment. Other formulas would be required for every other distinctive policy and some of these formulas would be complex and unwieldy. But the retrospective system disregards the form of the insurance almost entirely. Given two policies for the same amount with the same premium, the reserve would be the same after the same number of years, call them what you will. The simple arithmetical form of computing the reserves by this plan will answer as well as any ; but it has been developed algebraically as follows : Let jV x be the value at end of one year, C T be the dis- counted risk of one dollar at age x and P x be the premium paid. Then, x :Ci : : I __ i v x :c x - c z ( ,V X ) The last factor in this proportion is thus the discounted PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 171 value of the actual risk run by the company. Subtracting this value from the net premium P x and accumulating the remainder at the assumed rate of interest, i, we have: 1 V,= (i+i)[P,-^+c,( 1 V,)] ,V I= =(i +i) (P, c,)+ (. + i) c, ( ,V, ) We have q x = ^ = ^ ! '*' = i ^ And P* = i x i x i x -p-^, whence q x = i p x . Analyzing the value of c x , we find c x m v ^p = v q x = ' . and i (i + i) C E = i (i + i) ^' . =: i q x r= p x . Substituting this value, we have: Compute the value of : - and call it u x and the equation becomes Make a table of values of u x for convenience, by the following formula : i + i Multiplying numerator and denominator by v x+1 , which does not alter the value, we have ^ = (i +i)v^'l. ^ vM._ = D. The value at the beginning of the second year is the 172 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. value at the end of the first ( 1 V X ) plus the second pre- mium, P x . We thus get by analogy : ,V, = u x + 1 ( ,V r + P Y c i+ j) and, generally, n V x -u x+n _ 1 ( n _ 1 V x + P x -c x+n _ 1 ). This formula will accurately give the values for any form of insurance involving a level premium and a level benefit. Its only inconvenience is that to give the value in n years the value in n i years must be known. Practically this is usually no disadvantage at all, the course being com- monly to compute all the values in a series and tabulate them for future reference. The formula, however, requires nearly the same labor as the arithmetical formula first mentioned, almost its only superiority being that the actual insurance does not need to be computed for each year separately. It has come to be known as Wright's Accumulation Formula. Elizur Wright, the inventor of the formula, also extended it to cover reserves at the end of any month in the policy year, thus: To find the value for the nearest completed month, m, the formula becomes : n _^ v, = n _,v, + p, - ^ uv, + P.- .v, ) The point is that ( n _ 1 V x + P r n V x ) represents the net waste or deduction from the accumulation during the year, which may be positive if the mortality charge exceeds the interest or negative if the interest exceeds the mortality charge; the theory is that jV of this waste must be charged off the sum of the value at the beginning of the year to get at the intermediate value. The same result may be arrived at by adding to the value at the end of the year the portions of the waste not yet made, thus : PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 173 _ apV, = .V. + ^=^ ( n _, V, + P, - .V, ) Another formula which proceeds on a somewhat similar principle as Wright's Accumulation Formula, is known as Fackler's Formula, from the name of its inventor, David Parks Fackler. It might with propriety be called a distribution formula, being based upon the idea of dis- tributing or apportioning the aggregate reserve. Given l x+n lives with individual reserves at the beginning of the year of n V x each, making an aggregate of (1 X +J n V x ; and individual premiums of P x or an aggregate of l x+n P x . By the close of the year these aggregate sums will be increased by i interest and will then be diminished by d x+n death-claims, making the aggregate reserve at the end of the year stand as follows : U+, U v * ) = [ (W ( .v, + P, ) ] (i + i) - d, t +l v - ' -.+ . ( v + P ) - V x - V. n V * + r * ) Multiplying the numerator and denominator of T by v x+n which does not alter the value, we get , I+n V 4 IX + Q +1 Q which is, in commutation terms, ^ **" . Expressing this value by k x+n , which values may be tabulated, and substituting we get: TONTINE ACCUMULATION AND RETROSPEC- TIVE VALUATION. VALUATIONS on the retrospective principle may be made by reference to the ordinary commutation columns only, the formula being derived in a manner almost analogous to that employed by Mr. Fackler. In the language of the Institute of Actuaries' Text-Book, " the value of the policy consists of the portion assignable to that particular con- tract of the difference between the accumulated premiums and the accumulated claims in respect of all entrants. " This reasoning is very simple and plain, but a considerable road needs to be traversed before it can be applied. A simple expression for the accumulated premiums involves the following elements: ist, number of entrants, l x ; amount of premium, P x ; te^m already elapsed, n years; annual decedents, d x , d x+1 . . . d x+n _ ] ; inter- est at i rate; number of survivors, l x+n . Arithmetically, the operation would be to multiply P x by l x and the product by i-fi to get the accumulation at end of first year; then add to this P x x l^ +j for second year's pre- miums and multiply the same by i + i and continue the process for n years; the final product, divided by l x+n , gives each one his portion. In considering this tontine accumulation, called tontine because the benefits go to survivors only, we may proceed on the basis that P x = i in order to establish a general formula. Let us assign n T x as the value of an individual share of a fund started at age x by l x persons paying i at the beginning of each year, the fund being divided PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 175 among the l x+n survivors; and let us assign l x+n ( n l\ ) as the aggregate value. Then, T ._ U, and, generally: (.-.T. + Q] This, however, does not readily resolve into commuta- tion symbols. We may thus resolve it by considering this as an accumulation arising from an annuity granted to x, n years ago, but never drawn. He is now x + n years old. An immediate annuity from now on is worth to him a x ^ n N = T ! +n ~ 1 . This he still has, and, in addition thereto, the -L'x + n result of an accumulation of the temporary annuity | n a x which he has never drawn. We have the symbol Ja r to express the value of a deferred annuity that is, of one dated ahead n years and to begin at that time. But let us make n a negative quantity, viz.: n; we then have an expression for the value now of an annuity dated back n years, no part of which has been drawn. We have the formula for a deferred annuity : The value of n being made negative, this becomes ->.=%f But this expresses the value of an annuity dated back n 176 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. years from age x ; what we want is the value of an imme- diate annuity dated back n years from age x + n ; this becomes This value includes both the accumulation to the present time and the annuity beginning now ; so, to find the accu- mulation, we need to deduct the value of the annuity begin- ning now, a x+n , thus: It follows, therefore, that an accumulation of premiums, P, , for n years will be worth : P r T v== p . ( N .-.- N...-.) <- T - )= "IvT- The value of the accumulation of claims will be found, as follows: Consequently an insurance dated back instead of ahead n years, by making n minus, is worth : In dealing thus with it, it must be understood that the value would not exist if only one person, and he living-, were in question ; this value is an average of the aggregate values of a number of insurances attaching now as of n years ago on a number of persons, then x n years of age, the claims on account of the deaths already to be at once settled with interest. But we are dealing with men now aged x + n and who- PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 177 were aged x, n years ago. Therefore, the equation becomes: This value, however, includes not only the policy's por- tion of the unpaid claims that is, the value of the insur- ance already enjoyed but also the value of the policy's portion of all future claims i. e. , of the insurance yet to be enjoyed, viz., a whole life insurance from age x + n, which is worth : A M < + n ~~D^T This, then, must be deducted from the value already ascertained in order to arrive at the value of the policy's portion of the accumulated claims, thus : Since the reserve of the policy, issued at age x, after n years, premiums being P x , is found to be the difference between its portion of the accumulated premiums a~d its portion of the accumulated claims, it becomes : P. (N._ - N, +a _.) - (M.- M.+ .) ' D,+. This formula will answer for any policy with a level pre- mium and for a level amount, provided only the valuation is made during the premium-paying period. This last formula is especially useful in making valua- tions of a company's policies in a group and without calcu- lating each reserve separately. The following method for so utilizing it was invented by W. D. Whiting. Let S be the sum insured under any policy; then the formula becomes : S P, (N.-.- N.+._.) - S (M, - M.+ .) ~ ~ 178 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. Now as between any two policies, the insured tinder which are now x + n years of age, the factors D x + n , M x+n and N x+n _j are the same. The factors which are not the same are $! P x , N,_ T and M x , because x, the age of entry, was not the same, and also because the kinds of policies may not be the same. Suppose, then, we tabulate these factors for all premium-paying insurances in persons now aged x + n, in a convenient form for our calculations, thus: S P x N x _, | S P x | S (- M x ) | S Our formula splits up as follows : o v _ S P. N,_- S P x N,^- S M x + S M z+n ' "~ U. + . Adding our first column in the tabulation gives us for all the insurances the first term in the numerator. Add- ing the second column gives us S P x for all insurances, which, when multiplied by N s+n _ 1 , gives the second term in our numerator. Adding the third column gives us the third term. Adding the fourth gives us the aggregate sums insured, which being multiplied by M x+n , gives us the fourth term. From the first term, thus determined, deduct the second and third and add the fourth ; then divide by D x+ and the quotient is the aggregate value. VALUATION ADDITIONAL FORMULAS. SUPPOSE the reserve n+ i V P x to be known, to derive n V P x . It is clear that at the beginning of the (n+ i)th year, the premium having been paid, there was in hand n V P x + P x , which sum was just sufficient with its interest to cover the insurance cost and produce n+l V P x , the reserve at the end of the year. It follows that it is the equivalent of the present values of the insurance cost and the terminal reserve, thus : .V P. + P, = v [q,* ,+ (p, + J ( D+ ,V P. ) ] .V P. = v [q, + n + (p., J ( n+ ,V P, ) ] - P, This formula is general, applying to all annual premium policies. To get the reserve n V P x , in terms of the premium P x and the annuity, a x+n . If the insurance were payable at once, its value would be i; not being payable until death of insured now aged x + n, we must deduct v i (i + a x+ n ), the value of the interest that i would earn before death occurs and also the value of premiums still to be paid, P x (i -h a z+n ), thus: n VP x =i_(P x + vi) (i +a x+n ) Given the premium, P x , the premium, P x+n , and the single premium, A x+n , to get a formula confined to these terms. A person aged x + n, applying for insurance, would pay a premium P r+n for an insurance of i; a pre- p mium of P x would purchase an insurance of * . But the ^x+n insurance, issued n years ago, is for i; it follows that the reserve held to secure it must be sufficient to pay for an p insurance of i -=-^- , thus i8o PRACTICAL LESSONS IN ACTUARIAL SCIENCE. V P. = A, + (,- To get a formula in terms of a x and a x + n . Suppose a person aged x invests i in an immediate life annuity; it will buy - a year in advance. Of this amount v i i + a x is equivalent to the value of the interest and the remainder is equivalent to the i, which disappears or reverts to the insurance company on the death of x. Therefore, it is also equal to P x . The whole annuity, then, is v i + P x ; and if P x is paid from it each year for an insurance of i, the joint values of the insurance and annuity will alwavs be i, since they secure, first, the equivalent of the interest on i until death, and upon death, i. After n years, then, the value of the insurance is i, less the remaining annuity, thus: i -fa, To get a formula in terms of A x and A x+n . Substi- tuting in the foregoing equation the values of a x+n and a, , as per the formula : i A T a x = - i, we get: v i A + I + r^ n I V p=i ^L = I _L=At, i A x i A x i + - i v i _A x+p A x i-A x To get a formula in terms of P x and P X+B . Substi- tuting in the same equation the values of a x+n and a x , as per the formula : i PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 181 VP.= i f '*- T I Si To find the values of net reserve D V P z and of gross reserve n V'P x in terms of the other. One of our simplest formulas for the ,net reserve is : V P = (P I+n - P, ) (i + a, +B ) The interpretation, as hitherto explained, is that to make good the difference between the net premium at age attained and the net premium at age at entry, we must have the value of an immediate annuity equal to this dif- ference. In this formula we may make a complete separa- tion, thus: P z+n (i + a x + n ) = the value of the future insurance and P x (r+a x + n ) = the value of the future premiums. By gross valuation, the value of future insur- ance is not altered, but the value of future premiums is made P' x (i + a x + n ), the value of an immediate annuity equal to the office premium instead of the net premium. The formula thus becomes : .V'P.= (P, + .-P' I ) (i+a^J Let P' x = P x + $ and substituting, we get : .V'P,= (P i+> _P,_)(i + a, + J = .VP,-*(i H-a^J In words, the gross reserve equals the net reserve, less an immediate annuity of the difference between the net and gross premiums. This is on the basis that the gross premium exceeds the net ; otherwise the sign of $ is changed to+. Conversely : VALUATION RETURN PREMIUM INSURANCES. IN the case of a single premium whole life insurance with return of premium, the insurance becomes i + A r instead of i, and the reserve is the same as for any other whole life policy for a like sum, and is found by multiply- ing V A x by i + A x , thus : v(i.+ A.)..vX". Policies with an annual premium present further diffi- culties. They can best be dealt with on the retrospective basis, /. e. , by progressing a fund composed of the net pre- miums and another fund composed of the insurance costs for the whole amount insured. We have found that for all insurances for level amount this formula becomes: v . P.(N._,-N.^)-(M.-M.Q in which the first term expresses the value of the accumu- lated premiums and the second the value of the accumu- lated insurance costs. Substituting TT, the net premium of the return premium policy, for P in the first term, we still have an expression for the value of the accumulated pre- miums. But the insurance being not i, but i plus an increasing insurance of ?/, the value of the accumulated insurance costs is not the second term, but the second term plus the value of accumulated costs of the increasing insur- ance of ;/. The value of such an increasing insurance at the beginning or age x was, as we know from the formula for TT' (R x R x+ nM + ) temporary increasing insurance, - =- - ; for its value at the end of the period, we need merely to substitute D x + n as the denominator, thus: ^(R x R x+n nM x+n ) PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 183 Our formula for valuation, then, becomes : - (N,_- N.O - - (M. - M. + .) - - V (R, - R, + . v _ ~" M. + .) __ D*. This formula will answer for the valuation during the premium-paying- period of all annual premium return-pre- mium insurances where the return-premium benefit is pay- able at death, whether it also be payable at maturity or not. If the return-premium benefit of an endowment be only payable at maturity, the form ula re verts to its original form : being no additional insurance cost to be deducted. After the premium-paying period is completed, in the case of a life policy the insurance is paid up for i + t TT' X (t being the number of premiums paid) and the value becomes: V = i + t TT'( n V A x ) i + tj (A x+n ) In the case of an endowment with return of premiums, either at death or maturity : V = i + t/ ( V a JE f ) = i + t / ( m _.^ 1+ J If with premiums returnable only at death during the remainder of the endowment period, then: In the cases of policies with tontine dividend periods with return-premium benefits only during the periods, this becomes, after premium-paying period is completed: Returning to an endowment policy, premiums all paid, with return-premium benefit only on maturity, we have : VALUATIONTRUE RESERVES. IN dealing with premiums and loading, it was pointed out that the customary system of computing the expense provisions is crude and inaccurate and, especially, non- actuarial and unscientific, in that it makes no adequate provision for the initial expense. It was suggested that this could be remedied by increasing the computed net premium by an annuity equal to the sum required to cover this expense and then deducting the sum required from the first of these net premiums, leaving the remainder only as a real net premium for the first year. In premium-computations this might not seem important since it is easy to add this annuity in making up the gross or office premiums; and then, since the premiums are ade- quate, we need not bother about their division into net pre- miums and loading. But when it comes to computing reserves the case is different. These are computed on the basis of the net pre- miums, the loading not merely being accounted adequate to cover the expenses, but also able to cover them when they are incurred, i. e., not only equivalent to them, but actually discharging them. But this is just what it will not do when the customary system is employed. The formulas for reserves are based on this idea, viz., that the net premium is progressed at interest and the cost of insurance deducted, the remainder being the reserve. This gives a reserve at the end of the first and of every other year but the last, larger than the actual premiums, after providing for the foreseen expenses and losses, will accumulate on the basis of the interest and mortality assumptions on which both premiums and reserves are PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 185 computed. If companies had no salvage on mortality or expense estimates and no excess interest, they could not qualify under such requirements; and a new company with a large business cannot do so, anyhow. Since the reserve is intended to accurately measure the liability on the sup- position that there is no gain or salvage, it follows that it is not a fair test. Thomas Bond Sprague, president of the Faculty of Actuaries of Scotland, presented this matter at the Con- gress of Actuaries in 1895, and it was practically the unani- mous view of the distinguished actuaries there present that the theoretical reserve system needs such modification as will make it square with the facts. The true reserves, actual net premiums having been determined by the formulas providing for initial cost, are not more difficult to compute than other reserves. They may be readily calculated by the retrospective accumula- tion plan by simply dealing with the first year net premium ( 'P' 2 P' F ) and thereafter with the subsequent net premiums ( a P'=P+f ). The same values of u x will apply. On the prospective plan the formulas require no altera- tion except the substitution of the new subsequent net pre- mium ( Z P'= P'+ f ) for the old or flat net premium. Or the true reserve may be derived from the theoretical reserves by the following simple process: deduct from the ascertained net theoretical reserve the value of the imme- diate annuity added to the net premium as an equivalent to the initial cost, thus : n TV x = n V x -f (i +a, + n ) If the premiums be payable for a limited term only, the formula becomes ,TV, = .V.-f (i + L-A+J 186 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. In another respect the theoretical reserves often differ from the true reserves, viz., in the case of all paid-up insurances and also of all insurances with limited pre- miums during the premium-paying period as well. This for the reason that an intelligently computed loading should provide for expenses all through the life of the policy and not merely while premiums are being paid. The net reserve system which takes into account only death losses does not make this provision. Let be the annual provision for expenses after pre- mium-paying ceases and ' be the part of the loading dur- ing the premium-paying period intended to accumulate this provision. Then to find the true reserve after the premium-paying period, the theoretical reserve must be increased by the value of an annuity of $. In the case of whole life insurance this becomes : n T V,= A x+n +*(i + a x+n ). In the case of any insurance terminating in a less period, the formula is: .TV, = .V, + (i+ L_A + ,,). During the premium - paying period the theoretical reserve is increased by the amount of the accumulation of N 3 + , u o i. ' m Temp. Addition, = . m r = ^ | = * m ^ m ' |, A x + m d x + m d *+ m 208 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. These formulas cover all cases of applying surplus to increase the insurance; it may, however, be desired to apply the surplus to buy an annuity either for life or a term of years. The same principle applies. Suppose the dividend at end of m years to be applied to buy a life annuity; the formula is: T> Annuity = -7-^- or if the annuity desired is an imme- a x+ m diate one, then : Annuity = a x+ m It sometimes occurs, as in term, limited payment and endowment policies, that the annuity is desired for the remaining years of the payment period only, Suppose such a policy with n payments of which m payments have been made, to find what annuity to reduce future premiums the surplus will buy ; the formula becomes : T> Annuity = = m / ' > - PARTICIPATION OF ANNUITIES. ANNUITIES are nearly always issued on a non-participat- ing basis; but there is in the nature of things no reason why they might not be issued on a participating basis, and some consideration of plans for distributing surplus thereto is proper at this place. In life insurances there is a salvage from mortality esti- mates when less lives fail than were expected to fail ; in annuities there is, on the contrary, a salvage when more lives fail than were expected to fail. Deaths in life insur- ances cause the payment of the insurance ; deaths among annuitants stop the payment of the annuities. So far as gains from excess interest over the hypothetical rate used in computing premiums are concerned, the cases of annuities and insurances are analogous. Likewise so far as salvage from loading is concerned. In the cases of both these items there is never any diffi- culty in apportioning the gains or salvages on the contri- bution plan, the excess interest according to the mean annuity fund and the salvage on the loadings according to the loadings. A variation of the apportionment of this expense salvage would be to treat a part or all the loading as the present value of an additional annuity to be employed to defray expenses throughout the term and to apportion the salvage each year according to these additional annuities available for expenses. But it is the ascertaining and apportioning gains from mortality salvages which offer the greatest difficulties. The simplest method of ascertaining the amount of this salvage is as follows : Make up the amount of net present 2io PRACTICAL LESSONS IN ACTUARIAL SCIENCE. values of annuities expected according to the mortality table to be released by deaths during the year; compare with this the actual amount so released. This will show the salvage or excess from which the percentage of gain or loss may be computed. The apportionment requires a fuller explanation, during which we will investigate the effect of the mortality expe- rience upon each annuity value. Suppose l x annuities at age x, the value of which is l x a x . During the year this amount is increased by i interest and then diminished by l x+ 1 annuity payments, and the remaining fund is the value of 1, + I annuities at age x+ i or l x+1 a x+ ,. But 1 I+1 = 1 X d x , and it is evident that the fund has been modified by the d x deaths in two ways : first, there have been l x d x annuity pay- ments, and, second, there are only l x d x annuities still in force. If there were more than d x deaths, there would be two gains: first, because less annuity payments to make, and, second, because less annuity values to provide for; and, on the contrary, if there were less than d x deaths, there would be two losses: first, because more annuity payments to make, and, second, because more annuity values to provide for. To return to our analysis, we have "discovered that _ rT **x+l - 1 J.+ l U a. (i + i) ^ PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 211 Now, having ascertained the rate of actual interest and the percentage of gain or loss by deaths upon the values expected to be released by deaths, proceed as follows : first, make a corrected table of deaths by modifying the figures d" d' in the d column so that ^- = j~ for all ages. Then i x i z F * = a * T + i '-i To find the surplus deduct the value of the annuity : ,B a x F a x+I a x+1 or generally n Ba x =Fa x+n _ a x+n SURRENDER VALUES. HAD the first plans of insurance been scientific and the first policies mutual and participating, no doubt the reason- ing about surrender values would have been about like this: The fund remaining from the premiums of a policy have not been needed to pay for insurance already enjoyed ; as the policy is now to be discontinued, this fund will not be needed to pay for future insurance; therefore it may, and indeed should, be returned upon discontinuance. Later, it is conceivable that reasons might have been discovered why some deduction from this value might be made in some cases or some addition to it in other cases, according as it was less or more desirable to have the life off the books. But it is clear that the burden of proof lies upon him who would make any change in the primary proposition which has every appearance of being just. But the first plans of insurance were neither scientific nor mutual ; and, moreover, the patrons did not understand that there was any accumulation of unearned premium. The cupidity of insurers caused them to grant grudgingly only those surrender values which, for business reasons, it seemed wisest to give. At the bottom no consideration of what ought to be given entered into the matter. Partly under the stress of public demand, partly because of legislation or the fear of legislation and partly because of competition among companies, the various plans for computing surrender values have been evolved. In many cases they are purely arbitrary. In others they have at least the semblance of scientific accuracy. When the subject of surrender values was first mooted by the Massachusetts commissioners, the companies even PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 213 forfeited the dividend additions or reversions insurance that was fully paid for upon failure to pay premiums on the original policy. The form of surrender values first proposed by the com- missioners and adopted as the first law of Massachusetts on the subject was to apply the reserve to extend the insur- ance, The premium for this term insurance might be loaded twenty- five per cent; or what was the same thing, eighty per cent of the reserve was to be so applied as a net single premium. We have elsewhere given formulas for computing the time of insurances, the premium and amount insured being given. At the time when the law was pro- posed, 1860, such a formula was not generally in use, and one objection raised against the system was the labor it involved. Elizur Wright, one of the commissioners, silenced this objection by computing net single premiums for all terms and all ages and publishing the same. This table is reproduced in the appendix of this book. To com- pute extension terms from it, find first the even number of years for which the pre mium will pay and also the remainder of premium; then ascertain the difference between the premium for that term and a term one year longer ; let A represent this difference and the days in excess of the even years may be ascertained by the following proportion : Days : 365 : : remainder of premium : A. This is not quite exact, as it does not take into account that the risk of death during the year is an increasing one ; but nearly all ordinary actuarial computations are in this respect inaccurate. Before this law was enacted, the New York Life Insur- ance Company introduced another system which has been more generally adopted, viz. , paid-up insurances. In the case of life policies payable in ten premiums, it 214 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. agreed that after a certain period a paid-up policy should be granted for as many tenths of the original amount as annual premiums had been paid. This idea has been applied to all limited payment life and endowment policies. It is too simple to require exemplification. It is also unscientific. The values granted in the earlier years are higher than the reserves would buy, and those granted in later years much lower. Some companies have always avoided this plan in spite of its simplicity and popu- larity. In the case of whole life insurances with continuous premiums, it is plainly inapplicable. The better and more scientific rule has been to apply the reserve to buy paid-up insurance. Because the premiums have already contributed a full loading it has by many been considered fairer to apply the reserve as a net premium ; but such is not the universal practice. This is also too simple to require a formula. The payment of cash surrender values is of yet later adoption. The determination of the amount to be paid has been a disputed matter even to this day. Against the proposition that the whole policy fund remaining unex- pended should be allowed, the following objections, among others, have been interposed: it would increase the sur- renders; the companies would suffer from adverse selec- tion, those lives conscious of being in superior condition retiring and those conscious of impairment persisting at all hazards; the insured being interested in the surrender value and only the beneficiary in the ultimate proceeds, the former is tempted to betray the interests of the latter. The last of these objections is not actuarial and the first presents no real disadvantage, even if true and the facts are against it unless the second be valid. That there PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 215 might be adverse selection, especially if the soundness of a company is suspected, is certainly reasonable; but no evidence exists to prove that there is any considerable adverse selection in a going company with a good reputa- tion. The insured contributes to the payment of losses other than his own, according to his age and the actual insurance, or, as we have seen, according to the " costs of insurance." Hence it follows that the effect of adverse selection on the part of those who withdraw is some func- tion of all the " costs of insurance "to be paid by them; this, on the principle that being better lives than the aver- age they would, if they did not discontinue, pay a little more for their insurance than the value of the benefits received. Consequently the loss by adverse selection can be offset by making a surrender charge against the reserve of a sufficient percentage of all future costs of insurance. This was the idea which Elizur Wright introduced into the Massachusetts statutes when he caused the cash sur- render law to replace that providing for extended insur- ance; the new law provided for a surrender charge of eight per cent of the value of the costs of insurance, called the * ' insurance value. ' ' Another plan has been to allow a surrender value on the basis of some mortality or interest standard different from the one actually employed in computing the reserves. Thus, for instance, the Australian Mutual Provident Society formerly reserved on the basis of the H m6 table and four per cent, on policies more than five years old, but allowed as cash values the reserves on the H m table with four per cent, the difference being in effect a surrender charge. A somewhat similar plan, long employed by an Ameri- can company, was based on the idea that the company should forfeit nothing of its advantage to the discontinuing 216 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. policyholder. Thus this company gave as a surrender value non-participating paid-up insurance and allowed as a cash value for that a sum believed to approximate the actual cost of furnishing this insurance. To do this a mortality table nearly corresponding to the company's actual expe- rience and a rate of interest at least equal to the actual were employed to compute the value of the paid-up insur- ance, which sum was allowed as a surrender value. The first cost of procuring insurances is commonly, together with other expenses chargeable, more than the loading on first premiums. Consequently, the reserve at the end of the first year is usually not to be had from the premium itself; in other words, the actual and theoretical reserves do not agree. Therefore to offset this the Aus- tralian Mutual Provident Society deducts a diminishing percentage of the reserve on surrender during the first five years. Other companies deduct a fixed sum per $1000 insured from all surrender values. Evidently the sum which should be paid is the " true reserve " as ascertained by the formulas given in the chapter entitled " Valuation True Reserves," beginning on page 184. The proper deduction from the theoretical reserve is there given; it is just adequate to cover the part of the initial cost, not yet made good. INSURANCE VALUE. REFERENCE to the value of all future insurance costs, known as 4i insurance value," has already been made. The symbol n A K, followed by symbols indicating the kind of policy, has been adopted to express the idea; thus n A K P x means the " insurance value " after n years of an insurance issued at age x for life at an annual net premium of P,. Let us find the value n A K P x . We have the two conversion formulas for premiums for whole life insurance: and P x = - -- d = ~ -- d = ^ -- d We find that the reserve at the end of n years is the then single premium, less the value of all future annual pre- miums, thus: .V, = A I+n - (i + a I+n ) P, ST.*.-,/ D, Since the " actual insurance " upon which the " cost of insurance " is computed is the face of the policy, less the reserve, we have : ,_ v - 1 -r I -'^ which formula is general. The " cost of insurance" in each case, therefore, will be 218 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. K P _ .*.-. /, _ y ) _.^-, + ._ I i, + (I v - ' " u._, L D I+B IOJ Now a series of these values from one year to the last year of the table are the " costs of insurance." The first is .(,- V ^---* i. ( ' rj ' i._D. + A*W_ but is due at the end of the year ; consequently its present value is less than that by the discount or equals : vd,|-N, /D z \"j i, LD, viwJ The cost of insurance the next year will be : This sum will be due only at the end of the second year, and not then unless the life shall have survived the first year; consequently we must multiply by both v 1 and to get the present value, thus : ' generally > The total insurance value, which we will designate by the symbol 8 A K P r (meaning the insurance value of a policy issued at age x after s years), will be the present value of this series beyond s years, thus when s = o, i. e. , when the insurance begins : PRACTICAL LESSONS IN ACTUARIAL SCIENCE. D x It will be observed that N_ 219 are common factors. We may substitute for D i+l , D x+2 , etc., among the denominators their value v x+1 l j+1 , v*+ a l x+a , which gives us: Let us multiply numerators and denominators by v x , which will not alter the value, thus: D * P - '" H.-.L Let us compute all these values N x (?-/ from the begin- ning to the end of the table and make a commutation col- umn of them which we may call rf x and then sum these values from each age to the end of the table to make a new column A X . We then have : 220 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. In the case of a single premium insurance, the actual N _ \ insurance becomes : i n V x = i ( i d '* n ~ 1 ) J-v*. / N = d '" f "~ l > and the general formula for cost of insurance becomes : , = =' (i- .V,) = =- *>! *S+*-I Without going through all the intermediate stages, it is at once evident that in the sum of the series of these costs we will have no factor * which was brought in by the JN X _ 1 annual premium but that d will appear in its stead, the other factors remaining unchanged. This gives us : To compute insurance values of temporary insurances, limited payment insurances and endowments, it is neces- sary to consider first of all that we will be dealing with (D x+n + D^^ + D^ +n _ 2 + ...... ), instead of N x _j simply. These values are known as ** curtate com- mutations ' ' and are expressed by the general symbol N x _ a 3, or, in American symbols, *+ n N x , which means N x _, N x+n _ 1 or the values of D beginning with D x and ending with D I+n . These curtate values on the basis of the Actuaries' table and four per cent interest have been computed and tabulated for most ages and terms by Elizur Wright and copyrighted by him. The summation of these curtate values gives us new values for A, which we may denominate T+n A x . Without going into details concerning each form of policy, I append the following formulas taken from the Book of PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 221 Formulas compiled by David H. Wells and published by the Connecticut Mutual Life Insurance Company for all ordinary level premium level insurances (using American symbols for curtate commutations) : A _ v A A T7- I A _ *+n+l +a i+n 8 AK| n A x _ AK E ='* l> * lA ' + ' *^2 A K P - x * + ' A F *-N^M D/ A \ A A ., _ -. * /x + n *+.! -L. + n + i z+. +n "D / A \ A A-rr -D T7 ^r + n / ^^n x+ 8 \ ^ + n + i + -f K f n ^ z - -^ - I -^T -^r i + n- N x-! \ -L'x+s / ^H-. "D / A \ A A A V P T? _ "^z+n /+t ^.\ ++! *+'i ^--n '+* ^^^"^N^l D. + . r D + . The intricacy and difficulties of these computations, as well as the circumstances that commutation columns r + n * r are necessary for every combination of ages required in 222 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. computing have combined to make the surrender charge based on insurance values an unpopular thing with com- panies and the idea has never been applied, openly at least, except in Massachusetts, where the system is embodied in the laws. The insurance values for most ordinary policies on the Actuaries' table and four per cent have been com- puted by Elizur Wright and are included in his ' ' Valua- tion Tables;" the values by the American Experience Table and four and one-half per cent he has also computed and published in the volume entitled " Insurance Value Tables." The idea of a surrender charge based upon these values is unquestionably the right one if a deduction is to be made to cover possible adverse selection. Any other surrender charges will not be proportionate, even, to the damage done, without speaking of equivalent to that damage. A rough method of arriving at such a surrender charge would answer all practical purposes for a company, not compelled by law to be so exact. An approximation which will always give the value a little too high is made as fol- lows : Find the actual insurance for this year; find the actual insurance for the last year of the term originally contem- plated (which is zero, if the policy be either an endowment or a whole life insurance) ; add these together and divide by 2, which gives an approximation of the average insurance from this day forward; find the single premium for this insurance for the remainder of the term, which will approximate the insurance value. In a formula this becomes, letting n be the original term and x + s the present age, T V 4- T -- V .A K, = i - V ' , V ' (I.- A, + .) PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 223 This formula will apply best to policies with continuous premiums or paid-up policies ; but will give results not so very far out of the way even during the premium-paying period of limited premium policies. INDIVIDUAL ACCOUNTS. MANY life insurance companies keep no other individual account with a policy than a memorandum of the age, amount and kind of policy, the date of issue, frequency and amount of premiums and the fact of their payment. The reserves on these policies they can ascertain when they will by reference to tables or by a short computation. The surplus they ascertain at least as often as the divi- dend period requires, by group computations, thus making out a scale of dividends for all forms of policies at all ages. The reserve and surplus added together at any time shows the whole fund. For a company with a large number of policies this sys- tem is decidedly less laborious, and, since when it is faith- fully carried out, it yields the same result as would any system based on individual accounts and on the same prin- ciples, it is satisfactory. Its sole disadvantage is that it is not comprehensible by the insured, who consider the matter one of mystery in consequence. Because of this, Elizur Wright devised a system of accounting which is readily comprehensible. It is fully set forth in his ' ' Savings Bank Life Insurance Tables. ' ' The idea is to divide the premiums as they are paid in into three different portions, viz., loading, cost of insurance and deposit for reserve, sending each to its own fund and crediting each individual policy with the premium and charging against it the loading and the cost of insurance. The loading is easily ascertained by deducting the net premium from the gross. The cost of insurance in this case is the cost at the end PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 225 of the year discounted; as the amount is deducted on receipt of the premium. When this cost is taken out of the net premium it is clear that the remainder constitutes a deposit for reserve. This, together with any previous accumulations, constitutes the net credits of the individual account, which credits are at the close of the year improved at the actual rate of inter- est. The salvage on the cost of mortality and the salvage on the loading for expenses are also then added and the total brought down. The terminal reserve for that year is deducted and carried forward; and the remainder is surplus, either carried for- ward also as a part of the fund or withdrawn or applied. Mr. Wright, in the book mentioned, divided the pre- miums for each year of many usual forms of policies at all ordinary ages of issue into these three parts, which are sometimes mistakenly called " elements." The mistake is often also made of supposing that the cost of insurance and the deposit for reserve are the same ratios all through as on the first year of insurance ; instead, the line of demar- cation is constantly shifting. This system of accounting offers several advantages. It causes the company to keep its expense and mortuary pro- visions separate from its reserves, so that it cannot trench upon the latter without knowing it. The plan also facili- tates the calculation of the saving on mortality and on loading. Notwithstanding which facts it is somewhat intricate and involved as a method of individual accounting and subject, though in far less degree, to the same objections as the other, viz., that it is not understood by the insured. The system is only objectionable because amounts are first charged to the individual and then some part of the 226 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. sams amounts credited back. The account may easily be so simplified as to be comprehensible to any business man without explanation by merely crediting the premiums, charging out the actual expenses and mortality at the end of the year and progressing the remaining fund at inter- est. This system also does away with the necessity for extensive tabulation. In practice, the process may be car- ried forward as follows : Credit the premiums; at the end of the year charge for the actual expense in proportion to the loading; charge for the mortality by computing the cost at the actual rate for the age attained, of a one year's insurance for the net. actual insurance (i. e., the face of the policy, less the fund remaining after expenses are de- ducted); credit interest on the remainder at the actual rate earned and carry forward the balance. Credit the new premium and proceed as before, charging for any dividend paid as well as for other items as before. INSURED INSTALMENT LOANS OR ADVANCED ENDOWMENTS. WE have yet to consider the case of a company engaging, for a consideration of an annual payment for a fixed term but contingent upon the survival of the payer, to pay an immediate sum to him. Of course no such engagement would be undertaken unless the annual payment were secured, which is usually done by real estate mortgage. The transaction thus becomes one of an insured instalment loan. It is evident that this is precisely equivalent to the com- pany's buying a temporary annuity equal to the instalment on the life of the payer. Let S be the sum advanced, n the term and x the age. Then, to find the payment, first find the value of | n a x and divide S by it, thus, designating the payment by -, the symbol for unusual annual premiums: -N ' This is for the case of payments due at the end of the year; if payments are due at the beginning of the year, divide by i + | n _ t a T , the value of an immediate temporary annuity, thus: - SD r If the payments are to continue throughout life, a very unusual case, the payments become 228 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. SD, = -=!-.= N r = " ' and a * T5T N * Q sp, respectively. Equivalent formulas may be arrived at in another man- ner. If such a loan is made, it is evident that the company must collect the interest each year on the whole amount advanced and must also collect a sum equal to the net pre- mium for an endowment insurance for the amount advanced, the maturity of the endowment insurance at death or in n years canceling the loans. The formula for an advance of i thus becomes: This covers only the case of an advance with payments yearly in advance, interest also collected yearly in advance. If, however, payments are at the end of the year, the for- mula may be employed by altering the denominator of the fraction, thus: ~_ n M.-M... + D.+. N, - N, + . These formulas will give the same results as the formulas based on considerations of a temporary annuity; which fact is but another phase of the equivalents explained m the chapter on " Correspondences and Equivalents." A variation from these last formulas is found in forms which use one rate of interest in determining the endow- ment premium and another in computing interest on the advance, as i and i, respectively, thus: PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 229 f M x M T+ + D x . = I + P . J = I + ^-N^' when payments are yearly in advance and _, M r M x+n + D. +n N,-N z+n when payments are at the end of the year. The effect of this is to increase or diminish the actual rate of interest realized, according as i, the rate at which the sinking fund is progressed, is less or greater than i, the rate of interest charged on the loan. This form is, in practice, only employed to get an increased profit. Another variation of a similar character is to add a load- ing to the endowment premium, as, for instance, by using the office premium, P' x -, thus: TT { + P' - - L n A x uj The valuation of such a contract offers no difficulties. The simplest form is to treat it as an endowment, pur- chased by T i, and to find the value of the endowment which is the amount of the principal debt that has been discharged. Where the endowment premium is net (P x ^ and not P' x -,) and at the same rate of interest as the loan, this method will give precisely the same result as a partial pay- ment account, the actual insurance being charged for at the tabular rate of cost. The net value of such a contract is an asset and not a liability of the company. It is the present value of the future premiums, thus: .VT_=(I + |._ m _, a, ) _N.+._. N,*.-,^ SD. D, ' N, N,,. _(N.-N. t .)(K.^.-N.^^,) S, D x ' EXPECTATION OF LIFE. THE probable life of an individual in a group such as in group l x , is until just one-half the group will have died, his chances being even to be among those who have died or those who survive. Probably this should have been called <4 expectation," but that name was given to some- thing slightly different. In dealing with annuities and insurances based on one life, we have not needed to employ these terms; but before approaching joint-life problems it is necessary to consider them. Suppose, then, a group of l x persons. Of these d x will die during the year, and, on the hypothesis that deaths are uniform throughout the year, these will have lived in all one-half d x years. The next year d x+l .will die, having survived the first year ; these will have lived in all one and one -half d x+1 years; and so on. The whole group, then, taken together, will live a series of years, computed as fol- lows: d x +fd x + l + |d, + a + ......... or V(d x +d I+l + d x + 2 ) . . +d l+1 + 2 d x + 2 + 3 d x+3 + . . But the sum of the series d x = l x ; that is, in the end all the group and no more die ; and d x+l 4- 2 d x+ a + 3 d x+3 4- . . . breaks up into series, d x +, + d x4 . 2 + d x+3 + ...... etc. which are respectively equal to l x+ l + I 1+a + l x+3 + . . . Therefore the whole series of years takes the form Since this is the whole number of years that all the lives of the group will live, it follows that the average for each PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 231 will be found by dividing by l x . This average is called the complete expectation of life and the formula becomes: - - i 4- T The curtate expectation of life is the same, less the half- year which on the average the members of the group will live during the year in which death occurs, thus : _ The average for each being e x , it follows that the average age of the individuals of the group at death will be x+e, . There may be computed also an expectation during the term of n years or after the term of n years, | ;e x and n | e, respectively, whose values are seen to be, using curtate expectations only: 1, These two, taken together, are equal to e x . To simplify, we may write these formulas as follows : We have seen that -y^ = p x , the probability of surviv- ^x ing one year and generally -^ = n p x , the probability of IT 232 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. surviving the nth year. Consequently our formula takes the form: The curtate expectation, then, is the sum of the probabil- ities of surviving each year of life. From this fact we may proceed to discover the value of the joint expectation of x-j-y, i. e., the average number of years that will pass while both are living. Out of l x -f 1 persons, the total number of pairs of persons, one being aged x and one y, will be l x X l y , which we may designate as l x . In one year the total number will be l x + i + 1 +1 and the number of pairs l x + l : y+J , which we may desig- nate as \ y . The total years of joint life, then, will be the sum: The average number of years of joint life or the joint life curtate expectation may be found by dividing by l xy , the original number of pairs, thus: '1 + 2 L + 3 L + . 6 ,y = *y The probability of joint life for one year is clearly ^ and generally np xy = -^ ; so that our formula may be written : e =. i, p ^zy J nl xy The expectation of x after the death of y is the sum of the compound probabilities that x will survive each year and y die in each year, thus : e y |*= 2 n p x (i n p y ) = 2 ( n p x n p, y ) JOINT LIFE ANNUITIES. WE have seen that where one life is involved, a x = E x + 2 E X + 3 E X + _ v l x+1 + y 2 l x+2 + v 3 l x+3 + . . . N g U x We have also seen that and The value of an endowment for one year, payable only if both x and y survive the year, becomes : " - 1 or, generally, _ B - - 1. 1, A joint-life annuity, i. e., a payment of i at the end of each year that both x and y survive, is equal to a series of joint-life endowments, thus: _vl.+.l,+ . + v'l.+.l,+ . + v ''.+.*!,*. 1,1, From the same reasoning, it is clear that J=0 +a x + 1 : y+1 ) E 17 By the use of these formulas, the values of a iy may be computed, beginning at the extreme of the mortality table for the older life and finding the value at those ages and working backward to the ages at outset; in this manner 234 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. the successive values of joint-life annuities at all intervals between the ages may be determined. This was done according to the Carlisle table by David Chisholtn. The following methods of employing commutation col- umns have been employed: Griffith Davies, inventor of the D x and N x columns, multiplies both terms of the fraction y U U. + v' i.+. U, + ........... by v x , that is, by the value of i discounted as many years as the age of the older of the two lives, thus: _ V i, l, _ N. y Another form, preferred because not involving the ques- El tion: Which life is the older? is to multiply by v 2 , thus: .. a * 1 i _.~~2 r 2 1 a -, v^U, Having arrived at these commutation columns, we may develop the formulas for temporary, deferred and deferred temporary annuities as follows: PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 235 vl.+, 1,+. + v'U, 1,+, + . . . .v-l.+.U. P., '.1, Similar formulas will apply to any number of lives; but the construction of commutation columns for all such com- binations will be found too laborious a task to be under- taken. There may also be annual premiums for each of these deferred annuities, thus: N - NT * I + l 1N x n-i + n I "NT "NT 1N * + n j+n ^ + ' j + n+ N,-, -,-, -N JOINT LIFE ANNUITIES EQUAL AGES FORMULAS. IF one were to attempt to construct joint life annuities for all intervals between ages, according to the formulas given in the last chapter, he would need to proceed in one of two ways, viz., compute the annuities successively down- ward from the highest age in the table of the two lives from the elder to the lowest age in the table for the younger, by the formula: a xy = (i -4- a* +j . y+] ) E Yy ; or com- pute D Xy and N* y col Limns for all values of x and y. In either case, a complete set of computations, from the highest age in the table for the older to the lowest age in the table for the younger, would be required for each set of annuities. Labor of just this kind has been performed and the results are obtainable often in printed tables. Moreover, abso- lutely accurate results can be obtained in no other manner. The system becomes yet more complicated and the labor more grievous when three or more lives are involved. The labor is greatly lightened by formulas which discover the approximate equivalent of the two lives in lives of equal ages. Suppose that an age w be found, such that: X + * r = 2 * . If the force of mortality be thus the same, it follows that p xy = p ww and therefore E Xy = E^. The first payment under the two annuities has thus the same value. Pay- ments after the first will have approximately the same values; and the annuity becomes PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 237 _vUU. + v'i.+.U.+ N This will apply in like manner to all temporary, deferred and deferred temporary annuities, thus: I n a *, = I n a N" _ "%+n w + n D ww a x = I a x y n I m ww N 4. : , N This rule is also of very general application and may be applied to any number of lives, as m, so long as ^ x + <" y + ^, + = ni ^ w The same formulas will cover these problems by adding w or w + n wherever w or w -f- n appears, for each addi- tional life. The difference between Davies' and De Morgan's for- mulas for D xy disappear when the ages become equal, thus: v x l x l x = v^ l x 1, . Commutation columns for any number of lives may be formed, thus: D==VU, D xxx = v* l x l x l x , etc., and N XI , N X3tx , etc., columns are made by summing all the values of D above D IX , D xrx , etc. 238 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. Thus from the D xx and N rx , D xir and N r , etc., columns the values of a xx , a xxx , etc. , may be found for all integral or " whole number " ages, as a ai . ai , etc. But the formula /" x -I- ^ + /", . . . . = m/ u w will almost always give an intermediate or fractional age, as 21.63, 32.14, etc. It is plainly out of the question to construct D xx , N, x , etc., columns for each of these possible fractional ages. What we may do, then, is: when we have found the fractional age, then get the values of the annuity at the ** whole number " ages next younger and next older than the fractional age. Then multiply the difference between these values by the fraction and add the result to the value of a for the " whole number " age next younger than the fractional age. Considerable labor is saved, so far as annuities for the whole term of life are concerned, by computing once for all the annuities for " whole number " ages and tabulating them for reference. JOINT LIFE INSURANCES. SINGLE PREMIUMS. WE have seen that where one life is involved, v d x + v 2 d x+l + v 3 d x+2 + ...... ~~ M We have also seen that From which it follows that one year's insurance would be worth: * T 7~i * +l y and_the formula for joint life insurance becomes: A _ v (1. \ -1.+ . '^ .) + v' (1. '^ . ~ 1.^. 1.*.) + - 1.1, We may construct commutation columns on Davies' plan by multiplying by v z , giving the form: A ~ In a like manner commutation columns may be con L structed on De Morgan's plan, by multiplying by v a , giving values: 240 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. The same formula for A zv applies. The formulas for A jy , n |A jT and J m A xy take the corresponding forms: A _ ~ _, +n;; l y ~ ~ Similar formulas will apply to any number, as m lives, the values of the commutation symbols becoming for three lives: C,. = V*. (1,1,1, -l, + ,l, tl l. + ,) according as Davies' or De Morgan's plan is employed. M xyz in either case equals * C Xyz . M xj , M iyi , etc., are seldom tabulated. We have seen that "c x = v^ 1 (l x - 1, +1 ) = v (v x l x ) - v** 1 l x+1 -vD x -D x+l and, consequently, M x =vN x -N z+1 We now find that C xy = V+ 1 (U 1 7 - l x + 1 1 7 , ,) = v (v UV ) v x+1 l x+1 i y+ , = vD Xy -D x + i:T + 1 and, consequently M.^vN^-N,.,^, Consequently, by substituting this value for M sr wher- PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 241 ever found, we may work the problems of joint life insur- ances by reference to the N and D columns only. The same principle of equivalent equal ages applies as in joint life annuities. Thus when + = 2 P M - M _ ww w + n Iw + n - D w M + . + vN (v N w+n : w+n - N w+n+l : w+n+1 ) -- (v N w+n Wn+m - Precisely similar formulas are applicable to the case of any number of lives, as m lives, when ^ + ^ + ^ + . . . = m t* w As in the case of annuities, it is seldom found that the age thus arrived at is a "whole number" age. It is usually a fractional age, as 21.62, etc. To get the value of the insurance, first compute the values for the ' ' whole number " ages next lower and higher. Find the difference between these and multiply by the fraction ; add this pro- duct to the value for the next lower " whole number " or integral age. Where tables of annuity values have been prepared, the value of a joint life insurance is most readily computed by the conversion formula, hereinafter given in the special chapter on 4< Conversion Formulas for All Cases." JOINT LIFE INSURANCES. ANNUAL AND LIM- ITED PREMIUMS. AN immediate joint life annuity (i + a xy ) has the following value: But this is the value of a premium of i to be paid annually while both x and y survive. Consequently M *.,= Or in the case of lives of equal ages into which others may be converted, 2 p w = /" x + f\ , we have _vN ww -N.. i: ,+, N._, :W _, An immediate temporary joint life annuity is Consequently | t _,a ly PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 243 or, when 2 /* w = /" x + /" y t p_ = vN CT -N.^,,... In like manner, the annual and limited premiums for temporary, deferred and deferred temporary joint life insurances are found: M ww - M w+ . iw+ . A P. = + I.-, ^ " N._. ; ,_.-N, + ._. ! , +c _. I p ._ CT -N, + .,. +1 )-(vN w+ .,. + .-N. + . + . :w+ . + .) l n V M M 1M W _ 1:W _! IM^JJ : w + t + ] _ JA I *y , M w+n:w+n = vN w+B8w+ , N, !PX __ vN w+n:w+n N w+n + i;w+n+1 'I iw + n + l 244 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. M w+n . w+n M w+n+m . w+n+m D ww M w+p . w+n M w+n+m . w+ (v N w+n:w+n N w+B+1!w+n+1 ) (v N w+n+ N w+n+m+i;w+n+m+1 ) N "NT "-^W 1 .- W 1 1N W+ _J ; w+ _! JOINT LIFE ENDOWMENTS AND ENDOWMENT INSURANCES. WE have seen that P* y = P* P y = '^ 1 y ' and that or, generally, F _ v U. V. !/,. ; ; D ww The annual premium for this becomes P E 5y = P E ww = - And the limited premium becomes p P P "P n ww D w+n:w+ , PWW -Pw + n ; w + n N _ 4 _ ]sj ~ N w _! . w _, N w+ _! . + _ 246 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. As in cases involving but one life, an endowment insur- ance is simply an endowment and an insurance, thus: D w+n;w+D + M ww M w+Biw+ D w+n:w+n + (vN wvv N W+1SW+1 ) -- (vN w+nSW+n - The annual premium for a joint life endowment insurance becomes: p ^ y = P n ^: ww = i + n | Q J aww D w+n!w+n + (v N ww N w+1 . w+1 ) (v N w+n:w+n N w _ 1 . w _ 1 - - N w+B _ 1!w+n _ 1 D ww D w+n : w +n + (v N ww N w+1 . w+1 ) (v N w+n: w+n - N w+n+i;w+n+1 ) _ 'N^^., N.^/,.4^ And a limited premium becomes (vN ww N w + i:w + 1 ) (v N w+n + liw+n+l ) D ww (vN ww - N w+i:w+1 )-(v N w+n:w+n N w+n+i;w+p+1 ) JOINT LIFE NET VALUATIONS. THE principles of valuations of joint life contracts are precisely the same as when one life only is involved. Thus, if n years of a joint life annuity has passed, the value is that of an annuity for a term n years less than the original term, on lives n years advanced. As, for instance, n Va Xy = n Va ww = a w+n:w+n . V L a *y= n V J *ww= | m - n a w + n:w + n n V m I a *y = a V m I 3ww = m - n I a w + n : w+n Thus also if n years of a joint life insurance has passed, the value is that of an insurance for a term n years less than the original term, on lives n years advanced. As, for instance, The case of annual premium policies is entirely analo- gous to policies on one life. Valuations may be made by either the prospective or the retrospective method. First on the prospective method: n V P xy = n V P ww = A w+n:w+n P ww (i -f- a w+n:w+n ) 'XCX This is the same in form as the prospective formula for values of a single life insurance. The retrospective method, involving the accumulated premiums, less the accumulated losses, gives us: 248 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. P.. (N w _ t ; w _ x N w+p _, : w+n _, ) (M ww - M w+n ; w+n ) D w + n:w + a P ww (N w _, !M - N^^ : w+n _ t ) - [(v N ww - N w+l Sw+i ) - This retrospective formula is general and applicable to all forms of insurance, including endowment insurance, during the premium-paying period, provided the benefit is not an increasing sum. The prospective formula will not apply to other forms of insurance ; but other formulas may be derived to correspond to formulas for single lives, thus: .V, P v = .V, ? = A. +o :W+B - ,? (i * |,_ n _, a. + . !w+ J This applies during the premium-paying period only; after that the value is the single premium at the ages attained; in other words : t P ww (i + | t _ B _, a w+n . w+n ) becomes zero. The formula for the value of a temporary insurance becomes: (v N w+n . w+ll N w+p + 1 .+,+, ) (v If premiums are limited to t years during the premium- paying period the formula will be the same, except that in the last term of the numerator t will be substituted for m. PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 249 The formula for the value of a pure endowment is: V m P E sy = n V m P E ww SB m _ Q E w _ n : w+n - m P E ww Again, where premiums are limited to t years during the premium-paying period substitute t for m in the last term of the numerator. The formula for an endowment insurance becomes: n V m P J xy = V m P JE WW = m _ n JE w+u : w+n m P D w+m:w+m + (v N w+r _ :w+n N w+n + 1 : w+n+1 ) (v N w+m : w+m = N w+n+|iw+B + l ) -.P^wCN w+ .- |iw+ . +I - N w+ , + Iiw+ ,_,) -D w+n : w+n Again, where premiums are limited to t years, during the premium-paying' period, substitute t for m in the last term of the numerator. JOINT LIFE VALUATIONS. RETROSPECTIVE CONTINUOUS METHODS. FORMULAS similar to Elizur Wright's and David Parks Fackler's accumulation formulas for single life insurances may also be evolved by a similar course of reasoning. As to the first, we have for the first year of insurance that the net premium, less the discounted probability of dying, _ www _ " UU buys a one-year pure endowment equal to the terminal reserve. We have found the value of such an endowment to be E = vp,^l%^ = D l w l w L-'ww Its value, therefore, at the end of the year is D ww p =Uww- The value of the insurance, then, becomes: 1 VP ww =u ww (P ww c ww ) The next year the conditions change only by adding the second premium to the reserve before deducting c w+i . w+i , thus: YP 11 / v P 4- P -c 2 v r ww u w + l . w + l ( , v r ww + r ww c w + l . w + J ; And, generally, PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 25 1 This formula will apply to any insurance, including endowment insurance, with level premiums and not involv- ing a varying benefit, during the premium-paying period only ; it will also answer after the premium-paying period has elapsed, P ww in that case becoming zero. The Fackler formula is equally applicable, thus: the fund at the end of the first year is (UUP^) (i + i) d^ (l w l w P ww ) (i + i) - (l w l w - l w+I l w+1 ) ; and this is divided among l w+1 l w+1 pairs, thus: U, U, ( ,VP WW ) = 0.1. P-.) (i + i) -0.1.- W.+.) Consequently v P - i v r ww r+1 l w +l 1 +1 1 Multiplying both terms of fractions by v w+1 we get: t y p^ -_^ PWW p ww _ Q Adopting k ww as a symbol for =; - and substituting u^ for , we have U w +i : w + i V P = u wvv P ww k ww Evidently the only change in this formula the second year is to add this first year reserve to the premium before the other operations, thus: VP 11 / V P _i_ P \ 1' .. ^ww U w+1 . w + 1 ( t V F ww + F ww ) -- k w + 1 . w+i And, generally, n V P W(V - u^^ : w+n _ t ( D _, V P ww + P ww ) k^^ . w+B _, PROBABILITIES OF SURVIVORSHIP. THERE is one probability that remains to consider, viz. , the probability that x will die during any given year and y be alive at the moment of his death ; this is called the probability of survivorship. We give it the symbol ql y or, more generally, n _ l \ qi y , which latter means the probability that x will die during the nth year, y being still alive. This probability cleaves into two probabilities, viz. , that x will die in the nth year and y survive that year, and that x will die in the nth year and y die in that year also, but atter the death of x. The first of these probabilities we know to be n _ Jq, ( .Py ) = (n-i P* nPx ) nPy - The second of these probabilities is arrived at as follows: The probability that both x and y will die in the nth year is n _, | q x (,_! q y ) = ( B _ 1 p x n p x ) (._, p y n p y ). Each, on the supposition of uniform deaths throughout the year, is as likely to die in one part of the year as in another; con- sequently the chance that either one will die first is equal, or, in other words, y, ( a _, p x p z ) ( n _ x p y n p y ). Thus the entire probability becomes: n _ 1 1 q xy = ( n _ 1 p x n p x ) n p y + y 2 (_! p x n p x )(_! p y n p y ) = (_, P X n p x ) n p y + ^( n _, p x n p x X-, p y - ^ (.-, Px nPx) nPy = % (_, p x n p x ) ( n _ 1 p y 4- n p y PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 253 When n = i, the formula becomes: qiy = # ( oPx Px ) (p y + O p y ) But o p z and o p y , the probability of x and y respectively being alive at the moment the term begins, are by hypoth- esis both equal to certainty or i. Substituting, our for- mula becomes qiy=^ (i Px) (i+P, ) We may further develop the formula for 1,^ q^ y as fol- lows: But we have seen that n p x ( n _j p y ) = : y ~* and, therefore, Py i '^ y : *"" 1 , as developed in the les- sons on life probabilities. Substituting, we have: SURVIVORSHIP INSURANCE. AN insurance on the life of x which is payable only if y survive him, is called a contingent or survivorship insurance. It maybe further involved by being also payable to z if y does not survive x and z does survive him, etc. ; or by being a joint-life insurance on the lives of x and y, payable to z if z survives the first of them to die. Thus the number of the insured and the number of successive beneficiaries may in theory be indefinitely multiplied. In practice more than two lives are seldom dealt with and more than three lives almost never. We shall here undertake to deal with two lives only. Suppose a survivorship insurance for i year on the life of x, payable to y if he survive x. Remembering the for- mula for the probability of x dying within the year, y sur- viving him, i. e. , qiy = % (.-, P, D Px ) (.-, Py + B Py ) ^ (I Px) (l+Py) we have for the one year's insurance, which we will desig- nate by the symbol | ^Jy, the following formula: I i A iy = V % ( O p, - p. ) ( O p y + p y ) = V J/2 (I " Px) (I + Py) The value, now, of the second year's insurance will be v 2 Vz (Px 2 Px ) (p y + 2 p y ) an d of the nth year's insurance V " % C-i Px -~ D Px ) C-l Py + D Py ) Consequently, an insurance for two years will be worth: ( o p x - p x ) ( o p y p y ) y 2 ( Px - 2 p x ) ( Py + 8 p y ) PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 255 And, generally: | n A xy = v y 2 ( o p x p z ) ( o p y + p y ) + V % (p x 3 p x ) (p y + 8 p y ) + v % ( B _ 1 p x n p x ) (_! p y + n p, ) It follows that a survivorship insurance extending throughout the joint lives is worth: A *y = V % ( O p x p x ) ( P T -f p y ) + v a ^ ( Px - ,p x ) (p y + 2 p y ) = 2 v n y 2 ( a _ i Px nPz ) c^, Py + n p y ) But in considering probabilities of survivorship, we found that L-i qiy = % (.-i Px nP- ) C~ , Py + .Py ) Substituting, we have: This looks much more complex than the other, but is really more readily convertible into parts which we can solve separately. First of all, the series: 2 v n ( n _j p xy np*y) 2 v n n q xy A xy . Then the series: 2 v n n p I _ 1 :y = a,., :y and 2 v n u p x : ^ = a x ;y _, . Substituting, we have: A; ? -> 256 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. Rnt A . M y . - vN_. !r _, N I; *** : y i a _ = N x _ 1;y Substituting again, we get: But p y _ t D x . y _, = i_ (v 1 i x 1^) = v * l x l y and p x _! D x _, . y = p- (v 31 - 1 l x _, l y ) = v 1 - 1 l x l y This latter value needs but to be multiplied by v to also become v* l x l y = D xy . We may thus multiply it in the N _ quantity ** : - by also multiplying the numerator by v, which will not alter the value, thus: v N x _ t . y _ v N x _ t . y v(p x _ 1 D x l lsy ) = ~D^ Substituting these, we have: Aj y = y 2 ( vN '-' :> -'~ N "p iy N ' :r "' + vJ ^) This applies only to commutation tables according to Davies' formula (D sy = v x l x l y ) when x > y. When x < y, the formula becomes: , v (N,_, ; y - f - N x ; T . , ) + (N^, ; y - N xy ) 2 ., PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 257 With commutation tables, according to De Morgan's formula (D, r = v V l x l y ), 1 / + y P* i -Dx i : y ^,, or y > x, the formula becomes: - N., For annual premiums, we have Ai p, _ xy " i + a xy In other words, substitute N x _! . y _ , for D xy in any formula for A^ y and it will give Piy. In the same way limited premiums may be computed by substituting N x _ x :T _, N x+t _i : ^ +i l instead of D Tv . EQUIVALENT EQUAL AGES IN SURVIVORSHIP r INSURANCES. THE theory of equivalent equal ages is that since m /" w ^ + P 7 + r m + - . . -, N ww = N Xy , N www = N xyi , etc. This theory is not quite accurate, especially where tables are not graduated by Makeham's formula; but it answers for all practical purposes when using any well graduated table. This being true, it is possible to solve survivorship insurance problems by finding equivalent equal age values for the values of N and D in the formulas. Thus, when x > y, we have, by Davies' plan of commu- tation columns: A , v (N._. , ,_, + N M , , ) - (N., + N, , ^.) A ' y = 2U ly Let us find equivalent age values, thus: N ww = N xy , Nj; = N^ :y _ n N" = N x _, :y and N" = N: Y -,. Substituting, we have: v (NJJ + N W 2 4) (N ww + N w 3 ^) iy = When x < y, we have the formula: v (N _ N x . . ) 4- (N _ . N ) Al, = i x ' y * * ' '* ^ -zy 2 L>Xy Making here the same substitutions, we have: v (N^w N^) 4- (N w 2 ^ N w ) * = 2 D In the case of equal ages commutation tables by Davies' and De Morgan's formulas are the same. Applying the PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 259 values of N and D to the formula for A^ y by De Morgan's commutations, whether x > y or x < y, thus: we get : v Mil __ M i v # (N 22 _ N 33 v -l^ww -^ ww T v \- L>l ww -^ww * , These formulas may be made to answer for annual pre- miums, thus: :: N WW + v^ (N^ N W 3 W 3 ) v Nj; N ww + v 1 ^ (N W 2 J N w 3 ^) N w _ 1:w _ x For limited payments for t years, make the denominator "NT N IM W _J :w _i iN w+t _ 1 : w + t ,. The foregoing has been submitted to Mr. Emory McClin- tock, who says of it: '* Your formulas appear at once to be sound and useful as applied to tables graduated on the Make- ham system," and adds that as to other graduations, while " it seems probable that the approximations would be fairly close," the matter '* can only be resolved, one way or the other, by actual trial in a number of different cases." His comments are at once clear and cogent. A way of testing the application may be suggested, viz., test as to any given table, the correctness of the equivalent equal 260 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. age formulas for joint-life insurances or annuities. If the approximation is close there, it will answer as well in sur- vivorship formulas. If a table is too irregular to admit of such substitution, the use of equivalent age symbols must either be avoided, or, if the benefits are for long periods as the whole of life, a graduation by Makeham's system may be made from the facts developed in connection with the poorly-graduated table. This will not, however, be so likely to answer in the case of short term contracts. Concerning the use of equivalent equal ages in joint-life computations, the Institute of Actuaries' Text- Book says; " The values so found will correspond to the mortality table as constructed by Makeham's formula; but as that table adheres very closely to the original facts, the values of joint-life annuities so found may be used without much loss of accuracy along with values of annuities taken from the volume of ' Institute of Actuaries' Life Tables,' these being based on the same original facts, but graduated by a different formula." This agrees notably with Mr. McClintock's comments. When N and D at equivalent equal ages closely approxi- mates N xy and D Xy , they may be substituted in any formula involving joint life or survivorship problems, as, for instance, in computing the special M columns. SURVIVORSHIP INSURANCE. SPECIAL C. AND M. COLUMNS. ONE phase of the formula which we obtained for B _, was: n 1 | -ixy - Where n = i, this becomes: UL u, This is the probability that x will die during the first year, y being alive at the moment of his death. Now if i be then payable to y at the end of the year, its value becomes j 1 I Al | i-tt-xv i*V v* I, l y Should it not be thought advisable to use the joint-life commutation columns, the calculation of survivorship insurances may be facilitated by computing C^ y and M^ y columns as follows: c; y = v+- a, i, +K Cx + l:y + , = v "+" d I+1 1 J+1+ ^, etc. Miy = c; y + c, Wi + . . . It is evident that vM x l y 262 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. : M' M And, generally, r 1 _i- r 1 _L r A i l-xy + ^x+! ; y+l "I" ^x + n 1 ; y+ n i I n ^^y r> J-'xy By the same analogies A l xy = ~^ M xl n pi "^xy I .Pi, = w ^ i : y + n J SUCCESSIVE AND SURVIVORSHIP ANNUITIES. THERE may be an annuity for the life of x, with succes- sion to y for his life if he survive x, etc. Such an annuity is called an annuity for successive lives. The symbol a x77T7 * s gi ven to this value. The value of an annuity for the life of x and thereafter through the life of y, if he survive x, is evidently greater than a x and less than a x + a y . To consider it first in comparison with a x + a y we find that it differs therefrom only in the circumstance that a x 4 a y pays 2 during the joint lives of x and y, while the annuity we are considering pays but i or i less than a x + a y . But i for the joint lives is worth a xy , therefore a~ h worth a xy less than a x + a y , thus : *~ y = a x + a y a xy N xy _ _ " D y D xy We may consider it, second, with reference to a x . It is greater than a x by the value of an annuity to y during his after lifetime if he survive x. This latter annuity we may call a x | y . We have, then, the formula : a ,77 a x +a x | y It will be noted that the first term of these two formulas is the same quantity, viz. , a x . We have a = a x + a y n xy = a x + a x | y .-. a x | y =a y a xy 264 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. That is: a survivorship or reversionary annuity on the life of y beginning at the death of x if y survive, equals a life annuity on the life of y, less an annuity for the joint lives of x and y. The following is a formula for an annual premium for this survivorship annuity: NZ. N^ pra | )= D, D H _ = N,D. T N. y V.- I y / AT T) /"NT \ -NT - L>I T i : y i *^y \ L ^xi : yij J -^x i : y i REVERSIONARY OR SURVIVORSHIP ANNUITIES. WE have already developed the formula for a survivor- ship annuity (a x | y ) in terms of other annuities. But it will pay to consider this from the standpoint of the insurance involved in it; for this is really a survivorship insurance on the life of x of an annuity for y's after-lifetime. In order to deal with it in this aspect, let us adopt an alterna- tive symbol expressing that idea, viz., AJy (a y ). Suppose an insurance for one year on the life of x of a sur- vivorship annuity for the after-lifetime of y. Its value is: The value of a similar insurance for the second year is: I A ' (A \ - - * +1 ; y +1 /liliiL i i|i A*y (fy+t) -- pT ~ l-jx I LJ xy \JJy +2 / The value of the two years' insurance is: D xy is a common factor in these denominators. Separ- ating it out, we may compute a special commutation series which we will call Ci y , thus: r , / N y+2 \ :y + i WMSJT+I I n /' \ ^y+a / Our insurance now becomes: pi , pi I Al ~ \ ^xy + ^* + l ;y-H 266 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. Summing these C values from each age to the end of the table, we have a series of special commutation columns: Mi r Our insurance assumes, then, the general form : By analogy, we see that: pi _,_ pi / X ^-^TV l~ V-/TT4-1 " V 4- 1 ~T" (a y )ora x y = ~^- , " The annual premiums are: pi / Q , _ M xy These are for an annuity first payment i year after the end of year in which x dies If an immediate annuity is wanted, add A; y to A^ y (a y ), P^ y to P.l y (a, ), etc. REVERSIONARY ANNUITIES. THE SURVIVORSHIP INSURANCE OF A TEM- PORARY OR DEFERRED ANNUITY. WE may also have the insurance to y, if he survive x, of a temporary or a deferred annuity, A y ( | m a y ) and A^ y ( m |a y ). The insurance itself may also be temporary or deferred. Let us consider the case of an insurance for i year of an annuity for m years. Its value is i v ( I a ^ C ' y x Ny+1 ~ N y+ m +i ! i ***? v ' "sr / - r) A n U *7 U , + i The value of two years' insurance would be: .,-N.^ + \ ^D^T Let us now construct new C values: ^J+a And also M values : Our two years' insurance now becomes: .:^. . or generally 268 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. '"Ml. The formulas for whole life and deferred insurances become : Al,(La y )= y_ xy Im Altering the denominators gives us annual premiums : I M 1 xy N - N "aitjr-l 1>( x + t i : y +t i Im Im M 1 M ' _ ** x i : y i ^ x + n i : y + n J Let us now consider an insurance for i year and then for 2 years of an annuity deferred m years, thus : C 1 N - PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 269 Let us again construct new C values : And also M values: Our two years' insurance now becomes: mi^ ml I. Ai, ( . ! a, ) = "* T' :> *'; or. generally, ml The formulas for whole life and deferred insurances become : ml Altering the denominators gives us annual premiums 'x i :y i 270 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 'M- P xy mly "M REVERSIONARY ANNUITIES. COMBINATIONS. WE may have the case of an insurance on the life of x of an annuity certain for m years, plus an annuity to y for his after-lifetime if he survive more than m years after x. This annuity = A x (a-,) + A 1 ( m |a y ) _ M x ( aj ) M xy D x D xy This is for an annuity, first payment year after end of year in which x dies ; if there are to be m 4- i payments in the annuity certain, one at end of year in which x dies and m thereafter, add to the value A x . The annual premium = _ + i + a x i -4- a xy X '(aa) '~M xy This gives a premium of two separable parts, the first collectible for the life of y, the second only for the joint lives, i. e., ceasing if x outlives y. This, also, is for an annuity first payment in i year from end of year in which x dies. If there are to be m + i pay- ments in the annuity certain, the first payment at end of the year in which x dies, then add P x to the premium. This would not cease upon death of y. The limited annual premium = A, (aa) < A!, (.|a.) i + |,_, a, i + |,_, a.,, 272 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. M a Vi The first part of this also is payable for the life of x or t years, and the second for the joint lives of x and y or t years. Add t P x to get an annuity certain of m + i pay- ments, the first due at end of year in which x dies. The value of a temporary insurance of this type is For an annuity certain of m + i payments, the first due at end of year in which x dies, add | n A x . The annual premiums for temporary insurance is : i + | D _, a x i 4 |._, a,, _(M.-M.+.)(aj) , "X-Xl.:,*, The first part is for life of x or n years, the second for the joint lives only or n years. For an annuity certain of m + i payments, the first due at end of year in which x dies, add n P x . The value of a deferred insurance is : y - n n *^a * The annual premium for a deferred insurance is : , = ,|A.(aa) JA' r (Ja,) i + | n _, a, i + | n _, a ly PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 273 The first part of this annual premium is also payable for life of x or for n years; the second part for the joint lives or n years. For an annuity certain of m + i payments, the first due at end of year in which x dies, add n A x and JP X , re- spectively. In computing such premiums the numerator of the sec- ond part may be derived from one mortality table, repre- senting experience among annuitants, and the other factors from another mortality table representing experience among insured lives. CONVERSION FORMULAS, JOINT LIFE, SUC- CESSIVE LIVES AND SURVIVORSHIP ANNUITIES. WE have seen that there are subtle relations between all of these benefits. It remains to gather them together as far as may be. Thus we have a _ = a x 4- a y a xy z (j) = a x +a x | y a x | y = a _ a x a,| 7 or Ai y (a, ) = A^ (| .a, ) + A', ( ml a y ) Ai y (| m a y ) =A; j .(a y )-A; y (> y ) A iy ( l a , ) = A iy ( a y ) ~ A ij (L a y ) Pa.l.or P; y (a, ) Pi y (| m a, ) + Pl y ( Ja y p; y (l m a y ) = p; y (a y ) - p; y (> y ) P iy (JaJ = P; y (a y j-P; y (| m a y ) The inter-relations : A= I.A + JA P= I.P + J P = P-P PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 275 all hold, the sorts of insurances being throughout the for- mulas strictly the same. The Institute of Actuaries' Text-Book gives the follow- ing additional formulas, which are not especially useful and which are here mentioned because the student might misunderstand their significance when he encounters them. They are : L a *l y = L a , L a *y Ip I I . . I a n I * x | y n ! <** n I o-xy The first of these signifies an annuity limited to n years from now, anyhow, beginning only upon the death of x during the life of y ; the second an annuity deferred n years from now, anyhow, but never accruing unless x dies dur- ing the lifetime of y and, if x dies after n years from now, the annuity begins at once. VALUATIONS. SURVIVORSHIP CONTRACTS. V ,Pi, = A ; m;ytm - ,Pi, (i + | V | Pi, = | rt A - P V.I Pi, = n - m: . + -J Pi, (i ,V a_ - a. *,y x + m>y VPa,|, = P a ,|, m V Pi, (a y ) = A t .V, Pi, (a, ) = A - It + m a x + m : m V I. Pi, (a, ) = A I _ a .V j Pi, (a ") = _" _ (a y+n ) - [Pi y (a y )] (i + a, +m:j +m (a,;;) - [ ,P; y (a y )] (i + m ) !j+ _ (a, + J - [|. Pi y (a, )] (i + ) ; m ; ^ (a, + J - [ J Pi y (a y )] i + V I. Ai, ( | . a, ) = _- A, +m - , , + ., ( |. a PRACTICAL LESSONS IN ACTUARIAL SCIENCE. .V .1 Al y (I. a y ) = ,_J A +. . ,.- ( | m a, + .0 'V P; y '( |. a y ) = A I+ .. +., (|. a y + 0/ ) - [Pi, ( | m a y )] A ( I. a, ) = A ;- . !T+- , ( I. a y+m ,) - [ ,Pl y (|. a y )] |. a, ) = |._ m . A io/ . j+m , (Ua y+m ,)- [!?;, (|a (i + L-m' a +m ' :yfm ') | m a y ) = n _ m ,| A t- ,. j ^(| m a, + m O-[.|P 1I (la y )] (i + !_ a l4 . m / :y+m ,) , ( m a y = n _ m , iin/!j+m , n | a y+ln , ) i y ( m | a, )=,_,! A | m/:7+m , ( m |a y+m ,) m ,V P^ ( J a y ) = A.. m , :j+m , (J a y+m ,) - fPJ y (J a y )] (i + a I+m .. y+ln .) 'V .Pl, (J a y ) = A^^XJ a y+m ,) - [,PJ y (J a y )] C 1 + l t m' **x+ m' ',+ m' ) m 'V| n Pi y CJ-S ) = | n - m ' A |m/r ^ m/ ( m | a y+ffi ,) - [ n P^ y (m| a y )] (i + l-*f* t + m >:, + m ') .'V J Pi y (J a y ) = n _ m ,j A x | m/: ^ m/ (J a y . ffi O - [J P^ y (m| a y )] (i -I- | D _ V a x+m , :y+m /) The combination of a survivorship insurance of an annuity certain and of a deferred annuity to y to begin when the annuity certain ends, may be dissolved into the two parts and the reserves found separately and then added together. These formulas are not developed into commutation sym- bols, but the student has, of course, by this time learned 270 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. how to do this himself by substituting the equivalent values in commutation symbols for the corresponding values in these formulas. However, in order to assist him it may be suggested that the formulas are the same as for reserves on other joint- life policies, the special M values being substituted thus, generally : V A ^HL ^ o +m M. _ M /N+._. - N + ._.\ D + _ ' ' N_, - N +1 _, ^ D +m VI p _ M-H. M+. M M+. /N +n _, N + \ D +m " N_, N +B _, V D + = v ip_. M +n M +n /N +m _. N + ._.\ n ' n "TCT N "n ! J-' + in ,. 1N ] - iN +n 1 V L'+m Thus, for instance, by substitution, V Pi - - M T j m : y + m M^y /N x+m _ 1 ;y + m A m v A xy -- p. - ^ I T~) -"^x+mry + M -^x 1 : y 1 \ -^x + m :y + m / - L ^x+ m : y + m m Mjy- -M x+p;y+n /N x+m _,. y+m _ 1 N x+t _ l:y+t _ Mjy- -M x+p;y+n / N x _, : y _, N^-i :,*.-, \ SICKNESS INSURANCE. SUPPOSE in a group of m persons aged x, there are r weeks of illness in one year. Then, expressing the rate of sickness by z x , we have : If each person ill is to receive i during his illness z x must be contributed by each person. But if contributions are at the beginning of the year, v^ z x will suffice, since on the average sick benefits are payable at the middle of the year. The value at age x of the second year's insurance is E x (v% z x+1 ) = ' +1 |V Z * +1 >' and generally the value of the nth year's insurance is * +p ~ 1 x+n "" 1 . An insurance for two years, then, becomes: v^ z x -;- ^-=- which, for uniformity's sake, we may reduce to a common denominator, thus : - Thus, also, an insurance for the whole of life, which we will call s x , becomes : *- D x The values (D x v^ z x , etc.) may be summed from each age through the table and given the symbol K x , K x+1 , etc. Substituting, our formula becomes: 280 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. We also develop formulas for temporary and deferred insurances, as follows: _ D x v 1 ^ z x + D x+1 v^ z x +1 D x Annual premiums in advance are developed thus; P~.= s - = K - ^ N - = K - . N N x 4 1> x+t 1 x K x -K x+n N x _ x D x D x x K x+n * D x K x+n NN x 1 J-^x + n 1 Suppose, however, as is frequently the case, the com- pany engages for a payment of one sum during the first six months of illness, then for a smaller sum for six additional months, and then for a smaller for the remaining duration of the illness. PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 281 Then we must make v^ z x = the value of the benefit for the first six months, v& z' x = the value of the benefit for the second six months and v a z" = the value of the benefit for the remaining illness, a being the average period elapsing before benefits are paid. Then we must compute D x v^ z x , D x v^ z' x and D x v a z" x values for each age and sum them to set K x , K' x and K" x columns. Then our formulas become : _ Of course, such columns can be made for any combina- tion. Disability in old age is hardly an illness, arising, as it does, from age itself. Consequently, and also because life has little earning value in old age, it is customary to have the insurance stop at age 65 or 70 and often to have an annuity begin at that age. Suppose an illness insurance of i a week to age 70 and an immediate annuity of 50, beginning at 70. The value is: K x K + N The annual premium is : p ri T 4. - (J 7 0-* S * + f 70 The reserves on sickness policies are computed on the same principle as life insurance reserves, viz., by deducting from the value of future benefits the value of future pre- miums, thus: 282 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. rr V c c *^* +m m v x x + m ' r\ L'x-t-iii m V P s x = s x+m P s x (i + a x+m ) K x+m K x /N x+m _A D x+m " N x _> V D x+m 7 m V t P s x = s x+m - t P s x (i + 1^,,^ a x+m ) K. +m K x /N.^-, - N x+t _.\ D x+m " N M - N x+t _, V D x+m TT _ TT VI c _ I c _ X^x + m ^x+ n m vi n s x i n _ m s x+m - =- L'x + m m V P | n s x = ! n _ m s x+m -- P| n s x (i + !,,_,_, a x+m ) _ K x+m - - K x+p ^ K x K x+n /^, - v D x+ jr V n | S r = n _ m I S x + m = -=r L^x + m V PJ s x = n _ m | s x+m P J s x (i + !_,_, a x+m ) K x+n K x+m D x+m N x _ : N^.^ D x+ CONVERSION TABLES. THE student must have observed that there is a certain relation between annual and single premiums and between both of these and annuities which holds whatever be the nature of the insurance. Thus: A = i d (i + a) = i vi (i + a) P^-l d = ^ vi i + a . i H a This holds good whether the insurances be on one life or on many lives, the annuity being of the same order. Thus : A x = i d (i +a x ) 1 + ln a * A xy = i d (i + a xy ) A' y = i d ( i + a__) V * y/ Consequently, if a of a certain sort or " status " be known, we may easily find either A or P of the same status, provided we also know the rate of interest ; for the other factors in the second part of the equations are known quantities. A and P, then, may be found by reference to these formulas (a and i being known;, without knowing anything about the mortality table even, not to speak of not knowing the form of policy. These formulas are so simple and so easily applied that it would seem that no one would care for anything more or better. But the ingenuity and industry of the older actuaries have spared little for their successors to do in this as in many other matters. 284 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. William Orchard, in a work published in 1856, gave to the world tables from which to one-thousandth of the unit (dollar or pound) the single or annual premium may be ascertained (the annuity being known and the rate of interest). Orchard's table gives premiums corresponding to annuity values differing by hundredths of the unit, and he furnishes differences for making the values approximate even smaller fractions. So extensive tables are hardly requisite, as, indeed, are no such tables, since computation from the formulas offers no difficulties. The Institute of Actuaries' Text-Book gives a conversion table, corresponding only to unit values of the annuity, with tables of differences corresponding to the fractions in the annuity value. The computation of the premiums in these tables would offer no difficulties, being direct from the formulas; but the differences were determined by separate computations by a different method. First as to single premiums: Suppose A to be known, then to find by how much A should be increased or dimin- ished, a having been increased by A a. We have A = i d (i + a) A + A A = i d (i + a + A a ) Substituting the former equation from the latter, we have A A = d (A a) That is to say, the difference is subtractive, which means that A diminishes as a increases a thing plainly true since a large annuity value means small decrement by deaths, which in turn means low insurance premiums. Making A a = i, successively, we find that A A becomes PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 285 a fixed quantity the moment the value of i is given. Thus when i = 4%, A a = . i, then A A .038^6154 x .1 = .003846154 For every . i, then, that the annuity is more than the whole number value in the table, .003846154 is subtracted from A; and for every . i that the annuity is less than the whole number value, .003846154 is added to A. Likewise for every .01, more or less, .0003846154 is subtracted or added, respectively. The difference columns in the table are mere tabulations of these differences corresponding to differences of .1 and .01 in the annuities. Annual premium conversion tables offer no greater diffi- culties, except that the differences are on quite another basis. Thus: i -f a i + a + A a Subtracting the former equation from the latter, we have A P _ i + a + a i + a That is to say, assigning A a = . i , the differences will be i i A = i +a+ .1 It is evident, also, that this difference will be minus or subtractive because the first fraction, having the larger denominator and the same numerator, is* the smaller. The series of differences constructed in this manner will vary as a varies as is seen in the table of differences accompanying 286 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. the conversion table. They do not, however, involve or depend upon the rate of interest, and are applicable what- ever the rate of interest. In applying the single premium conversion table, for every difference of .01 in the annuity subtract one-tenth as much from the premium as if the difference were .1, etc. In computing annual premiums from the conversion table a different course must be pursued. The table of differ- ences only gives the differences proper to . i, .2, etc. It will be observed that the difference for .2 is not twice .1, but an irregular amount more. Consequently for fractions smaller than . i, one must proportion. For example: given a = 1 3. 842 to find P, interest three per cent. When a = 1 3, P = . 04230. To this add. 00386 for the .8. If this were .9, we would add .0043^ or .00046 more. Being .842 instead of either .8 or .9, we will increase what we add for 8 by rfo (.00046) = .00019. We thus have P for 13. = .04230 Add for .8 .00386 Add for .042 .00019 Total P = .04635 CORRESPONDENCES AND EQUIVALENTS. THE idea of accumulation is so essential to a proper understanding of the principles of actuarial computations and at the same time in some respects so puzzling and intricate, that a few words upon the subject of equivalents are needed. We have familiarized ourselves with the idea that the value of the insurance and the value of the premiums must be equal, i. e., that the two must be equivalents. This brings forward the conception that a sum in hand and instalments to be paid in future may be equivalent, and that two sets of instalments to be paid in future may be equivalent to the same sum in hand and so to each other; a conception which may be illustrated further. Suppose a man who wished to borrow a sum of money and provide for its repayment, including interest thereon, by a certain annual payment for a fixed time. He might do this in either of four ways: 1. He might borrow the amount of one person, the inter- est to be compounded and the whole sum to be paid at the close of the period ; and then invest a sufficient sum annu- ally at compound interest at the same rate to accumulate the amount required to pay the whole. 2. He might borrow the amount of one person, paying the interest annually and the principal at the end of the period; and invest annually at compound interest at the same rate a sum sufficient to accumulate an amount equal to the principal. 3. He might borrow the amount with the privilege of annual partial payments, and each year after paying the 288 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. interest on the unpaid balance reduce the principal by pay- ing thereon the remainder of his annual sum after meeting the interest. 4. He might find a man or company willing to give him the amount in return for an annual payment of the sum for the agreed period. These four are equivalent, and the same annual sum. applied in any way of the four, will equally well discharge the debt. In the same way an endowment insurance is equivalent to a term insurance for the same period, and a pure endow- ment also for the same period, and these in turn are equiv- alent to the premiums progressed through the period at interest, less the expected expense and mortuary costs. There is no difference in its real nature, whatever way you compute it ; and reasoning based on one equivalent which does not result the same when based on any other equiv- alent, is necessarily misleading. These remarks are called forth because the issue has been raised that, although these ways are equivalent, yet, somehow, the insurance value is bigger when you look at it one way than when you compute it in the ordinary way. The student should beware of being contused and misled by fallacies. COMPARATIVE NOTATION. IN this work, except where the fact that the notations are otherwise is noted, the notation throughout is that of the Institute of Actuaries' Text- Book, adopted in 1895 by the International Congress of Actuaries as the universal standard. The Actuarial Society of America has also adopted the same notations, with the following exceptions: Since all American books and tables have printed the commutation column, N, and formulas based thereon on the basis that D x + D x+1 + D I+3 -f = N x instead of D x + 1 -f D x+2 + D x + 3 + = N x , as in the universal notation, the society adopts the symbol N x , printed in heavy-faced type, as equivalent to N x _, of the universal system. The society also adopts S = sums insured as an alternative for (w d) and n single premium as an alternative for A; the expressions K x , K x and u x are all distinctively American. The adoption of this notation by the Actuarial Society of America was, in 1890, and it has hitherto been employed in no American book or tables, excepting in those papers and tables published by members of the society and mostly included in the society's publications. The following is a partial comparison of the English symbols with the sym- bols formerly used in America and contained in almost all American books and tables, so far as they differ: 290 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. UNIVERSAL STANDARD CORRESPONDING AMERICAN NOTATION. SYMBOLS. Symbols for Premiums. A TT P i + a x or a x A x ' + 1.** A ,n 1 + J a , A ,|. A 9 n xn n A E x or A x -, e n in n n x Symbols for Commutation Columns. N x _, N x or N x N x N x+1 or N xn - n N 2 M in -, n M x Symbols for Valuation. V H .V, HU Q V t P x !H x+n V|P x tH s+n n A K A, I+n I. SHORT METHODS OF COMPUTATION. FROM the mathematical viewpoint actuarial science con- sists mainly in " short cuts." The commutation and val- uation columns are illustrations of this. Short methods of computation are therefore welcomed. Subtraction is the sole elementary calculation that in no case admits of shortening. In addition, considerable is gained by carrying hundreds, at least, automatically on the fingers; this greatly relieves the memory, and one soon gets so he can pick out hundreds very readily. Thus two columns may also be carried for- ward nearly as easily as one. Carrying smaller totals, as twenties or forties, in this manner may also be practiced with the result of a great gain in speed. One learns to pick out these smaller totals more readily. Multiplication may be greatly shortened where the prod- uct is only desired to a certain number of figures, as, for instance, extended to a certain number of decimals. Let us analyze the ordinary multiplication. Take, for instance, 684.7869 X .3182, with product desired to be extended to three decimals only. We have : 684.7869 .3182 13695738 54782952 6847869 20543607 217.89919158 292 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. Of this product the last five figures are not needed. In practice the first foreshortening is not to add any beyond the fourth decimals and to add them, mentally only, to find what to carry into the third decimals. But we may shorten the process much more by not mak- ing the unnecessary figures ; thus : 684.7869 .3182 5478 6848 205436 217.899 We do this, knowing that 680 X .0002 gives us three decimals; 684 X .008 gives also three decimals, etc. In each case we are careful to carry one where the figure in the multiplier multiplied by the figure in the multiplicand next right of the one we start with indicates that we should carry. This short system may be varied thus for convenience: 684.7869 2813 205436 6848 5478 217.899 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 293 This consists simply in discovering which figure in the multiplicand furthest toward the right needs to be multiplied by the figure in the multiplier furthest to the right to give the required number of decimals. Then set down this figure there and reverse your multiplier, writing the figures to the right instead of the left of this figure. Then multiply in the ordinary way, setting down the result as in foregoing illustrations, and beginning in each case by multiplying the figure by the one immediately over it in the multiplicand. There is also a short system of division based on the same idea. Let us first consider the usual form : 2I7 .8 99 | 684.786 19092 26997 25020 22274 27460 25456 20040 19092 948 294 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. Now the adding of all these ciphers seems of little use. Suppose we shorten it, thus: .3182 | 2I7 .8 99 684.786 + 19092 27 2 5 In this example, after we reach the second figure in the quotient, we divide each successive time with a diminishing divisor, thus: .318, .32, .3, taking care, however, to watch for " carries " in multiplying back. When dealing with very large numbers, multiplication and division become very formidable operations. This is especially true when numbers are being raised to powers, as " squared," " cubed," etc. No means of shortening these operations has been devised that compare for speed and accuracy with the use of logarithms. No attempt will be made in this book to explain logar- ithms. Text-books on their computation and use are many. Their use, moreover, is easily explained even though the user does not understand the wherefore. Suffice it here to say that by their use multiplication is performed by a single addition and division by a single subtraction. Raising to PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 295 powers is also performed by a single multiplication and extracting a root by a single division. The most formida- ble computations become simple by their use. The following is a list of logarithmic tables and charts which the author has found useful, with some particulars about each: ABRIDGED LIST OF USEFUL LOGARITHMIC TABLES. Professor E. L. Richard's Logarithmic Tables A very small, compact book containing six place logarithms with differences only, and other tables. To natural number 10000. Crockett's Tables A somewhat larger compilation, six place logarithms with proportional parts ; also other tables. To natural number 10000. Jones's Tables- Four place, six place and ten place logarithms. Anti- logarithms. Addition- Subtraction logarithms. Natural logarithms. Squares, cubes, square roots, cube roots. Reciprocals. Binomial coefficients and Bessel's coeffi- cients. A low-priced, beautifully printed and very accurate and complete book. To natural number 10000. Von Vega's Logarithms. A famous work. Seven place logarithms to natural number 100,000. Some editions go to 180,000. Lang's Logarithms Eleven places. Peter Gray's Logarithms Twenty-four or any less number of places. 296 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. Scott's Logarithms A very complete work. Five place logarithms to natural number 100,000. Also anti-logarithms. In- dexed so as to be readily handled. Logarithm and Anti-Logarithm Card Four places. Convenient. 'Schroen's Logarithms Seven places. Natural numbers to 180,000. Edited by Professor de Morgan. Tables des Logarithms a Twenty-seven Decimales. French. By M. Fedor Thoman. CALCULATING MACHINES AND TABLES. THE labor of computations may be greatly lessened by the use of calculating machines or tables. Two machines which are especially useful for multiply- ing and dividing, each giving the product to fifteen figures and the quotient to nine, are the *Tate and the *Odhner The former is an English machine and a favorite with actuaries. It is very solidly made, and consequently expensive. Its weight prevents it from being portable. The Odhner costs about half as much, and is light enough to carry in the hand. It is manufactured by the inventor, W. T. Odhner. Those who have used the machines in this country express satisfaction with them. By these machines you set up the multiplier and multi- plicand and by a mechanical motion obtain the product; or by reversing the operation, a division is performed. Pri- marily the action of the machine is addition and subtrac- tion; multiplication and division are performed as continued additions and subtractions, respectively. The machines tend to increase accuracy. As compared with ordinary calculations, the risk of error is not more than half. Less competent persons can also be trusted to compute by using the machines. They are especially adapted to continued computations, such as : 186 * 9.1434 -H 14-32 9.8217 + 27.342, where the next step is a direct modification of the result of the previous process. They are least adapted to cases where two sets of continued computations need to be combined, as: (17.342 + 9.817) 4- 834.6 -=r (96.8421.2). The diffi- culty is that, when the first set of computations are com- * The Spectator Company, New York, agents for the United States. 298 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. plete, the second set cannot be begun without taking the result of the first off the machine. Perhaps the best machine for continuous additions is the Comptometer, where the additions are effected by the manipulation of keys like those of the typewriter. The accuracy may be tested by continued subtractions, with considerable speed. There are several prepared tables and charts for the per- formance of multiplications and divisions. Most of these run only to multiplications of two figures by three. Others go so far as three figures by three, some three by four. Thsre is a considerable loss in speed in using such tables, even when excellently indexed. The following are note- worthy tables: Bourne's Multiplication Cards The work is but partly done; but some aid is given in reaching an accurate result. New Calculation Tables for Multiplication and Division A complete and well-indexed work giving in conven- . ient form up to three by four figures. Crelle's Rechentafeln A complete and well-indexed work giving up to three by three figures. Robinsonian Unique Calculator . A complete work giving up to three by three figures, including mixed numbers. INTERPOLATION. FOR a description of the mathematical process and formula of interpolation, see page 62 ; for an illustration of interpolation without formula, see page 48. The formula for interpolation is really arbitrary in a slight degree; often even better results may be attained in simple interpolations by following the data closely. Much labor may be rendered unnecessary by resort to interpolation. For instance, in computing rates of pre- mium, actual computations at five years intervals of ages are all that are commonly required. The premiums at other ages can be approximated by interpolation. The following is an illustration of how this may be done. Suppose that one has carefully computed the ordinary life premiums by the Actuaries' Table and four per cent, at five-year intervals, thus: Age. Premium. Age. Premium. 20 I2 -95 45 28.85 25 14-72 5 35-78 3 l6 -97 55 45-3 35 19.87 60 57.56 40 23.68 Now merely to get the series started, let us also compute the premiums for ages twenty-one and twenty-two, thus: Age. Pre-nium. Age. Premium. 21 13.27 22 13.61 It will be observed that the differences are: .32 and .34, respectively. The total difference to age twenty-five is 1.77. Guided by this, let us construct a column of differ- 300 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. ences, starting with thirty-two and increasing at an aug- menting rate, the sum of which will be 1.77 : 32 34 35 37 39 i.77 Applying these differences, we get: Age. Premium. 20 I2 >95 21 13.27 22 13.61 23 I 3-9 6 24 14-33 25 I4.72 Proceeding again, we construct differences joining in our previous series, so that the sum of the new differences is 2. 25, the difference between the premiums at age twenty- five and at age thirty, thus: .41 43 45 47 49 2.25 Applying these, we get : Age. Premium. 25 14.72 26 15.13 27 IS-S^ 28 16.01 29 16.48 30 16.97 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 301 Proceed again with series of differences, to get a sum of 2.90, difference between ages thirty and thirty-five, thus: 52 55 58 .61 .64 2.90 Applying these, we get : Age. Premium. 3 16.97 31 17-49 32 18.04 33 18.62 34 19-23 35 J 9-87 Proceeding with differences, their sum to be 3.81, thus: .67 7i 75 79 83 3-75 We find that we are six under. It becomes evident that the second difference, four, must, somewhere, have been altered to five. Let us try again : .68 72 .76 .80 -85 302 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. Applying this, we get: Age. Premium. 35 19-87 3<5 20.55 37 21.27 38 22.03 39 22.83 40 23.68 Comparing this with the figures obtained by actual com- putation, it will be found that each, for ages thirty-six to age thirty-nine, inclusive, is too high by .01. The real differences were : .67 .72 .76 .80 .86 3.81 This immediately suggests the disabilities of the inter- polation system ; it can only approximate, when the series is not perfectly regular. Proceed with the next differences, sum to be 5. 17: .91 .96 1.03 1. 10 1.17 5-17 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 303, Here we also have an irregular series; had we joined in our own previous series, we should have got: .91 97 1.03 1. 10 1.16 It is not necessary to proceed further to illustrate both the uses and the defects of interpolation by differences. Where, as is commonly the case, the series is irregular. differences will give but approximate results, reliable for ordinary purposes, but not for the basis of continuous oper- ations like the computation of reserve values by the retro- spective formulas. THE USE OF PREPARED TABLES. WORK which has once been well done by actuaries, it is certainly foolish to be continually duplicating. Interest and discount tables for all sorts of terms and all sorts of rates, both integral and fractional, may now be had. The following are a few such : Depreciation, Appreciation, Discount, etc. Tables. How to Value Bonds. (Giving fractional rates.) Horatio J. Croad. Robinsonian Building-Loan Interest Tables. 11 Universal Interest Tables. Convenient and valuable. Tables of Compound Interest, three-fourth per cent to ten percent. Lieutenant-Colonel W. H. Oakes. A valuable work. There are many valuable prepared tables giving life insurance computations. In using the familiar standard tables, it is now rarely really necessary to make original computations unless something unusual is desired. The following are but a few such tables: For Actuaries' Tables. PRINCIPLES AND PRACTICE OF LIFE INSURANCE. Gives mortality, commutation table, annuities at from three per cent to eight per cent, net single premiums three per cent to eight per cent; and at four per cent net single premiums for endowment insurance for any term, net annual premiums, term, limited payment and endowment, terminal reserves for most common policies, costs of insur- ance. PRACTICAL LESSONS IN ACTUARIAL SCIENCE. 305 WRIGHT'S LIFE INSURANCE VALUATION TABLES. Gives exhaustive tables of monthly values, terminal values, premiums, actual insurances, costs of insurance and insurance values. WRIGHT'S SAVINGS BANK LIFE INSURANCE TABLES. Gives tables of analysis of the premiums into cost of insurance and deposit for reserve, including actual insur- ances and insurance values. MEAN ANNUAL AND MONTHLY NET RESERVES. Published by the New York Insurance Department for use of the department and of companies reporting. WRIGHT'S REPORTS, 1859-65. Gives various commutations and valuation tables, includ- ing five per cent and six per cent, not elsewhere to be had. Also a table of net single premiums for all integral tempor- ary insurances. TERM ANNUITIES. Net values of term annuities for all ages and terms. William E. Starr, Actuary State Mutual Life Insurance Company. American Experience Tables. PRINCIPLES AND PRACTICE OF LIFE INSURANCE. Gives mortality table, commutation tables at four and one-half per cent, four per cent, three and one-half per cent and three per cent. Also common net single and annual premiums and terminal reserves. AMERICAN EXPERIENCE TABLES. Published by the Mutual Life Insurance Company. AMERICAN EXPERIENCE THREE PER CENT TABLES. Published by the Connecticut Mutual Life Insurance Company. 306 PRACTICAL LESSONS IN ACTUARIAL SCIENCE. WRIGHT'S INSURANCE VALUE TABLES. American Experience and Four and One-half Per Cent. H M , H M (s) and H F Tables. INSTITUTE OF ACTUARIES' TABLES. Gives a valuable collection of tables on the H M , H M and H F tables (Woolhouse). INSTITUTE OF ACTUARIES' TEXT-BOOK, VOL. II. Gives valuable tables H M (King & Hardy). BOWSER'S VALUATION AND OTHER TABLES. COLQUHON'S " " HARDY'S VALUATION TABLES. KING & WHITALL'S VALUATION AND OTHER TABLES. MANLY' s TABLES. (Accelerating maturities.) For information about books of prepared tables additional to these and especially for books dealing with other mor- tality tables, the reader is referred to The Spectator Com- pany's latest revised catalogue. TABLES. AMERICAN EXPERIENCE MORTALITY TABLE. INTRODUCTION. THIS table is an adaptation of the Seventeen Offices or Actuaries' Table to the early experience of the Mutual Life Insurance Company of New York, assuming age ninety-five instead of 100 as the final or extreme limit of life. It was adopted by the State of New York as its legal standard, interest being taken at four and one-half per cent, and this was followed by its adoption in several other States. When New York changed its interest assumption to four per cent, it also changed to the Actuaries' Table, which is now the legal standard there. The American Experience Table is, however, largely employed by New York companies in their calculations, and notably in more recent departures, such as three per cent and three and one-half per cent reserves. For permission to use the three per cent tables in this series we are indebted to the Connecticut Mutual Life Insurance Company, which computed them and has copy- righted them. For assistance in supplying other tables thanks are due John Tatlock, Jr., assistant actuary of the Mutual Life Insurance Company of New York. AMERICAN EXPERIENCE TABLE. TABLE I. AMERICAN EXPERIENCE TABLE OF MORTALITY. * Number Num- Yearly Proba- Yearly Proba- ft, Num- -L Num- b_~ Yearly Proba- Yearly Proba- bo ^ PER CENT. 329 TABLE \\l.Coni. COMMUTATION TABLES. ACTUARIES 1 TABLE, THREE AND ONE-HALF PER CENT. Age. D N S M R 55 9568.466 109844 512 1031757.012 5530.3468 80484.520 56 9044 613 100799.899 921912.500 5330.0643 74954.173 57 8536.663 92263.236 821112.601 5127.9699 69624.109 58 8044.429 84218.807 728849.365 4924.4150 64496.139 59 7567.316 76651 491 644630.553 4719-3355 59571.724 60 61 7104.894 6656.385 69546.597 62890.212 567979.067 498432.470 4512.8134 4304.5676 54852.388 50339 575 62 6221.555 56668.657 435542.258 4094.8323 46035.007 63 5800.049 50868 608 378873.601 3883.7174 41940.175 64 539I-860 45476.748 328004.993 3671.6667 38056.458 65 4996.846 40479.902 282528.245 3458.9849 34384.791 66 4615.050 35864.852 242048.343 3246.1637 30925.806 67 4246.677 31618.175 206183.491 3033-8549 27679.642 68 3891.866 27726.309 174565-316 2822 6526 24645 788 69 3551.075 24175234 146839.007 2613.4700 21823.135 70 3224.832 20950.402 122663.773 2407.3118 19209 665 71 72 2913.463 2617.449 18036.939 15419.490 101713 371 83676.432 2204.9952 2007.5041 16802353 I4597-358 73 23^7 229 13082.261 68256.942 1815.7987 12589.854 74 2073 285 11008.976 55174.681 1630.8898 10774.055 75 1825.959 9183.017 44165.705 1453-6734 9*43-1653 76 159=5 621 7587 306 34982.688 1285.0851 7689 4919 77 1382 596 6204.800 27395.292 1126 0173 6404 4068 78 1186.937 5017.863 21190.492 977.1121 5278-3895 79 1008.672 4009.191 16172.629 838 9866 4301.2774 80 847.806 3161.385 12163.438 712 2303 3462.2908 81 704.125 2457.260 9002.0527 597.2182 2750,0605 82 577.2904 1879.9697 6544-7926 494-1945 2152.8423 83 466 7443 1413.2254 4664.8229 403 1702 1658.6478 84 371.6310 1041.5944 3 2 5i'5975 323-8407 1255.4776 85 290 9572 750.6372 2210.0031 255-734I 931.6369 86 223 4621 527.I75I 1459.3659 198.0782 675.9028 87 167.8707 359.3044 932.1908 150.0435 477.8246 88 89 122.9051 87 24784 236.39934 149.15150 572.8864 336.48707 1107546 79.25367 327.7810 217.0264 90 59.65039 89.50111 187.33557 54.60661 137.7727 91 38.97561 50.52550 97.83446 35.94903 83 16612 92 24.06371 26.46179 47.30896 22.35512 47.21712 93 13.82760 12.63419 20.84717 12.93277 24.86200 94 7.25145 5-38274 6.21298 6.82421 11.92923 95 3.38888 1.99386 2 83024 3.20686 5.10502 96 1.36122 0.63264 0.83638 1.29379 1.89816 97 0.46209 0.17055 0.20374 0.440/0 0.60437 98 0.13737 0.03318 0.03318 O.I3l6l 0.16367 99 0.03318 o.ooooo o.ooooo 0.03206 0.03206 33 COMMUTATION TABLES ACTUARIES' 3 PER CENT. TABLE IV. COMMUTATION TABLES. ACTUARIES' TABLE, THREE PER CENT. Age. D N S M R 10 74409 391 I737895-587 34875249.57 21623.808 743735-36 11 7I753-769 1666141.818 33I37353-98 2II35-452 722111.55 12 69191.125 1596950.693 31471212 16 20662.722 700976.10 13 66718 250 1530232 443 29874261 47 20205.122 680313.37 14 6433L39I 1465901.052 28344029.03 19761.512 660108.25 15 62026 971 1403874 08 1 26878127.98 19330.823 640346.74 16 59802.215 1344071.866 25474253 90 18912.678 621015.92 17 57653-833 1286418 033 24130182.03 18506.107 602103 24 18 55579.278 1230838.755 22843764.00 18110.790 583597-I3 19 53575-521 1177263.234 21612925.25 17725.847 565486.34 20 51640.230 1125623.004 20435662 02 17351.009 547760.50 21 49770.613 1075852.391 19310039.02 16985.475 530409.49 22 47964.530 1027887.861 18234186.63 16629.022 513424.01 23 46219.915 981667.946 17206298.77 16281.432 490794.99 24 44534.270 937T33-676 16224630.82 15941.998 480513-56 25 42905.695 894227.981 15287497.14 15610.539 464571.56 26 41332.357 852895.624 14393269.16 15286.880 448961.02 27 39812.019 813083.605 13540373-54 14970.398 433674.14 28 38342.995 774740.610 12727289.93 14660.947 418703.74 29 36923.226 737817.384 11952549.32 I4357-964 404042.80 30 35551.161 ! 702266.223 11214731 94 14061.333 389684.83 31 34224.900 668041.323 10512465.72 13770.543 375623.50 32 32943.018 635098.305 9844424.394 13485-503 361852.96 33 31703.760 603394.545 9209326.089 13205.750 348367-45 34 30505.816 572888.729 8605931.544 12931.216 335161.70 35 29347.917 543540.812 8033042.815 12661.835 322230.49 36 28228.483 515312.329 7489502.003 12397.196 309568.65 37 27146.347 488165.982 6974189.674 12137.249 297171.46 38 26100.374 462065.608 6486023.692 11881.946 28,034.21 39 25089.146 436976.462 6023958.084 11630.922 273152.26 40 24111.615 412864.847 5586981.622 11384.144 261521.34 41 23166.768 389698.079 5174116.775 11141.577 250137.19 42 22253.327 367444.752 4784418.696 10902 897 238995.62 43 21369.797 346074.955 4416973.944 10667.521 228092.72 44 20513-953 325561.002 4070898.989 10434.099 217425.20 45 19683.489 305877.513 3745337-987 10201.128 206991.10 46 18876.809 287000.704 3439460.474 9967.7548 196789.97 47 48 18091 700 17327-356 268909.004 2^1581 648 3152459.770 2883550.766 9732.4545 9495-0537 186822.22 177089.76 49 16582.791 23-1998.857 2031969.118 9255.1695 I67594-7I 50 15857-320 219141.537 2396970.261 9012.6917 I58339-54 51 15150.075 203991.462 2177828.724 8767.3105 149326.85 52 14460.256 189531.206 1973837-262 8518.7557 140559 54 53 13787.122 175744.084 1784306.056 8266.7941 13-2040.78 54 13129.988 162614.096 1608561 972 8011.2270 123773-99 COMMUTATION TABLES ACTUARIES' 3 PER CENT. 331 TABLE IV.-Cont. COMMUTATION TABLES. ACTUARIES' TABLE, THREE PER CENT. Age. D N S M R 55 12488.615 150125.481 1445947.876 7752.2815 115762.76 56 11862.195 138263.286 1295822.395 7489.6069 108010.48 57 11250.356 127012.930 II57559-I09 7223.2693 100520.87 58 106^3.112 1163598179 1030546.179 6953.7048 93297.603 59 10069.9246 106289.8933 914186.361 6680.8029 86343-898 60 9500.4702 96789.4231 807896.468 6404.6472 79663.095 61 8943-945I 87845.4780 711107.045 6124.8348 73258.448 62 8400 2602 7944=; 2178 623261.567 5841.6530 67133.613 63 7869.1640 71576.0538 543816.349 55^5 2249 61291.960 64 7350.8705 64225.1833 472240.295 5266.1305 55736.735 65 6845.4050 57379-7783 408015.112 4974.7681 50470.605 66 6353 0557 51026.7226 350635.334 4681.7995 45495.837 67 5874.3331 45152.3895 299608.611 4388.1174 40814.037 68 5409.6665 39742.7230 254456.222 4094.5478 36425.920 69 4959.9295 34782.7935 214713.499 3802.3741 32331.372 70 4526.1183 30256 6752 179930.706 3513.0269 28528.998 71 4108.9560 26147.7192 149674.031 3227 6930 25015.971 72 3709.3972 22438.3220 123526.312 2947.8127 21788.278 73 3328.3564 19109.9656 101087.990 2674.8128 18840.465 74 2966.8145 16143.1511 81978.024 2410.2132 16165.652 75 2625.5795 I35i7-57i6 65834 873 2155.3903 13755-439 76 2305-5I34 11212.0582 52317 301 1911.7973 11600.049 77 2007.4097 9204.6485 41105.2426 1680.8446 9688.2516 78 1731.6945 7472.9540 31900.5941 I463-5977 8007.4070 79 1478.7588 5994.1952 24427.6401 1261.0997 6543.8093 80 1248.9556 4745-23959 18433.4449 1074.3672 5282.7096 81 1042.3246 3702.91499 13688.2053 904.1136 4208.3424 82 858.71807 2844.19692 9985 2903 750.8660 3304.2288 83 697.65114 2146.54578 7141.0934 614.8103 2553.36<28 84 558.18032 1588.36546 4994.5476 495-6595 1938.5525 85 439.13164 1149.23382 3406.1821 392-8685 1442.8930 86 338.00088 810.33294 2256.9483 305.42803 1050.0245 87 255 82730 554.50564 14466154 232.22536 744-5965 88 188.21086 366.29478 892.ioq8 172.06020 512.3711 89 134.25576 232.03902 525-8150 123.58696 340.3109 90 91 92.23475 60.55882 139.80427 79-24545 293.7760 I 53-97i7 85-47631 56 48683 216 7239 131.2476 92 37 57077 41.67468 74.72624 35.26264 74.76080 93 21.69390 19.98078 33-05156 20.48007 39.49816 94 11.43190 8.54888 13.07078 10.84993 19.01809 95 5-36850 3.18038 4.52190 5.11950 8.16816 96 2.16684 i-oi354 I-34I52 2.07420 304866 97 0.73915 0.27439 0.32798 0.70962 0.97446 98 0.22080 0-05359 0.05350 0.21281 o 26484 99 0-05359 0.00000 o.ooooo 0.05203 0.05203 332 ACTUARIES' TABLE. TABLE V. ACTUARIES' TABLE. VALUATION COLUMNS. Age. THREE PER CENT. THREE AND ONE-HALF PER CENT. FOUR PER CENT. Ux. Cx. Ux. c x . Ux- Cx. 10 1.03701 .006563 1.04204 .006531 1.04708 .006500 11 1.03704 .006588 1.04207 .006556 1.04710 .006525 12 .03706 .OO66I3 1.04210 .006582 1.04/13 .006550 13 .03710 .006664 1.04214 .006617 1.04717 .006585 14 03715 .006695 1.04219 .000663 1.04722 .006630 15 .03720 .006741 1.04224 .006709 1.04727 .006677 16 .03726 .006799 1.04230 .006766 i 04733 .006733 17 .03732 .006857 1.04236 .006824 1.04740 .006791 18 1.03740 .006926 1.04244 .006892 1.04747 .006860 19 1.03748 .006996 1.04251 .006963 1-04755 .006929 20 1.03756 .007078 1.04260 .007044 1.04764 .OO7OIO 21 1.03765 .007162 1.04269 .007127 1.04773 .007093 22 1.03775 .007247 1.04278 .007212 1.04782 .007177 23 I-03785 007344 1.04289 .007308 1.04793 .007273 24 1.03796 .007443 1.04300 .007407 1.04803 .007371 25 1.03807 .007544 1.04310 .007507 1.04814 .007471 26 1.03819 .007657 i-043 2 3 .007620 1.04827 .007583 27 1.03831 .007773 1-04335 007735 1.04839 .007698 28 1.03845 .007902 1.04349 .007864 1.048=54 .007826 29 1.03859 .008034 1-04364 .007995 1.04868 007956 30 1.03875 008l79 1.04379 .008110 1.04884 .008101 31 1.03891 .008329 1.04396 .008288 1.04900 .008248 32 1.03909 .008492 1-04413 .008451 1.04918 .008410 33 1.03927 .008659 1.04431 .008617 1.04936 .008576 34 1-03945 .008831 1.04450 .008788 1-04955 .008746 35 1.03966 .009017 1.04470 .008974 1.04975 .008Q3I 36 1.03986 .009209 1.04491 .009164 1.04996 .009122 37 1.04007 .009405 1.04512 .009359 1.05017 .009314 38 1.04031 .009618 1.04536 009571 1.05040 .009525 39 1.04054 .009836 1-04559 .009788 1.05064 .009741 40 1.04078 .010060 1.04584 .010011 1.05089 .009963 41 1.04105 .010303 1.04610 .010253 1.05116 .OIO2O4 42 1.04134 .010577 1.04640 .010526 1.05146 .OIO476 43 1.04172 .010923 1.04678 .010870 1.05183 .OI08l8 44 1.04219 .011357 1.04725 .011302 1.05231 .011247 45 1.04273 .011856 1.04779 .011799 1.05286 .011742 46 1-04339 .012465 1.04846 .012405 1-05353 .012345 47 1.04411 .OT3T22 1.04918 .013059 1.05425 .OI2996 48 1.04490 .013844 1.04997 .013777 1.05504 .013711 49 1.04575 .014622 1.05083 .014552 1.05590 .014482 50 51 1.04668 1.04770 .015474 .016406 1.05176 1.05279 .015400 .016327 1.05684 1.05788 .015326 .016248 52 1.04882 .017425 I-0539I .017340 1.05901 .017257 53 1.05005 018537 I.055I5 .018447 1.06024. .018359 54 1.05136 .OI9722 1.05646 .019626 1.06156 .019532 ACTUARIES' TABLE. 333 TABLE V.-Cont. ACTUARIES' TABLE. VALUATION COLUMNS. Age. THREE PER CENT. THREE AND ONE-HALF PER CENT. FOUR PHR CENT. u*. c. Ux- Cx u x . c x . 55 1.05281 .021033 i 05792 .020932 1.06303 .020831 56 1.05438 .022453 105950 .022344 1.06462 .022237 57 1.05606 .023960 1.06119 .023845 1.06632 .023730 58 105791 .025617 1.06305 .025493 1.06819 .025371 59 1.05994 .027423 1.06508 .027291 1.07023 .027160 60 1.06222 .029453 1.06738 029310 1.07254 .029169 61 1.06472 .031661 1.06989 .031509 1.07506 031357 62 1.06749 .034098 1.07267 1.07786 .033770 63 I 07051 .036738 1.07570 .036560 1.08090 .036384 64 1.07384 .039636 1.07905 309445 1.08427 039255 65 1.07750 .042798 1.08273 .042591 1.08796 .042386 66 1.08149 .046227 1.08674 .046004 1.09199 .045782 67 1.08589 049975 1.09117 049733 1.09644 .049494 68 1.09067 .054010 1.09597 053749 1.10126 053490 69 1.09585 .058337 1.10117 .058055 1.10648 057776 70 I IOI53 .063042 i 10687 .062737 1. 1 1222 .062436 71 1.10772 .068115 1.11309 .067786 1.11847 067460 72 1.11448 073597 1.11989 .073241 I.I2530 .072889 73 I.I2I86 .079498 1.12731 .079114 I.I3275 .078734 74 1.12997 .085891 i-i3545 .085476 I.I4093 .085065 75 1.13883 .092777 i.i4435 .092329 I.I4988 .091885 76" 1.14850 .100174 1.15408 .099690 I-I5965 .099211 77 1.15922 .108222 1.16484 . 107700 I.I7047 .107182 78 1.17105 .116936 1.17673 .116371 1.18242 .115812 79 1.18400 .126276 1.18974 .125666 I.I9549 .125062 80 1.19824 .136317 1.20406 135658 1.20987 .135006 81 1.21381 .147025 1.21971 .146315 1.22560 .145611 82 1.23087 .158441 1.23684 .157671 1.24282 .156917 83 1.24987 .170788 1-25593 .169961 1.26200 .169146 84 1.26890 .184153 1.27727 .183256 1.28344 .182383 85 I.2957S .199121 1.30204 .198159 1.30833 .197207 86 1.32472 .216000 1.33116 .214956 1-33759 .213923 87 1.35926 .235179 1.36585 .234042 1.37246 .232917 88 1.40188 .257548 1.40869 .256303 I.4I549 .255071 89 1.46643 .283866 1.46265 .282495 1.46972 281137 90 1.52306 .3I43 I 1.53045 .312782 1.53785 .311279 91 1. 61186 .350473 i 61968 .348780 I.6275I .347103 92 1.73186 393459 i 74027 .391558 1.74867 .389676 93 1.89766 .443910 1.90687 .441765 1.91609 .439641 94 2.12944 .501266 2.13981 .498841 2.I5OII .496446 95 2-47757 .567252 2.48959 .5645 1 I 2 50162 .561798 96 2.93153 .629756 2.94579 .626714 2-95999 .623701 97 3-34760 .672144 3-36383 .668897 3-37999 .665680 98 4.12017 .728155 4.14014 .724638 4-15999 721154 99 .970874 .966183 .961538 INTRODUCTION TO CRAIG'S MODIFIED ACTU- ARIES' TABLE FOR IMPAIRED LIVES. No experience table for impaired or sub-standard lives being available, James M. Craig, actuary of the Metropoli- tan Life Insurance Company, has modified the Actuaries' Table so as to answer in his judgment for impaired life purposes. The modification consists in doubling the rate of mortality to age sixty, after which it gradually declines toward the Actuaries' Table ratios. Mr. Craig developed the table, with commutation columns at four per cent, on this basis and published it in the papers of the Actuarial Society of America, reporting the pro- ceedings of the Society at its meeting on October 23 and 24, 1890. We do not understand that this table is in actual use. CRAIG'S IMPAIRED TABLE. 337 TABLE No. I. CRAIG'S MODIFIED ACTUARIES' TABLE FOR IMPAIRED LIVES. AGE. Number Living. Number Dying. . Deaths Per 1000. AGE. Number Living. Number Dying. Deaths Per 1000 10 100,000 1352 I3-520 55 40,071 1736 43-328 11 12 98,648 97,309 1339 1326 I3-572 13.624 56 57 36^62 1773 1805 46-252 49-358 13 95-983 1315 I3-696 58 34,757 X ol 4 52.772 14 94,668 1306 13.792 59 32,923 1860 56.492 15 93,362 1297 13.888 60 31,063 1847 59-458 16 92,065 1289 14.006 61 29,216 1829 62.615 17 00,776 1282 14.124 62 27,387 1808 66.027 18 8q,494 1277 14 268 63 25,579 1781 69.626 19 88,217 1271 14.412 64 23,798 1749 73487 20 86,946 1268 14.582 65 22,049 1711 77-584 21 22 85,678 84,414 1264 1260 14754 14.928 66 67 20,338 18,673 1665 1614 86^76 23 83.154 1258 15.128 68 17,059 1556 9I-233 24 81,896 1256 15332 69 15,503 1490 96.139 25 80,640 1253 15.540 70 14,013 1419 101.295 26 79>3&7 1252 15-774 71 12,594 1343 106.640 27 78,135 1251 16.012 72 11,251 1262 112.191 28 76,884 1252 16.278 73 9,989 1177 117.912 29 75.632 1252 16.550 74 8,812 1091 123.855 30 74.38o 1253 16.850 75 7,721 1003 129.962 31 73. 12 7 1255 17.156 76 6,718 914 136,198 32 71,872 1257 17.494 77 5,804 828 142.680 83 70,615 1260 17.838 78 4,976 743 I49'350 34 69.355 1262 18.190 79 4,233 661 156.078 35 68,093 1266 18.576 80 3-572 58i 162.871 86 66,827 1268 18 970 81 2,991 507 169.608 37 65,559 1270 19-374 82 2,484 437 176.250 38 64,289 1274 19.812 83 2,047 374 182.950 39 63,015 . 1277 20.262 84 1,673 317 189.678 40 61,738 1279 20.724 85 1.356 278 205.095 41 60,459 1283 21.224 86 1,078 240 222.480 42 59-176 1289 21.788 87 838 202 242.234 43 57,887 1303 22.502 88 636 168 265.274 44 56,584 1324 23.394 89 468 136 292.382 45 5^,260 1350 24.424 90 332 107 322.892 46 53,910 1384 25-678 91 225 81 360.000 47 52,526 1420 27.032 92 144 58 402.778 48 51,106 1458 28.520 93 86 39 453-488 49 49,648 1495 30.122 94 47 24 510.638 50 48,153 1535 31.876 95 23 13 565.217 51 46,618 1575 33-796 96 10 6 600.000 52 45,043 1617 35.894 97 4 3 750.000 53 43426 1658 38.186 98 i i 1000.000 54 41,768 1697 40.626 338 CRAIG'S IMPAIRED TABLE. TABLE II. COMMUTATION COLUMNS, CRAIG'S MODIFIED ACTUARIES' TABLES. FOUR PER CENT. Age. D x . N x . Cx. M x . 10 67,556.40 1,137,420.01 878.2322 21,211.1591 11 64,079.77 I >73-340.24 8363354 20,332.9269 12 60,778.91 1,012.561.33 796.3611 19,496.5915 13 57,644.89 954,91644 759.3796 18,700.2304 14 54,668.40 900,248.04 725.1761 17,940.8508 15 51,840 65 848,407.39 692.4787 17,215 6747 16 17 49.I54-24 46,601.95 799,253.15 752,651.20 661.7378 632.8311 16,523.1960 15,861.4582 18 44,176.74 708,474.46 606.1178 15,228.6271 19 41,871.49 666,602.97 580.0679 14,622.5093 20 39.68l.02 626,921.95 556.4415 14,042.4414 21 37,598-42 589,323.53 533-35" i3,485-9999 22 35,618.91 553,704.62 511.2148 12,952.6488 23 33-737-74 519,966.88 490.7722 12,441.4340 24 31,949.35 488,017.53 471.1470 11,950.6618 25 30,249.43 457,768.10 45L9433 11,479.5148 26 28,634 O2 429,134.08 434-2149 11,027.5715 27 27,098.55 402,035.53 417.1797 io,593-3566 28 25,639-05 376,396.48 401.4551 10,176.1769 29 24,251.48 352,145.00 386.0154 9,774.7218 30 22,932.77 329,212.23 371.4644 9,388.7064 31 21,67923 307,533-oo 357-7478 9,017.2420 32 20,487 69 287,045.31 344.5362 8,659.4942 33 J 9.355 15 267,690.16 332.0755 8,314.9580 34 18,278.65 249,411.51 319.8097 7,982.8825 35 36 17,255.79 16,283.67 232,155.72 215,872.05 308.4850 297.0886 7,663.0728 7,354-5878 37 15,360.28 200,511 77 286.1120 7,057.4992 38 IM83-35 186,028.42 275.9752 6,771-3872 39 13,65037 172,378.05 265.9851 6,495.4120 40 12,859.35 159,518.70 256.1556 6,229.4269 41 12,108.61 147,410.09 247.0737 5,973.2713 42 11,395.82 136,014 27 238.6816 5,726.1976 43 10,718.82 125,295.45 23L9939 ^,487.5160 44 10,074.55 115,220 90 226.6662 5,255.5221 45 9,460.401 105,760.50 222.2289 5,028.8559 46 8,874.341 96,886.154 219 0637 4,806 6270 47 8,3i3-973 88,572.181 216.1169 4,587.5633 48 7,778.078 80,794.103 213.3652 4,371.4464 49 7,265.538 73,528.565 210.3659 4,158.0812 50 6,775-753 66,752.812 207.6870 3.947.7I53 51 6,^07.462 60,445.350 204.9028 3,740.0283 52 5,859 959 54,585-39* 202 2754 3-535-1255 53 5,432.289 49,153.102 199.4276 3,3328501 54 5,023.939 44,129.163 196.2682 3,133.4225 CRAIG'S IMPAIRED TABLE. TABLE II. Cont. 339 COMMUTATION COLUMNS, CRAIG'S MODIFIED ACTUARIES- TABLES, FOUR PER CENT. Age. D x . NX. Cx. iVi x . 55 4,634.452 39,494.711 193-0554 2,937.1543 56 4,263.120 35,231.591 189.5869 2,744.0989 57 3,909.575 31,322.016 185.5847 2,554.5120 58 3.573.610 27,748.406 181.3147 2,368 9273 59 3,254.867 24.493-539 176.8116 2,187.6126 60 2,952.849 21,540.690 168.8232 2,010.8010 61 2,670.459 18,870.231 160.7490 1,8419778 62 2,407.016 16,463.215 152.7905 1,681.2288 63 2,161.630 14,301.585 144.7205 1,528.4383 64 I,933-778 12,367.807 136.6546 1,383.7178 65 1,722.755 10,645.052 128.5440 1,247.0632 66 1-527.953 9,117.0987 120.2763 1,118.5192 67 1,348.900 7,768.1987 112.1084 998.2429 68 1,184.918 6,583.2807 103.9221 886.1345 69 1,035.414 5,547.8667 95-6863 782.2124 70 899.9008 4,647.9659 87.6218 686.5261 71 72 777.6659 668.0169 3,870.3000 3,202.2831 79-7393 72.0488 598.9043 519.1650 73 570.2820 2,632,0011 64.6114 447.1162 74 483.7347 2,148.2664 57.5873 382.5048 75 407.5453 1,740.7211 50.9063 324.9175 76 340.9654 1,399-7557 44.6041 274.0112 77 283.2410 1,116.5147 38-8531 229.4071 78 233-493 8 883.0209 33-5242 190.5540 79 190.9930 692.0279 28.6768 157.0298 80 154.9676 537.0603 24.2370 128.3530 81 124.7726 412.2877 20.3363 104 1 1 60 82 99-6357 312.6520 16.8547 83-7797 83 78.9507 233-7013 13 8698 66.9250 84 62.0432 171.6581 11.3039 53.0552 85 48.3536 123.3045 9.53179 41.75126 86 36.9614 86.34306 7.91256 32.21947 87 27.6280 58.71506 6.40360 24.30691 88 20.16 i 8 38.55326 5.12081 17.90331 89 14.2651 24.28816 3.98602 12.78250 90 9-7359 14-55757 3- OI 547 8.79648 91 6.34095 8.216624 2.19494 5.78101 92 3.90211 4-3I45I4 1-51125 358607 93 2.24082 2.073694 .977067 2.07482 94 1.17749 .896204 .578160 1.09775 95 554070 .342134 .301119 .519591 96 .231630 .110504 .133632 .218472 97 .089088 .021416 .064248 .084840 98 .02:416 .000000 .020592 .020592 NORTHAMPTON TABLE OF MORTALITY. THE most commonly employed of the mortality tables embracing unselected lives has been the Northampton, adapted from mortality statistics of the English town of that name by Dr. Price. It is now used only for computa- tions involving impaired lives or in computing the values of annuity interests, dowers, etc., in the courts. For the latter purpose it is the standard in New York at five per cent interest and in many other States at six per cent or higher. Therefore we depart from our custom and give annuity tables at from three per cent to eight per cent interest, as well as commutation tables at three per cent and four per cent. Joint-life annuities at several rates of interest are also given. These will be found useful to attorneys in computing values of life remainders, etc. We also give annuity commutation tables at five per cent, computed especially for this work by Mr. G. L. Plumley, assistant actuary of the Home Life Insurance Company of New York, and kindly contributed by him. The commutation table at four per cent is furnished by the kindness of Samuel C. Beckley, superintendent of Actuarial Department, New York Life Insurance Com- pany, having been computed under his supervision for the use of the office. The three per cent commutation tables and annuity tables are from ' * David Jones. " NORTHAMPTON TABLE. 343 TABLE I. NORTHAMPTON TABLE OF MORTALITY. i! M Living. Decre- ments. Propor- tion Which Die. Propor- tion Which Survive. 1 Living. Decre- ments. Propor- tion Which Die. Propor- tion Which Survive. o 11,650 3,000 .2575" .742489 49 2,936 79 .026908 .973092 1 8,650 1-367 158035 .841965 50 2,857 81 .028351 .971649 2 7,283 502 .068928 .931072 51 2,776 82 029539 .970461 3 6,781 335 .049403 950597 52 2,694 82 .030438 .969562 4 6,446 197 .030562 .969438 53 2,612 82 .031394 .<,686o6 5 6,249 184 .029445 970555 54 2,530 82 .032411 957589 6 6,065 140 .023084 .976916 55 2,448 82 033497 .-.66503 7 5,925 no .018565 981435 56 ' 2,366 82 .034658 965342 8 5.815 80 013757 .986243 57 2,284 82 .035902 .964098 9 5.735 60 .010462 989538 58 2,202 82 .037239 .962761 10 5.675 52 .009163 .990837 59 2,120 82 038679 .961321 11 5.623 50 .008892 .991108 60 2,038 82 .040235 959765 12 5-573 50 .008972 .991028 61 ,956 82 .041922 .958078 13 5,523 So .009053 .990947 62 ,874 81 .043223 956777 14 5,473 So .009136 .990864 63 ,793 81 .045176 .954824 15 5.423 So .009220 .990780 64 ,712 80 .046729 953271 16 5,373 .009864 .990136 65 ,632 80 .049020 .950980 17 5-320 58 .010902 .989098 66 ,552 80 .051546 .948454 18 5,262 63 .011972 .988028 67 ,472 80 .054348 945652 19 5-199 67 .012887 .987113 68 ,392 80 .057471 942529 20 5,132 72 .014030 .985970 69 ,312 80 .060975 .939025 21 5,060 75 .014822 .985178 70 ,232 80 064935 .935065 22 4,985 75 OI504S 984955 71 ,152 80 .069444 .930556 23 4,910 75 015275 .984725 72 ,072 80 .074627 925373 24 25 4.835 4,76o 75 75 015512 .015756 .984488 .984244 73 74 992 912 80 80 .080645 .087719 .919355 .912281 26 4-685 75 .016009 .983991 75 832 80 096154 .903846 27 28 29 4,610 4-535 4.460 75 75 75 .016269 .016538 .Ol68l6 983731 .983462 .983184 76 77 78 752 77 11 .102393 .108148 .112957 .897607 .891852 887043 30 4,385 75 .017104 .982896 79 534 65 .121723 .878277 81 4-310 75 .017401 .982599 80 469 63 .134328 .865672 32 4,235 75 .017710 .982290 81 406 60 147783 .852217 33 4,160 75 .018029 .981971 82 346 57 .164740 .835260 34 35 4,085 4,010 75 75 .018360 .018704 -981640 981296 83 84 289 234 3 .190311 .205128 .809689 .794872 36 3,935 75 .019060 .980940 85 186 41 .220430 779570 37 3,86o 75 019430 .980570 86 145 34 .234483 .765517 38 3.785 75 .019815 .980185 87 in 28 252252 .747748 39 75 .020216 .979784 88 83 21 .253012 .746988 40 3,635 76 .020908 .979092 89 62 16 .258065 741935 41 3,559 77 .021635 978365 90 46 12 .260869 .739131 42 3,482 78 .022401 977599 91 34 10 .294118 .705882 43 3,404 78 .022914 .977086 92 24 8 333333 .666667 44 3,326 78 .023452 976548 93 16 7 437500 .562500 45 3,248 78 .024015 975985 94 9 5 .555556 .444444 46 3-170 78 .024606 975394 95 4 3 .750000 .250000 47 3.092 78 .025227 974773 96 i i 48 3-014 78 .025879 .974121 344 NORTHAMPTON TABLE. TABLE II. NORTHAMPTON, FOUR PER CENT. Age. Dx N x Sx o 11650. 120320.4448446 2010147 3392293 1 8317.3076800 112003.1371646 1889826.8943847 2 6733.5428774 105269.5942872 1777823.7572201 3 6028.2843172 99241.3099700 1672554.1629329 4 5510.0678087 93731.2421613 1573312.8529629 5 5136.2225104 88595 0196509 1479581.6108016 6 4793-257634S 83801.7620164 1390986.5911507 7 4502.5130253 79299.2489911 1307184.8291343 g 4248.9635712 75050.2854199 1227885.5701432 9 4029.3349539 71020 9504660 1152835.2947233 10 11 3833.8266658 3652.5935694 67187.1238002 63534-5302308 1081814.3442573 1014627.2204571 12 3480.8793597 60053.6508711 951092.6902263 13 3316.9706991 56730.6801720 891039.0393552 14 3160.5211128 53576 1590592 834302.3591832 15 3011.1993835 50564 9596757 780726.2001240 16 2868.6886511 47696.2710246 730161.2404483 17 18 2731.1456900 2597.4711674 44965.1253346 42367.6541672 682464.9694237 637499.8440891 19 2467.6659416 39899.9882256 595132.1899219 20 2342.1778274 37557-8I03982 555232.2016963 21 2220.4980160 3S337.3I2322 517674.3912981 22 2103.4476192 32233.8647630 482337.0789159 23 1992.1162803 31241.7484827 450103.2141529 24 1886.2373075 2 9355 5III752' 418861.4656702 25 1785.5559680 27569-9532072 389505.9544950 26 1689.8290426 25880.1261646 361936.0012878 27 28 29 1598 8243877 1512.3203265 1430. 1052886 24281.3017769 22768.9814504 21338.8761618 336055.8751232 311774.5733463 289005.5918959 30 I35I-9773680 19996.8987938 267666.7157341 31 1277.7437206 18709.1550732 247669.8169403 32 1207.2203759 17501.9346973 228960.6618671 33 1140.2317472 16361.7029501 211458.7271698 34 1076.6102877 15285.0926624 195097.0242197 35 1016.1960347 14268 8966277 179811.9315573 36 958.8364132 13310.0602145 165543.0349296 37 904.3858410 12405 6743735 152232.9747151 38 852.7053526 11552 9690209 139827.3003416 39 803.6624631 10749.3065578 128274.3313207 40 757.1306604 9992.1758974 117525.0247629 41 712.7891529 9279.3867445 107532.8488655 42 670.5459063 8608.8408382 98253.4621210 43 630.3125528 7978.5282854 89644 6212828 44 45 592.1821601 556.0524357 7386.3461253 6830.2936896 81666.0929974 74279.7468721 46 521.8259362 6308.4677534 67449-453I825 47 489.4096755 5819.0580779 61140.9854291 48 458.7150066 5360.3430713 55321.9273512 49 429.6575283 4930.6855430 49901.5842799 NORTHAMPTON TABLE. 345 TABLE II. Cont. COMMUTATION TABLE, NORTHAMPTON, FOUR PER CENT. Age. C, M, R. 2884.6153800 6574.2136525 49543.2222343 1 1263.8683391 3689.5982725 42969.0085618 2 446.2761727 2425.7299334 39279.4103093 3 4 286 3594037 161.9196407 I979-4537607 1693.0943570 36853.6803759 34874.2266152 5 145.4178735 I53I.I747I63 33181.1322582 6 106.3884934 1385 7568428 31649.9575419 7 80.3759231 1279.3683494 30264.2006991 8 56.2069392 1198.9924263 29984.8323497 9 40-5338502 1142.7854871 27785.8399234 10 11 33.7782084 31 2298526 II02.25I6369 1068.4734285 26643 0544363 25540.8027994 12 30.0287045 1037.2435759 24472.3293709 13 28.8737540 I007.2I487I4 234350857950 14 27.7632250 978.34III74 22427.8709236 15 26.6954090 950.5778924 21449.5298062 16 27.2087823 923.8824834 20498.9519138 17 28 6304310 896.673701! 19575-0694304 18 29.9024725 868.0432701 18678.3957293 19 30.5779257 838.1407976 17810.3524592 20 31.5960192 807.5628719 16972.2116616 21 31.6466543 775.9668527 16164.6487897 22 30.4294748 744.3201984 15388.6819370 23 29.2591103 713.8907236 14644.3617386 24 28.1337600 684.63X6133 13930-4710150 25 27.0516923 656.4978533 13245.8394017 26 26.0112428 629.4461610 12589.3415484 27 28 25.0108103 24.0488558 603.4349182 578.4241079 II959-8953874 11356.4604692 29 23.1239003 5543752521 10778.0363613 30 22.2345195 53I.25I35I8 10223.6611092 31 32 21-3793455 20.5570628 509.0168323 487.6374868 9692.4097574 9183.3929251 33 19.7664068 467.0804240 8695.7554383 34 19.0061603 447.3140172 8228.6750143 35 18.2751540 428.3078569 7781.3609971 36 17.5722638 410.0327029 7353.0531402 37 16.8964073 392 4604391 6943.0204373 38 16.2465458 375.5640318 6550.5599982 39 15.6216781 359-3I74860 6I74-9959664 40 15.2211227 343.6958079 5815.6784804 41 14.8282696 328.4746852 5471.9826725 42 14.4431196 313.6464156 5I43-5079873 43 13.8876153 209.2032960 4829.8615717 44 13 3534760 285.3156807 4530-6582757 45 12 8398811 271.9622047 4249.3425950 46 12.3460397 259.1223236 3973.3803903 47 II.87H9I2 246.7762839 3714.2580667 48 11.4140074 234.9050927 3467.4817828 49 II.II62970 223.4904853 3232.5766901 346 NORTHAMPTON TABLE. TABLE U.Cont. NORTHAMPTON, FOUR PER CENT. Age. Dx NX Sz 50 402.0159553 4528.6695877 45030.8987369 51 375-5944378 4153-0751499 40502.2291492 52 350.4805637 3802.5945862 36349.1539993 53 54 55 326.7429160 304.3127769 283.1246885 3475 8516702 3171 5388933 2888.4142048 32546.5594131 29070.7077429 25899.1688496 56 263.1162825 2625.2979223 23010 7546448 57 244.2281657 2381.0697566 20385.4567225 58 226.4037607 2154.6659959 18004.3869659 59 209.5891784 1945.0768175 15849.7209700 60 I93-7330952 I75I-3437223 13904.6441525 61 178.7866739 1572.5570484 12153.3004302 62 164.7033863 1407.8536621 10580.7433818 63 151.5234716 1256.3301905 9172.8897197 64 65 139 "37474 1275125990 1117.2164431 989.7038441 7916.5595292 6799-3430861 66 116.5980662 873-1057779 5809.6392420 67 106.3344685 766.7713094 4936.5334641 68 96.6879024 670.0834070 4169.7621547 69 87.6260922 582.4573148 3499.6787477 70 79.1183008 5Q3-339 OI 4o 2917.2214329 71 7i- I3533I 8 432.2036822 2 113. 8824189 72 03 6494104 368.5542718 1981.6787367 73 56.6340835 311.9201883 1613.1244649 74 50.0642491 261.8559392 1301.2042766 75 76 43.9160134 38.1666546 217.9399258 I79-77327I2 1039-3483374 821.4084116 77 32.9409923 146.8322789 641.6351404 78 28.2485430 "8.5837359 494.8028615 79 24.0939198 94.4898161 376.2191256 80 20 3472508 74-1425653 281.7293095 81 16.9365742 57.2059911 207.5867442 82 13 8784925 43.3274986 150.3807531 83 11.1462994 32.1811992 107.0532545 84 8.6779134 23-5032858 74.8720553 85 6.6325275 16.8707583 51.3687695 86 4.9716527 11.8991056 34.4980112 87 3 6595057 8.2395999 22.5989056 88 2.6311415 5.6084584 14-3593057 89 1.8898376 3.7186208 8.7508473 90 91 1.3482094 0.9581754 2.3704114 1.4122360 92 0.6503453 .7618907 1.2495791 93 0.4168880 3450027 .4876884 94 0.2254803 .1195224 .1426857 95 0.0963591 .0231633 .0231633 96 0.0231633 .0000000 .0000000 NORTHAMPTON TABLE. 347 TABLE II. Cont. COMMUTATION TABLE, NORTHAMPTON, FOUR PER CENT. Age. Cx M. Ri 50 10.9593478 212.3741883 3009.0862048 51 10.6679310 201.4148405 2796.7120165 52 10.2576260 190.7469095 2595.2971760 53 9.8631019 180.4892835 2404.5502665 54 9.4837518 170.6261816 2224.0609830 55 9.1189920 161.1424298 2053.4348014 56 8.7682616 152.0234378 1892.2923716 57 58 8.4310211 8 1067512 143.2551762 134.8241551 1740.2689338 1597.0137576 59 7.7949528 12 6. 7174039 1462.1896025 60 7.4951469 118.9224511 1335.4721986 61 7.2068718 111.4273042 1216.5497475 62 6 8451764 104.2204324 1105.1224433 63 6.5819004 97-375256o 1000.9020109 64 6.2506176 90-7933556 903.5267549 65 6.0102096 84.5427380 812-7333993 66 5.7790472 78.5325284 728.1906613 67 5.5567760 72.7534812 649.6581329, 68 5.3430544 67.1967052 576.9046517 69 5.1375520 61.8536508 509.7079465 70 4-9399536 56.7160988 447.8542957 71 4.7499560 51.7761452 391.1381969 72 4.5672648 47.0261892 339.3620517 73 4.3916008 42.4589244 292.3358625 74 75 4.2226936 4.0602824 38.0673236 33.8446300 249.8769381 211.8096145 76 3-7577I3 2 29.7843476 177.9649845 77 3.4254878 26.0266344 148.1806369 78 3.0681396 22.6011466 122 1540025 79 2.8199816 i9-5337o 99.5528559 80 2.6280891 16.7130254 80.0198489 81 2.4066751 14.0849363 63.3068235 82 2.1984051 11.6782612 49.2218872 83 2.0396805 9.4798561 37.5436260 84 1.7116200 7.4401756 28.0637699 85 1.4057777 5.7285556 20.6235943 86 1.1209297 4 3227779 14.8950387 87 .8876140 3.2018482 10.5722608 88 .6401063 2.3142342 7.3704126 89 .4689424 1.6741279 5.0561784 90 .3381796 1.2051855 3.3820505 91 .2709772 .8670059 2.1768650 92 .2084440 .5960287 1.309859! 93 1753736 .3875847 .7138304 94 .1204489 .2122111 .3262457 95 .0694898 .0917622 .1140346 96 .0222724 .0222724 .0222724 348 NORTHAMPTON TABLE. TABLE III. COMMUTATION COLUMNS, NORTHAMPTON TABLE, THREE PER CENT. Age. P. N. S. M. R. 1 11650.000 8398.058 142947.351 134549.293 2719587-3 2576639.9 7147.166 4234.544 70883.26 6373609 2 6864.926 127684.367 2442090.7 2946.015 59501-55 3 6205.576 121478.791 2314406.3 2486.614 56555-53 4 5727.188 115751.604 2192927.5 2188.971 54068.92 5 5390.442 110361.161 2077175-9 2019.037 5I879-95 6 5079-342 105281.819 1966814.7 1864.940 49860.91 7 4817.567 100464.252 1861532.9 1751.107 47995-97 8 4590415 95873.837 1761068.7 1664.272 46244.87 9 4395.400 91478.437 1665194.8 1602.958 44580.59 10 11 4222.733 4062.175 87255.705 83I93-530 I5737I6.4 1486460.7 IS58.3I3 1520.747 42977.64 41419.32 12 3908.790 79284.740 1403267.2 1485.678 39898.58 13 3760.894 75523-846 1323982.4 1451.630 38412.90 14 3618.298 71905.548 1248458.6 1418.574 36961.27 15 3480.817 68424.731 II76553-0 1386.481 35542.69 16 3348.276 65076.455 1108128.3 I355-323 34156.21 17 3218.688 61857.767 1043051.8 I323-257 32800.89 18 3000.870 58766.897 981194.1 1289.188 31477.63 19 2964.917 55801.980 922427.2 1253.260 30188.44 20 2841.464 52960.516 866625.2 1216.164 28935.18 21 2719 999 50240.516 813664.7 1177.460 27719.02 22 2601.634 47638.882 763424.2 1138.318 26541.56 23 2487.856 45151.026 715785-3 1100.317 25403 24 24 2378.500 42772.526 1063.422 24302 93 25 2273 402 40499.124 627861.7 1027.601 23239.50 26 2172.410 38326.714 587362.6 992.8241 22211.90 27 2075.372 36251.343 5490359 959.0599 21219.08 28 1982.143 34269.199 512784.5 926.2791 20260.02 29 1892.585 32376.615 4785I5-3 894-4S3I 19333-74 30 1806.562 30570.053 446138.7 863.5541 18439.29 31 32 I723-945 1644.607 28846. 108 27201.501 415568.7 386722.6 833-555I 804.4298 17575-73 16742.18 33 1568.429 25633-072 35952I.I 7761528 15937-75 34 1495.293 24I37.779 333888 o 748.6994 15161.59 35 1425.087 22712.691 309750.2 722.0456 ! 14412.90 36 I357.703 21354.988 287037.5 696.1682 13690.85 37 1293.034 20061.954 2656825 671.0445 i 12994.68 38 1230.981 18830.973 245620.6 646.6525 12323.64 39 1171.446 17659.528 226789.6 622.9710 11676.98 40 "14-334 16545.194 209130.1 599.9792 11054.01 41 1059.258 15485.936 192584.9 577-3595 1045403 42 1006.156 14479.780 177099.0 555.1096 9876.67 43 954.968 13524.811 162619.2 533-2273 9321.57 44 905.008 12618.903 149094.4 511.9823 8788.34 45 858.897 11760.007 I3647S.5 8276.36 46 813.855 10946.152 I247I5.5 47L3307 7785.00 47 770.708 10175.444 113769.3 451.8885 7313-67 NORTHAMPTON TABLE. 349 TABLE III. Cont. COMMUTATION COLUMNS, NORTHAMPTON TABLE, THREE PER CENT. Age. D. N. S. M. R. 48 49 729.384 689.814 9446.059 8756.245 103593-9 94147.80 433.0126 414.6865 6861.780 6428.768 50 651.702 8104.543 8539I-56 396.6660 6014.081 51 614,782 7489-762 77287.01 378.7275 5617.415 52 579244 6910.517 69797.25 361.0965 5238.688 53 6365.262 62886.73 343.9790 4877.59I 54 512.756 5852.506 56521.47 327.3600 4533-612 55 481.686 5370.820 50668.97 311-2251 4206.252 56 451.991 4918.829 45298.15 295.5601 3895.027 57 423.618 4495-2H 40379.32 280.3512 3599.467 58 396.514 4098.697 35884.11 265.5855 3319.116 59 60 370.629 345.9I6 3728.068 3382.152 31785-41 28057.34 251.2498 237.3317 3053-530 2802.281 61 322.328 3059.824 24675.19 223.8190 2564.949 62 299.821 2760.004 21615.37 210.6998 2341-130 63 278.506 2481.497 18855.36 198.1180 2130.430 64 258.179 2223.217 16373.87 185.9027 1932.312 65 238.946 1984371 14150.55 174.1896 1746.409 66 220.615 1763-756 12166.18 162.8177 1572.220 67 203.149 1560.608 10402.42 151.7770 1409.402 68 186.512 1374.095 8841.81 141.0579 1257-625 69 170.673 1203.422 7467.72 130.6510 1116.567 70 I55-598 1047.824 6264.30 120.5472 985.916 71 141.257 906.5667 5216.47 110.7377 865.369 72 127.619 778.9479 4309.91 101.2139 754.631 73 74 "4655 102.339 664 2927 56i.954o 3530.96 2866.67 91 9675 82.9904 653.417 561.450 75 90.6425 2304.71 74.2748 478.460 76 79.5405 391.7710 1833.40 65.8130 404.185 77 69.3167 322.4543 1441-63 57.9058 338.372 78 60.0196 262.4347 1119.18 50.6277 280.466 79 51.6893 210.7454 856.74 44-0455 229.838 80 44.0751 166.6703 646.00 37.9370 185.793 81 37-0434 129.6269 479-33 32.1889 147.856 82 30.6495 98.9774 349-70 26.8740 115.667 83 24.8547 74.1227 250.72 21.9718 88.793 84 I9-5384 54-5843 176.60 17.3795 66.821 85 ' 15.0781 39.5062 122.01 13.4883 49.442 86 11.4121 28.0941 82.51 10.2614 35-953 87 8.4817 19.6124 54-41 7.66344 25.692 88 6.1574 I3-4550 34.80 5-58622 18.029 89 4.4655 8.9895 21-35 4.07368 12442 90 3.2166 5-7729 12.36 2.95484 8.369 91 2.3083 3.4846 6.584 2.14015 5.414 92 I.58I9 1.8827 3.100 1.48101 3-274 93 1.0238 8589 I.2I7 .969059 1-793 94 5591 .2998 3583 .534150 .8236 95 96 .2413 0585 0585 .0000 .0585 .0000 .232548 .056858 .2894 .0569 35 NORTHAMPTON TABLE. TABLE II. NORTHAMPTON TABLE, COMMUTATION COLUMNS, FIVE PER CENT. (Annuities only.) Age. Dx N a Age. D, N< 1 8238.0952 95264.0796 49 268.8316 2807.6873 2 6605.8957 88658.1839 50 249.1411 2558.5462 3 5857.6828 82800.5011 51 230.5501 2327.9962 4 5303.1401 77497-36io 52 213.0856 2114.9106 5 4896.2550 72601.1060 53 196.7616 1918.1490 6 4525.7964 68075.3096 54 181.5091 1736.6399 7 4210.7869 63864.5227 55 167.2630 1569.3769 g 3935.8209 59928.7018 56 153.9622 1415.4147 9 3696.8322 56231.8696 57 141.5488 1273.8660 10 3483-9577 52747.9119 58 129.9685 "43 8975 11 3287.6516 49460.2603 59 119.1701 1024.7275 12 3103.2549 46357.0054 60 109.1054 915.6221 13 2928.9648 43428.0406 61 99.7290 8158930 14 2764 2369 40663.8037 62 90.9983 724.8948 15 2608.5557 38055.2480 63 82.9191 64I-9757 16 2461.4332 35593.8148 64 75-4030 566.5727 17 2321.0984 33272.7164 65 68.4567 498.1160 18 2186 4697 31086.2467 66 62.0009 436.1152 19 2057.4209 29028.8258 67 56.0047 380.1104 20 1934.1968 27094.6290 68 50.4391 329.6714 21 1816.2483 25278.3807 69 45.2764 284.3949 22 1704.1216 235742591 70 40.4911 243.9038 23 IS98 5551 21975.7040 71 36.0589 207.8449 24 1499.1783 20476.5257 72 3I-9569 175.8880 25 1405.6412 19070 8845 73 28.1639 147.7241 26 1317.6128 I7753- 2 7I7 74 24-6596 123.0644 27 1234.7808 16518.4909 75 21.4252 101.6392 28 29 1156.8497 1083.5406 15361.6412 14278.1006 76 77 18.4430 15.7662 83.1962 67.4300 30 1014.5901 13263.5105 78 13.3916 54.03843 31 949-7493 12313.7611 79 11.3132 42 72520 32 888.7832 1x424.9779 80 9.46300 33.26220 33 831.4698 I0593-5o8i 81 7.80177 25.46044 34 777-5994 9815.9088 82 6.33218 19.12825 35 726.9740 9088.9348 83 5.03716 14.09109 36 6794069 8409.5279 84 3.88432 10 20677 37 634-72I5 7774.8063 85 2.94051 7.26626 38 59 2 -75i3 7182.0550 86 2.18317 5.08309 39 553-339 6628.7161 87 1.59167 3-49I4T5 40 516.3361 6112.3800 88 I-I3350 2.357919 41 481.4672 5630.9128 89 .806388 I.55I53I 42 448.6196 5182.2932 90 .569798 981733 43 417.6858 4764.6075 91 .401100 580633 44 388.6808 43759267 92 .269647 .310986 45 361.4911 4014.4356 93 .171204 .1397818 46 336.0094 3678.4262 94 .0917167 .0480652 47 312.1350 3366.2912 95 .0388219 .0092433 48 289.7723 3076.5189 96 .0092433 .0000000 NORTHAMPTON TABLE. TABLE V. NORTHAMPTON TABLE OF MORTALITY. Value of Annuities, first payment due in one year, computed at different rates of interest. 6 Three Per Cent. Four Per Cent. Five Per Cent. Six Per Cent. Seven Per Cent. a, < Three Per Cent. Four Per Cent. Five Per Cent. Six Per Cent. Seven Per Cent. i 16.02 13-47 11.56 10. 1 1 8.96 49 12.69 11.48 10.44 9-56 8.80 2 18.60 I5-63 13.42 11.72 10.39 50 12.44 11.26 10.27 9-42 8.68 3 19.58 16.46 14.14 12-35 10.94 51 12. 18 II. 06 10. io 9-27 8.56 4 20.21 17.01 14.61 12.77 11.32 52 H-93 10.85 9-93 9-13 8-44 5 20-47 I7-25 14.83 12.96 11.49 53 11.67 10.64 9-75 8.98 8.31 6 20.73 1748 15.04 13.16 11.67 54 11.41 10.42 9-57 8.83 8.18 7 20.85 17.61 15-17 13.28 11.78 55 11.15 IO.2O 9-38 8.67 8.05 8 20.89 17.66 I5-23 13-34 11.84 56 10.88 9.98 9.19 8-51 7.91 9 20.81 17.63 15.21 13-34 11.85 57 10.61 9-75 9.00 8-34 7-77 10 20.66 I7-5 2 I5-I4 13.29 II.8I 58 10.34 9-52 8.80 8.17 7.62 11 20.48 17-39 15.04 13.21 H-75 59 10.06 9.28 8.60 8.00 7-47 12 20.28 I7-25 14.94 13.13 11.69 60 9-78 9.04 8-39 7.82 7-3 1 13 20.08 17.11 I4-83 I3-04 11.62 61 9-49 8.80 8.18 7.64 7-i5 14 1987 16.95 14.71 12-95 11-55 62 9.21 8-55 7-97 7-45 6.99 15 1966 16.79 14-59 12.86 11.47 63 8.91 8.29 7-74 7-25 6.82 16 19.44 16.63 14 46 12 76 11.38 64 8.61 8.03 7-5i 7-05 6.64 17 19.22 16.46 14-33 12.66 11.30 65 8.30 7.76 7.28 6.84 6-45 18 19.01 16.31 14.22 12.56 11.23 66 7-99 7-49 7-03 6.63 6.26 19 18.82 16.17 14.11 12.48 ii. 16 67 7.68 7.21 6-79 6.41 6.06 20 18.64 16.04 14 oi 12.40 11.09 68 7-37 6-93 6-54 6.18 5-86 21 18.47 15-91 13.92 12.33 11.04 69 7-05 6.65 6.28 5-95 5-65 22 18.31 15.80 13-83 12.27 10.99 70 6-73 6.36 6.02 5-72 5-43 23 i8.it; 15.68 1375 12. 2O 10.94 71 6.42 6.08 5-76 5-48 5-22 24 17.98 15.56 13.66 12.13 10.89 72 6.10 5-79 5-50 5-24 5.00 25 17.81 1544 13-57 12 06 10.84 73 5-79 5-51 5-25 5-00 4.78 26 17.64 I5-3 2 13-47 11.99 10.78 74 5-49 5-23 4-99 4-77 4-57 27 17-47 15.19 I3-38 11.92 10.72 75 5.20 4.96 4-74 4-54 4-35 28 17.29 15.06 13.28 11.84 10.66 76 4-93 4.71 4-51 4-33 4-15 29 17.11 14.92 13.18 11.76 lo. 60 77 4-65 4.46 4.28 4.11 3-95 30 16.92 14.78 13.07 11.68 10-54 78 4-37 4.20 4-04 3-88 3-74 31 16.73 14.64 12.97 11.60 10.47 79 4.08 3-92 3-78 3-64 3-51 32 16.54 14.50 12.85 11.51 10.40 80 3-78 3-64 S-S 2 3-39 3.28 33 16.34 14-35 12-74 11.42 10.33 81 3-5 3-38 3-26 3-i6 3.06 34 16.14 14 20 12.62 H-33 10.26 82 3-23 3.12 3-02 2-93 2.84 35 15-94 14.04 12.50 11.24 10.18 83 2.98 2.89 280 2.71 2.63 36 15-73 13.88 12.38 ii 14 10. IO 84 2-79 2.71 2.63 2-55 2.48 37 15-52 13.72 12.25 11.04 10.02 85 2.62 2-54 2.47 2.40 2-34 38 15-3 13-55 12.12 10.93 9-94 86 2.46 2-39 2-33 2.27 2.21 39 15.08 1338 11.98 10.82 9-85 87 2.31 2.25 2 19 2.14 2.O9 40 14.85 13.20 11.84 10.71 9-75 88 2.IQ 2.13 2.08 2.03 1.98 41 14.62 13.02 11.70 10.59 9.66 89 2.01 1-97 1.92 1.88 1.84 42 14-39 12.84 n-55 10.47 9-56 90 1.79 1.76 1.72 1.69 1.66 43 14.16 12.66 11.41 10.36 9-47 91 1.50 1.47 i-45 1.42 1.40 44 13-93 12.47 11.26 10.24 9-37 92 I.I9 1.17 i.iS 1.14 1. 12 45 1369 12.28 ii. ii 10. 1 1 9.26 93 .84 -83 .82 .81 .80 46 13-45 12 09 10.95 9.98 9-15 94 54 -53 5 2 5^ 51 47 13.20 11.89 10.78 9-85 9.04 95 .24 .24 24 .24 23 48 12.95 11.69 10.62 9.71 893 352 NORTHAMPTON TABLE. TABLE VI, JOINT LIFE ANNUITIES TWO LIVES. AGES. Four Per Cent. Five Per Cent. Six Per Cent. AGES. Four Per Cent. Five Per Cent. Six Per Cent. Older. Younger. Older. Younger 1 1 8.252 7.287 6.515 17 7 13-599 12.083 10.849 12 13.480 12.009 10805 2 2 11.107 9-793 8.741 17 13-019 11.630 10.489 3 3 12.325 10.862 9.689 18 3 I2 -53i 11.134 9.998 g 13-569 12.070 10.847 4 4 I3-I85 11.621 10.365 13 13-303 11.864 10 685 18 12.841 11.483 10.365 5 5 J3-59I 11.984 10.691 19 4 12.876 11.447 10.284 6 1 10.741 9-479 8-467 9 13.482 12.006 10.799 6 14.005 12.358 11.031 14 13-130 11.723 10.568 19 12.679 "-35I 10-255 7 2 12.581 II. 100 9.911 7 14.224 12.596 11.251 20 5 12.993 11.561 10.391 10 13-355 11.906 10.719 8 3 I3-3I9 "755 10.498 15 12.961 H-585 10-453 8 14-399 12.731 11.382 20 12.535 11.232 10.156 9 4 9 13-775 I4-396 12.165 12.744 10.869 11.404 21 1 6 10.053 13.121 8.961 11.685 8.070 10.510 11 13.217 11.797 10.631 10 5 13-933 12.315 II.OIO 16 12.799 11.452 10.342 10 14.277 12.665 "345 21 12.409 11.131 10.074 11 1 10782 9-544 8-547 22 2 11.605 10.344 9.313 9 14.068 12.447 11.136 7 13.178 11.748 10.576 11 I4I33 12.546 11.249 12 13.078 11.686 10.541 17 12.646 11.327 10.239 12 2 12.438 II.OIO 9-857 22 12.293 11.042 10.002 7 14.111 12.498 11.192 - 12 13.966 12.411 11.139 23 3 12.161 10.843 9764 8 13.178 11.761 10.597 13 3 13.019 11528 10.324 13 12.934 11.570 10.446 8 14.089 12.492 11.197 18 12.500 11.209 10.140 13 I3-789 12.268 11.023 23 12.179 10.951 9.928 14 4 13-374 11.850 10.617 24 4 12.511 11.163 10.057 9 13.992 12.421 11.144 9 13.112 II 715 10.566 14 13.604 12.118 10.899 14 12.784 11.450 10.348 19 12.361 11.096 10.048 15 5 13-479 "954 10.716 24 12.062 10.858 9-853 10 13-841 12.302 11.048 15 I34H 11.960 10.767 25 5 12.633 11.281 10.170 10 12.998 11.627 10-497 16 1 10.406 9243 8.301 15 12.630 11.324 10.244 6 13-578 12 052 10 8l2 20 12 229 10.989 9.960 11 13-664 12.158 10.929 25 11.944 10.764 9.776 16 13.212 H-793 10.626 26 1 9.770 8.742 7.897 17 2 11.981 10.642 9-555 6 12-754 11.400 10.285 NORTHAMPTON TABLE. 353 TABLE VLCont. JOINT LIFE ANNUITIES TWO LIVES. AGES. Four Per Cent. Five Per Cent. Six Per Cent. AGES. Four Per Cent. Five Per Cent. Six Per Cent. Older. Younger. Older. Younger. 26 11 12861 "519 10.410 33 13 12.125 10.932 9-934 16 12.470 II 193 10.135 18 11.750 10 613 9.600 21 12.105 10.890 9.879 23 11.485 iQ-393 9-474 26 11.822 10.667 9.697 28 11.225 10 181 9299 33 10.902 9.919 9.082 27 2 11.264 I0.o8o 9.104 7 12.798 11.452 10.341 34 4 II 651 10.488 9.518 12 12.715 11.402 10.314 9 12.234 11.024 10.012 17 12.311 11.063 10.027 14 "959 10.796 9 822 22 11.987 10.796 9.803 19 n-595 10.486 9-554 27 11.699 10.567 9.616 24 H-352 10285 9-386 29 11.088 10.069 9.207 28 3 11.790 10-555 9-537 34 10.759 9.801 8.984 8 12.786 n-455 10.354 13 12.564 11.280 10215 35 5 11.732 ^572 9.602 18 12.158 10.939 9.924 10 12.098 10.916 9-925 23 ii 866 10.699 9.724 15 11787 10.655 9-703 28 n-573 10.466 9-533 20 n-445 10.363 9-451 25 11.217 10.175 9-295 29 4 12.116 10.855 9.813 30 10 948 9-954 9.112 9 12.710 11.401 10.315 35 10.612 9.680 8.883 14 12.408 H-I53 10.110 19 12.013 10.820 9.826 36 1 9.047 8.173 7.442 24 H-743 10.600 9- 6 43 6 11.812 10.656 9.687 29 "445 10.362 9.448 11 11.941 10.788 9.820 16 11.609 10.507 9579 30 5 12.220 10.959 9-9I3 21 11.302 10.246 9354 10 12.586 11.304 10.239 26 ii 078 10.062 9.201 15 12.246 11.021 10.001 31 10 805 9-837 9.014 20 II.873 10.707 9-732 36 10.462 9-555 8.778 25 11.618 10499 9.561 30 "313 10.255 9.360 37 2 10.392 9-3QO 8-551 7 11.819 10 676 9715 31 1 9-438 8.483 7.691 12 11.773 10.651 9-707 6 11 12.322 12.441 11.062 11.188 10.015 10.144 17 22 It. 430 11.163 10.358 10.132 9-454 9.260 16 12.078 10.883 9.886 27 10936 9.946 9105 21 11.742 10.600 9.644 32 10.659 9.716 8.913 26 11.489 10.396 9.476 37 10.307 9.427 8670 31 11.179 10.146 9.270 38 3 10.838 9.800 8.928 32 2 10.865 9.767 8.855 8 11.772 10.648 9.701 7 12.350 II. 100 10.060 13 11.000 10.509 9.588 12 12.286 11.062 10.042 18 11257 10.214 9333 17 11.911 10.746 9.771 23 II.O2O 10.015 9.163 22 11.615 10 498 9561 28 10.791 9.826 9005 27 "359 10.289 9389 33 10.508 9-591 8.808 32 ii 042 10.034 9.178 38 10.149 9.294 8-558 33 3 n-355 10.213 9.263 39 4 11.097 10043 9-157 8 12.323 11.090 IO.OOI 9 II.605 10.565 9- 6 37 354 NORTHAMPTON TABLE. TABLE Vl.Cont. JOINT LIFE ANNUITIES TWO LIVES. AGES. Four Per Cent. Five Per Cent. Six Per Cent. AGES. Four Per Cent. Five Per Cent. Six Per Cent. Older. Younger. Older. Younger. 39 14 11.420 10.360 9.464 44 29 10.117 9.267 8.536 19 11.089 10.074 9-215 34 9.869 9.058 8.358 24 10.874 9.895 9-063 39 9-550 8 787 8.127 29 10.642 9.703 8.902 44 9.160 8-457 7.843 34 10.354 9463 8.701 39 9.986 9.158 8.442 45 5 10.500 9-571 8.778 10 10.851 9900 9.088 40 5 11.150 10.102 9.219 15 10.607 9.690 8.905 10 II-5T3 10.442 9-537 20 10.330 9.448 8.692 15 11.234 10.205 9333 25 10.160 9.304 8.569 20 10.924 9-937 9.100 30 9-959 9-135 8.424 25 10.725 9.771 8.960 35 9.706 8.921 8.242 30 10.490 9.576 8-795 40 9.381 8.643 8003 35 10.196 9-331 8.589 45 8.990 8.312 7.718 40 9.820 9.016 8.322 46 1 8.071 7-379 6-787 41 1 8.585 7.800 7-135 6 10.528 9.609 8.823 6 11.203 10.163 9283 11 10.697 9-774 8.962 11 11.342 10.302 9.420 16 10.408 9.522 8.762 16 11.044 10.046 9.198 21 10.165 9.310 8-574 21 10768 9.809 8.992 26 10.000 9.170 8455 26 10.574 9.647 8-855 31 9-797 8.998 8.309 31 10.336 9.448 8.688 36 9-540 8.781 8.122 36 10.037 9.198 8.476 41 .210 8-497 7.878 41 9-654 8.876 8.202 46 .815 8.162 7.589 42 2 9.839 8.942 8.182 47 2 9-221 8-435 7.760 7 11.190 10.165 9.296 7 IO.49I 8.815 12 11.165 10.156 9.298 12 10.481 9-59 2 8.827 17 10.856 9.889 9.065 17 10.208 9-353 8.617 22 10.619 9-685 8.889 22 10.001 9-173 8.458 27 10.423 9.522 8.751 27 9.836 9.032 8338 32 10.182 9.320 8.580 32 9631 8.858 8.189 37 9877 9.062 8.362 37 9-370 8.636 7.998 42 9.491 8-737 808 3 42 9-037 8350 7-751 47 8.637 8.008 7455 43 3 . 10.242 9-3 T 5 8.528 8 11.130 10 124 9.270 48 3 9.566 8-759 8.063 13 10.985 10.007 9- I 73 8 10 404 9-524 8.767 18 10.677 9-739 8938 13 10.284 9425 8.686 23 28 10.470 10.272 9.562 9-396 8.785 8.645 18 23 IO.OII 9833 9.186 9.031 . 8-473 8.338 33 10.027 9.190 8.471 28 9.667 8.890 8.217 38 9716 8.927 8 246 33 9.461 8.714 8.066 43 9.326 8-599 7-965 38 9-195 8487 7.870 43 8862 8.200 7.621 44 4 10.468 9-531 8-733 48 8-453 7.849 7.316 9 II. 012 10.031 9.197 14 10.799 9852 9.042 49 4 9-744 8.932 8.230 19 10.502 9-592 8.814 9 10.263 9.409 8.673 24 I03I7 9-435 8.670 14 10.080 9252 8.538 NORTHAMPTON TABLE. 355 TABLE Vl.Cont. JOINT LIFE ANNUITIES TWO LIVES. AGES. Four Per Cent. Five Per Cent. Six Per Cent. AGES. Four Per Cent. Five Per Cent. Six Per Cent. Older. Younper. Older. Younger. 49 19 9.818 9.021 8.332 53 43 8.308 7.724 7.208 24 9661 8.88b 8.214 48 7.965 7.424 6945 29 9-495 8-744 8.092 53 7-544 7.056 6 620 34 9.286 8-565 7.038 39 44 9.015 8683 8-333 8.046 7-737 7.488 54 4 9 8-957 9.442 8.269 8.718 7.668 8.085 49 8.266 7.686 7-173 14 8.586 7.970 19 9.063 8.383 7.788 50 5 9.742 8.941 8.248 24 8-934 8.270 7.688 10 10085 9.260 8.548 29 8-799 8.153 7.586 15 9.872 9 076 8.386 34 8.629 8 005 7-457 20 9.630 8.861 8.195 39 8.406 7.810 7.286 25 9.488 8-739 8.089 44 8 130 7.569 7-073 30 9.321 8.596 7.966 49 7.780 7 262 6.802 35 9.110 8.415 7 809 54 7.362 6.897 6.480 40 8.834 8.177 7.602 45 8-503 7.891 7353 55 5 8.931 8.256 7-665 50 8.081 7.522 7.030 10 9.256 8.560 7-951 15 9077 8. 4.03 7.812 61 1 7-479 6.885 6.370 20 8.869 8.216 7643 6 9-745 8.956 8.271 25 8-754 8.116 7-555 11 9.894 9.100 8.411 30 8.619 7-999 7-453 16 9.665 8.899 8-234 35 8.448 7.849 7.322 21 9-454 8.712 8.067 40 8.221 7651 7.146 26 8-595 7.966 45 7.948 7.411 6-935 31 9.151 8.451 7.841 50 7-593 7.098 6.658 36 8-937 8.267 7.681 55 7.179 6-735 6-336 41 8.658 8.025 7.470 46 8.326 7-737 7.219 56 1 6.843 6.346 5-9" 51 7.900 7.366 6.893 6 8 902 8.241 7.662 11 9.052 8.386 7.801 52 2 8.520 7.848 7264 16 8.858 8.214 7.648 7 9.090 8.919 8.248 21 8.679 8-053 7.502 12 9.698 8934 8.270 26 8.570 7.958 7.419 17 9461 8.724 8.083 31 8.436 7.841 7.316 22 9.284 8.568 7-944 36 8.264 7.690 7-183 27 9.148 8.451 7.842 41 8-035 7.489 7.005 32 8.980 8.306 7.716 46 7-763 7.249 6793 37 8.763 8.119 7-553 51 7.409 6936 6515 42 8483 7.875 7-340 56 6-993 6.571 6.190 47 8.147 7-582 7.084 52 7.724 7.213 6.758 57 2 7.756 7.199 6.709 ^ 8817 8.176 7.612 53 3 8.815 8.128 7-529 12 8.839 8.203 7.643 8 9591 8.841 17 8.639 8.024 7.481 13 9-497 8.763 8 123 22 8.491 7.891 7.362 18 23 9.260 9.111 8.552 8.421 7-934 7.818 27 32 8-383 8.250 7-797 7.680 7.279 7175 28 8-975 8.304 7.716 37 8076 7-527 7.041 33 8 806 8.157 7.588 42 7.848 7.326 6.862 38 8.586 7966 7.421 47 7-574 7.084 6.648 356 NORTHAMPTON TABLE. TABLE Vl.Cont. JOINT LIFE ANNUITIES TWO LIVES. AGES. AGES. Four Per Five Per Six Per Four Per Five Per Six Per Cent. Cent. Cent. Cent. Cent. Cent. Older. Younger. Older. Younger. ~^~ 52 7.225 6-774 6.371 61 46 7.076 6.648 6.263 57 6.805 6.404 6.041 51 6-7Q5 6-395 6-035 56 6.465 6.100 5-770 58 3 7.986 7.421 6.922 61 6.030 5.712 5-420 8 8.691 8-073 7-527 13 8622 8.015 7-479 62 2 6.894 6-452 6.059 18 8.422 7.835 7.316 1 7.828 7.319 6.865 23 8.299 7.725 7.218 12 7863 7-357 6.905 28 8.193 7-632 7-135 17 7.700 7.208 6.770 33 8.060 7-5I5 7.031 22 7-58o 7.100 6.670 38 7.884 7.360 6.894 27 7-499 7.027 6.605 43 48 7.660 7-382 7.162 6.915 6.718 6.498 32 37 7-397 7.265 6-937 6.819 6524 6.418 53 7-039 6.609 6.225 42 7.088 6.660 6.276 58 6.614 6.234 5-890 47 6.875 6.469 6.104 52 6.600 6 222 5.880 69 4 8.075 7.514 7-017 57 6.270 5.6I3 9 8.519 7.927 7403 62 5-83I 5-533 5-259 14 8-399 7.821 7.310 19 8.207 7-648 7-153 63 3 7.048 6.605 6.209 24 8.104 7.556 7.070 8 7.669 7.184 6.750 29 7.999 7.464 6.988 13 7.625 7-147 6719 34 * 7-346 6.884 18 7-462 6.998 6-583 39 44 7.689 7.469 7.189 6-994 6-744 6.570 23 28 7365 7.286 6.910 6.839 6-503 6-439 49 7.186 6-742 6-344 33 7.186 6.750 6359 54 6.850 6.442 6.076 38 7-053 6.631 6.252 59 6.421 6.062 5.735 43 6.881 6-477 6.II2 60 5 8.011 7.466 6.982 48 53 6.667 6-399 6.283 6.042 5-937 5.7I9 10 8.314 7.750 7.250 58 6.070 5-744 5450 15 8.170 7.622 7-135 63 5.626 5-347 5.089 20 7-995 7-463 6.990 25 7-383 6.919 64 4 7076 6.641 6.251 30 7.802 7.292 6.837 9 7.470 7.010 6.598 35 7.669 7-174 6.732 14 7-381 6.931 6.527 40 7.490 7-oi5 6.590 19 7.226 6789 6.396 45 7-274 6.822 6.418 24 7-147 6.717 6-331 50 6.989 6.568 6.189 29 7.069 6.648 6.268 55 6.659 6.272 5-924 34 6.971 6-559 6.189 60 6.226 5.888 5-579 39 6.838 6.440 6.081 44 6.671 6.289 5 944 61 1 6.123 5.725 5-372 49 6-454 6.093 6 7-944 7.415 6.945 54 6.196 5.860 5-555 11 8.092 7-557 7.081 59 5.867 5.56i 5.284 16 21 7-935 7.787 7.416 7.281 6-953 6.830 64 5-4I7 5158 4.917 26 7.704 7.207 6-764 65 5 6.963 6.546 6.171 31 7.601 7.116 6.682 10 7.236 6.803 6.414 36 7.469 6.998 6-577 15 7.127 6-705 6-325 41 7.290 6.838 6-434 20 6.986 6.576 6.205 NORTHAMPTON TABLE. 357 TABLE Vl.Cont. JOINT LIFE ANNUITIES TWO LIVES. AGES. Four Per Cent. Five Per Cent. Six Per Cent. AGES. Four Per Cent. Five Per Cent. Six Per Cent. Older. Younger. Older. Younger. 65 25 6.920 6.515 6.151 68 58 5-341 5-084 4.849 30 6.844 6-447 6.089 63 5-017 4.786 4.576 35 6.747 6.360 6.010 68 4-537 4.I7I 40 6 614 6.240 5.901 45 6453 6.094 69 4 5-924 5.6II 5-332 50 6.236 5-897 5-59 9 6.262 5-929 5.626 55 5-986 5-67I 5.384 14 6.202 5.876 5-578 60 5-372 5.112 19 6.084 5.766 65 5.201 4.960 4-736 24 6.207 5713 5.427 29 5-973 5-664 5.383 66 1 6 5-295 4.996 6-447 4.728 6.087 34 39 5813 5-603 5-518 5-249 11 6.987 6.581 6.215 44 5696 5-4II 5.I50 16 6.866 6.472 6.115 49 5-54 1 5.268 5.019 21 6-749 6.364 6.015 54 5357 5.100 4.864 26 6.689 6.309 5.966 59 5-I2I 4-883 4-665 31 6615 6.243 5-905 64 4.798 4-390 36 6.520 6.156 5-827 69 4312 4.140 3-977 41 6.388 6.037 5-718 46 6.230 5.894 5-588 70 5 5.768 5-472 5-209 51 6.019 5-701 5.412 10 6.008 5.700 5.418 56 5-774 5-479 5-209 15 5-933 5.63I 5-355 61 5-447 5.180 4.938 20 5-532 5.262 66 4.982 4-759 4-551 25 5.780 5-223 30 5-729 5-442 5 180 67 2 5-896 5-569 5-276 35 5-663 5.382 5-125 7 6.684 0.306 5-963 40 5-571 5.298 5.047 12 6.730 6.351 6.009 45 5.460 5-195 4-953 17 6.604 6.236 5-93 50 5-306 5-054 4.822 22 6.512 6.151 5-824 55 5-132 4.893 4-674 27 6-454 6 098 5776 60 4.900 4.680 4.478 32 6.382 6.033 5.7I7 65 4-573 4-378 4.199 37 6.288 5-639 70 4.087 3-930 3-78I 42 6.159 5-831 5-S3 2 47 6.004 5-690 5-403 71 1 4-380 4.169 3-976 52 5.801 5.504 5-233 6 5.610 5-331 5-084 57 5-559 5-283 5-031 11 5-744 5-460 5199 62 5-285 4.986 4.760 16 5.660 5382 5-127 67 4.760 4-555 4.363 21 5-572 5-300 5-050 26 S-SS 2 5-263 5.016 68 3 5-965 5641 5-352 31 5483 5.218 4-974 8 6.490 6i34 5.811 36 5.419 5159 4.920 13 6.468 6.116 5.796 41 5-329 5.076 4.844 18 6-343 6.001 5-689 46 5-222 4.978 23 6.271 5-934 5.628 51 5.074 4.841 4.626 28 33 6.215 6.146 5-883 5.820 5-524 56 61 4.905 4.679 4-685 4.476 4.482 4.289 38 6.052 5-735 5.446 66 4-349 4.169 4.005 43 5929 5.622 5-343 71 3862 3-7I9 3.584 48 5-774 5.481 5-213 53 558o 5-303 5-050 72 2 4.814 4-588 4-380 358 NORTHAMPTON TABLE. TABLE Vl.Conf. JOINT LIFE ANNUITIES TWO LIVES. AGES. Four Per Cent. Five Per Cent. Six Per Cent. AGES. Four Per Cent. Five Per Cent. Six Per Cent. Older. Younger. Older. Younger. 72 7 5.418 5-157 4.929 75 25 4.589 4-396 4.216 12 5.478 5.216 4.976 30 4-557 4-365 4.188 17 5.389 5-133 4.899 35 4.516 4-327 4.152 22 5-3 2 l 5.070 4.840 40 4-457 4.272 4.101 27 5-283 5-035 4.807 45 4-386 4.206 4.040 32 5-236 4.992 4.767 50 4.285 4.112 3-951 37 5-174 4-934 4-7I4 55 4.171 4.006 3852 42 5.087 4-854 4.640 60 4.021 3.866 3.721 47 4-983 4-758 4-551 65 3.806 3-665 3-533 52 4.845 4-630 4-43 70 3471 3347 3-236 57 4.679 4-477 4 289 75 3-015 2.917 2.827 62 4-458 4.272 4099 67 4.124 3.960 3.811 76 6 4-403 4.221 4-053 72 3639 3-5io 3.387 11 4.487 4.301 4.148 16 4-452 4-270 4.101 73 3 4.811 4-591 4.389 21 4-391 4.212 4.046 8 5.204 4-963 4.752 26 4.365 4.188 4.024 13 5.212 4-972 4.751 31 4-335 4.160 18 5-123 4.889 4.673 36 4-295 4123 3.962 23 5.072 4841 4.628 41 4.238 4.069 3.912 28 5-036 4.808 4-597 46 4.171 4.006 3-853 33 4.991 4.766 4-559 51 4.074 3.916 3.768 38 4-93 4.710 4-507 56 3-815 3-674 43 4.848 4-634 4436 61 3821 3-679 3-546 48 4.746 4-539 4-348 66 3.606 3-477 3-357 53 4.614 4.417 4-234 71 3-270 3-159 3-059 58 4-455 4269 4.096 76 2 .833 2.750 2.668 63 4.236 4.066 3.908 68 3.901 3-752 3-616 1 77 7 4.222 4-055 3.899 73 3-421 3-304 3-193 12 4.368 4 I9S 3-943 17 4.210 4.045 3.892 74 4 4.726 4.516 4.323 22 4.164 4.001 3-850 9 4.969 4-747 4-556 27 4.140 3-979 3.829 14 4-950 4-731 4.528 32 4.111 3-952 3.804 19 4.866 4.651 4-453 37 4-073 3.916 3-770 24 4.827 4-6i5 4.419 42 4.019 3-865 3.722 29 4.792 4-583 4-390 47 3-954 3-805 3.666 34 4-749 4-543 4-353 52 3864 3.720 3.586 39 4.690 4.488 4.301 57 3.761 3-623 3-494 44 4.613 4.417 4-235 62 3.621 3-492 3371 49 4-5" 4.322 4.146 67 3405 3.289 3.180 54 4-389 4.208 4.040 72 3.070 2.971 2.882 59 4-234 4064 3.906 77 2.656 2-583 2511 64 4.019 386 4 3-7I9 69 3-683 3-547 3-423 78 8 4.016 3-864 3-722 74 3.211 3-105 3-005 13 4.022 3-871 3-729 18 3-964 3-8i5 3-677 75 5 4-557 4.362 4.181 23 3-93 3-783 3-646 10 4-725 4.522 4-350 28 3.908 3.762 3.626 15 4-695 4-495 4.310 33 3.881 3-737 3.602 20 4.619 4.424 4242 38 3.844 3.702 3570 UNIVERSITY NORTHAMPTON TABLE. TABLE VLCont. JOINT LIFE ANNUITIES TWO LIVES. 359 AGES. AGES. Four Per Five Pe Six Per Four Pe Five Per Six Per Cent. Cent. Cent. Cent. Cent. Cent. Older. Younger Older. Younger. 78 43 3-794 3-655 3-525 81 61 2.870 2.782 2.699, 48 3-731 3-596 3-469 66 2.746 2.664 2-587 53 3648 3-518 3-396 71 2.542 2.470 2.402 58 3-549 3-424 3-308 76 2.258 2.195 2.147 63 3-4I4 3-297 3.188 81 1.869 1.827 1.786 68 3-199 3-095 2.996 73 2.869 2.780 2.701 82 12 3.020 2.924 2-833 78 2.470 2.410 2.346 17 2.987 2.893 2.804 22 2.958 2.865 2.777 79 9 3-775 3-638 3-510 27 2-945 2-853 2.765 14 3-759 3.624 3-497 32 2.929 2.838 2-751 19 3-704 3-571 3-447 37 2.909 2.818 2-733 24 3- 6 79 3-548 3-424 42 2.878 2.789 2.705 29 3-659 3-528 3.406 47 2.843 2.756 2.673 34 3- 6 33 3-505 3-384 52 2.792 2.707 2.627 39 3-598 3-471 3-352 57 2-733 2.651 2-574 44 3-552 3.428 3-312 62 2.656 2.578 2.504 49 349 3-369 3-256 67 2-533 2.461 2-393 54 3.416 3-299 3.189 72 2334 2.271 2. 211 59 3.222 3-210 3-105 77 2.077 2.013 1-975 64 3.192 3.088 2.990 82 1.681 1.642 69 2.979 2.887 2.799 74 2.659 2.580 8.511 83 13 2 -794 2.709 2.628 79 2.271 2.217 2.161 18 2.760 2.677 2.598 23 2.740 2.657 2.579 80 10 3-517 3-39:; 3.281 28 2.728 2.646 2.568 15 3-49 2 3-372 3- 2 59 33 2.713 2.632 2.555 20 3-443 3-325 3.214 38 2.694 2.613 2.537 25 30 3.425 3.406 3-308 3.290 3.198 3.181 43 48 2.666 2.632 2.587 2.511 2.554 2.481 35 3-383 3-268 3.160 53 2-585 2.510 2 - 4 l! 40 3-349 3-236 3-130 58 2-530 2-457 2.388 45 3-308 3-197 3-093 63 2-457 2.387 2.321 50 3-247 3.140 3-039 68 2-336 2.272 2. 211 55 3.180 3.076 2.978 73 2.141 2.085 2.032 60 3.092 2.992 2.899 78 1.899 1.838 I.8IO 65 2.965 2.873 2.786 83 1.510 1.472 1.441 70 2-757 2.675 2.598 75 2.448 2.381 2.323 84 14 2.622 2-545 2.472 80 2.068 2.018 1.969 19 2.589 2-513 2.442 24 2-574 2.499 2.429 81 11 3264 3-I56 3-54 29 2-563 2.489 2.418 16 3- 2 35 3.128 3.028 34 2-549 2.476 2.406 21 3-IQ5 3.091 2.992 39 2.530 2-457 2.388 26 3.181 3-77 2.979 44 2-505 2-433 2-365 31 3.i 6 4 3.060 2.963 49 2470 2.400 2-334 36 3-I42 3.040 2-944 54 2.428 2.360 2295 41 3.109 3.009 2.914 59 2.376 2.310 2.247 46 3.072 2-973 2.881 64 2.305 2.242 2.182 51 3-oi5 2.920 2.829 69 2.183 2.126 2.071 56 2-953 2861 2.774 74 1.991 1.941 1.894 360 NORTHAMPTON TABLE. TABLE Vl.Cont. JOINT LIFE ANNUITIES-TWO LIVES. AGES. Four Pe Cent. Five Pe Cent. Six Per Cent. AGES. Four Per Cent. Five Per Cent. Six Per Cent. Older. Younger Older Younger 84 79 1.751 1-750 1.672 88 18 2.061 2.OI2 1.965 84 1.387 1-357 L330 23 28 2.048 2.041 1.999 1.092 1953 1.946 85 15 2.462 2-393 2-327 33 2-033 1.985 1-939 20 2-431 2.364 2299 38 2.022 1.974 1.929 25 2.421 2-354 2.290 43 2.006 1-959 1.914 30 2.411 2344 2.280 48 1.987 1.941 1.895 35 2.398 2-33 r 2.268 53 1.960 1.914 1.870 40 2-379 2-313 2.251 58 1.928 1.883 1.841 45 2.356 2.291 2.230 63 1.886 1.843 1.802 50 2.322 2.258 2.198 68 1.817 1.777 1-737 55 2.284 2.222 2.164 73 1.697 1.660 1.625 60 2.234 2.174 2.118 78 1.546 1-514 1.483 65 2.163 2.107 2053 83 1.259 1-235 1. 212 70 2.042 I.99I 1.941 88 1.030 1.063 1.044 75 1.856 1.811 1.769 80 1. 608 1573 '539 89 19 1.904 1.862 1.822 85 1-339 1.256 1232 24 i 895 1.854 1.814 29 1.889 1.848 1. 808 86 16 2-315 2.253 2.194 34 1.882 1.841 1.802 21 2.290 2.229 2.171 39 1.872 1.832 1.792 26 2.282 2.221 2.163 44 1.859 i. 818 1-779 81 2.272 2.212 2-154 49 " 1.840 1.800 I.76l 36 2.260 2.200 2-143 54 1.817 1.778 1-740 41 2.241 2.182 2.126 59 .788 i 750 I.7I3 46 2.221 2 162 2.107 64 I.75I 1.714 1.678 51 2.188 2.I3I 2.077 69 1.685 1.650 1.616 56 2.153 2.097 2.044 74 1-570 1.538 1.508 61 2.105 2.051 2.OOO 79 1.427 1.400 1373 66 2-035 1.984 1.936 84 1.164 I.I45 1.124 71 I.9I4 1.867 1.823 89 1.015 I.OOI .984 76 1-739 1.699 I.66I 81 1.478 1.447 I.4I7 90 20 1.704 1.670 1.638 86 I-I95 I.I7I I.I49 25 1.699 1.665 1-633 30 1,694 1.660 1.628 87 17 2.177 2.I2I 2.069 35 1.688 1.654 i 622 22 2.158 2.104 2051 40 1.679 1.646 1.614 27 2.151 2.096 2.044 45 1.668 1-635 1.604 32 2.142 2.088 2 036 50 1.651 1.619 1.590 37 42 2.130 2.113 2.077 2.060 2.026 2.OO9 55 60 1-633 i. 608 1.601 1-577 1-570 1-547 47 2.093 2.041 1991 65 1575 1-544 I-5I5 52 2.063 2.012 1.963 70 1515 1.486 1-459 57 2.030 1.980 1.932 75 1-413 1-387 i 361 62 1.985 1-937 1.891 80 1.278 1.255 i 234 67 I-9I5 1.870 1.826 85 1-054 1.038 I 021 72 1.794 1-753 I-7I3 90 .922 .909 .895 77 82 :$ 1-597 1.329 1.562 I-303 91 21 I 432 1.407 1.382 87 1.124 1.098 1.078 26 1.429 i 404 i 379 31 i 425 i 400 I-376 NORTHAMPTON TABLE. 361 TABLE Vl.Cont. JOINT LIFE ANNUITIES TWO LIVES. AGES. Four Per Cent. Five Per Cent. Six Per Cent. AGES. Four Per Cent. Five Per Cent. Six Per Cent. Older. Younger. Older. Younger. 91 36 1.420 I -395 I-37I 93 68 .760 750 .740 41 I-4I3 1.388 1.364 73 733 .723 .714 46 1405 1.380 1-356 78 .697 .688 .679 51 I-39I 1.367 1-343 83 .614 .606 599 56 1-377 1-353 1-330 88 554 547 541 61 1-358 1-334 1.311 93 365 .361 357 66 1-33 1.307 1.285 71 76 1.280 I 200 1.259 1.180 1.238 1.160 94 24 29 520 519 514 513 508 507 81 1.078 1.061 1.044 34 .518 512 .506 86 .902 .892 .879 39 517 511 505 91 756 .748 737 44 515 .509 503 49 512 507 .501 92 22 1.142 1.124 1.107 54 509 .503 498 27 1.140 1. 122 i 105 59 505 499 .494 32 I-I37 I.II9 1. 102 64 .500 494 .489 37 I-I34 X.ilO 1.099 69 .491 485 .480 42 1.128 I. Ill 1094 74 474 .469 .464 47 1. 122 1.105 1.089 79 453 .448 443 52 I.II3 1.095 1.079 84 403 3Q8 394 57 1. 102 1.085 1.069 89 373 369 .365 62 1.088 1.071 1-055 94 .201 .199 .197 67 1.067 1.050 J -035 72 1.028 I.OI2 997 95 25 .236 234 .232 77 .970 955 .942 30 .236 234 -231 82 .864 852 .840 35 235 233 231 87 .738 734 725 40 235 233 .231 92 .583 576 569 45 234 232 .230 50 233 .231 .229 93 23 .809 .798 .788 55 .232 230 .228 28 .808 797 .786 60 .230 .228 .226 33 .806 795 .785 65 .228 .226 .224 38 .80 4 793 .783 70 .224 .222 .220 43 .800 .790 779 75 .217 215 213 48 797 .786 776 80 .208 .206 .204 53 .790 .780 .770 85 .187 .185 -I8 3 58 .784 773 763 90 .177 175 .174 63 774 764 754 95 .000 059 .058 INTRODUCTION to H M AND H F TABLES. THE Healthy Male and Healthy Female Tables, some- times known as the Institute of Actuaries' Tables, which must not be confused with the Actuaries' Tables, and often referred to as the Twenty Offices' Tables, because the mortality statistics of twenty British offices was employed in making it, was published in 1869. It was the work of a number of English and Scotch actuaries, under the joint control of the Institute of Actuaries of London and of the Faculty of Actuaries of Scotland. The famous Woolhouse had charge, and his method of graduation, the most per- fect adaptation of the system of graduating by differences, was adopted; indeed, it was invented for the work. Very full tables, commutation and derived, were published by the Institute of Actuaries in connection with this table. Just as it thus reached perfection, the old system of graduating was ready to be discarded. What had formerly seemed nearly a chimera, viz., the idea of discovering a law of mortality and applying that law in graduation, was realized. Makeham's formula became recognized as offer- ing great advantages over all that preceded it. The con- sequence was that from the same data another set of mor- tality and commutation tables were evolved by applying Makeham's formula instead of Woolhouse's. These tables were published in the Text-Book of the Institute of Actu- aries, which appeared in 1887. Messrs. King and Hardy did the work, and the graduation is known by their name. In the following pages the results of both systems of graduation for male lives are presented. We also present a mortality table known as H M5 which illustrates the mor- tality, the first five years of insurance being eliminated. 3 6 4 H M (WOOLHOUSE) TABLE. TABLE I. INSTITUTE OF ACTUARIES' MORTALITY TABLES. MALE LIVES, HM WOOLHOUSE'S FORMULA. Age. No. Living. Decrement. Probability of Surviving. Probability of Dying, 10 IOO OOO 490 .995 loo o .004 900 o 11 99 5io 397 .996 oio 5 .003 989 5 12 99 H3 .996 680 6 .003 319 4 13 98 784 288 .997 084 5 .002 915 5 14 98 496 272 997 238 5 .002 761 5 15 98 224 282 .997 129 o .002 871 16 97 942 3i8 996 753 2 .003 246 8 17 97 624 379 .996 117 8 .003 882 2 18 97 245 466 .995 208 o .004 792 o 19 96 779 556 .994 255 o .005 745 o 20 96 223 609 .993 671 o .006 329 o 21 95 614 643 993 275 o .006 725 o 22 94 97i 650 993 155 8 .006 844 2 23 94 321 638 993 235 9 .006 764 I 24 93 683 622 993 36o 6 .006 639 4 25 93 061 617 993 369 9 .00$ 630 i 26 92 444 618 993 3M 9 .006 685 I 27 91 826 634 993 95 6 .006 904 4 28 91 192 654 .962 828 3 .007 171 7 29 90538 673 .992 566 7 007 433 3 30 89865 694 .992 277 3 .007 722 7 31 89 171 706 .992 082 6 .007 917 4 32 88 465 717 .991 895 i .008 104 9 33 87 748 727 .991 714 9 .008 285 i 34 87 021 740 .991 496 3 .008 503 7 35 86 281 757 .991 226 3 .008 773 7 36 85 524 779 .990 891 4 .009 108 6 37 84745 802 990 536 3 .009 463 7 38 83 943 821 .990 219 6 .009 780 4 89 83 122 838 .989 918 4 .010 081 6 40 82 284 848 .989 694 2 .010 305 8 41 42 81 436 80 582 Sg .989 513 2 .989 265 6 .010 486 8 .010 734 4 43 79 717 887 .988 873 i .on 126 9 44 78 830 911 988 443 5 on 556 5 45 77 9*9 950 .987 807 9 .012 192 I 46 76 969 996 .987 059 7 .012 940 3 47 75 973 1041 .986 297 8 .013 7O2 2 48 74 932 1082 .985 560 2 .014 439 8 49 73 850 1124 .984 780 o 015 220 50 72726 1160 ,984 049 7 .015 950 3 51 7i 566 "93 .983 330 i .016 669 9 52 70 373 1235 .982 450 7 .017 549 3 53 69 138 1286 .981 399 5 .018 600 5 54 67 852 1339 ,980 265 9 .019 734 i H M (WOOLHOUSE) TABLE, 365 TABLE l.Cont. INSTITUTE OF ACTUARIES' MORTALITY TABLES. MALE LIVES, HM WOOLHOUSE'S FORMULA. Age. No. Living. Decrement. Probability of Surviving. Probability of Dying. 55 66513 1399 .978 966 5 .021 033 5 56 65 114 1462 977 547 i 022 452 9 57 63652 1527 .976 010 2 .023 989 8 58 62 125 1592 974 374 2 .025 625 8 59 60 533 1667 .972 461 3 .027 538 7 60 58 866 1747 .970 322 4 .029 677 6 61 57 "9 1830 .967 961 6 .032 038 4 62 55 289 1915 965 3 6 3 8 .034 636 2 63 53 374 2001 .962 509 8 .037 490 2 64 5i 373 2076 959 589 7 .040 410 3 65 49 297 2141 956 569 4 .043 430 6 66 47 156 2196 953 43i 2 .046 568 8 67 44 960 2243 .950 III 2 .049 888 8 68 42 717 2274 .946 765 9 053 234 T 69 4 443 2319 .942 660 o .057 340 o 70 38 124 2371 .937 80S 2 .062 191 8 71 35 753 2433 .931 949 8 .068 050 2 72 33 320 2497 .925 060 o .074 940 o 73 74 30 823 28 269 2554 2578 .917 139 8 .908 804 7 .082 860 2 .091 -195 3 75 25 691 2527 .901 638 7 .098 361 3 76 23 164 2464 .893 628 i .106 371 9 77 20 700 2374 .885 314 o .114 686 o 78 1 8 326 2258 .876 787 i .123 212 9 79 16 068 2138 .866 940 5 133 59 5 80 13 930 2015 .855 348 i .144 651 9 81 " 915 I88 3 .841 963 9 .158 036 i 82 10 032 1719 .828 648 3 .171 351 7 83 8 313 1545 .814 146 5 185 853 5 84 6768 1346 .80I 122 9 .198 877 i 85 5 422 1138 .790 114 3 .209 885 7 86 4 284 941 780 345 5 .219 654 5 87 3 343 773 .768 770 6 .231 229 4 88 2 570 6i5 .760 700 4 .239 299 6 89 i 955 495 .746 803 i .253 196 9 90 i 460 408 .720 547 9 .279 452 i 91 i 052 329 .687 262 3 312 737 7 92 723 254 .648 686 o .351 314 o 93 469 195 .584 221 7 .415 778 3 94 274 139 .492 700 8 .507 299 2 95 135 86 .362 962 9 .637 037 I 96 49 40 .183 673 5 .816 326 5 97 9 9 .000 000 I.OOO OOO O 98 o 366 H M (WOOLHOUSE) TABLE. TABLE II. COMMUTATION COLUMNS. HM TABLES, WOOLHOUSE'S FORMULA. FOUR AND ONE-HALF PER CENT. Age. Dx N, Sx MX Rx 10 64 392.8 I I 88 642 20 229 255 10 434-33 327 960.1 11 61 317.9 I 127 324 19 040 613 10 132.39 317 525-8 12 13 58 443-4 55 74i-o I 068 881 I 013 140 17 913 289 16 844 408 9 898.29 9 712-65 307 393-4 297 495-1 14 53 185 2 959 954-7 15 831 268 9 557-14 287 782.5 15 So 754-3 909 200.3 14 871 313 9 416.59 278 225.4 16 48 429.3 860 771 o 13 962 113 9 277-15 268 808.8 17 46 193.4 814 577.6 13 101 342 9 126 68 259 53 T -6 18 44 032.6 770 545.1 12 286 764 8 955-07 250 404.9 19 41 934-5 728 610.6 II 5l6 219 8 753- *5 241 449.9 20 39 898.2 688 712.4 10 787 609 8 522.61 232 696.7 21 3 7 938.4 650 773-9 10 098 896 8 280.96 224 174.1 22 36 060.6 614 7I3-3 9 448 122 8 036 8 1 215 803.2 23 34 271-5 580 441.8 8 833 409 7 800.64 207 856.3 24 32 573-9 547 867.9 8 252 967 7 578.80 200 055.7 25 30 964-2 516 903.7 7 705 099 7 371-84 192 476.9 26 29 434 4 487 469.3 7 188 196 7 175-39 185 105.1 27 27 978.6 459 490.7 6 700 726 6 987.09 177 929-7 28 26 588.9 432 901.8 6 241 236 6 802.23 170 942.6 29 25 261.5 407 640.3 5 808 334 6 619.76 164 140.4 30 23 994-0 383 646.3 5 400 694 6 440.07 157 520.6 31 22 783.4 360 862.9 5 017 047 6 262.75 151 080.5 32 21 629.7 339 233-2 4 656 185 6 090.13 144 817.8 33 20 530.5 318 702.7 4 316 95i 5 922.38 138 727.6 34 19 483.7 299 219.1 3 998 249 5 759-6 132 805.3 35 18 486.1 280 733.0 3 699 030 5 601.05 127 045.7 36 17 534-8 263 198.2 3 4i8 297 5 445-85 121 444.6 37 I 6 626.9 246 571-3 3 155 098 5 293.01 115 998.8 38 15 760.3 230 810.9 2 908 527 5 142-43 no 705.8 39 14 934 2 215 876.8 2 677 716 4 994-93 105 563-3 40 14 147.0 201 729.8 2 461 839 4 850.85 zoo 568.4 41 42 13 398.3 12 686.9 188 331.5 175 644.7 2 260 110 2 071 778 4 7U-33 4 576.88 95 717-54 91 006.21 43 12 010.2 163 634.5 8 9 6 I 33 4 446.56 86 429.33 44 II 365-I 152 269.3 732 499 4 318.68 81 982.77 45 10 750.1 I4i 519.3 580 230 4 192.99 77 664.09 46 10 161.7 131 357-6 438 710 4 067.57 73 47i-io 47 9 598.29 121 759-3 37 353 3 941-74 69 403-53 48 9 059-11 112 700.2 185 594 3 815 88 65 461.79 49 8 543.82 IO4 156.4 072 893 3 690.70 61 645.91 50 8 051.47 96 104.90 968 737.0 3 566 27 57 955-20 51 7 581.86 88 523.03 872 632.1 3 443-37 54 388 94 52 7 134.43 81 388.61 784 109.1 3 3 2 2.43 50 945-56 53 6 707.39 74 681.22 702 720.4 3 202 6 1 47 623.14 54 6 299.17 68 382.05 628 039.2 3 083.23 44 420.52 H M (WOOLHOUSE) TABLE. 367 TABLE IL Cont. COMMUTATION COLUMNS. HM TABLES. WOOLHOUSES'S FORMULA. FOUR AND ONE-HALF PER CENT. Age. D x N x s* M x R 55 5 908.95 62 473-10 559 657-2 2 964.27 4i 337.29 56 5 535-57 56 937-53 497 184.1 2 845.34 38 373-02 57 5 178.26 5i 759-28 440 246.5 2 726.40 35 527-69 58 4 836.39 46 922.89 388 487.3 2 607-52 32 801.29 59 4 509-53 42 4^.36 341 564.4 2 488.92 30 193.76 60 4 i9 6 5 38 216.86 299 151.0 2 370.09 27 704.84 61 3 896.61 34 320.25 260 934.2 2 250.91 25 334-75 62 3 609.35 30 710.90 226 613.9 2 131.44 23 083.85 63 3 334-29 27 376.61 195 903.0 2 OII.8I 20 952.40 64 3 071.09 24 305-52 168 526.4 I 892.19 18 940.59 65 2 820.08 21 485.44 144 220.9 773-43 17 048.40 66 2 581.44 18 904.00 122 735.4 656.23 15 274.97 7 2 355-24 16 548.77 103 831.4 54I-I9 13 618.74 68 2 141.38 14 407.39 87 282.66 428.75 12 077.55 69 I 940.08 12 467.31 72 875.28 319.66 10 648.80 70 I 75 -08 10 717.23 60 407.97 213.21 9 329-138 71 I 570.57 9 146.663 49 690.74 109.06 8 115-927 72 I 400.66 7 746.005 40 544.08 006.78 7 006.870 73 I 239.90 6 506.108 32 798-07 906.337 6 000.088 74 I 088.19 5 417-918 26 291.96 808.023 5 093.751 75 946.366 4 47L552 20 874.04 7I3.059 4 285.728 76 816.536 3 655.016 16 402.49 623.981 3 572.669 77 698.258 2 956.758 12 74748 540.865 2 948.688 78 59 x -558 2 365.200 9 790.717 464-233 2 407.823 79 496.335 I 868.865 7 425.517 394-484 I 943.590 80 411.764 I 4=57.102 5 556.652 331.286 I 549.106 81 337-035 I 120 067 4 099.550 274.289 I 217.820 82 271-551 848.516 2 979-482 223.319 943-531 83 2I5-33I 633.186 2 130.966 178.792 720.213 84 167.761 465.424 I 497.781 140.495 541.421 85 128.610 336.814 I 032.357 108.568 400.926 86 97.240 8 239.573 695543 82.736 8 292.358 87 72.613 8 166.960 455-969 62.297 2 209.622 88 89 53-4I9 4 38.886 3 "3-540 74.654 289.010 175.470 46.229 8 33-997 o I47.324 101.095 90 27.789 9 46.864 IO0.8l6 24-575 i 67.097 91 19.161 7 27.702 53-952 17.143 6 42522 92 12.602 o I5.IOO 26.250 11.409 i 25-379 93 7.822 7 7.278 11.150 7.172 4 13970 94 4-373 4 2.904 3.872 4.060 o 6-797 95 96 2.062 o .716 2 .842 .126 . 9 68 .126 1.936 9 .679 9 2-737 .800 97 .125 9 .000 .000 .120 5 .121 368 H M (WOOLHOUSE) TABLE. TABLE III. .COMMUTATION COLUMNS. HM TABLES, WOOLHOUSE'S FORMULA. FOUR PER CENT. Age. D x . N s, MX Rx 10 6 7 55 6 -4 356 297 24 434 671 12 792.8 429 295.1 11 64 639.8 291 658 23 078 374 12 474-5 416 502.2 12 61 905.7 229 752 21 786 716 12 226.6 404 027.7 13 59 327.1 170 425 20 556 964 12 029.0 391 801.2 14 56 879 o "3 546 19 386 540 II 862.6 379 772.2 15 54 540-3 059 005 18 272 994 II 7II.6 367 909.6 16 52 292.0 006 713 17 213 988 II 56l.I 356 198.0 17 50 117 6 956 595 8 16 207 275 ii 397 8 344 636.9 18 48 002.9 908 593.0 15 250 679 n 210.7 333 239-1 19 45 935-4 862 657.6 14 342 086 10 989.5 322 028.4 20 43 9I4-9 818 742.6 13 479 429 10 735-8 311 038 8 21 41 958.6 776 784.0 12 660 686 10 468.5 300 303.1 22 40 073.5 736 710.5 II 883 902 10 197.2 289 834.5 23 38 268.5 698 442.0 li 147 192 9 933-49 279 637-3 24 3 6 547-8 661 894.2 10 448 750 9 684.60 269 703.8 25 34 908.7 626 985.5 9 786 55 9 451-27 200 OIQ.2 26 33 343-6 593 641.9 9 159 870 9 228.73 253 567.9 27 31 846.8 561 795.1 8 566 228 9 014.40 241 339.2 28 30 410.5 531 384-7 8 004 433 8 802.97 232 324 8 29 29 031 i 502 353.5 7 473 048 8 593-27 223 521.8 30 27 707.1 474 646 5 6 970 695 8 385.77 214 928.6 31 26 435-7 448 210.8 6 496 048 8 180.02 206 542.8 32 25 217.7 422 993.2 6 047 838 7 978.77 198 362.8 33 24 051.2 398 94*-9 5 624 844 7 782.25 190 384 o 34 22 934.6 376 007.4 5 225 902 7 590-64 182 601.8 35 21 864.9 354 142.4 4 849 895 7 403.12 175 on. i 36 20 839.5 333 302.9 4 495 753 7 218 66 167 608.0 87 19 855-5 3^3 447-4 4 162 450 7 036.14 160 389.3 88 18 911.1 294 536-3 3 849 002 6 855.46 153 353-2 39 18 005.9 276 530.4 3 554 466 6 677.62 146 497.7 40 17 138.9 259 391-5 3 277 936 6 503-07 139 820.1 41 42 16 309 8 IS 5i8.i 243 081.7 227 563 6 3 018 544 2 775 463 6 333-24 6 168.78 133 3i7.o 126 983.8 43 14 761.1 212 802 5 2 547 899 6 008.61 120 815 44 14 035.4 198 767.2 2 335 096 5 850.68 114 806.4 45 13 339-6 185 427.5 2 136 329 5 694.72 108 955.7 46 12 670.2 172 757.4 I 950 902 5 538.34 103 261.0 47 12 025.2 160 732.2 i 778 144 5 380.69 97 722.69 48 II 404.3 149 327.9 I 617 412 5 222.25 92 342.01 49 10 807.3 138 520.6 i 468 084 5 063.91 87 119.76 50 10 233.5 128 287 2 i 329 564 4 905.75 82 055.85 51 9 682.92 118 604.2 I 201 276 4 748.80 77 150.10 52 53 9 I55-30 8 648.68 109 448.9 loo 800.3 I 082 672 973 223.3 4 593-59 4 439-H 72 401.30 67 807.70 54 8 161.36 92 638.91 872 423.0 4 284.42 63 368.60 H M (WOOLHOI/SE) TABLE. 369 TABLE \ll.-Cont. COMMUTATION COLUMNS. HM TABLES, WOOLHOUSE'S FORMULA. FOUR PER CENT. Age. D. N x s, MX R* 55 7 692 59 84 946.32 779 784.1 4 129.56 59 084.17 56 7 241-15 77 705-17 694 837.8 3 973-98 54 954.61 57 6 806.31 70 898.86 617 132.6 3 817.65 58 6 387.53 64 5"-33 546 233.8 3 660.65 47 162.98 59 5 984-46 58 526.87 481 722.4 3 503-26 43 502.34 60 5 595.83 52 931.04 423 195-6 3 344-79 39 999.08 61 5 220.92 47 710.12 370 264.5 3 185.11 36 654.29 62 4 859-28 42 850.85 322 554.4 3 024.27 33 469.18 63 4 5io.55 38 340.30 279 703-6 2 862.44 30 444.91 64 4 174-47 34 165.83 241 363.3 2 699.84 27 582.47 65 3 851-71 30 314.12 207 197.4 2 537.64 24 882.62 66 3 542 72 26 771.40 176 883.3 2 376.79 22 344-99 67 3 247-83 23 523-58 150 111.9 2 2l8.l6 IQ 968.20 68 2 967.11 20 556.47 126 588.3 2 062.36 17 750.04 69 2 701.11 17 855.35 106 031.9 I 910.48 15 687.68 70 2 448.30 15 407.05 88 176.53 I 761.56 13 777.20 71 2 207.73 I 3 199.32 72 769.47 I 6I5.I5 12 015.65 72 978.36 II 220-97 59 570.15 I 470.69 10 400.50 73 759-71 9 461 258 48 349.18 I 328.13 8 929.809 74 551.83 7 909-431 38 887.92 I 187.93 7 601.675 75 356.07 6 553-3 6 6 30 978.49 I 051.86 6 413.742 76 175-66 5 377-7" 24 425-13 923.602 5 361.886 77 OlO.ig 4 367-521 19 047.42 803.355 4 438-284 78 859-938 3 507-583 14 679.90 691.957 3 634.928 79 724.983 2 782.600 ii 172.31 590.076 2 942.972 80 604.344 2 178.256 8 389-713 497-321 2 352-895 81 497-043 I 681.213 6 211.457 413.264 i 855.575 82 402.396 I 278.817 4 530.244 337-734 i 442.311 83 32O.62O 958.197 3 3 251-427 27I-435 i 104.577 84 250.992 707.205 3 2 293 230 214.138 833- J 43 85 I93-342 513.863 5 i 586.024 166. 142 619.004 88 146.887 366.976 9 i 072.161 127 123 452 863 87 II0.2I4 256.763 i 705.184 96.099 3 3 2 5-740 88 89 81.470 3 59.590 8 175.292 8 115.702 o 448.421 273.128 71.^94 8 52.848 8 229.641 158.046 90 42.791 o 72.911 o 157.426 38-340 9 105.197 91 29 647 I 43-263 9 84-5I5 26.842 8 66.856 92 I9-59I 7 23.672 2 41.251 17.927 7 40.013 93 12.220 11.452 2 17-579 11.309 6 22.086 94 6.864 6 4.587 6 6.124 6.424 2 10.776 95 3-252 I 1-335 5 1.536 3-075 7 4-352 96 I.I35 o .200 5 .201 1.083 6 1.276 97 .200 5 .000 o .OOO .192 7 193 370 H M (WOOLHOUSE) TABLE. TABLE IV. COMMUTATION COLUMNS. HM TABLES, WOOLHOUSE'S FORMULA. THREE AND ONE-HALF PER CENT. Age. Dx N x Sx MX R* 10 70 891.9 556 354 29 720 786 15 864.2 567 167.5 11 63 159.0 488 195 28 164 432 15 528.6 551 303-3 12 65 '91-3 422 604 26 676 237 15 265.9 535 774 6 13 63 162.9 359 44i 25 253 633 15 055.5 520 508.7 14 60 849.0 298 592 23 894 192 14 877.6 505 453-2 15 58 629.0 239 963 22 595 600 14 7r5 3 490 575-6 16 56 483-7 183 479 21 355 638 14 552.6 475 860.3 17 54 396.5 129 083 20 172 159 14 375-4 461 307.7 18 52 352-9 i 076 730 19 043 076 14 171.4 446 932.3 19 50 340.2 I 026 389 17 966 346 13 929.0 432 760.9 20 43 358.4 978 031.1 16 939 957 13 649.6 418 831 9 21 46 427.4 931 603.7 15 961 926 13 353 9 405 182.3 22 44 555-7 887 048.0 15 030 322 13 052.2 391 828.5 23 42 754-4 844 293.7 14 143 274 12 757.6 378 776 3 24 41 029.1 803 264 5 13 298 981 12 478.1 366 018.7 25 39 378.5 763 886.0 12 495 716 12 214.9 353 540.5 26 37 794 6 726 091.5 ii 731 830 II 962.7 34i 325-6 27 36 272.4 689 819.1 ii 005 739 II 718.6 329 362.9 28 34 803 8 655 015.2 10 315 919 II 476.6 317 644.3 29 33 385 7 621 629.5 9 660 904 II 235.4 306 167.7 30 32 017.0 589 612 6 9 039 275 TO 995-7 294 932.3 31 30 695 4 558 917.2 8 449 662 10 756.8 283 936.6 32 29 422.6 529 494-6 7 890 745 10 522.0 273 179.8 33 34 28 197.2 27 017.9 501 297.5 474 279 5 7 361 250 6 859 953 10 291 6 10 065.9 262 657.8 252 366.3 35 25 882.3 448 397.7 6 385 673 9 843-87 242 300.4 36 24 787-7 423 609.6 5 937 276 9 624.46 232 456.6 37 23 731-3 > 399 878.3 5 5i3 667 9 406.32 222 832.1 38 22 711. 8 377 166 5 5 "3 788 9 I89-33 213 425.8 39 21 729.1 355 437-4 4 736 622 8 974.71 204 236.4 40 20 782.7 334 654-7 4 381 185 8 763.05 195 261.7 41 IQ 872.9 314 781.7 4 046 530 8 556 ii 186 498.7 42 18 999.6 295; 782 2 3 73* 748 8 354-76 177 942.6 43 18 160 o 277 622.2 3 435 966 8 157-71 169 587.8 44 17 350-7 260 271.5 3 158 344 7 962.48 161 430.1 45 16 570 2 243 701.3 2 898 072 7 768.74 153 467 6 46 15 814.7 227 886.7 2 654 371 7 573 5i 145 698.9 47 15 082 I 212 804.5 2 426 484 7 375 82 138 125.3 48 14 372.4 198 432.1 2 213 68D 7 176.15 130 749.5 49 13 685.9 l8 4 746.2 2 015 248 6 975-64 123 573-3 50 13 021.8 171 724.4 830 5OI 6 774.38 116 597.7 51 12 380.8 159 343-5 658 777 6 573-70 109 823.3 52 II 762.7 147 580 8 499 434 6 374-29 103 249.6 53 ii 165 5 136 4I5-3 35i 853 6 174-85 96 875.33 54 10 587.3 125 828.1 215 437 5 974-19 90 700.49 H M (WOOLHOUSE) TABLE. TABLE IV.Cont. COMMUTATION COLUMNS. HM TABLES, WOOLHOUSE'S FORMULA. THREE AND ONE-HALF PER CENT. Age. D x NX S, M x Rz 55 10 027.4 115 800.7 i 089 609 5 772.32 84 726.30 56 9 484-51 106 316.2 973 808.7 5 568.54 73 953-98 57 8 958.02 97 358.14 867 492.6 5 362.79 73 385-44 58 8 447.46 88 910.68 770 134.4 5 155-16 68 022.65 59 7 952.65 80 958.04 681 223.7 4 946.00 62 867.49 60 7 472.ii 73 485-93 600 265.7 4 734-4 57 921.49 61 7 005.18 66 480.75 526 779.8 4 520.15 53 187.08 62 6 551-45 59 929.30 460 299.0 4 303-30 48 666.94 63 6 110.66 53 818.65 400 369.7 4 084.06 44 363-63 64 5 682.67 48 135-97 346 55I-I 3 862.72 40 279.57 65 5 268.63 42 867.34 298 415.1 3 640.84 36 416.85 66 4 869.38 37 997-96 255 547-8 3 4*9.76 32 776.01 67 4 485-63 33 512-34 217 549.8 3 200.67 29 356.25 68 4 II7-72 29 394.61 184 037.5 2 984-45 26 155.58 69 3 766.6 9 25 627.93 154 642.9 2 772.66 23 171-13 70 3 430-63 22 197.30 129 014.9 2 563-99 20 398.46 71 3 108.48 19 088.82 106 817.6 357.84 17 834.48 72 2 798.98 16 289.84 87 728.81 I53-46 15 476.63 73 2 501.67 13 788.18 71 438-97 950.80 13 323-I7 74 2 216.79 ii 571-39 57 650.80 750.52 II 372.37 75 I 946.50 9 624.883 46 079.41 555-20 9 621.842 76 I 695.69 7 929-191 36 454-53 370.21 8 066.643 77 I 464.08 6 465.116 28 525-34 195-94 6 696.430 78 I 252.34 5 212.781 22 060.22 033-71 5 500.491 79 I 060.90 4 151.881 16 847.44 884.622 4 466.784 80 888.634 3 263.247 12 695.56 748-233 3 582.162 81 734.388 2 528.859 9 432-3I4 624.037 2 833.929 82 597.419 i 931.440 6 93-455 511.902 2 209.892 83 478.309 i 453-I3I 4 972.015 412.995 I 697.990 84 376.245 i 076.886 3 518-884 327.106 I 284.995 85 291.226 785.660 o 2 441.998 254.809 957.889 5 86 222.321 563-339 5 i 656.338 I95.752 703.080 2 87 167.620 395-7I9 4 I 092.998 148.570 507.327 9 88 124.504 271.21=; 7 697.279 III. 122 358-758 o 89 9I-507 3 179.708 4 426.063 82.335 7 247.636 r 90 64.027 o 113.681 4 246.355 59-949 9 165.300 3 91 45.966 8 67.714 6 132-673 42-122 5 105-350 5 92 30.522 9 37.191 7 64-959 28.233 i 63.228 oo 93 19.130 2 18.061 5 27.767 17.872 5 34.994 95 94 10.798 4 7.263 i 9.706 10.187 6 17.122 41 95 5-140 5 2.122 6 2-443 4.894 8 6-934 83 96 1.802 7 319 9 .320 1.730 9 2.040 oo 97 319 9 .000 .000 309 i .309 09 37 2 H M (WOOLHOUSE) TABLE. TABLE V. COMMUTATION COLUMNS. HM TABLES. WOOLHOUSE'S FORMULA. THREE PER CENT. Age. D x N x Sx MX RX 10 74 409-4 I 796 867 36 413 646 19 906.2 756 181.7 11 71 888.1 724 979 34 616 779 19 552.2 736 275.5 12 69 SI5-9 655 463 32 891 800 19 273.8 716 723.3 13 67 267.1 588 196 31 236 337 19 049.7 697 449-6 14 65 H7-5 523 079 29 648 141 18 859.3 678 399.8 15 63 046.2 460 032 28 125 062 18 684.7 659 540.5 16 61 034.2 398 998 26 665 030 18 509.0 640 855.8 17 59 064.1 339 934 25 266 032 18 316.6 622 346.7 18 57 I2i- 2 282 813 23 926 098 18 094 o 604 030.1 19 55 I9I-7 227 621 22 643 285 17 828.2 585 936.1 20 53 276.3 174 345 21 415 664 17 520.4 568 107 9 21 5i 397-2 122 947 20 241 319 17 193.0 550 587-5 22 49 564-7 073 383 19 118 371 16 857.4 533 394-5 23 47 791-7 025 59i I 8 044 988 16 528.1 516 537.0 24 46 085.8 979 505-3 17 019 397 16 214.2 500 008.9 25 26 44 446.5 42 865.8 935 058.8 892 193.0 16 039 892 15 104 833 15 917.2 IS 631.1 483 794-7 467 877-5 27 41 339-1 850 854.0 14 212 640 15 352.9 452 246.5 28 39 857-9 810 996.1 12 361 786 IS 75-8 436 893.6 29 38 4I9-5 772 576.6 12 550 790 14 798.2 421 817.9 30 37 023.2 735 553-4 II 778 214 14 521.0 407 019.6 31 35 667.3 699 886.1 II 042 660 14 243.4 392 498.7 32 34 354-2 665 531.9 10 342 774 13 969.2 378 255.3 33 33 083.3 632 448.6 9 677 242 13 698.9 364 286.1 34 3i 853.6 600 595.0 9 044 794 13 432.8 350 587-2 35 30 662.8 569 932.2 8 444 199 13 169.8 337 154-5 36 29 508.6 540 423.6 7 874 266 12 908.6 323 984-7 37 28 388.1 512 035.5 7 333 843 12 647.6 311 076.1 38 27 33-5 484 735-o 6 821 807 12 386.8 298 428.4 39 26 246.1 458 489.0 6 337 072 12 127.6 286 041.6 40 25 224.7 433 264.2 5 878 583 II 870.7 273 914.1 41 24 237.6 409 026.6 5 445 3*9 II 618.3 262 043.4 42 23 284.9 385 741-7 5 036 292 II 371-5 250 425.1 43 22 364.0 363 377-6 4 650 551 II 128.8 239 053-6 44 21 471.1 341 906.6 4 287 173 10 887.3 227 924.7 45 20 604 8 321 301.8 3 945 267 10 646.3 217 037.5 46 19 760.8 301 541.0 3 623 965 10 402.4 206 391.1 47 18 936.9 282 604.1 3 322 424 10 154.2 195 088.7 48 18 *33-5 264 470.7 3 039 820 9 902.27 185 831.5 49 17 35i-i 247 119.6 2 775 349 9 648.05 175 932.2 50 16 589.3 230 530.3 2 528 229 9 391-66 166 284.2 51 15 849.2 214 681.0 2 297 699 9 I34-76 156 892.5 52 15 131-1 199 549-9 2 083 018 8 878.25 147 757-8 53 14 432.6 185 117-3 I 883 468 8 620.44 138 879.5 54 13 751-6 171 365.8 I 698 351 8 359-81 130 259.1 H M (WOOLHOUSE) TABLE. 373 TABLE V.Cont. COMMUTATION COLUMNS. HM TABLES. WOOLHOUSE'S FORMULA. THREE PER CENT. Age. Dx N Sx M, Rx 55 56 57 58 59 13 087.6 12 439.1 11 805.7 II 186.8 10 582.7 158 278.2 145 839.1 134 033-4 122 846.6 112 263.9 I 526 985 I 368 707 I 222 868 I 088 834 965 987.7 8 096.34 7 829.08 7 557-92 7 282.95 7 004.63 121 899.2 113 802.9 105 973-8 98 415.91 91 i3 2 -95 60 61 62 63 64 9 991-St 9 412.61 8 845.67 8 290.57 7 747-34 102 272.4 92 859.77 84 OI4.IO 75 723-52 67 976.19 853 723-8 751 451-4 658 591. e> 574 577-5 498 854.0 6 721.69 6 433.80 6 141.02 5 843.56 5 54i.8o 84 128.32 77 406.64 70 972.84 64 831.82 58 988.26 65 66 67 68 69 7 217-74 6 703-I7 6 204.86 5 723.60 5 261.08 60 758.45 54 055-28 47 850 42 42 126.82 36 865.74 430 877.8 370 119.4 316 064.1 268 213.7 226 086.9 5 237-85 4 933-50 4 630.44 4 329.90 4 034.08 53 446.26 48 208.62 43 275-" 38 644.67 34 314.77 70 71 72 73 74 4 814.96 4 3 8 3 99 3 966.66 3 562.52 3 172-17 32 0=50.78 27 666.79 23 700.13 20 I37.6I 16 965.44 189 221. 1 157 170.3 129 503.5 105 803.4 85 665.80 3 741-20 3 450.47 3 160.83 2 872.23 2 585-63 30 280.69 26 539-49 23 089 01 19 928.18 17 055.96 75 76 77 78 79 2 798.91 2 45O.IO 2 125.71 I 827.11 I 555-3 2 14 166.53 ii 716.43 9 590.717 7 763.609 6 208,285 68 700.36 54 533.83 42 817.40 33 226.68 25 463.07 2 304 77 2 037.49 I 784.46 I 547-77 i 329.20 14 470.32 12 165.55 10 128.06 8 343.609 6 795-842 80 81 82 83 84 I 30 9 .10 I 087.12 888.659 714.938 565.IH 4 899.184 3 812 061 2 923.402 2 208.464 I 643.354 19 254.79 14 355-6i 10 543-54 7 620.14 5 411-68 i 128.28 944 429 777.628 629.790 500.787 5 466.642 4 338.365 3 393-937 2 616.309 I 986.518 85 86 87 88 89 439-537 337-170 255-445 190.659 140.810 I 203.817 866.647 6II.2O2 420.543 279-733 3 768.32 2 564-51 i 697.86 i 086.66 666.11 391.672 302.107 230.203 172.857 128.561 I 485-732 I 094.060 791-953 561.750 388.893 90 91 92 93 94 102.095 71.421 4 47.655 6 30.013 i 17.023 .6 177-638 106.217 58.561 28.548 ii-525 386.38 208.74 102.53 43-97 15.42 93-947 o 66247 5 44 561 9 28.307 4 16.192 i 260.331 166.384 IOO.I37 55-575 27.268 95 96 97 8.143 2 2.869 6 5" 7 3-38i .512 .000 3-89 5i .00 7.807 6 2.771 i .4968 11.076 3.268 497 H M < 5 > TABLE. 375 TABLE I. HM (5) MORTALITY TABLE (WOOLHOUSE). 1) bJC 2 865.664 10 235.09 100 645 7 438.712 6 802.809 53 816.732 9 4'8 353 90 410.61 423-075 6 364 097 54 769 683 8 6-18.671 80 992 26 407.438 5 941-022 H M < 5 > TABLES. 377 TABLE ll.Cont. HM(5) COMMUTATION TABLES, FOUR PER CENT. Age. D. NX Sx MX RX 55 56 57 724.582 681.256 639.656 7 924.089 7 242.833 6 603 178 72 343-59 64 419-50 57 176.67 391-941 376 483 361.085 5 533.583 5 141-643 4 765.160 58 599-631 6 003.547 So 573-49 345.662 4 404.075 59 561.244 5 442.303 44 569-94 330.339 4 058.413 60 61 S24-3S3 488.738 4 9I7-950 4 429.211 39 127.64 34 209.69 315-034 299.587 3 728.074 3 413.040 62 454-472 3 974 739 29 780.48 284.118 3 "3-454 63 421.443 3 553-296 25 805.74 268.569 2 829.336 64 389.632 3 163.663 22 252.44 252.967 2 560.767 fiC 359.020 2 804.644 19 088.78 237.340 2 307.800 uO fifi 329.810 2 474-833 16 284.13 221.939 2 070.460 DO 67 3 OI -955 2 172 878 13 800.30 206.769 I 848.520 68 275-547 I 897.332 ii 636.42 191.974 I 641 751 69 250.523 I 646.809 9 739-092 177.548 I 449-777 70 226 887 I 419.922 8 092.283 163. 548 i 272.228 71 204.452 i 215.470 6 672.361 149.840 I 108.680 /I yo 183.111 I 032.359 5 456-892 136.362 958.840 m 70 162.766 869.592 8 4 424-533 123.060 822.478 /o 74 143.441 726.152 i 3 554-940 109.995 699.418 75 76 125.203 108.410 600.949 2 492 539 6 2 828.788 2 227.839 97-273 9 85.296 i 589-424 492.150 77 78 92.966 8 70.067 8 399-572 8 320.505 o I 735-299 I 33S-726 74.023 o 63.699 6 406.853 332-831 79 66.551 6 253.953 5 I 015.221 54.224 4 269.131 sn 55-358 4 198.595 i 761.268 45-591 o 214.906 oU 81 45.470 i 153.125 o 562.673 37-831 8 169.316 82 36.782 o 116.342 9 409.548 30.892 6 131.484 83 29-273 5 87.069 4 293-205 24.798 8 100.591 84 22.918 6 64.150 9 206.135 19.569 8 75-792 3 85 17.651 I 46.499 8 141.984 15-183 7 56.222 6 86 13 406 3 33-093 4 95-485 11.617 9 41.038 8 87 10.055 4 23.038 o 62.391 8.782 6 29.420 9 88 7.417 92 15.620 i 39-353 6.531 8 20.638 4 89 5 395 18 10 224 9 23-733 4-794 4 14.106 5 90 3-839 47 6.385 5 13-508 3-446 2 9.312 i 91 2.649 07 3-736 4 7.133 2-403 5 5.86^ 9 92 1-734 25 2.002 I 3.386 1-590 5 3.462 4 93 1.068 28 -933 9 1-3*4 991 3 1-87! 9 94 .576 23 357 6 450 540 3 .880 6 95 03 .264 99 .092 (5 .092 7 .000 093 .000 .251 2 .089 I 340 3 .089 i 378 H M < 5 > TABLE. TABLE III. HM(5) COMMUTATION COLUMNS, THREE AND ONE-HALF PER CENT. A., D x N, Sx MX *, 10 7 089.19 152 122.0 2 851 473.6 705-23 57 400.61 11 6 822.06 145 300.0 2 699 351.6 677.84 55 695.38 12 6 568.86 138 73I.I 2 554 051 6 655-34 54 017.54 13 6 327-54 132 403.6 2 415 320.5 636.15 52 362.21 14 6 095.65 126 307.9 2 282 9I7.O 618.24 50 726.06 15 5 871.61 120 436.3 2 156 609.1 600.33 49 107.82 16 5 654.60 114 781.7 2 036 172.8 581.88 47 507.49 17 5 44277 109 338.9 I 921 39I.I 561.26 45 925-61 18 5 235.56 104 103.4 i 8 12 052.1 538.11 44 364-35 19 5 030.95 99 072.42 I 707 948.8 510.54 42 826.24 20 21 4 828.15 4 626.03 94 244.27 89 618 24 I 608 876.4 I 514 632.1 477.88 439-03 41 315.70 39 837.82 22 4 426.44 85 191.80 I 425 013 9 395.87 3 8 398.79 23 4 232.78 80 959.02 I 339 822 i 35I-90 37 002.93 24 4 045.85 76 913.17 i 258 863.0 308.10 35 651.03 25 3 866.72 73 046.45 I 181 949.9 265.79 34 342.92 26 3 696.71 69 349.74 i 108 903 4 22654 33 077.13 27 3 535-76 65 813.99 i 039 553.7 190.59 31 850.59 28 3 382.22 62 431-77 973 739 7 156.63 30 660.00 29 3 236.13 59 I95-63 911 307.9 I24. 9 I 2 9 503.37 30 31 3 097-13 2 964.86 56 098.51 53 I33-65 852 112.3 796 013.8 095-34 067.81 28 378.46 27 283.11 32 2 838.32 50 295 33 742 880. i 04I-53 26 215.31 33 2 716.95 47 578.38 692 584.8 016.14 25 173-78 34 2 600.86 44 977-52 645 006.4 991 927 24 157-63 35 2 489.21 42 488.31 600 028 9 968.229 23 165.71 36 2 380.98 40 107- 3 3 557 540.6 944-173 22 197.48 37 2 276.66 37 830.^8 517 433-3 920.370 21 253.31 38 2 176.13 35 654 55 479 602.6 8 9 6 8 3 I 20 332.93 39 2 079.28 33 575 27 443 948.0 873.566 19 436.10 40 I 986.48 31 588.79 410 372.8 851.087 18 562.54 41 42 I 897.59 1 812.67 29 691 20 27 878.53 378 784.0 349 092 8 829 368 808 619 17 711-45 16 882.08 43 I 73I.IO 26 147.44 321 214.2 788.345 16 073 46 44 I 652.75 24 494.69 295 066.8 768.535 15 285.12 45 I 577.29 22 917.39 270 572.1 748.971 14 516.58 46 I 504.23 21 413.16 247 654.7 729.246 13 767.61 47 I 433-51 19 979.65 226 241.5 709.394 13 038.37 48 I 3 6 5-09 18 614.57 206 261.9 689.446 12 328.97 49 I 298.72 17 315.84 187 647.3 669.246 " 639 53 50 i 234 39 16 081.45 170 331-5 648.834 10 970.28 51 i 172.24 14 909.21 154 250 o 628.420 10 321.45 52 I 112 20 13 797.01 139 340 8 608.028 9 693.026 53 I 054.41 12 742.60 125 543-8 587.841 084.998 54 998.466 II 744.13 112 801.2 567.557 497.157 H M & TABLE. 379 TABLE lll.Cont. HM(5) COMMUTATION COLUMNS, THREE AND ONE-HALF PER CENT. Age. D x N x Sx MX Rx 55 944.500 10 799-634 10 i 057 09 547-355 7 929.600 56 892-313 9 907.321 QO 257.45 527.108 7 382.245 57 58 841.873 793.007 9 065.448 8 272.441 80 35O.I3 71 284.68 506.843 486.446 6 855.137 6 348.294 59 745.827 7 526.614 63 012.24 466.083 5 861.848 60 700.170 6 826 444 55 485-63 445.646 5 395-765 61 655-766 6 170.678 48 659.19 424.920 4 950.118 62 612.735 .5 557-943 42 488.51 404.065 4 525-198 63 570.949 4 986 994 36 930.57 382.999 4 121.133 64 530.403 4 456.590 31 943-57 361.761 3 738.134 65 491.092 3 965.499 27 486.98 340.386 3 376.373 66 453-3*6 3 512.182 23 521.48 319.217 3 035-987 67 4I7-035 3 095.147 20 009 30 298.266 2 716.770 68 382.400 2 712.746 16 914.15 277-734 2 418.504 69 349-352 2 363 395 14 201.41 257.616 2 140.770 70 317.921 2 045.474 ii 838.01 237-999 883.154 71 287.869 i 757.605 9 792-539 218.698 645-154 72 259.065 i 498.540 8 034.934 199.629 426.456 73 23I-394 i 267.146 6 536.395 180.719 226 827 74 204.905 i 062.240 5 269.249 162.055 046.108 75 179-717 882.523 8 4 ^07.008 143.796 884.053 3 76 156.363 726.160 5 3 324-485 126.519 740.257 8 77 134-737 591.423 2 2 598.324 110.181 613-738 3 78 II5-I47 476.276 2 2 006.901 95.147 2 503-557 i 79 97.387 8 378.888 4 I 530.625 81.281 8 408.409 9 80 81.399 7 297.488 7 I I5I-736 68.587 o 327.128 i 81 67.182 8 230.305 9 854.248 57.122 8 258.541 i 82 54.608 6 175.697 4 623 942 46.820 4 201.418 3 83 43.671 o 132.026 4 448.244 37 729 5 T 54-597 9 84 34-355 7 97-670 7 3l6.2l8 29.891 i 116.868 4 85 26.587 4 71.083 3 218.547 23.284 5 86 977 3 86 20.291 2 50.792 2 147.464 17.887 4 63.692 8 87 15.292 9 35-499 3 96.672 13-575 3 45-805 5 88 11.336 i 24.163 i 6I.I73 10.135 7 32.230 2 89 8.284 8 15-878 3 37.009 7.467 7 22.094 5 90 5-924 3 9954 o 21.131 5-387 4 14.626 8 91 4-107 3 5-846 7 11.177 3.770 7 9.239 4 92 2 701 9 3.144 8 5-330 2.504 2 5.468 7 93 1.672 4 1.472 4 2.186 1.566 o 2.964 6 *4 .906 4 .566 o .713 .856 6 J.398 5 95 .418 9 .147 2 .147 399 7 541 9 96 .147 2 .000 o .000 .142 2 .142 2 380 H M(5) TABLE. TABLE IV. HM(5) COMMUTATION COLUMNS, THREE PER CENT. Age. D x N x Sx M x Rx 10 7 440.94 175 322.2 3 487 413.4 2 117-74 75 864.78 11 7 195.32 168 126.9 3 312 091.2 2 088 84 73 747.04 12 6 961.90 161 165.0 3 143 964.3 2 065.00 71 658.19 13 6 738.69 154 426.3 2 982 799-4 2 044.57 69 593-19 14 6 523-25 147 903.0 2 828 373.1 2 025.40 67 548.62 15 6 314.00 141 589 o 2 680 470.1 2 006.14 65 523-23 16 6 110.15 135 478.9 2 538 881.0 I 986.20 63 517-09 17 5 909-80 129 569.1 2 403 402.2 I 963.81 61 530.89 18 5 712.41 123 856 7 2 273 833 I I 938.56 59 567 07 19 5 5i5-8i 118 3409 2 149 976.4 I 908.33 57 628.52 20 5 3i9-i6 113 021 7 031 635.6 I 872.34 55 720.19 21 5 121.23 107 900.5 918 613.9 i 829 34 53 847-85 22 4 924.06 I 02 976 4 810 713.4 I 781.32 52 018.51 23 4 731-49 98 244.92 707 737.0 i 732.17 50 237.18 24 4 544-48 93 700.44 609 492.1 i 682.98 48 505.01 25 4 364-36 89 336.08 5*5 791-6 i 635.22 46 822.03 26 4 192.73 85 143-35 426 455-5 i 590 71 45 186.81 27 4 029 64 8t 113.71 341 312 2 i 549-74 43 596.io 28 3 873-37 77 240.33 260 198.5 i 510.84 42 046.36 29 3 724-06 73 516.27 182 958.2 i 474-35 40 535 52 30 3 581-40 69 934.87 109 441.9 i 440.15 39 061.18 31 3 445-09 66 489.78 039 507-0 i 408.15 37 621.03 32 3 314-07 63 I75-7I 973 OI 7-2 i 377-47 36 212.88 33 3 187.76 59 987-95 909 841.5 i 347.69 34 835-40 34 3 066.36 56 921.60 849 853.6 i 319.14 33 487-72 35 2 948.97 53 972.63 792 932.0 I 291.06 32 168.58 36 2 834.44 51 138.18 738 959-3 I 262.42 30 877.52 37 2 723-41 48 414.77 687 821.2 i 231-95 29 615.10 38 2 615.79 45 798-98 639 406.4 i 205.65 28 381.15 39 2 5H.50 43 287.47 593 607.4 I 177-55 . 27 175-49 40 2 411.07 40 876.41 550 3I9-9 I 150.27 25 997-94 41 2 3I4.36 38 562.05 509 443-5 i 123.78 24 847-67 42 2 221.52 36 340.53 470 881.5 i 098.35 23 723.89 43 2 I3r-85 34 208.69 434 54* -o I 073.38 22 625 54 44 2 045.24 32 163.45 400 332.3 I 048.87 21 552.15 45 I 961.34 30 202. 10 368 168.8 I 024.54 20 5O3-28 46 I 879.57 28 322.54 337 966.7 999-895 19 478.74 47 I 799.90 26 522.64 309 644.2 974.969 18 478.85 48 I 722 31 24 800.33 283 121.5 949.801 17 503.88 49 I 646.53 23 153-80 258 321.2 924.192 16 554.08 50 I 572.57 21 581.23 235 167.4 898.188 15 629.88 51 I 500.63 20 080.60 213 586.2 872-055 14 731.70 52 I 430.70 i 8 649.90 193 S05-6 845.823 13 859.64 53 I 362.93 17 286.97 174 855-7 819.730 13 013.82 54 I 296.89 15 990 08 157 568.7 793-383 12 194.09 TABLE. TABLE IV.Cont. HM(5) COMMUTATION COLUMNS, THREE PER CENT. Age. DX NX S* M x RI 55 I 232.75 14 757.34 141 578 6 767 016 ii 400.71 56 I 170.29 13 587.05 126 821.3 740.462 10 633.69 57 I 109.49 12 477.56 113 234.2 7I3.754 9 893.228 58 I 050 17 II 427.39 100 756.7 686.743 9 179474 59 992.482 10 434.91 89 329.28 659- 6 45 8 49 2 -73i 60 936.248 9 498.661 78 894.37 632.318 7 833 085 61 881.129 8 617.532 69 395-71 604.469 7 200.767 62 827.307 7 790.225 60 778.18 576.3 6 596.298 63 774.630 7 015.596 52 987.95 547 730 6 019 987 64 723.113 6 292.483 45 972 35 518.775 5 472.257 65 672.769 5 619.714 39 679.87 489.493 4 953482 66 624.033 4 995.680 34 060.16 460.352 4 463.989 67 576.876 4 418.805 29 064.48 43L370 4 003.637 68 531-534 3 887.271 24 645.67 402.831 3 572.267 69 487-954 3 399-3I7 20 758.40 374-732 3 169.436 70 446.209 953-109 17 359-08 347-199 2 794-704 71 405.991 547.118 14 405.98 319.978 2 447-504 72 367.142 179.976 II 858.86 292.954 2 127.526 73 329.519 850 457 9 678 882 266.024 i 834.572 74 293.214 557-243 7 828.425 239-3 1 ? i 568.548 75 258.418 298.825 6 271.181 213.061 i 329.231 76 225.929 072.896 4 972.356 188.099 i 116.169 77 195.627 877.269 2 3 899.460 164.378 928.070 78 167.995 709.274 2 3 022.191 142.443 763-693 79 142.775 566.499 5 2 312.917 122.116 621.249 80 119.915 466.584 7 I 746.417 103.415 499-133 81 99-451 5 347-133 2 I 299.833 86.444 2 395.7I8 82 81.230 i 265.903 I 952.699 71.119 4 309.274 83 65.275 8 200.627 3 686.796 57.531 o 238-155 84 51.601 4 149.025 9 486.169 45-757 9 180.624 85 40.127 4 108.898 5 337.143 35-786 9 134.866 86 30-773 4 78.125 i 228.245 27.601 6 99-079 87 23-305 7 54.819 4 150.120 21.030 2 71-477 83 17-359 6 37-459 8 9530 15.762 9 50.447 89 12.748.5 24.711 3 57.840 "657 5 34-684 90 9.160 5 15-550 7 33-129 8.440 8 23.027 91 6.381 8 9.169 o 17-578 5.928 8 14-586 92 4.218 5 4-950 5 8.409 3-951 4 8.657 93 94 2 623 8 1.4^9 o 2.326 8 .897 8 3-459 1.132 2.479 6 1.361 2 4.706 2.226 95 -6635 234 3 234 637 4 .865 96 234 3 .000 o .000 227 4 .227 H M (KING & HARDY) TABLE. TABLE I. INSTITUTE OF ACTUARIES' TABLES HM KING & HARDY'S GRAD- UATIONS BY MAKEHAM'S FORMULA. Age. No. Living. No. Dying. Probabilities of Surviving. ?* = (1 - A) //, o 127 283 14 358 .88720 .11280 .15920 1 112 925 3 962 .96492 .03508 .07901 2 108 963 2 375 .97821 .02179 .02366 3 106 588 I 646 .98456 .01544 .01787 4 104 942 i 325 .98737 .01263 .01379 5 103 617 i 061 .98976 .01024 .01142 6 102 556 852 .99170 .00830 .00925 7 10 i 704 683 .99328 .00672 .00748 g 101 021 557 .99449 00551 .00607 9 ioo 464 ' 464 99538 .00462 .OO5O2 10 IOO OOO 408 99591 .00409 .00428 11 99 59 2 369 .99630 .00370 .00388 12 99 223 346 .99653 .00347 00359 13 98 877 337 .99658 00342 .00342 14 98 540 337 .99658 .00342 .00540 15 98 203 360 .99635 .00365 00353 16 97 843 384 .99607 .00393 .00378 17 97 459 425 .99563 .00437 .00414 18 97 034 465 .Q9522 .00478 .00458 19 96 569 508 99474 .00526 .00504 20 96 06 1 548 .99428 .00572 .00550 21 95 513 582 99392 .00608 .00592 22 94 93i 609 99357 .00643 .00629 23 94 3 22 631 99332 .00668 .00659 24 93 691 647 .99309 .00691 .00682 25 93 44 658 .99293 .00707 .06701 26 92 386 664 .99280 .00720 .00716 27 91 722 673 .99268 .00732 .00729 28 91 049 678 99254 .00746 .00742 29 90 371 686 .99241 .00759 0075=; 30 89685 691 .99229 .00/71 .00768 31 88 994 700 .99213 .00787 .00782 32 88 294 709 .99197 .00803 .00798 33 87 585 719 .99179 .00821 .008I5 34 86 866 729 .99161 .00839 .00833 35 86 137 742 .99138 .00862 .00854 36 85 395 756 99 II 5 .00885 .00876 37 84 639 770 .99090 .00910 .OOgOl 38 83 869 786 .99063 .00937 .00928 39 83 083 806 .99031 .00969 00957 40 82 277 823 .98999 .OIOOI .00 9 90 41 81 454 846 .98962 .01038 .01025 42 80 608 871 .98919 .01081 .01064 43 79 737 895 .98878 .01122 .OIIO6 44 78 842 924 .98828 .01172 OII53 45 77 918 954 .98776 .01224 .01204 46 76 964 986 .98719 .OI28l .01200 47 75 978 021 98655 01345 .01321 48 74 957 061 98585 .01415 .01388 49 73 896 101 .98510 .01490 .01462 50 72 795 144 .98428 .01572 .01542 51 71 651 193 98335 .01665 .01631 H M (KING & HARDY) TABLE. 383 TABLE l.Cont. INSTITUTE OF ACTUARIES' TABLES HM KING & HARDY'S GRAD- UATIONS BY MAKEHAM'S FORMULA. Age. No. Living. No. Dying. Probabilities of Surviving. - Px 52 70 458 I 243 98236 .01764 .01727 53 69 215 I 296 ,98127 .01873 01833 54 67 919 i 353 .98008 .01992 .01950 55 66 566 i 414 .97877 .02123 .02077 56 65 152 i 475 97735 .02265 .02216 57 63 677 i 54i .97580 .02420 .02369 58 62 136 i 612 .97407 02593 .02536 59 60 524 i 682 .97221 .02779 .02719 60 58 842 i 755 .97017 .02983 .02920 61 57 087 i 830 .96794 .03206 .03140 62 55 257 i 906 .96549 03451 .03381 63 53 35i i 983 .96283 .03717 03645 64 5i 368 2 059 95993 .04007 03934 65 49 309 2 T33 95673 .04327 .04251 66 47 176 2 204 95328 .04672 04599 67 44 972 2 273 94947 05053 .04979 68 69 42 6qg 40 365 2 3H 2 388 94534 .94083 .05466 05917 05396 05853 70 37 977 2 434 93590 .06410 06355 71 35 543 2 468 93057 .06943 .06901 72 33 75 2 4C,0 .92472 .07528 .07502 73 30 585 2 496 .91840 .08160 .08160 74 28 089 2 487 .91144 .08856 .08881 75 25 602 2 459 .90396 .09604 .09671 76 23 143 2 4 I2 . 89578 .10422 .10536 77 20 731 2 343 .88697 .11303 .11485 78 18 388 2 255 87738 . 12262 .12523. 79 16 133 2 I 4 6 .86696 13304 .13662 80 13 987 2 Ol8 85574 . 14426 .14909 81 ir 969 I 873 84351 15649 .16275 82 10 096 I 712 .83042 .i6gs8 .17772 83 8 384 I 540 .81632 .18368 .19412 84 6 844 I 361 .80114 .19886 .21209 85 5 483 i 180 .78478 .21522 .23177 86 4 33 I 002 76715 23285 25343 87 3 3i 830 74855 25145 88 2 471 671 72845 27155 .30286 89 I 800 527 70723 .29277 33123 90 I 273 402 .68421 3 I 579 .36230 91 8 7 I 296 .66016 33984 39635 92 575 209 .63652 36348 43366 93 366 144 .60655 39345 47453 94 95 222 I2 9 II .58109 55039 .41891 .44961 51930 .56836 96 97 71 37 3 .52112 5I35I .47888 .48649 .62211 .68100 98 19 10 47369 .52631 74552 99 9 5 44445 55555 .81621 100 101 4 I Q 3 I .25000 .00000 .75000 I.OOOCO .89366 97851 102 3 8 4 H M (KING & HARDY) TABLE. TABLE II. COMMUTATION COLUMNS-H* TABLE, MAKEHAM'S FORMULA, FOUR AND ONE-HALF PER CENT. Age. D x NX Sx Q. M, R. 127 280 2 016 855 36 266 604 13 739- 34 951- 490 091. 1 108 060 908 795 34 249 749 3 628.1 21 211 8 455 139-7 2 99 781 809 014 32 340 954 2 O8l.2 J 7 583.7 433 927.9 3 4 93 43 87 999 715 611 627 612 30 531 940 28 816 329 I 380.3 I 063.3 IS 502.5 14 122.2 416 344.2 400 841.7 5 83 148 544 464 27 188 717 814.74 13 058.90 386 719.54 6 78 752 465 712 25 644 253 626.08 12 244.16 373 660.64 7 8 74 736 71 036 390 976 319 940 24 178 541 22 787 565 480.28 374-81 II 618.08 II 137.80 361 416.48 349 798.40 9 67 602 252 338 21 467 625 298.79 10 762.99 338 660.60 10 64 393 187 945 20 215 287 251.41 10 464.20 327 897.61 11 61 368 126 577 19 027 342 217-59 10 212.79 317 433.41 12 58 57 068 070 17 900 765 I95-24 9 995-20 307 220.62 13 55 794 012 276 i 6 832 695 181.97 9 799-96 297 225.42 14 53 208 959 068 15 820 419 174.14 9 617-99 287 425.46 15 5o 743 908 325 14 861 351 178.01 9 443-85 277 807.47 16 48 380 859 945 13 953 026 I8I.70 9 265.84 268 363.62 17 46 US 813 830 13 093 081 192.44 9 084.14 259 097-78 18 43 937 769 893 12 279 251 201.48 8 891.70 250 013.64 19 41 844 728 049 II 509 358 210.63 8 690.22 241 121.94 20 39 831 688 218 10 781 309 217.44 8 479-59 232 431.72 21 37898 650 320 10 093 091 220.98 8 262.15 223 952.13 22 36 045 6f4 275 9 442 771 221.28 8 041.17 215 689.98 23 34 272 580 003 8 828 496 219.40 7 819.89 207 648.81 24 32 577 547 420 8 248 493 215.27 7 600.49 199 828.92 25 30 959 516 467 7 701 067 20 9 .5I 7 385-22 192 228.43 26 29 416 487 051 7 184 600 202.32 7 I75-7I 184 843.21 27 27 947 459 104 6 697 549 196.23 6 973-39 177 667.50 28 26 547 432 557 6 238 445 189.17 6 777.16 170 694.11 29 25 215 407 342 5 805 888 183.16 6 587.99 163 916.95 30 23 946 383 396 5 398 546 176.56 6 404.83 157 328.96 31 22 738 360 658 5 015 150 I/I- 15 6 228.27 150 924.13 32 21 588 339 070 4 654 492 165.89 6 057.12 144 695.86 33 20 492 318 578 4 315 422 160.98 5 891.23 138 638.74 34 19 449 299 129 3 996 844 156.19 5 730-25 132 747.51 35 1 8 455 280 674 3 697 7^5 152.13 5 574-o6 127 017.26 36 17 508 263 i 66 3 417 041 148.33 5 421-93 121 443-20 37 16 606 246 560 3 153 875 144-57 5 273.60 116 021.27 38 15 746 230 814 2 907 315 I4I.2I 5 129-03 no 747.67 39 14 927 215 887 2 676 501 138.58 4 987.82 105 618.64 40 14 146 2OI 741 2 460 614 I35-40 4 849-24 IOO 630.82 41 13 401 188 340 2 2 5 8 873 133.20 4 7I3-84 95 781-58 42 12 691 175 649 2 070 533 I3L23 4 580.64 91 067.74 43 12 013 163 636 I 894 884 129.03 4 449.41 86 487.10 44 II 367 152 269 I 731 248 I27. 4 8 4 320.38 82 037.69 45 10 750 141 519 i 578 979 12595 4 192-90 77 7I7-3I 46 10 161 131 357-7 i 437 459-6 I24.57 4 066.95 73 524.41 47 9 598.9 121 758.8 i 306 101.9 I2 3-44 3 942.38 69 457-46 48 9 002.1 112 696.7 i 184 343.1 122.75 3 818.94 65 5i5-o8 49 8 549 i 104 147.6 i 071 646.4 121.89 3 696.19 61 696.14 H M (KING & HARDY) TABLE. 385 TABLE II. Con t. COMMUTATION COLUMNS HM TABLE, MAKEHAM'S FORMULA, FOUR AND ONE-HALF PER CENT. Age. Dx NX Sx c x MX R x 50 8 059.2 96 088.4 967 498.8 121.20 3.574-30 57 999 95 51 7 590-8 88 497.6 871 410.4 120.95 3 453-io 54 425.65 52 53 7 143-a 6 714 9 81 354.6 74 639 7 782 912.8 701 558 2 120.59 120.32 3 332.15 3 211.56 50 972.55 47 640.40 54 6 3054 68 334-3 626 918.5 1 2O 2O 3 091.24 44 428.84 55 5 9I3-6 62 420.7 558 584.2 120.21 2 971.04 41 337.60 56 5 538.9 56 881.8 496 163.5 II9-99 2 850.83 38 36656 57 5 180.2 51 701.6 439 281.7 119.97 2 730.84 35 515-73 58 4 837-3 46 864.3 387 58o.i 120.09 610.87 32 784-89 59 4 508.9 42 355-4 340 7I5-8 119.91 490.78 30 17402 60 4 194.8 38 160.6 298 360 4 "9-73 370.87 27 683.24 61 3 894-5 34 266.1 260 199 8 119.46 251.14 25 3*2.37 62 3 607.3 30 658.8 225 933-7 119.07 131.68 23 061.23 63 3 332-8 27 326.0 195 274.9 118.54 012.61 20 929 55 64 3 070.8 24 255.2 167 948.9 117.79 894.07 18 916.94 65 2 8208 21 434.4 143 693-7 116.76 776.28 17 022.87 66 2 5825 18 851.9 122 259.3 115.46 659-52 15 246.59 67 2 355-9 16 496.0 103 407.4 "3-94 544.06 13 587-07 68 2 140.5 14 355-5 86 911.4 111.96 430.12 12 043 01 69 I 936.4 12 419.1 72 555.9 109.62 318.16 10 612.89 70 i 743-3 10 675.8 60 136.8 106.92 208.54 9 294.73 71 i 561.3 9 "4-5 49 461.0 103.75 I 101.62 8 086.19 72 i 390.4 7 724-1 40 346.5 100. 16 997.87 6 984.57 73 i 230.3 6 493.8 32 622.4 96.079 897.707 5 986.699 74 i 081.3 5 412.49 26 128.57 91.614 801.628 5 088.992 75 943.08 4 469-41 20 716.08 86.680 710.014 4 287.364 76 8i5-79 3 653-62 16 246.67 81.364 623 334 3 577-350 77 699.31 2 954 31 12 593-05 75-631 541.970 2 954.016 78 593-55 2 360.76 9 638.74 69.656 466.339 2 412.046 79 498.34 i 862.42 7 277-98 63.435 396.683 I 945.707 80 4I3-45 i 448.97 5 415.56 57.082 333248 I 549.024 81 338.56 i 110.41 3 966.59 50.699 276.166 I 215.776 82 273.28 837-13 2 856.18 44-346 225.467 939.610 83 217.17 619.96 2 019.05 38.172 181 121 7I4-I43 84 169.64 450.32 I 399.09 32283 142 949 533-022 85 130.06 320.260 948.768 26.784 110.666 390073 86 97.672 222.588 628.508 21.765 83.882 279.407 87 71.703 150886 405.920 17.252 62.117 195 525 88 5i -3 62 99.524 255-034 13-347 44-865 133-408 89 35-803 63.721 I55-5IO 10.031 3I-5I8 83-543 90 24.230 39-491 91.789 7.3223 21.4866 57-0253 91 15-865 23 626 52.298 5-1593 14.1643 35-5387 92 10.022 13.604 28.672 3.4861 9.0050 21 -3744 93 6.105 7-499 15.068 2.2984 55189 12.3694 94 3-543 3-956 7.569 1.4205 3.2205 6.8505 95 1.970 1.986 3 613 .8478 1.8000 3.6300 96 1.038 .948 1.627 4755 .9522 1.8300 97 .518 430 .679 .2409 .4767 .8778 98 254 .176 .249 .1281 2358 .4011 99 100 "5 .049 .061 .012 073 .012 .0613 .03^2 .1077 .0464 .1653 .0576 101 .012 .... .... .0112 .0112 .0112 3 86 H M (KING & HARDY) TABLE. TABLE III. COMMUTATION COLUMNS HM TABLE, MAKEHAM'S FORMULA. FOUR PER CENT. Age. D x NX Sx Cx If, Rx 127 280 2 204 737 42 280 578 13 806. 37 590- 616 157. 1 108 580 2 096 157 40 075 841 3 663.0 23 784.0 578 567-3 2 100 740 995 4i7 37 979 684 2 III-3 20 121. 554 783-3 8 94 757 900 660 35 984 267 I 407.0 18 009.7 534 662 3 4 89 706 810 954 34 083 607 I 089.1 16 602.7 516 652 6 5- 85 165 725 789 32 272 653 838.53 15 513-57 500 049.89 6 81 051 644 738 30 546 864 647.46 14 675.04 484 536.32 7 77288 567 450 28 902 126 499.06 14 027.58 469 861.28 8 73 814 493 636 27 334 676 391-35 13 528.52 455 833.70 9 70 585 423 051 25 841 040 3I3-47 13 137.17 442 305 18 10 67 557 355 494 24 417 989 26503 12 823.70 429 168.01 11 64 692 290 802 23 062 495 230.48 12 558.67 416 344 31 12 61 974 228 828 21 771 693 207.80 12 328.19 403 785.64 13 59 384 169 444 20 542 865 I94.6I 12 120-39 391 457-45 14 56 904 112 540 19 373 42i 187.12 II 925.78 379 337-06 15 54 528 058 012 18 260 881 192.21 II 738.66 367 411.28 16 52 240 005 772 17 202 869 197.13 " 546.45 355 672.62 17 50 032 955 740 i 6 197 097 20979 II 349-32 344 126.17 18 47 898 907 842 15 241 357 220.71 II 139-53 332 776-85 19 45 836 862 006 14 333 515 231.84 10 918 82 321 637.32 20 43 841 818 165 13 47i 509 240.48 10 686.98 310 718.50 21 41 914 776 251 12 653 344 245-58 10 446.50 300 031.52 22 40 057 736 194 II 877 093 247.09 IO 2OO 92 289 585.02 23 38268 697 926 ii 140 899 246.17 9 953.83 279 384.10 24 3 6 55i 66 r 375 10 442 973 242.70 9 707 66 269 430.27 25 34 903 626 472 9 78i 598 237-33 9 464.96 259 722.61 26 33 323 593 149 9 155 126 230.29 9 227.63 250 257.65 27 31 810 56i 339 8 561 977 224-43 8 997-34 241 030.02 28 3 3 6 3 53 976 8 ooo 638 217.40 8 772.91 232 032.68 29 28 977 501 999 7 469 662 211.50 8 555-51 223 259.77 30 27 652 474 347 6 967 663 204.86 8 344-01 214 704.26 31 26383 447 964 6 493 316 199-54 8 I39-I5 206 360.25 32 25 169 422 795 6 045 352 194-33 7 939.61 198 221. IO 33 24 007 398 788 5 622 557 189.50 7 745-28 190 281.49 34 22 894 375 894 5 223 769 184.74 7 555 78 182 536.21 35 21 828 354 066 4 847 875 180.80 7 37I-04 174 980.43 36 20 808 333 258 4 493 809 I77.I3 7 190.24 167 609.39 37 19 831 313 427 4 160 551 173-47 7 013.11 160 410 .15 38 39 18 894 17 997 294 533 276 536 3 847 124 3 552 59i 170.26 167.88 6 839.64 6 669 38 153 406 04 146 566.40 40 17 138 259 398 3 276 055 164.83 6 501.50 139 897 02 41 16 313 243 085 3 016 657 162.92 6 336.67 133 395- C 2 42 15 523 227 562 2 773 572 l6l.28 6 173-75 127 058.85 43 14 765 212 797 2 546 010 I 59-35 6 012.47 I2O 885.10 44 14038 198 759 2 333 213 158.19 5 853.12 114 872.63 45 13 340 185 419 2 134 454 157-04 5 694.93 109 019.51 46 12 670 172 749 i 949 03^ 156.07 5 537.89 103 324.58 47 12 026 160 723 I 776 286 155-39 5 381-82 97 786.69 48 II 408 149 315 i 615 563 155-27 5 226.43 92 404.87 49 10 814 138 501 i 466 248 I54-92 5 071.16 87 178.44 H M (KING & HARDY) TABLE. 387 TABLE Ul.Cont. COMMUTATION COLUMNS HM TABLE, MAKEHAM'S FORMULA, FOUR PER CENT. Age. D x NX Sx Cx M z Rx 50 10 243. 128 258.0 i 327 747.3 154-79 4 916.24 82 107.28 51 9 694-4 118 563.6 i 199 489 3 155-21 4 76i.45 77 191.04 52 9 166.4 109 397.2 i 080 925.7 155-49 4 606.24 72 429.59 53 8 658.2 100 739.0 971 528.5 I55-89 4 450.75 67 823.35 54 8 169.4 92 569.6 870 789 5 15648 4 294.86 63 372 60 55 7 6987 84 870.9 778 219.9 I57-25 4 138 38 59 077 74 56 7 245-4 77 625.5 693 349- 157-72 3 981-13 54 939.36 57 6 808.9 70 816.6 615 723 5 158.44 3 823.41 50 958.23 58 6 388.7 64 427.9 544 906.9 159-37 3 664 97 47 134 82 59 5 983-6 58 444 3 480 479.0 I59-89 3 50560 43 469.85 60 5 593-6 52 850 7 422 034.7 160.42 3 345-71 39 964.25 61 5 218. i 47 632.6 369 184.0 160.83 3 185.29 36 618 54 62 4 856.5 42 776 I 321 551-4 161.07 3 024 46 33 433 25 63 4 5o8 6 38 267.5 278 775 3 161 14 2 863 39 30 408 79 64 4 I74-I 34 093-4 240 507.8 160.88 2 702.25 27 545.40 65 3 852.7 30 240.7 206 414 4 16025 2 54*-37 24 843 15 66 3 544-2 26 696.5 176 173.7 159.21 2 38I.I2 22 301.78 67 3 248.7 23 447-8 149 477.2 15788 2 221.91 19 920.66 68 2 965.9 20 481 9 126 029.4 155 88 2 064.03 17 698.75 69 2 695.9 17 786 o 105 547-5 T5336 908.15 15 634-72 70 2 438.9 15 347.1 87 761.5 150.30 75479 13 726.57 71 2 194.7 13 152.4 7 2 4I4-4 14654 60449 II 971. 78 72 I 963.8 II 188.6 59 262.0 142.16 45795 10 367.29 73 I 746.1 9 44 2 -5 48 073 4 137.02 3!5-79 8 909.34 74 I 541-9 7 900.6 38 630.9 131 27 178.77 7 593-55 75 I 351-4 6 5492 30 730.3 124 80 047.50 6 414-78 76 I 174.6 5 374-6 24 181.1 117.71 922.70 5 367-28 77 I OII.7 4 362.91 18 806.47 109.94 804.99 4 44458 78 862.84 3 500.07 14 443 56 101.75 69505 3 639-59 79 727.93 2 772.14 10 943.49 93.102 593299 2 944 536 80 606 8 1 2 165.33 8 171 35 8}. 182 500.197 2 35i 237 81 49930 i 666 03 6 006 02 75-129 416015 i 851 040 82 404.97 i 261.06 4 339-99 66028 340.886 i 435 025 83 323-36 937.70 3 078 93 57-1" 274 858 i 094.139 84 253- ST 683.89 2 141-23 48.532 217.747 819.281 85 I95-5 2 488.37 i 457 34 40459 l6g. 2IS 601.534 86 147-54 340-83 968.97 33-035 128756 432.319 87 108.83 232.001 628.136 26.312 95.721 303-563 88 78.332 153.669 396.135 20 453 69.409 207.842 89 54.866 98.803 242.466 15 .146 48.956 138.433 90 37-310 6i.493 143.663 it. 329 33510 89477 91 24-547 36.946 82.170 8.0208 22.I8II 55.9668 92 15-581 21.365 45-224 5-4457 14.1603 33-7857 93 9.536 11.829 2^.859 3-6077 8.7146 19.6254 94 5562 6.267 12.030 2.2403 5.1069 10.9108 95 3-io8 3-159 5-763 , 13435 2.8666 5-8039 96 1.645 1514 2.604 7573 1-5231 2-9373 97 .824 .690 1.090 385^ .7658 1.4142 98 99 .407 -185 I .400 .117 .2059 .0990 3803 .1744 .6484 .->68i 100 079 .019 .019 05/1 0754 .0337 101 .019 .0183 .0183 .0183 3 88 H M (KING & HARDY) TABLE. TABLE IV. COMMUTATION COLUMNS HM TABLE, MAKEHAM'S FORMULA. THREE AND ONE-HALF PER CENT. Age D r N s. Cx MX Rx 127 280 2 425 772 49 703 849 13 872 40 948. 785 908 1 109 no 2 316 662 47 278 077 3 698.5 27 075.5 744 959.6 2 101 720 2 214 942 44 961 415 2 142.1 23 377-0 717 884.1 3 96 137 2 118 805 42 746 473 i 434-4 21 2349 694 507.1 4 91 451 2 027 354 40 627 668 i II5-6 19 800 5 673 272.2 5 87 243 I 940 III 38 600 314 863 14 18 684.94 653 471.69 6 83 433 i 856 681 36 660 203 669.67 17 821.80 634 786 75 7 79 939 I 776 742 34 803 522 518.68 17 152 13 616 964.95 8 76 717 i 700 025 33 026 780 408 70 16 633.45 599 812.82 9 73 714 i 626 311 31 326 755 3 28 94 16 224 75 583 179-37 10 70 892 i 555 4i9 29 700 444 27946 15 895.81 566 954.62 11 68 215 i 487 204 28 145 025 24420 15 616.35 551 058.81 12 65 664 i 421 540 26 657 821 221 24 15 372.15 535 44 2 -46 13 63 224 i 358 316 25 236 281 208.19 15 150.91 520 070 31 14 60 877 i 297 439 23 877 9 6 5 201.15 14 942.72 504 919.40 15 58615 i 238 824 22 580 526 207.6l 14 741-57 489 976 68 16 56 426 i 182 398 21 341 702 213 96 14 533 96 475 235.11 17 54 3^4 i 128 091 20 159 304 228.80 14 320.00 460 701.15 18 52 238 i 075 856 19 031 2IO 241.87 14 091.20 446 381 15 19 50 231 i 025 625 17 955 354 255-30 13 849-33 432 289.95 20 48 277 977 348 16 929 729 266.09 13 594.03 418 440 62 21 46 378 930 970 IS 95 ' 38i 273.04 13 327.94 404 846.59 22 44 537 886 433 15 021 411 276.05 13 054.90 39i 51865 23 42 754 843 679 T 4 I3| 978 27635 12 778.85 378 463.75 24 41 33 802 646 13 2 9 I 299 273 77 12 502.50 365 684 90 25 39 37i 763 275 12 488 653 269.02 12 228.73 353 182.40 26 37 77i 725 504 ii 725 37*3 262.20 II 959.71 340 953 67 27 36 231 689 273 10 999 874 256.86 II 697.42 328 993 96 28 34 750 654 523 10 310 601 250.01 II 440.56 317 296.54 29 33 324 621 199 9 656 078 244.41 II I9 -55 305 855-98 30 3i 953 589 246 9 034 879 237.86 10 946.14 294 665.43 31 30 634 558 612 8 445 633 232 81 10 708.28 283 719.29 32 29 366 529 246 7 887 021 227.83 10 475.47 273 on. 01 33 28 145 501 101 7 357 775 223.23 io 247.64 262 535.54 34 26 970 474 131 6 856 674 218 6) 10 024.41 252 287 90 35 25 839 448 292 6 382 543 215 06 9 805.72 242 263.49 36 24 750 4 2 3 542 5 934 251 211 70 9 590.66 232 457-77 37 23 702 399 840 5 510 709 20833 9 378.96 222 867.11 38 22 692 377 148 5 no 869 205.47 9 170.63 213 488.15 39 21 719 355 429 4 73^ 721 203.58 8 965 16 204 317.52 40 2O 781 334 648 4 378 292 2OO.84 8 761.58 IQ5 352 36 41 19 877 3 r 4 77i 4 043 644 19947 8 560.74 186 590.78 42 19 OO6 295 765 3 728 873 198.42 8 361.27 178 030.04 43 18 165 277 600 3 433 108 196.99 8 162.85 169 668.77 44 17 353 260 247 3 155 503 196.49 7 965.86 161 505.92 45 16 570 243 677 2 895 26l I96.O2 7 769-37 153 540.o6 46 15 814 227 863 2 651 584 195-74 7 573-35 145 770.69 47 15 083 212 780 2 423 7^1 19583 7 377-6i 138 197.34 48 14 377 198 403 2 2IO 94! 19663 7 181.78 130 819.73 49 13 694 184 709 2 012 538 I 9 7.I4 6 985.15 123 637.95 H M (KING & HARDY) TABLE. 389 TABLE IV.Cont. COMMUTATION COLUMNS HM TABLE, MAKEHAM'S FORMULA. THREE AND ONE-HALF PER CENT. Age. Dx NX Sx Cx M x Rx 50 13 034- 171 675. I 827 829. 197.91 6 788.01 116 652.80 51 12 395- 159 280. I 656 154. 199.41 6 590.10 109 864.79 52 il 717. 147 503. I 496 874. 200.74 6 390.69 103 274.69 53 ii 178. 136 325. i 349 37i. 2O2 22 6 189.95 96 884.00 54 10 598. 125 727. i 213 046. 203.98 5 987.73 90 694.05 55 10 035. 115 691.8 i 087 319.3 205.96 5 783-75 84 706.32 56 9 490.1 1 06 201.7 971 627.5 207.58 5 577-79 78 922.57 57 8 961.5 97 240.2 865 425.8 209.54 5 370-21 73 344.78 58 8 448.9 88 791.3 768 185.6 211 78 5 160.67 67 974-57 59 7 951-5 80 839.8 679 394-3 2I3.5I 4 948.89 62 813.90 60 61 7 469.! 7 001.3 73 370.7 66 369.4 598 554-5 525 183.8 21524 216 85 4 735-38 4 520.14 57 865.01 53 129.63 62 6 547-7 59 821.7 458 814.4 2I8.2I 4 303-29 48 609.49 63 6 108.0 53 7I3-7 398 992.7 219.35 4 085.08 44 306.20 64 5 682.! 48 031.6 345 279.0 22O.O6 3 865.73 40 221.12 65 5 270.0 42 761.6 297 247.4 220.26 3 645-67 36 355-39 66 4 871-5 37 890.1 254 485.8 219.89 3 425.41 32 709.72 67 4 4868 33 43-3 2t6 595.7 219.11 3 205.52 29 284.31 68 4 116.0 29 287.3 183. 192.4 217.38 2 986.41 26 078.79 63 3 7595 25 527-8 153 905-I 214.89 2 769-03 23 092 38 70 3 4I7-4 22 II0.4 128 377.3 211.62 2 554-14 20 323.35 71 3 090.2 19 020.2 106 266.9 207.32 2 342.52 17 769.21 72 2 778.4 16 241.8 87 246.7 2O2 09 2 135 20 15 426.69 73 2 482.3 13 759 5 71 004.9 195-73 i 933-H 13 291.49 74 2 202.7 ii 556.8 57 245.4 188.43 1 737.38 ii 358 38 75 I 939 7 9 617.1 45 688.6 iSo.OI I 548.95 9 621.00 75 i 694.1 7 9 2 3- S 6 071.5 170.60 I 368.94 8 072 05 77 i 466.3 6 456.7 28 148.5 160.11 I 198.34 6 703.11 78 79 i 256 6 ! 065.2 5 200.1 21 691.8 16 491.72 I4 8.8 9 136.90 I 038.23 889.34 5 504.77 4 466.54 80 892.26 3 242 63 12 356.83 124-38 752.44 3 577-20 81 737.72 2 504-9 1 114.20 in 54 628.06 2 824.76 82 601.23 i 903.68 609.29 98503 516.521 2 196.695 83 482.39 i 421.29 4 705-61 85.611 418.018 I 680.174 84 380.47 i 040.82 3 284.32 73.102 332.407 I 262.156 85 294.50 746.32 2 243.50 61.236 259-305 929.749 86 223.31 523-01 I 497-iS 50241 198.069 670.444 87 165.52 357-49 974-17 40.210 147.828 472.375 88 119.71 237.784 616.680 3I-407 107.618 324 547 89 84.252 I53.532 378.896 23-833 76.211 216.929 90 57-571 95.961 225.364 17-565 52.378 140.718 91 38.058 57.903 129.403 12.496 34.813 88.340 92 93 24-275 14.929 33.628 18.699 71.500 37.872 8.5251 5-675 1 22.3170 31.2102 94 8.749 9-950 I9-I73 35412 8.1168 17.4183 95 4.912 5-038 9.223 2.1338 4-5756 9-30I5 96 2.612 2.426 4.185 1.2086 2.4418 4-7259 97 L3I5 i. in 1-759 .6183 1.2332 2.2841 98 -653 -458 .648 3317 .6149 1.0509 99 .299 159 .190 .1603 .2832 4360 100 .128 .031 .031 .0930 .1229 .1528 101 .031 .0299 .0299 .0299 39 H M (KING & HARDY) TABLE. TABLE V. COMMUTATION COLUMNS HM TABLE, MAKEHAM'S FORMULA. THREE PER CENT. Age. Dx N x Sx Cx MX R* 127 283 2 688 021 58 493 170 13 943- 45284. i 016 517. 1 109 6^6 2 578 385 56 255 149 3 734-6 31 344.2 971 233.2 2 102 708 2 475 77 53 676 764 2 173-5 27 609.6 939 889.0 3 97 544 2 378 133 51 201 087 I 4624 25 436.1 912 279.4 4 93 240 2 284.893 48 822 954 I 143-0 23 973-7 886 843.3 5 89 380 2 195 5*3 42 538 06 i 888.59 22 830.68 862 869.57 6 85 83 9 2 109 624 44 342 548 692.76 21 942.09 840 038.89 7 82 695 2 O26 929 42 232 924 539-16 21 249.33 818 096.80 8 79 746 I 947 183 40 205 995 426.89 2O 7IO 17 796 847.47 9 76 996 I 870 187 38 258 812 345-26 20 283.28 776 I37-30 10 74 41 I 795 777 36 388 625 294-75 19 938.02 755 854.02 11 71 947 I 723 830 34 592 848 258 81 19 643.27 735 916.00 12 69 592 I 654 238 32 869 018 235-61 19 384.46 716 272.73 13 67 332 I 586 906 31 214 780 222.80 19 148.85 696 888.27 14 65 146 I 521 760 29 627 874 216.31 i 8 926.05 677 739-42 15 63 032 I 458 728 28 106 114 224.34 i 8 709 74 658 813.37 16 60 972 i 397 756 26 647 386 232.33 i 8 485.40 640 103 63 17 58 964 i 338 79 2 25 249 630 249.64 18 253.07 621 618.23 18 S 6 997 I 281 795 23 910 838 265.18 18 003.43 603 365 1 6 19 55 072 i 226 723 22 629 043 281.27 17 73 8 - 2 5 585 361.73 20 53 188 i 173 535 21 402 320 29458 17 456 98 567 623 48 21 5i 343 I 122 192 20 228 785 30374 17 162.40 550 166.50 22 49 544 I 072 648 19 106 593 308.57 16 858.66 533 004 10 23 47 791 I 024 857 i 8 033 945 310.41 16 550.09 516 145.44 24 46 090 978 767 17 009 088 309.01 16 239.68 499 595 35 25 44 439 934 328 i 6 030 321 305.11 IS 930-67 483 355-67 26 42 839 891 489 19 095 993 298.92 15 625.56 467 425.00 27 41 291 850 198 14 204 504 294.16 15 326 64 451 799.44 28 39 796 810 402 13 354 306 287.71 15 032.48 436 472.80 29 38 349 772 053 12 513 94 28262 14 744-77 421 440 32 30 36 949 735 104 ii 771 851 276.39 14 462 15 406 695.55 31 35 597 699 507 ii 036 747 271.84 14 185.76 392 233.40 32 34283 6-35 219 10 337 240 26731 13 913.92 378 047.64 33 33 022 632 197 9 672 021 263.18 13 646.01 3 6 4 133 72 34 31 797 6 TO 400 9 039 824 259.08 13 383 43 350 487-11 35 30 612 569 788 8 439 424 256.01 13 124.35 337 103.68 36 29 464 540 324 7 869 636 253-24 12 868.34 3 2 3 979-33 37 28 352 511 972 7 329 312 25043 12 6I5.IO 311 110.99 38 27 277 484 695 6 817 340 248 1 8 12 364.67 298 495.89 39 26 234 458 461 6 332 615 247.09 12 116 49 286 131.22 40 25 223 433 238 5 874 184 244-95 ii 869.40 274 014.73 41 24 243 408 995 7 440 946 24446 ii 624.45 262 145.33 42 23 293 385 7^2 5 031 951 244-35 ii 379 99 250 520.88 43 22 370 3 6 3 33 2 4 646 249 243-77 ii i35- 6 4 239 140.89 44 21 474 341 858 4 282 917 244-34 10 891.87 2^8 005.25 45 20 604 3 2 i 254 3 94i 059 244-93 10 647.53 217 H3 38 46 19 760 3i 494 3 619 805 245-77 10 402.60 206 465 85 47 18 938 282 556 3 3i8 311 247.08 10 156.83 196 063 25 43 18 139 2H 417 3 35 755 249.29 9 909-75 185 906.42 a 17 362 247 055 2 77i 338 251-15 9 660.46 175 996.67 H M (KING & HARDY) TABLE. TABLE V.Cont. COMMUTATION COLUMNS HM TABLE, MAKEHAM'S FORMULA THREE PER CENT. Age. D x NX Sx Cx MX Rx 50 51 16 605. 15 868. 230 450. 214 582. 2 524 283. 2 293 833. 253.36 256.51 9 409-31 9 155-95 166 336 21 156 926 90 52 IS 149- 199 433- 2 079 251. 259.48 8 899.44 147 770.95 53 14 449. 184 984. I 879 818. 262.66 8 639.96 138 871 51 54 55 13 765. 13 098. 171 219. 158 121. I 694 834. i 523 615. 266.23 270.13 8 377-30 8 111.07 133 2 31 . 55 121 854.25 56 12 447. 145 674. i 365 494- 273-57 7 840.94 113 743 18 57 ii 810. 133 864. I 219 820. 277.49 7 567.37 105 902.24 58 ii 189. 122 675. i 085 956. 281.82 7 289.88 98 334 87 59 10 581. 112 093.8 963 280.6 285.50 7 008.06 91 044.99 60 9 987.6 102 106.2 851 186.8 289 21 6 722.56 84 036.93 61 9 407-4 92 698.8 749 080.6 292.78 6 433-35 73 3I4.37 62 8 840.6 83 858 2 656 381.8 29605 6 140.57 70 881.02 63 8 2869 75 571-3 572 523-6 29905 5 844.52 64 740.45 64 7 746.6 67 824.7 496 952-3 301 47 5 545-47 58 895.93 65 7 219-5 6O 605 2 427 127.6 303.20 5 244.00 53 350-46 66 6 705.9 53 899 3 368 522.4 304-I7 4 940.80 48 106.46 67 6 206.5 47 692.8 314 623.1 3 4-56 4 636.63 43 165-66 68 5 721.2 41 971.6 266 930.3 303.62 4 332-07 38 529.03 69 70 5 251.0 4 79^3 26 720.6 3i 924-3 224 958.7 188 238.1 301-59 298.46 4 028 45 3 726.86 34 196 96 30 168.51 71 4 3582 27 566.1 156 3I3-8 293-81 3 428.40 26 441.65 72 3 937-5 23 628.6 128 747.7 287.79 3 134-59 23 013.25 73 3 5350 20 093.6 105 119.1 280.09 2 846.80 19 878.66 74 3 152 o 16 941.6 ST 025.5 270.95 2 566 71 17 031.86 75 2 789.2 14 152.4 68 083.9 260.09 2 295 76 14 465.15 76 2 447.9 ii 704.5 53 931-5 247.69 2 035.67 12 169.39 77 2 128.9 9 575-6 42 227.0 233.60 I 787.98 10 133.72 78 I 833.3 7 742-3 32 651.4 218.28 I 554.38 8 345-74 79 I 561.6 6 180.7 24 909.1 201.67 I 336.10 6 791-36 80 I 3H-4 4 866.3 18 728.4 184.12 I 134-43 5 455 26 81 I 0^2.0 3 774-27 13 862.06 165.92 950.31 4 320-83 82 894-33 2 879.94 10 087.79 147-23 784.39 3 370.52 83 721.04 2 158.90 7 207.85 128.58 637.16 2 586.13 84 571-45 I 587-45 5 048 95 110.33 508.58 i 948.97 85 444-49 I 142.96 3 .461-50 92.871 398.246 i 440.392 86 33866 804.30 2 318-54 76-565 305-375 i 042.146 87 252.24 552.06 I 514-24 6i.574 228.810 736.771 88 183-31 368.75 962.18 48.329 167.236 507.961 89 12965 239.100 593-425 36.852 118.907 340-725 90 89.018 150.082 354-325 27.292 82.055 221.818 91 59-133 90.949 204.243 19.511 54-763 I39-763 92 37.901 53-048 113.294 13-375 35-252 85.000 93 23.422 29.626 60.246 8.9466 21.8765 49-7476 94 13-793 I5-833 30.620 5.6097 12.9299 27.8711 95 7.781 8.052 14.787 3-3967 7.3202 14.9412 96 4.158 3-894 6-735 I-933 2 3-9235 7.6210 97 2.104 1.790 2.841 .9936 1.9903 36975 98 1.049 .741 1.051 -5359 .9967 1.7072 99 .482 259 .310 .2602 .4608 .7105 100 .208 .051 .051 .1515 .2006 .2497 101 .051 .0491 .0491 .0491 392 H F (WOOLHOUSE) TABLE. TABLE I. INSTITUTE OF ACTUARIES' TABLES. FEMALE LIVES, HF. WOOLHOUSE'S FORMULA. Aee. No. Living. Decrement. Probability of Surviving. Probability of Dying. 10 IOO OOO 314 .996 860 .003 140 o 11 99 686 420 .995 786 8 .004 213 2 12 99 266 510 .994 862 3 .005 137 7 13 98 756 581 .994 116 8 .005 883 2 14 98 175 632 993 562 5 .006 437 5 15 97 543 667 .993 162 o .006 838 o 16 96 876 683 .992 949 8 .OO7 OjO 2 17 96 193 680 992 93 9 .OO7 069 I 18 95 513 659 .993 loo 4 .006 899 6 19 94 854 635 993 305 5 .006 694 5 20 94 219 648 993 122 4 .006 877 6 21 93 57i 682 .992 711 4 .007 288 6 22 92 889 736 .992 076 6 .007 923 4 23 92 153 813 .991 177 7 .008 822 3 24 9i 340 899 .990 157 7 .009 842 3 25 < 90 44i 978 ,989 186 3 .010 813 7 26 89 463 ici8 .988 621 o .on 379 o 27 88 445 i45 .988 184 7 .on 815 3 28 87 400 1050 .987 986 3 .012 013 7 29 86 350 1032 .988 048 6 oil 951 4 30 85318 IOII .988 T.^,0 2 .on 849 8 31 84 37 987 .988 2g2 8 .Oil 707 2 32 83 320 964 .988 430 2 .oil 569 8 33 82 356 960 988 343 3 .on 656 7 34 8 1 396 954 .988 279 5 .on 720 5 35 80 442 946 .988 240 o .on 760 o 36 79 496 946 .988 loo o ,011 900 o 37 78 550 946 987 956 7 .012 043 3 38 77 604 946 .987 809 9 .012 190 I 39 76 658 946 987 659 5 .012 340 5 40 75 7^2 950 987 452 5 .012 547 5 41 74 762 953 .987 252 9 .012 747 i 42 73 809 955 .987 061 2 .012 938 8 43 72 854 958 .986 850 4 .013 149 6 44 71 896 962 .986 619 6 .013 380 4 45 70 934 966 .986 381 7 .013 618 3 46 69968 963 .986 236 6 .013 763 4 47 69 005 958 .986 116 9 .013 883 i 48 68 047 953 985 995 o .014 005 o 49 67 094 95 .985 840 8 .014 159 2 50 66 144 956 985 546 7 014 453 3 51 65 188 975 985 043 3 ,014 956 7 52 64 213 1003 .984 380 i .0 5 619 9 53 63 2IO 1037 .983 594 4 .016 405 6 54 62 173 1081 .982 613 o .017 387 o H F (WOOLHOUSE) TAHLE. 393 TABLE l.Cont. INSTITUTE OF ACTUARIES' TABLES. FEMALE LIVES, HF. WOOLHOUSE'S FORMULA. Age. No. Living. Decrement. Probability of Surviving. Probability of Dying. 55 61 092 Ill6 .981 732 5 .018 267 5 56 59 9/6 1144 .980 925 7 019 074 3 57 58 832 1170 .980 112 9 .019 887 I 58 57662 1196 .979 258 4 .020 741 6 59 56466 1231 .978 199 3 .021 800 7 60 55 235 1308 .976 319 4 .023 680 6 61 53 927 1395 .974 131 7 .025 868 3 62 5 2 532 1495 .971 541 2 .028 458 8 63 5i 037 1601 .968 630 6 .031 369 4 64 49 436 1706 .965 490 7 34 509 3 65 47 730 1784 .962 623 i .037 376 9 66 45 946 1846 .959 822 4 .040 177 6 67 44 100 1914 .956 598 6 .043 401 4 68 42 186 1982 953 017 6 .046 982 4 69 40 204 2050 .949 oio o .050 990 o 70 38 154 2123 944 357 i 055 642 9 71 72 36 031 33 799 2232 2338 938 053 3 .930 826 4 .061 946 7 .069 173 6 73 31 461 2425 .922 920 4 .077 079 6 74 29 036 2490 .914 244 4 085 755 6 75 26 546 2518 .905 145 8 .094 854 2 76 24 028 2500 895 954 7 .104 045 3 77 21 528 2363 .890 236 o .109 764 o 78 19 165 2205 .884 946 5 "5 053 5 79 16 960 2024 .880 660.4 .119 339 6 80 14 936 1819 .878 213 7 .121 786 3 81 13 117 1621 .876 419 9 .123 580 i 82 ii 436 ISM .868 302 o .131 698 o 83 9 982 1450 854 738 5 .145 261 5 84 8 532 1389 .837 201 I .162 798 9 85 7 143 1326 .8r4 363 8 .185 636 2 86 5 817 1234 .787 863 2 .212 136 8 87 4 583 1086 763 37 3 .236 962 7 88 3 497 93 .741 778 6 .258 221 4 89 -2 594 707 .727 448 o .272 552 o 90 i 887 519 .724 960 3 275 039 7 91 i 368 368 .730 994 2 . .269 005 8 92 I OOO 232 .768 ooo o .232 ooo o 93 768 138 .820 312 5 .179 687 5 94 630 117 .814 285 7 .186 714 3 95 513 107 .791 423 o .208 577 o 96 406 104 743 842 4 .256 157 6 97 302 102 .662 251 6 337 7+8 4 98 2OO 100 .500 ooo o .500 ooo o 99 100 100 .000 ooo o I. COO OOO O 394 COMMUTATION COLUMNS, H 1 TABLE XII. COMMUTATION COLUMNS, HF. FOUR AND ONE-HALF PER CENT. A?e. D, NX Sx M x RX 10 64 392.8 I 148 675 19 196 516 12 155 4 334 I85-9 11 61 426.4 I 087 248 i 8 047 842 ii 961.9 322 030 4 12 58 533-6 I 028 715 I 6 960 594 " 7I4.3 310 068.5 13 55 72S- 2 972 989.5 15 93i 879 ii 426.5 298 354.2 14 53 on.8 919 977.7 14 958 889 II II2.8 286 927.8 15 50 402.5 869 575-2 14 038 912 10 786.2 275 815.0 16 47 902.2 821 673.0 13 169 336 10 456.4 265 028.8 17 45 516.3 776 156-7 12 347 663 10 I 33 2 254 572.4 18 43 2 48.3 732 908.4 II 571 507 9 825.30 244 439-2 19 41 100.4 691 808.0 10 838 598 9 539.76 234 613.9 20 39 067.2 652 740.8 10 146 760 9 276.46 225 074.1 21 37 127-8 615 613.0 9 494 050 9 019 34 215 797.7 22 35 270.0 580 342 9 8 878 437 8 760.39 206 778 3 23 33 483-8 546 859.1 8 298 094 8 492 96 198 017.9 24 3i 759-2 515 099.9 7 75i 234 8 210.28 189 525.0 25 30 092.5 485 007.4 7 236 135 7 9II-I5 181 314.7 26 28 485.2 456 522 2 6 75i 127 7 599-76 173 403-5 27 26 948.4 429 573-8 6 294 605 7 289.58 165 803 8 28 25 483-3 404 090.5 5 865 031 6 984.89 158 514.2 29 24 092.9 379 997-5 5 46o 941 6 691.92 151 529-3 30 31 22 779-9 21 540.6 3-7 217.6 335 677-0 5 080 943 4 723 726 6 416.38 6 158.07 144 837.4 138 421.0 32 20 371.7 315 305-2 4 388 049 5 9i6-74 132 263.0 33 10 268.9 296 036.3 4 072 743 5 691.20 126 346.2 34 I 8 224.2 277 812. i 3 776 707 5 476 26 120 655.0 35 17 235.1 260 577.0 3 498 895 5 271.86 115 178 8 36 16 298.9 244 278.1 3 238 318 5 077.90 109 906.9 37 IS 4II-5 228 866.7 2 994 040 4 892 3 104 829.0 38 14 570.2 274 296.5 2 765 173 4 714.68 99 936.71 39 13 772.8 200 523.7 2 550 8 77 4 544 72 95 222.03 40 13 017.1 187 506.6 2 350 353 4 382.08 9 677.3 1 41 12 3OO.2 175 206.4 2 162 846 4 225 78 86 295.24 42 43 It 6205 10 976.2 163 585-9 152 609.7 987 640 824 054 4 075 74 3 93i 86 82 069.46 77 993-72 44 10 365-4 142 244 2 671 445 3 793-74 74 061.87 45 9 786.37 132 457-8 529 200 3 651-02 70 268.13 48 9 237-41 123 220 4 396 743 3 533-48 66 607.12 47 8 717.96 114 5025 273 5 2 2 3 411.82 63 073-63 48 8 226.73 106 275 8 159 020 3 296.00 59 661.82 49 7 762 21 9 8 5I3-54 052 744 3 i 8 5-74 56 365-82 50 7 322.78 91 190.76 954 230.3 3 080.57 53 180 07 51 6 906.16 84 284.60 863 039.5 2 979-29 50 099.51 52 6 509 92 77 774.67 778 754 9 2 88044 47 1 20. 22 53 6 132.29 71 6;2.39 700 980.3 2 783.I4 44 239-77 54 5 771-94 65 870.44 629 337 9 2 686.87 41 456.64 COMMUTATION COLUMNS H F . 395 TABLE XII. Cont. COMMUTATION COLUMNS, HF. FOUR AND ONE-HALF PER CENT. Age. DX N x Sz M x KX 55 5 427-36 60 443-09 563 467.4 2 590.83 38 769.77 56 5 098.77 55 344-32 503 024.4 2 495-96 36 178.94 57 4 786.14 50 558.18 447 680.0 2 402.89 33 682.98 58 59 4 488.95 4 206.55 46 069.23 41 862.68 397 121.9 351 052.6 2 311 80 2 222.71 31 280.10 28 968.29 60 3 937.65 37 925-03 309 189.9 2 134-95 26 745-59 61 3 678.85 34 246.18 271 264 9 2 045.72 24 610.64 62 3 429-37 30 816.81 237 018.7 I 954.65 22 564.92 63 3 188.30 27 628.52 2~>6 201.9 I 861 26 20 610.27 64 2 955-29 24 673.22 178 573 4 I 765.55 18 749.01 65 2 730-44 21 942.78 153 900.2 I 667.96 16 983 46 66 2 515.20 19 427.58 131 957.4 I 57030 15 315.50 67 2 310.19 17 117.40 112 529.8 i 473-59 13 745-21 68 2 II476 15 002.64 95 412 42 i 377.64 12 271.02 69 I 928.61 13 074.02 80 409.78 i 282.57 10 893.97 70 I 75I-4 6 ii 322.57 67 335.76 i 188.46 9 611.403 71 I 582.78 9 739-789 56 013.19 i 095.20 8 422-941 72 I 420.79 8 318.996 46 273 40 i 001.38 7 3 2 7.739 73 I 265.56 7 053 434 37 954.41 907.327 6 526 362 74 I 117.72 5 935.719 30 900.97 813.979 5 4I9-035 75 977-861 4 957.858 24 965.25 722.256 4 605.056 76 846.992 4 110.866 20 007.39 633-496 3 882.800 77 78 726.188 618.640 3 384-677 2 766.037 15 896.53 12 511.85 549.166 472.889 3 249-303 2 700.138 79 523.889 2 242.148 9 745.815 404.777 2 227.249 80 441.500 I 800.648 7 503.667 344-949 I 822.472 81 371.035 I 429.613 5 703-019 293.495 I 477-5 2 4 82 SII.ISO I 118.4^3 4 273.406 249.617 I 184.028 83 258.563 859.871 3 154-973 210.400 934.411 84 211.486 648384 2 295.102 174-458 7240II 85 169.432 478.952 I 646.718 141.5" 549-552 86 132.038 34&-9I5 I 167.766 111.413 408.041 87 99.548 o 247.367 820.851 84.609 I 296.628 88 72.687 9 174.679 573-45 62.035 8 212.019 89 5I-596 5 123.082 398.806 ' 44-074 4 149.983 90 35.917 5 87.165 275.724 30.617 3 105.909 91 24-917 5 62.247 188.559 21 164 75.29I 92 17.430 2 44.817 126.312 14719 7 54-1*7 93 12.809 9 32.007 81-495 10 880 o 39-378 94 10.055 6 21.952 49.488 8.677 3 28.498 95 7.835 6 J4.II6 27.536 6.890 3 19.820 96 5-934 2 8.182 13.420 5-326 3 12 930 97 4.224 o 3-958 5.239 3-871 7 7.604 98 2.676 9 I.28I I.28l 2.506 5 3-732 99 i.k8o 8 000 .000 1.225 7 1.226 H F (WOOLHOUSE) TABLE. TABLE III. COMMUTATION COLUMNS, HF. FOUR PER CENT. Age. D x N. Sx M Rx 10 67 556.4 308 776 23 182 517 14 620.6 431 760 9 11 64 754.1 244 021 21 873 741 14 416.6 417 140.3 12 62 001 2 182 020 2O 629 720 14 154-3 402 723.7 13 59 3io-3 122 7IO 19 447 700 13 848.0 388 569.4 14 56 693.6 066 016 i 8 324 990 13 512.5 374 721-5 15 54 162.2 i on 854 17 258 973 13 161.5 361 209.0 16 51 722.9 960 I3I.2 16 247 119 12 805.4 348 047.5 17 49 3 82 -9 910 748.3 15 286 988 12 454-8 335 242 o 18 47 147-9 863 600 4 14 376 240 12 Ilg.I 322 787 2 19 45 021.7 818 578.7 13 512 639 II 806.3 310 668.1 20 43 000.3 775 578.4 12 694 061 II 516.5 298 861.8 21 41 062.1 734 516.3 II 918 482 II 232.2 287 345-3 22 39 195-0 695 321 3 II 183 966 10 944.4 276 113.1 23 37 388.9 657 932.4 I 488 645 10 64^.8 265 168 7 24 35 633-7 622 298.7 9 830 712 10 328.6 254 522.9 25 33 925-9 588 372-7 9 208 414 9 991.38 244 194-3 26 32 268.4 556 104.4 8 620 041 9 638.62 234 202 9 27 30 674.2 525 430-2 8 063 937 9 285.56 224 564.3 28 29 145-9 496 284.2 7 538 507 8 937.08 215 278.7 29 27 688.3 468 596.0 7 042 222 8 600.40 206 341.7 30 26 305.1 442 290 9 6 573 626 8 282.21 197 741-3 31 24 993 7 417 297.2 6 131 335 7 982 49 189 45Q.i 32 23 751-0 393 546.2 5 714 038 7 701.14 181 4766 33 22 573 3 370 972.9 5 320 492 7 436-91 173 775 4 34 21 452.1 349 520.8 4 949 519 7 183 90 166 338.5 35 20 385 3 329 135 5 4 599 998 6 942.14 159 154 6 36 19 370.7 309 764 8 4 270 863 6 711.63 IS2 212.5 37 18 404.0 291 360.8 3 961 098 6 489 99 I 4 5 500.9 38 17 483-1 273 877-8 3 669 737 6 276.87 139 010-9 39 16 605.7 257 272.1 3 395 860 6 071.94 132 734-0 40 15 770.0 241 502.1 3 138 588 5 874.90 126 662.1 41 14 973.2 226 528 9 2 897 085 5 684.64 I2O 787.2 42 14 213.8 212 315.1 2 670 557 5 501-11 115 IO2.5 43 13 490-3 198 824.9 2 458 24 t 5 324.28 109 601.4 44 12 800.8 186 024.1 2 259 417 5 153-71 104 277.1 45 12 143 8 173 880.3 073 392 4 989.02 99 123.41 46 II 5177 162 362.6 899 512 4 830.00 94 134.40 47 10 922 3 I5i 440 3 737 ISO 4 677-57 89 304.40 48 10 356.4 141 083 9 585 709 4 531-77 84 626 82 49 9 818.01 131 265.3 444 625 4 392.31 80 095.05 50 9 307.30 121 958.0 313 360 4 258.63 75 702.74 51 8 819.98 113 138 o 191 402 4 129.28 71 444 ii 52 8 3^3.90 104 784.1 078 264 4 002.44 67 314-83 53 7 907-13 96 876.96 973 480.2 3 876.97 63 312.39 54 7 478 28 8 9 398.69 876 603 2 3 752-24 59 435.42 H F (WOOLHOUSE) TABLE. 397 TABLE Ul.Cont. COMMUTATION COLUMNS. HF. FOUR PER CENT. Age. DX NX Sx M x Rx 55 7 065.63 82 333.06 787 204.5 3 627.22 55 683.18 56 6 669.77 75 663.30 704 871.5 3 503-11 52 055.96 57 6 290.91 69 372.39 629 208.2 3 380.78 48 552.85 58 5 928.65 63 443-74 559 835 8 3 260.48 45 172.07 59 5 582.39 57 861.35 496 392.0 3 142-24 41 911.59 60 5 250.66 52 610.69 438 530.7 3 025.22 38 769 35 61 4 929.16 47 681.53 385 920.0 2 905 67 ; 35 744.12 62 4 616.97 43 064.56 338 238.5 2 783-06 32 838.45 63 4 3I3-05 38 75I-5I 295 173-9 2 656.72 : 30 055.39 64 4 017.07 34 734 44 256 422.4 2 526.63 27 398.67 65 3 729-27 ' 31 005.16 221 688.0 2 393-33 24 872.04 66 67 2 451.81 3 185.70 27 553-35 24 367.65 190 682.8 163 129.5 2 259 31 2 125.96 22 478.70 20 219.40 68 2 930 23 21 437-42 138 761.8 i 993-01 ' 18 093 44 69 2 685.15 1 3 752.27 117 324.4 i 860.64 16 100.43 70 2 450.23 16 302.05 98 572.11 i 728.99 14 239.80 71 2 224.89 14 077.15 82 270.07 i 597.89 12 5I0.8l 72 2 006.80 12 O7O 36 68 192.92 1 465 37 10 912.92 73 I 796.13 10 274.22 56 122.56 i 331 89 9 447-551 74 i 593-93 8 680.290 45 848.34 i 198.77 8 115.602 75 i 401.20 7 279.095 37 168.05 i 067.34 6 916.893 76 T 219.51 6 059.589 29 888.96 939-541 5 849-555 77 I 050.60 5 008.991 23 829.37 8I7-537 4 910.014 78 899.308 4 109.684 18 820.38 706.654 4 092.477 79 765.230 3 344-454 14 710.69 607.165 3 385-823 80 647.988 2 696.465 ii 366.24 5I9-355 2 778.658 81 547.185 2 I49.28I 8 669.773 443-475 2 259 302 82 461.119 I 688.162 6 520 493 378.454 I 815.828 83 384.991 I 303 171 4 832.331 320.062 I 437-373 84 316.4X0 986.761 3 529-161 266.288 I II7.3I2 85 254-7II 732.050 2 542.400 216.758 851.024 86 199.449 532.601 i 810.350 171.293 634265 87 I5I-095 381.506 i 277.749 130.610 462.972 88 110.857 270.650 896.243 96-183 4 332.362 89 76.068 4 191.581 625.593 68.658 8 236.179 90 55-305 9 136.275 434-012 47 937 4 167 520 91 38.552 5 97-723 297.737 33-3 11 I II9-583 92 27x97 7 70.625 200.014 23-339 i 86.271 93 20.010 6 50615 129.389 I7-294 3 62.932 94 15 783 6 34.83I 78.774 13.836 9 45.638 95 12.358 i 22-473 43-943 11.018 4 31.801 96 9404 3 13 069 21.470 8-539 9 20.783 97 6.726 3 6.342 8.402 6.223 6 12.243 98 4.283 i 2.059 2.059 4 039 2 6.019 99 2.059 2 .OOO .000 i 980 o I 980 H F (WOOLHOUSE) TABLE. TABLE IV. COMMUTATION COLUMNS, HF. THREE AND ONE-HALF PER CENT. Age. D x NX Sx MX R* 10 70 891.9 499 667 28 199 634 17 781.2 563 837.1 11 68 279.5 431 387 26 699 967 17 566.1 546 055.9 12 65 692.6 3 6 5 695 25 268 580 17 288.2 528 489.8 13 63 145-0 302 550 23 902 885 16 962.1 511 201.6 14 60 650.7 241 899 22 600 335 16 603.1 494 239-6 15 58 222.5 183 677 21 358 436 16 225 9 477 636.4 16 55 869.0 127 808 20 174 759 15 841.3 461 410.5 17 53 599-1 074 208 19 046 952 15 460.7 445 569-2 18 51 420.5 I 022 788 17 972 743 15 094.6 430 108.6 19 49 33 8 -9 973 449-1 16 949 955 14 751-8 415 014.0 20 47 351-3 926 097.9 15 976 506 14 432.7 400 262.2 21 45 435-4 880 662.5 15 050 408 14 118.0 ' 385 829.5 22 43 578.9 837 083.6 14 169 746 13 798.1 371 711.4 23 41 771.6 705 312.0 13 332 662 13 464 5 357 9134 24 40 003.0 755 309-0 12 537 350 13 108.4 344 44 8 -9 25 38 269.8 717 039.1 ii 782 041 12 728.0 331 3405 26 36 575-9 680 463.3 ii 065 002 12 328.1 318 612.5 27 34 936-"9 645 526.4 10 384 539 II 926.0 306 284 4 28 33 356.6 612 169.8 9 739 012 II 527-2 294 358 4 29 31 841.4 580 328.4 9 126 843 II 140.0 282 831 2 30 30 397.0 549 931-5 8 546 514 10 772.3 271 691.2 31 29 021.0 520 910.4 7 996 583 10 424.3 260 918.9 32 27 711.4 493 I99-I 7 475 672 10 096.0 250 494.5 33 26 464.5 466 734.6 6 982 473 9 786.27 240 398.5 34 25 27L5 441 463.1 6 515 739 9 488.21 230 6l2.2 35 24 130 7 417 332.3 6 074 276 202 03 221 124.0 36 23 040.5 394 291 8 5 656 943 927-85 211 922.0 37 21 996.5 372 295 3 5 262 651 8 662.94 202 994 i 38 20 996.7 351 298.6 4 890 356 8 406 99 194 331-2 39 20 039.4 331 259.2 4 539 058 8 159.70 185 924.2 40 19 122.8 312 136-5 4 207 798 7 920 76 177 764.5 41 42 18 244.3 17 402.6 293 892.2 276 489.6 3 895 662 3 601 770 7 688.93 7 464.24 169 843.7 162 154.8 43 16 596.6 259 893.0 3 325 280 7 246.68 154 690.6 44 15 824.5 244 068.5 3 065^ 387 7 035.82 147 443-9 45 15 084.8 228 983.8 2 82t 319 6 831.24 140 408 i 46 14 376.2 214 607.6 2 592 335 6 632.76 133 576.8 47 13 698.9 200 908.7 2 377 727 6 441.59 126 944.1 48 13 05I-9 187 856.9 2 176 818 6 257.84 120 502.5 49 12 433-9 175 423.0 I 988 962 6 081.23 114 244.6 50 ii 843-3 163 579-7 i 813 539 5 9ii-i3 108 163.4 51 II 277.4 152 302.3 i 649 959 5 745-74 102 252.3 52 10 733 i 141 569.2 I 497 657 5 582 77 9 6 506.54 53 10 208.2 131 361 o I 356 087 5 420.79 90 923.76 54 9 701.14 121 659.9 i 224 726 5 258.98 85 5^2 97 H F (WOOLHOUSE) TABLE. 399 TABLE FJ.Cont. COMMUTATION COLUMNS, HF. THREE AND ONE-HALF PER CENT. Age. D x N, s. MX R, 55 t2IO 12 112 449 8 I 103 067. 5 096.01 80 243.99 56 736.11 103 713.7 990 616.8 4 933.46 75 147-98 57 8 279.68 95 433-97 886 903.1 4 772 46 70 214.52 58 7 840 60 87 593-37 791 469.1 4 613.37 65 442.07 59 7 4i 8 -33 80 175.04 7C3 875-8 4 456.24 60 828.70 60 7 Oil. 22 73 163.82 623 700.7 4 299.98 56 372.46 61 6 613.71 66 550.11 55 536 9 4 139-57 52 072.48 62 6 224.75 60 325.36 483 986.8 3 Q74-27 47 932.91 63 5 843-10 54 482.26 423 661.4 3 803.11 43 958.65 64 5 468.41 49 013-85 369 I79- 2 3 626.01 40 155-54 65 5 101.16 43 9 I2 -/o 320 165.3 3 443-68 36 529-53 66 4 744-44 39 168.26 276 252.6 3 259.47 33 085.84 67 4 399 82 34 768.44 237 084.4 3 075-29 29 826.38 68 4 066.54 30 701.90 202 315.9 2 890.79 26 751.09 69 3 744-43 26 957.48 171 614.0 2 7C6.20 23 860.30 70 3 433-33 23 524-I5 144 656.5 2 521.72 21 154.10 71 3 132-65 20 391.50 121 132.4 2 337- T4 18 632.38 72 2 839.22 17 552.28 ioo 740.9 2 14965 16 295.23 73 2 553-45 14 998-83 83 188.62 I 959.89 14 I45-58 74 2 276 94 12 721. gj 68 189.79 I 769.73 12 185.69 75 2 on 28 10 710.62 55 467-89 I 581.07 10 415-96 76 i 758.9 \ 8 9=51.676 44 757,28 i 396 75 8 834.888 77 78 i 522 64 i 30967 7 429-037 6 119.368 35 805 60 28 376 56 I 2I9.Q2 I 058.45 7 438.I43 6 218.219 79 i 119.79 4 999-574 22 257.19 912.859 5 159-774 80 952.810 4 046 764 17 257 62 783.742 4 246.915 81 808.474 3 238.290 13 210 86 671.627 3 463-172 82 6-4.602 2 553-688 9 972.568 575-094 2 791-545 83 84 574 339 474- 39 i 979 348 i 5-5-039 7 418.880 5 439-532 487-983 407.375 2 216451 I 728.468 85 383.664 i 121.375 3 934-493 332769 I 321.094 86 301 876 819.499 2 8I3.II7 263.955 988.325 87 229.794 589 705 I 993-618 2O2.O82 724.369 88 169.412 420 292 I 403.9X4 149.471 522.288 89 121.417 298 876 983.621 I07.2G4 372.8T7 90 85-337 6 213 538 684 746 75-230 7 265.613 91 59 774 3 IS3-764 471 208 52.55.3 2 190.382 92 42 217 o "1-547 3I7-444 37-017 3 137.829 93 31-326 3 80.220 205898 27-554 2 IOO 8 12 94 24.828 3 55-392 125 667 22.115 6 73.258 95 19-533 7 35.858 70.285 17.660 5 51.142 96 14.936 6 20 922 34.427 13.724 o 33-482 97 10.734 8 10.187 I3.5 5 10.027 3 19758 98 6.868 7 33i8 3-3I8 6.524 2 9-730 99 3-318 2 .000 .000 3.206 o 3.206 400 H F (WOOLHOUSE) TABLE. TABLE V. COMMUTATION COLUMNS, HF. .THREE PER CENT. Age. D, NX s, M x KX 10 74 49 4 729 35- 34 5^2 827. 21 881.9 744 232.8 11 72 015.3 657 O2O. 32 833 792. 21 655.0 722 350.9 12 69 623 2 587 397- 31 176 772. 21 360.5 700 695.9 13 67 248.0 520 149. 29 589 375- 21 OI^ 2 679 335-4 14 64 905.2 455 244. 28 069 226. 20 629.1 658 322.3 15 62 609.1 392 634. 26 613 982. 20 223.4 637 693.2 16 63 369.9 S3 2 265. 25 221 348. 19 807.7 617 469.8 17 58 198.4 274 066. 23 889 083. 19 394 5 597 662.1 18 56 103 8 217 962. 22 615 017. 18 995.1 578 267.5 19 54 093-9 163 863. 21 397 055. 18 619.3 559 272.4 20 52 166.8 in 702. 20 233 186. 1 8 267.7 540 653.2 21 50 299.0 061 403. 19 121 484. 17 919.4 522 385.5 22 4 3 478.1 012 925. 18 060 082. 17 563-4 504 466.1 23 46 693 2 966 23 r. 4 17 047 157. 17 190.5 486 902.7 24 44 933- 2 921 298.2 I 6 080 926. 16 790.6 469 712.2 25 43 195 i 878 103.1 15 159 628. 16 361.2 452 921.6 26 41 483-5 836 619.6 14 281 524. 15 907.7 436 560.4 27 39 8 7.0 796 802 6 13 444 905. 15 449-4 420 652 7 28 3 8 203 5 758 602.1 12 648 102. 14 992.7 405 203.3 29 36 642.3 721 959.8 II 889 500. 14 547-1 390 210.7 30 35 149-9 686 809.9 II 167 540. 14 121.9 375 663 6 31 33 721-7 6 3 088.2 10 480 731. 13 7I7-5 361 541.6 32 32 3S6-3 620 731.9 9 827 642. 13 334-3 347 824.1 33 31 050.4 589 681.6 206 910. 12 97O.8 334 489-8 34 29 794.6 559 887.0 617 229. 12 619.4 321 519.0 35 28 587.8 531 299.2 8 057 342. 12 280.4 308 899.6 36 27 428 7 503 870.5 7 526 043 II 954.0 296 619.3 37 26 312.9 477 557-6 7 022 172. II 637.1 284 665.3 38 25 238.9 452 3^8 7 6 544 615- II 329.4 273 028.2 39 24 205.0 428 113 7 6 092 296. II 0307 261 698.8 40 23 210.0 404 903.7 5 664 182. 10 740.7 250 668.1 41 22 251.3 382 652.4 5 259 279. 10 4s8.o 239 927.4 42 21 327.8 361 324.6 4 876 626. 10 182.6 229 460 5 43 20 438 7 340 886.0 4 SiS 3^2. 9 9H-65 219 286.9 44 I 9 582.4 321 303-5 4 174 416. 9 653-72 209 372 3 45 1 8 757-7 302 545-8 3 853 112. 9 399-33 199 718.5 46 17 963-3 284 582.5 3 550 566. 9 I5I-33 190 319.2 47 17 200. 1 267 382.4 3 265 984. 8 911.29 181 167.9 48 1 6 467.3 250 915.1 2 998 601. 8 679.45 172 256.6 49 15 763.8 235 I5I-3 2 747 686. 8 455-55 163 577-1 50 15 087 9 220 063 4 2 512 535. 8 238.85 155 121. 6 61 14 436 7 .205 626.7 2 292 47?. 8 027.13 146 882.7 52 13 806.6 191 82O.I 2 086 845. 7 817-49 138 85=5.6 53 13 195 i 178 624.9 , i 895 025. 7 608. 1 1 131 038.1 54 12 600.6 166 024.3 I 716 400. 7 397-94 123 430.0 H F (WOOLHOUSE) TABLE. 401 TABLE V.Cont. COMMUTATION COLUMNS. HF. THREE PER CENT. Age. P. NX S, MX Rx 55 12 020.9 154 003.4 I 550 376. 7 185-24 116 032.1 56 II 457.6 142 545.9 i 396 372. 6 972.04 108 846.8 57 10 911.7 131 634.2 i 253 826. 6 759-86 101 874.8 58 to 383.2 121 25I.O I 122 192. 6 549.18 95 "4-93 59 9 871.67 i" 379-3 i ooo 941. 6 340.09 88 565.75 60 9 375-21 102 004.1 889 561.8 6 131.15 82 225.66 61 8 886.60 93 "7-50 787 557-7 5 915-60 76 094.51 62 8 404.58 84 712.92 694 440.2 5 692.42 70 178.91 63 7 927-57 76 785.35 609 727.3 5 460.20 64 486.49 64 7 455.23 69 330.12 532 942.0 5 218.76 59 026.29 65 6 988.31 62 341.82 463 611.8 4 968.98 53 807.53 66 6 531.17 55 810.65 401 270.0 4 715.39 48 838.555 67 6 086.18 49 724.47 345 459-4 4 460.62 44 123.16 68 5 652.45 44 072.02 295 734-9 4 204.17 39 662.54 69 5 229.99 38 842.03 251 662.9 3 946.34 35 458.37 70 4 818.75 34 023.28 212 820.9 3 687.43 31 512.04 71 4 418.08 29 605.20 178 797.6 3 427-11 27 824.61 72 4 023.68 25 581.52 149 192.4 3 161.39 24 397-50 73 3 636.26 21 945.26 123 610.9 2 891.17 21 236.11 74 3 258.23 18 687.03 101 665.6 2 619.05 18 344.94 75 2 892.06 15 794-97 82 978.57 2 347.78 15 725-89 76 2 541-49 13 253.48 67 183.61 081.44 13 378.li 77 2 210.74 H 042.74 53 930.13 824.71 ii 296.67 78 79 I 910.76 I 641.67 9 131.980 7 490.313 42 887.40 33 755-42 589.12 375.69 9 471-955 7 882.832 80 81 I 403.64 I 196.79 6 086.671 4 889.877 26 265.10 20 178.43 185.48 019.51 6 507.145 5 321-668 82 I 018.34 3 87L533 15 288.55 875.920 4 302.156 83 858.476 3 013.058 II 417.02 745.713 3 426.236 84 7I2.40O 2 300.658 8 403.964 624.641 2 680.523 85 579.05I i 721.607 6 103.307 512.041 2 055.882 86 I 263.784 4 381-700 407.679 i 543.841 87 350.196 13.588 3 117-916 I 136.161 88 259.430 54.158 2 204.329 232.820 822.774. 89 186.835 467-323 i 550.171 167.781 589-954 90 I3L954 335-370 i 082.848 118.342 422.173 91 92.875 o 242.495 747.478 83.106 9 303-830 92 65.913 6 176.581 504.984 58.850 7 220.723 93 49-147 3 127.434 328.403 44.004 i 161.873 94 39.141 9 88.292 200.969 35-430 2 117.869 95 30.944 3 57-347 112.677 28.372 7 82.438 96 23.776 7 33571 55-330 22.106 4 54.066 97 17.171 o 16.400 21-759 16.193 2 31-959 98 11.040 3 5-359 5-359 10.562 7 15.766 99 5-359 4 .000 .000 5-203 3 5-203 INTRODUCTION TO PARK'S ENGLISH LIFE TABLES No. 3. THE English Life Tables No. 3, Male and Female, are from the population experience of Great Britain, and were graduated by Dr. William Farr and published in the 60 's. This table (males), with five per cent interest, was made the first standard for valuations in the State of New York in 1865, but was abandoned after one year for the Ameri- can Experience Table and four and one-half per cent. PARK'S ENGLISH TABLE No. 3. MALES. 405 TABLE I. FARR'S ENGLISH LIFE TABLE, No. 3. MALES. Proba- Proba- Proba- Proba- Age Living. Dying. bility of Living. bility of Dying. Age. Living. Dying. bility of Living. bility of Dying. 5"745 83719 .83640 .16359 55 209539 5144 97545 .02455 1 428026 27521 93570 .06428 56 204395 5281 .97416 .02584 2 400505 14215 .96451 .03549 57 199114 5428 97274 .02726 3 386290 9213 97615 02385 58 193686 5584 .97117 .02883 4 377077 6719 .98218 .01782 59 188102 5752 .96942 .03058 5 370358 5033 .98641 01359 60 182350 5929 .96748 .03252 6 365325 3953 .98918 .01082 61 176421 6118 96532 .03468 7 361372 3310 .99084 .00916 62 170303 63*4 .96293 .03707 8 358062 2734 .99236 .00764 63 163989 6515 .96027 03973 9 355328 2297 99354 .00646 64 157474 6720 95733 .04267 10 353031 1983 .99438 .00562 65 150754 6921 .95409 .04591 11 351048 1776 99494 .00506 66 143833 7H5 .95053 .04947 12 13 14 349272 3476o6 345969 1666 1637 1679 99523 99529 .99515 .00477 .00471 .00485 67 68 69 136718 129421 121963 7297 7458 7593 .94663 .94237 93774 .05337 05763 .06220 15 344290 1781 99483 .00517 70 114370 7695 .93272 .06728 16 17 342509 340581 1928 2112 99437 .99380 .00563 .00620 71 72 106675 98919 7756 7770 .92729 .92144 .07271 .07856 18 338469 2320 .99314 .00686 73 91149 7733 .91516 .08484 19 336149 2541 99244 .00756 74 .90843 20 333608 2764 .99172 .00828 75 75777 7483 .90124 09157 21 330844 2801 .99153 .00847 76 68294 7268 .89359 .10641 22 328043 2836 .99136 .00864 77 61026 6990 88545 455 23 325207 2868 .99118 .00882 78 54036 6655 .87684 .12316 24 25 322339 319442 2897 2926 .99101 .99084 .00899 .00916 79 80 4738i 4IH5 6266 5832 86775 .85816 13225 .14184 26 316516 2954 .99067 .00933 81 35283 5361 .84807 I5I93 27 3T3562 2981 .99049 00951 82 29922 4862 .83750 .16250 28 310581 3009 .99031 .00969 83 25060 4349 .82644 .17356 29 307572 3038 .99012 .00988 84 20711 3834 .81488 .18512 30 304534 3068 .98993 .01007 85 16877 3328 .80285 I97I5 31 301466 3100 .98972 .01028 86 *3549 2840 .79035 .20965 32 298366 3134 .98950 .01050 87 10709 2384 77737 .22263 33 295232 3171 .98926 .01074 88 8325 1965 76395 .23605 34 292061 3211 .98901 .01099 89 6360 1590 .75009 .24991 35 288850 3254 .98873 .01127 90 4770 1260 .73580 .26420 36 285596 3300 .98844 .01156 91 979 .72110 .27890 37 282296 3352 .98813 .01187 92 2531 744 .70601 .29399 38 278944 3406 .98779 .01221 93 1787 553 .69056 30944 39 275538 3465 98742 .01258 94 1234 401 .67476 40 41 272073 268544 3529 3596 .98703 .98661 .01297 01339 95 96 833 548 1 .65863 .64221 34137 35779 42 264948 3668 98615 01385 97 352 .62552 .37448 43 261280 3746 .98567 .01433 98 220 86 .60859 39HI 44 257534 3826 .98514 .01486 99 134 55 59144 .40856 45 253708 3912 .98458 .01542 100 79 33 57412 .42588 46 249796 4OOI .98398 .01602 101 46 21 .5S66 4 44336 47 48 245795 241700 4095 4192 .98266 .01666 01734 102 103 25 14 II 7 .53905 .52137 .46095 .47863 49 237508 4292 .98193 .01807 104 7 3 .50364 .49636 50 233216 4395 .98116 .01884 105 4 2 .48589 .51411 51 228821 4626 .97978 .02022 106 2 I .46816 .53184 52 224195 4758 .97878 .02112 107 I I .45048 .54952 53 219437 4885 97774 .O2226 108 .43288 .56712 54 214552 5013 .97664 02336 406 FARR'S ENGLISH LIFE TABLE No. 3, TABLE II. FARR'S ENGLISH LIFE TABLE, No. 3, COMMUTATION COLUMNS. FIVE PER CENT. MALES. Age. Dx N, Sx M, Rx 5"745 6497341 106086554 177979.04 1623582.49 1 407644 6089697 99589213 98246.67 1445003.45 2 363270 5726427 93499516 73284.31 1347356.78 3 333692 5392735 87773089 61004.86 1274072.47 4 310222 5082513 82380354 53425.30 1213067.61 5 290185 4792328 77297841 48160.79 1159642.31 6 7 272611 256820 45I97I7 4262897 72505513 67985796 44405.09 4*595-77 1111482.52 1067076.43 8 242351 4020546 63722899 39355-43 1025480.66 9 229048 3791498 59702353 37593-07 986125.23 10 216730 3574768 36182.91 948532.16 11 205251 3369517 52336087 35023.49 912349.25 12 194488 3175029 48966570 34034-55 877325-76 13 184343 2990686 4579I54I 33*5103 843291.21 14 174738 2815948 42800855 32324.23 810140.18 15 165609 2650339 39984907 31516.60 7778*5-95 16 156908 2493431 37334568 30700.70 746299.35 17 148594 2344 8 37 34841137 29859.52 715598.65 18 140641 2204196 32496300 28981.94 685739-13 19 133025 2071171 30292104 28063.84 656757-19 20 125733 1945438 28220933 27106 16 628693.35 21 "8754 1826684 26275495 26114.04 601587.19 22 112141 I7I4543 24448811 25*56.52 575473-15 23 105878 1608665 22734268 24233 20 550316.63 24 99947 1508718 21125603 23343.93 526083.43 25 94332 1414386 19616885 22488.44 502739.50 26 89017 1325369 18202499 21665 53 480251.06 "27 83987 1241382 16877130 20874.31 458585-53 28 79227 1162155 IS635748 20113.88 437711.22 29 74723 1087432 14473593 19382.85 417597-34 30 70462 1016970 13386161 18679.93 398214.49 81 66431 950539 12369191 18003.87 379534-56 32 62617 887922 11418652 I7353-28 361530.69 83 59009 828913 10530730 16726.88 344177.41 34 55595 773318 9701817 16123 26 327450.53 35 52366 720952 8928499 I554I-I4 311327.27 36 671642 8207547 I4979-3I 295786.13 87 46419 625223 7535905 14436.67 280806.82 88 4368 4 581539 6910682 i39 r *-73 266370.15 39 41096 540443 6329143 I3403- 73* 252458.42 40 38647 501796 5788700 12911.54 239054.69 41 36329 465467 5286904 12434.13 226143.15 42 34136 43I33I 4821437 11970.82 213709.02 43 32060 399271 4390106 11520.74 201738.20 44 30096 369175 3990835 11082.98 190217.46 45 28237 340938 3621660 10657.16 179134.48 46 26478 314460 3280722 10242 50 168477.32 47 24813 289647 2966262 9838.60 158234.82 48 23238 266409 2676615 9444.90 148396.22 49 21747 244662 2410206 9061.06 138951.32 60 20337 224325 2165544 8686.78 129890.26 61 19004 205321 1941219 8321.77 121203.48 52 17733 187588 1735898 7955-87 112881.71 63 16530 171058 1548310 7597-45 104925 84 54 15393 155665 1377252 7246.99 97328.39 FARE'S ENGLISH LIFE TABLE No. 3. 407 TABLE \\.~Cont. FARR'S ENGLISH LIFE TABLE, No. 3, COMMUTATION COLUMNS, FIVE PER CENT. MALES. Age. Dx NX Sx MX Rx 55 I43I7 141348 1221587 690447 90081.40 56 I330I 128047 1080239 6569.74 83176.93 57 12340 115707 952192 6242.45 76607.19 58 11432 104275 836485 5922.07 70364.74 59 10574 93701.3 732210.1 5608.18 64442.67 60 9762.2 83939-1 638508.8 5300.24 58834.49 61 8995.0 74944-1 554569.7 4997.94 53534-25 62 8269.6 66674.5 479625.6 4700.86 48536-31 63 7583.9 59090.6 412951.1 4408.86 43835.45 64 6935^ 52I54-9 353860.5 4121.91 39426.59 65 6323.6 458313 301705.6 3840.03 35304.68 66 5746.0 40085.3 255874.3 3563-54 31464.65 67 5201.6 34883.7 215789.0 3292 84 27901.11 68 4689.6 30194.1 180905.3 3028.43 24608.27 69 4208.9 25985.2 150711.2 2771.06 21579.84 70 3758.9 22226.3 124726.0 2521.51 18808.78 71 3339-1 18887.2 102499.7 2280 65 16287.27 72 2948.8 15938.4 83612.5 2049.44 14006.62 73 2587.8 13350 6 67674.1 1828.84 11957.18 74 2255-5 11095 i 54323.5 1619.75 10128.34 75 I95L4 9I43.7 43228.4 1423.03 8508.59 76 1674.9 7468.8 34084 7 1239-51 7085.56 77 1425-4 6043.4 26615.9 1069.75 5846.05 78 1202.0 4841.4 20572.5 914.26 4776-30 79 1003.8 3837.62 I573LI4 773-27 3862.04 80 829.57 3008.05 11893.52 646 84 3088.77 81 678.00 2330.05 888547 534-77 2441.93 82 547-61 1782.44 655542 436.66 1907.16 83 436.79 1345 65 4772.98 35L92 1470.50 84 34379 1001.86 3427.33 279.73 111858 85 266.81 735-05 2425.47 219.12 838.85 86 204.01 1690.42 169.01 61973 87 I53-56 377-48 1159-38 128.29 450.72 88 113.69 263793 781.900 95-73 322.43 89 82.715 181.078 518.107 70.17 226.70 90 59.089 121 989 337.029 50-47 156.53 91 41.407 80.582 215.040 35-6i 106.06 92 93 28.437 19.121 5 2 145 33.024 I34-458 82.313 24.61 16.65 70.45 45.84 94 12-575 20.4485 49.2892 II.OI 29.19 95 8.0812 12.3673 28.8407 7.12 18.18 96 5.0691 7.2982 16.4734 4-49 II. 06 97 3.1004 4-I978 2.76 6-57 98 99 1.8470 1.0706 2.3508 1.2802 III '65 .96 I'.i6 100 .6030 .6772 13464 54 1.20 101 .3297 -3475 .6692 30 .66 102 .1748 .1727 -3217 .16 36 103 .0897 .0830 .1490 .09 .20 104 .0446 .0384 .0660 05 .11 105 .0214 .0170 .0276 03 .06 106 .0099 .0071 .0106 .02 03 107 .0044 .0027 -0035 .01 .01 108 .0019 .0008 .0008 109 .0008 4 o8 PARK'S ENGLISH TABLE No. 3. MALES. TABLE III. COMMUTATION COLUMNS, FARR'S ENGLISH LIFE TABLE NO. 3 MALES, FOUR PER CENT. Age. D, N x S, M x Rx 5"745 7673088 I50I39344 196943.65 2410246.79 1 4H563 7261525 141954511 116444.61 2213303.14 2 370289 6891236 134281423 90999.90 2096858.53 3 343410 6547826 127019898 78362.82 2005858.63 4 322327 6225499 120128662 70487.51 1927495.81 5 304407 5921092 113580836 64964.98 1857008.30 6 288721 5632371 IQ7355337 60987.33 1792043.32 7 274613 5357758 101434245 57983.38 I73I055-99 8 261632 5096126 95801874 55564.79 1673072.61 9 249649 4846477 90444116 53643.92 1617507.82 10 238495 4607982 85347990 52092.15 1563863.90 11 228034 4379948 80501513 50804.03 1511771.75 12 218154 4161794 75893531 49694.75 1460967.72 13 208763 395303* 7I5I3583 48694.19 1411272.97 14 199789 3753242 67351789 47748.86 1362578.78 15 191172 3562070 63398758 46816.57 1314829.92 16 182869 3379201 59645516 45865.68 1268013.35 17 174845 3204356 56083446 44875.90 1222147.67 18 167078 3037278 52704245 43233-36 1177271.77 19 159550 2877728 49499889 42732.19 1133438.41 20 152254 2725474 46462611 41572.51 1090706 22 21 145186 2580288 43584883 40359-57 I049I33-7I 22 138419 2441869 40859409 39177.67 1008774.14 23 24 13*945 I2575I 2309924 2184173 38279121 35837252 38027 03 36908.16 969596 47 931569.44 25 119828 2064345 33527328 35821.45 894661.28 26 114164 1950181 3I343I55 34766.07 858839.83 27 108749 1841432 29278810 33741-57 824073.76 28 103572 1737860 27328629 32747.47 790332.19 29 98623 1639237 25487197 31782.63 757584.72 30 93894 1545343 23749337 30845.96 725802.09 31 89373 1455970 22IIOIOO 29936.42 694956 13 32 85052 1370918 20564757 29052.74 665019.71 33 80921 1289997 19108787 28193.73 635966.97 34 76973 1213024 17737869 27358.01 607773.24 35 73199 1139825 16447872 26544.29 580415.23 36 37 69591 66141 1070234 1004093 15234848 14095023 2575I-39 24978.21 553870.94 528119.55 38 62842 941251 13024789 24223.05 503141.34 39 59687 881^64 12020696 23485.24 478918.29 40 56670 824894 11079445 22763.52 455433-05 41 53784 771110 10I9788I 432669.53 42 51022 720088 9372987 21364.24 410612.79 43 48381 671707 8601877 20685.04 389248.55 44 45853 625854 7881789 20018.08 368563.51 45 43434 582420 72IOO82 19363.07 348545.43 46 41120 541300 6584228 18719 10 329182.36 47 48 38905 36785 502395 465610 6201802 5460508 18085.81 17462.57 310463.26 292377.45 49 34757 430853 4958H3 16849.11 274914.88 50 32816 398037 4492503 16245.17 258065.77 51 30960 367077 4061650 15650.52 241820.60 52 29167 337910 3663613 15048.69 226170.08 53 27450 310460 3296536 14453.50 211121.39 54 25807 284653 2958626 13865.95 196667.89 PARK'S ENGLISH TABLE No. 3. MALES. 409 TABLE III. Cont. COMMUTATION COLUMNS, FARR'S ENGLISH LIFE TABLE No. 3 MALES, FOUR PER CENT. Age. Dx N x s. M x R, 55 24234 260419 2648166 13286.14 182801.97 56 22730 237689 2363513 12714.09 169515.83 57 21291 216398 2 103094 12149.39 156801.74 58 19914 196484 186^405 11591.30 144652.35 59 18596 177888 1649007 11039.25 133061.05 60 17334 160554 1452523 10492.46 122021.80 61 16126 144428 1274635 9950.52 111529.34 62 14968 129460 III4O8I 9412.82 101578.82 63 13858 115602 969653 8870.23 92165.00 64 12796 102806 840193 8349.83 83286.77 65 66 1 1779 10806 91027 80220 7 724591 621785 7824.78 7304.82 74936.94 67112.16 67 9876.2 70344 5 530757-5 6790.85 59807 34 68 8989.5 6i355 450536.8 6284.00 53016.49 69 8I45-7 53209.3 380192.3 5785.89 46732.49 70 7344-8 45864.5 318837.3 5298.27 40946.60 71 6587-1 39277.4 265628.0 4823.11 35648.33 72 5873-3 33404-I 219763.5 4362.60 30825 22 73 5203.7 28200.4 180486.1 3919.00 26462.62 74 4579- ! 23621.3 147082.0 3494-5 22543.62 75 3999.8 19621.5 118881. 6 3091.29 19049.12 76 3466.1 i6i554 95260.3 3711.50 I5957-83 77 2978.2 13177.2 75638.8 2356.81 13246.33 78 2535-6 10641.6 59483-4 2028.81 10889.52 79 2137.8 8503.8 46306.2 1728.54 8860.71 80 1783-7 6720. i 35664.6 1456.69 7132.17 81 1471.9 5248.2 27160.8 1213.40 5675.48 82 1200.2 4047.99 20440.7 99836 4462.08 83 84 966.53 768.05 3081.46 2313-41 15192-54 ii 144 55 810 84 649.56 3463-72 2652.88 85 60I.80 1711.61 8063 09 512.84 200332 86 464-S7 1247.04 5749-68 398.73 1490.48 87 353-5 893.99 4038.07 305-1 I09L75 88 263.90 630.09 2791.03 229.53 786.65 89 193.85 43624 1897.04 169.63 557-12 90 139-81 296.429 1266.95 123.03 387.49 91 98.917 197.512 830.713 8752 26446 92 68.585 128.927 534.284 6099 176.94 93 46560 82.367 336.772 41.60 "595 94 30.916 5i-45i 207.845 2775 7435 95 20.058 5i-393 125.478 18.09 46.60 96 12.703 18.6904 74027 11.49 28.51 97 7.8442 10.8462 42-6343 7.12 17.02 98 4.7180 6.1282 23-9439 4-29 9.90 99 2.7609 3-3673 13.0977 2.52 5-6i 100 1.5701 1.7972 6.9695 1-43' 101 .8668 '934 3.6022 .80 1.66 102 .4639 .4665 1.8050 .42 86 103 .2405 .2260 .8747 .23 .46 104 .1205 1055 .4081 .11 .21 105 .0584 .0471 .1821 .06 .10 106 .0273 .0198 .0766 .03 .04 107 .0123 .0075 .0295 .01 .01 108 .0053 .0022 .0097 .00 .00 109 .0022 .0022 .00 .00 4 io FARR'S ENGLISH TABLE No. 3 MALES. TABLE IV. FARR'S ENGLISH LIFE TABLE No. 3, COMMUTATION COLUMNS, THREE PER CENT. MALES. Age. D, NX Sx MX Rx 5 "745 9288491 206018049 226301.22 37997ii-5o 1 4ISSS9 8872932 196217813 145020.64 3573410.28 2 377514 8495418 186929322 119079.46 3428389.64 3 3535io 8141908 178056390 106070.72 3309310 18 4 335028 7806880 169560972 97885.09 3203239.46 5 3 J 9474 7487406 161419064 92089.22 3105354.37 6 35953 7181453 153612184 87874.16 3013265.15 7 293828 6887625 146124778 84660 oi 2925390.99 8 282657 6604968 138943325 82047.07 2840730.98 9 272329 6332639 132055700 79951.69 2758683.91 10 262688 6069951 125450732 78242.51 2678732.22 11 253605 5816346 119118093 76809.95 2600489.71 12 244972 5571374 113048142 75564.30 2523679.76 13 236703 5334671 107231796 74429.84 2448115.46 14 228726 5105945 101660422 73347-59 2373685.62 15 220987 4884958 96325751 72269.90 2300338.03 16 213441 46715*7 91219806 71160.04 2228068.13 17 18 206057 198815 4465460 4266645 86334848 81663331 69993.57 68752.99 2156908.09 2086914.52 19 191701 4074944 77197871 67429.93 2018161.53 20 184711 3890233 72931226 66023.04 1950731-60 21 177845 3712388 68856282 64537-25 1884708.56 22 171203 3541185 64966049 63075-43 1820171.31 23 164780 3376405 61253661 61638.45 1757095.88 24 158570 3217835 57712476 60227.58 1695457-43 25 152567 3065268 54336071 58843.96 1635229 85 26 146767 2918501 51118236 57487.19 1576385-89 27 141162 2777339 48052968 56i57 33 1518898.70 28 135748 2641591 45134467 54854 40 1462741.37 29 130517 2511074 42357128 53577-54 1407886.97 30 125464 2385610 39715537 52325 92 I354309-43 31 120583 2265027 37204463 51098.76 1301983.51 32 115867 2149160 34818853 49894.92 1250884 75 33 111310 2037850 32553826 48713-32 1200989.83 34 106907 1930943 30404666 47552.59 1152276.51 35 102653 l8282C,0 28366816 46411.45 1104723.92 86 98540 1729750 26435873 45288.71 1058312 47 37 94564 1635186 24607583 44183.27 1013023.76 38 90720 1544466 22877833 43093-" 968840.49 39 87002 1457464 21242647 42017.65 92574738 40 83406 1374158 19698181 40955.43 883729 73 41 79926 1294132 18240717 39905.10 842774.30 42 76559 1217573 16866659 38866 oo 802869 20 43 73300 1144273 15572521 37836.97 764003 20 44 70145 1174128 14354954 36816.67 726166.23 45 67090 1007038 13210681 35804.93 689349 56 46 64132 942906 12136553 34800.58 653544.63 47 61267 881639 11129515 33803 30 618744 05 48 58491 823148 10186609 32812 31 584940 75 49 55803 767345 9304970 31827 40 552128.44 50 53 1 9 * 7141-7 8481822 30848.36 520301.04 51 50676 663471 7714477 29875.03 489452.68 52 53 48205 45808 615266 569458 7000330 6336859 28880.38 27887.15 459577-65 430697.27 54 43483 525975 5721593 26897.11 402810.12 PARK'S ENGLISH TABLE No. 3 MALES. 411 TABLE IV.Cnnt. FARR'S ENGLISH LIFE TABLE No. 3, COMMUTATION COLUMNS, THREE PER CENT. MALES. Age. D, N. Si MX Rx 55 41230 484745 5I52I35 25910.72 375913-01 56 445698 4626160 24928.03 350002.29 57 36930 408768 4141415 23948.55 325074.26 58 34877 37389* 3695717 22971.13 301125.71 59 32885 34IOO6 3286949 21994.91 278154.58 60 30951 310055 2913058 21018.61 256159.67 61 62 29072 27247 280983 253736 2572052 2261997 20041 57 19062.75 235141.06 215099.49 63 25472 228264 1981014 18082.00 196036.74 64 23748 204516 1727278 17099.50 J 7795474 65 22072 182444 1499014 16115.60 1608 1:5. 24 66 20446 I6I998 1294498 I5I3 1 -79 144739.64 67 18868 I43I30 1112054 14149.86 129607.85 68 I734I 125789 1950056 13172.14 "5457-99 69 15866 109923 806926 12201.96 102285 85 70 14445 95478 681137 11242.98 90083.89 71 13080 82398 571214 10299.43 78840.91 72 II776 70622 475736 9376.10 68541.48 73 10535 60086.9 393338 8478.04 59165.38 74 9360.4 50726.5 322716.4 7610.29 50687.34 75 82555 42471.0 262629.5 6778.06 43077-05 76 7223.5 35247-5 211903.0 5986.57 36298.99 77 62669 28980.6 169432.0 5240.21 30312.42 78 5387.4 23593-2 134184-5 4543-30 25072.21 79 IQ006.9 105203.9 3899.12 20528.91 80 3863.8 IS 143- I 81610 7 3310.26 16629 79 81 32I 9 .2 II923.9 62603.8 2778-15 I33I9-53 82 2650.6 9273-3 47460.7 2303.26 10541.38 83 2155-2 7II8.I 35536.8 1885.12 8238.12 84 1729.3 5388.8 ' 26263.5 1521.99 6353-00 85 I368.I 4020.7 19145.4 1211.18 4831.01 86 1066.4 2954.30 13756.6 949-25 3619.83 87 8l828 2136.02 9735-92 732.24 2670.58 88 89 61758 458.06 1518.44 1060.38 6781.62 4645.60 :is 1938.34 1382.96 90 333-58 726.80 3127.16 302.66 969.11 91 488.50 2066.78 217.12 666.45 92 166.83 321.67 I339-98 152.59 449-33 93 "4-35 207.323 851.48 104.98 296.74 94 76.668 130.655 529 8n 70.62 191.76 95 50.225 80.430 322.488 46.43 121.14 96 32.117 48.313 191-833 29.74 74-71 97 20.025 28.288 111.403 18.60 98 12.161 I6.I27I 63090 11.31 26.37 99 7.1856 8.9415 34.8024 6.70 15.06 100 4.1261 4-8I54 18.6753 3-84 8.36 101 2.2999 2.5155 97338 2.17 4-52 102 1.2429 1.2726 4.9184 1.14 2-35 103 6505 .6221 2.4029 .62 1. 21 104 .3293 .2928 I.I303 30 59 105 .1610 .1318 .5082 17 29 106 0759 0559 2154 .08 .12 107 0345 .O2I4 .0836 .04 .04 108 .0151 .0063 .0277 109 .0063 .0000 .0063 FARR'S ENGLISH TABLE No. 3. FEMALES. 413 TABLE I. Cont. FARR'S ENGLISH LIFE TABLE, No. 3. FEMALES. Age. Living. Dying. Proba- bility of Living. Proba- bility of Dying. Age. Living. Dying Proba- bility of Living. Proba- bility of Dying. 488255 65774 .86529 .13471 55 211576 4439 .97902 .02098 1 422481 26159 .93808 .06192 56 207137 4628 .97766 .02234 2 396322 14023 .96462 .03538 57 202509 4817 .97621 02379 3 382299 9243 .97582 .02418 58 197692 5009 .97466 .02534 4 373056 6596 .98232 .01768 59 192683 5206 .97298 .02702 5 366460 4866 .98672 .01328 60 187477 5409 .97115 .02885 6 361594 3815 .98945 .01055 61 182068 5619 .96914 .03086 7 357779 3249 .99092 .00908 62 176449 5835 .96693 .03307 8 9 354530 351806 2724 2328 .99232 99338 .00768 .00662 63 64 170614 164557 6057 6282 .96182 .03550 .03818 10 349478 2045 99415 .00585 65 158275 6509 .95888 .04112 11 347433 1861 .99464 .00536 66 151766 6731 .95565 04435 12 345572 1765 .99489 .00511 67 145035 6947 .95210 .04790 13 343807 1745 99493 .00507 68 138088 7149 .94823 .05177 14 342062 1789 99477 .00523 69 130939 .94401 05599 15 340273 1888 99445 .00555 70 123607 7489 .93941 .06059 16 338385 2029 .99400 .00600 71 116118 7613 93444 .06556 17 336356 2205 99345 .00655 72 108505 7698 .92905 .07095 18 334I5I 2400 .99282 .00718 73 100807 7736 .92325 07675 19 33I75I 2609 .99214 .00786 74 93071 7724 .91702 .08298 20 329142 2819 .99144 .00856 75 85347 7653 91033 .08967 21 326323 2867 .99121 .00879 76 77694 7521 .90319 .09681 22 323456 2912 .99100 .00900 77 70173 7329 89557 .10443 23 320544 2952 .99079 .00921 78 62844 7071 .88747 "2S3 24 317592 2989 .99059 .00941 79 55773 6755 .87889 .12111 25 314603 3024 .99039 .00961 80 49018 6382 .86981 .13019 26 3U579 .99020 .00980 81 42636 5959 .86023 .13977 27 308524 3084 .99000 .01000 82 36677 5496 .85015 .14985 28 305440 3112 .98981 .01019 83 31181 5003 .83956 .16044 29 302328 3138 .98962 .01038 84 26178 4490 .82847 I7I53 30 299190 3163 .98943 .01057 85 21688 3972 .81688 .18312 31 296027 3187 .98924 .01076 86 17716 3458 .80480 .19520 32 292840 3209 .98904 .01096 87 14258 2962 .79223 .20777 33 289631 3223 .98884 .01116 88 11296 2494 .77919 .22081 34 286398 3255 .98863 .01137 89 8802 2063 .76567 23433 35 283143 3279 .98842 .01158 90 6739 1673 75170 .24830 36 279864 33^1 .98820 .01180 91 5066 1331 73729 .26271 37 276563 3326 .98797 .01203 92 3735 1037 .72246 27754 38 273237 3350 .98774 .01226 93 2698 79 .70722 .29278 39 269887 3376 .98749 .01251 94 1908 588 .69160 .30840 40 266511 3402 .01277 95 1320 428 67563 .3 2 437 41 263109 3431 .98696 .01304 96 892 304 .65931 .34069 42 259678 3459 .98668 .01332 97 588 210 .64269 35731 43 256219 3490 .98638 .01362 98 378 142 .62579 37421 44 252729 3522 .98606 .01394 99 236 92 .60864 39136 45 249207 3555 98573 .01427 100 144 59 .59126 .40874 46 245652 3591 98538 .01462 101 85 36 57370 .42630 47 242061 3627 .98502 .01498 102 49 22 .55598 .44402 48 238434 3665 .98463 .01537 103 27 12 53814 .46186 49 234769 3705 .98422 .01578 104 15 7 .52021 47979 50 231064 3746 98379 .01621 105 8 4 .50222 .49778 51 227318 3788 98334 .01666 106 4 2 .48422 51578 52 223530 3832 .98286 .01714 107 2 I .46624 .53376 53 219698 3876 .98236 .01764 108 I I .44831 55169 54 215822 4246 98033 .01967 4 i4 FARR'S ENGLISH LIFE TABLE No. 3. TABLE II. FARR'S ENGLISH LIFE TABLE No. 3, COMMUTATION COLUMNS, FOUR PER CENT. FEMALES. Age. D, NX s M x Rx 488255 7593795 149289213 177406.81 2340157-77 1 406231 7187564 141207163 114162.57 2162750.96 2 366422 6821142 133613368 89977.11 2048588.39 3 339862 6481280 126425804 77510.72 1958611.28 4 318890 6162390 119604662 69609.77 1881100.56 5 301203 5861187 113123382 64188.34 1811490.79 6 285773 55754*4 106960992 60342.67 1747302.45 7 271882 5303532 101099805 57443.58 1686959 78 8 259052 5044480 95524391 55069.57 1629516.20 9 247174 4797306 90220859 53155 72 1574446.63 10 236094 4561212 85176379 5*583-01 1521290.91 11 225686 4335526 80379073 50254.62 1469707.90 12 215843 4119683 75817861 49092.25 1419453.28 13 206481 3913202 71482335 48032.24 1370361.03 14 197532 3715670 67362652 47024.55 1322328.79 15 188942 3526728 63449450 46031.18 1275304.24 16 180667 3346o6i 59733780 45023.16 1229273.06 17 172676 3173185 56207052 43981.53 1184249.90 18 164947 3008438 52860991 42893.08 1140268.37 19 157463 2850975 49687606 41753.94 1097375.29 20 150216 2700759 46679168 40563.23 1055621.35 21 143202 2557557 43828193 39326.16 1015058.12 22 136484 2421073 41127434 38116.41 975731-96 23 130053 2291020 38569877 36934-93 937615-55 24 123899 2167121 36148804 35783 29 910680.62 25 118013 2049108 33857784 34662.07 864897.33 26 112383 1936725 31690663 33571.35 830235.26 27 107001 1829724 29641555 32511-83 796663.91 28 101857 1727867 27704830 31483.39 764152.08 29 96942 1630925 25875106 30485-52 732668.69 30 92246 1538679 24147239 29518.02 702183.17 31 87760 1450919 22516314 28580.32 672665.15 32 83477 1367442 20977635 27671.84 644084.83 33 79386 1288056 19526716 26792.27 616412 99 34 75481 1212575 18159274 25940.21 589620.72 35 71753 1140822 16871218 25115-34 563680.51 36 68194 1072628 15658643 24316.35 538565.17 37 64798 1007830 14517821 23542.94 514248.82 38 61556 946274 I3445I93 22793.64 490705.88 39 58463 887811 12437363 22067.96 467912.24 40 555" 832300 11491089 21364.78 445844.28 41 52695 779605 10603278 20683 43 424479.50 42 50008 729597 9770978 20022.71 403796.07 43 47444 682153 8991373 19382.21 383773-36 44 44998 637155 8261776 18760.83 364391.15 45 46 42664 40438 594491 554053 6942468 18157.87 17572.67 345630.32 327472.45 47 383H 515739 6347977 17004 28 309899.78 48 36288 479451 5793924 16452.27 292895.50 49 34356 445^95 5278185 I59I5-93 276443.23 50 32514 412581 4798734 15394 59 260527.30 51 30756 381825 4353639 14887.75 245132.71 52 29081 352744 3941058 14394.94 230244 96 53 27483 325261 3559233 I39I558 215850.02 54 2595y 299302 3206489 13449-37 201934.44 PARK'S ENGLISH LIFE TABLE No. 3. TABLE II. Cont. FARR'S ENGLISH LIFE TABLE No. 3, FOUR PER CENT. COMMUTATION COLUMNS, FEMALES. Age. Dr N Sx M, Rx 55 24470 274832 2881228 12958.30 188485.07 56 23035 251797 2581926 12464.65 175526.77 57 21654 230143 6307094 11969.78 163062.12 58 20326 209817 2055297 11474.51 151092.34 59 19049 190768 1825154 10979.31 139617.83 60 17822 172946 1615337 10484 43 128638.52 61 16642 156304 1424569 9990.02 11815409 62 1.5508 140796 1251623 9496.17 108164.07 63 14418 126378 1095319 9003.06 98667 90 64 13372 113006 954523 8510.88 89664.84 65 12366 100640 828145 8020.05 81153.96 66 11402 89238 715139 753I-04 73I33-9I 67 10477 78761.2 614499 7044.81 65602.87 68 9591-5 69169.7 525261.2 6562.27 58558.06 69 8745.3 60424.5 446500.0 6084.80 51995 79 70 7938.0 52486.5 377330.3 5613.94 45910.99 71 7170.2 453i6.3 316905.8 5I5I-50 40297.05 72 64424 38873-9 264419.3 4699.48 35145 55 73 5755-2 33118.7 219103.0 4259.99 30446 07 74 5109.1 28009.6 180229.1 3835-32 26186.08 75 4504.9 23504-7 147110.4 3427.62 22350.76 76 3943-3 19561.4 119100.8 3039.20 18923.14 77 3424-5 16136.9 95596.1 2672.16 15883.94 78 2948.9 13188.0 76034.7 2328.25 13211.78 79 2516.5 10671.5 59897.8 2009 21 10883.53 80 2126.6 85449 467098 1716.15 8874-32 81 1778.6 6766.3 36038.3 1449.92 7158.17 82 1471.2 5295-1 27493 4 1210.90 5708.25 83 12O2.6 4092.49 20727.1 998.93 4497-35 84 970.82 3121 67 15432 04 8I3.39 3498.42 85 773-36 2348-31 i 1339 5 5 653.28 2685.03 86 607.45 1740.86 8217.88 5I7-09 2031.75 87 470.07 1270.79 586957 403.08 1514.66 88 358.08 912.71 4128.71 309.18 1111.58 89 268.28 644.43 2857.92 233.16 802.40 90 197.52 446.91 1945.21 172.70 56924 91 142.76 304-I5 1300 78 12555 396.54 92 1OI.2I 202 938 85387 89.48 270.99 93 94 70.307 47.810 132.631 84 821 S49-7I8 346.780 62.46 42.67 181.51 119-05 95 31-794 53-027 214.149 28.51 76.38 96 20.655 32.372 129.328 18.60 47.87 97 I3-094 19.2778 76.301 11.83 29.27 98 80918 II. 1860 43 9 2 93 7-33 17-44 99 4.8690 6.3170 24-6515 4.41 10.11 100 2.8495 34675 13 4655 2-59 5-70 101 I.620O I-847S 7-1485 1-47 3-" 102 .8936 3.6810 .81 1.64 103 4777 .4762 1-8335 .42 83 104 .2472 .2290 8796 .22 .41 105 .1236 .1054 4034 .11 .19 106 0597 0457 .1744 05 .08 107 .0278 .0179 .0690 .02 03 108 .0125 .0054 .0233 .01 .01 109 .0054 .0000 .0054 .00 .00 416 PARK'S ENGLISH TABLE No. 3. FEMALES. TABLE III. FARR'S ENGLISH LIFE TABLES No. 3, COMMUTATION COLUMNS, THREE PER CENT. FEMALES. Age. D x N x Sx V x RX 488255 9203701 205605825 205965.03 3703432 41 1 4ioi75 8793526 195913865 142106.77 3497467.38 2 373571 8419955 186710168 117449.40 3355360.61 3 349858 8070097 177916642 104616.37 3237911.21 4 331456 7738641 169496687 96404.08 3133294.84 5 316112 7422529 161426590 90714.31 3036890.76 6 302830 7119699 153687949 86639.11 2946176.45 7 290907 6828792 146265420 83537.17 2859537.34 8 9 279870 269630 6548922 6279292 139145721 132316929 80972.38 78884.66 2776000.17 2695027.79 10 260044 6019248 125768007 77152.4^ 2616143.13 11 250993 5768255 119488715 75675.06 2538990.72 12 242377 55fc5878 113469467 74369.79 2463315.66 13 234116 5291762 107701212 73167 91 2388945.87 14 226143 5065619 102175334 72014.26 2315777.96 15 218408 4847211 96883572 70865.97 2243763.70 16 210870 4636341 9i8i7953 69689.43 2172897.73 17 203501 4432840 86970742 68461.85 2103208.30 18 196279 4236561 82334401 67166.64 2034746.45 19 189193 4047368 77901561 65797.95 1967579.81 20 182238 3865130 73665000 64353.4I 1901781.86 21 I754I5 3689715 69617632 62838.06 1837428.45 22 168809 3520906 65752502 61341.79 1774590 39 23 162417 3358489 62062787 59866.30 1713248.60 24 156234 3202255 58541881 58414- ii 1653382.30 25 150256 3051999 55183392 56986.55 1594968.19 26 27 144478 138894 2907521 2768627 51981137 48929138 55584.34 54209.01 1537981.64 1482397.30 28 133501 2635126 46021617 52861.07 1428188.29 29 128292 2506834 43252990 51540.50 1375327.22 30 123262 2383572 40617864 50247.69 1323786.72 31 118407 2265165 38110030 48982.53 1273539.03 32 113721 2151444 35727458 47744.90 1224556.50 33 109198 2042246 33462293 46535.02 1176811 60 34 35 104835 100624 1937411 1836787 31310849 29268603 45351.60 44194.83 1130276.58 1084924.98 36 96562 1740225 27331192 43063.47 1040730.15 37 92644 1647581 25494405 41957.69 997666.68 38 88864 I5587I7 23754180 40875.99 955708.99 39 85218 1473499 22100599 39818.22 14833.00 40 81701 1391798 20547882 38783.28 75014.78 41 78309 I3I3489 19074383 37770.75 836231.50 42 75036 1238453 17682585 36779-33 798460.75 43 71880 1166573 16369096 35808.93 761681.42 44 68836 1097737 15130643 34858.35 725872.49 45 65900 1031837 13964070 33927.00 691014.14 46 63068 968769 12866333 330I4-30 657087.14 47 60336 908433 11834496 32119.21 624072.84 48 57701 850732 10865727 31241.48 591953 63 49 55159 795573 9957294 30380.39 560712.15 50 52707 742866 9106562 29535.25 530331 76 51 50343 692523 8310989 28705.65 500796.51 52 53 48062 45862 644461 598599 6875600 27891.18 27091.25 472090. 86 444199.68 54 43741 554858 6231139 26305.70 417108.43 FARR'S ENGLISH TABLE No. 3. FEMALES. 417 TABLE III Cont. FARR'S ENGLISH LIFE TABLES No. 3, COMMUTATION COLUMNS, THREE PER CENT. FEMALES. Age. Dx NX Sx MX Rx 55 41631 513227 5632540 25470.23 390802.73 56 39571 473656 5077682 24622.22 365332-50 57 37560 436096 4564455 23763.86 340710.28 58 400498 4090799 22896.46 316946.42 59 33686 366812 3654703 22020.76 294049.96 60 61 31821 30003 334991 304988 3254205 2887393 21137.13 20245.78 272029.20 250892.07 62 28230 276758 2552402 19346.80 230646.29 63 26501 250257 2247414 18440.45 211299.49 64 24816 225441 1970656 17527.02 192859 04 65 23174 202267 1720399 16607.25 175332.02 66 21573 180694 1494958 15682.00 158724.77 67 20016 160678 1292691 14753-06 143042.77 68 18502 142176 in 1997 13822.24 128289.71 69 17033 125143 95i3 T 9 12892.25 114467.47 70 15611 109532 809143 11966.24 101575.22 71 72 14238 12917 95294 82377 684000 574468 11047.95 10141.64 89608.98 78561.03 73 11651 70726 479174 9251.91 68419.39 74 10444 60281.5 396797 8383.83 59167.48 75 76 77 8217.9 7206.1 509833 42765.4 35559-3 326070.7 265789.2 214805.9 7542.34 6732.87 5960.53 50783-65 43241-31 36508.44 78 62656 29293.7 172040.5 5229.83 30547.91 79 5398.6 23895-1 136481.2 4545.38 25318.08 80 4606.6 19288.5 107187.5 3910.56 20772.70 81 82 3890.1 3248.9 15398.4 12149-5 83292.4 64003.9 3328.27 2800.41 16862.14 I3533.87 83 2681.6 94679 48605.5 2327.74 10733.46 84 2185.8 7282.1 36456.0 191000 8405.72 85 1758.1 5524-0 26988.1 1546.02 6495.72 86 1394-4 4129.6 19706.0 1233.41 4949.70 87 1089 5 3040.10 14182.0 969.18 3716.29 88 838.00 2202.10 10052.37 749-44 2747.11 89 633.94 I568.I6 7012.27 569.81 1997 67 90 471-25 1096.91 4810.17 425.55 1427.86 91 343.9 2 752.99 3242.01 3H.97 1002.31 92 246.18 506.81 2145.10 224.24 690.34 93 172.68 334-13 I3Q2 II 157.88 466.10 94 118.56 215-574 885.30 108.80 308.22 95 79611 I35-963 55I-I73 73-33 199.42 96 52.221 83-742 335-599 48.26 126.09 97 33427 50.315 199.636 30.98 77.83 98 20.858 29-457 115.894 19-39 46.85 99 100 12.672 7.4882 16.7848 9.2966 65579 36.1217 11.78 6-99 27.46 15-68 101 4.2985 4.9981 19-3369 4.01 8.69 102 2-3942 2.6039 10.0403 2.24 4.68 103 1.2924 i-3"5 5.0422 1.19 2.44 104 .6752 .6363 2.4383 .64 1-25 105 .3410 2953 1.1268 33 .6! 106 .1663 .1290 495 .16 .28 107 .0782 .0508 .1952 .08 .12 108 0354 .0154 .0662 .04 .04 109 .0154 .0154 UNIVERSITY INTRODUCTION TO CARLISLE TABLES. THE Carlisle Table was constructed by the philosopher and scientist, Milne, from the parish records of two parishes of Carlisle, Scotland, covering a period of but eight years, viz., from 1780 to 1787, inclusive. This table was pub- lished by him in 1816 in a volume which was very largely given over to discussions of the virtues of Swedish govern- ment tables. By a strange chance the incident became the important thing; the Carlisle Tables have been widely adopted and employed, while the Swedish Tables have slumbered. The Carlisle Table slowly displaced the Northampton in the favor of the British companies. Neither the Seven- teen Offices' or Actuaries' Table nor even Dr. Farr's Eng- lish Life Tables were able to displace the Carlisle, which was not discarded by most British companies until after the H M and H F Tables appeared, and has not been dis- carded by all even at this date. The Commutation Tables given herein are taken from. 44 David Jones," with the exception of the M and R col- umns of the four per cent tables where " David Jones" was found to be wrong, and therefore figures from 4i David Chisholm's Tables" were substituted. CARLISLE TABLE. 421 TABLE I. CARLISLE TABLE OF MORTALITY. Age. Living. Decre- ments. Propor- tion Which Die. Propor- tion Which Survive. Age. Living. Decre- ments. Propor- tion Which Die. Propor- tion Which Survive. 10000 1539 .153900 .846100 53 4211 68 .016148 .983852 1 8461 682 .080605 9*9395 54 4M3 70 .016896 .983104 2 7779 505 .064918 .935082 55 4073 73 .017923 .982077 3 7274 276 037943 .962057 56 4000 76 .019000 .981000 4 6998 2O I .028723 .971277 57 3924 82 .020897 .979103 5 6797 121 .017802 .982198 58 3842 93 .024206 975794 6 6676 82 .012283 .987717 59 3749 106 .028274 .971726 7 6594 58 .008796 .991204 60 3 6 43 122 .033489 .966511 8 6536 43 .006579 .993421 61 3521 126 .035785 .964215 9 6493 33 .005082 .994918 62 3395 127 .037408 .962592 10 6460 29 .004489 .9955" 63 3268 125 038250 .961750 11 6-J3 1 3i .004820 .995180 64 3143 125 .039771 .960229 12 6400 3 2 .005000 .995000 65 3018 124 .041087 958913 13 6368 33 .005182 .994818 66 2894 123 .042502 .957498 14 6335 35 005525 994475 67 2771 I2 3 .044388 .955612 15 6300 39 .006191 .993809 68 2648 123 .046450 953550 16 6261 4 2 .006708 69 2525 124 .049109 .950891 17 6219 43 .006914 .993086 70 2401 124 .051645 .948355 18 6176 43 .006962 993038 71 2277 134 .058849 .941151 19 6i33 43 .007011 .992989 72 2143 146 .068129 .931871 20 6090 43 .007061 .992939 73 1997 156 .078117 .921883 21 6047 42 .006946 993054 74 1841 166 .090168 .909832 22 6005 42 .006994 .993006 75 1675 160 .095522 .904478 23 5963 42 .007043 992957 76 I5IS 156 . 102970 .897030 24 5921 42 .007093 .992907 77 1359 146 .107432 .892568 25 5879 43 .007314 .992686 78 1213 132 .108821 .891179 26 5836 43 .007368 .992632 79 1081 128 .118409 .881591 27 5793 45 .007768 .992232 80 953 116 .121721 .878279 28 5748 So .008699 .991301 81 837 112 .133811 .866189 29 5698 56 .009828 .990172 82 725 102 . 140690 .859310 30 5642 57 .010103 .989897 83 623 94 .150883 .849117 31 5585 57 .010206 .989794 84 529 84 .158790 .841210 32 5528 56 .010130 .989870 85 445 78 .175281 .824719 33 5472 55 .010051 .989949 86 367 71 .193461 806539 34 5417 55 .010153 .989847 87 296 64 .216216 783784 35 53 62 .010257 989743 88 232 5i .219828 .780172 36 5307 56 .010552 .989448 89 181 39 .215470 784530 37 5251 57 .010855 .989145 90 142 37 .260563 739437 38 5194 58 .011167 .988833 91 105 30 .285714 .714286 39 5I3 6 61 .011877 .988123 92 75 2t .280000 .720000 40 5075 66 .013005 .986995 93 54 *4 259259 .740741 41 5009 69 013775 .986225 94 40 10 .250000 .750000 42 4940 7i 014373 .985627 95 30 7 233333 .766667 43 4869 7i .014582 .985418 96 23 5 .217391 .782609 44 4798 7i .014798 .985202 97 18 4 .222222 777778 45 4727 70 .014809 .985191 98 14 3 .214286 785714 46 4657 69 .014816 .985184 99 II 2 .I8l8l8 .818182 47 4588 67 .014603 985397 100 9 2 .222222 .777778 48 49 4521 4458 6 3 61 013935 .013683 .986065 .986317 101 102 7 5 2 2 .285714 .400000 .714286 .600000 50 4397 59 .013418 .986582 103 3 2 .666666 333334 51 4338 62 .014292 .985708 104 i I 52 4276 65 .015201 .984799 422 CARLISLE TABLE. TABLE II. COMMUTATION COLUMNS, CARLISLE TABLE, FOUR PER CENT. Age. D r N x Sx MX Rx 10000.0000 142816.4335 2661123.5878 4122.4446 44587.9711 1 8135-5769 134680.8566 2518307.1543 2642.6369 40465.5259 2 7192.1228 127488.7339 2383626.2977 2OI2.O895 37822.8887 3 6466.5595 121022.1744 2256137.5638 1563.1464 35810.7989 4 5981.9197 115040.2547 2135115.3894 1327.2204 34247.6523 5 5586.6386 109453.6161 2020075.1347 II62.0I3I 32920.4316 6 7 5276.1398 5010 8980 104177.4763 99166 5782 1910621.5186 1806444.0423 1066.3850 1004.0718 31758.41825 30692.03297 8 4775.7912 94390.7871 1707277.4641 961.6917 29687.96095 9 4561.8957 89828.8914 1612886.6770 931.4805 28726.26896 10 4364.1445 85464.7469 1523057-7856 909.1869 27794.78820 11 4177.4550 81287.2919 1437593-0387 890.3490 26885 60106 12 3997.4211 77289.8708 1356305-7468 870.9865 25995-25I77 13 3824.4558 73465.4150 1279015.8760 851.7682 25124.26499 14 3658.3046 69807.1103 1205550.4610 832.7115 24272.49658 15 3498.1664 66308.9440 1135743-3507 813.2772 23439-78484 16 3342.7991 62966. 1449 1069434.4067 792.4548 22626.50735 17 3192.6682 59773-4766 1006468.2618 770.8931 21834.05228 18 3048.6473 56724.8294 946694.7852 749.6671 21063.15888 19 2910.9820 53813.8474 889969.9558 729-2575 20313.49149 20 2779.3965 51034.4509 836156.1084 709.6329 19584 23373 21 2653.6268 48380.8241 785121.6575 690.7630 18874.60061 22 2533.8421 45846.9820 736740.8334 673.0409 18183.83734 23 2419.3461 43427.6359 690893.8514 656.0004 17510.79619 24 2309.9092 41117.7267 647466.2155 639-6I53 16854.79554 25 2205.3117 38912.4150 606348.4888 623.8604 16215.17999 26 2104.9823 36807.4327 567436 0738 608.3507 15591.31935 27 2009.1084 34798.3243 530628.6411 5934376 14982.96835 28 1916.8285 32881.4958 495830.3168 578.43" I 43 8 9-5346 29 1827.0717 31054.4240 462948.8210 562.3986 13811.09906 30 I 739-5339 29314.8901 431894.3970 545-I327 13248.70023 31 I655-7306 27659-1596 402579.5069 528.2345 12703.56724 32 1575-8003 26083.3593 374920.3473 511.9862 12175.33249 33 1499.8433 24583.5160 348836.9880 496.6369 11663.34604 34 35 36 1427.6617 1358.8137 1293.1499 23155.8543 21797.0406 20503.8907 324253-4720 301097.6177 279300.5771 482.1415 468.2037 454.8019 11166.70886 10684.56705 10216.36309 37 1230.2928 19273-5979 258796 6864 441.6813 9761.56091 38 1170.1325 18103.4654 239523.0885 428.8400 93I9.87935 39 "12.5635 16990.9019 221419.6231 416.2760 8891.03906 40 1057.0669 15933-8350 204428.7212 403.5704 8474.76277 41 1003 1921 14930.6430 188494.8862 390.3520 8071.19211 42 951.3202 13979-3228 173564.2432 377.0644 7680.83979 43 901.5840 13077-7388 159584.9204 363.9174 7303-775I4 44 854.2664 12223.4724 146507.1816 351.2761 6939.85743 45 809.2549 11414.2176 134283.7092 339.I2II 6588.58101 46 766 6067 10647 6108 122869 49 X 6 327-598I 6249.45967 47 726.2004 9921.4104 II222I 8808 316.6766 5921.86130 48 688.0725 9233-3379 102300.4704 306.4795 5605.18443 49 652.3887 8580.9492 93067.1325 297.2600 5298.704615 40 618.7134 7962.2358 84486 1833 288.6766 5001.444289 51 586.9340 7375.3019 76523.9475 280.6938 4712.767431 52 556.2936 6819 0083 69148.6456 272.6278 4432-073309 CARLISLE TABLE. 423 TAfeLE II. Cont. COMMUTATION COLUMNS, CARLISLE TABLE, FOUR PER CENT. Age. D x N. Sx M x Rx 53 526.7666 6292.2417 62329.6373 264.4968 4159.445184 54 498.3272 5793.9I45 56037.3956 256 3176 3894.948105 55 471.0649 5322.8496 50243.4811 248.2218 3638.630184 56 444.8289 4878.0207 44920.6315 240.1036 3390.408149 57 58 59 4I9-5934 395.0242 370.6367 4458.4273 4063.4031 3692.7664 40042.6108 35584.1835 31520 7804 231.9770 223.5459 214.3517 3150.304242 2918 327019 2694.780818 60 346.3050 3346.4614 27828.0140 204.2753 2480.428859 61 3 2 I 8343 3024.6271 24481.5526 193.1240 2276.153302 62 298.3821 2726.2451 21456 9255 182.0500 2083.029060 63 276.1733 2450.0718 18730.6804 171.3174 1900.978792 64 255.3940 2194.6778 16280.6086 161.1602 1729.661083 65 235.8046 1958 8732 14085.9308 151.3936 1568.500628 66 217.4193 1741.4540 12127.0576 142.0778 1417.106762 67 200.1717 1541.2822 10385.6036 133-1925 1275.028720 68 183.9293 I357.3530 8844.3214 124.6489 1141.835964 69 168.6402 1188.7128 7486.9684 116.4340 1017.186752 70 154.1908 1034.5220 6298.2556 108.4708 900.752486 71 140.6034 8939186 5263.7336 100.8139 792.281425 72 127.2395 766 6792 4369.8150 92.8577 691.467293 73 114.0103 652.6688 3603.1358 84.5224 598.609336 74 101.0617 551 6071 2950.4670 75.9588 514.086639 75 88.4126 463.1945 2398.8599 67.1967 438.127564 76 76.8916 386.3028 1935.6654 59.0762 370.930579 77 66.3212 319.9817 1549.3626 514631 311.854158 78 56.9194 263.0623 1229.3809 44.6121 260.390767 79 48.7744 214.2878 966.3186 38.6563 215.778351 80 41.34527 172.9426 752.0308 33.1031 I77.I2I735 81 3491604 138.02654 579.08815 28.26413 144.018314 82 29.08066 108.94588 441.06161 23.77167 II5.7539I4 83 24.02818 84.91770 332.11573 19.83768 91.981974 84 19.61802 65.29968 247.19803 16 35168 72.144021 85 15.86814 49 43154 181.89835 13-35635 55.792068 86 12.58342 36.84812 132.46681 10.68194 42.435450 87 9-75868 27.08943 95.61869 8.34118 3L753238 88 7-35452 19.73492 68.52926 6.31235 23.411791 89 5.5I7II 14.21781 48.79434 4.7578o 17.099176 90 4.16186 10.0^595 34.57653 3-6I475 12.341105 91 2.95907 7.09688 24.52058 2.57203 8.726o8l 92 2.032329 5.064547 17.423696 1.75910 6.153777 93 1.406997 3 657550 12.359149 1.21194 4.394405 94 1.002135 2 655416 8.701599 .861190 3.182198 95 .722693 1.932722 6.046183 .620292 2.320738 96 .532755 1.399967 4.113461 .458149 I.700I77 97 .400902 .999065 2.713494 .346787 I.24I757 98 .299820 .699245 1.714429 .261124 .894700 99 .226512 .472733 1.015184 .199618 .633305 100 .178200 .294532 .542451 .160018 .433687 101 .133270 .161263 .247919 .121941 .273669 102 09I53I .069731 .086656 .085329 .I5I72 7 103 .052806 .016925 .016925 .050124 .066399 104 .016925 .000000 .000000 .016274 .010274 424 CARLISLE TABLE. TABLE III. COMMUTATION COLUMNS, CARLISLE TABLE. THREE AND ONE-HALF PER CENT. Age. D x N x s, M x Rx 10000.0000 156719.2811 3126762.5941 4362.1499 55413-1307 1 8174.8792 148544.4019 2970043.3130 2875.1934 51050.9808 2 7261.7707 141282.6222 2821498.9111 2238.5391 48175.7874 3 6560.7312 I3472I.89IO 2680216.2889 1783.0580 45937.2484 4 6098.3527 128623.5383 2545494-3979 1542.5400 44154.1903 5 5722.8916 122900.6466 2416870.8596 I373-3034 42611.6504 6 5430.9303 117469.7164 2293970.2130 1274.8698 41238.3470 7 5182.8244 112286.8920 2176500.4966 1210.4185 39963.4772 8 49635140 I07323-3780 2064213.6046 1166.3726 38753.0587 9 4764.1152 102559.2629 1956890.2266 1134.8222 37586.6861 10 4579 6i55 97979.6473 1854330.9637 1111.4279 36451.8639 11 4404.8859 93574-76I5 I75435i-3 l6 4 1091.5645 35340.4360 12 4235.4131 89339-3484 1660776.5549 1071.0492 34248.8715 13 4071 7256 85267.6227 1571437-2065 1050.5883 33177.8223 14 3913.6476 8I353-975I 1486169.5838 1030.2015 32127.2341 15 3760.4109 77593-5642 1404815.6087 1009.3103 3 1097.0326 16 3610.7557 73982.8085 1327222.0445 986.8188 30087.7223 17 3465.2503 70517 5582 1253239.2360 963.4162 29100.9036 18 3324.9184 67192.6398 1182721.6778 940.2667 28137.4874 19 3190.1149 64002.5249 1115529.0380 9179000 27197.2207 20 3060.6262 60941.8987 1051526.5131 896.2896 26279.3208 21 2936.2472 58005.6515 900584.6144 875.4101 25383-0311 22 2817.2495 55188.4020 932578.9629 855-7058 24507.6210 23 2702.9422 52485.4597 877390-5609 836.6678 23651-9153 24 2593.1442 49892.3156 824905.1012 818.2736 22815 2475 25 2487.6811 47404.6344 775012.7856 800.5014 21996.9740 26 2385.9766 45018.6578 727608.1512 782.9214 21196.4726 27 2288.3059 42730.3519 682589.4934 7659358 20413-5512 28 2193.7491 40536.6028 639859.1415 748 7614 19647.6154 29 2101.1270 38435-4758 599322.5387 730.3340 18898.8540 30 2010.1228 36425-353 560887.0629 710.3724 18168.5309 31 1922.5265 34502 8265 524461.7099 600.7513 17458.1576 32 I 8 38.5559 326642706 489958.8834 67I-7937 16767.4063 33 1758 3873 30905.8833 557294 6128 653-7985 16095.6127 34 1681.8488 29224.0345 426388.7295 636.7223 15441.8142 35 1608.4759 276I5-5586 397164.6950 620.2236 14805.0919 36 1538. ! 4 22 26077.4164 369549.1364 604.2828 14184.8684 37 1470.4460 24606.9704 343471.7200 588.6010 13580.5856 38 1405.2987 23201.6717 318864.7496 573-I790 12991.9846 39 1342.6146 21859.0571 295663.0779 558.0170 12418.8057 40 1281.8053 20577.2518 273804.0208 542.6101 11860.7887 41 1222.3531 19354.8987 253226.7690 526.5040 11318.1786 42 1164.7488 18190.1499 233871.8703 510.2353 10791.6745 43 1109 1869 17080.9630 215681 7204 494.0611 10281.4393 44 1056.0509 16024 9121 198600.7575 478.4338 9787.3782 45 1005.2402 15019.67 rg 182575.8453 463-3350 9308 9444 46 956.8639 14062.8080 167556.1731 448.9522 8845.6094 47 910.8083 13151.9998 I53493-3654 435-2544 8396.6572 48 867.1570 12284.8428 140341.3656 422.4033 7961.4028 49 826.1576 11458.6852 128056.5528 410.7282 7538.9995 50 787.2977 10671.3875 116597.8376 399.8059 7128.2713 51 750.4672 9920.9203 105926.4501 389.5990 6728.4654 52 7I4-7259 9206 1945 96005.5298 379-2358 6338.8664 CARLISLE TABLE. 425 TABLE III. Cent. COMMUTATION COLUMNS, CARLISLE TABLE, THREE AND ONE-HALF PER CENT. Age D, N x Sx M x RX 53 54 680.0592 646.4516 8526.1353 7879.6836 86799-3353 78273.2000 368.7386 358.1282 5959.6305 5590.8920 55 614.0379 7265.6457 70393.5164 347.5752 5232.7637 56 582.6402 6683.0056 63127.8707 336.9420 4885.1886 57 552-2415 6130.7640 56444.8651 326.2462 4548.2466 58 522.4167 5608.3473 50314.1012 315.0962 4222.0004 59 492.5324 5115.8149 44705-7538 302 8781 3906.9042 60 462.4217 4653-3932 39589.9389 289 4231 3604.0261 61 431.8219 4221.5713 34936.3457 274.4608 3314.6030 62 402.2889 3819.2824 30714.9744 259.5305 3040.1422 63 374-I450 3445-1373 26895.6920 244.9906 2780.6117 64 347.6658 3097 4715 23450.5547 231.1636 2535.6212 65 322.5496 2774.9219 20353.0832 217.8042 2304.4576 66 67 298.8377 276.4605 2476.0842 2199.6237 17578.1613 15102.0771 204.9998 192.7282 2086.6534 1881.6536 68 255-2550 1944.3687 12902 4534 180 8715 I68R.92S4 69 235-1675 1709.2013 10958.0847 169.4159 1508.0539 70 216.0567 1493.1446 9248.8834 158.2576 1338.6380 71 197.9695 1295.1751 7755-7388 147.4766 1180.3804 72 180.0184 1115.1567 64605637 136.2202 1032.9038 73 162.0812 953-0756 5345.4070 124.3705 896.6835 74 144.3670 808.7086 4392.3314 112.1374 772.3130 75 126.9079 681.8007 3583.6228 99.56024 660.1756 76 110.9037 570.8970 2901.8221 87.84764 560.6154 77 96.1197 474-7773 2330-9251 76.81403 472.7677 78 82.8922 391.8851 1856.14780 66.83690 395-9537 79 71-3737 320.51141 1464.26271 58.12152 329.1168 80 00.79459 259.71682 1143.75130 49.95604 2709953 81 51.589005 208.127819 884.034478 42.80631 221.0392 82 43.174700 164.953119 675.906659 36.13656 178 2329 83 35.845863 129.107256 510.953540 30.26774 142.0964 84 29.4080^1 99.699205 381.846284 25.04211 111.8286 85 23.901782 75.797423 282.147079 20.53031 86.78652 86 19.045660 56.751762 206.349656 16.48246 66.25621 87 14.841618 41.910145 149.597893 12.92248 49-77375 88 11.239247 30.670898 107.687748 9.821995 36.85128 89 8.472029 22.198869 77.016850 7.434849 27.02928 90 6.421801 15.777068 54-817981 5.671114 19-59443 91 4-587937 11.1891305 390409134 4-0544I3 13-92332 92 3 160278 8.0228525 27.8517829 2.787902 9.868906 93 2.2026281 5.8202244 19.8289304 i.93i3 2 4 7.081005 94 i 5764024 4.2438220 14.0087060 I-379583 5.149681 95 1.1423205 3.1015015 9.7648840 .9988096 3.770098 96 .8461633 2 2553382 6.6633825 .7412817 2.771288 97 .6398212 I.6I55I70 4.4080443 .5635536 2.030006 98 .4808101 I.I347069 2.7925273 .4261792 1.466453 99 .3650043 .7697026 1.6578204 .3266326 1.040274 100 .2885410 .4811616 .8881178 .2625124 .713641 101 .2168317 .2643300 .4069562 .2005605 .451129 102 .1496423 .1146877 .1426262 .1407036 .250568 108 .0867491 .0279385 .0279385 .0828708 .109865 104 .0279285 .0000000 .OOOOOOO .0269938 .026994 426 CARLISLE TABLE. TABLE IV. COMMUTATION COLUMNS, CARLISLE TABLE, THREE PER CENT. Age. Dx ! NX Sx MX Rx 10000.000 173198.234 3702049.698 4664.129 70035.663 8214.563 104983.671 3528851.464 3169.954 6537I-534 7332.454 6656.740 157651.218 150994.477 3363867 793 3206216.575 2527.104 2064.057 62201.580 59674.476 6217.632 144776.845 3055222.098 1819.735 57609.519 5863.152 138913.693 2910445.253 1646.351 55789.784 5591-045 133322.648 277i537-56o 1545-015 54143.434 7 5361.525 127961.123 2638208.912 1478.341 52598.419 8 5159-579 122801.544 2510247.790 1432.556 51120.078 9 4976.344 117825 200 2387446.246 1399.600 49687.522 10 4806.847 II30I8.353 2269621.046 1375.045 48287.922 11 4645.891 108372.462 2156602.693 1354.095 46912877 12 4488.831 103883.631 2048230.230 1332.352 45558-783 13 4336.298 99547-333 1944346 599 1310.561 44226.431 14 4188.181 95359-I52 1844799 266 1288.744 42915.870 15 4043-730 91315.421 1749440.115 1266.279 41627.126 16 17 3901.648 3762.597 87413773 83651.176 1658124.694 1570710.920 1241.976 1216.565 40360.847 39118.871 18 3627.749 80023.427 1487059.745 1191.307 37902.306 19 3497.564 76525.862 1407036.318 1166.785 36710.999 20 3371.885 73153 977 1330510.456 1142.977 35544.214 21 3250.560 69903.417 1257356.478 1119.862 34401.238 22 3133-964 66769 452 1187453.062 1097.943 33281.376 23 24 3021.403 2912.740 63748.049 60835.310 1120683.609 1056935.560 1076.662 1056.000 32183.434 31106.773 25 2807.843 58027.467 996100.251 1035.741 30050.773 26 2706.122 55321.344 938072.784 1016.002 29014.832 27 2607.945 527I3-399 882751.440 996.6439 27998 830 28 2512.317 50201.082 830038.041 976.9755 27002.186 29 2417.926 47783.156 779836.959 955-7581 26025.211 30 2324.429 45458.727 732053.803 932.6869 25069.453 31 2233 928 43224.799 686595.076 909.8876 24136.766 32 2146.727 41078.072 643370.278 887.7524 23226.879 33 2063.088 39014.984 602292.206 866.6389 22339.127 34 1982.865 37032.119 563277222 846.5065 21472.488 35 1905.566 35126.553 526245.103 826.9604 20625.982 36 1831.087 33295.466 491118.550 807.9836 19799.022 37 1758.995 31536.470 457823.085 789.2245 18991.039 38 39 1689.225 1621.710 29847 246 28225.536 426286.614 396439.368 770.6867 752.3729 18201.815 17431.129 40 I555-776 26669.760 368213.833 733-6730 16678 756 41 1490.819 25178.941 341544.073 714.0295 15945.083 42 1427.459 23751.482 316365.132 694.0913 15231.054 43 1365-964 22385 519 292613.650 674 1728 14536 963 44 1306.840 21078.670 270228.131 654.8344 13862.790 45 1250.001 19828.678 249149.452 636.0593 13207.956 46 1195.622 18633.056 229320.774 618.0877 12571.897 47 "43599 17489457 210687.718 600.8888 11953.810 48 1094.077 16395.380 193198.262 584.6749 11352.921 49 1047.408 15347 972 176802.882 569.8731 10768.246 50 1002.987 14344.985 161454.911 555.9585 10198.373 51 960.7073 13384.277 147109.926 542.8922 9642.415 52 9I9-3947 12464.883 133725-648 529.5614 9009.522 53 879.0475 II585-835 121260.766 515.9926 8569.9610 CARLISLE TABLE. 427 TABLE IV.Cont. COMMUTATION COLUMNS, CARLISLE TABLE, THREE PER CENT. Age. DX N x S x MX Rx 54 839.6626 10746.173 109674.930 502.2111 8053.9683 55 801.4327 9944.7400 98928.7577 488.4374 7551-7573 56 764.1444 9180.5956 88984.0177 474.4917 7063.3199 57 727.7919 8452.8038 79803.4221 460.3959 6588.8282 58 691.8283 7760.9755 71350.6183 445.6301 6128.4323 59 60 65S-4I93 618.3376 7105.5562 6487.2186 63589.6429 56484.0867 429.3714 411-3797 5682.8022 5353-4308 61 580.2235 5906.9951 49996.8681 391.2754 4842.0511 62 543- 16 5* 5363.8300 44089.8730 371.1167 4450-7757 63 507.6178 4856.2121 38726.0431 351.3898 4079.6590 64 473.9822 4382.2300 33869.8310 332.5391 3728.2692 65 441.8752 3940.3548 29487.6010 314.2374 33Q5.730I 66 411.3786 3528.9762 25547.2462 296.6110 3081.4927 67 382.4216 3146.5546 22018.2700 279-6359 2784.8817 68 354.8025 2791.7521 18871.7155 263.1553 2505.2458 69 328.4679 2463.2842 16079.9634 247.1547 2242.0905 70 303.2400 2160.0442 13616.6792 231-4938 I994.9358 71 279.2030 1880.8412 11456.6350 216.2891 1763.4420 72 255.1185 1625.7227 9575-7937 200.3367 1547.1529 73 230.8132 1394.9095 7950.0710 183.4621 1346.8161 74 206.5852 1188.3243 6555.i6i5 165.9567 1163.354! 75 182.4832 1005.8411 5366.8372 147.8718 997-3973 76 160.2447 845-5965 4360.9960 130.9483 849.5255 77 139-5575 706.0390 35I5-3996 114.9285 718.5772 78 120.9365 585.1025 2809.3606 100.3722 603.6487 79 104.6369 480 4656 2224.2581 87.5951 503.2765 80 89.56018 390.9054 I743-7925 75.5660 415.6814 81 76.36780 314.5376 1352.8871 64.9822 340.H53 82 64.22226 250.3153 1038.3495 55.0610 275-I33 1 83 53-57947 196.7359 788.03419 46.2887 220.0722 84 44.17014 152.5657 591.29831 38.4400 173.7834 85 36 07413 116.4916 438.73257 31.6305 135-3435 86 87 28.88449 22.61795 87.60711 64.98917 322.24097 234-63386 25-4915 20.0663 103.7130 78.22142 88 17.21124 47.77792 169.64470 15-3184 58.15517 89 13.03664 34.74128 121.86677 11.6450 42.83681 90 91 9.929746 7.128560 24.81154 17.68298 87.12549 62.31395 8.91786 6.40589 3LI9I77 22.27390 92 4.943523 12.73946 44.63097 4.42849 15.86801 93 3.455667 9.28379 31-89152 3.08461 11-43952 94 2.485197 6.79859 22.60773 2.21480 8-35491 95 1.809610 4.98898 15.80914 1.61159 6.14011 96 1.346959 3.64202 10.82016 1.20165 4-52852 97 98 1.023438 .772823 2.61858 1.84576 7.17813 4-55955 .917360 .696554 3.32687 2.40951 99 .589532 1.25623 2.71379 .535772 1.71296 100 .468296 .787934 1.45756 .431706 1.17718 101 .353621 .669626 .330672 .745480 102 .245230 .189083 .235313 .232580 .414809 103 .142852 .046231 .046231 .137345 .182229 104 .046231 .000000 .000000 .044884 .044884 INTRODUCTION TO AMERICAN TROPICAL EXPERIENCE TABLE. THIS table was graduated by Makeham's formula from the experience of the New York Life Insurance Company on lives in the tropics. It is the work of Messrs. Jones and Robertson of the New York Life' s then actuarial force. This table, with four per cent interest, has already been adopted as its standard by at least one South American company. AMERICAN TROPICAL EXPERIENCE TABLE. TABLE I. AMERICAN TROPICAL EXPERIENCE TABLES. 8> Living. Dying. Per Cent Surviv- ing. Per Cent Dying. 1 Living. Dyinjr. Per Cent Surviv- ing. Per Cent Dying. 20 21 100 000 98 826 174 168 .98826 .98818 O.OTI74 .01182 59 60 47 925 45 977 I 948 I 999 95935 95652 0.04065 .04348 22 97 658 163 .98809 .01191 61 43 978 2 048 95343 .04657 23 9 6 495 158 .98800 .01200 62 41 930 2 093 .95008 .04992 24 95 337 155 .98789 .01211 63 39 837 2 136 .94638 .05362 25 94 182 I5i .98778 .01222 64 37 701 2 I 73 .94236 05764 26 93 031 149 .98765 01235 65 35 528 2 203 93799 .06201 27 91 882 148 .98751 .01249 66 33 325 2 227 93317 .06683 28 90 734 148 .98735 .01265 67 31 098 2 240 .92797 .07203 29 89 586 147 .98720 .OI28O 68 28 858 2 243 .92227 .07773 30 88 439 153 .98700 .01300 69 26 615 2 234 .91606 .08394 31 87 289 15 1 .98681 .01319 70 24 381 2 211 .90931 .09069 32 85 138 15* .98658 .01342 71 22 170 2 I 74 .90194 .09806 33 84 982 i6r .98634 .01366 72 19 996 2 120 .89398 .10602 34 83 821 167 .98608 .01392 73 17 876 2 050 .88532 .11468 35 82 654 175 .98578 .01422 74 15 826 I 965 .87584 .12416 36 8c 479 184 .98547 01453 75 13 861 I 862 .86567 .13433 37 80 295 195 .98512 .01488 76 ii 999 I 744 85465 .14535 38 79 loo 207 .98474 .01526 77 10 255 i 614 .84261 15739 39 77 893 222 .98431 .01509 78 8 6\i I 472 .82965 17035 40 76 671 237 .98387 .Ot6l3 79 7 169 i 320 .81587 .18413 41 75 434 2 5 6 98335 .01665 80 5 849 i 166 .80065 Z 9935 42 74 178 2 7 6 .98280 .01720 81 4683 i 009 78454 .21546 43 72 902 2 9 8 .98220 .01780 82 3 674 855 .76728 .23272 44 71 604 322 .98154 .01846 83 2 819 708 .74885 .25115 45 70 282 349 .98080 .01920 84 2 III 573 .72856 .27144 46 68 933 377 .98002 .01998 85 I 538 449 .70806 .29194 47 67 556 409 979H .02086 86 I 089 343 .68503 .31497 48 66 147 443 .97818 .02182 87 746 253 .66086 .33914 49 64 704 479 977H .02286 88 493 179 .63692 .36308 50 63 225 5i8 97599 .02401 89 3*4 123 .60828 .39172 51 6: 707 559 97474 .02526 90 191 80 58115 .41885 52 60 148 602 97337 .02663 91 in 49 55856 .44144 53 58 546 647 .97187 .02813 92 62 30 51613 .48387 54 56 899 694 .97023 .02977 93 3 2 18 43750 .56250 55 55 205 744 .96841 .03159 94 14 9 35714 .64286 56 53 46i 794 .96644 .03356 95 5 4 .20000 .80000 57 51 667 845 .96429 03571 96 i i .00000 I.OOOOO 58 49 822 897 .96192 .03808 97 Constant. Number. Napierian Logarithm. Common Logarithm. c g s k 1.0964780 0.9986194 0.9890916 125,620. + o 0921034 0.0013816 0.0109684 + 11.7410196 -! 0400000 o. 0006000 0.0047635 + 5.0990600 lug / x = log k -f x log s 4- c* ^ ^ - _L \ log s 4- (log c log^) c* 43 2 AMERICAN TROPICAL EXPERIENCE TABLE. TABLE IV. COMMUTATION COLUMNS AMERICAN TROPICAL EXPERIENCE TABLES, FOUR PER CENT. Age. D x NX Sx C x MX Rx 20 45 639 769 547 n 648 619 5I5-I9 14 285.36 335 809.33 21 43 368 726 179 10 879 072 492.84 13 770.17 321 523-97 22 41 207 684 972 10 152 893 471.86 13 277-33 307 753-80 23 39 iSi 645 821 9 467 921 451.76 12 805.47 294 476.47 24 37 193 608 628 8 822 loo 433-26 12 353.71 281 671.00 25 35 3 2 9 573 299 8 213 472 415.15 II 920.45 269 317 29 26 33 555 539 744 7 640 173 398.49 II 505.30 257 396-84 27 31 866 507 878 7 loo 429 382.83 ii 106.81 245 891.54 28 30 258 477 620 6 592 551 368.11 10 723.98 234 784-73 29 28 726 448 894 6 114 931 353-64 10 355-87 224 060.75 30 27 267 421 627 5 666 037 340.93 10 002.23 213 704.88 31 25 878 395 749 5 244 410 328.10 9 661.30 203 702.65 32 24 554 37i 195 4 848 661 316.85 9 333 20 194 041.35 33 23 293 347 902 4 477 466 305-98 9 016.35 184 708.15 34 22 09! 325 811 4 129 564 295.74 8 710.37 175 691.80 35 36 2O 946 19 854 304 865 285 on 3 803 753 3 498 888 286.31 277.41 8 414-63 8 128.32 166 981.43 158 566.80 37 18 813 266 198 3 213 877 269.22 7 850 91 150 438.48 38 17 820 248 378 2 947 679 261.46 7 58i.6 9 142 587.57 39 16 873 23 i 505 2 699 301 254-53 7 320.23 135 005.88 40 15 970 215 535 2 467 796 247-74 7 065.70 127 685.65 41 15 108 200 427 2 252 26l 241.87 6 817.96 120 619.95 42 14 285 I 86 142 2 051 834 236.27 6 576-09 113 801.99 43 13 499 172 613 I 865 692 231.10 339.82 IO7 225.90 44 12 749 159 894 I 6 93 49 226.32 6 108.72 100 886.08 45 12 032 147 862 i 533 155 222.06 5 882.40 94 777.36 46 ii 347 I3 6 515 i 385 293 217.96 5 660.34 88 894.96 47 10 693 125 822 i 248 778 214.44 5 442-38 83 234.62 48 10 067 "5 755-Q I 122 955.7 211.17 5 227.94 77 792-24 49 9 468.9 106 286.1 I 007 200-7 208. 1 1 5 016.77 72 564.30 50 8 896.6 97 3890 9OO 914.6 20539 4 808.66 67 547-53 51 8 349.0 89 040.5 80 3 525.1 202.82 4 603.27 62 738.87 52 7 825.1 8r 215.4 714 484.6 200.40 4 400.4=; 58 135-60 53 7 3237 73 891-7 633 269.2 198 10 4 200.05 53 735-15 54 6 843.9 67 047.8 559 377-5 195-92 4 001.95 49 535-10 55 6 384.8 60 663 o 492 329.7 I93-9S 3 806.03 45 533-15 56 5 945 2 54 7i7 8 431 666.7 191.83 3 612.08 41 727.12 57 5 524-8 49 193-0 376 948.9 189.70 3 420.25 38 115 04 58 59 5 122.6 4 738.0 44 070.4 39 332.4 327 755-9 283 685.5 187.54 185.18 3 2 30-55 3 043.01 34 694.79 31 464.24 60 4 370.6 34 961 .8 244 353-1 182.72 2 857.83 28 421.23 61 4 019.8 30 942 o 209 391.3 180.00 2 675.11 25 563 40 62 3 685.2 27 256.8 178 449.3 176.88 2 495-11 22 888.29 63 3 366.6 23 890.2 151 1925 173-57 2 318-23 2O 393.18 64 3 063.5 20 826.7 127 302.3 169.78 2 144.66 I 8 074.95 AMERICAN TROPICAL EXPERIENCE TABLE. 433 TABLE IV.Cont. COMMUTATION COLUMNS AMERICAN TROPICAL EXPERIENCE TABLES, FOUR PER CENT. Age. Dx NT s, C, M x R* 65 2 775-9 I 8 050.8 106 475.6 165-51 I 974.88 15 930.29 66 2 503.6 15 547-2 88 424.8 160.87 I 809.37 13 955-41 67 2 246.5 13 300.7 72 877.6 155-59 I 648.50 12 146.04 68 69 2 004.5 I 777.6 ii 296.2 9 518.6 59 576.9 48 280.7 149.81 143-47 I 492.91 i 343-10 10 497.54 9004.63 70 I 565-7 7 952-9 38 762.1 136.53 i 199.63 7 661.53 71 I 369.0 6 583-9 30 809.2 129.08 i 063.10 6 461.90 72 73 74 I 187.3 I O2O.6 868.77 5 396.6 4 375-97 3 507-20 24 225.3 18 828.74 M 452.77 121.03 "2-53 103.72 934-02 812.99 700.46 5 398.80 4 464-78 3 651.79 75 73i- 6 3 2 775-57 io 945-57 94-503 596.740 2 95L332 76 608.99 2 166.58 8 170.00 85.110 502.237 2 354 592 77 78 500.46 405-47 I 666.12 I 260.65 6 003.42 4 337-30 75-736 66.416 417.127 34I-39I i 852.355 i 435.228 79 323-46 937-19 3 076.65 57-267 274-975 I 093.837 80 253-75 683-44 2 139.46 48.641 217.708 818.862 81 195-35 488.09 i 456.02 40.472 169.067 601.154 82 83 147-37 108.72 340.72 231.997 967-93^ 627.206 32.976 26.256 128.595 95-6i9 432.087 303-492 84 78.287 153-710 395-209 20.432 69-363 207.873 85 54-843 98.867 241.499 15-395 48.931 138.510 86 37-339 61.528 142.632 11.308 33-536 89.579 87 36.933 81.104 8.0202 22.2281 56.0433 88 15.628 21.305 44.171 5-456I 14.2079 33-8I52 89 9-571 "734 22.866 3-6050 8.7518 19.6073 90 5-598 6.136 11.132 2.2545 5-1468 10.8555 91 3.128 3.008 4.996 1.3278 2.8923 5-7087 92 1.680 1.328 1.988 0.7817 1.5645 2.8164 93 0.834 0.494 0.660 .4510 0.7828 1.2519 94 351 143 .166 .2168 .3318 0.4691 95 .120 .023 .023 .0927 .1150 1373 96 .023 .000 .000 .0223 .0223 .0223 Constant. Number. Logarithm. i .04 7.6020600 (i ~l~ *') 1.04 0.0170333 V 1.0198039 .9615385 0.0085167 ^9829667 vX .9805807 T.99I4833 d 6 .0384615 .0392207 .0396078 *. 5850267 ^.5935156 7.5977807 434 AMERICAN TROPICAL EXPERIENCE TABLE. TABLE III. COMMUTATION COLUMNS AMERICAN TROPICAL EXPERIENCE TABLES, THREE AND ONE-HALF PER CENT. Age. Dz N x Si C x M T Rz 20 21 50 257 47 9 8 7 913 139 86 S 152 14 491 634 13 578 495 570.06 547-97 17 678.03 17 107.97 440 761.19 423 083.16 22 45 816 819 336 12 713 343 527-17 16 560.00 405 975-19 23 24 43 740 41 754 775 596 733 842 ii 894 007 n 118 411 507-15 488.73 16 032.83 15 525.68 389 415.19 373 382.36 25 39 853 693 989 10 384 569 470.57 IS 036.95 357 856.68 26 38 035 655 954 9 690 580 453.87 14 566.38 342 819.73 27 36 295 619 659 034 626 438.14 14 112.51 328 253.35 28 34 62 9 585 030 414 967 423-32 13 674-37 314 140.84 29 33 035 55i 995 7 829 937 408.65 13 251.05 300 466.47 30 31 32 3i 509 30 048 28 649 520 486 490 438 461 789 7 277 942 6 757 456 6 267 01 8 395-86 382.81 371-47 12 842.40 12 446.54 12 063.73 287 215.42 274 373-02 261 926.48 33 27 308 434 4 8 i 5 805 229 360.46 II 692.26 249 862 75 34 26 024 408 457 5 370 748 350.07 II 331.80 238 170.49 35 24 794 383 663 4 962 291 34055 10 981.73 226 838.69 36 23 615 360 048 4 578 628 33156 10 641.18 215 856.96 37 22 485 337 563 4 218 580 323-32 10 309.62 205 215.78 38 21 401 316 162 3 881 017 31553 9 986.30 194 906.16 39 2O 362 295 800 3 564 855 308.64 9 670.77 184 919.86 40 19 3 6 5 276 435 3 269 055 301.87 9 362.13 175 249.09 41 1 8 408 258 027 2 992 62O 296.14 9 060.20 165 886.96 42 17 490 240 537 2 734 593 29068 8 764.12 156 826.70 43 1 6 608 223 929 2 494 056 285.69 8 473-44 148 062.58 44 15 76o 208 169 2 270 127 281.14 8 187.75 139 589.14 45 14 946 193 223 061 958 277.18 7 906.61 131 401.39 46 14 164 179 059 868 735 273.36 7 629.43 123 494-78 47 13 4 11 165 648 689 676 270.26 7 356-07 H5 865.35 48 12687 152 961 524 028 267.42 7 085.81 108 509.28 49 II 991 140 970 371 067 264.82 6 818.39 101 423.47 50 ii 321 129 649 230 097 262.61 6 553-57 94 605.08 51 10 675 118 974 loo 448 260.58 6 290.96 88 051.51 52 10 054 108 919.8 981 473.6 258.72 6 030.38 81 760.55 53 9 454-9 99 464.9 872 553-8 256.99 5 771-66 75 730.17 54 8 878.2 90 586.7 773 088.9 255-38 5 5I4-67 69 958.51 55 8 322.6 82 264.1 682 502.2 254-03 5 259-29 64 443-84 56 7 787-1 74 477 o 600 238.1 252.48 5 005.26 59 I84-55 57 7 271-3 67 205.7 525 761.1 250.87 4 752.78 54 179.29 58 6 7746 60 431.1 458 555-4 249.22 4 501.91 49 426.51 59 6 296.2 54 134-9 398 124.3 247.27 4 252.69 44 924.60 60 5 836.1 48 298.8 343 989-4 245.16 4 005.42 40 671.91 61 5 393-5 42 905-3 295 690 6 242.68 3 760.26 36 666.49 62 37 936.8 252 785-3 239.62 3 5I7-58 32 906.23 63 4 560.8 33 376-o 214 848.5 236.28 3 277.96 29 388.65 64 4 170.3 29 205.7 181 472.5 232 24 3 041.68 26 110.69 AMERICAN TROPICAL EXPERIENCE TABLE. 435 TABLE lll.Cont. COMMUTATION COLUMNS AMERICAN TROPICAL EXPERIENCE TABLES, THREE AND ONE-HALF PER CENT. Age. D x N x Sx Cx MX Rr 65 3 797-1 25 408.6 152 266.8 227.48 2 809.44 23 069.01 66 3 441-2 21 96 7-4 126 858.2 222.19 2 581.96 20 259.57 67 3 102.6 18 86 4.8 104 890.8 215.93 2 359-77 17 677.61 68 2 781.8 16 083.0 86 026:0 208.90 2 143-84 15 317.84 69 2 478.8 13 604.2 69 943.0 201.03 i 934-94 13 174.00 70 2 194.0 II 410.2 56 338.8 192.23 i 733-91 II 239.06 71 I 927-5 9 482.7 44 928.6 182.62 i 541.68 9 505.15 72 I 679.7 7 803.0 35 445-9 172.06 i 359-06 7 963-47 73 I 450.9 6 352.1 27 642 9 160.76 i 187.00 6 604.41 74 I 241.0 5 in. i 21 290.8 148.88 i 026.24 5 417.41 75 I 050.2 4 060.93 16 179.67 136.31 877-36 4 39I.I7 76 878.37 3 182.56 12 118.74 123.35 74I-05 3 5i3-8i 77 725-32 2 457 24 8 936.18 110.30 617.70 2 772.76 78 590-50 i 86674 6 478.94 97.190 507.402 2 I55-057 79 473-34 i 393-40 4 612.20 84.207 410.212 i 647.655. 80 373-12 i 020.28 3 218.80 71.867 326.005 i 237.441 81 288.64 731-64 2 198 52 60.087 254.138 911.438 82 218.79 512-85 i 466 88 49-195 194-051 657.300 83 162.20 350.65 954-03 39-359 144-856 463.249 84 "7-35 233.297 603.382 30.777 105.497 318.393 85 82.609 150.688 370.085 23.301 74.720 212.896 86 56-514 94.174 219.397 17.198 5I-4I9 138.176 87 37-45 56.769 125.223 12.257 34-221 86757 88 23883 32.886 68.454 8-3784 21.9636 52 5355 89 14.697 18.189 35-568 5-5625 13-5852 30.5719 90 8.638 9-551 17-379 3-4956 8 0227 16.9867 91 4.850 4.701 7.828 2.06 86 4-527I 8.9640 92 2.617 2.084 3.127 1.2237 24585 4.4369 93 1.305 779 1-043 .7094 1-2348 1.9784 94 552 .227 .264 .3427 .5254 7436 95 .190 037 -037 .1472 .1827 .2182 96 .037 .000 .000 0355 0355 -0.355 Constant. Number. Logarithm. i 035 -z 5440680 (!+') 1-035 0.0149403 (x-M)K 1.0173495 0.0074702 v .9661836 T-9850597 v% .9829465 T.9925298 d .0338164 7.5291277 A .0344014 ^.5365764 *'(*) .0346990 3.5403170 436 AMERICAN TROPICAL EXPERIENCE TABLE. TABLE IV. COMMUTATION COLUMNS AMERICAN TROPICAL EXPERIENCE TABLE, THREE PER CENT. Age. Dx NX Sz Cx M x Rx 20 55 368 I 088 402 18 124 163 631.08 22 053.87 582 569.81 21 22 53 124 50 067 I 035 278 984 311 17 035 761 i 6 ooo 483 609.57 589.28 21 422.79 20 813.22 560 5I5-94 539 093.15 23 48 893 935 4i8 15 016 172 56966 20 223.94 518 279.93 24 46 899 888 519 14 080 754 55I-63 19 654.28 498 055.99 25 44 982 843 537 13 192 235 533-71 19 102.65 478 401.71 26 43 138 800 399 12 348 698 51727 18 568.94 459 299.06 27 41 364 759 035 II 548 299 501.76 18 051.67 440 730-12 28 39 658 719 377 10 789 264 487-I5 17 549-91 422 678.45 29 38 015 681 362 10 069 887 472.55 17 062.76 405 128.54 30 36436 644 926 9 388 525 459-99 16 590.21 388 065.78 31 34 9i4 610 012 8 743 599 446.98 16 130.22 371 475-57 32 33 45i 576 561 8 133 587 435-84 15 683.24 355 345-35 33 32 040 544 521 7 557 026 424.98 15 247.40 339 662.11 34 30 682 513 839 7 012 505 4I4-73 14 822.42 324 414.71 35 29 374 484 465 6 498 666 405.41 14 407-69 309 592.29 36 28 113 456 352 6 014 201 396.62 14 002.28 295 184 60 37 26 897 429 455 5 557 849 388.65 13 605.66 281 182.32 38 25 725 403 730 5 128 394 381.11 13 217.01 267 176.66 39 24 595 379 135 4 724 664 374-61 12 835.90 254 359-65 40 23 504 355 631 4 345 529 368.17 12 461.29 241 523.75 41 22 451 333 180 3 989 898 362.93 12 093.12 229 062.46 42 21 434 311 746 3 656 718 357-97 II 730.19 216 969.34 43 20 452 291 294 3 344 972 353-54 II 372.22 205 239.15 44 19 503 271 791 3 053 678 349-59 II 018.68 193 866.93 45 18 585 253 206 2 781 887 346.34 10 669 09 182 848.25 46 17 698 235 5o8 2 528 681 343-23 10 322.75 172 179.16 47 16 839 218 669 2 293 I 73 340.98 9 979-52 161 856.41 48 16 008 202 661 2 074 504 339-03 9 638.54 151 876.89 49 15 202 187 459 I 8 7 I 8 43 337-37 9 299-51 142 238.35 50 14 422 173 037 684 384 336-18 8 962.14 132 938.84 51 13 666 159 37i 5ii 347 335-21 8 625 96 123 976.70 52 12 933 146 438 35i 976 334-42 8 290.75 "5 350 74 53 12 221 134 217 205 538 333-8o 7 956.33 107 059 99 54 II 532 122 685 071 321 333.32 7 622.53 99 103.66 55 10 863 III 822 948 636 333-17 7 289.21 91 481.13 56 10 213 ioi 609.1 836 813.8 332-74 6 956.04 84 191 92 57 9 582.8 92 026.3 735 204.7 332.23 6 623.30 77 235.88 58 59 8 9714 8 378.5 83 054-9 74 676 4 643 178.4 56o 123.5 33 J -64 330-64 6 291.07 5 959-43 70 612.58 64 321.51 60 7 803.8 66 872.6 485 447.1 329.41 5 628.79 58 362.08 61 7 247.1 59 625.5 4i8 574-5 327.66 5 299.38 52 733-29 62 6 708.4 52 917-1 358 949 o 325 ii 4 971-72 47 433-9 1 63 6 187.9 46 729.2 306 031.9 322.12 4 646.61 42 462 19 61 5 685.5 4i 043.7 259 302.7 318,16 4 324 49 37 815.58 AMERICAN TROPICAL EXPERIENCE TABLE. 437 TABLE IV.Cont. COMMUTATION COLUMNS-AMERICAN TROPICAL EXPERIENCE TABLE, THREE PER CENT. Age. B. N. Sx c x MX Rx 65 5 201.8 35 841.9 218 259.0 313.15 4 006.33 33 491.09 66 4 737-1 31 104.8 182 417.1 307-34 3 693.18 29 484.76 67 4 291.8 26 813.0 151 312.3 300.14 3 385-84 25 79I-58 68 69 3 866.7 3 462.2 22 946.3 19 484.1 124 499.3 101 553- 291.78 282.15 3 085.70 2 793-92 22 405.74 19 320.04 70 3 079-3 16 404.8 82 068.9 271.11 2 511.77 16 526.12 71 718.5 13 686.3 65 664.1 258.81 2 240.66 14 014.35 72 ii 305.8 51 977-8 2 45-03 I 981.85 it 773-69 73 74 066. i 775-9 9 239.7 7 463-8 40 672.0 31 432-3 230.04 214.08 I 736.82 I 506.78 9 791-84 8 055.02 75 510.1 5 953-7 23 968.5 196.95 I 292.70 6 548.24 76 269.2 4 684.5 18 014.8 179.09 I 095.75 5 255-54 77 053-1 3 631.38 13 330.32 160.92 916.66 4 159-79 78 79 861.51 693-93 2 769.87 2 075.94 9 698.94 6 929.07 142.48 124.05 755-74 613.26 3 243-13 70 11 549-67 427.28 i 526.27 i 098.99 4 853-I3 3 326.86 106.39 89.380 489.21 382.822 i 874.13 i 384.916 82 325.45 773-54 2 227.87 73-532 293.442 i 002.094 83 84 242.44 176.26 354-84 i 454-33 923.23 59.116 46.451 219.910 160.794 708.652 488.742 85 124.68 230.160 568389 35-338 "4-343 327.948 86 85.709 144.451 338.229 26.209 79.005 213.605 87 57-003 87.448 193.778 18.769 52-796 134.600 88 36.574 50.874 106.330 12.893 34.027 81.804 89 22.616 28.258 55.456 8.6011 21.1344 47.7765 90 91 I3-356 7-536 14.902 7-366 27.198 12.296 5.4313 3.2298 125333 7.1020 26.6421 14 1088 92 4.087 3-279 4-930 1.9198 3-8722 7.0068 93 2.048 1.231 1.651 1.1183 1.9524 3-I346 94 .870 -301 .420 54 2 9 .8341 1.1822 95 302 -059 -059 2343 .2912 .3481 96 059 .OOJ .000 .0569 .0569 .0569 Constant. Number. Logarithm. i 03 .4771213 (i + *') 1.03 0.0128372 (i + M 1.0148891 0.0064186 V* .9708738 1.9871628 v% d fi 9853293 .0291262 .0295588 T-99358I4 7.4642838 7.4706867 C) .0297783 .4738999 * Value $1, due at end of year. INTRODUCTION TO TWENTY-THREE GERMAN OFFICES' TABLE. THE Twenty-three German Offices' Table is, as its name implies, a graduation from the experience of twenty-three German companies. It is standard now throughout the German Empire. The commutation tables have not been before published. They are furnished by the kindness of Messrs. Weeks and Frankland of the New York Life Insurance Company's actuarial force, and were computed for the use of that office. TWENTY-THREE GERMAN OFFICES' TABLE. 441 TABLE I. TWENTY-THREE GERMAN COMPANIES' MORTALITY TABLE. Age. lx 4. Age. lx dx 17 102787 909 59 57792 1900 18 101878 936 60 55892 1976 19 100942 942 61 53916 2038 20 IOOOOO 919 62 51878 2097 21 99081 908 63 49781 2I 49 22 23 97286 887 861 64 65 47632 45435 2197 2246 24 25 96425 95590 III 66 67 43*89 40887 2302 2355 26 94774 804 68 38532 2399 27 93970 797 69 36i33 2432 28 93*73 795 70 33701 2452 29 92378 800 71 31249 2455 30 91578 808 72 28794 2436 31 90770 818 73 26358 2406 32 89952 8 3 i 74 23952 2360 33 89121 841 75 21592 2299 34 88280 856 76 19293 2210 35 87424 873 77 17083 2103 36 86551 889 78 14980 1982 37 85662 906 79 12998 . I8 4 8 38 84756 928 80 11150 1730, 39 83828 950 81 9420 1599 40 82878 975 82 7821 1443 41 81903 1006 83 6378 1264 42 43 80897 79862 1035 1063 84 85 54 4034 1080 8 9 6 44 78799 1092 86 3138 715 45 77707 1117 87 2423 587 46 76590 1140 88 1836 487 47 7545 1169 89 1349 394 48 74281 1204 90 309 49 73077 1246 91 646 50 71831 1303 92 413 167 51 70528 1362 93 246 "3 52 69166 1425 94 133 69 53 67741 1490 95 64 37 54 66251 1556 96 27 18 55 64695 1621 97 9 6 56 63074 1691 98 3 2 57 61383 1759 99 i I 58 59624 1832 442 TWENTY -THREE GERMAN OFFICES' TABLE. TABLE II. TWENTY-THREE GERMAN COMPANIES' COMMUTATION TABLES, FOUR PER CENT. Age. D z N, s, C x M x Rx 17 52768 966826 15945462 448.71 I3553-09 367092.66 18 50290 916536 14978643 444.27 13104.38 353539 57 19 47911 868625 14062114 429.92 12660.11 340435.19 20 45639 822986 I3I93489 403.29 12230.19 327775.08 21 43480 779506 12370503 383-14 11826.90 315544.89 22 41425 738081 11590997 359-88 11443.76 303717.99 23 39472 698609 10852916 335-90 11083 88 292274.23 24 37618 660991 10154307 313.22 10747.98 281190.35 25 35857 625134 294.32 10434.76 270442.37 26 34184 590950 8868182 278.84 10140.44 260007.61 27 32590 558360 8277232 265.78 9861.60 249867.17 28 31071 7718872 254.92 9595-82 240005.57 29 29621 497668 7191583 246.66 9340.90 230409.75 30 28235 469433 6693915 239-54 9094.24 221068.85 31 26910 442523 6224482 233-18 8854.70 211974.61 32 25642 416881 5781959 227.77 8621.52 203119.91 33 24428 392453 5365078 221.65 8393-75 194498.39 34 23266 369187 4972625 216.92 8172.10 186104.64 35 22155 347032 4603438 212.72 7955-iS 177932.54 36 21090 32594 2 4256406 208.29 7742.46 169977.36 37 20070 305872 3930464 204.11 7534-17 162234.90 38 19094 286778 3624592 2OI.O2 7330.06 154700.73 39 18159 268619 3337814 197.88 7129.04 14737067 40 17263 251356 3069195 195.27 6931.16 140241.63 41 16403 234953 2817839 193-73 6735-89 133310.47 42 43 15579 14788 219374 204586 2582886 2363512 $3 6542.16 6350.51 126574 58 12003242 44 14030 190556 2158926 186.95 6161.25 113681.91 45 13303 177253 1968370 183.87 5974.30 107520.66 46 12608 164645 1791117 180.44 5790-43 101546.36 47 11942 152703 1626472 177.92 5609.99 95755-93 48 H305 141398 ' 1473769 176.20 5432-07 90145.94 49 10694 130704 1332371 175-33 5255.87 84713.87 50 10108 120595.6 1201667.0 176.30 5080.54 79458.00 51 9542.5 111053.1 1081071.4 177.19 4904.24 74377.46 52 53 8998.3 8474.0 102054.8 9358o.8 970018 3 867963 5 178.26 179.22 4727.05 4548.79 69473.22 64746.17 54 7968.8 85612.0 774382.7 179.96 4369-57 60197.38 55 7482.4 78129.6 688770.7 180.27 4189.61 55827.81 56 70I4-3 7iii5-3 610641.1 180.82 4009.34 51638.20 57 6563.7 6455I-6 539525-8 180.86 3828.52 47628.86 58 6130.4 58421.2 474974.2 181.12 3647.66 43800.34 59 57I3-5 52707.7 416553-0 180.62 3466.54 40152.68 60 53I3.I 473946 363845 3 180.62 3285.92 36686.14 61 4928.2 42466.4 316450.7 179.12 3105.30 33400.22 TWENTY-THREE GERMAN OFFICES' TABLE. 443 TABLE II. Cont. TWENTY-THREE GERMAN COMPANIES' COMMUTATION TABLES, FOUR PER CENT. Age. D t N x Sx C x M, RX 62 4559-5 37906.9 273984-3 177.22 2926.18 30294.92 63 4206.9 33700.0 236077.4 174.62 2748.96 27368.74 64 3870.5 29829.5 202377.4 171.66 2574-34 24619.78 65 355o.o 26279.5 172547.9 168.74 2402.68 22045 44 66 3 2 44-7 23034.8 146268.4 166.29 2233.94 19642.76 67 29536 20081.2 123233.6 163.58 2067.65 17408.82 68 2676 4 17404.8 103152.4 160.23 1904.07 I534LI7 69 24I3-3 I499I-5 85747.6 156.18 1743.84 I3437.IO 70 2164.3 12827.2 70756 I 151.41 1587.66 11693.26 71 1929.6 10897.6 57928.9 145-77 !436.25 10105.60 72 1709.6 9188.0 47031-3 139.07 1290.48 8669.35 73 1504.8 7683.2 37843-3 132.08 1151-41 7378.87 74 ^4.9 6368.3 30160.1 124.57 1019.33 6227.46 75 "39-7 5228.55 23791.78 116.68 894.76 5208.13 76 979.19 4249.36 18563.23 107.85 778.08 43I3-37 77 833-68 3415.68 14313.87 98.68 670.23 3535.29 78 702.93 2712.75 10898 19 89-43 571-55 2865.06 79 586.47 2126.28 8185.44 80.17 482.12 2293.51 80 483.74 1642.54 6059.16 72.17 401.95 1811.39 81 392.96 1249.58 4416.62 64.14 329.78 1409.44 82 3I3-7I 935.87 3167.04 5565 265.64 1079.66 83 245.99 689.88 2231.17 46.88 209.99 814.02 84 189.65 500.23 1541.29 38.51 163.11 604.03 85 I43-85 356.38 1041.06 30.72 124.60 440.92 86 107-59 248.786 684.675 23.57 9388 316.32 87 79.883 168.903 435.889 18.61 70.31 222.44 88 58-203 110.700 266 9 6 14.84 51-70 152.13 89 90 41.120 27.990 69.580 41.590 156.286 86.706 11-55 8708 36.86 2 53 r 3 100.43 63.566 91 18.205 23-385 45 116 6-314 16.005 38-253 92 93 11.191 6.410 12.194 5.784 21.731 9-537 4-351 2.831 10.291 5-940 21.648 "357 94 3-332 2-452 3-753 1.662 3.109 5.417 95 96 1.542 0.625 0.910 0.285 1.301 0.391 0.8570 0.4009 1.4474 0.5904 2.3081 0.8607 97 O.2OO 0.085 0.106 0.1285 0.1895 0.2703 98 0.064 O O2I 0.021 0.0412 0.0610 0.0808 99 O.O2I O.OOO 0.000 0.0198 0.0198 0.0198 444 TWENTY-THREE GERMAN OFFICES' TABLE. TABLE III. TWENTY-THREE GERMAN COMPANIES' COMMUTATION TABLES, THREE AND ONE-HALF PER CENT. Age. D x NX Sx Cx MX Rx 17 57273 1138563 19777475 489-37 16834.60 486597.22 18 54847 1083716 18638912 486.87 16345.23 469762.62 19 52506 1031210 I7555I96 473-42 15858.36 453417-39 20 50257 980953 16523986 446.24 15384.94 437559-03 21 48111 932842 15543033 425.99 14938.70 422174.09 22 46058 886784 14610191 402.07 14512.71 407235-39 23 44098 842686 13723407 377.08 14110.64 392722.68 24 25 42230 40449 800456 760007 12880721 12080265 353-33 333-6i 13733-56 13380.23 378612.04 364878.48 26 38747 721260 11320253 317.59 13046.62 351498.25 27 37U9 684141 10599998 304.18 12729.03 338451.63 28 35560 648581 99H857 293.16 12424.85 325722.60 29 34064 6I45I7 9266276 285.02 12131.69 313297.75 30 32627 581890 8651759 278.14 11846.67 301166.06 31 31246 550644 8069869 272.06 11508.53 289319.39 32 29917 520727 7519225 267.04 11296.47 277750.86 33 28638 492089 6998498 261.11 11029.43 266454.39 34 27409 464680 6506409 256.78 10768.32 255424.96 35 26225 438455 6041729 253.02 10511.54 244656.64 36 25086 4 J 3369 5603274 248.95 10258.52 234145.10 37 23988 389381 5^89905 245.13 10009.57 223886.58 38 22932 366449 4800524 242.59 9764-44 213877.01 39 21914 344535 4434075 23994 9521.85 204112.57 40 20933 323602 4089540 237.93 9281.91 194590.72 41 I 99 8 7 303615 3765938 237.19 9043.98 185308.81 42 19074 284541 3462323 235.78 8806.79 176264.83 43 18193 266348 3177782 233-97 8571.01 167458.04 44 17344 249004 2911434 232.23 8337-04 158887.03 45 16525 232479 2662430 229.51 8104.81 150549-99 46 15737 216742 2429951 226.31 7875.30 142445.18 47 14978 201764 2213209 224.22 7648.99 134569.88 48 14248 187516 2011445 223.13 7424-77 126920.89 49 13543 173973 1823929 223.10 7201.64 119496.12 50 12862 161111 1649956 225.42 6978.54 112294.48 51 1 2201 148910 1488845 227.66 6753.12 105315.94 52 H56l 137349 1339935 230.13 6525.46 98562.82 53 10940 126408.9 1202586 232.49 6295.33 92037.36 54 10337.6 116071.3 1076177 234-58 6062.84 85742.03 55 9753-4 106317.9 960105.9 236.12 5828.26 79679.19 56 9187.4 97130.5 853788.0 237-98 5592.14 73850.93 57 58 8638.8 8107.4 88491.7 80384.3 756657.5 668165.8 239-18 240.68 5354.16 5114.98 68258.79 62904.63 59 7592.6 72791.7 587781-5 241.18 4874.30 57789-65 60 7094.7 65697 o 514989.8 242.34 4633.12 52915-35 61 6612.4 59084.6 449292.8 241.49 4390.78 48282.23 TWENTY-THREE GERMAN OFFICES' TABLE. 445 TABLE III. Cont. TWENTY-THREE GERMAN COMPANIES' COMMUTATION TABLES, THREE AND ONE-HALF PER CENT. Age. D, NX Sx Cx M x Rx 62 6147.3 52937.3 390208.2 240.08 4149.29 4389L45 63 5699-3 47238.0 337270.9 237.72 3909.21 39742.16 64 5268.9 41969.1 290032.9 234.81 3671.49 35832.95 65 4855-9 37113.2 248063.8 231-93 3436.68 32161.46 66 4459-8 32653-4 210950.6 229.67 3 2 04.75 28724.78 67 4079.3 28574.1 178297.2 227.01 2975.08 25520.03 68 37I4-3 24859.8 149723.1 223.43 2748.07 22544.95 69 3365.3 21494.5 124863.3 218.85 2524.64 19796.88 70 3032.6 18461.9 103368.8 213.19 2305.79 17272.24 71 2716.9 15745.0 84906.9 206.23 2092.60 14966.45 72 73 74 2418.8 2I 39-3 1878.3 13326.2 11186.9 9308.6 69161.9 44648^8 197.71 188.67 178.81 1886.37 1688.66 1499.99 12873.85 10987.48 9298.82 75 1636 o 7672.6 35340.2 168.30 I32I.I8 7798-83 76 1412.3 6260.3 27667.6 156.31 1152.88 6477.65 77 1208.3 5052.02 21407.3 143.71 996.57 5324 77 78 1023.73 4028.29 16355-3 130.86 852.86 4328.20 79 858.21 3170.08 12327.0 117.89 722.OO 3475-34 80 711.30 2458.78 9156.9 106.63 604.11 2753-34 81 580.61 1878.17 6698.16 95-22 497.48 2149.23 82 465-76 1412.41 4819.99 83.03 402.26 1651.75 83 366.98 1045-43 3407.58 70.27 3I9-23 1249.49 84 284.30 761.13 2362.148 58.01 248.96 930.26 85 21668 544-45 1601.018 46.50 190.95 681.30 86 162.85 381.60 1056.568 35-85 *44-45 490.35 87 121.49 260 ii 674.968 28.44 108.60 345-90 88 88.95 171.16 414.858 22.80 80.16 237.30 89 63.14 108.02 243.698 17.82 57.36 I57-I4 90 43-19 64.83 135-678 13-50 39-535 99-78 91 28.23 36.60 70.848 9.837 2-6.035 60239 92 17.44 19.16 34.248 6.812 16.198 34204 93 10.03 9.128 15.088 4-453 9.386 18.006 94 5.242 3.886 5.960 2.627 4-933 8.620 95 2.437 I.44Q 2.074 1.361 2.306 3.687 96 0-993 0.456 0.625 O64.O 0945 1.381 97 0.320 0.136 0.169 0.206 0.305 0436 98 0.103 0.033 0.033 0.0664 0.0985 0.1306 99 0.033 0.000 0.000 0.0321 0.0321 0.0321 446 TWENTY-THREE GERMAN OFFICES' TABLE. TABLE IV. TWENTY-THREE GERMAN OFFICES COMMUTATION TABLES THREE PER CENT. Age. Dx NX S x c x MX Rx 17 62188 1347790 24679861 55394 21120.77 650080.37 18 59843 1287947 23332071 533-79 20586.83 628959.60 19 57566 1230381 22044124 521-56 20053.04 608372.77 20 55368 1175013 20813743 494.01 19531.48 588319.73 21 53261 1121752 19638730 473-88 19037.47 568788.25 22 51236 1070516 18516978 449-44 18563.59 549750.78 23 49294 102 I 222 17446462 423-56 18114.15 531187.19 24 47435 973787 16425240 398.80 17690.59 513073.04 25 26 45654 43946 928133 884187 I545I453 14523320 37838 361.95 17291.79 16913.41 495382.45 478090.66 27 42304 841883 I3639I33 348.35 16551.46 461177.25 28 40724 80II59 12797250 337.36 16203.11 444625 79 29 39200 761959 11996091 329-59 15865.75 428422.68 30 37729 724230 11234132 323-I9 15536.16 412556.93 31 36307 687923 10509902 317.66 15212.97 397020.77 32 33 34932 336oi 652991 619390 9821979 9168988 3I3-3I 307-84 14895-31 14582.00 381807.80 366912.49 34 32315 587075 8549598 304.21 14274.16 352330.49 35 31069 556006 7962523 301.21 1396995 33805633 86 29863 526143 7406517 297.80 13668.74 324086.38 37 28695 497448 6880374 294.66 13370.94 310417.64 88 27565 469883 6382926 293.02 13076.28 297046 70 39 26469 443414 5913043 291.23 12783.26 283970.42 40 254 7 418007 5469629 290.19 12492.03 271187.16 41 24377 393630 5051622 290.69 12201.84 258695.13 42 23376 370254 4657992 290.36 11911.15 246493.29 48 22405 347849 4287738 289-53 11620.79 234582.14 44 21463 326386 3939889 288.77 11331.26 222961.35 45 20549 305837 3613503 286.78 11042.49 211630.09 46 19664 286173 3307666 284.16 10755.71 200587.60 47 18807 267366 3021493 282.90 10471.55 189831.89 48 17976 249390 2754127 282.88 10188.65 179360.34 49 17170 232220 2504737 284.22 9905-77 169171.69 50 16385 215835 2272517 288.57 9621.55 159265.92 51 15619 200216 2056682 292.85 9332-98 149644.37 52 53 14872 14141 185344 I7I203 1856466 1671122 297.47 301.98 8742.66 140311.39 131271.26 54 13427 157776 1499919 306.17 844068 122528.60 55 12730 145046 I342I43 309.67 8134-51 114087 92 56 12049 132997 1197097 3I3-63 7824.84 I05953-4I 57 "385 I2l6l2 1064100 316.74 7511.21 98128.57 58 10737 II0875 942487.7 320.28 7194.47 90617.36 59 10104 100770 . 6 831612.7 32249 6874.19 83422.89 60 9486.8 91283 8 730842 . i 325.62 6551-70 76548.70 61 8884.8 82399.0 639558.3 326,06 6226.08 69997.00 TWENTY-THREE GERMAN OFFICES' TABLE. 447 TABLE IV.-Cont. TWENTY-THREE GERMAN OFFICES COMMUTATION TABLES, THREE PER CENT. Age. Dx NX Sx Cx M, RX 62 8300.0 74099.0 557159-3 325-73 5900.02 63770.92 63 7732.5 66366.5 483060.3 324.08 5574-29 57870.90 64 7183.2 59183.3 416693.8 321.67 5250.21 52296.61 65 6652.3 52531-0 357510.5 319.27 4928.54 47046.40 66 6I39-3 46391.7 304979.5 317.70 4609.27 42117.86 67 5642.8 40748.9 258587.8 315.55 429I-S7 37508.59 68 5162.9 35586.0 217838 9 312.08 3976.02 33217.02 69 70 4700.4 4256.4 30885.6 26629.2 182252.9 151367-3 307.16 300.66 3663.94 3356.78 29241.00 25577-06 71 383I-7 22797.5 124738.1 292.26 3056.12 22220.28 72 3427-9 19369.6 101940.6 281.55 2763.86 19164.16 73 3 46.5 16323.1 82571.0 269.99 2482.31 16400.30 74 2687.8 13635-3 66247.9 257.11 2212.32 13917.99 75 2352.4 11282.9 52612 6 243-17 1955-21 11705.67 76 2040.7 9242.2 41329.7 226.95 1712.04 9750.46 77 1754-3 7487.9 32087.5 209.67 1485.09 8038.42 78 1493-5 5994-4 24599.6 191-85 1275.42 6553-33 79 1258.2 4736.2 18605.2 173.67 1083.57 5277.91 80 1047.8 3688.39 13869.05 157.85 909.90 4194-34 81 859.48 2828.91 10180.66 141.64 752.05 3284.44 82 692.81 2136.10 7351-75 124.10 610.41 2532.39 83 548.53 I587.57 5215-65 105.54 486.31 1921.98 84 427.01 1160.56 3628.08 87-55 380.77 1435.67 85 86 327.02 246.98 833-54 586.56 2467.52 1633.98 70.52 54.63 293.22 222.70 *%& 87 88 185.15 136.21 401.41 265.203 1047.42 646.006 43-55 35-08 168.07 124.52 538.98 370.91 89 97.136 168.040 380.803 27-55 89.44 246.39 90 66.781 101.259 212.763 20.98 61.89 156.95 91 43.858 57-401 111.504 15-36 40.91 95-06 92 27.222 30.179 54-103 10.69 25-55 54.15 93 15-743 14-4359 23-9243 7.021 14.863 28.600 94 8.2633 6.1726 9.4884 4.162 7.842 13-737 95 3-8605 2.3121 3.3158 2.167 3.680 5-895 96 1.5812 0.73092 1.00371 1.023 I-5I3 2.215 97 0.51172 0.21920 0.27279 0.3312 0.4904 0.7016 98 0.16561 0.05359 0.05359 0.1072 0.1592 0.2II2 99 0-05359 0.00000 o.ooooo 0.0520 0.0520 0.0520 THE FRENCH ACTUARIES' (A. F. AND R. F.) TABLES. THE French Actuaries' Tables are drawn from the com- bined experience of four French companies and were com- puted and adjusted by a committee of French actuaries. The tables were at first prepared according to Woolhouse' s formula, but this work was discarded and a new gradua- tion made by Makeham's formula. The tables were pub- lished in complete form in " Tables de Mortalite," in 1895. In making the tables, the sexes were not separated; but tables were separately computed on the lives of insured and annuitants. The former is called A. F. (Assurances Fran- gais or French Insurance) and the latter R. F. (Rentiers Frangais or French Annuitants.). Commutation columns are given at three per cent, three and one-half per cent and four per cent on both the A. F. and the R. F. tables. FRENCH ACTUARIES' TABLE ASSURANCES. 45 l TABLE I. FRENCH ACTUARIES' MORTALITY TABLE, ASSURANCE (A F). 1, dx q* ;"* 60 < lx d x qx it, 1000000 36015 .03602 .04181 52 607659 11270 01855 .01813 1 963985 26497 .02749 .03186 53 596389 H795 .01978 i933 2 937488 J 9549 .02085 .02415 54 584594 12348 .02112 .02064 3 9*7939 J 4453 01575 .01821 55 572246 12924 .02259 .02207 4 903486 10721 .01187 .01370 56 559322 J 3525 .02420 .02364 5 892765 Son .00897 .01032 57 545797 14148 .02592 02534 6 884754 6078 .00687 .00783 58 53^49 14788 .02782 .02721 7 878676 4744 .00540 .00605 59 516861 15444 .02988 .02924 8 873932 3876 .00443 .00485 60 501417 16110 .03213 .03146 9 870056 3372 .00388 .OO4IO 61 485307 16782 .03458 03389 10 866684 3155 .00364 .00372 62 468525 !745o .03725 03653 11 863529 3158 .00366 .00362 63 451075 18111 .04015 .03942 12 860371 3328 .00387 .00374 64 432964 18750 .04331 .04258 13 857043 3617 .00422 .00403 65 414214 19363 .04674 .04602 14 853426 3980 .00466 .00444 66 394851 19933 .05048 .04978 15 849446 4377 00515 .00491 67 374918 20450 05455 .05388 16 845069 4771 .00565 .00542 68 354468 20901 .05896 .05836 17 840298 5125 .00610 .00590 69 336567 21268 .06376 06325 18 835173 54" .00648 .00632 70 312299 21540 .06897 .06859 19 829762 5603 .00675 .00665 71 290759 21697 .07462 .07442 20 824159 5688 .00690 .00687 72 269062 21729 .08076 .08078 21 818471 5662 .00692 .00695 73 247333 21619 .08741 .08773 22 812809 5538 .00681 .00690 74 225714 21355 .09461 09531 23 807271 5345 .00662 .00670 75 204359 20929 .10241 10359 24 801926 5140 .00641 .00663 76 !83430 20334 .11085 .11263 25 796786 5006 .00628 .00625 77 163096 19566 .11997 .12250 26 791780 5067 .00640 .00636 78 J43530 18634 .12982 13327 27 786713 5*35 .00653 .00648 79 124896 17542 .14045 .14502 28 781578 5210 .00667 .00662 80 J o7354 16307 .15190 -I5786 29 776368 5293 .00682 .00676 81 91047 14953 .16424 .17187 30 771075 5385 .00698 .00692 82 76094 13506 17749 .18717 31 765690 54 8 7 .00717 .00710 83 62588 12000 .19172 .20387 32 760203 5597 .00736 .00729 84 50588 10470 .20698 .22210 33 754606 5719 .00758 .00750 85 40118 8959 22330 .24200 34 748887 5851 .00781 .00772 86 3H59 7501 .24074 26373 35 74303 6 5997 .00807 .00797 87 23658 6135 25933 .28745 36 37 737039 730884 6i55 6328 .00835 .008^6 .00824 .008^4 88 89 17523 12632 4891 3791 .27911 .^0010 .31334 .34161 38 724556 6514 .00899 .00886 90 8841 2849 .32231 37247 39 718042 6718 .00936 .00921 91 5992 2072 34575 .40615 40 711324 6938 .00975 .00959 92 3920 1452 37043 .44293 41 704386 7176 .01019 .01001 93 2468 978 .39630 .48308 42 697210 7433 .01066 .01047 94 1490 6 3 I 42333 .52691 43 689777 7710 .01118 .01097 95 859 388 45146 57476 44 682067 8009 .01174 .01152 96 47i 226 .48060 .62699 45 674058 8329 .01236 .01211 97 245 125 .51065 .68401 46 665729 8673 01303 .01276 98 120 65 .54148 .74626 47 657056 9041 .0:376 01347 99 55 32 .57292 .81422 48 648015 9434 .01456 .01425 100 23 14 .60478 .88841 49 638581 9854 01543 .01509 101 9 6 .63686 .96940 50 628727 10298 .01638 .01602 102 3 2 .66891 1.05782 51 618429 10770 .01742 .01703 103 i I .700 I-I5434 45 2 FRENCH ACTUARIES' TABLE ASSURANCES. TABLE II. COMMUTATION TABLES, FRENCH ACTUARIES' ASSURANCE (A. F.). FOUR PER CENT. Age, D, NX S z Cx M x RX 1000000.0 18979926.0 362935557 353I5-6 236128 5356451 1 926908.7 18053017.3 343955631 24983.1 200812 5120324 2 866760.2 17186257.1 325902613 17723.2 175829 4919512 3 816044.4 16370212.7 308716356 12599.2 158106 4743683 4 772303 5 15597909.2 292346143 89864 145506 4585578 733787.9 14864121.3 276748234 6456.6 136520 4440072 6 699234.0 14164887.3 261884113 4710.3 130063 4303552 7 667721.4 13497165.9 247719226 3535-o 125353 4173489 8 638573-6 12858592.3 234222060 2777.2 121818 4048136 9 611289.7 12247302.2 221363467 2323.1 119041 3926319 10 585500-7 11661801.9 209116165 2090.0 116718 3807278 11 560931.9 11100870.0 197454363 2011.5 114628 3690561 12 537385.1 10563484.9 186353493 20383 112616 3575933 13 514717.8 10048767.1 175790008 2130.1 110578 3463317 14 492832.2 9555934-9 165741241 2253.7 108448 3352739 15 471667.2 9084267.7 156185306 2383.2 106194 3244291 16 45"89 3 8633078.4 147101038 2497.8 103811 3138097 17 431386.5 8201691.9 138467960 2580.0 101313 3034286 18 412265 o 7789426.9 130266268 2619.2 98732.9 2932973 19 393840.2 73955867 122476841 2607.8 96113.7 2834240 20 37&135-3 7019451.4 115081255 2545-5 935059 2738126 21 359172.6 6660278.8 108061803 2436.4 90960.4 2644620 22 342969.2 6317309.6 101401524 2291.4 88524.0 2553660 23 3 2 753i-i 5989778 5 95084215 2126 5 86232 6 2465136 24 312848.5 5676930.0 89094436 1966.3 84106.1 2378903 :25 298887.8 5378042.2 83417506 1841.3 82139 8 2294797 :26 285586.6 5092455.6 78039464 1792.16 80298.53 2212657 27 272845.1 4819610.5 72947J08 1746.22 78506.37 2132359 28 260638.7 4S5897I.8 68127398 1703.75 76760.15 2053853 29 248943-5 4310028.3 63568426 1664.44 75056.40 1977092 30 237736.7 4072291.6 59258398 1628.09 73391.96 1902036 31 226996.5 3845295.1 55186106 1594.89 71763.87 1828644 32 216701.9 3628593.2 51340811 1564.55 70168.98 1756880 33 206833.1 3421760 i 47712218 I537-05 68604.43 1686711 34 197370.8 3224389.3 44290458 1512.26 67067.38 1618107 35 188296.7 3036092.6 41066069 1490.23 65555-12 1551039 36 179593-3 2856499 3 38329976 1470.66 64064.89 1485484 37 171243.7 2685255.6 35173477 1453-75 62594.23 1421419 38 163231.9 2522023.7 32488221 I439-I3 61140.48 1358825 39 155542.7 2366481.0 29966197 1426.94 59701-35 1297685 40 148160.9 2218320.1 27599716 1417.05 58274.41 1237983 41 141072.9 2077247.2 25381396 1409.29 56857-36 1179709 42 134265 T 1942982.1 23304149 1403-58 55448.07 1122052 43 127724.8 1815257.3 21361167 1399.96 54044.49 1067403 44 121439 5 1693817.8 19545910 1398.22 52644.53 1013359 45 II5397-7 1578420.1 17852092 1398.28 51246.31 960714.4 46 109588.2 1468831.9 16273672 1399.98 49848-03 909468.1 47 104000.5 1364831 36 14804840 1403.27 48448.05 859620.1 48 98624.45 1266206.91 13440008 1407.96 47044.78 811172.0 49 93450-57 1172756.34 12173802 I4I3-93 45636.82 764127.2 FRENCH ACTUARIES' TABLE ASSURANCES. 453 TABLE II. Cont. COMMUTATION TABLES, FRENCH ACTUARIES' ASSURANCE (A. F.). FOUR PER CENT. Age D x N T Sx C x MX Rx 50 88469.88 1084286.46 11001045 1420.98 44222.89 718490.4 51 83673.81 1000612.65 9916758.7 1428.96 42801.91 674267.5 52 79054.36 921558.29 8916146.1 1437.69 41372.95 631465.6 53 74604.07 846954.22 7994587.8 1446.86 39935.26 590092.7 54 70315.90 776638.32 7147633.6 1456.31 38488.40 550I57-4 55 66183.44 710454.88 6370995-2 1465.74 37032.09 511669.0 56 62200.64 648254.24 5660540.4 1474.91 35566.35 474636.9 57 58362.05 589892.19 5012286.1 1483.41 34091.44 439070.6 58 54662.76 535229.43 4422393.9 1490.96 32608.03 404979.1 59 51098.34 484131.09 3887164.5 1497.22 31117.07 372371.1 60 47664.90 436466.19 3403033-4 1501.73 29619.85 341254.0 61 44359.08 392107.11 2966567.2 1504.11 28118.12 311634.2 62 41178 06 350929.05 2574460.1 1503-94 26614.01 283516.1 63 38119.56 312809.49 2223531.1 1500.75 25110.07 256902.1 64 35181.82 277627.67 1910721.6 1494.07 23609.32 231792.0 65 32363-63 245264.04 1633093-9 1483-45 22115.25 208182.7 66 29664.23 215599.81 1387829.9 1468.45 20631.80 186067.4 67 27083.37 188516.44 1172230.1 1448.63 19163.35 I65435-6 68 24621 22 163895.22 983713.6 1423.55 17714.72 146272.3 69 22278.35 141616.87 819818.4 1392.91 16291.17 128557.5 70 20055.64 121561.23 678201.5 1356.395 14898.26 112266.4 71 I7954- 2 3 103607.00 556640.3 1313-823 13541-87 97368.11 72 I5975-38 87631.62 453033-3 1265.103 12228.04 83826.25 73 14120.41 73511.21 365401.7 1210.276 10962.94 71598.21 74 75 12390.55 10786.79 61120.66 50333-870 291890.5 230769.8 1149-537 1083.259 9752.663 8603.126 60635.27 50882.61 76 9309.700 41024.170 180435.9 1011.949 7519.867 42279.48 77 7959-343 33064.827 139411.762 936.357 6507.918 34759-61 78 6735.048 26329.779 106346.935 857.385 5571-561 28251.69 79 5635-279 20694.500 80017.156 776.116 4714.176 22680.13 80 4657.498 16037.002 59322.656 693.756 3938.060 17965.96 81 3798082 12238.920 43285.654 611.670 3244.304 14027.90 82 3052.218 9186.702 31046.734 531.226 2632.634 10783.59 83 2413.917 6772.785 21860.032 453.819 2101.408 8150-959 84 1876.072 4896.713 15087.247 380.766 1647.589 6049.551 85 1430.546 3466.167 10190.534 313.244 1266.823 4401.962 86 1068 365 2397.8021 6724.3672 252.210 953-579 3135.139 87 779.9668 1617.8353 4326.5651 198.342 701.369 2181.560 88 555-4772 1062.3581 2708.7298 152.031 503.027 1480.191 89 385-0372 677.3209 1646.3717 113-305 350.996 977.164 90 259.1249 418 1960 969.0508 81.895 237.691 626.168 91 168.8525 249-3435 550.8548 57.248 I55-796 388.477 92 106.2221 143.12136 301.51127 38.5*5 98.548 232.681 93 64.30267 78.81869 158.38991 24.988 59-963 I34-I33 94 37.32667 41.49202 79.57122 15-494 34-975 74.170 95 20.69731 20.79471 38.07920 9.163 19.481 39-195 96 10.91669 9.878020 17.284485 5-145 10.318 19.714 97 5.452030 4.425990 7.406465 2.730 5-173 9.396 98 2.565322 1.860668 2.980475 1.363 2-443 4.223 99 1.131017 0.729651 1.119807 0.636 1.080 1.780 454 FRENCH ACTUARIES' TABLE ASSURANCES. TABLE III. COMMUTATION TABLES, FRENCH ACTUARIES' ASSURANCE (A F), THREE AND ONE-HALF PER CENT. Age. D NX S x Cx MX Rx /' gooooo.o 20876419.6 426306796 35400.8 264733.4 6837104 1 931386 6 19945033.0 405430376 25164.4 229332.6 6572371 2 3 875155.0 827928.4 18241949.6 385485338 366415460 17938.0 12813.5 204168.2 186230.2 6343038 6138870 4 787335-7 17454613.9 348173511 9183.4 173416.7 5952640 5 751684.4 16702929.5 330718867 6330.0 164233-3 5779224 6 719748.0 15983181.5 314015967 4860.1 157603.3 5614990 7 690631.4 15292550 o 298032786 3665.2 152743-1 5457387 8 663674.0 14628876.1 282740236 2893.3 149078.0 5304644 9 638387 o 13990489.1 368111360 2432.0 146184.7 5155566 10 614408.6 13376080.5 254120871 2198.5 143752.7 5009382 11 591470.5 12784610.0 240744790 2126.2 141554.1 4865629 12 569379.2 12215230.8 227960180 2164.9 139427.9 4724075 13 547996.9 11667233.0 215744949 22733 137263.0 4584647 14 527231.1 11140002.8 204077715 2416.9 134989-7 4447384 15 507026.4 10632976.4 192937713 2568.0 132572.8 4312395 16 487356.3 10145620.1 182304736 2704.6 130004.8 4179822 17 468217.3 9677402.8 172159116 2807.0 127300.3 4049817 18 449624.7 9227778.1 162481713 2863.4 J24493-3 3922517 19 431605.4 8796172 7 153253935 2864.7 121629.9 3798023 20 414194.2 8381978.5 144457763 2809.9 118765.1 3676393 21 397425.8 7984552.7 136075784 2702.4 H5955.3 3557628 22 381329.8 7603222.9 128091231 2553-9 113252.9 3441673 23 365924.4 7237298.5 120488008 2381.5 110699.0 3328420 24 351209.2 6886089.3 113250710 2212.7 108317.5 3217721 25 337157-7 6548931.6 106364621 2082.1 106104.8 3109404 26 323709.6 6225222.0 99815689 2036.30 104022.7 3003299 27 310761.4 5914460.6 93590467 1993.68 101986.4 2899276 28 298292.7 5616167.9 87676006 1954-59 99992 74 2797290 29 286284.3 5329883.6 82059839 1918.73 98038.15 2697297 30 274717.2 5055166.4 76729955 1885.88 96119.42 2599259 31 263573.6 4791592.8 71674789 1856.35 94233-54 2503139 32 252835.8 4538757-0 66883196 92337.19 2408906 33 242487.2 4296269 8 62344439 1806.36 90547.36 2316529 34 232511.6 4063758.2 58048169 1785.81 88741.00 2225981 35 222893.5 3840864.7 53984411 1768.29 86955 19 2137240 36 213617.9 3627246.8 50143546 1753.51 85186.90 2050285 37 204670.5 3422576.3 46516299 1741,72 83433-39 1965098 38 196037.3 3226539 o 43093723 1732.52 81691.67 1881665 39 187705 o 3038834 o 39867184 1726 15 79959-15 1799973 40 179660.8 2859173 2 36828350 1722 47 78233.00 1720014 41 171892.3 2687280.9 33969177 1721.31 76510.53 1641781 42 164387.5 2522893.4 31281896 1722.62 74789.22 1565271 43 157135-3 2365758.! 28759002 1726.48 73066 60 1490481 44 150124.5 2215633.6 26393244 1732.66 71340.12 1417415 45 143344.7 2072288.9 24177611 1741.11 69607.46 1346075 46 136785.9 I935503-0 22105322 1751 64 67866.35 1276467 47 130438.5 1805064.5 20169819 1764 24 66114.71 1208601 48 124293.4 168077I.I 18364754 1778.69 64350.47 I 142486 49 118341.9 1562429.2 16683983.0 1794.87 62571.78 1078136 FRENCH ACTUARIES' TABLE ASSURANCES. 455 TABLE III Cont, COMMUTATION TABLES, FRENCH ACTUARIES' ASSURANCE (A F), THREE AND ONE-HALF PER CENT. 8. D x NX Sx Cx MX R* 50 H2575-8 14498534 15121554.0 1812.53 60776.91 1015564 51 106987.2 1342866.2 13671700.6 1831.51 58964.38 954786.9 52 101569.0 1241297.16 12328834.5 1851.60 57132.87 895822.6 53 96314.36 1144982.80 "087537.3 1872.42 55281.27 838689.7 54 91216.85 1053765.95 1893.75 53408.85 783408.4 55 86270.80 967495.15 8888788.6 1915.21 51515.10 729999.6 56 81470.87 886024.28 7921293.4 1936.51 49599.89 678484.5 57 76812.36 809211.92 7035269.1 1957.08 47663.38 628884.6 58 72291.13 736920.79 6226057.2 1976.55 45706.30 581221.2 59 67903.67 669017.12 5489136.4 1994.42 43729.75 535514.9 60 63647.03 605370.09 4820119.3 2010.10 41735.33 491785-1 61 59518.90 545851.19 4214749.2 2023.02 39725.23 450049.8 62 55517.67 490333-52 3668898.0 2032.56 37702.21 410324.6 63 64 51642.38 47892.74 438691.14 390798.40 3178564.5 2739873-4 2038.05 2038.78 35669.65 33631.60 372622.4 336952.7 65 44269.19 346529.21 2349075-0 2034.07 31592.82 303321.1 66 40772.79 305756.42 2002545.8 2023.23 29558.75 271728.3 67 37405-29 268351.13 1696789.3 2005.55 27535.52 242169.6 68 34169.05 234182.08 1428438.2 1980.36 25529.97 214634.0 69 31067.01 203115.07 1194256.1 1947.09 23549-61 189104.1 70 28102.56 175012.51 991141.1 1905.209 21602.52 165554.5 71 25279.52 149732.99 816128.6 1854.326 19697.32 ^43951-9 72 22601.97 127131.02 666395.6 1794.188 17842.99 124254.6 73 20074.08 107056.94 539264.5 1724.724 16048.80 106411.6 74 75 17699.95 15483.40 89356.99 73873-59 432207.6 342850.6 1646.082 1558.669 14324.08 12678.00 90362.82 76038.74 76 13427.74 60445.85 268977.0 1463 096 11119.33 63360.75 77 H535-53 48910.320 208531.179 1360.344 9656.230 52241.42 78 9808.304 39102.016 159620.859 1251.631 8296.886 42585-19 79 8246.349 30855.667 120518.843 1138.467 7044.255 34289-31 80 6848.444 24007.223 89663.176 I02257I 5905.788 27245.05 81 5611.728 18395-495 65655-953 905 935 4883.217 21339.26 82 4531.486 13864.009 47260.458 790.591 3977.282 16456.05 83 3601.145 10262.864 33396.449 678.654 3186.691 12478.77 84 2812.295 7450.569 23I33.585 572.159 2508.037 9292.074 85 2154.794 5295 775 15683.016 472.972 1935.878 6784.037 86 1617.026 3678 749 10387.241 382.654 1462.906 4848.159 87 1186.223 2492.5255 6708.4928 302.379 1080.252 3385-253 88 848.8859 1643.6396 4215.9673 232.896 777.873 2305.001 89 591.2605 1052.3791 2572.3277 174.411 544-977 1527.128 90 399.8326 652.5465 1519 9486 126.670 370.556 982.152 91 261.8001 390.7464 867.4021 88.975 243.896 611.586 92 165.4892 225.25719 476.65578 60.259 154.921 367.690 93 100.66467 124.59252 251.39859 39-212 94.662 212.769 94 58.71653 65.87599 126.80607 24.432 55450 118.107 95 32.71509 33.16090 60.93008 14.518 31.018 62.657 96 17.33876 15.822135 27.769184 8.191 16.499 31639 97 8.701182 7.120953 11.947049 4368 8.308 15-140 98 4.113912 3.007041 4.826096 2.191 3-94* 6.832 99 1-822533 1.184508 1.819055 1.027 1-750 2.891 45 6 FRENCH ACTUARIES' TABLE ASSURANCES. TABLE IV. COMMUTATION TABLES, FRENCH ACTUARIES' ASSURANCE (A F), THREE PER CENT. Age. D x NX Sx Cx MX Rx IOOOQOO 23124696.2 505068944.1 35486.6 301766.4 8841014 1 935907-9 22188788.3 481944247 9 25347-8 266279.8 8539247 2 883672.4 21305115.9 459755459-6 18156.5 240932.0 8272968 3 840044.1 20465071.8 438450343.7 13032.5 2227755 8032036 4 802735.5 19662336.3 417985271.9 9385-7 209743.0 7809260 5 770107.0 18892229.3 398322935.6 68090 200357.3 7599517 6 740967.8 18151261.5 379430706.3 5015.6 195548.3 7399160 7 714444.0 17436817.5 361279444.8 3800.7 188532.7 7205611 8 689890.0 16746927.5 343842627.3 3014.9 184732.0 7017079 9 666825.5 16080102.0 327095699 8 2546.4 I8I7I7.I 6832347 10 644894.4 15435207 6 311015597.8 2319.8 I79I70.7 6650630 11 623831.7 14811375.9 295580390.2 2241.4 I76857-5 6471459 12 13 603447. 1 583604.6 14207928.8 13624324.2 280769014.3 266561085.5 2299.9 2426.9 174609.6 172309.7 6294601 6119992 14 564215.2 13060109.0 252936761.3 2592.6 169882 8 5947682 15 545227.2 12514881.8 239876652.3 2768.2 167290.2 5777799 16 526619.3 11988262.5 227361770 5 2929.5 164522.0 5610509 17 508394.3 11479868.2 215373508.0 3055-2 161592.5 5445987 18 490576.2 10989292 o 203893639.8 3131-8 158537.3 5284395 19 473201.7 10516090.3 192904347 8 3148.4 155405.5 5125857 20 456516 9 10059773.4 182388257.5 3103.1 152257.1 4970452 21 439968 6 9619804.8 172328484.1 2999.0 149154.0 4818195 22 4241990 9195605.8 162708679.3 2847.8 146155.0 4669041 23 409037-5 8786568.3 I535I30735 2668.5 143307.2 4522886 24 394494-5 8392073.8 144726565.2 2491.4 140638 7 4379579 25 380549.6 8011524.2 I3633443I-4 2355-7 138147.3 4238940 26 367144.4 76443798 128322907.2 23 r 5-i 13579 r - 6 4100793 27 354169 8 7290210.0 120678527.4 2277.7 133476.5 3965001 28 3416098 6948600.2 113388317.4 2243.9 131198.8 3831524 29 329449 i 6619151.1 106439717.2 2213.4 128955.0 3700326 30 317672.6 6301478.5 99820566. 1 2186.0 126741.6 357I37I 31 306266.1 5995212.4 93519087.6 2162.3 124555-5 3444629 32 295215.1 5699997.3 87523875 2 2141.7 122393 3 3320074 33 284506.4 5415490 9 81823877.9 2124.5 120251.6 3197680 34 274126 4 5141364 5 76408387.0 2110.5 118127 o 3077429 35 264062.6 4877301.9 71267022.5 2IOO.O 116016.5 2959302 36 254302.4 4622999.5 66389720.6 2092.5 113916.5 2843285 37 244833-7 4378165.8 61766721 I 2088.5 111824.0 2729369 38 235644-7 4142521.1 53788555-3 2087.6 109735 5 2617545 39 226724.2 3915796.9 53246034.2 2090.0 107647.9 2507809 40 218061.3 3697735-6 49330237.3 2095.7 105557-8 2400161 41 209645.0 3488090.6 45632501 7 2104.5 103462.2 2294603 42 201465 3 3286625.3 42144411.1 2116.3 101357.7 2191141 43 193512.1 30Q3II3 2 38857785 8 2I3L3 99241.4 2089784 44 185775-8 2007337-4 3576-1672.6 2149.3 97110.1 1990542 45 178247.1 2729090.3 3 2 8S733S- 3 2170.3 94960.8 1893432 46 170917.0 2558I73-3 30128244 9 2194.0 92790-5 1798471 47 163777.0 2394396 3 27570071.6 2220.5 905965 1705681 48 156818.9 2237577.4 25I7S675-3 2249.6 88376.0 1615084 49 150034.8 2087542.6 22938097.9 228I.I 86126.4 1526708 FRENCH ACTUARIES* TABLE ASSURANCES. 457 TABLE IV.Cont. COMMUTATION TABLES, FRENCH ACTUARIES' ASSURANCE (A F), THREE PER CENT. I Dx NX S* c, MX Rx 50 143417.3 1944125 3 20850555.3 2314.67 83845-4 1440582. 51 I36959-3 1807166.0 18906430.0 2350.28 815307 1356736- 52 130654.4 1676511.6 17099264.0 2387-58 79180.4 1275206. 53 124496.4 1552015.2 15422752.4 2426.15 76792.8 1196025. 54 1184797 1433535-5 13870737-2 2465.70 74366.7 1119233. 55 112599.4 1320936.1 12437201.7 2505-76 71901.0 1044866 o 56 106850.7 1214085.4 11116265 6 69395.2 972964.9 57 101230.1 1112855.28 9902180.2 2585.46 66849 3 903569-7 58 95734-07 1017121.21 8789324 93 2623.85 64263.8 836720.4 59 90360.35 926760.86 7772203.72 2660.43 61639.9 772456.6 60 85107.12 841653.74 6845442.86 2694.36 58979-5 710816.6 61 79973-44 761680.30 6003789. 12 2724.84 56285.2 651837.1 62 74959.26 686721.04 5242108.82 2750.97 53560.3 595551-9 63 70065.37 616655.67 4555387-78 2771.80 50809.4 541991.6 64 65293-50 551362.17 3938732.11 2786.25 48037 6 491182.2 65 60646.39 490715.78 3387369.94 2793.30 452SL3 443144.6 66 56127.66 434588.12 2896654. 16 2791.91 67 5I74T.95 382846.17 2462066 04 2780.95 39666. 1 355435 3 68 47494-77 335351.40 2079219.87 2759-35 36885.2 315769.2 69 43392.56 291958.84 1743868.47 2726.16 34125.8 278884.0 70 39442.55 252516.29 1451909.63 2680.46 3I399-7 244758.2 71 35652.60 216863.69 H99393 34 2621.55 28719.2 213358.6 72 32031.09 184832.60 982529.65 2548.84 26097.6 184639.4 73 28586.70 156245.90 797697.05 2462.05 23548.8 I5854I-7 74 25328.16 130917.74 641451.15 2361.20 21086.7 134992.9 75 22263.89 10865385 510533.41 2246.663 187255 113906.2 76 19401.74 89252.11 401879.56 2119.141 16478.9 95180 42 77 16748.60 72503-51 312627.45 1979 880 14359-7 78701.59 78 79 . I4309-95 12089 5 2 5819356 46104.04 240123.94 181930.38 1830.498 1673.079 12379 9 10549.4 64342.00 51962 08 80 10088.87 36015.169 135826.34 1510.056 8876.28 41412.75 81 8307.113 27708.056 99811.169 1344.310 7366.23 32536.49 82 6740 582 20967.474 72103.113 1178.848 6021.92 25170.28 83 6382.701 I5584-773 5II35-639 1016.849 4843.07 19148.37 84 4223-998 11360.775 35550.866 861.446 3826.22 14305-31 85 3252.160 8108.615 24190.091 7I5-566 2964-77 10479.09 86 2452.371 5656.244 16081.476 581.734 2249.21 7514.32 87 1807.751 3848493 10425.232 461.927 1667.47 5265.11 88 1299 944 2548.5485 6576 7388 357.509 1205-55 3597.64 89 909.8240 1638.7245 4028.1903 269.031 848.036 2392.09 90 618.2440 1020.4805 2389.4658 196337 579006 I544-05 91 406.7752 613-7053 1368.9853 138.581 382.669 965.048 92 258.3793 355.3260 755.2800 94.310 244 101 582 379 93 157.9313 I97-39470 399.95404 61.668 149.779 338.291 94 92.56666 104.82804 202.55934 38.611 88.ni 188.512 95 51-82573 53.00231 97-73 T 3o 23.055 49.500 100.401 96 27.60060 25.40171 44.72899 13.070 26.445 50.901 97 13.91816 H.483S53 19.327278 7.003 13-375 98 6.61239 4.871114 7-843725 3-530 6.372 11.080 99 2.943644 1.927470 2.972611 1.663 2.842 4-703 FRENCH ACTUARIES TABLE ANNUITANTS. 459 TABLE I. FRENCH ACTUARIES' MORTALITY TABLE ANNUITIES (R. F.). Age. lx dx qx Ka Age. l x d. qx u* IOOOOCO 36015 .03602 .04181 53 622913 9419 .01512 .01478 1 963985 26497 .02749 .03186 54 613494 9860 .01607 .01570 2 937488 19550 .02085 .02415 55 603634 10332 .01712 .01672 3 917939 14453 01575 .01821 56 593302 10837 .01826 .01783 4 10721 .01187 .01370 57 582465 "373 01953 .01906 5 2765 8011 .00897 .01032 58 571092 "943 .02091 .02040 6 4754 6078 .00687 .00783 59 559149 12545 .02244 .02189 7 878676 4744 .00540 .00605 60 546604 13*77 .0241 i .02352 8 873932 3876 .00443 .00485 61 533427 13839 .02594 02531 9 10 870056 866684 3372 315? .00388 .00364 .00410 .00372 62 63 519588 505060 14548 15240 .02796 .opi7 .02729 .02946 11 863529 3158 .00366 .00362 64 489820 15969 .03260 03185 12 860371 3328 .00387 .00374 65 473851 16712 03527 .03448 13 857043 3617 .00422 .00403 66 457139 17459 .03819 .03737 14 853426 3980 .00466 .00444 67 439680 18202 .04140 .04056 15 849446 4377 00515 .00491 68 421478 18929 .04491 .04406 16 845069 4771 .00565 .00542 69 402549 19630 .04876 .04791 17 840298 5125 .Oo6lO .00590 70 382919 20289 .05298 05215 18 19 835173 829762 54" 5633 .00648 .00675 .00632 .00665 71 72 362630 ?4I74t 20889 21413 .05760 .06266 .05681 .06193 20 824159 5688 .00690 .00687 73 320328 21244 .06819 06758 21 818471 5662 .00692 .00695 74 298484 22159 .07424 .07378 22 812809 5538 .00681 .00690 75 276325 22341 .08085 .08061 23 807271 5345 .00662 .00671 76 253984 22366 .08806 .08812 24 801926 SHo .00641 .00665 77 231618 2222O 09593 .09638 25 796786 4969 .00624 .00622 78 209398 21886 .10452 .10547 26 791817 4990 .00630 .00629 79 187512 21350 .11386 "547 27 786827 5016 .00638 .00636 80 166162 20609 .12403 .12647 28 781811 5047 .00646 .00643 81 145553 19662 .13508 13857 29 776764 5083 .00654 .00652 82 125891 I85I7 .14708 .15189 30 771681 5125 .00664 .00661 83 107374 17189 .16009 .16653 31 766556 5173 .00675 .00672 84 90185 15708 .17417 .18265 32 761383 5227 .00687 .00683 85 74477 I4I05 .18939 .20037 33 756156 5290 .00699 .00695 86 60372 12425 .20581 .21987 34 750866 5358 .00714 .00709 87 47947 I07I5 .22348 .24132 35 745508 5438 .00729 .00724 88 37232 9028 .24248 .26492 36 740070 5525 .00747 .00740 89 28204 7413 .26283 .29089 37 734545 5623 .00766 .00759 90 20791 5917 .28460 3 I 945 38 728922 5732 .00786 .00779 91 14874 4578 .30780 35087 39 723190 5852 .00809 .00801 92 10296 3423 .33246 .38544 40 717338 5980 .00834 .00825 93 6873 2465 .35857 .42347 41 711352 6i33 .00862 .00851 94 4408 1702 .38612 46530 42 705219 6294 .00893 .00881 95 2706 "23 .41506 5"33 43 698925 6473 .00926 .009 [3 96 1583 705 4453 2 .56196 44 692452 6668 .00963 .00948 97 878 419 .47681 .61766 45 685784 6882 .01003 .00987 98 459 234 50939 .67893 46 678902 7H5 .01048 .01030 99 225 122 .54289 74635 47 671787 7370 .01097 .01078 100 103 59 .57711 .82051 48 664417 7647 .01151 .01129 101 44 27 .61181 . .90209 49 656770 7947 .OI2IO .01187 102 17 II .64670 .99184 50 648823 8275 .01275 .01249 103 6 4 .68147 .09058 51 640548 8627 01347 .01319 104 2 I .71578 .19920 52 631921 9008 .01427 01395 105 I I 74927 .31870 460 FRENCH ACTUARIES' TABLE ANNUITANTS. TABLE II. COMMUTATION TABLES, FRENCH ACTUARIES' ANNUITY (R. F.), FOUR PER CENT. Age. D, N, s, Cx MX K, IOOOOOO 19139904.6 373016522 353I5-6 229853 5II79I9 1 926908.7 18212995.9 353876617 24983.1 194537 4888066 2 866760.2 17346235.7 335663622 17723.2 169554 4693529 3 816044.4 16530191.3 318317386 12599.2 I5I83I 4523975 4 772303.5 15757887.8 301787195 8986.4 139232 4372144 5 733787-9 150240999 286029307 6456.6 130245 4232912 6 699234.0 14324865.9 271005208 4710.3 123789 4102667 7 667721.4 13657144.5 256680342 3535-Q 119078 3978878 8 9 638573 6 611289 7 13018570.9 12407281.2 243023198 230004627 2777.2 2323.1 H5543 112766 3859800 3744257 10 585500 7 11821780.5 217597346 2090.0 110443 363I49I 11 560931.9 11260848.6 205775565 2011.5 108353 3521048 12 537385-I 10723463.5 194514717 2038.3 106341 3412695 13 514717.8 10208745 7 183791253 2130.1 104303 3306354 14 492852.2 97I59I3-5 173582508 2253-7 I02I73.0 3202051 15 471667.2 9244246.3 163866594 2383.2 99919.0 309 9 878 16 451189.3 8793057-0 154622348 2497.8 97535-8 2999959 17 431386.5 8361670.5 145829291 2580.0 95038.0 2 9 02423 18 412265.0 7949405.5 137467620 2619.2 92458.0 2807385 19 393840.2 755556S-3 129518215 2607.8 89838.8 2714927 20 376I35-3 7179430.0 121962649 2545-5 87231 o 6625088 21 359172.6 6820257.4 114783219 2436.4 84685.5 2537857 22 342969.2 6477288.2 107962962 2291.4 82249.1 2453172 23 32753I-I 6149757.1 101485674 2126.5 79957-7 2370923 24 312848 5 5836908.6 953359 I 7 1966.3 77831.2 2290965 25 298887.8 5538020.8 80499008 1827.65 75864.93 22I3I34 26 285600.0 5252420.8 83960987 1764.96 74037.28 2137269 27 272884.7 4979536.1 78708567 1705.92 72272.32 2063231 28 260716.3 4718819.8 73729030 1650.48 70566.40 1990959 29 249070.3 4469749-5 69010211 1598.19 68915.92 1920393 30 237923 5 4231826.0 64540461 1549.42 67317.73 I85M77 31 227253 3 4004572.7 60308635 150375 65768.31 I784I59 32 217038.3 3787534-4 56304062 1461.12 64264.56 I7I839I 33 207257.9 3580276.5 52516528 1421 64 62803 44 I654I26 34 197892.4 3882384.1 48936251.5 I384-93 61381.80 I59I323 35 188923.1 3193461.0 45553867.4 I35I-I9 59996.87 I52994I 36 180331.9 3013129.1 42360406.4 1320.20 58645-68 1469944 37 172101.5 2841027.6 39347277-3 1291.87 5732S.48 I4II298 38 164215.5 2676812 I 36506249.7 1266.20 56033.61 1353973 39 156657.8 2520154.3 33829437.6 1243.11 54767-41 1297939 40 149413.6 2370740.7 31309283.3 1222 53 53524.30 I243I72 41 142468.1 2228272.6 289385426 1204.42 52301.77 1189748 42 135807.6 2092465.0 26710270.0 1188.69 51097.35 II37446 43 129418.6 1963046.4 24617805.0 1175.29 49908.66 1086348 44 123288.5 I839757-9 22654758.6 1164.14 48733-37 1036340 45 117405.1 1722352.8 20815000.7 1155.28 47569.23 987606.4 46 111756.7 1610596. i 19092647.9 1148.49 46413.95 940037.1 47 106332.2 1504263.9 17482051.8 1143.86 45265.46 893623.2 48 IOII20.8 1403143.11 I5977787-9 1141.22 44121.60 848357.7 49 96112.48 1307030.63 14574644.8 1140.51 42980.38 804236.1 FRENCH ACTUARIES' TABLE ANNUITANTS. 461. TABLE II. Cont. COMMUTATION TABLES, FRENCH ACTUARIES' ANNUITY (R. F.), FOUR PER CENT. Age. D, N x Sz Cx MX R 50 91297.51 1215733-12 13267614.2 1141.67 41839.87 761255.8 51 86666.56 1129066.56 12051881.1 1144.63 40698.20 719415.9 52 82210.85 1046855.71 10922814.5 1149.21 39553-57 677717.7 53 77922.00 968933.71 9875958.8 1155-36 38404.36 639164.1 54 73792.07 895141.64 8907025.1 1162.93 37249.00 600759.8 55 69813.58 825328 06 8011883.4 1171 77 36086.07 563510.8 56 57 6597945 62283.01 759348.61 697065.60 7186555.4 6427206.8 1181.71 1192.56 349I4-30 33732-59 527424.7 492510.4 58 58718.12 638347.48 5730141.2 1204.10 32540.03 458777.8 59 55279-OI 583068.47 5091793.7 1216.13 3I335-93 426237.8 60 51960.41 531108.06 4508725.2 1228.31 30119.80 394901.8 61 48757.47 482350.59 3977617.2 1240.41 28891.49 364782.0 62 45665.86 436684.73 3495266.6 1252.04 27651.08 335890.5 63 42681.76 394002.97 3058581.8 1262.86 26399 04 308239.5 64 39801.83 354201.14 2664578.9 1272.46 25136.18 281840.4 65 37023.25 517177.89 2310377-7 1280.40 23863.72 256704.2 66 34343.76 282834.13 1993199.8 1286.18 22583.32 232840.5 67 31761.65 251072.48 1710365.7 1289.33 21297.14 210257.2 68 29275.76 221796.72 1459293.2 1289.33 20007.81 188960.1 69 26885.48 194911.24 1237496.5 1285.60 18718.48 168952.3 70 24590.80 170320.44 1042585.3 1277.610 17432.88 150233.8 71 22392.21 147928.23 872264.8 1264.844 16155 27 132800.9 72 20290.70 127637-53 724336.6 1246.745 14890.42 116645.6 73 18287.76 109349.77 596699.1 1222.866 13643.68 101755.2 74 16385.27 92964.50 487349.3 1192.827 12420.81 88111.52 75 14585.42 78379.08 394384.8 1156.307 11227.98 75690.71 76 12890.60 65488.48 316005.7 1113-143 10071.68 64462 73 77 11303.29 54185.191 250517.24 1063.323 8958.533 5439 I -5 78 9825.882 44359.309 196332.05 1007.021 7895.210 45432.51 79 8460.500 35898.809 151972.74 944.628 6888.189 37537-30 80 7208.820 28689.989 116073.93 876.764 5943.56i 30649.12 81 6071.825 22618.164 87383.95 804.287 5066.797 2470555 82 5049.631 17568.533 64765.78 728.298 4262.510 19638.76 83 4141.268 13427.265 47I97.25 650.104 3534.212 15376.25 84 3344-5I 2 10082.753 33769 98 571.207 2884.108 11842.04 85 86 2655.766 2069.996 7426.987 5356.991 23687.23 16260.24 493.210 417.746 2312.901 1819.691 8957.927 6645.026 87 1580.749 3776.242 10903.25 346.410 1401.945 4825335 88 1180.269 2595-9727 7127.010 280.633 1055.535 3423-390 89 859.6944 1786.2783 4531.0376 221.568 774.902 2367-855 90 609.3636 1126.9147 2794-7593 170 057 553-334 1592.953 91 419.1732 707.7415 1667 8446 126.517 383-277 1039.619 92 278.9919 428.7496 960.1031 90.952 256.760 656.342 93 179.0761 249.6735 531-3535 6^.967 165.808 399-582 94 110.4472 139.22632 281.6800 41.816 102.841 233-774 95 65.19406 74.03226 142.45372 26535 61.025 130.933 96 36.66811 37-364I5 68.42146 16.011 34.490 69.908 97 I9-5S673 17.807419 3I-0573I 9144 18.479 35-418 98 9.838400 7.969019 13.249892 4914 9-335 16.939 99 4.641161 3-327858 5.280873 2.472 4.421 7.604 462 FRENCH ACTUARIES' TABLE ANNUITANTS. TABLE III. COMMUTATION TABLES, FRENCH ACTUARIES' ANNUITY (R F), THREE AND ONE-HALF PER CENT. Age. D, N x s, Cx M Rx o IOOOOOO.O 21093571.6 440158875 35400.8 257262.5 6573996 1 931386.6 20162185.0 419065304 25164.4 221861.7 6316733 2 3 875155-0 827928.4 19287030.0 18459101.6 398903119 379616089 17938.0 12813.5 196697.3 178759.3 6094872 5898174 4 787335.7 17671765.9 361156987 91834 I65945-8 5719415 5 751684.4 16920081.5 343485221 6630.0 156762.4 5553469 Q 719748.0 16200333.5 326565140 4860 I 150132.4 5396707 7 690631.4 15509702. 1 310364806 3665.2 145272.3 5246574 8 663674.0 14846028.1 294855104 28933 141607 I 5101302 9 638387.0 14207641.1 280009075 2432.0 138713.8 4959695 10 614408.6 13593232.5 265801435 2198.5 136281.8 4820981 11 591470.5 13001762.0 252208202 2126.2 134083 3 4684699 12 569379.2 12432382.8 239206440 2164.9 I3I957.I 4550616 13 547996.9 11884385.9 226774058 2273.3 129792.2 4418659 14 527231.1 11357154.8 214889672 2416.9 127518.9 4288867 15 507026.4 10850128.4 203532517 2568.0 125102.0 4161348 16 487356.3 10362772.1 192682389 2704.5 122534.0 4036246 17 468217.3 9894554.8 182319016 2807.0 119829.5 3913712 18 449624 7 9444930.1 172425062 28634 117022.5 3793882 19 431605.4 9013324 7 162980132 2864.7 114159.1 3676860 20 414194 2 8599130.5 153966807 2809.9 111294.4 3562701 21 22 397425-8 381329.8 8201704 7 7820374 9 145367676 I37I65972 27024 25539 108484.5 105782.1 3451406 3342922 23 3659244 7454450.5 129345597 2381.5 103228.2 3237140 24 351209.2 7103241.3 121891146 2212 7 100846.7 3133912 25 337157-7 6766083.6 114787905 2066.64 98634.05 3033065 26 323724.8 6442358.8 108021821 2OO5 . 40 96567.41 2934431 27 310806.4 6131552.4 101579463 1947.68 94562.01 2837863 28 298381.5 5833170.9 95447910 1893.48 92614.33 2743301 29 286430.2 5546740.7 89614739 1842.35 90720.85 2650687 30 274933.2 5271807.5 84067999 1794.76 88878.50 2559966 31 263871.8 5007935.7 78796191 1750 28 87083.74 2471088 32 253228.1 4754707.6 73788255 1708.87 85333-46 2384004 33 242985.2 4511722.4 69033548 1670.73 83624.59 2298671 34 233126.0 4278596.4 64521825 ^35 -44 81953.86 2215046 35 223635.0 4054961.4 60243229 1603.31 80318.42 2133092 36 214496.5 3840464.9 56188268 1574.10 78715.11 2052774 37 205605.8 3634769.1 52347803 1547-77 77141.01 1974059 38 197218.5 3437550.6 48713034 1524-34 75593-24 1896918 39 189050.9 3248499.7 45275483 1503-78 74068.90 1821324 40 181179.7 3067320 o 42026983 1486.02 72565.12 1747255 41 173592.2 2893727.8 38959663 1471.08 71079.10 1674690 42 166276.0 2727451.8 36065935 1458.88 69608.02 1603611 43 159219.2 2568232.6 33338484 1449.40 68149.14 1534003 44 152410.2 2415822.4 30770251 1442.60 66699.74 1465854 45 145838.3 2269984.1 28354429 1438.53 65257-14 I399 T 54 46 139492.5 2130491.6 26084445 1436.98 63818.61 1333897 47 133362.9 1997128.7 23953953 1438.11 62381.63 1270079 48 127439-5 1869689 2 21956824 1441.72 60943.52 1207697 49 121712.8 1747976.4 20087135 1447.78 59501.80 H46753 FRENCH ACTUARIES' TABLE ANNUITANTS. 463 TABLE III. Con t. COMMUTATION TABLES, FRENCH ACTUARIES' ANNUITY (R F), THREE AND ONE-HALF PER CENT. & D x NX s, Cx M x Rx 50 116173.9 1631802.5 I8339I59 1456.25 58054.02 1087252 51 110813.9 1520988.6 16707356 I 1467.08 56597-77 1029198 52 10 ,624.5 1415364 10 15186367.5 1480.07 55130 69 972599.9 53 100597.82 1314766.28 13771003.4 1495 18 53650 62 917469.2 54 95726.29 1219039.99 12456237.1 1512.24 52I55.44 863818.6 55 91002.73 1128:37.26 11237197.1 I53 X -09 50643.20 811663.1 56 8642034 1041616.92 10109159.8 I55L55 49112.11 761019.9 57 81972.88 959644.04 9067542.9 I573-36 47560.56 711907.8 58 77654-32 881989.72 8107898.9 1596.26 45987.20 664347.2 59 73459 3 1 808530.41 7225909.1 1619.99 44390.94 618360.0 60 69382.81 739147.60 64173787 1644.13 42770.05 573969.1 61 65420.47 673727-I3 5678231.1 1668.34 41126.82 531198.2 62 61568.30 612158.83 5004504.0 1692 12 39458.48 490071.3 63 57^23.03 554335-So 4392345.2 1714.99 37766.36 450612.9 64 54i8i.93 500153.87 3838009.4 1736.37 36051.37 412846.5 65 50642.94 449510-93 3337855-5 I755-64 34315.00 376795.1 66 47204.68 402306.25 2888344.6 1772.09 32559-36 342480.1 67 43866.55 358439.70 2486038.3 1785.01 30787.27 309920.8 68 40628.56 317811.14 2127598.6 I793-63 29002.26 279I33-5 69 37491.62 280319.52 1809787.5 1797.09 27208.63 250131.2 70 34457.36 245862.16 1429468.0 I794-546 254H.54 222922.6 71 31528.20 21433396 1283605.8 1785.197 23616.99 197511.1 72 28707.30 185626.66 1069271.8 1768.153 21831.79 173894.1 73 25998.54 150628.12 883645.2 1742.666 20063.64 152062.3 74 23406.42 136221.70 724017.1 1708 071 18320 98 131998.6 75 20935.98 115285.72 587795.4 1663.775 16612.90 113677.7 76 18592.60 96693.12 472509.6 1609.404 14949.13 97064.76 77 16381.93 80311.19 3758i6.5 1544.802 13339-73 82115.63 78 14309.51 66001.68 295505.3 1470.073 11794.92 68775.91 79 12380.62 53621.063 229503.65 I385-652 10324.85 56980.98 80 10599.940 43021.123 175882.59 1292.318 8939 198 46656.13 81 8971.220 34049 903 132861.47 II9I,2l8 7646.880 37716.94 82 7496.955 26552.948 98811.56 1083.882 6455.662 30070.06 83 6178.051 20374.897 72258.61 972.183 537i.78o 23614.39 84 50I3-532 15361-365 51883.72 858.325 4399-597 18242.61 85 4000.314 11361.051 36522.35 744.704 354I-272 13843.02 86 3I33-044 8228.007 25161.30 633.807 2796.568 10301.74 87 2404.103 5823.904 16933.29 528.114 2162.761 7505.176 88 1803 699 4020.205 11109.39 429.902 1634.647 5342.415 89 1320.141 2700.0638 7089.1851 341.060 1204.745 3707.768 90 940 2549 1759.8089 4389.1213 263.034 863685 2503.023 91 649.9136 1109.8953 2629.3124 196.633 600.651 1639.338 92 434.6570 675-2383 1519.4171 142.042 404.018 1038.687 93 280.3404 394.8979 844.1788 98.812 261.976 634-669 94 I73-7384 221.1595 449.2810 65-937 163.164 372.693 95 103.0486 118.11087 228.12146 42.043 97.227 1209.529 96 58.23921 59.87166 110.01059 25.491 55-184 112.302 97 31.21161 28.66005 50-13893 14.629 29693 57.118 98 15-77747 12.882581 21.478881 7.899 15.064 27-425 99 7.478814 5-403767 8.596300 3-992 7165 12.361 464 FRENCH ACTUARIES' TABLE ANNUITANTS. TABLE IV. COMMUTATION TABLES, FRENCH ACTUARIES' ANNUITY (R. F.), THREE PER CENT. Age. Dx N x Sx Cx MX R* 1000000.0 23420995.6 524197257.1 35486.6 293008.1 8657535 1 935907.9 22485087.7 500776261.5 25347.8 257521.5 8274527 2 883672.4 21601415.3 478291173.8 18156.5 232173.7 8017006 3 840044.1 20761371.2 456689758.5 13032-5 214017.2 7784832 4 802735.5 19958635.7 435928387.3 9385.7 200984.7 7570815 5 770107.0 19188528 7 4I596975I-6 68090 191599.0 7369830 6 740967.8 18447560.9 396781222.9 SOI5-6 184790.0 7178231 7 714444.0 17733116.9 378333662.0 3800.7 179774-4 6993441 8 689890.0 17043226.9 360600545 i 30I4-9 175973-7 6813367 9 666825.5 16376401.4 343557318.2 2546.4 172958.8 6637693 10 644894.4 I573I507-0 327180916.8 23198 170412.4 6464734 11 623831.7 15107675.3 3114494098 2241.8 168092.6 6294321 12 603447.1 14504228.2 296341734.5 2299.9 165851 I 6126229 13 583604.6 13920623 6 281837506.3 2426.9 163551-2 5960378 14 564215.2 13356408.4 267916882.7 2592.6 161124.3 5796826 15 545227.2 I28lll8l 2 254560474.3 2768.2 I5853I-7 5635702 16 526619.3 12284561.9 241749293.1 2929-5 I55763-5 5477170 17 508394.3 11776167.6 229464731.2 3055-2 152834 o 5321407 18 490576.2 II28559I.4 217688563.6 3I3I-8 149778.8 5168573 19 473201.7 10812389.7 206402972.2 3148.4 146647.0 5018794 20 456316.9 10356072.8 195590582.5 3103.1 143498.6 4872147 21 4399686 99l6l04.2 185234509.7 2999.0 140395.5 4728648 22 424199.0 9491905.2 175318405-5 2847.8 137396.5 4588253 23 409037-5 9082867.7 165826500.3 2668.5 134548.7 4450856 24 394494.5 8688373.2 156743632.6 2491.4 131880.2 4316308 25 380549 6 8307823.6 148055259.4 2338.27 129388.8 4184428 26 367161.6 7940662.0 139747435-8 2279.99 127050 5 27 354221.1 . 7586440.9 131806773.8 2225.11 124770.5 3927988 28 341711.4 7244729.5 124220332.9 2173.70 122545.4 3803218 29 329616.9 6915112 6 116975603.4 2125.27 120371.7 3680672 30 317922.3 6597190.3 110060490.8 2080.42 118246 4 3560301 31 306612.6 6290577.7 1034633005 2038.70 116166. o 3442054 32 2956734 5994904.3 97172722.8 2003.14 114127.3 3325888 83 285090.7 5709813.6 911778185 1964.99 112127.2 3211761 34 274851.0 5434962.6 85468004.9 1932.82 110162.2 3099634 35 264941 . i 5170021.5 80033042 3 1904.05 108229.4 2989472 86 255348.2 4914673.3 74863020.8 1878.44 106325.3 2881242 87 246060. i 4668613.2 69948347 5 185598 104446.9 2774917 38 39 237064.5 228349.9 4431548.7 4203198.8 65279734.3 60848185.6 1836.76 1820.78 102590.9 100754.1 2670470 2567879 40 219904.8 3983294.0 56644986.8 1808.01 98933-36 2467125 41 211718.4 3771575 6 52661692.8 1798.52 97125.35 2368192 42 203779-5 3567796.1 48890117.2 1792.26 953 2 6 83 2271066 43 196078.5 3371717.6 45322321.1 1789.26 93534-57 2175739 44 188604.4 3183113.2 41950603.5 1789.51 9i745'3i 2082205 45 181347.8 3001765.4 38767490 3 1793.12 89955.80 1990460 46 174299.0 2827466.4 3576572 \ 9 1799.89 88162.68 1900504 47 167448 9 2660017.5 32938258.5 1810.04 86362.79 1812341 48 160788.3 2499229.2 30278241.0 1823.40 84552.75 1725978 49 154308 4 2344920.8 27779011.8 1839-95 82729.35 1641425 FRENCH ACTUARIES' TABLE ANNUITANTS. 465 TABLE IV.Cont. COMMUTATION TABLES, FRENCH ACTUARIES' ANNUITY (R. F.), THREE PER CENT. M < D, N Sx C x M x R, 50 148001.1 2196919.7 25434091.0 1859.70 80889.40 1558696 51 141857.9 2055061.8 23237171.3 1882 63 79029.70 1477807 E2 135871.2 1919190.6 21182109.5 1908 51 77147.07 1398777 53 130033.2 1789157.4 19262918.9 I937-36 75238.56 1321630 54 124336.9 1664820 5 17473761.5 1968.98 73301 20 1246391 55 118775.4 1546045 i 15808941.0 2003 20 71332 22 1173090 56 113342.1 1432703.0 14262895.9 2039.82 69329.02 1101758 57 108031.0 1324672 o 12830192.9 2078.52 67289.20 1032429 58 102836 5 1221835.53 11505520 85 2119.01 65210.68 965139-8 59 97753-30 11.24082.23 10283685.32 2l6o 96 63091.67 899929.1 60 92770-87 1031305-36 9159603 09 2203.80 60930.71 836837.4 61 87903.16 943402.20 8128297.73 2247.12 58726 91 775906.7 62 83128.74 860273.46 7184895.53 2290 20 56^79.79 717179.8 63 78450.89 781822.57 6324622 07 2332-43 54189.59 660700 o 64 73867.72 707954 8 5 5542799 50 2372 97 51857 i6 606510.4 65 69378.08 638576.77 4834844 65 2410.96 49484.19 5546;3-3 66 6498 c 79 573594 98 4196267.88 2445-36 47073 23 505169.1 67 60679.65 512915 33 3622672.90 2475-I5 44627.87 458095.8 68 56473 43 456441.90 3109757.57 2499.17 42152 72 413468.0 69 52366.09 404075 8 I 2653315.67 2516.14 39653 55 37I3I5 2 70 48361.64 355714.17 2249239. 86 2524.776 37I37-4I 331661.7 71 44465-32 312248.85 1893525.69 2523.817 34612.64 294524.3 72 40683.44 270565 41 1582276.84 2511-857 32088.82 2599II.6 73 37023.49 233541.92 1311711.43 2487.667 29576.96 227822.8 74 33493-97 200047 95 1078169.51 2450.115 27089 29 198245.9 75 30104.26 16994369 878121.56 2398.162 24639.18 I7II56.6 76 26864.46 143079 23 708177.87 233 r - 053 22241 02 146517.4 77 23785-15 119294.08 565098 64 2248.344 19509 96 1242764 78 20877.03 98417.05 445804 56 2149 969 17661.62 104366.4 79 18150.54 80266.51 347387.51 2036.341 15511-65 86704.79 80 15615.42 64651.09 267121.00 1908 398 13475-31 7II93.I4 81 13280.21 51370.88 202469.91 1767.639 11566 91 57717.83 82 11151-71 40219.169 151099 034 1616 172 9799.273 46150.92 83 9 2 34-453 30984.716 110879.865 1456.655 8183 101 3635 1 . 65 84 7533-20S 234545H 79895 149 1292.300 6726.446 28168.55 85 6037.542 17416.909 56440.638 1126.675 5434 146 21442 10 86 4751-554 12665.415 39023.669 963 552 4307 47i 16007.96 87 3663 745 9001.670 26358.254 806.769 3343 919 11700.48 88 2762.101 6239.569 17356.584 659 925 2537 150 8356.565 89 2031 416 4208.153 11117.015 526 088 1877.225 58I9-4I5 90 i453- 8 76 2754 2 77 6908.862 407.702 I35I-I37 3942.190 91 1009 812 1744.4651 4I54-5855 306.261 943 435 2591053 92 678.6325 1065.8326 2410.1204 222 307 637.174 1647 6t 8 93 439.8^17 626 0109 1344.2878 155.400 414.867 1010.444 94 273.8987 352-1122 718.2769 104.200 259.467 595 577 95 163.2448 188.86739 366.16469 66.764 155.267 336.110 96 92.70774 96.15965 I77-29730 40.676 88.503 180.843 97 49.92520 46 23445 81.13765 23-457 47827 92.340 98 25-3597I 20.87474 34 93 o 12.728 24-370 44-5*3 99 12 07932 8.79542 14.02846 6.464 11.642 20 143 DESPARCIEUX TABLE OF MORTALITY. ( Former French Standard for Insurances. ) THE Desparcieux Table of Mortality takes its name from its author, who published it in 1746 in a volume entitled, " Essai sur les probabilities de la vie humaine ' ' (An Essay on the Probabilities of Human Life). Two tables were in fact published, one drawn from the experience of the ' ' ton- tine funds " of 1689, 1696 and 1734, and one from the mor- tuary registers of fourteen monasteries covering 8700 male lives and twenty-six convents covering 1519 female lives. The latter table, somewhat adapted, became the French standard for life insurances, and is known as the Despar- cieux Table practically to the exclusion of the other. The tables here published are taken from " Theorie Mathematique des Livs Assurances," by Dennoy. The columns headed D x and N x are not by our formula. The values are progressed to the end of the table at compound interest instead of discounted to the age at compound dis- count. But they may be used precisely as D r and N x . The French symbols are: T a equivalent to D x . S a equivalent to N z+1 . 468 DESPARCIEUX TABLES FRENCH. TABLE I. DESPARCIEUX TABLE OF MORTALITY. 1 Number Living. Num- ber Dying. Probabil- ity of Dying. Probability of Living. tc Number Living. Num- ber Dying. Probabil- ity of Dying. Prob- ability . of Living. L359 267 .19647 80353 48 599 9 .01502 .98498 1 1,092 49 .04487 95513 49 590 9 01525 98475 2 1.043 43 .04123 .95877 50 581 10 .01721 .98279 3 1,000 30 .03000 .97000 51 571 ir .01926 .98074 4 970 22 .02268 .97732 52 560 ii .01964 .98036 5 948 18 .01899 .98101 53 549 ii .02004 6 930 IS .01613 .98387 54 538 12 .02231 .97769 7 915 13 .01421 98579 55 526 12 .02281 .97719 8 902 12 .01330 .98670 56 514 12 02335 .97665 9 890 10 .01124 .98876 57 502 12 .02590 .97410 10 880 8 .00909 .99091 58 489 13 .02659 97341 11 872 6 .00688 .99312 59 476 13 02731 .97269 12 866 6 .00693 99307 60 463 13 .02808 .97192 13 860 6 .00698 .99301 61 450 13 .02889 .97111 14 854 6 .00702 .99298 62 437 13 03204 .96796 15 848 6 .00708 .99292 63 423 14 03310 .96690 16 842 7 .00831 .99169 64 409 14 03423 .96577 17 83=5 7 .00838 .99162 65 395 15 .03797 .96203 18 828 7 .00845 99^55 66 380 10 .04211 95789 19 821 7 .00853 .99147 67 3^4 17 .04670 95330 20 8! 4 8 .00983 .99017 68 347 18 .05187 .94813 21 806 8 .00993 .99007 69 329 19 05775 .94225 22 798 8 .01002 .98908 70 310 19 .06129 93871 23 790 8 .01013 .98987 71 291 20 06873 .93127 24 782 8 .01023 .98077 72 271 2O .07380 .92620 25 774 8 .01034 .98966 73 251 2O .07968 .92032 26 766 8 .01044 .98956 74 231 2O .08658 .91342 27 758 8 01055 989-15 75 211 19 .09005 90995 28 750 8 .01067 98933 76 192 19 .09896 .90104 29 742 8 .01078 .98922 77 173 19 .10983 .89017 30 31 734 726 8 8 .01090 .01102 .98910 .98898 78 79 3 18 18 .n688 13255 .88312 .86745 32 718 8 .01114 .98886 80 118 17 .14407 85593 33 710 8 .OII27 .98873 81 101 16 .15841 .84159 34 702 8 .OII4O .98860 82 85 14 .16471 83529 35 694 8 .OII53 .98847 83 12 .16901 .83099 36 686 8 .01166 .98834 84 59 II .18664 .81336 37 678 7 .01032 .98968 85 48 10 .20833 .79167 38 671 7 .01043 .98957 86 38 9 .23684 39 664 7 .01054 .98946 87 29 7 .24138 .75862 40 657 7 .01065 98935 88 22 6 27273 .72727 41 650 7 .01077 .98923 89 16 5 3 I2 5o .68750 42 643 7 .01089 90 II 4 36364 .63636 43 636 7 .01101 .98899 91 7 3 .42857 57142 44 629 7 .01113 .98887 92 4 2 .50000 .50000 45 622 .01125 .98875 93 2 I .50000 .50000 46 615 8 .01301 .98699 94 I I I.OOOOO .00000 47 607 8 .01318 .98682 DESPARCIEUX TABLES FRENCH. 469 TABLE II. DESPARCIEUX TABLE OF MORTALITY COMMUTATION! TABLES, FOUR PER CENT.* Age. D, N, Age. D, N T 54244.23 763202.83 48 3638.82 47583.04 1 41910.53 721292.30 49 3446.29 44136.75 2 38490.34 682801.96 50 3263.20 40873 ^ 3 35484- 647317.85 51 3083.68 37789.87 4 33095-7S 614222.10 52 2907.96 34881.91 5 31101.09 583121.01 53 2741.19 32140.72 6 29337-07 553783-94 54 2582.95 29557-77 7 27753-75 526030.19 55 2428.21 27129.56 8 26307.14 499723.05 56 2281.56 24848 oo 9 24938.81 474764.24 57 2142.57 22705.43 10 23729.21 451035.03 58 2006.82 20698.61 11 22609.12 428425.91 59 1878.34 18820.27 12 2I589-95 406835.96 60 1756.77 17063.50 13 20615.74 386220.22 61 1641.77 15421.73 14 19684 52 366535.70 62 1533-03 13888.70 15 18794.46 347721.24 63 1426.84 12461.86 16 17943-73 329797.51 64 1326.55 "135.31 17 17110.15 312687.37 65 1231.87 9903-44 18 16314.13 296373.24 66 "39-51 8763.93 19 I5554-05 280819.19 67 1049.54 771439 20 14828.31 265990.88 68 962.05 6752.34 21 14117.86 251873.02 69 877.06 5875-28 22 13340.13 238432.89 70 704.63 5080.65 23 12793.64 225639.25 71 717.23 4363-42 24 12177.01 213462.24 72 64225 3721.17 25 11588.87 201873.37 73 571-97 3149-20 26 11027.97 190845.40 74 506.15 2643.05 27 10493.08 180352.32 75 444-54 2198.51 28 9983.01 170369.31 76 388.96 1809.55 29 9496.66 160872.65 77 336.99 1472.56 30 9032.95 151839.70 78 288.44 1184.12 31 8^90.87 143248.83 79 244-93 939-19 32 8169.41 135079.42 80 20434 734.85 33 7767.69 127311.72 81 168.17 566.68 34 7384.78 119926.94 82 136.09 430.59 35 7019.83 112907.11 83 109.30 321.29 36 6672 03 106235.08 84 87-34 23395 37 6340.59 99894.19 85 68.32 165.63 38 6033.78 93860.71 86 52.bt 113.62 39 574I-I9 88119.52 87 38.16 75-46 40 5462.18 82657.34 88 27.84 47.62 41 5196-13 77461.21 89 19.47 28.15 42 594 2 -47 72518.74 90 12.87 15.28 43 4700.65 67818 09 91 7.87 7.41 44 4470.10 63347-99 92 4-33 3.08 45 4250-34 59097.65 93 2.08 1.00 46 4040.88 55056.77 94 1. 00 o.oo 47 3834 9i 51221.86 95 o.oo o.oo (* See Introduction.) DUVILLARD TABLE OF MORTALITY. (Former French Standard for Annuities.) THE Duvillard Table of Mortality takes its name from its author, who published it in 1806 in a work entitled " Analyse de 1'influence de la petite vesole sur la Mor- talite " (Analysis of the influence of smallpox upon the mortality). This table was constructed to show the nor- mal mortality by way of contrast. It was drawn from population statistics embracing 101,542 deaths in a popula- tion of 2,920,672. The columns headed D x and N x are not by our formula. The values are progressed to the end of the table at com- pound interest instead of discounted to the age at entry at compound discount. But they may be used precisely as D r and N x . The French symbols are: T a equivalent to D x . S a equivalent to N x+1 . 472 DUVILLARD TABLES FRENCH. TABLE I. DUVILLARD TABLE OF MORTALITY. Age. Number Living. Number Dying. Probability of Dying. Probability of Living. 1,000,000 232,475 .23248 76752 1 767,525 95.691 .12467 87533 2 671,834 47,166 .07020 .92980 3 624,668 25,955 04155 95845 4 598,713 15,562 .02599 .97401 5 583.151 10,126 .01736 .98264 6 573,025 7,187 .01254 .98746 7 565,838 5,593 .00988 .99012 8 9 560,245 555-486 4,759 4,364 .00849 .00786 99I5I .99214 10 551,122 4-234 .00768 .99232 11 546,888 4,258 .00779 .99221 12 542,630 4,375 .00806 .99194 13 538-255 4,544 .00844 .99156 14 533,711 4,742 .00889 99III 15 528,969 4,949 .00936 .99064 16 524,020 5,157 .00984 .99016 17 518,863 .01033 .98967 18 513-502 51553 .01081 .98919 19 507,949 5,733 .01129 .98871 20 502,216 .01175 .98825 21 496,317 6,050 .01219 .98781 22 490,267 6,184 .01262 .98738 23 484,083 6,306 .01303 98697 24 477,777 6,411 .01342 .98658 25 471,366 6,503 .01380 .98620 26 464,863 6,581 .01416 .98584 27 458,282 6,647 .01450 98550 28 451,635 6.703 .01484 .98516 29 444-932 6749 .01517 98483 30 438,183 6,785 .01548 98452 31 43L398 6,815 .01580 .98420 32 424,583 6,839 .Ol6ll .98389 33 417-744 6,858 .01642 98358 34 410,886 6,874 .01673 .98327 35 404,012 6,889 .01705 98295 36 397-123 6,904 .01739 .98261 37 390,219 6,919 01773 .98227 38 383-300 6,937 .OlSlO .98190 39 376,363 6,959 .01849 .98151 40 369-404 6,985 .01891 .98109 41 362,419 7,019 .01937 .98063 42 355-400 7,058 .01985 .98015 43 348,342 7,107 .02040 .97960 44 341-235 7,163 .02099 .97901 45 334-072 7,229 .02164 .97836 46 326,843 7,304 .02235 97765 47 3 I 9-539 7.391 02313 .97687 48 312,148 7,486 02398 .97602 49 304,662 7,592 .02492 .97508 50 297,070 7.709 02595 97405 51 289,361 7,834 .02707 .97293 52 281,527 7967 .02830 .97170 53 273-560 8,110 .02965 97035 54 265,450 8,257 .03111 .96889 DUVILLARD TABLES FRENCH. 473 TABLE l.Cont. DUVILLARD TABLE OF MORTALITY. Age. Number Living. Number Dying. Probability of Dying. Probability of Living. 55 257.193 8,411 .03270 .96730 56 248,782 8,568 .03444 96556 57 240,214 8,726 03633 .96367 58 59 231,488 222,605 8,883 9.038 03837 .04060 .96163 .95940 60 213,567 9,187 .04302 .95698 61 204,380 9-326 04563 95437 62 195.054 9-454 .04847 95153 63 185,600 9565 05154 .94846 64 176,035 9.658 .05487 94513 65 166,377 9,726 .05846 94154 66 156,651 9,769 .06236 93764 67 146,882 9,78o .06658 93342 68 137,102 9.755 .07115 .92885 69 127,347 9,691 .07610 .92390 70 117,656 9-586 .08148 .91852 71 108,070 9-433 .08729 .91271 72 73 98,637 89,404 9-233 8,981 .09361 .10045 90639 89955 74 80,423 8,678 .10791 75 71.745 8,321 .11598 .88402 76 63.4 2 4 7.913 .12476 87524 77 55,5" 7-454 .13428 .86572 78 48,057 6,950 .14462 .85538 79 41,107 6,402 15574 .84426 80 34.705 5 9i9 .16767 83233 81 28 886 5.206 .18023 .81977 82 23,680 4-574 .19316 .80684 83 19,106 3931 20593 .79407 84 15,175 3,289 .21674 .78326 85 11,886 2662 .22383 77617 86 87 9,224 7-165 2,059 .22322 .20865 .77678 .79135 88 89 5,670 4,686 856 17355 .18267 .82645 81733 90 3,830 737 .19243 80757 91 3-093 627 .20272 .79728 92 2,466 528 .21411 .78589 93 1.938 438 .22652 77348 94 1-499 359 23949 .76050 95 1,140 290 25351 .74649 96 850 229 . -27145 72855 97 621 179 .28710 .71290 98 442 135 30545 69455 99 307 ICO 32573 67427 100 207 72 34783 .65217 101 135 51 37778 .62222 102 84 33 .39286 .60714 103 51 22 43137 56863 104 29 13 .44828 55172 105 16 8 .50000 .50000 106 8 4 .50 oo .50000 107 4 2 .50000 .50000 108 i I 1. 00000 .00000 109 474 DUVILLARD TABLES FRENCH. TABLE II. DUVILLARD TABLE OF MORTALITY COMMUTATION FOUR PER CENT.* TABLES Age. D, NX Age. Dx NX 1 71,884,311 53,050,964 835,173,161 782,122,197 56 57 1,988,776 1,846,426 18,772,299 16,925,873 2 44,650,806 737,471,391 58 1,710,916 15,214,957 3 39-919,330 697,552,061 59 1,581,983 13-632,974 4 36,789,119 660,762,942 60 1.459,378 12,173.596 5 34,454,706 626,308,236 61 1,342,885 10,830,711 6 32 554,239 593,753.997 62 1,232,316 9.598,395 7 30,909,560 562,844,437 63 1,127,487 8,470,908 8 29,426,953 533,417,484 64 1,028,252 7,442 656 9 28,054,793 505,362,691 65 934-459 6,508,197 10 26,763,840 478,598,851 66 845.993 5,662,204 11 25,536,753 453,062,098 67 762,727 4,899,477 12 24>363,393 428,698,705 68 684,559 4,214-919 13 23;237,459 405,461,246 69 611,396 3,603,523 14 22,155,082 383,306,164 70 543-143 3,060,380 15 21,113,694 362,192,470 71 479.703 2,580,677 16 20,111,685 342,080,785 72 420,992 2,159,686 17 19,147,850 322,932,935 73 366,908 1,792,778 18 18,221,159 304,711,776 74 317,357 1,475,421 19 17,330,880 287,380,896 75 272,223 1,203,198 20 16,476,231 270,904,665 76 23 I >395 971,803 21 15,656,442 255,248,223 77 194,736 777,067 22 14,870,764 240,377,459 78 162,101 614,966 23 14,118,448 226,259,011 79 133.326 481,640 24 13-398,592 212,860,419 80 108,233 373,407 25 12,710,389 200,150,030 81 86,621 286,786 26 12,052,917 188,097,113 82 68,278 218,508 27 28 11,425,276 10,826,500 176,671,837 165,845,337 83 84 52,971 40-454 165,537 I25,08| 29 10,255,596 155,589,741 85 30,467 94,6l6 30 145.878,170 86 22,735 7I,88l 31 9,193,452 136,684,718 87 16,980 54-901 32 8,700,212 127, 984, co6 88 12,921 41,980 33 8,230,840 119,753.666 89 10,268 31,712 34 7,784,341 111,969,325 90 8,069 23,643 35 7,359.723 104,609,602 91 6,266 17-377 36 6,955,989 97,653,613 92 4804 12,573 37 6,572,170 91,081,443 93 3,630 8,943 38 6,207,361 84,874,082 94 2,700 6,243 39 5,860,582 79,013,500 95 1,974 4,269 40 5,530,980 73,482,520 96 L4I7 2,852 41 5.217,687 68,264,833 97 1,859 42 4,919,843 63.344,990 98 680 1,179 43 4,636,670 58,708,320 99 454 725 44 4,367.377 54,340,943 100 295 430 45 4,111,250 50,229,693 101 185 245 46 3,867,582 46,362,111 102 no 135 47 3-635,724 42,726,387 103 65 70 48 3,415,028 39, 3", 359 104 35 35 49 3,204.931 36,106,428 105 19 16 50 3,004,870 33,101,558 106 9 7 51 2,814,321 30,287,237 107 4 3 52 2,632,816 27,654,421 108 2 53 2,459,912 25,194,509 109 I 54 2,295,178 22,899,331 110 o 55 2,138,256 20,751,075 * See Introduction. INTRODUCTION TO COMPANY TABLES. MANY companies have published statistics in relation to their mortality experience. Few have converted their experience into actual mortality tables. Two of the most recent of such company tables are those of the Australian Mutual Provident Society of Sydney, N. S. W., and the Canada Life Assurance Company of Hamilton, Ontario. The former covers an experience from 1849-88 and the latter from 1847-93. The table of the Australian company was graduated by Woolhouse's method and covered 114,781 lives. The Canadian company dealt with 35,287 lives, and graduated its table by Makeham's formula. It also pre- pared a second table giving the mortality on lives, leaving out years of insurance before the sixth, also graduated by Makeham's formula. Both are here given. 47 6 AUSTRALIAN EXPERIENCE TABLE. TABLE I. AUSTRALIAN MUT. PROVIDENT SOCIETY'S EXPERIENCE TABLE. Age. Living. Dying. Prob- ability of Dying. Prob- ability of Living. Age. Living. Dying. Prob- ability of Dying. Prob- ability of Living. 100,000 2,842 .028420 .971580 50 72,262 831 .011500 .988500 1 97-158 2,249 .023148 .976852 51 71,431 863 .012082 .987918 2 94,909 I.73 2 .01824^ .981751 52 70,568 88 5 .012541 .987459 3 93.177 1,291 013855 .986145 53 69,683 90S .012987 .987013 4 91,886 929 .010110 .989890 54 68,778 .013580 .986420 5 90957 641 .007047 992953 55 67,844 969 .014281 .985719 6 90,316 432 .004783 .995217 56 66,875 1,013 .015148 .984852 7 89,884 299 .003327 .996673 57 65,862 1,064 016155 983845 8 89.585 243 .002713 997287 58 64,798 1,125 .017362 .982638 9 89,342 205 .002295 997705 59 63.673 1,186 .018626 .981374 10 89.137 190 .002132 .997868 60 62,487 1,246 .019940 .980060 11 88,947 160 .001799 .998201 61 61,241 1,315 .021473 .978527 12 88,787 181 .002039 .997961 62 59.926 1,397 .023312 .976688 13 88,606 202 .002280 .997720 63 58,529 1,526 .026073 973927 14 88,404 2 3 2 .002624 .997376 64 57,003 1,674 029367 970633 15 88,172 260 .002949 .997051 65 55.329 1,848 .033400 .966600 16 17 87,912 87,621 2 9 I 2 9 6 .003310 .003378 .996690 .996622 66 67 53,481 51,490 1,991 2,169 .037228 .042125 .962772 957875 18 87-325 34 .003481 .996519 68 49,321 2,203 .044667 955333 19 87.021 304 003493 .996507 69 47,118 2,229 .047307 952t)93 20 86,717 302 .003483 .996517 70 44,889 2,247 .050057 949943 21 86,415 294 .003402 .996598 71 42,642 2,276 053375 .946625 22 86,121 2 9 6 .003437 996563 72 40,366 2,141 .053040 .946960 23 24 85,825 85,528 297 297 .003461 .003473 .996539 .996527 73 74 38,225 36,015 2,210 2,274 .057816 .063140 .942184 .936860 25 85,231 302 003543 996457 75 33,741 2,238 .066329 933671 26 84,929 312 .003674 .996326 76 3 r .503 2,273 .072152 .927848 27 84,617 325 .003841 .996159 77 29,230 2,553 .087342 .912658 28 84,292 33 2 .003939 .996061 78 26,677 2,672 .100161 .899839 29 83,960 351 .004181 .995819 79 24,005 2,744 .114310 .885690 30 83,609 369 .004413 .995587 80 21,261 2,898 . 136306 .863694 31 83,240 392 .004709 .995291 81 18,363 2,905 .158199 .841801 32 82,848 407 .004913 .995087 82 15.458 2,662 .172209 .827791 33 82,441 433 .005252 .994748 83 12,796 2,396 .187246 .812754 34 82,008 450 .005487 994513 84 10,400 2,111 .202981 .797019 35 81.558 463 .005677 .9943 2 3 85 8,289 1,759 .212209 .787791 36 37 81,095 80,622 473 491 .005833 .006090 .994167 .993910 86 87 6,530 5.095 1,435 1,182 219755 .231992 .768008 38 80,131 503 .006277 .993723 88 3,913 965 .246614 753386 39 79,628 527 .006618 .993382 89 2,948 775 .262890 .737110 40 79,101 555 .007016 .992984 90 2,173 615 .283019 .716981 41 78,546 587 .007473 .992527 91 i,558 487 .312580 .687420 42 77-959 617 .007914 .992086 92 1,071 378 352941 .647059 43 77,342 648 .008378 .991622 93 693 282 .406926 593074 44 76,694 675 .008801 .991199 94 411 199 .484185 515815 45 76,019 699 .009195 .990805 95 212 129 .608491 391509 46 75.320 722 .009586 .990414 96 83 72 . 867470 .132530 47 74.598 75i .010067 989933 97 II ii I.OOOOOO .000000 48 73,847 77Q OTOC 1O QSCMsI 98 o 49 73^68 //y 806 . 156 227 ooo 39 482 174 044 70 38 174 ooo 118 053 32 165 ooo 134 562 ooo 71 39 350 87 703 25 810 102 397 72 23 740 63 963 20 367 76 587 73 1 8 243 45 720 15 783 56 220 74 13 749 31 971 ooo II 991 40 437 75 10 141 ooo 21 830 o 8 911 500 28 446 300 76 1 307 3 14 522 7 6 467 7 19 534 8 77 5 132 2 9 390 5 4 573 6 13 067 i 78 3 503 7 5 886 8 3 142 5 8 493 5 79 2 318 9 3 567 900 2 092 5 5 35i o 80 I 484 200 2 083 65 i 347 ooo i 258 500 81 9*4 85 I 168 80 834 7i i 911 46 82 54i 43 62737 496 48 i 076 75 83 306 49 320 88 282 36 580 27 84 165 26 155 622 152 92 297 91 85 84 349 71 273 78363 144 985 86 40 662 30 611 37 921 66 622 87 1 8 348 12 262.9 17 170 28 701 88 7 704-8 4 558.1 7 233.1 530.9 89 3 005-3 i 552.8 2 830.0 4 297.8 90 91 i 069.2 347-23 483.61 136-38 I 009.5 328.63 i 467-8 458.29 92 104.16 32.216 98.918 129.658 93 26.681 5-535 25.442 30.740 94 4.910 .625 4^975 5.2980 95 .602 .023 .5782 .6005 96 .023 .000 .0223 .0223 Constant. Number. Logarithm. i .04 ?. 6020600 (i 4- ) x 1.04 1.0198039 0.0170333 0.0085167 V 9615385 7-9829667 d .9805807 .0384615 I -9914833 7.5850267 ft .0392207 ?-5935^56 /( .0396078 7-5977807 H M (KING & HARDY) JOINT LIFE TABLE. 491 TABLE I. HM (KING & HARDY) JOINT LIFE ANNUITIES EQUAL AGES, FOUR AND ONE-HALF PER CENT. Age. Two Lives n Three Lives rat Four Lives Age. Two Lives Three Lives a zxx Four Lives 11.952 9.165 7.106 51 9-053 7-5T2 6-453 1 14.868 12.714 10.986 52 8.784 7.2=;6 6.2II 2 15.686 13.788 12.243 53 8.512 6.9^8 5-970 3 16.132 14-394 12.973 54 8.237 6.740 5-729 4 16.390 14.761 13427 55 7.962 6.482 5-488 5 6 16.569 1 6 674 15.024 I5.I93 13.763 13.988 56 57 7.685 7.407 6.224 5-967 5-249 5-012 7 16.717 15.278 14.113 58 7.129 5-7" 4.776 8 16.707 15.292 14.151 59 6.852 5-457 4-544 9 16.653 15.248 14.119 60 6-575 5.206 4.316 10 16.564 I5-I57 14.029 61 6300 4-957 4.091 11 16.452 15.035 I3.903 62 6.027 4.712 3.870 12 16.321 14.887 I3.746 63 5.756 4471 3.654 13 16.174 14.720 13.566 64 5.489 4-235 3-442 14 16.018 14542 I3-372 65 5-225 4.003 3-237 15 15.854 14-353 13.167 66 4.965 3-777 3-037 16 15.689 14.165 12.962 67 4.709 3.556 2.843 17 15.525 I3-978 12.761 68 4-459 3-341 2.656 18 15-366 13.800 12.570 69 4.214 3.133 2-475 19 15.212 I3-630 12.390 70 3-975 2.932 2.301 20 15-065 I3-470 12.224 71 3-742 2737 2-134 21 14925 13.320 12.070 72 3-5 l6 2-549 1.974 22 14788 13.177 11.925 73 3.296 2.369 1.822 23 I4-653 I3.039 11.787 74 3.084 2.196 1.676 24 14.520 12.902 11.652 75 2.879 2031 1-538 25 14-385 12.766 11.518 76 2.682 -873 1.406 26 14.247 12.628 "383 77 2-493 723 1.283 27 14-105 12.485 11.244 78 2.312 -580 i 165 28 I3.958 12.338 II.IOI 79 2.138 445 1055 29 13806 12.186 10953 80 973 317 952 30 13.649 12.029 10.800 81 .815 .197 .855 31 13.486 11865 10.640 82 .666 .084 765 32 13.317 11.696 10.476 83 525 .978 .682 33 13.142 11.522 10.307 84 391 .879 .604 84 12.961 11.341 10.132 85 .265 .786 53 2 35 12.776 11.155 9-950 86 .146 .700 .466 36 12583 10.964 9-765 87 .036 .621 .406 37 12.385 10.767 9-573 88 .932 547 350 38 12.181 10.564 9-377 89 .835 .480 .300 39 11.972 10.357 9-175 90 744 .417 255 40 11.756 10.143 8.969 91 .661 .361 .214 41 ".536 9-925 8-757 92 .585 .180 42 11309 9.701 8.541 93 .510 .261 .145 43 11.077 9-473 8.322 94 449 .222 .119 44 10.840 9.240 8.098 95 .389 .I8 4 .094 45 10.598 9.003 7.871 96 342 .155 .076 46 10-351 8.763 7.640 97 .317 .144 .070 47 10.099 8.519 7.407 98 .258 .110 .050 48 9.844 8.271 7171 99 .200 .085 037 49 9-584 8.021 6-933 100 .060 .015 .004 50 9.320 7.768 6694 492 HM (KiNG & HARDY) JOINT LIFE TABLE. TABLE II. HM (KING & HARDY) JOINT LIFE ANNUITIES EQUAL AGES, FOUR PER CENT. Age. Two Lives a Three Lives axxx Four Lives a'rai Age. Two Lives a Three Lives a x xx Four Lives axxxx 12.938 9.850 7-594 51 9.407 7.764 6.644 1 16.095 13.669 11.747 52 9.117 7.492 6.389 2 16978 14.824 13-093 53 8.826 7219 6.135 3 17-453 I5-470 13.871 54 8-533 6-945 5-882 4 17-725 15-859 14-353 55 8.238 6.673 5-630 5 17.908 16.134 I4-705 56 7-944 6.401 5-380 6 18.012 16.305 14.936 57 7.649 6.131 S-lS 2 7 18.047 16.386 15.060 58 7-354 5.862 4.887 8 18.024 16391 15.091 59 7.061 5-597 4.646 9 17-953 16.332 15-045 60 6.769 5-334 4.408 10 17.844 16.222 14-940 61 6479 5-075 4-175 11 17.711 16.079 14-794 62 6.193 4.820 3-946 12 17-557 15.910 14.616 63 5-909 4-569 3-723 13 15.720 14.414 64 5-629 4-324 3-505 14 17.207 I55I8 14.197 65 5-353 4.084 3-293 15 17.018 15-305 13.969 66 5.082 3-850 3.088 16 16,829 I5-093 I3-742 67 4.816 3.622 2.888 17 16.641 14.884 I3-5I9 68 4.556 3401 2.696 18 19 I6. 4 |8 16.281 14.683 14.492 13.308 13.108 69 70 4.302 4-054 3.186 2,979 2.511 2-333 20 16.112 14,312 12.923 71 3-814 2.779 2.163 21 15-950 14.142 12.751 72 3-58o 2-587 1-999 22 I5-79 1 13-979 12.589 73 3-354 2.402 1.844 23 I5-637 13-823 12-435 74 S-iS 6 2.226 1.695 24 15.481 13.668 12.283 75 2.925 2.057 1-555 25 i5-3 2 5 I3-5I3 12.134 76 2723 1.896 1.421 26 15.166 I3-356 ii 982 77 2-530 1-743 1-295 27 15.002 13-I94 11.827 78 2-344 1.598 I-I77 28 I4-833 13.028 11.666 79 2.166 1.460 1.065 29 30 14.659 14.479 12.857 12.681 11.502 11-332 80 81 1.998 1837 3-331 1.208 .960 .862 31 14-293 12497 11,156 82 1.685 1.094 -771 32 14.102 12.309 10.975 83 I-54I .987 .687 33 I3-905 12 114 10.788 84 1.406 .886 .608 34 13.701 11.914 iQ-595 85 1.278 793 .536 35 I3-49I II.708 10.397 86 1-157 .706 .469 36 13.276 11.497 10. 194 87 1.045 .626 .408 37 I3- 54 11.286 9-985 88 .940 551 353 38 12.827 11.057 9.772 89 .842 -483 -302 39 12.594 10.829 9-552 90 '-750 ,420 .256 40 12.355 10.596 9-329 91 .666 -363 .216 41 12. Ill 10.358 f.IOI 92 590 .313 .181 42 11.860 IO.II5 .868 93 .514 .263 . I4 6 43 11.606 9.868 8-633 94 452 .224 .120 44 11.346 9.616 8-393 95 392 .185 -095 45 11.082 9.360 8.150 96 345 .156 .076 46 10.812 9 101 7.904 97 319 -145 .070 47 10538 8.839 7-655 98 .259 .in .050 48 10.260 8,573 7.404 99 .201 .086 .038 49 50 9-979 9-695 8.305 8.036 7-I52 6.899 100 .060 .015 .004 H M (KING & HARDY) JOINT LIFE TABLE. 493 TABLE III. HM (KING & HARDY) JOINT LIFE ANNUITIES EQUAL AGES, THREE AND ONE-HALF PER CENT. Age. Two Lives a xx Three Lives a xxx Four Lives a xxxx Age. Two Lives a xx Three Lives a xxx Four Lives a XXXX 14.079 10.633 8.146 51 9.785 8.030 6.845 1 17.513 14.760 12.608 52 9-474 7.740 6-577 2 18.467 16.004 14.053 53 9.161 7.451 6.309 3 4 18.974 19.260 16.696 17.107 14.885 15-396 54 55 8.847 8-532 7.161 6.873 6.043 5-779 5 19-447 17-393 15-765 56 8.218 6.587 5.517 6 19.546 17.567 16.003 57 7.904 6.303 5-258 7 19-570 17.642 16.124 58 7-592 6.O2I 5.002 8 19530 17.633 16.145 59 7.282 5-742 4-751 9 19-439 17-555 16.084 60 6-973 5.468 4-504 10 19.307 17.424 15-958 61 6.668 5-197 4.262 11 19.146 I7.257 15789 62 6.366 4.921 4.025 12 18.964 17.060 15586 63 6.068 4.671 3-794 13 18.765 16.843 15358 64 5-775 4.416 3-570 14 18555 16.612 15-115 65 5-486 4.167 3-351 15 18.337 16372 14.860 66 5.204 3.925 3.140 16 18.118 16.132 14.608 67 4.927 3-690 2-935 17 17.901 15.895 14-359 68 4-656 3.462 2.738 18 17.690 15-669 14.124 69 4-393 3-24I 2.548 19 17.486 15452 13901 70 4-136 3.028 2.366 20 17.289 15.248 13.694 71 3.887 2.823 2.191 21 17.101 I5-C55 13-502 72 3.646 2.626 2 025 22 16.917 14.870 13.320 73 3413 2-437 1.866 23 16.736 14.691 13.146 74 3.189 2.256 L7I5 24 16.555 14-514 12.976 75 2973 2.083 1.5-2 25 16.374 H-337 12.808 76 2.765 1.919 1.436 26 16.189 I4-I59 12.637 77 2.567 764 1.308 27 16.000 13976 12.464 78 2-377 .6l6 1.188 28 15-805 13-787 12.284 79 2.195 476 1.075 29 15-605 13-593 12.100 80 2.023 344 30 15-399 13-394 11.912 81 1.859 .220 .869 31 15.187 13-189 II.7I6 82 1-705 .104 777 32 14.968 12.978 "515 83 1.558 995 .692 33 14-744 12.761 11.309 84 1.420 .894 613 34 14.513 12.538 11.097 85 1.290 799 540 35 14.277 12.309 10.879 86 1.168 .711 472 36 14035 12.075 10.656 87 1-055 630 .411 37 I3-787 11.836 10.428 88 .948 555 355 38 13-532 11.590 10.19^ 89 .849 .486 304 39 13.272 H-339 9.958 90 .756 258 40 13.007 11.084 9-7I5 91 .671 .366 .217 41 12.736 10.824 9.468 92 594 .315 .182 42 12.459 10.559 9.217 93 .517 .264 .147 43 12.179 10.291 8.964 94 455 .225 .121 44 11.893 10.018 8.706 95 394 .186 095 45 11.603 9-742 8.446 96 347 157 .076 46 11.308 9.462 8.183 97 .321 .145 .071 47 II. 010 9.180 7.918 98 .261 .112 . .050 48 10.709 8.895 7651 99 .202 .086 .038 49 10.404 8.608 7.383 100 .060 .015 .004 50 10.096 8.320 7.II4 494 (KiNG & HARDY) JOINT LIFE TABLE. TABLE IV. HM (KING & HARDY) JOINT LIFE ANNUITIES-EQUAL AGES, THREE PER CENT. Age. Two Lives tn Three Lives Km Four Lives axixx Age. Two Lives a Three Lives Four Lives axuz 15.406 "534 8.774 51 10.190 8.312 7.057 1 19.161 16 013 13.587 52 9-854 8.004 6.774 2 20.197 I 7-3S8 I5-I43 53 9-5I7 7.696 6.491 3 20.740 18.100 16.035 54 9.180 7.389 6.2II 4 5 21.038 21.227 18.534 18.833 16.577 16.965 55 56 8.844 8.509 7.085 6.783 5-934 5.660 6 21.318 19 006 17.208 57 8-175 6-483 5-389 7 21.327 19.072 I7-3 2 5 58 7-843 6.187 5.122 8 21.265 19.046 17-333 59 7-5I4 5 895 4.861 9 21.147 18.946 I7-253 60 7.188 5.607 4.604 10 20.984 18.787 17.103 61 6866 5-325 4-353 11 20.791 18.589 16.906 62 6.548 5.048 4.107 12 20.575 18.361 16.674 63 6-235 4-777 3.869 13 20.339 18.110 16.414 64 4.512 3-636 14 20.094 17.846 16.141 65 5.626 4-254 3-4" 15 19.839 I7-572 I5-854 66 5-331 4.004 3.I93 16 19-583 17.300 I5-57I 67 5.042 3-76o 2.983 17 I9-332 17.030 I5-293 68 4-76l 3-525 2.780 18 19.087 16.773 15.029 69 4.487 3-297 2.586 19 18.848 16.526 14.780 70 4.221 3.078 2-399 20 18619 16.293 I4-548 71 3-964 2.868 2.221 21 18.398 16.073 14-331 72 3-715 2.665 2.051 22 18.183 15.861 14.126 73 3-475 2.472 .889 23 17.972 15-656 I3-930 74 3- 2 43 2.287 735 24 17.761 15-453 13-738 75 3.021 2. Ill .589 25 I/-549 15-251 I3-548 76 2808 *-943 451 26 17-334 15046 I3-356 77 2605 1.784 .322 27 17.114 I4-837 13.160 78 2 410 1.634 .199 28 16.888 14.623 12.959 79 2.225 1.492 .084 29 30 16.657 1 6 420 14.404 14.179 12-754 12-543 80 81 2.049 .882 I.358 1232 f 31 16.176 13-947 12.325 82 .724 1.114 .784 32 I5-927 13.710 12.102 83 576 1.004 .697 33 15671 13-467 II 874 84 435 .901 .617 34 15.410 13.218 11.640 85 33 .806 543 35 15.142 12.964 II 400 86 .179 .717 475 36 14.869 12.704 II 156 87 .064 635 .414 37 14.589 12.439 10.906 88 .956 559 357 38 14.304 12.167 10.6^2 89 .8^6 .490 -306 39 14 013 11.892 10.392 90 .762 425 259 40 13.718 11. 612 10.130 91 .676 .368 .218 41 I3-4I7 11.327 9.862 92 -598 317 .183 42 13.110 11.037 9-590 93 521 .266 .147 43 12.800 10.746 9-3I7 94 458 .226 .121 44 12.485 10.449 9.040 95 397 .187 .096 45 12.167 10.150 8.761 96 349 157 .077 46 11.844 9.847 8.479 97 323 .146 .071 47 11.518 9543 8.196 98 .262 .112 .051 48 11.189 9-237 7.912 99 .203 .087 .038 49 10.858 8.930 7.627 100 .061 .015 .C0 4 50 10.525 8.621 7-342 49<5 CONVERSION TABLE. TABLE I. SINGLE PREMIUM CONVERSION TABLE. For finding by inspection the Value of A from that of a. Value VALUE OF A. AT VARIOU s RATES OF INTEREST. of a z = .03 / = .035 i = .04 * = .045 i = .05 * = .06 O .97087 .96618 96154 .95694 .95238 94340 I 94I7S 93237 .92308 .91388 .90476 .88679 2 .91262 89855 .88462 .8708! 85714 .83019 3 .88350 .86473 .84615 82775 .80952 77358 4 85437 .83092 .80769 .78469 .76190 .71698 5 .82524 .79710 76923 74 l6 3 .71429 .66038 6 .79612 76329 .73077 .69856 .66667 60377 7 8 .76699 .73786 .72947 .69565 69231 65385 65550 .61244 61905 57143 54717 49056 9 .70874 .66184 61538 56938 52381 43396 10 ii .67961 .65049 .62802 .59420 .57692 53846 .52632 48325 .47619 42857 .37736 32075 12 .62136 56039 .50000 .44019 38095 26415 13 59223 52657 .46154 39713 33333 20755 14 563" 49275 .42308 35407 28571 .15094 15 16 53398 50485 .45894 .42512 .38462 34615 .31100 .26794 .23810 .19048 .09434 03774 *7 47573 39 I 3o 30769 .22488 .14286 .00000 18 19 .44660 .41748 35749 .32367 .26923 .23077 .18182 .13876 .09524 .04762 20 38835 .28986 .19231 .09569 .00000 .... 21 .35922 .25604 .15385 .05263 .... 22 33010 .22222 .11538 .00957 .... 23 .30097 .18841 .07692 .00000 24 .27184 15459 .03846 2 .24272 21359 .12077 .08696 .00000 27 .18447 05314 28 15534 .01932 .... 29 .12621 .00000 3 .09709 - CONVERSION TABLE. 497 TABLE II. SINGLE PREMIUM CONVERSION TABLE. Table of Differences. DIFFERENCE OF A (SUBTR ACTIVE). Difference of a i = .03 i = .035 i = .04 i = .045 * = .05 * = .06 .1 .00291 .00338 .00385 .00431 .00476 .00566 .2 .00583 .00676 .00769 .00861 .00952 .01132 3 .00874 .01015 .01154 .01292 .01429 .01698 4 .01165 01353 01538 .01722 .01905 .02264 5 .01456 .01691 .01923 .02153 .02381 .02830 .6 .01748 .02029 .02308 .02584 .02857 .03396 7 .02039 .02367 .02692 .03014 03333 .03962 .8 .02330 .02705 .03077 .03810 .04528 9 .02621 .03044 .03462 .03876 .04286 .05094 .01 .00029 .00034 .00038 .00043 .00048 .00057 .02 .00058 .00068 .00077 .00086 .00095 .00113 3 .00087 .00101 .00115 .00129 .00143 .00170 .04 .00117 .00135 .00154 .00172 .00190 .00226 05 .00146 .00169 .00192 .00215 .00238 .00283 .06 .00175 .00203 .00231 .00258 .00286 .00340 .07 .00204 .00237 .00269 .00301 00333 .00396 .08 .00233 .00271 .00308 00344 .00381 00453 .09 .00262 .00304 .00346 .00388 .00429 00509 .001 .00003 .00003 .00004 .00004 .00005 .00006 .002 .00006 .00007 .00008 .00009 .OOOIO .OOOII .003 .00009 .00010 .00012 .00013 .00014 .00017 .004 .00012 .00014 .00015 .00017 .00019 .00023 .005 .00015 .00017 .00019 .00022 .00024 .00028 .006 .00017 .00020 .00023 .00026 .00029 .00034 .007 .OCO2O .00024 .OOO27 .00030 .00033 .00040 .008 .00023 .00027 .00031 .00034 .00038 .00045 .009 .00026 .00030 .00035 .00039 .00043 .00051 49 8 CONVERSION TABLE. TABLE III. ANNUAL PREMIUM CONVERSION TABLE. For finding by inspection the Value of P from that of a. Value of a VALUE OF P AT VARIOUS RATES OF INTEREST. / = .03 i = .035 i = .04 i = .045 / = .05 / = .06 O I .97087 .47087 .96618 .46618 .96154 .46154 95694 .45694 95238 45238 .94340 44340 2 3 4 .30421 .22087 .17087 .29952 .21618 .16618 .29487 .21154 .16154 .29027 .20694 .15694 .28571 .20238 15238 .27673 .19340 .14340 5 13754 .13285 .12821 .12361 .11905 .11006 6 .11373 .10904 .10440 .09980 .00524 .08625 7 .09587 .09118 .08654 .08194 .07738 .06840 8 9 .08199 .07087 .07730 .06618 .07265 .06154 .06805 .05694 .06349 .05238 05451 .04340 10 .06178 .05709 05245 .04785 04329 03431 ii .05421 .04952 .04487 .04027 03571 .02673 12 .04780 .04311 .03846 .03386 .02930 .02032 T 3 .04230 .03761 .03297 .02837 .02381 .01483 14 03754 .03285 .02821 .02361 .01905 .01006 15 03337 .02868 .02404 .01944 .01488 .00590 16 .02970 .02501 .02036 .01576 .OII2I .00222 17 .02643 .02174 .01709 .01249 .00794 .00000 18 02351 .01882 .01417 .00957 .OO5OI 19 .02087 .01618 .01154 .00694 .00238 20 .01849 .01380 .00916 .0041:6 .00000 .... 21 .01633 .01164 .00699 .00239 .... .... 22 01435 .00966 .00502 .00042 .... 23 .01254 .0078=; .00321 .00000 24 .01087 .00618 .00154 25 00934 .00465 .00000 .... .... 26 .00791 .00322 .... 27 .00659 .00190 28 .00536 .00067 29 .00421 .00000 30 .00313 CONVERSION TABLE. 499 TABLE IV. ANNUAL PREMIUM CONVERSION TABLE. Table of Differences. Value of a I DIFFERENCE OF P (SUBTRACTIVE). *= ' *= A a = -3 A = .4 A = .5 A. = .6 4.-J o .09091 .16667 .22077 .28571 33333 37500 .41176 44444 47368 I .02381 045*5 .06522 08333 . 100 JO ."538 .12063 .14286 2 .01075 .02083 .03030 .03921 .04762 05555 .06306 .07017 07692 3 .00610 .OligO .01744 .02273 .02778 .03261 .03723 .04167 .04592 4 .00392 .00769 .OII32 .01481 .01818 .02143 .02456 .02759 03051 s .00274 .00538 .00794 .01042 .01282 01515 .01742 .01961 .02174 6 .00201 .00397 .00587 .00772 00953 .01128 .01299 .01465 .CI628 7 .00154 .00305 .00452 00595 or V35 .00872 .01006 .01136 . .01264 8 .00122 .00241 .00358 .00473 .00585 .00694 .00802 .00907 .OIOIO 9 .00099 .00196 .00291 .00385 .00476 00566 .00654 .00741 .00826 10 .OOO82 .OOl62 .00241 00319 .00395 .00470 .00544 .00616 .00688 ii .00068 .00136 .OO2O3 .00268 00333 .00396 00459 .00520 .00581 12 .00058 .OOIl6 00173 00229 .00285 .00339 .00393 .00446 .00498 13 .OC05I OOIOI .OOI5O .00199 .00246 .00294 .00340 .00386 .00432 14 .00044 .00088 .00131 00173 .00215 .00257 .00298 .00338 .00378 15 .00039 .00077 .00115 .OOT52 .00189 .00226 .00262 .00298 00333 16 .00034 .00068 .00102 00135 00168 .00200 .00232 .00264 .00295 17 .OOO3I .00061 .OOOgr .OOI2I .00151 .OOlSo .00208 .00237 .00265 18 .OOO27 .00055 .00082 OOIOS 00135 .00l6l .00187 .OC2I2 .00238 19 .00025 .00049 .00074 .OOOgS .00122 .00146 .00169 .OO192 .00215 20 .00023 .00045 .00^67 .00089 .00111 .00132 .00154 .00175 .00106 21 .0002 I .00041 COO62 .00082 .00102 .00121 .00141 .00160 .00179 22 .O3OI9 .00038 .00056 .00074 .00093 .00111 .00129 .00146 .00164 23 .OOOlS .00035 .00052 .00069 .00085 .00102 .OOllS .00135 .00151 24 .OOOl6 .00032 .00047 .00063 .00078 .00094 .00109 .00124 .00139 25 .00015 .00029 .00044 .00058 .00072 .00087 .OOTOI .00115 .00128 26 .OOOI4 .00027 .00041 .00054 .00068 .OOOSl .00094 .00107 .00120 27 .00012 .00025 .00037 .00050 .00062 .00074 .00087 .00099 .ooiir 28 .00012 .00023 .00035 .00047 .00058 .00070 .OOOSl .00002 .00103 29 .OOOII .00022 .00033 .00043 .000^4 .00065 .00 76 .00086 .00097 3 .OOOIO .00021 .00031 .00041 .00051 .00061 .00071 .OOOSl .00091 x ITTu 'M/^^ fir OF TH ^ f UN X2 IVEB SITY ) GENERAL LIBRARY UNIVERSITY OF CALIFORNIA BERKELEY RETURN TO DESK FROM WHICH BORROWED This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. SFebbi LD 21-100m-l,'54(1887sl6)476 \ S UNIVERSITY OF CALIFORNIA LIBRARY