NA TURE 



457 



THURSDAY, MARCH 15, i! 



LIFE CONTINGENCIES. 



Institute of Actuaries' Text-book of the Principles of 

 Interest, Life Annuities and Assurances, and their 

 Practical Applicatiott. Part II. Life Contingencies 

 (including Life Annuities and Assurances). By 

 George King. (London : C. and E. Layton, 1887). 



SOME years ago the Council of the Institute of 

 Actuaries came to the conclusion that the students 

 of actuarial science were subjected to great inconvenience 

 and loss of time in consequence of the number of different 

 books and scientific papers to be consulted in acquiring 

 a knowledge of the subject. Persons actively engaged in 

 the work, and wishing to refresh their memory as to the 

 best methods of solving some special question, frequently 

 felt the same sort of inconvenience. The Council, with 

 that consideration for the students which has always 

 been characteristic of them, resolved to provide what 

 was wanted. They accordingly authorized the compila- 

 tion and publication — the cost to be borne by the 

 Institute — of a " Text-book of the Principles of Interest, 

 Life Annuities and Assurances, and their Practical Appli- 

 cation." The first volume, entitled Part I., and treating 

 of the principles of interest (including annuities-certain), 

 has been before the public since 1882. The second part, 

 which is concerned with " Life Contingencies," has now 

 been issued. The editing or authorship of this portion of 

 the text-book was intrusted to Mr. George King, the 

 Actuary of the Atlas Insurance Company, and formerly 

 of the AUiance, whose practical acquaintance with assur- 

 ance calculations^ well-known devotion to his work, and 

 experience as a lecturer at the Institute, qualified him, in 

 a high degree, for undertaking the task. 



In the opening chapters of the present volume, the 

 author deals with the ordinary mortality table, its con- 

 struction from different kinds of data, and its varied 

 application by the actuary and the statist, including the 

 determination of the probable numbers dying or sur- 

 viving in a community, or in an annuity or other society. 

 Such a table, showing out of a certain number of persons 

 born how many attain to each year of age, may obviously 

 be formed from records of the duration of life in a great 

 number of individual cases ; always provided the cases 

 constitute a fair selection. Here, however, arises great 

 practical difficulty, and mortality tables are, in conse- 

 quence, usually constructed from observations yielding 

 the probability of living one year at each year of age. 

 This is so important a fact, at least to students com- 

 mencing the study, that we should have been glad if the 

 "elementary illustrations" given by the author had in- 

 cluded a numerical illustration in brief detsil reproducing 

 the process underlying one or other of the standard 

 tables. The author has proceeded wisely, we think, in 

 first collecting the elementary formuUe of the doctrine of 

 chances, and then showing how these may be applied to 

 the numbers of the mortality table in order to solve the 

 many and important questions arising in connection with 

 single or joint lives. He points out two fallacies which it 

 Vol XXXVII. — No. 959. 



is desirable the public should recognize as such. This 

 is one : — 



" It will be found . . . that the higher the age from 

 which we count, the greater will be the average age at 

 death. Thus, at age 10, the average age at death is 

 60-257 years; at age 20, it is 62"ioi ; at age 30, it is 

 64726 ; and at age 60, it is 73"8o8. ... It is frequently 

 stated by shallow reasoners that some professions, such as 

 that of the lawyer, must be conducive to longevity . . . 

 because the average age at death of the members of that 

 profession is much higher than that of the general popu- 

 lation. But the general population starts from age o ; 

 and starting from age o the average age at death, if the 

 mortality were to follow the table, would be only 47785 

 years, whereas ... a lawyer does not enter the profes- 

 sion until he reaches manhood ; and usually it is not until 

 many years later that he attains sufficient eminence for 

 his death to be commented upon. Therefore, even if the 

 rate of mortality among lawyers be not more favourable 

 than among the general population, the average age at 

 death of those whose deaths attract notice must be 

 greater." 



Much attention is given in this portion of the book to 

 attempts which have been made to embody the law 

 of mortality in a mathematical formula which should 

 readily lend itself to the purposes of calculation. Two 

 such attempts are introduced to our notice : the hypo- 

 thesis of De Moivre, and the hypothesis of Gompertz. 

 De Moivre, in his treatise on " Annuities on Lives," pub- 

 lished in 1725, made the assumption, now well known, 

 that, out of eighty-six births, one person dies every year 

 until they are all extinct. Gompertz, in a paper contri- 

 buted to the Royal Society in 1825, just a century later, 

 observed : " It is possible that death may be the con- 

 sequence of two generally co-existing causes : the one, 

 chance, without previous disposition to death or deterior- 

 ation ; the other, a deterioration, or increased inability to 

 withstand destruction." It would appear, however, that he 

 did not pursue this twofold notion to its conclusion, but 

 contented himself with investigating the effect of supposing 

 " the average exhaustion of a man's power to avoid death 

 to be such that at the end of equal infinitely small inter- 

 vals of time he lost equal portions of his remaining power 

 to oppose destruction which he had at the commencement 

 of these intervals." The words now quoted, taken alone, 

 perhaps do not give a very precise idea of what was in- 

 tended, but they really cover the assumption that the 

 force of mortality increases in geometrical progression, 

 and may be represented, as Mr. King says, by ^c^, where 

 B and c are constants, and x the age. From this, the 

 equivalent of the differential coefficient of the log of the 

 number living, we find the number living at age x may 

 be expressed in the form k{gY^. By judiciously choosing 

 values for the constants k, g, and c, the results approxi- 

 mate more or less closely to fact for a greater or smaller 

 extent of life, but it was left to Mr. Makeham, the present 

 Actuary to the Church of England Assurance Company, 

 to perfect the formula, and render it an exponent of the 

 effect of the two co-existing causes of death originally 

 contemplated by Gompertz. The final shape of the 

 formula then became ks^{gY^ ,\\''\\qx€\t). a fourth constant, 

 s, is introduced. In this shape, although there still re- 

 mains a difficulty with the youngest ages of life, the 

 formula has been used for adjusting crude observations 



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