JOURNAL

OF THE

INSTITUTE OF ACTUARIES

AND

ASSURANCE MAGAZINE.

On the Integral of Gompertz's Function for expressing the Values of Sums depending upon the contingency of life. By W. M. MAKEHAM, Fellow of the Institute of Actuaries.

GOMPERTZ'S Function for expressing the probability of a person

1

surviving any given period, x, is 99. For the purpose of

9

adapting this expression to the entire period of adult life I have added the factor € ̄-ax. -αx The two expressions become identical in form when the factor e- representing the effect of discount is introduced, and both may therefore be correctly designated by the name of "Gompertz's Function."”

The interest excited by the publication of Gompertz's celebrated hypothesis, and the attention which has in consequence. been bestowed upon it, have already resulted in the discovery of some curious and important properties of the function in question; and, as Mr. Woolhouse has pointed out, it is not improbable that the disclosure of others may reward the industry of those who choose to cultivate the field. The principal of those already discovered―viz., the property of uniform seniority-we owe to De Morgan. By the aid of this important property the number of

VOL. XVII.

X

tables required in the computation of Life Contingencies became (as I have shown on a former occasion) materially reduced. For each mortality table, and each rate of interest, a complete set, from a single life up to any required number of joint lives, could be comprised in a comparatively moderate space, while under the old system the task would be attended with insuperable difficulties.

A further step in the same direction will be found in a paper by Mr. Woolhouse (see Journal of the Institute of Actuaries for July 1870), who suggested a contrivance by which the number of tables for any given rate of mortality could be reduced to a single series for successive rates of interest,—the annuity values for any given combination of ages being found by a process of double interpolation. It will be seen also, from the Editor's note at the end of the paper in question, that a similar idea had occurred to Mr. Meech, an eminent American actuary, and was communicated to me by that gentleman in December 1868.

By the last mentioned improvement we should be able to deduce the value of an annuity on any given combination of lives and at any given rate of interest, by means of a single table of double entry calculated for specific values of the constants g and q of the formula. But the table in question would be of no use for deducing the values of annuities according to a mortality table derived from other values of these constants. The table, however, which I now submit, and which, like that last referred to, is a single table of double entry, has this further important advantage, viz., that it is applicable to all values of the constants g and q whatever, and is therefore available for determining the values of annuities, not only for any given combination of lives and for any given rate of interest, but also for any given rate of mortality. That is to say, the advantage obtained by the law of uniform seniority (as modified according to the idea of Messrs. Meech and Woolhouse) in reference to a single given mortality table, is, by means of a contrivance which I have now to explain, extended so as to comprise all mortality tables whatever constructed according to Gompertz's Function.

The value of a continuous annuity on any given life or any combination of joint lives expressed in terms of Gompertz's Function, is

Now it will be seen that in order to tabulate this function

completely we should require a table of no less than four variables -corresponding to the three constants g, q, and (a+8), together with a, the inferior limit of integration. Three variables, altho' a somewhat troublesome matter, are at least practicable (as I have shown in a paper on this subject in vol. 16 of the Journal) by reason of the fact that space has three dimensions; but four involve a difficulty which would puzzle the genius of Euclid himself to surmount.

Among the contrivances available for facilitating the tabulation of definite integrals none are of more important use than that of changing the independent variable. In the case in hand, I put

and substituting these values in the given function, the integral becomes

in which it will be seen that by a very simple transformation we have succeeded in eliminating two of the arbitrary constants, and thus reducing the process of tabulation to one of double entry only. The following is a proof of the transformation just effected.

from which we get the transformed integral above given, as the constant c from the way in which it is involved evidently disappears. On a former occasion (see Journal vol. xiii, p. 349) I proposed the following transformation of the same integral, which would answer all the purposes of that just described, but which I think would be found more troublesome to calculate.

(logem

(loge}) . qa.logeq=v.logeq. Hence, substituting, we have:

= qmx =

Again,

the second member of which equation (omitting the factors outside the symbol of integration) is the form of the well known gammafunction, or second Eulerian Integral.

Now it is a well known property of the last named integral that
Jε-v.vm.dv= − € ̄v vm +m√ e ̄v vm−1 d v

whence the following equation is easily deduced:

loge 4

+

∞

€ ̄v.vm-2dv

e

V

By means of these equations, having found the value of an annuity corresponding to any given value of m, we may with facility determine the annuity corresponding to m+1. It is by virtue of this property that we are able to limit the tabulated integral to values

of n from 1 to 2,—all other values being easily deduced from those found in the table.

Comparing the expression for the integral tabulated, viz.—

with that representing the logarithmic values of annuities, viz.—

it will be seen that the two are identical in form. This identity suggests a simple method of computing the former integral by the rules laid down by Mr. Woolhouse for the calculation of continuous annuities, which is the course I have found most convenient in the construction of the accompanying table.

The figures in antique type in the table are the divided differential coefficients of the function; and the process of interpolation is performed in accordance with the rules given in my paper on this subject in vol. xvi of the Journal (see page 111). The following examples will be sufficient to illustrate the process::

Example I. Required the value of a continuous annuity at 5 percent on a life aged 30; according to the values of the constants deduced by Mr. Woolhouse from the new HMF experience (see Journal, vol. xv, p. 405.)

Nearest tabular value (corresponding to z=3·8 and n=1·3)