An essay on probabilities and their applications to life contingencies and insurance offices

ON UNIFORMLY CHANGING ANNUITIES. XXVii
of this application of the calculus, which appear to us of much value, nor are we at all sanguine in expecting any.^{" The tendency of such an assertion is to encourage those who study the subject, to stop short of the differential calculus in their mathematical studies. Now I assert, 1. That the calculus aforesaid may, as evidenced in the results of chapter IV., lead to most valuable rules in the estimation of complicated proba2. That if the calculus be not serviceable in the deduction of the law of mortality, it is from defect of observed data. As soon as larger and more correct tables of the numbers living are obtained, the differential calculus is ready to furnish methods for correcting those now in use. 3. That the differential calculus may be made to give important simplifications of processes, and to render the tables already constructed immediately available for purposes to which no one now dreams of applying them.
If v be the present value of 11., to be received at the end of a year, and 9v be the present value of a conannuity of 11., then that of an annuity which is to be 11. at the end of the first year, 21., 31., &c., at the end of the second, third, &c. years, is v9'v, where v is the differential coefficient of 9v. Now 1 + r being the amount of 11. in one year, we have}

01 x ¢c = ^Opv — A2$v + ^Q30^{v — A^4~bv}+ &e.

Take out the value of an annuity of 11. at the given rate of interest, and at several successive higher rates : take the successive differences, the difference of the differences, and so on. To the first difference add half of the second difference, one-third of the third, and so on : the sum of these, multiplied by the amount of 1001. in one year at the first named rate, is the value of the annuity required.

I take examples from the Northampton tables, at 4 per cent., because Mr. Morgan has given a table of the annuities required, which will serve to find verifications. First suppose the age to be 5 years.

PROBLEM. A life annuity is ,^{'m at the end of the first year, and diminishes n every year, until nothing is due, after which it ceases entirely. Required its present value.}

be extinguished during the tabular life of the party, from the value of an annuity of ^{'(m+n) subtract n times that of an increasing annuity of 11. found as already described. But when the annuity can be extinguished during the life of the party (say in t years exactly, so that m=nt), then to the preceding result add n times the value of an increasing annuity of I1. on a life t+ 1 years older than the party, multiplied by the chance of his living _{t+1 years, and by the present. value of X1 due t + 1 years hence.
PROBLEM. Required the present value of lm to be received at the end of the year in which A dies, if in a year, or , ^{'(m—n) ^{if in the second year, and so on.
RULE. When n is so small that the sum insured cannot be extinguished during the tabular life of the party, to the value of a perpetuity of I'm, add that of an increasing annuity of Win,2n, in, &c., and subtract the value of a simple life annuity, of which the yearly payment is :m, increased by the product of a perpetuity due, and the value of a simple annuity of
: divide the difference by the value of a perpetuity due, and the quotient is the present value required. But if the insurance be extinguished in t years, or if m=nt : find the product of an annuity due of £1 on a life t+1 years older than the given life, and of a perpetuity ; subtract the value of an increasing annuity of 11. on that life, and having multiplied the difference by the chance of the first life surviving t+ 1 years, and by the present value of n due t + 1 years hence, add the result to the dividend in the first part of the rule, before dividing by the value of a perpetuity due of 1.
Various other questions will present themselves, which can be easily reduced to practice by aid of the expeditious rule for finding the values of increasing annuities. This rule may be applied to the Carlisle tables (for which Mr. Milne has deduced the values of annuities on single lives, at rates of interest from S}}}}to 8 per cent.,

both included), and also to joint lives, though not with so much correctness, on account of the tables not containing so many rates of interest.
The following is an instance in which a deduction from the calculus of differences will supply in a rough manthe deficiencies of tables. There are none of these for determining the mean duration of the joint existence of two lives, but the defect may be supplied with sufficient accuracy for many purposes, and particularly at the middle and older ages, by the following RULE. Let (3), (4), &c. stand for the values of an annuity on a single life, or on two joint lives, at 3, 4, &c. per cent.: from twice (3) subtract (6) and reserve the remainder : from (4) subtract (5), and having halved the remainder, to it add the tenth part of (5), and multiply the result by 9. Subtract the last product from the reserved remainder, and multiply the difference by 10. The result increased by 5 in the case of a single life, or by 25 in that of two joint lives, will be something under the mean duration required. For example, and to take a very unfavourable case, let the Carlisle table be used, the life being 10 years old.