ON UNIFORMLY CHANGING ANNUITIES. XXIX 



be extinguished during the tabular life of the party, 

 from the value of an annuity of (m -f n) subtract n 

 times that of an increasing annuity of II. found as 

 already described. But \vhen the annuity can be ex- 

 tinguished 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 IL 

 on a life -f 1 years older than the party, multiplied by 

 the chance of his living t -f 1 years, and by the present 

 value of 5^ 1 due t-\-l years hence. 



PROBLEM. Required the present value of m 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 j^Pra, add that of 

 an increasing annuity of n, ^g%n, 3n, &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 

 n : 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 + 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 l. 



Various other questions will present them selves, which 

 can be easily reduced to practice by aid of the expedi- 

 tious 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 3 to 8 per cent., 



