206 ESSAY ON PROBABILITIES. 



At the end of tl years, if A be then alive (of 

 which find the chance), he enters upon an annuity of 

 11. y being then n -f t 1 years of age. The value of an 

 annuity at that age, multiplied by the chance of attain- 

 ing that age, and by the present value of I/., to be 

 received tl years hence, is, on the principles ex- 

 plained in p. 189-, tne present value of the annuity. 



Example w = 45, =11 (Carlisle tables 5 per cent.) 

 10fa 45 = $|f=-862 9(1 =-614 |A 55 =10-3 

 862 x *614 x 10-3 = 5-45 5 '45 Answer. 



PROBLEM. An annuity of II. is granted to A, aged n, 

 for t years, provided he live so long : what is its value ? 

 \A n t=A n -t\A n 



The last equation is obvious ; for the whole annuity 

 on A's life is made up of a contingent annuity for 

 t years, and a contingent annuity to commence payment 

 in t -f 1 years. Consequently, from the whole value of 

 an annuity for A's life subtract that of an annuity to 

 commence payment in t -f 1 years, if he should be then 

 alive, and the remainder is the value of an annuity for 

 t years, if he should live so long. 



Example w = 45, t\Q 



By last problem 10|A 45 = 5'5, |A 45 =12-6. 

 Therefore |( A 10)= | A 45 - 10|A 45 = 7'1 7 '1 Answer. 



PROBLEM. To find A m I B w? the value of an annuity 

 on the life of B, aged n y the first payment of which is 

 to be made at the end of the year in which the life of 

 A, aged m, fails. This is called a survivorship annuity, 

 since it can never be paid unless B survive A. To 

 give this annuity is evidently to give a complete annuity 

 to B, on condition that he shall restore it as long as A 

 is alive ; that is, 



A|B=|B-|AB; 



or, from the value of an annuity on B's life subtract 

 that of an annuity on the joint lives. Thus (Carlisle 

 tables 4> per cent.), the value of an annuity on the life 



