by means of Tables of Single and Joint Lives. 41 



the case of all dead. lip=^n, the expression becomes ^(1 — 1) 

 or <^(0) or 1, the event being certain. 



By referring to the well-known process for obtaining the value 

 of an annuity which depends upon the existence of any status of 

 a contingent nature, the reader will without any difficulty at 

 once perceive the correctness of the subjoined solution of the 

 following problem. 



Problem. — To determine the value of an annuity during the 

 continuance of any status which can be made to depend upon the 

 life or death of any number of persons out of the set A|, Ag . . A^. 



Mode of solution. — Find the probability (by the formulae above 

 given) that the status will be in existence after the lapse of any 

 number of years ; for 5^ s^. . . write their values «i + «2 + ^3 + • • ^ 

 «i«2 + «i«3 -f . . . j and if necessary, multiply the expression out, 

 so as to get rid of all brackets ; then for each small letter write 

 the corresponding large letter ; and the result is the value of the 

 annuity in question expressed in values of annuities on single 

 and joint lives, on the assumption that A;^ means the value of an 

 annuity on the life of A„ ; A^ A^ the value of an annuity on the 

 joint lives of A^ and A„ ; and so on for any number of persons. 

 The most convenient notation is to consider these values as 

 framed on the assumption that unity denotes the value of a pre- 

 sent perpetuity ; so that A„ means the ratio of the value of an 

 annuity on the life of A^ to the value of a perpetuity, or the 

 value of the annuity on the a'ssumption that a perpetuity is 1. 

 Calling A„ the value of such a status, the value of a perpetuity 

 expectant on the determination or failure of the status is 1 — A^. 

 The preceding rule, therefore, determines also the value of a 

 perpetuity expectant on the failure of any status formed out of n 

 lives, or payable only when the status does not exist. 



The above mode of solution will be rendered plain by a few 

 examples. 



Let us take every contingency that can be made from the 

 lives of Ai and Ag. Here 



which give the following probabilities and values : — 



Probability that both are dead is <^(1) or \ — [a^-\-a^ +^1^2} 

 Value of perpetuity expectant on death of survivor, 



1 — (A,+A2)+AiA2; 



Probability that one or both are living, a^-^a^^a^a^; 



Value of annuity for two lives and lifeof survivor, Aj + Ag— A^Ag. 



These two instances are complementary to each other. 



Probability that one only is living is —^'(1) or «i H-flg— 2«|«2; 



