44 Mr. C. J. Hargreave on the Valuation of Life Contingencies 



and after the multiplications, to change the letters into the cor- 

 responding names. The result would be given in this case in 

 terms of no less than 165 annuities on joint lives. 



The preceding formulae may be adapted to the case of contin- 

 gencies involving not merely lives, but also a fixed term of years, 

 it being assumed that the term, as well as the lives, begins to run 

 inprasenti. Thus, if T denote the value 'of an annuity for a 

 term, and AjT the value of an annuity for so much of the term 

 as A, shall live, all the results above given remain applicable if 

 we substitute T for one of the lives. Thus Aj +T— A^T is the 

 value of an annuity for the term and the life, whichever shall 

 last the longest; Ai + T--2AiT is the value of an annuity for 

 the residue of one period (the term or the life) after the other 

 hasexpired. So Ai + A2 + T-(A,A2 + AiT+A2T) ^-AAgTis the 

 value of an annuity for two lives and a terra, whichever shall last 

 the longest. 



In these investigations A„ denotes the present value of an 

 annuity which is payable at intervals of a year from the present 

 time, provided the life be in existence on the day of payment ; 

 the complement of this, or 1 — A„, must therefore be understood 

 as denoting a perpetuity expectant on the death of A„ , on the 

 assumption that it commences in point of interest at the begin- 

 ning of the year in which the life drops, or that its first payment 

 takes place at the beginning of the year ensuing the fall of the 

 life. Unless this assumption be made, the two added together 

 will not constitute a present perpetuity. In like manner, an 

 annuity commencing at the death of Aj and continuing until the 

 death of A^ is represented by Ag— AiAg, on the assumption that 

 it begins in interest at the commencement of the year in which 

 Aj dies. So, when we represent by M(l— A„) the value of a 

 sum M payable on the death of A„ , we mean that it is payable 

 at the commencement of the year of A„'s death ; and if it be not 

 payable until the end of that year, we must take a year's discount 

 from it. The diminution of value occasioned by payment being 

 postponed until the fall of a life or other status may be regarded 

 as a particular mode of discounting the sum, the rate of discount 

 depending on the nature of the status. Thus the value of M 

 pounds payable at the death of the survivor of Aj A^ . . A„ is 

 M(1-2;A + 2;(AA)-S(AAA)+ ..); and the value of M pounds 

 payable after the decease of n lives where they are not coexistent, 

 but each life (or rather a life of that value) is nominated on 

 the death of his predecessor, (or more strictly at the beginning 

 of the year in which he dies), is 



M(l-A,)(l-A,)...(l-A„). 



If all the lives have the same value P, the value of M pounds 



