188 ESSAY ON PROBABILITIES. 



ment (as is usually done) at the time of contracting, 

 then the premiums altogether constitute an annuity due 

 upon A's life. 



All problems relating to this subject might be solved 

 from first principles, as will be shown ; but, in such case, 

 even the most simple of them would be attended with 

 laborious calculation. Tables are therefore constructed 

 of such results as will most facilitate the solutions of all 

 problems : and these tables give the values of annuities 

 on single lives, and on two joint lives. The annuity 

 presumed is always II. per annum, from which the 

 value of any other annuity is found by simple multi- 

 plication. 



By the value of an annuity is meant the sum which 

 must be paid down in order to enable the grantor of 

 the annuity to pay it as it becomes due during the 

 term of its continuance. If money made no interest, 

 the average duration of such sets of circumstances as 

 are conditions for the payment of the annuity would 

 immediately point out its value. For instance, accord- 

 ing to the Carlisle tables, persons aged 40 live, one 

 with another, 27*6 years. Now, remembering that 

 half a year was added (p. 163.), as being the part of 

 their last year which, one with another, they pass 

 through, but which will not (in our meaning of the 

 term annuity) entitle them to any payment, we see that 

 27*1 is the average number of payments which such 

 annuitants, aged 40, will receive; that is, one hundred 

 with another, 100 annuitants will receive 2710 pay- 

 ments. Consequently, money making no interest, each 

 of them must pay 27*1 01. for a life annuity of I/., 

 or 27*10 years' purchase. But if each had been 

 entitled to his fraction of the sum which would have 

 become due had he lived to the end of the year, then 

 2 7' 60 years' purchase would have been necessary. 



Let us now suppose money to make interest. It 

 has before now seemed, to more than one writer, that 

 the value of a life annuity must be the same as that of 

 an annuity certain during the average duration of the 

 life. Many of my readers will not see the fallacy of 



