216 



ESSAY ON PROBABILITIES. 



to commute their interests for interests due : that is, the 

 perpetuitant, instead of 11. a year hence and so on, 

 desires to receive a fraction of a pound now, and the 

 same fraction at the end of every year ; and the same 

 for the annuitant. Say the value of money is four per 

 cent.., then the perpetuitant desires to change an interest 

 which is worth twenty-five years' purchase into an 

 equivalent interest worth twenty-six years' purchase 

 (or income) ; consequently his year's income (now due, 

 &c.) must be only -||/. Say that the annuity is worth 

 ten years' purchase; then by the same reasoning the 

 yearly income of the annuitant (now due, c.) must 

 be only ^l. The second is less than the first ; whereas 

 the original incomes were the same, both I/. But 

 there must be some consideration which the com- 

 mutation gives to the annuitant, and for which this 

 greater diminution of his income is the payment ; and 

 it is as follows : Since the commutation forestalls each 

 successive payment, giving it (or the substitute for it) 

 a year before it becomes due, the annuitant would 

 receive, if his income were made equal to that of the 

 perpetuitant, the II. which, had he lived, would have 

 become due at the end of his last year, but which his death 

 hinders from becoming due. This difference of income 

 (If "~~TrX * s therefore equivalent to preventing his 

 receiving M. at the end of the year in which he dies, 

 and it is taken from him now and in every succeeding 

 year of his life. Consequently it is the premium which 

 such an annuitant should pay to receive 11. at the end 

 of the year in which he dies ; and it is also the result 

 of the first preceding rule. 



The second rule may receive an explanation of a 

 similar kind. I now reverse the problem, and ask the 

 following 



QUESTION. If an office charge the premium p for 

 insuring 11. at the end of the year in which a life (or 

 other terminable status) drops, what should we infer 

 that they suppose to be the greatest possible value of 

 an annuity to continue during the remainder of that life 



