ON ANNUITIES. 189 



supposing that, money making 4 per cent., l6'7l., the 

 value of an annuity certain for 28 years, must be but very 

 little more than the value of an annuity on a life aged 

 40. Such a rule, if it be absolutely true, must be true 

 in any extreme case, however physically impossible. 

 Suppose, then, that of 101 persons aged 40, one lives for 

 ever, and the rest die between the ages of 41 and 42 : 

 whence 100 of them will receive only one payment, 

 and the remaining one will receive a perpetuity of \L 

 The present value of an annuity of 11. for one year, at 

 4 per cent., is *96/., and $6l. increased by the value 

 of I/, for ever, is 121 1., the present value of all pay- 

 ments. If, then, it be not known which is to have the 

 perpetuity, -j-y/. is what each should pay, or 1*2 II. 

 very nearly. This result is undoubtedly correct : but 

 the average life of the 101 persons is infinitely great, 

 since there is no number of years which is not ex- 

 ceeded by them all put together. Each, then, would 

 have to pay for a perpetuity, if the preceding fallacy 

 were admitted ; or all together would pay 25 x 101/._, 

 that is 2525/., more than 200 times the truth. It is 

 true that, applied to any actually existing law of life, 

 the incorrect notion cannot produce results so grossly 

 false as the preceding: but it is quite sufficient for 

 us to know that a rule is incorrect in its principle, to 

 make it necessary to apply correct reasoning before we 

 can attempt to say how far it is wrong. 



The rule for calculating the real value of an annuity 

 is made up of a collection of individual cases,, not 

 more complicated than the following. A is to receive 

 one pound ten years hence, if a halfpenny which is 

 to be thrown up give a head; what is the present 

 worth of his chance ? If he were to receive the money 

 immediately, in case of winning, its value, on the prin- 

 ciples in chapter V., would be '51. ; but the present 

 value of II. deferred for ten years at 4 per cent, is 

 ?09/. : whence that of -51. similarly deferred is '3551. 

 This simple process contains the method of valuing 

 the sum which must be now paid down to secure the 

 tenth payment of an annuity to a life belonging to a 



