22 ESSAY ON PROBABILITIES. 



perpetuity should be treated as that of the value of a 

 perpetuity due at the end of the year in which the life 

 drops. 



PROBLEM. Required the present value of II. to be re- 

 ceived at the end of the year in which A dies, provided 

 that event take place before the expiration of t years 



from the present time : signified by A| 1 t. 



Suppose a person to have the certain reversion of a 

 perpetuity due at the end of t years, or sooner, if A die 

 before t years are expired : the reversion of the perpetuity 

 after the failure of the joint existence of A and t years, 

 is | | A ty which can be found from page 206, But 

 this is more than that fraction of a perpetuity due at the 

 end of the year in which A dies, which will pay for the 

 chance of entering on it before t years are expired : for 

 part of it expresses the value of a perpetuity due, which, 

 though A should be alive, is to be entered on by the 

 failure of t years. If t'\~ a be the chance that A is 

 alive at the end of t years, then t] a X t\ must be 

 deducted, as being expressed twice in the preceding : 

 consequently 



|-|A t t ta X t\ 



is the present value of a perpetuity due, or -|, to be 

 entered upon at the end of the year in which A dies^ if 

 before t years. The preceding then divided by -j gives 

 the present value of 11. to be received under the condi- 

 tions of the question. But a perpetuity created at the 

 end of t years, or t\, divided by a perpetuity now due^ 

 gives the present value of I/, to be received at the end of 

 t+1 years; which gives the following 



RULE. From the value of a perpetuity subtract that of 

 an annuity on the given life for t years, and divide by 

 the value of a perpetuity due. From the quotient sub- 

 tract the present value of one pound to be received at 

 the end of t -j- 1 years, if the life be in being at the end 

 of t years: the difference is the present value of 11. to 

 be received at the end of the year in which the life drops, 

 if before t years have expired. The present value just 



