ON THE VALUE OF REVERSIONS. 227 



found, divided by that of an annuity due on the given 

 life for t years, gives the requisite premium. 



And since the present value of such an insurance as 

 the preceding, together with the present value of II. to 

 be received at the end of the year in which A dies, if 

 after t years, make up the present value of IL to be re- 

 ceived in any case at the death of A, the third diminished 

 by the first will give the second. But it will be prefer- 

 able to make an independent investigation of this case. 



PBOBLEM. Required t : A 



1 the present value of I/. 



to be received at the end of the year in which A dies, 

 provided t years shall have previously expired. 



If from the present value of a perpetuity deferred for 

 t years, we deduct that of an annuity on the life of A de- 

 ferred for t years, we have the value of a deferred perpe- 

 tuity, further suspended during the term by which A out- 

 lasts t years, and to commence at the end of the year in 

 which A dies : or not to be suspended at all if A should 

 die in less than t years. Take away the value of a per- 

 petuity beginning from the end of t years, if A should 

 have died in the interval, and we have remaining the 

 present value of a perpetuity due at the qnd of the year 

 in which A dies, if that be deferred beyond t years. 

 This last is therefore 



t\ t\A. -(l-t t a) X t\ 



where t t a is the chance of A living t years. This can 

 be reduced to 



t t a x t\ *|A 



Divide this by the value of a perpetuity due, and we 

 have the present value of 1 /. receivable on the same con- 

 ditions. But t\ divided by -| gives t \ I, as before; 

 whence the following 



RULE. Multiply the present value of II. receivable at 

 the end of t -f- 1 years by the chance which A has of 

 living t years; and from the product subtract the quotient 



