ON THE VALUE OP REVERSIONS. 



223 



beginning, or |AB..., is evidently |AB increased by AB|l, 

 the present value of an insurance of one pound on the 

 joint lives. 



PROBLEM. Required the present value of ^l to be 

 paid at the end of any year, provided that both A and 

 B die in that year ; which has been signified by A : B|l. 



Grant the following annuities ; 



|AB, on the joint lives of A and B. 



|(AB...) the same as |AB, but to be also paid at 

 the end of the year in which the joint existence fails. 



Take in exchange the following annuities : 



|(A...B) and |(B... .A) two annuities to be paid 

 during the joint lives, and also at the end of the year 

 in which the joint existence fails, provided B in the first, 

 and A in the second, be alive at the end of the year. 



The balance of this transaction will be^l to be paid 

 at the end of any year, provided B and A both die in 

 the year. For as long as both are alive, two annuities 

 are payable each way ; if A die and B remain alive 

 till the end of the year, |AB has ceased, |(AB...) is 

 payable, but |(B...A)has ceased, and (A...B) is payable ; 

 similarly in the case of B dying and A remaining alive. 

 If both die in one year |AB has ceased, but |(AB...) 

 is payable, while JB...A and |A...B have both ceased. 

 Consequently, the only possible payment which the 

 grantor has to make, over and above those which he 

 receives, is the ^1 in the question proposed ; or 



We are now in a condition to solve the final PROBLEM. 

 Required the value of \ to be paid at the end of the 

 year in which A dies, if B should have been alive at the 

 moment of A's death. This is denoted by A|1B. 



When A dies before B, either B survives till the 

 end of the year, or dies in the intermediate time. The 

 insurance on the first risk is worth A| IB determined 

 in p. 222.; on tbe second it is worth half the result of 

 the last problem, if it be considered that the chances of 



