ON ANNUITIES. 207 



of B, aged 30, to commence after the death of A aged 

 35, is l6'9 13-5,, or 3'4 years' purchase. 



PROBLEM. To find AB|A:B, the value of an 

 annuity on the life of the survivor, whichever it may 

 be, after the death of the other. 



AB|A:B=|A + |B-2(|AB) 



This is the same thing as giving an annuity to both, 

 on condition that both restore it as long as both are 

 alive. From the sum of the annuities, therefore, on 

 the lives of A and of B, subtract twice the value of an 

 annuity on the joint lives. 



PROBLEM. To find ABC|AB : BC : CA, or 

 ABC|(ABC) , the present value of an annuity to 

 begin payment at the end of the year in which one of 

 the three dies, and to continue as long as both of 

 the other two are alive. Give each pair an annuity on 

 their joint existence, and withdraw all three annuities 

 as long as all three are alive. 



ABC|(ABC) 2 =|AB + |BC + |CA-3(|ABC) 



If the annuity be to commence immediately, without 

 waiting for the death of one, this is an additional grant 

 of |ABC or 



|AB:BC: CA=|AB + |BC + |CA-2(|ABC) 



PROBLEM. To determine |A : B, the present value 

 of an annuity to be continued as long as eitlier A or B 

 shall be alive. Give both an annuity, but withdraw it 

 from one as long as both shall be alive. 



|A:B=|A + |B-|AB 



PROBLEM. To determine |A : B : C, the present 

 value of an annuity to be continued so long as any one 

 of the three shall be alive. Give each an annuity, and 

 withdraw one of the annuities from any pair as long as 

 that pair shall be both alive : but as this would take 

 away the annuity during the joint continuance of the 



