ON ANNUITIES. 



201 



I now proceed to some further instances : 



| n means the present value of an annuity of IL to last 

 n years. 



m-\n the present value of an annuity of I/., which is to 

 commence payment at the end of m years, and then 

 to last n 1 years, or n I more payments. 



n\ the present value of a perpetuity of II. , com- 

 mencing at the end of n years, or first paid at the 

 end of n -f 1 years. 



n-\ the present value of a perpetuity, first paid at 

 the end of n years. 



| A the present value of an annuity on the life of A. 



JAB the present value of an annuity on the joint 

 lives of A and B, to cease with either. 



t \ A the present value of an annuity on the life of A, 

 to begin in t years ; that is, the first payment to 

 be made at the end of t -j- 1 years, if A should then 

 be alive. 



\At the present value of an annuity on A's life, or 

 t years, whichever drops first. 



A | the present value of U. for ever, to be first received 

 at the end of the year in which A dies. 



AB| the present value of IL for ever, to be first re- 

 ceived at the end of the first year in which A or B 

 dies. 



A 1 1 the value of an annuity for t years, payment to 

 begin at the end of the year in which A dies. 



AB|C the present value of an annuity which begins 

 payment at the end of the year in which either 

 A or B dies (the first), provided C be then alive, 

 and which continues during the life of C. 



| A : B signifies the present value of an annuity which 

 is to be paid as long as either A or B is alive. 



AB : C|D : E the present value of an annuity which is 

 not to be paid as long as A and B are both alive, 

 nor as long as C is alive, but which begins when 

 the joint existence of A and B, and that of C, are 

 both terminated; and continues as long as either 

 D or E are alive. 



