100 



BULLETIN 136, U. S. DEPARTMENT OF AGBICULTUEE. 



Fundamental relations between the present value and the 

 amount of an annuity. — Since a^ and s^ are the values of the 

 same annuity upon two dates differing by n years, it follows by the 

 principle of reduction of values from one date to another, explained 

 on page 95, that 



S^ = (l+i) w «y7|, 



and in like manner that 



tt ( J = ^), 



8 i& = (l + i)«a&l. 

 As tables are not published giving the values of a { §) and s%\ when 

 p is different from 1, it is desirable for purposes of computation to 

 express a relation between these functions and the tabulated func- 

 tions dn\ and Sjtj- This can easily be done by accumulating to the 

 end of each year the p payments of \jp which in a ( ^ and s^ are 

 distributed at equal intervals through the year. By formula (11) 

 this accumulation to the end of each year will be equal to 



,<« 



o Tl = 



'I 



p[(l+i)Vr-l] 



This converts the annuity into one with annual rent s ( jj payable at 

 the end of each year for n years. Therefore 



M>) _ «(£)*_,. (24) 



The most frequent intervals in practice are semiannual, quarterly, 

 and monthly, and to meet this requirement the values of 8&\, sQ, 

 and s'lf are given below for various rates of interest. 







Values of s ( *> = 



ll 





i 







HJ(p) p[ 



l + /)l/7 1] 



V 



V/2% 



l 3 i% 



-/a 



%H% 



2 1 2% 



2M% 



3% 



2 

 4 

 12 



1. 00373604 

 1. 00560755 

 1.00685652 



1. 00435603 

 1.00653878 

 1.00799571 



1.00497525 

 1. 00746906 

 1.00913389 



1. 00559371 

 1. 00839839 

 1.01027107 



1.00621142 

 1. 00932677 

 1.01140725 



1.00682837 

 1.01025422 

 1.01254243 



1.00744458 

 1.01118072 

 1.01367662 



P 



33 i% 



4% 



VAX 



5% 



514% 



6% 



7% 



2 

 4* 

 12 



1.00867475 



1.01303094 

 1. 01594203 



1.00990195 

 1.01487744 

 1. 01820351 



1.01112621 

 1.01672026 

 1.02046109 



1.01234754 

 1.01856942 

 1. 02271479 



1.01356596 

 1.02039495 

 1.02496465 



1.01478151 

 1.02222088 

 1.02721070 



1.01720402 

 1.02588002 

 1. 03169143 



Example 13. — What is the present value of an annuity for 30 years at effective 

 rate 3% , payable in monthly installments of $25? 

 By formula (23) with n=30, p=12, £=.03, 



a 151 =s( ^ ) • %i=1.01367662X19.6004413=19.86850909. 



Therefore the present value of a similar annuity of $25 per month, or with annual 

 rent of $300, is 



19.86850909X$300=$5,960.55. 



