100 BULLETIN 136, U. S. DEPARTMENT OF AGRICULTURE. 
Fundamental relations between the present value and the 
amount of an annuity.—Since a; and sz 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 
Oni aa Oe Sis 
Sz = (1 +4) "da; 
and in like manner that 
a® sel WG. 
Ss = (1+7)"a®. 
As tables are not published giving the values of a and s, when 
p is different frem 1, it is desirable for purposes of computation to 
express a relation between these functions and the tabulated func- 
tions a, and s;. This can easily be done by accumulating to the 
end of each year the p payments of 1/p 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 
core v tee 
1 pli +17] i@ 
This converts the annuity into one with annual rent s® payable at 
the end of each year for n years. Therefore 
(p) (Pp) 9 
Loree sah Aira) ee 
(yp) _ .(p) 2 | 
Sri = S77 Sni° (24) ; 
The most frequent intervals in practice are semiannual, quarterly, 
and monthly, and to meet this requirement the values of s@, s%, 
(12) 
and Ss are given below for various rates of interest. 
Te Ser em ee uo EY 
Values of S)=@/)(p) PId+oip—T] 
D 1A eZ 2% 214% 244% | 234% 3% 
2 | 1.00373604 | 1.00435603 | 1.00497525 | 1.00559371 | 1.00621142 | 1.00682837 | 1.00744458 
4} 1.00560755 | 1.00653878 | 1.00746906 | 1. 00839839 | 1.00932677 | 1.01025422 | 1.01118072 
2 
12 | 1.00685652 | 1.00799571 | 1.00913389 | 1.01027107 | 1.01140725 | 1.01254243 | 1.01367662 
- = _ = — _ — 
| 
p | 3%% AU Ni ALZOr 5% 54% | 6% 7% 
- 2 | 1.00867475 | 1.00990195 | 1.01112621 | 1.01234754 
4 | 1.01303094 | 1.01487744 | 1.01672026 | 1.01856942 
12 | 1.01594203 | 1. 01820351 | 1.02046109 | 1.02271479 
. 01356596 | 1.01478151 | 1.01720402 
.02039495 | 1.02222688 | 1.02588002 
. 02496465 | 1.02721070 | 1. 03169143 
oa’ 
} 
Example 138.—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, 1=.03, 
aG=SF) - Gq =1.01367662 X 19.6004413=19.86850909. 
Therefore the present value of a similar annuity of $25 per month, or with annual 
rent of $300, is 
19.86850909 X $300 =$5,960.55. 
