98 BULLETIN 136, U. S. DEPARTMENT OF AGRICULTURE. 
credited with interest semiannually. Table 6, on page 15, was cal- 
culated by formula (17). 
Example 9.—To find the annual payment which will accumulate in 20 years to 
$100,000 when interest is 34 per cent compounded semiannually. 
Taking n=20 and j=.035 and consulting Tables 32 and 33 with 12 per cent interest 
for values of sy and 1/sgy, respectively, there results: 
i 
Sz] ° —- =2.0175X .0174721=.0352500. 
2] S40} 
Hence the annual payment to sinking fund is 
0352500  $100,000=$3 ,525.00. 
Example 10.—To find the sinking fund, which set aside semiannually and accumu- _ 
lated as received, with 3% compound interest, will amount in 50 years to $1,000,000. 
Here formula (16) is used with p=2, m=1, j=.03, n=50, and 
1 @3208)2 1 
SS — 00439999: 
er eenyicen ne 
The required sinking fund is therefore 
.00439999 X $1,000,000 =$4,399.99. 
In the special case when the frequency of conversion coincides 
with the number of payments per annum, or m=p, the amount of 
each payment to the sinking fund is 
i j/pP i 
am Ji np ts : (18) 
pss At+s/p)yr?—1 Sap 
where Sz) 1s to be taken at rate 7/p. 
Example 11.—To find a sinking fund which, set aside semiannually and credited 
with a nominal rate of 3% convertible twice a year, will accumulate in 30 years to 
$1,000,000. 
Here apply formula (18) by substituting p=2, 7=.03, and n=30; this gives 
a 02034 
where I/sgj is taken at 15%. Then the sinking fund which would accumulate to 
$1,000,000 is 
0103934 $1,000,000 =$10,393.40. 
Four important cases of sinking funds are illustrated in the pre-_ 
ceding examples. They arise from the fact that payments to a 
sinking fund may be annual or semiannual and interest on a sinking 
fund annual or semiannual. Formula (16) covers all of them when 
p and m are properly chosen. The following schedule illustrates 
this fact: 
j 
| hate anes ItHlus- 
la Sinking-fund ' Interest on sinking : 
| Case. Ls payments. He fund. ae 
| 
Sool Ss Ammaeal 1 | Annual 8 
2 haa Annual 2 Semiannual 9 
3 2 Semiannual 1 Annual 10 
4 2 Semiannual 2 Semiannual 11 
{ 
Peas Pa 
ORD PS RY Ae 
s 
aa ‘ 
