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 3J per cent compounded semiannually. 



Taking a=20 and ;'=.035 and consulting Tables 32 and 33 with If per cent interest 

 for values of s-g-| and !/•%], respectively, there results: 



sy, • — =2.0175 X .0174721=. 0352500. 



1 S 40| 



Hence the annual payment to sinking fund is 



.0352500x$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 £>=2, m=l, j=.03, ?i=50, and 



1 (1 



2.23 



~ l = . 00439999. 



(1+.03) 50 -! 

 The required sinking fund is therefore 



.00439999 X$l,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 



1 JIP = _i 



On 



ps%] 



1 



(18) 



(I+J/P)" 1 ' 

 where s^l is to be taken at rate j/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, j—.OS, and n =30; this gives 



2sS S; 



= .0103934, 

 Then the sinking fund which would accumulate to 



where l/s^ is taken at \\%. 



$1,000,000 is 



.0103934X11,000,000=$] 0,393.40. 



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: 



Case. 



V 



Sinking-fund 

 payments. 



m 



Interest on sinking 

 fund- 



Illus- 

 trated in 

 example. 



1 



2 

 3 



4 



1 



1 

 2 

 2 



Annual 

 Annual 

 Semiannual 

 Semiannual 



1 



2 

 1 

 2 



Annual 

 Semiannual 

 Annual 

 Semiannual 



8 



9 



10 



11 



