HIGHWAY BONDS. 97 
The special case where interest is converted with the same fre- 
quency as the payment of annuity installments, or when m=p, 
deserves particular mention. Formula (13) then reduces to 
s® = Ch!) eee ea S—) 
ni ; “np! 
pane fa JI/P De 
where sj 1s to be taken at the effective rate j/p. 
(14) 
Example 7.—What will be the accumulation in 47 years of an annual sinking fund 
of 1% of $1,000,000, paid in semiannually, if the fund is credited with a nominal rate 
of 3% convertible twice a year? 
This is an application of formula (14) where n=47, p=2, and soq 1s taken at 13%; 
hence 
1 s>5 X$10,000=$1,017,764.25.2 
Sinking fund which will amount to 1.—An annuity with 
annual rent of 1 will amount in 7 years to sj; it follows that an 
annuity with annual rent of 1/s,; will amount in 7 years to 1. The 
quantity 1/s, is the sinking fund which will accumulate to 1 in n 
years. 
Values for this important function 
1 ) 
Ele ee 
are given for various rates of interest and for terms ranging from 1 
to 60 intervals or years in Table 33. 
Example 8.—To find an annual sinking fund, which, credited with 3% compound 
interest, will accumulate in 50 years to $1,000,000. 
Applying formula (15) where n=50, and 71=.03, there results 1/s3>, =.0088655. 
Therefore the required sinking fund is 
.0088655 X $1,000 ,000=$8, 865.50. 
In like manner 1/s%) is the annual rent of an annuity which, at 
the nominal rate 7 convertible m times a year, will accumulate to 
1 in ” years. The annual rent is payable in p installments during 
each year; hence each installment is equal to 1/ps®. The install- 
ments may be regarded as the sinking fund, payable at the end of 
every pth part of a year, which in 7 years will accumulate to 1. 
The amount of each payment to the sinking fund is 
1 (1+ 9/m)™p—1 
ps® ~ Ce jimyrt 
When p=1 and m=2, formula (16) gives the value of the annual 
sinking fund which, improved at compound interest semiannually, will 
accumulate in n years to 1. 
The formula simplifies to the following: 
149/2)?—-1 j/2 1 
TE Mac aes ee 
This formula is of considerable practical importance because pay- 
ments to the sinking fund are usually made annually and the fund 
(15) 
(16) 
(17) 
1 For calculation of s55 see Example 22, page 113. 
92448°—15——7 
