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(P)= iM+Jipy^ii = i. s __, (14) 



** p j/p p p 



where s^\ is to be taken at the effective rate j/p. 



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 sj^\ is taken at \\%\ 



hence 



is?7lX$10,000=$l,017,764.25. 1 



Sinking fund which will amount to 1. — An annuity with 

 annual rent of 1 will amount in n years to s^; it follows that an 

 annuity with annual rent of 1/ss] will amount in n years to 1. The 

 quantity 1/s^ is the sinking fund which will accumulate to 1 in n 

 years. 



Values for this important function 



J_ i a5 ) 



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 i=.03, there results l/sy^i =.0088655. 

 Therefore the required sinking fund is 



.0088655X$1,000,000=$8, 865.50. 

 In like manner l/s ( f/ is the annual rent of an annuity which, at 

 the nominal rate j convertible m times a year, will accumulate to 

 1 in n 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 pih part of a year, which in n years will accumulate to 1. 

 The amount of each payment to the sinking fund is 



1 _ (l-\-j/m) ni 'p-l 

 ps ( g ~ (l+j/m) mn -l ' (16) 



When p=l 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: 



j/2 _ (1+^/2)^-1-^ 5i=j at ^ 2 %- (17) 



This formula is of considerable practical importance because pay- 

 ments to the sinking fund are usually made annually and the fund 



1 For calculation of s-^ see Example 22, page 113. 

 52448°— 15 7 



