HIGHWAY BONDS. 101 



The annuity which 1 will purchase. — The present value a-^ of 

 an annuity may be viewed as the principal which, invested at the 

 effective rate of interest i, will provide a payment of 1 at the end of 

 each year and will not be exhausted until the end of the nth. year; 

 in other words, a^ is just sufficient to purchase an n year annuity 

 of annual rent 1 payable at the end of each year. By proportion 

 it appears that 1 will purchase an n year annuity of annual rent 

 l/a,i; payable at the end of each year. This quantity may be 

 described as the annuity which 1 will purchase, and its value is 



_L = _1 — = L_ (25) 



This function is of great importance in annuity bond calculations, 

 and its values are given for 60 terms and different rates of interest in 

 Table 36, on pages 126 and 127. 



Example 14. — To find the uniform annual payment which in 20 years will dis- 

 charge a loan of $100,000, including both principal and interest, at 5 per cent com- 

 pounded annually. 



In this case n=20, i=.05; employing formula (25) and referring to Table 36, it 

 follows that a loan of 1 will be discharged, both principal and interest, by an annual 

 payment of 



— = .0802426; 



fl 20] 



hence the loan of $100,000 will be likewise discharged by an annual payment of 

 .0802426X$100,000=$8,024. 26. 



By similar reasoning it follows that 1 will purchase an immediate 

 annuity-certain with annual rent l/a ( fp payable in p installments each 

 year. The value of each 'periodical installment is 



1 (l+j/mj m i*—l 



paW l-(l+j/m)- 



(26) 



where interest is at the nominal rate j with frequency of conversion m. 

 Whenm = 1, the nominal rate j^ becomes the effective rate i. When 

 the conversion of interest occurs with the same frequency as the 

 periodical payment, that is, when m = p, formula (26) reduces to the 

 important particular case 



pa^ 1 - ( 1 +j/p)~ n P a™? ' 



where a^ is to be taken at j/p per cent. 



Example 15. — To find the half yearly payment at 5% compounded semiannually 

 which will discharge both principal and interest on a loan of $100,000 in three years. 



By formula (27) with n=3, p=2, a loan of 1 will be discharged, both principal and 

 interest, in three years by a semiannual payment of 



1 =.1815500, 



aj\ (taken at 2\f ) 



and the loan of $100,000 will be discharged in like manner by 



.1815500X$100,000=$18,155.00. 



