HIGHWAY BONDS. 99 



In most cases in the illustrative tables in the body of this bulletin, 

 for simplicity of presentation, annual payments and annual interest 

 are assumed, whereas in practice usually annual payments and semi- 

 annual interest are employed. 



Present value of an immediate annuity-certain. — The present 

 value of an immediate annuity-certain for n years, with annual rent 

 1 payable at the end of each year, is designated by the symbol a^. 



It is equal to the sum of the present values of 1, due at the suc- 

 ceeding yearly intervals. By formula (5) the present value of 1, 

 due at the end of one year at the effective rate of interest i, is 

 v = 1/(1 +i) ; at the end of two years, v 2 = 1/(1 +i) 2 , etc. .......; 



at the end of n years, v n = 1/(1 +i) n . Hence 



a^ = v J r v 2 + +V 11 



1 1 



~l+i + (l+i) 2 + 



The sum of this geometric series is 



*-^-^ ' d9) 



and its values are given in Table 35. 



Example 12. — To find the present value at 3% of an annual payment of $56,325 

 at the end of each year for thirty years. 



Referring to Table 35, it is seen that a^ at 3% is 19.6004413, and therefore the 

 equired present value is 



19.6004413X156,325=11,103,994.86. 



While the above demonstration relates to an annuity of 1 per 

 annum, payable at the end of each year, the same principles apply 

 to finding the present value of an annuity of 1 per annum, payable 

 in p installments during each year. The present value of such an 

 annuity is designated by the symbol a l $, and its value is repre- 

 sented by the following formula: 



«' p[(i+iyip-i] p[(i+iyip-i] y J 



In formulas (19) and (20) the values of the annuities are expressed 

 in terms of the effective rate i. If (1+i) is replaced by (l+j/m) m 

 in accordance with formula (2), there result the present values of 

 the same annuities expressed as follows in terms of the nominal rate 

 of interest j, with frequency of conversion m: 



_ l-(l+j/m)-^ r . 



and 



to, = l-q+j/m)-™ , (22) 



nl p[(l+j/m)™ip-l] y } 



