104 BULLETIN 136, U. S. DEPARTMENT OF AGRICULTURE. 



number, it would be necessary only to multiply each item by L/a» ! to 

 obtain the corresponding schedule for a loan of L. 



If in the preceding discussion year is replaced by interval, the 

 schedule may be made to apply to loans repaid by equal install- 

 ments at the end of each interval. 



Relation between annuity which 1 will purchase and 

 sinking fund which will amount to 1.— The important relation 



-L = -L+i (29) 



On, *S| ■ 



can easily be verified by substitution of the values of l/afl and 

 l/s"l expressed in terms of i, by formulas (25) and (15). 



The relation (29) merely expresses the fact that the annual rent, 

 1/a"' on the annuity which 1 will purchase, must include, not only the 

 interest i on the unit so invested, but also a sinking fund, 1/s" 1 , which 

 will accumulate to the invested unit at the end of the term of the 

 annuity. 



Application to bond calculations. — An important application 

 of the theory of compound interest and annuities arises in the valua- 

 tion of bonds. First to determine the value of a bond issue redeem- 

 able in one sum on a given date, with interest, or dividends, on the 

 outstanding bonds at rate g, and all computed, or valued, so as to 

 yield the purchaser a given effective rate of interest i. Consider an 

 issue of $100,000 highway bonds, denomination $500, dated January 

 1, 1914, maturing January 1, 1948, interest 5 per cent, payable 

 annually. 



The annual interest, or dividends, on these bonds is 5 per cent, 

 and the bonds are redeemed at the end of 34 years. Suppose an 

 intending purchaser desires to pay a price which will yield a net 

 income of 3 per cent on his investment; how much ought he to bid ? 

 This is the nature of the general problem. If the purchaser desires 

 to realize 5 per cent on his investment, he must bid $100,000 for the 

 bonds, or $1 for each dollar to be redeemed. If, however, he is 

 content with 3 per cent, more than $100,000 must be paid for the 

 bonds, that is, more than $1 for each dollar to be redeemed. In 

 this case the bonds are said to be bought at a premium; if less 

 than $1 is paid for each dollar to be redeemed, the bonds are said 

 to be bought at a discount. 



In the general case, let C denote the price to be paid on redemption; 

 i, the effective rate of interest employed in the valuation of the bonds, 

 which is the net income rate to the purchaser; g, the ratio of the 

 dividend per annum to O; n, the number of years after which the 

 bonds are redeemed; K, the present value of C, due n years hence, 



