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 — 
An) Sn 
) 
+4 (29) 
can easily be verified by substitution of the values of 1/a” and 
1 /s"| expressed in terms of 7, 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 7 on the unit so invested, but also a sinking fund, 1/s“, 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 7. 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 Q; n, the number of years after which the 
bonds are redeemed; K, the present value of C, due m years hence, 
