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



where A is the value of each unit of the sum to be redeemed, and Kis 

 the present value of the various parts of the unit at effective rate i 

 due in t?!, n 2 , . . . . n r years. Letting .4 — 1 = I', formula (38) becomes 



k=(l-K) (g-i)/i. (39) 



The premium is positive if g is greater than i, and negative, or a dis- 

 count, if g is less than i; for the first factor (1 — K) can not be nega- 

 tive, as K by definition is the present value of a series of future pay- 

 ments whose sum is 1 , and hence their present discounted value must 

 be less than 1. This shows in all cases that a bond issue must be 

 bought at a premium, if it is valued at a hirer rate i than the rate 

 of dividend g; and at a discount, if it is valued at a higher rate i than 

 the rate of dividend g. 



Serial bonds. — To apply the general formula (39) to the case of 

 a bond issue redeemed by n equal annual installments, consider a 

 unit of the total sum to be redeemed. Since this unit is to be 

 redeemed in n equal installments over n years, the annual portion 

 redeemed is 1/n. 



1 ri 1/n 1/n 1/n 



1 yr. 2 yrs. 3 yrs. n yrs. 



The present value, K, of these n installments is clearly the value of 

 an annuity of annual rent 1/n; hence 



K=-a,n\ X 1/n = a»/n. 



Substituting this value of K in formula (39). the following formula 

 is obtained: 



k=(l-a^/n) (g-i) I. (40) 



Example 19. — What is the bid on 8100,000 highway if c serial bonds maturing in 

 20 equal annual installments, to net the purchaser an effecth'e rate of 3 ■ , ? 



Here n=20, gr=.04, i=.03, and a ^ =14.8774749; consequently 



£=(1-14.877 L749 20)(.04-.03)/.03 



=(1-. 743873745) X1/3 = .256126255X1/3=.085375418. 



Hence the bid on 8100,000 is 



1.085375418X8100,000=8108,537.54. 



Extension of formulas to case when dividends are payable 

 and interest is convertible in times per annum. — Formula (36) 

 assumes that dividends are payable once a year and that the effective 

 rate of interest is i per annum. Replacing year by interval and assum- 

 ing dividends to be paid at the end of each interval and the rate of 

 interest realized by the investor a nominal rate convertible m times 

 a year, formula (36) still applies, if the present value K of the several 



