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 7 
dueinn,,7,,....n,years. Letting A—1 =k, formula (38) becomes 
k=(1—K) (g—4)/4. (39) 
The premium is positive if g is greater than 7, and negative, or a dis- 
count, if g is less than 2; 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 lower rate 2 than the rate 
of dividend g; and at a discount, if 1t is valued at a higher rate 2 than 
the rate of dividend g. 
Serial bonds.—To apply the general formula (89) 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/n 1/n 
| | | 
1 yr. 2 yrs. 
es 
| 
n yIs. 
1/n 1/n 
| 
3 ve 
The present value, K, of these n installments is clearly the value of 
an annuity of annual rent 1/n; hence 
Le Gy Ol 1: 
Substituting this value of A in formula (89), the followmg formula 
is obtained: 
k=(1-—a,/n) (g—?@)/é. (40) 
Example 19.—What is the bid on $100,000 highway 4% serial bonds maturing in 
20 equal annual installments, to net the purchaser an effective rate of 3%? 
Here n=20, g=.04, 1=.03, and ast =14.8774749; consequently 
k=(1—14.8774749/20)(.04—.03)/.03 
=(1 — .743873745) 1/3 =.256126255 & 1/3 =.085375418. 
Hence the bid on $100,000 is 
1.085375418 X $100,000 =$108 537.54. 
Extension of formulas to case when dividends are payable 
and interest is convertible m times per annum.—Formula (36) 
assumes that dividends are payable once a year and that the effective 
rate of interest is? per annum. Replacing year by vnterval 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 XK of the several 
ee 
