106 BULLETIN 136, U. S. DEPARTMENT OF AGRICULTURE. 
If in formula (31) the total sum to be redeemed is regarded as unity, 
then C=1 and K=v", the present value of 1 due in n years, and 
there results 
A=14 9294) = 14 G-Day, (32) 
In this formula az is taken at 7 per cent, and gives the bid ona 
bond where the sum to be redeemed is 1. Denoting the excess of A 
over 1 by k, which is called the premium, formula (82) becomes 
k=(g—-t)a (33) 
n\? 
where the 7 per cent over the symbol az means that the function is 
to be taken from the 7 per cent annuity table. 
This is the fundamental formula in bond calculations. It admits 
of a very simple interpretation, for it states that the premium on a 
bond is equal to the present value of an n year annuity at 2 per cent 
whose annual rent is the excess (g—7) of the nominal rate of dividend 
of the bond over the effective rate of interest 7, desired to be realized 
by the investor. 
Unit redeemed, 1 
U v 
Unit invested, 1 1 
Leys. 2 yrs. n—1 yrs. n yrs. 
| | | 
l gJ a v g is, g : v 
The dividend paid each year on each unit of the bond to be redeemed 
is g, which may be divided into two parts,7 and g—7. For the first 
part the investor pays 1 and in return receives interest of 7 each year 
and the 1 is redeemed at the end of n years. For the second part the 
investor pays the premium, k=(g—1)aq, and this is repaid, both 
principal and interest at rate 7, in n equal annual installments of 
(g—v). A portion ‘of each installment goes toward the repayment of 
the premium k which is eventualty reduced to zero. This is called 
the amortization or writing off of the premium. 
It is thus seen that, if & is positive, the bond is bought at a premium; 
and if & is negative, it is bought at a discount. Since az is always 
positive, it appears from formula (33) that the sign of k will be posi- 
tive when g is greater than 7, or when the rate of dividend is greater 
than the rate of interest used in valuation; conversely, when g is less 
than 2, & 1s negative. 
Example 16.—To find the bid on the highway bond mentioned on page 104, on 
the hypothesis that the purchaser wishes to realize 3% on his investment. 
Consider a dollar (unit) of the loan C=100,000. Here n=34, g=.05, 1=.03, and by 
formula (33), 
Premium, ke g 
k=(.05— 03)a34° = .0221.1318367 =.422636734, 
or the premium is slightly over 42 cents on the dollar. Since for each dollar of the 
loan the purchaser must pay $1.422636734, for the whole loan of $100,000 he must pay 
1.422636734 X $100, 000=$142 263.67. 
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