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



If in formula (31) the total sum to he redeemed is regarded as unity, 

 then 0=1 and K=v n , the present value of 1 due in n years, and 

 there results 



A = 1 + (1 ~*'" ) (ff -i) = l+(g- i) am • (32) 



In this formula a^ is taken at i per cent, and gives the bid on a 

 bond where the sum to be redeemed is 1. Denoting the excess of A 

 over 1 by Jc, which is called the premium, formula (32) becomes 



k=((/-i)a^, (33) 



where the i per cent over the symbol a^ means that the function is 

 to be taken from the i 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 i per cent 

 whose annual rent is the excess (g — i) of the nominal rate of dividend 

 of the bond over the effective rate of interest i, desired to be realized 



by the investor. 



Unit redeemed, 1 



Unit invested, 1 i i i i 



1 yr. . 2 yrs. n — 1 yrs. n yrs. 



Premium, /<• g — i g — i g—i g — i 



The dividend paid each year on each unit of the bond to be redeemed 

 is g, which may be divided into two parts, i and g — i. For the first 

 part the investor pays 1 and in return receives interest of i each year 

 and the 1 is redeemed at the end of n years. For the second part the 

 investor pays the premium, lc=(g — i)a^, and this is repaid, both 

 principal and interest at rate i, in n equal annual installments of 

 (g — i). A portion of each installment goes toward the repayment of 

 the premium Tc which is eventually reduced to zero. This is called 

 the amortization or writing off of the 'premium. 



It is thus seen that, if lc is positive, the bond is bought at a premium ; 

 and if £ is negative, it is bought at a discount. Since a^ is always 

 positive, it appears from formula (33) that the sign of ~k will be posi- 

 tive when g is greater than i, or wlien the rate of dividend is greater 

 than the rate of interest used in valuation; conversely, when g is less 

 than i, Jc is negative. 



Example 16. — To find the bid on the highway bond mentioned on page 104, on 

 the hypothesis that the purchaser wishes t< i realize 3 % < m his investment. 



Consider a dollar (unit) of the loan = 100,000. Here ?i=34. <7 = .05, i=.03. and by 

 f ormula ( 33 i . 



/fe=(.05-.03)agj' = .02X21.1318367=.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 8100.000 he must pay 



1.422636734X1100, 000=$142.263. 67. 



