HIGHWAY BONDS. 105 



at the effective rate of interest i; and A, the present value of ; or bid 

 .on, the bonds. 



In the above illustration G'=100,000, and n=34. The dividend or interest per 

 annum is 5,000. Hence £=5,000/100,000=. 05. 



Returning to the general problem, the value of the bonds, so far as 

 the purchaser or holder is concerned, consists of two parts: 



1 . The annual interest, or dividend, to he received. 



2. The sum to he redeemed at the end of n years. 



Hence, to find the present value, A, of the bonds, the present value 

 of each of these parts must be determined and added together. The 

 interest per unit of the redemption price Cis, by definition, g; if the 

 interest on 1 unit is g, the interest on O units is gC. Hence at the end 

 of every year for n years the holder will receive gC units. 



Dividend Redemption payment, C 



payments gC gC gC gC 



1 yr. 2 yrs. n— 1 yrs. n yrs. 



It is evident that these interest or dividend payments of gC at the 

 end of every year constitute an immediate annuity-certain of annual 

 rent gC and term of n years. The value of such an annuity with 

 annual rent 1 is a^; hence the value of the annuity with annual 

 rent gC 'is 



gC a^, 



where a^\ is to be taken at the rate of interest i to he employed in the 

 valuation of the bonds, a rate which in general is different from g, 

 the rate of dividend. 



By formula (5), the present value of the sum C, to be redeemed 

 in n years, is v n =C. 



Adding these parts together, the result is 



A=v n C+gCa^. 



Substituting in this relation the value of a„i given by formula 

 (19), it follows that 



A=v n C+i(0-v n C). 

 Since, by definition, K=v n C, the hid is given by 



A = K+ ( j(C-K) (30) 



and the premium by 



A-C=(C-K)t<ZrJtl. (31) 



