HIGHWAY BONDS. LO5 
at the effective rate of interest 7; and A, the present value of, or bid 
on, the bonds. 
In the above illustration C=100,000, and n=34. The dividend or interest per 
annum is 5,000. Hence g=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 cnterest, or dividend, to be received. 
2. The sum to be 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 CunitsisgC. 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 
| | | i 
1 ie 2 yrs. n—1 yrs: nm 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 aq; hence the value of the annuity with annual 
rent gC is 
gC Tri 
where az is to be taken at the rate of interest 2 to be 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"™=C. 
Adding these parts together, the result is 
A=v"C'+ qCaq. 
Substituting in this relation the value of aj given by formula 
(19), it follows that 
A=vr04 20-0). 
Since, by definition, K=v"C, the bid is given by 
A=K+"%(0-K) (30) 
and the premium by 
base eG) eee (31) 
