HIGHWAY BONDS. 109 
9, the fixed rate of dividend to be paid on 
the outstanding bonds; 
a, the effective rate of interest employed 
in the valuation of the bonds, which 
is the net income rate to the pur- 
chaser; 
>i Gad es the present values, at the effective rate 
i, of the separate installments with 
ther respective dividends. 
C, G C, 
sige | | 
N, YES. Day WARS N, yrs. 
Each installment redeemed may be regarded as furnishing a distinct 
problem under formula (30) so that, in order to value the entire bond 
issue, it may be treated as made up of r distinct issues and, after 
_ finding the value of each one, they may be added together for the value 
or bid on the total issue. 
By formula (30) in the case of a single issue of C, at net income rate 
2, dividend rate g, due in 7, years, the present value, or bid, A,, 1s: 
7 — HG (g/t) (EC, — Ke). 
Similarly, A,= K,+ (g/t) ((,— K,), 
Adding, 
Cees eA) (IG eK) 
+(g/)[(C,+OQ,+....4+¢6)-(K,+h4+....4+4K,)]. 
The total sum to be redeemed, C,+C,+.. ..+0,, is denoted by C; 
the total present value of C, in n, vears, C, in n, years, and so on, 
_ which by definition is equal to K,+K,+....+K,, by K; and the 
total value of the issue, A,+A,+....+A,, by A; then for the bid 
there results 
A = K+ (g/t) (C— 4K), (36) 
and for the premium, 
A-—C=(C-K) g-b/é. (37) 
It thus appears that formulas (30) and (31), which were derived before 
for the case of a bond issue redeemed in one sum, hold also for the 
more general issue redeemed in any number of installments. 
Installment bonds when total sum to be redeemed is 1.— 
When 1 is the total sum to be redeemed, that is, when O=1, formula 
(37) becomes 
A—1=(1—-K) (g—-#)/é, (38) 
