HIGHWAY BONDS. 100 We 
installments to be redeemed is calculated at the effective rate 7/m 
per interval, and the dividend per unit of the sum to be redeemed is 
taken at the rate g/m per interval. The formula is unchanged in form 
since m cancels out in the ratio g/m to 7/m. 
General formula for valuation of bonds.—Assume that: 
1. The bonds are redeemed in r equal installments. 
2. The first redemption of bonds is made at the end of f years. 
3. The remaining r—1 bond redemptions are made at intervals of 
t years. 
4. The annual rate of dividend is g paid in m equal installments. 
5. The bond issue is valued at the nominal rate jm). 
First find the present value, A, of an issue of the above type where 
O=1. The value of a similar total issue of Cis then found by mul 
tiplying A by @ Since the unit fund is redeemed in r equal install- 
ments, each one will be 1/r. 
Redemption payments 1/r 1/r ye Wie 
jf years | t yrs. | t yrs. 
The total term of the issue is seen to be f+ (r—1)t years. As in 
preceding extension of formulas when dividends are payable and 
interest is convertible m times per annum, apply formula (36) to each 
installment of 1/r in the unit issue and the formula for the value of &, 
the premium per unit of the total sum to be redeemed, may readily 
be obtained. Expressed in terms of annuities, it appears as follows: 
_ Am (F+try| — Amf| aN ie 2 7 
(es E a Co re At Jeg DJ at rate Jj/m. (41) 
The annuity present values in this formula must be computed at the 
rate of interest 7/m. The most common case in practice is where 
the dividends are paid semiannually. Here m=2, and formula (41) 
becomes: 
E= ) 1 — See Neg —j)[j at rate j/2. (42) 
The last two formulas are very general in their application and 
have the advantage that when employed in practical computations 
it is necessary to consult only a table of values of ap. 
Example 20.—To find the bid on $1,100,000 highway bonds, interest 5% payable 
semiannually, dated January 1, 1914, maturing $100,000 on January 1, 1922, 1924, 
1926, 1928, 1930, 1932, 1934, 1936, 1938, 1940, and 1942, to net the purchaser a nomi- 
nal rate of 4%, compounded semiannually, on his investment. 
Here j—s, t—2, r—11, g=.05, m—2, and 7(.,—.04. Accordingly, m(/-+-ir)=60, 
mf=16, and mt=4. Substituting in formula (42), 
\ 60 at Oo 
is 4 eee (.05 —.04)/.04 at 2%. 
