107 
HIGHWAY BONDS. 
Dividends payable and interest convertible semiannually.— 
When the net income interest rate desired by the investor is nominal, 
SAY Jum), and the dividends per unit of the sum to be redeemed are 
paid in m equal installments, g/m, during the year, it is evident that 
it is a case of m times n intervals with g/m as dividend and 7/m as 
the effective rate of interest per interval. Hence formula (33) becomes 
=P gi 
jlm% 
(34) 
In particular, if the net income is 7), and the dividend payments 
are semiannual, 
I12% ‘ 
Wan | = (35) 
This formula provides for the valuation of all bonds, redeemed in one 
sum at the end of a term of n years and with semiannual dividends. 
Particular attention is called to the fact that the annuity must be 
taken for the term 2n, and at the rate of interest 7/2. 
nai 
jp Ee. 
Example 17.—What is the bid on $100,000 highway 5% bonds maturing at the end 
of 3 years, interest payable semiannually, to net purchaser a nominal rate of 4% con- 
vertible half-yearly? 
Here n=3, g=.05, 7=.04, m=2, and formula (35) gives 
p= SG =? = 005 X5.6014309=.0280071545. 
Hence the premium on $100,000 is $2,800.72, and the corresponding bid is $102,800.72. 
The progress of this bond loan is illustrated in the following schedule. 
ScHEDULE III. 
Book value or Sep ANTE | Semiannual Amortiza- | Redemption 
oar principal at be- saaterest of dividend of | tionofpre- | payment at 
; ginning of hali- 907 227% on | mium at end end of half- 
year. 10° bonds. | of half-year. year. 
4 $102, 800. 72 $2, 056. 01 $2, 500. 00 $443. 99 0.00 
1 102, 356. 73 2, 047.13 2, 500. 00 452. 87 0. 00 
13 101, 903. 86 2, 038. 08 2, 500. 00 461. 92 0. 00 
2 101, 441. 94 2, 028. 84 2, 500. 00 471.16 0. 00 
2A 100, 970. 78 2, 019. 42 2, 500. 00 480. 58 0. 00 
3 100, 490. 20 2,009.80 | 2,500.00 490. 20 $100, 000. 00 
m) ‘Totals 609, 964. 23 12, 199. 28 15, 000. 00 2 800. 72 100, 000. 00 
_ At the outset the holder has an investment of $102,800.72 upon 
_ which, at 2 per cent, at the end of the first half-year, $2,056.01 interest 
is due; the dividend payment of $2,500.00 then made on the bonds 
provides for this interest and a balance of $443.99 remains, which is 
applied to amortize or write off the premium so that the book-value, or 
invested principal, is reduced to $102,356.73 at the beginning of the 
second half-year. This process continues for three years until the 
entire premium of $2,800.72 is written off and the bonds are redeemed 
by the payment of $100,000. The various columns are added and 
_ the checks upon these totals are obvious. 
