HIGHWAY BONDS. 113 
Example 22.—To find sg at 14% when the limit of the tables is 60 years or terms. 
Applying formula (47) there results 
Gon) 1 (1.015) (1.015) =1 
Soa] 60-434) 015 015 
__2.4432198 X 1.6589964 — 1 
=203 .552 
015 03 .5528568 
By formula (48) i, 
Saq)=Seo-¢aq}— (1.015): tS alate S3q] 
=1.6589964  96.2146517 +-43.93309 15 =203.5528523. 
The correct value of sgy at 15% to seven places of decimals is 203.5528497; so the 
above method may be regarded as giving the correct value to about five places of 
decimals. In most practical cases this will be sufficiently accurate. 
Valuation of serial bonds bearing semiannual dividends.— 
The most common type of serial bond bears semiannual dividends 
and is redeemed in equal annual installments, the first of which is 
paid at the end of the first year. Formula (42) lends itself directly 
to the valuation of this bond at a nominal rate of interest 7 convert- 
ible twice a year. In this case f=t=1, r=n, and 
" Ama — AF ONE ks ° 
k=| 1 — tra“ |g —J)/j at rate 7/2. (49) 
Formula (49) requires the use of a table of values of aq only. It 
can be put in another convenient form for computation involving 
the use of a table of values of aq and sq. For, by formula (46), 
Oar = Ay + V'dy, and, since v’/aq=1/(1 + 1)*aq=1/sq, after a simple 
reduction, there results 
| 1 ~ a lg jj at vate J/2. (50) 
Example 23.—$300,000 highway serial bonds bearing 4% interest payable semi- 
annually, dated January 1, 1914, mature $100,000 January 1, 1915, 1916, and 1917. 
What price should be paid to realize a net income of 3% compounded semiannually? 
Here n=3, g=.04, j(.)=.03, and by formula (49) 
az a5 F 
r—[1-“S | (.04—.03)/.03 at 14% 
305) 
= .0575373 X 1/3=.0191791, 
therefore the price to earn 3% compounded semiannually is 
1.0191791 X$300,000=$305, 753.73. 
The following schedule illustrates the progress of this loan. 
52448°—15——8 
