HIGHWAY BONDS. 113 



Example 22. — To find ggj] at \\% when the limit of the tables is 60 years or terms. 

 Applying formula (47) there results 



_ (1.015) 60+3t -l _ (1.015) 60 X (1.015) 34 -! 

 *m|-«8o+»|- Q15 _ 015 



= 2.4432198X1.6589964-1 =2035528568 



.015 

 By formula (48) 



S 9i| = S 60+3i| = (l-015)' 'Sgo.+ SgJl 



=1.6589964X96.2146517+43.9330915=203.5528523. 



The correct value of Sgji at \\% 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 j convert- 

 ible twice a year. In this case /=£=!, r = n, and 



fc -[ 1 " a? W 3 >-^ at rate J/2. 



(49) 



Formula (49) requires the use of a table of values of a^ only. It 

 can be put in another convenient form for computation involving 

 the use of a table of values of a^ and ssj. For, by formula (46), 

 Gtem = o,s\ + v 2 a^ b and, since v 2 /a~2i = 1/(1 + ifa-% = 1/s-ij, after a simple 

 reduction, there results 



fc = [l-|||](<7-i)/i at rate J/2. (50) 



Example 23. — $300,000 highway serial bonds bearing 4% interest payable semi- 

 annually, dated January 1, 1914, mature 8100,000 January 1, 1915, 1916, and 1917. 

 What price should be paid to realize a net income of 3% compounded semiannually? 



Here n=3, gr=.04, j( 2 )=-03, and by formula (49) 



* = [ 1- h£~] (- 04 -- 03 )/- 03 at H% 



= .0575373 X 1/3= .0191791, 



therefore the price to earn 3% compounded semiannually is 



1.0191791XS300,000=$305,753.73. 



The following schedule illustrates the progress of this loan. 

 52448°— 15 8 



