HIGHWAY BONDS. 



107 



Dividends payable and interest, convertible semiannually.— 



When the net income interest rate desired by the investor is nominal, 

 say j( m ), 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 j/m as 

 the effective rate of interest per interval. Hence formula (33) becomes 



{g ~ j) -aS°- (34) 



k 



m 



In particular, if the net income is ;/( 2 ), and the dividend payments 

 are semiannual, 



k= <£^Jl a m. (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 j/2. 



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='l, g==.0o, j=.0±, ?n=2, and formula (35) gives 



*= 



(.05 -.04) 2% 



a ^° =.005X5.6014309=. 0280071545. 



Hence the premium on §100,000 is $2,800.72, and the corresponding bid is 8102,800.72. 

 The progress of this bond loan is illustrated in the following schedule. 



Schedule III. 



Year. 



Book value or 

 principal at be- 

 ginning of half- 

 year. 



Semiannual 



interest of 



2%. 



Semiannual 



dividend of 



24% on 



bonds. 



Amortiza- Redemption 

 tion of pre- ] payment at 

 mium at end i end of half- 

 of half-year. year. 



i 



l 



14 

 2 



2i 

 3 



Totals 



8102, 800. 72 

 102, 356. 73 

 101, 903. 86 

 101, 441. 94 

 100, 970. 78 

 100, 490. 20 



S2, 056. 01 

 2, 047. 13 

 2, 038. 08 

 2, 028. 84 

 2, 019. 42 

 2, 009. 80 



$2,500.00 S443.99 0.00 

 2,500.00 452.87 0.00 

 2,500.00 461.92 0.00 

 2,500.00 471.16 0.00 

 2, 500. 00 480. 58 0. 00 

 2, 500. 00 490. 20 S100, 000. 00 



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 look-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. 



