108 



BULLETIN 136, L T . S. DEPARTMENT OF AGRICULTURE. 



Example 18. — What is the bid on $100,000 highway 3% 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, f/=.03, j=.04, m=2, and formula (35) gives 



(.03-.04) „2% 

 t= h "61 



■ .005X5.6014309= - .0280071545. 



Hence the discount on $100,000 is $2,800.72, and the corresponding bid is $97,199.28. 

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



Schedule IV. 



Year. 



Book value or 



principal at 



Beginning of 



half-year. 



Semiannual in- 

 terest of 2%. 



Semiannual 



dividend of 



1-t% on bonds. 



Accumulation 

 of discount 

 at end of 

 half-year. 



Redemption 

 payment 

 at.end of 

 half-year. 



1 

 1 



n 



2 



01 



3 



$97, 199. 28 

 97, 643. 27 

 98,096. It 



98, 558. 06 



99, 029. 22 

 99, 509. 80 



|1, 943. 99 

 1,952.87 

 1,961.92 

 1, 971. 16 

 1, 980. 58 

 1,990.20 



$1, 500. 00 

 1, 500. 00 

 1, 500. 00 

 1, 500. 00 

 1, 500. 00 

 1, 500. 00 



$443. 99 

 452. 87 

 461.92 

 471.16 

 480. 58 

 490. 20 



0.00 

 0.00 

 0.00 

 0.00 

 0.00 

 $100, 000. 00 



Totals 



590, 035. 77 



11,800.72 9,000.00 



2, 800. 72 



100, 000. 00 



In this case the holder has an initial investment of $97,199.28, and 

 at the end of the first half-year 2 per cent interest, or $1,943.99, is due. 

 The dividend payment of $1,500.00, then made on the bonds, is not 

 sufficient to provide for this interest, and the difference of $443.99 is 

 added to the principal and determines the booTc value at the beginning 

 of the second half-year. This is called the accumulation or writing 

 on of discount. By continuing this process for three years the 

 entire discount of $2,800.72 is written on the initial principal, and 

 the book value, $100,000, is then redeemed. The totals of the several 

 columns may be used to check the numerical work. 



Valuation of bonds redeemed in installments. — For the 

 valuation of bonds which are not redeemed in one sum, but in a series 

 of installments, first consider the simpler case where the dividend 

 payments are annual and the rate of interest is the effective rate i. 

 Let C 1} C 2 , .... C r , denote the successive installments by which 



the bonds are to be redeemed; 

 n t , n 2 , .... n r , the respective number of years after 



which the successive installments 

 become due; 

 K 1} K 2 , .... K r , the present values, at the effective rate 



of interest i, of 



C x due n x }7"ears hence, 

 C 2 due n 2 years hence, 



C r due n r years hence; 



