HIGHWAY BONDS. Ill 



installments to be redeemed is calculated at the effective rate j/m 

 per interval, and the dividend per unit of the sum to be redeemed is 

 taken at the rate g/m per interval. The formula is unchanged in form 

 since m cancels out in the ratio g/m to j/m. 



General formula for valuation of bonds. — Assume that: 



1. The bonds are redeemed in r equal installments. 



2. The first redemption of bonds is made at the end of /years. 



3. The remaining r— 1 bond redemptions are made at intervals of 

 t years. 



4. The annual rate of dividend is g paid in m equal installments. 



5. The bond issue is valued at the nominal rate j (m) . 



First find the present value, A, of an issue of the above type where 

 C= 1. The value of a similar total issue of (7 is then found by mul- 

 tiplying A by 0. Since the unit fund is redeemed in r equal install- 

 ments, each one will be 1/r. 



Redemption payments 1/r 1/r 1/r 1/r 



/years t yrs. t yrs. 



The total term of the issue is seen to be/+ (r—l)t years. As in 

 preceding extension of formulas when dividends are payable and 

 interest is convertible m times per annum, apply formula (36) to each 

 installment of 1/r in the unit issue and the formula for the value of Jc, 

 the premium per unit of the total sum to be redeemed, may readily 

 be obtained. Expressed in terms of annuities, it appears as follows: 



/•• 



i 1 - a ™ {f 7a^ a ^ ~ \ y - j)lj at rate jlni - (41) 



The annuity present values in this formula must be computed at the 

 rate of interest j/m. The most common case in practice is where 

 the dividends are paid semiannually. Here m = 2, and formula (41) 

 becomes : 



k =[ 1 - a2if+ ra^ a2fl ] ( « ~ j)lj at rate jl2 ' 



(42) 



The last two formulas are very general in their application and 

 have the advantage that when employed in practical computations 

 it is necessary to consult only a table of values of a^. 



Example 20. — To find the bid on $1,100,000 highway bonds, interest 5% payable 

 semiannually, dated January 1, 1914, maturing $100,000 on January 1, 1922, 1924, 

 1926, 1928, 1930, 1932, 1934, 1936, 1938, 1940, and 1942, to net the purchaser a nomi- 

 nal rate of 4%, compounded semiannually, on his investment. 



Here /=8, t=2, r=ll, g=.05, m=2, and j (2) = .04. Accordingly, m(f+tr)=6Q, 

 m/=16, and mt=4. Substituting in formula (42), 



fc=ri- "?~ g ^ 1(.05-.04)/.04 at 2%. 



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