HIGHWAY BONDS. 109 



g, the fixed rate of dividend to be paid on 



the outstanding bonds ; 

 i, the effective rate of interest employed 



in the valuation of the bonds, which 

 is the net income rate to the pur- 

 chaser; 



and A x , A 2 , Ay, the present values, at the effective rate 



i, of the separate installments with 

 their respective dividends. 



L/ x L 2 C j. 



n x yrs. n 2 yrs. n r yrs. 



Each installment redeemed may be regarded as furnishing a distinct 

 problem under formula (30) so that, in order to value the entire bond 

 issue, it may be treated as made up of r distinct issues and, after 

 finding the value of each one, they may be added together for the value 

 or bid on the total issue. 



By formula (30) in the case of a single issue of C x at net income rate 

 i, dividend rate g, due in n x years, the present value, or bid, A x , is: 



A x =K x + (g/i) (C\-K x ). 

 Similarly, A 2 = K 2 + (g/i) ( C 2 - IQ, 



A r =K r +(g/i) (C r -IQ. 

 Adding, 



(A x + A 2 + . . . .+A r ) = (K x + K 2 + . . . . + IQ 



+ (g/i)[(0 1 +0 2 + . . . . + C r )-(K X +K 2 + . . . . + IQI 



The total sum to be redeemed, C t +C 2 + . . . . + C r , is denoted by C; 

 the total present value of C x in n x years, C 2 in n 2 years, and so on, 

 which by definition is equal to Zj + Z 2 + . . . . + K r , by K; and the 

 total value of the issue, A x + A 2 + . . . .+A r . by A; then for the bid 

 there results 



A = K+(g/i) (C-K), (36) 



and for the premium, 



A-C=(C~K) (g-i)/i. (37) 



It thus appears that formulas (30) and (31), which were derived before 

 for the case of a bond issue redeemed in one sum, hold also for the 

 more general issue redeemed in any number of installments. 

 Installment bonds when total sum to be redeemed is 1.— 



When 1 is the total sum to be redeemed, that is, when C=\, formula 

 (37) becomes 



A-l = {l-K) {g-i)li, (38) 



