130 MATHEMATICAL THEORY OF INVESTMENT income to the buyer. If such expenses have been incurred, let the total amount be denoted by b. Then the amount realized by the seller is . . ' A, - b = CV + rCa^> - b. (6) Formula (6) may be modified to correspond with any one of the formulas (2), (3), or (4), as occasion may require. EXAMPLES 1. How much must be paid for a bond of $100 with dividend rate at 6% nominal, payable January 1 and July 1 and redeemable after 20 years at par, to yield 5%*nominal, convertible semiannually ? Solution. Since the conversion interval for the interest coincides with the dividend period, the solution will be given by formula (4). The present value of the redemption price will be $100 x (1.025)-t» =-- $37.243. The total dividend paid in a year is $6, and the semiannual dividend is $3. The present value of the annuity constituted by the dividend payments is therefore $3 x1-^5)-4^ $3X25.103 = $75.309. It follows that the purchase price of the bond will be $37.248 + $75.309 = $112.55. The premium to be paid is $12.55. 2. How much must be paid for a bond of $100 with dividend rate at 5% nominal, payable January 1 and July 1 and redeemable at par in 20 years, to yield 6% nominal, convertible semiannually? 49. The computation of the premium. PROBLEM. To find the premium that must be paid on a bond purchased to yield a given rate of interest. Of course the premium is known as soon as the purchase price is known. It turns out, however, that it is easier in most cases to compute the premium directly than to compute the purchase price, so that the best way to compute the purchase price is to compute the premium. THE VALUATION OF BONDS 131 Let P denote the premium and C the par value. Then, by the notation of § 48, j ==(7+p (1_) or, P=A^-C. (2) Replacing A^ by its value as given by (1) of § 48, P=Cva+rCaw-C (3) =^^P———^} ^[(i+zy-i] J or, P=C.——(1^?——. [r-p^(l+iy--l]}. (4) P [(1+0^-1] The second factor in the last expression is exactly a^, and by referring back to (3), § 23, we see that the expression i »[CI+I")P—I] is the nominal rate, convertible p times a year, corresponding to the effective rate i. Let it be denoted by j^y as in (3'), § 23. Using a^ and j^ instead of their equivalent expressions in (4), we obtain, finally, J'=Ca<^)'(r-j(p)). (5) Formula (4) is readily adapted to logarithmic computation as soon as o^ and j, . are known. The most important case is that in which the conversion interval for interest coincides with the dividend period. In this case m=p and / ,-\-.mi 1- 1+^- -<p) \ m/ "»1 ————————————In J(l-^)'-l] "LV w J / i\~"" l-(l+^\ _l \ P/ P 3_ P =^(<^atrate^. (See (5), § 32.) 132 MATHEMATICAL THEORY OF INVESTMENT Moreover, when m = p, the expression m ^)^[a+^-i]=^[(i+^)'-i] reduces to j. Consequently, equation (4) reduces to ^^"^(^atrate^). (6) Formulas (5) and (6) apply equally well to the computation of the discount when bonds are bought below par, the only difference being that for such bonds r—j is negative. The discount is then to be considered as a sort of negative premium. EXAMPLES 1. Find the purchase price of a bond for $100 with dividend rate at 6% nominal, payable January 1 and July 1 and redeemable after 20 years at par, to yield 5% nominal, convertible semiannually. Solution. We have C = $100, r = .06, j = .05, p = m = 2, and n = 20. Hence, by formula (6), -, 100 (.06 - .05) , P = ———————' (a^| at rate .025) = .50 x 25.103 = $12.55. 2. A bond for $100 pays 6% nominal, payable January 1 and July 1, and is to be redeemed at par after 5 years. At what price must it be bought to yield 4% nominal, convertible January 1 and July 1 ? 3. Prove that after the £th interest payment has been made, the value of a bond redeemable at- par after n years is given by the formula A t = C + C "^.(a——r, at rate f}, —„ P \ '"°~1'1 P/ where C denotes the par value. 4. When bonds, redeemable on a certain date but due at a later date, are sold at a premium, their value is found on the supposition that they will be redeemed at the earlier date. Why ? HINT. The debtor can redeem the bond at par at any time after the earlier date. The question to be considered is whether it is worth more or less than par.