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Unformatted text preview: ill sell at a premium, since it provides periodic income in the form of coupon payments in excess of that required by investors on other similar bonds. If the coupon rate is lower than the required return on a bond, the bond will sell at a discount since it provides insufficient coupon payments compared to that required by investors on other similar bonds. For premium bonds, the coupon rate exceeds the YTM; for discount bonds, the YTM exceeds the coupon rate, and for bonds selling at par, the YTM is equal to the coupon rate. Current yield is defined as the annual coupon payment divided by the current bond price. For premium bonds, the current yield exceeds the YTM, for discount bonds the current yield is less than the YTM, and for bonds selling at par value, the current yield is equal to the YTM. In all cases, the current yield plus the expected oneperiod capital gains yield of the bond must be equal to the required return. c. 17. A longterm bond has more interest rate risk compared to a shortterm bond, all else the same. A low coupon bond has more interest rate risk than a high coupon bond, all else the same. When comparing a high coupon, longterm bond to a low coupon, shortterm bond, we are unsure which has more interest rate risk. Generally, the maturity of a bond is a more important determinant of the interest rate risk, so the longterm, high coupon bond probably has more interest rate risk. The exception would be if the maturities are close, and the coupon rates are vastly different. 217 Solutions to Questions and Problems NOTE: All endofchapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem. NOTE: Most problems do not explicitly list a par value for bonds. Even though a bond can have any par value, in general, corporate bonds in the United States will have a par value of $1,000. We will use this par value in all problems unless a different par value is explicitly stated. Basic 1. The price of a pure discount (zero coupon) bond is the present value of the par. Remember, even though there are no coupon payments, the periods are semiannual to stay consistent with coupon bond payments. So, the price of the bond for each YTM is: a. P = $1,000/(1 + .05/2)20 = $610.27 b. P = $1,000/(1 + .10/2)20 = $376.89 c. P = $1,000/(1 + .15/2)20 = $235.41 2. The price of any bond is the PV of the interest payment, plus the PV of the par value. Notice this problem assumes a semiannual coupon. The price of the bond at each YTM will be: a. P = $35({1 – [1/(1 + .035)]50 } / .035) + $1,000[1 / (1 + .035)50] P = $1,000.00 When the YTM and the coupon rate are equal, the bond will sell at par. b. P = $35({1 – [1/(1 + .045)]50 } / .045) + $1,000[1 / (1 + .045)50] P = $802.38 When the YTM is greater than the coupon rate, the bond will sell at a discount. c. P = $35({1 – [1/(1 + .025)]50 } / .025) + $1,000[1 / (1 + .025)50] P = $1,283.62 When the YTM is less than the coupon rate, the bond will sell at a premium. We would like to introduce shorthand notation here. Rather than write (or type, as the case may be) the entire equation for the PV of a lump sum, or the PVA equation, it is common to abbreviate the equations as: PVIFR,t = 1 / (1 + r)t which stands for Present Value Interest Factor 218 PVIFAR,t = ({1 – [1/(1 + r)]t } / r ) which stands for Present Value Interest Factor of an Annuity These abbreviations are short hand notation for the equations in which the interest rate and the number of periods are substituted into the equation and solved. We will use this shorthand notation in the remainder of the solutions key. 3. Here we are finding the YTM of a semiannual coupon bond. The bond price equation is: P = $1,050 = $39(PVIFAR%,20) + $1,000(PVIFR%,20) Since we cannot solve the equation directly for R, using a spreadsheet, a financial calculator, or trial and error, we find: R = 3.547% Since the coupon payments are semiannual, this is the semiannual interest rate. The YTM is the APR of the bond, so: YTM = 2 × 3.547% = 7.09% 4. Here we need to find the coupon rate of the bond. All we need to do is to set up the bond pricing equation and solve for the coupon payment as follows: P = $1,175 = C(PVIFA3.8%,27) + $1,000(PVIF3.8%,27) Solving for the coupon payment, we get: C = $48.48 Since this is the semiannual payment, the annual coupon payment is: 2 × $48.48 = $96.96 And the coupon rate is the annual coupon payment divided by par value, so: Coupon rate = $96.96 / $1,000 = .09696 or 9.70% 5. The price of any bond is the PV of the interest payment, plus the PV of the par value. The fact that the bond is denominated in euros is irrelevant. Notice this problem assumes an annual coupon. The price of the bond will be: P = €84({1 – [1/(1 + .076)...
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 Spring '10
 eshmalwi
 Finance, Corporate Finance

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