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Ross5eChap06sm - Chapter 6 How to Value Bonds and Stocks...

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Chapter 6: How to Value Bonds and Stocks 6.1 The price of a pure discount (zero coupon) bond is the present value of the par. Even though the bond makes no coupon payments, the present value is found using semiannual compounding periods, consistent with coupon bonds. This is a bond pricing convention. So, the price of the bond for each YTM is: a. Price = $1,000/1 + 0.025 20 = $610.27 b. Price = $1,000/1 + 0.05 20 = $376.89 c. Price = $1,000/1 + 0.075 20 = $235.41 6.2. The price of any bond is the PV of the interest payment, plus the PV of the par value. Notice this problem assumes semi-annual coupon. The price of the bond at each YTM will be: a. P = $40({1 – [1/(1 + 0.04)] 40 } / 0.04) + $1,000[1 / (1 + 0.04) 40 ] P = $1,000.00 When the YTM and the coupon rate are equal, the bond will sell at par. b. P = $40({1 – [1/(1 + 0.05)] 40 } / 0.05) + $1,000[1 /1 + 0.05 40 ] P = $828.41 When the YTM is greater than the coupon rate, the bond will sell at a discount. c. P = $40({1 – [1/(1 + 0.03)] 40 } / 0.03) + $1,000[1 /1 + 0.03 40 ] P = $1,231.15 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 an annuity , it is common to abbreviate the equations as: t r Α = ({1 – [1/(1 + r)] t } / r ) which stands for Present Value Interest Factor of an Annuity This abbreviation is 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. 6.3. Here we are finding the YTM of a semiannual coupon bond. The bond price equation is: Price = $970 = $43 20 r Α + $1,000 /1+r 20 Since we cannot solve the equation directly for r, using a spreadsheet, a financial calculator, or trial and error, we find: r = 0.04531 or 4.531% Answers to End–of–Chapter Problems B–41
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Since the coupon payments are semiannual, this is the semiannual interest rate. The YTM is the APR of the bond, so: YTM = 2 x 4.531% = 9.06% 6.4. Here we are finding the YTM of semiannual coupon bonds for various maturity lengths. The bond price equation is: P = C t r Α + $1,000/1+r t Miller Corporation bond: P 0 = $40 26 03 . 0 Α + $1,000/1+0.03 26 = $1,178.77 P 1 = $40 24 03 . 0 Α + $1,000/1+0.03 24 = $1,169.36 P 3 = $40 20 03 . 0 Α + $1,000/1+0.03 20 = $1,148.77 P 8 = $40 10 03 . 0 Α + $1,000/1+0.03 10 = $1,085.30 P 12 = $40 2 03 . 0 Α + $1,000/1+0.03 2 = $1,019.13 P 13 = $1,000 Modigliani Company bond: P 0 = $30 26 04 . 0 Α + $1,000/1+0.04 26 = $840.17 P 1 = $30 24 04 . 0 Α + $1,000/1+0.04 24 = $847.53 P 3 = $30 20 04 . 0 Α + $1,000/1+0.04 20 = $864.10 P 8 = $30 10 04 . 0 Α + $1,000/1+0.04 10 = $918.89 P 12 = $30 2 04 . 0 Α + $1,000/1+0.04 24 = $981.14 P 13 = $1,000 All else held equal, the premium over par value for a premium bond declines as maturity approaches, and the discount from par value for a discount bond declines as maturity approaches. This is called “pull to par.” In both cases, the largest percentage price changes occur at the shortest maturity lengths. Also, notice that the price of each bond when no time is left to maturity is the par value, even though the purchaser would receive the par value plus the coupon payment immediately. This is because we calculate the clean price of the bond.
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