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Unformatted text preview: ‐year coupon bond with 10% annual coupon using forward rates and calculate its price. 1 1 The 2‐year yield is obtained using: (1+y2)2 = (1+y1) x (1+f2). If we consider a face value of 1000, the coupon payments are 100, then for the price of a 2‐year bond: [(1.05)(1.07)]1/2 ‐ 1 = 6%; FV = 1000, n = 2, PMT = 100, y2 = 6%, PV = $1,073.34 Part d. (3 points) You have purchased a bond for $973.02. The bond has a coupon rate of 6.4%, pays interest annually, has a face value of $1,000, 4 years to maturity, and a yield to maturity of 7.2%. The bond's duration is 3.6481 years. You expect that interest rates will fall by 0.3% later today. ‐ Use the modified duration to find the approximate percentage change in the bond's price. Find the approximate new price of the bond from this calculation. ‐ Do the regular present value calculations to find the bond's exact new price at its new yield to maturity. ‐ What is the amount of the difference between the two answers? Why are your answers different? Explain the reason in words and illustrate it graphically. Calculations are shown below. ‐ Find new price using modified duration: Modified duration = 3.6481/1.072 = 3.403 years. Approximate percentage price change using modified duration = ‐3.403 * (‐.0003) = 1.02%. New Price = $973.02 * 1.0102 = $982.94 ($982.96 if duration isn't rounded) ‐ Find new price by taking present value at the new yield to maturity: N=4, I=6.9%, PMT=64, FV=1000, CPT PV=983.03. ‐ The answers are different by $0.09. The reason is that using modified duration gives an approximation of the percentage change in price. It should only be used for small changes in yields because of bond price convexity. As you move farther away from the original yield, the slope of the straight line that shows the duration approximation no longer matches the slope of the curved line that shows the actual price changes. Part e. (3 points) Assume that you have a portfolio with duration of 3.2 and convexity of 6.7282. The current yield curve is flat at 2%. Construct a portfolio of zero coupon bonds with maturities of 1 and 2 years and cash, such that it matches the duration and convexity of your portfolio. 1 2 Denote the weights of the 1 and 2‐year zero coupon bonds by w1 and w2. Their durations are 1 and 2 respectively....
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