A 12.75-year maturity zero-coupon bond selling at a yield to maturity of 8% (effective annual yield) has convexity of 150.3 and modified duration of 11.81 years. A 30-year maturity 6% coupon bond making annual coupon payments also selling at a yield to maturity of 8% has nearly identical duration - 11.79 years-but considerably higher convexity of 231.2. Assume throughout that the face value is $1,000. For parts (a)-(d) you should use discrete time results. Part (e) deals with continuous time. (a) Suppose the yield to maturity on both bonds increases to 9%. (i) What will be the actual percentage capital loss on each bond? (ii) What percentage capital loss would be predicted by the duration-with-convexity rule? (b) Repeat part (a), but this time assume the yield to maturity decreases to 7%. (c) Compare the performance of the two bonds in the two scenarios, one involving an increase in rates, the other a decrease. Based on the comparative investment performance, explain the attraction of convexity. (d) In view of your answer to part (c), do you think it would be possible for two bonds with equal duration but different convexity to be priced initially at the same yield to maturity if the yields on both bonds always increased or decreased by equal amounts, as in this example? (Hint: Would anyone be willing to buy the bond with lower convexity under these circumstances?) (e) All of the above calculations are based on a discrete-time model. Suppose now that the yields (8% for zero-coupon bond and 6% for the 30 year coupon bond) are continuously compounded and that the maturity of the zero-coupon bond is 13 years. Using the definition explained in class, compute the duration and convexity of each bond. A 12.75-year maturity zero-coupon bond selling at a yield to maturity of 8% (effective annual yield) has convexity of 150.3 and modified duration of 11.81 years. A 30-year maturity 6% coupon bond making annual coupon payments also selling at a yield to maturity of 8% has nearly identical duration - 11.79 years-but considerably higher convexity of 231.2. Assume throughout that the face value is $1,000. For parts (a)-(d) you should use discrete time results. Part (e) deals with continuous time. (a) Suppose the yield to maturity on both bonds increases to 9%. (i) What will be the actual percentage capital loss on each bond? (ii) What percentage capital loss would be predicted by the duration-with-convexity rule? (b) Repeat part (a), but this time assume the yield to maturity decreases to 7%. (c) Compare the performance of the two bonds in the two scenarios, one involving an increase in rates, the other a decrease. Based on the comparative investment performance, explain the attraction of convexity. (d) In view of your answer to part (c), do you think it would be possible for two bonds with equal duration but different convexity to be priced initially at the same yield to maturity if the yields on both bonds always increased or decreased by equal amounts, as in this example? (Hint: Would anyone be willing to buy the bond with lower convexity under these circumstances?) (e) All of the above calculations are based on a discrete-time model. Suppose now that the yields (8% for zero-coupon bond and 6% for the 30 year coupon bond) are continuously compounded and that the maturity of the zero-coupon bond is 13 years. Using the definition explained in class, compute the duration and convexity of each bond
