1.Duration and Convexity

Your research department reports continuously compounded interest rates as

Maturity (Years) 0.5, 1.0, 1.5, 2.0

Interest Rate (%) 1.00, 1.50, 2.00, 2.00

(a) Use these rates to compute the prices P_z(0,1) and P_z(0,2) of one- and two-year zero coupon bonds, and the price P_c(0,2) of a two-year, 3% coupon bond. Coupons are paid semi-annually, and the face value of all bonds is 100.

(b) Obtain the coupon bond's duration and convexity.

(c) Suppose that the monthly changes in the interest rates have a mean of zero and a standard deviation of 0.5%. Obtain the monthly 95% Value at Risk and Expected Shortfall on the coupon bond.

(d) Construct a hedge portfolio of 1 coupon bond and k one-year zero coupon bonds that has zero duration. What is the value of k? What is the convexity of the hedge portfolio?

(e) Construct a hedge portfolio of 1 coupon bond, and k₁ one-year zero coupon bonds and k2 two-year zero coupon bonds, that has zero duration and convexity. What are the values of k₁ and k₂?

(f) Suppose that the yield curve shifts upward with dr = 1%. Recalculate P_c(0,2), P_z(0,1) and P_z(0,2) and use this to calculate the change in the values of the hedge portfolios constructed in (d) and (e). Comment on the result.