# Hedging the duration and convexity of a bond portfolio

I'm trying to work through this homework question, but not sure how to approach it.

You recently took over as the manager of a bond portfolio. Your total assets under management – all consisting of bonds – amount to $554 million dollars. At a recent risk committee meeting you committed to reducing the duration of your portfolio to 2 years and to reducing convexity to 30. Currently the portfolio has a modified duration of 4.78 years and convexity of 54. Rather than trade the underlying bonds to achieve these risk targets, you decide to risk manage the bond portfolio with derivatives. The following two derivative contracts are available: 1. Interest Rate Swap, notional \$1 million, duration 5 years, convexity 10.
2. Interest Rate Swaption, notional \\$1 million, duration 0.5 years, convexity 74.

Calculate the number of derivative contracts of each type you need to buy in order to achieve the duration and convexity risk targets.

I know I can achieve a perfect hedge for duration and complexity by applying the Taylor approximation for the bond price:

$$\Delta B_0 \approx -D \times \Delta y \times B_0 + \frac{1}{2} \times C \times (\Delta y)^2 \times B_0$$

I'm not sure how I can apply this here to achieve the target duration and complexity.

• Forget the formula. Find out how much duration (-2.78 on 554 million right?) and how much convexity (-24 right?) you need to sell to reach the target. Then figure out a package of the two derivatives that provides this duration and this convexity (two equations, two unknowns). – Alex C Nov 14 '19 at 14:33