Setting Bond Portfolio Duration and Convexity with Derivatives

Summary

The document asks how to use interest rate swaps and swaptions to adjust a bond portfolio to target duration and convexity. It gives the portfolio’s assets under management and current risk measures, along with each derivative contract’s notional, duration, and convexity. The requested result is the number of contracts to buy to reach both targets.

The poster recognizes that duration and convexity enter a second-order approximation of bond price changes as yields move, but is unsure how to apply that approximation to the position-sizing problem. The natural setup is to express the portfolio’s target risk exposures as equations, then add the exposures contributed by the derivative positions and solve for the contract quantities. However, the document supplies no solution, sign convention, or worked calculation. The figures and contract sensitivities belong to this exercise; applying them in practice would require checking instrument direction and how the quoted risk measures are defined.

Key ideas

• The exercise asks for derivative quantities that move a bond portfolio to specified duration and convexity targets.
• It provides current portfolio risk measures and sensitivities for two available contracts.
• The Taylor approximation describes how duration and convexity contribute to price changes as yields move.
• The document does not solve for contract quantities or specify the sign convention.

Tags

# Hedging the duration and convexity of a bond portfolio


# 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:

- Interest Rate Swap, notional \$1 million, duration 5 years, convexity 10.

- 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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.