Managing Yield-Curve Risk with Combination Hedges

The method presented here to evaluate and regulate the risk exposure of default-free bonds generalizes the traditional duration concept by taking nonparallel changes in the term structure of interest rates into account. The article shows how to immunize a default-free bond by hedging with two standard hedging instruments (a combination hedge) and how to replicate a diversified bond portfolio by using a few standard hedging instruments. This standard hedging instrument representation can provide comprehensive information about the risk structure of the portfolio, and trading the standard hedging instruments can help fine-tune a position. Managing the yield-curve risk of bond or swap portfolios consists of two tasks: (1) quantifying the risk exposure of the portfolio with respect to a change in the yield curve and (2) adjusting the risk in a predetermined way (i.e., the hedging problem). Useful methods for attacking the first problem, evaluating the risk exposure, are the three-factor approach based on a principal-components analysis of historical data, the concept of key-rate durations, and value at risk. But these methods are less helpful for solving the hedging problem; thus, many market participants (and electronic trading and information systems) still use the traditional duration concept to calculate a risk measure (basis point value) and hedge quantities. The duration concept has the severe weakness, however, that it considers only parallel yield-curve shifts. The approach proposed here enables traders and portfolio managers to quantify and regulate the risk in a single bond investment or a bond portfolio. The approach generalizes the duration concept by introducing a flexible model for nonparallel shifts of the yield curve, which are triggered by the yield changes of standard hedging instruments—for example, bond futures. In this article, I derive a combination hedge for a bond consisting of two neighboring standard hedging instruments with maturities, respectively, shorter than and longer than the maturity of the bond. For example, a 7-year bond would be hedged with a certain ratio of a 5-year hedging instrument and a 10-year instrument. The difference between the duration hedge and the hedge ratios based on this approach is that the yield spreads come into play in this approach, which immunizes against slope changes in the yield curve. An empirical study performed with several German government bonds shows that hedging a single bond with two instruments results in a better hedge than hedging with only one instrument. Furthermore, the analysis demonstrates that the hedge ratios can be used to price a single bond and thereby assess whether the market price of the bond allows arbitrage. A portfolio consisting of bonds whose maturities lie between the maturities of two standard hedging instruments can be hedged with those two instruments. The notional amount is given by the sum of the hedge quantities for a single bond. To hedge a portfolio of bonds with a wide range of maturities, three or more instruments typically have to be considered. I show how to represent a portfolio of default-free bonds by a set of at least two standard hedging instruments—that is, standard hedging instrument representation (SHIR). A historical analysis demonstrates that the profit/loss of a diversified portfolio consisting of German government bonds with maturities between 2 and 10 years is approximately equal to the P/L of the SHIR consisting of a 2-year, a 5-year, and a 10-year standard hedging instrument. The SHIR of the portfolio serves as a risk measure because the risk exposures of the standard hedging instruments are known from their durations or from other methods. The SHIR nicely represents what yield-curve movement traders and portfolio managers speculate on, and it enables them to fine-tune their positions by trading the liquid standard hedging instruments.