## CHAPTER 6

**6.1 Introduction**

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ou have now learned how fixed-income securities are valued, given the yield curve. The value of the security is the present value of the future cash flows from the security. Each cash flow is discounted using the interest rate for the appropriate maturity, with these rates given by the yield curve.

Of course, if the yield curve changes, then the value of the security also changes. Over time, interest rates for different maturities typically fluctuate, and fixed-income security prices change accordingly.

Therefore, a significant source of risk that any fixed-income position is exposed to is interest rate risk. This is the case even if the portfolio, or the position, consists of all default-free fixed-income securities.

This raises the question: How do financial institutions manage, or **hedge**, interest rate risk? A **hedge** is an investment strategy that is designed to reduce the variability of a position’s value that arises from some specific source of risk, such as interest rate risk.

In this chapter, we analyze three types of hedging strategies used to control interest rate risk. Before any hedging is possible, however, you must know how a position is exposed to changes in the yield curve. At a *macro *level, for the position as a whole, exposure to interest rate risk can be estimated by first computing the net cash flows as a function of time to maturity. In a large financial institution, this can require aggregating tens of thousands of different fixed-income securities, across a large number of different trading desks at a particular time!

As a result, responsibility for hedging is often delegated to a *micro*, trading desk level. That is, each desk is responsible for managing its exposure to changes in interest rates. This practice guarantees that exposure in the aggregate is controlled, but one advantage is that too __much__ hedging may be undertaken at this level, because the activities of one individual desk can provide a partial hedge for the normal activities of another desk.

In the case of interest rate risk, the hedging strategy reduces variability in a position’s value by introducing another security, or portfolio of securities, whose value also fluctuates with changes in interest rates. This security, or portfolio, is referred to as the **hedging instrument**. When contracts of the hedging instrument, in an appropriate number (i.e., the **hedge ratio**,) are introduced into the hedged portfolio, the variability in the value of the original (unhedged) position is reduced or even eliminated.

We can classify hedges in terms of whether the hedging instrument is defined directly in terms of the underlying position being hedged, or whether the underlying position is different. For example, suppose you want to hedge a position consisting of Treasury bonds. If you hedge using Treasury bond futures, then the underlying position is the same security. If you hedge using Eurodollar futures, then the underlying position is a different but correlated security.

In the first case, when the underlying security is the same, either a **long hedge **or a **short hedge **can be put in place. The description of the hedge depends upon whether the hedging instrument is held long or short. If the underlying security is different, then the hedge is called a **cross hedge**. Most hedges are cross hedges whenever some large non homogeneous position is hedged.

When either type of hedge is used, the trader looks to reduce the price risk associated with shifts in interest rates by substituting it for some (lesser) **basis risk**. Basis risk is the difference between the spot price of the position being hedged and the price of the instrument used for hedging.

If the hedging instrument is a perfect synthetic equivalent of the underlying position, then the basis risk is zero. In fact, any discrepancy would imply that an arbitrage opportunity exists. If a perfect synthetic equivalent does not exist, which is the usual case, then basis risk exists. For example, suppose Eurodollar futures are used to hedge a position in Treasuries. The basis risk is the spot value of the Treasury position minus the Eurodollar futures price.

When both the position and the hedging instrument are well-specified, the next step is to identify the appropriate **hedge ratio**. The hedge ratio is the number of contracts of the hedging instrument required to buy or sell to construct the hedge.

Computation of the hedge ratio can be straightforward or quite complex. There are three basic approaches to hedging that can be taken. As we describe them, we will assume you want to minimize interest rate risk. Of course, this is only one possible objective. More generally, you may want to measure the effect of hedging on the return from your position and hedge only some part of the risk. In practice, the hedge portfolio chosen will depend on each institution’s preferences for interest rate risk.

Approaches to Hedging Interest Rate Risk

The following three approaches to hedging interest rate risk are discussed in more detail in this chapter.

1. Interest rate risk can be managed perfectly by constructing the exact synthetic equivalent of the underlying position. This approach is called

** cash matching**.

2. Interest rate risk can be measured, and managed, by calculating a bond’s yield to maturity and then calculating the sensitivity of the bond’s price to changes in this yield. This approach reduces the complexity of the problem, since the many interest rates that affect the value of the bond are combined into a single yield to maturity. This approach goes by the name of **duration**. Duration allows you to calculate the hedge ratio analytically.

The importance of duration for hedging interest rate risk stems from the **bond immunization theorem**, which shows that portfolios with an appropriate duration are hedged. We apply this theorem to examples using standard fixed-income securities and futures contracts.

3. The third approach to hedging interest rate risk is the **minimum-variance** approach. Here, you estimate the hedge ratio numerically from the problem of minimizing the variance of the hedged returns. This approach can be applied empirically, to sample estimates of variance and covariance, or analytically, using estimates provided from some set of valuation models.

In the next topic, we the technique of *Cash Matching*.