# Duration Hedging

Hedging Interest-Rate Risk with Duration

Before implementing any kind of hedging method against the interest-rate risk, we need to understand how bond prices change, given a change in interest rates. This is critical to successful bond management.

5.1 Basics of Interest-Rate Risk: Qualitative Insights

The basics of bond price movements as a result of interest-rate changes are perhaps best summarized by the ﬁve theorems on the relationship between bond prices and yields. As an illustration (see Table 5.1), let us consider the percentage price change for 4 bonds with different annual coupon rates (8% and 5%) and different maturities (5 years and 25 years), starting with a common 8% yield-to-maturity (YTM), and assuming successively a new yield of 5%, 7%, 7.99%, 8.01%, 9% and 11%.

From this example, we can make the following observations. Using the bond valuation model, one can show the changes that occur in the price of a bond (i.e., its volatility), given a change in yields, as a result of bond variables such as time to maturity and coupon, and show that these observations actually hold in all generalities. For now, we simply state these “theorems.” More detailed comments about these elements will follow. We leave the proof of these theorems as an exercise to the mathematically oriented reader.

Table 5.1 Percentage Price Change for 4 Bonds, Starting with a Common 8% YTM. New yield (%)

Change (bps)

8%/25 (%)

8%/5 (%)

5%/25 (%)

5.00

7.00

7.99

8.01

9.00

11.00

−300

−100

−1

+1

+100

+300

42.28

11.65

0.11

−0.11

−9.82

−25.27

12.99

4.10

0.04

−0.04

−3.89

−11.09

47.11

12.82

0.12

−0.12

−10.69

−27.22

5%/5 (%)

13.61

4.29

0.04

−0.04

−4.07

−11.58

5.1.1 The Five Theorems of Bond Pricing

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Bond prices move inversely to interest rates. Investors must always keep in mind a fundamental fact about the relationship between bond prices and bond yields: bond prices move inversely to market yields. When the level of required yields demanded by investors on new issues changes, the required yields on all bonds already outstanding will also change. For these yields to change, the prices of these bonds must change. This inverse relationship is the basis for understanding, valuing and managing bonds.

•

Holding maturity constant, a decrease in rates will raise bond prices on a percentage basis more than a corresponding increase in rates will lower bond prices. Obviously, bond price volatility

164

Fixed Income Securities

can work for, as well as against, investors. Money can be made, and lost, in risk-free Treasury securities as well as in riskier corporate bonds.

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All things being equal, bond price volatility is an increasing function of maturity. Long-term bond prices ﬂuctuate more than short-term bond prices. Although the inverse relationship between bond prices and interest rates is the basis of all bond analysis, a complete understanding of bond price changes as a result of interest-rate changes requires additional information. An increase in interest rates will cause bond prices to decline, but the exact amount of decline will depend on important variables unique to each bond such as time to maturity and coupon. An important principle is that for a given change in market yields, changes in bond prices are directly related to time to maturity. Therefore, as interest rates change, the prices of longer-term bonds will change more than the prices of shorter-term bonds, everything else being equal.

•

A related principle regarding maturity is as follows: the percentage price change that occurs as a result of the direct relationship between a bond’s maturity and its price volatility increases at a decreasing rate as time to maturity increases. In other words, the percentage of price change resulting from an increase in time to maturity increases, but at a decreasing rate. Put simply, a doubling of the time to maturity will not result in a doubling of...

References: Bierwag, G.O., 1987, Duration Analysis: Managing Interest Rate Risk , Ballinger Publishing Company, Cambridge, MA.

Chambers, D.R., and S.K. Nawalkha (Editors), 1999, Interest Rate Risk Measurement and Management , Institutional Investor, New York.

Fabozzi, F.J., 1996, Fixed-Income Mathematics , 3rd Edition, McGraw-Hill, New York.

Fabozzi, F.J., 1999, Duration, Convexity and Other Bond Risk Measures , John Wiley & Sons,

Chichester.

Martellini, L., and P. Priaulet, 2000, Fixed-Income Securities: Dynamic Methods for Interest Rate

Risk Pricing and Hedging , John Wiley & Sons, Chichester.

Macaulay, F.R., 1938, The Movements of Interest Rates, Bond Yields, and Stock Prices in the United

States Since 1859 , Columbia University Press, NBER, New York.

5.4.2 Papers

Bierwag, G.O., 1977, “Immunization, Duration and the Term Structure of Interest Rates”, Journal

Bierwag, G.O., G.G. Kaufman, and A. Toevs, 1983, “Duration: Its Development and Use in Bond

Portfolio Management”, Financial Analysts Journal , 39, 15–35.

Chance D.M., and J.V. Jordan, 1996, “Duration, Convexity, and Time as Components of Bond

Returns”, Journal of Fixed Income , 6(2), 88–96.

Christensen P.O., and B.G. Sorensen, 1994, “Duration, Convexity and Time Value”, Journal of

Portfolio Management , 20(2), 51–60.

Fama, E.F., and K.R. French, 1992, “The Cross-Section of Expected Stock Returns”, Journal of

Finance , 47(2), 427–465.

Grove, M.A., 1974, “On Duration and the Optimal Maturity Structure of the Balance Sheet”, Bell

Journal of Economics and Management Science , 5, 696–709.

Ilmanen, A., 1996, “Does Duration Extension Enhance Long-Term Expected Returns?” Journal of

Fixed Income , 6(2), 23–36.

Ingersoll, J.E., J. Skelton, and R.L. Weil, 1978, “Duration Forty Years After”, Journal of Financial

and Quantitative Analysis , 34, 627–648.

Litterman, R., and J. Scheinkman, 1991, “Common Factors Affecting Bond Returns”, Journal of

Fixed Income , 1(1), 54–61.

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