Duration Hedging
5
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
•
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.
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
•
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.
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.