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This article proposes two extensions of current practice in applying duration gaps for macrohedging the equity position of a financial institution against interest rate risk. The first adjusts for the relative convexities of the asset and liability portfolios, which we call the convexity gap. When the second-order (convexity) condition is not satisfied, satisfying the first-order (duration) condition is not sufficient to achieve a perfect hedge against interest rate risk. The second extension shows how to modify duration gaps recognizing that banks and other financial institutions hold assets that are not default-free. Numerical examples demonstrate that ignoring this adjustment introduces considerable hedging error. [G20]
One of the great success stories of academic finance, duration analysis enjoys widespread practitioner application in at least two major areas. The first is bond portfolio management, where billions of dollars in fixed income funds are managed using duration as a measure of interest rate sensitivity. Second, treasury risk managers in banks, insurance companies, and other financial institutions use duration gaps to control their institutions' exposures to interest rate risk.
Both applications are based on the relationship between duration and price changes derived by a firstorder Taylor series approximation around the original interest rate. This approximation ignores two important features that require adjustments to the model. First, when interest rate moves are large, a second-order approximation (convexity) is also necessary. When fixed income securities are subject to default risk as well as interest rate risk, a second adjustment is required. These adjustments are recognized by bond portfolio managers and some of the researchers working on duration analysis.
In the case of large interest rate moves, Fong and Vasicek (1984) recognize that considering duration alone may not be sufficient to hedge against interest rate risk. They develop a risk measure called Msquared, which they argue should be minimized in order for duration strategies to be more effective hedges against interest rate fluctuations. Bierwag, Fooladi, and Roberts (1993) show that, contrary to the argument presented in Fong and Vasicek (1984), M-squared is not stochastic-process-free. They also show that the bullet portfolio recommended by Fong and Vasicek has a minimum M-squared only if a specific convexity condition holds.
Christensen and Sorenson (1994) refer to both duration and convexity as important tools for controlling...