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Home > Financial Risk Management Best Practice > Tools for Measuring Interest Rate Risk

Financial Risk Management Best Practice

Tools for Measuring Interest Rate Risk

by Steven V. Mann

This Chapter Covers

  • The two dimensions of interest rate risk: level risk and curve risk.

  • The two approaches for measuring interest rate risk: duration/convexity and full valuation.

  • The valuation of any security involves determining the present values of expected cash flows and figuring out how much the embedded options are worth.

  • Four attributes govern a bond’s response to a yield change: the remaining term to maturity, the level of the coupon rate, the level of the yields, and the presence of embedded options.

  • Duration can be defined as the average percentage price change for a 100 bp movement in the reference yield curve.

  • Convexity measures the magnitude of the error of linear approximation by duration of the price–yield relationship.

  • The full valuation approach performs a full valuation of a bond (or bond portfolio) for each of the chosen interest rate scenarios.


One can understand risk intuitively as the chance of an unpleasant surprise. Financial institutions must be able to manage their exposure to risk. To accomplish this task, they must be able to identify the risks to which they are exposed and measure that exposure. Numerous types of risk put a financial institution in harm’s way. One major exposure is to interest rate risk. Interest rate risk has two dimensions—level risk and curve risk. Level risk is the chance of an adverse movement in the price of a bond owing to a parallel shift in the reference yield curve. Curve risk is the prospect of an adverse movement in the price of a bond owing to a change in the shape (i.e. slope and curvature) of the reference yield curve. To manage their exposure to interest rate risk, financial institutions must be able to measure both level and curve risk. The purpose of this chapter is to introduce the tools for measuring both types of interest rate risk. Most of the time will be spent discussing level risk, while curve risk will be covered to a lesser extent.

We will discuss two approaches for assessing the interest rate risk exposure of a bond or bond portfolio. The first approach entails the computation of measures that approximate how a bond’s price or the portfolio’s value will change when interest rates change. The most commonly used measures are duration and convexity. We will discuss duration and convexity measures for bonds and bond portfolios. The second approach is the full valuation approach, which involves selecting possible interest rate scenarios for how interest rates and yield spreads may change and then revaluing the bond position.

Why Do Bond Prices Change?

Any security can be thought of as a package of one or more cash flows. A complicated security is a package of cash flows with one or more options attached. The option attached to the security gives its owner (either the holder or the issuer) the right to alter the security’s cash flows at an optimal time. That said, the valuation of any security involves determining present values and figuring out how much the embedded options are worth.

To determine present values, a discount rate is employed. Simply put, a discount rate is an exchange rate across time. A discount rate translates the value of a set of cash flows scattered across time and with different exposures to risk into a single sum. In general, discount rates are driven by two factors—a default-free benchmark interest rate and a risk premium. A default-free benchmark interest rate reflects the risk-free interest rate. Even in a world without risk, interest rates are positive. The reason is that human beings would prefer to consume now as opposed to later. Interest rates are bribes designed to quell our impatience to consume. Correspondingly, risk premiums are bribes that are designed to neutralize our risk-aversion. As a general description of how human beings behave, we operate under the assumption that investors are risk-averse. Risk-aversion does not mean that investors shun risk, merely that one must compensate them for taking it. The risk premium is the spread over the risk-free rate required to induce an investor to hold a risky security as opposed to a safe one. For a given level of risk, the risk premium is time-varying. The size of the risk premium depends on both the level of interest rates and investors’ willingness to bear risk.

Given this background, we can now address the question of why bond prices change. First, any factor that results in a change in the bond’s required yield will result in a change in the price of an option-free bond in the opposite direction. Second, the mere passage of time will push all bond prices to the par values. Bond prices that trade at a premium will be pulled to par without any change in required yield. Although we are cognizant of the second factor, our focus in this chapter is the first factor—specifically, the immediate response of the bond price to an instantaneous change in the bond’s required yield.

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Further reading


  • de La Grandville, Olivier. Bond Pricing and Portfolio Analysis: Protecting Investors in the Long Run. Cambridge, MA: MIT Press, 2001.
  • Fabozzi, Frank J., and Steven V. Mann. Introduction to Fixed Income Analytics: Relative Value Analysis, Risk Measures, and Valuation. 2nd ed. Hoboken, NJ: Wiley, 2010.
  • Fabozzi, Frank J., and Steven V. Mann (eds). The Handbook of Fixed Income Securities. 8th ed. New York: McGraw Hill, 2012.
  • Fabozzi, Frank J., Steven V. Mann, and Moorad Choudhry. Measuring and Controlling Interest Rate and Credit Risk. 2nd ed. Hoboken, NJ: Wiley, 2003.
  • Golub, Bennett W., and Leo M. Tilman. Risk Management: Approaches for Fixed Income Markets. New York: Wiley, 2000.
  • Ilmanen, Antti. Expected Returns: An Investor’s Guide to Harvesting Market Rewards. Chichester, UK: Wiley, 2011.

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