# CHAPTER 4

# BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT

In this chapter we review some basic refinements in interest rate risk management. The concept of duration discussed in Chapter 3 is a good first approximation to measure the risk embedded in fixed income instruments. However, it is possible to improve upon it, and this is accomplished in two ways: First, we realize that the relation between a bond price and the interest rate is not linear, which is the implicit assumption in the duration approximation. Second, we also realize that the term structure of interest rates does not move in a parallel fashion, which is a second important assumption of the duration concept. By generalizing the risk model along these two dimensions we are able to obtain more precise measures of risk, as well as improve upon risk management practices.

## 4.1 CONVEXITY

The relation between bond prices and interest rates is not linear. Assume for instance a flat term structure of interest rates at rates *r* = .01, *r* = .02, ..., *r* = .15. Figure 4.1 plots the values of 5-, 10-, 20- and 30-year zero coupon bonds against these interest rates. We see that as the interest rate *r* increases, the zero coupon bonds decrease, and they become flatter and flatter as the interest rate becomes higher and higher. This pattern is especially true for long-dated zero coupon bonds.

This observation has an impact on the practice of interest rate risk management. In Chapter 3 we explored the notion of duration, that is, ...