J V0

and D is as described in Equation (21.3).

As an illustration of the use of convexity let us return to the example presented in Table 21.1. Consider the 5-year pure discount bond, which pays \$1000 at maturity. Change the assumption in the table to a larger change in interest rates: in particular assume interest rates change from 10% to 12.2%. At a 10% interest rate the price of the 5-year pure discount bond is

whereas at a 12.2% interest rate the price is \$562.39. The rate of price change as interest rates rise from 10% to 12.2% is

Pw 620.92

The duration on the bond is 5 years, whereas Ai = 0.022/1.10 = .02. If we estimated the unexpected rate of return on the bond just using duration (Equation [21.1]), we would estimate it as

This is a 6.4% error. To obtain a better estimate we wish to apply Equation (21.4), which corrects the duration measure for convexity. The convexity on this bond is

5(6)1000

CamjugiL

Ru =-5(0.02) +15(0.02)2 =-0.10 + 0.006 = -0.094 or -9.4%

The convexity measure has produced an exact estimate in this case. In general, even using duration and convexity, the estimate will only be an approximation, though often a very good one.

As a second example consider Figure 21.1, which plots the actual price of a bond when different flat yield curves are assumed. The bond prices being plotted are for an eight-year bond with a 10% coupon that pays interest semiannually. Also plotted on the curve is the estimated price of the bond using duration alone (the straight line) and using duration plus convexity (the dashed curve). For small changes in the yield curve, the actual price change is closely matched by both, the estimate using duration alone and the estimate using duration plus convexity.5 For large price changes the introduction of convexity improves the estimation.

So far we have graphed only the relationship between price and yield for bonds without call features. For these bonds the relationship has the nice curved shape shown in Figure 21.1. The curved shape is known as the convexity, and for bonds without options such as those depicted in Figure 21.1, it is called positive convexity. When bonds have option features, the relationship between price and yield is not so simple. Figure 21.2 plots the relationship between price and yield for a callable bond. This relationship has negative convexity for yield below 10% but positive convexity for yields above. The reasons for the shape for yield below 10% is easy to understand. As the price of the bond exceeds the call price it pays the corporation to call. Investors knowing this will not pay much above the

Interest rate shift Figure 21.1 Actual price change and estimated price change.

5The plots of price using duration and using duration plus convexity were obtained as follows. When the yield changes, the bond price changes. The unanticipated return is the change in price divided by the preshift price or

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