Duration captures the first-order effect of a change of interest rates and convexity measures second-order effects. If convexity is important, matching durations does not make the EV immune to rate changes and the duration gaps are not a reliable measure of sensitivity.

FIGURE 27.3 EV-interest rate profile

Convexity

The difference, the EV, has a shape that depends upon the relative curvatures of the profiles of assets and liabilities. Convexity risk arises from the different convexities of the curves representing the relationship between asset and liability values to interest rates. When there are significant differences between the curvatures, or "convexities," of the asset/interest rate and liability/interest rate profiles, significant variations of the EV might occur, even when slopes are similar at the current rate. This is "convexity risk."

Graphically, the slope of the "market value/interest rate" profile relates to duration. When the interest rate moves, so does the duration. The curvature of the profile shows how the slope changes (Figure 27.4). The curvature means that the sensitivity to downward moves of the

FIGURE 27.4 Sensitivity and interest rates

interest rate is higher than the sensitivity to upward moves. The effect of a decrease in rates, from 9% to 8%, is bigger than the change generated by a move from 4% to 3%. "Convexity" measures the change in duration when the rates move. With significant variations of interest rates, the second-order effect, or convexity, should be considered.

Sources of Convexity

The first source of convexity is the non-linear relationship between market value and discount rates, due to discount factors, such as (1 + /)'. A fraction of convexity comes from this mathematical formula. Wide variations of interest rates make the convexity become non-negligible. Convexity also relates to options embedded in banks' balance sheets because of their "kinked" payoffs.

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