Economic Value and Convexity Risk

The techniques for managing the risk of economic values rely on the duration and the convexity of assets and liabilities. The duration is the sensitivity, or the slope of the curve, linking the relationship between economic value and discount rate. Convexity is the change of duration when interest rates change significantly, and measures the curvature of the relation. Since economic value depends on the asset and on the liability values, it depends also on the shapes of the curves representing how asset value and liability value change when interest rates vary, hence on their relative durations and convexities. Moreover, convexity increases in the presence of options. This suggests measuring the optional risk through differentials between embedded options and optional hedges. It is common to designate duration risk as "delta" risk and convexity risk as "gamma" risk following the generic terminology used for options.

The first part of the this chapter addresses, sequentially, duration properties, the condition on the duration gaps that make economic value immune to small variations of interest rates, and how to meet such conditions with derivative instruments. In the presence of significant volatility of interest rates, convexity effects become non-negligible. Accordingly, the second part of this chapter addresses, sequentially, the properties of convexity, the conditions on convexities of assets and liabilities that serves for protecting the economic value in presence of volatility of interest rates, and how to achieve such a result through a proper usage of derivatives. Convexity effects exist whenever interest rates are subject to large shocks. They also appear in the presence of options because these magnify convexity. The part of this chapter addresses optional risk and the related optional gap. Once all effects are covered, it becomes clear that interest rate gaps are not enough for managing interest rate risk. They should be supplemented by duration gaps, by convexity gaps and by optional gaps to avoid adverse effect on the economic value of the balance sheet.