Interest Rate Sensitivities Under the Vasicek and Cox-Ingersoll-Ross Models

Benchmark models for the term structure and dynamic of the interest rate, having the instantaneous rate as the only state variable, were introduced by Vasicek (V) and Cox-Ingersoll-Ross (CIR). Then the measure of a zero-coupon bond price change with respect to any change of the short-term interest rate movement is made by a sensitivity parameter referred as a stochastic duration. Therefore this last notion is theoretically superior to the Macaulay's duration under which the yield-curve is assumed to have made a parallel shift. However, empirical tests on bond immunization performance have not demonstrated any actual superiority of the stochastic duration when compared to the simple classical duration. Consequently in the present paper, we introduce alternative zero-coupon sensitivities with respect to the one-factor shock related to the considered V/CIR model. They lead to a high accuracy level for the approximation of the bond change with respect to the short-term interest rate movement. Our finding allows to get pointwise estimates of a bond hedging error. This is economically more sounding than the standard variance approach. Moreover our result has an implication on stress-testing framework. Indeed we get explicit expressions which enable us to map a view on shocks onto a view on the future bond change. The approach, we introduce here, is suitable for the perspective of hedging or managing a global portfolio or hybrid products.