On the significance of expected shortfall as a coherent risk measure

This article shows that any coherent risk measure is given by a convex combination of expected shortfalls, and an expected shortfall (ES) is optimal in the sense that it gives the minimum value among the class of plausible coherent risk measures. Hence, it is of great practical interest to estimate the ES with given confidence level from the market data in a stable fashion. In this article, we propose an extrapolation method to estimate the ES of interest. Some numerical results are given to show the efficiency of our method.