Approximating expected shortfall for heavy tailed distributions

The Basle Committee’s proposed move from Value at Risk to expected shortfall as the mandated risk measure in its market risk framework necessitates practical methods for evaluating it. This paper derives a saddlepoint approximation for this purpose. Unlike earlier results, the approximation does not require the existence of a moment generating function and is therefore applicable to the heavy tailed distributions prevalent in finance. We also establish a link between our proposed approximation and mean-expected shortfall portfolio optimization. Numerical examples include the noncentral t , generalized error, and α-stable distributions. Optimization of a portfolio of DJIA stocks is considered as an empirical application.