Saddlepoint approximation of expected shortfall for transformed means

Expected Shortfall, as a coherent risk measure, has received a substantial amount of attention in the literature recently. For many distributions of practical interest however, it cannot be obtained in explicit form, and numerical techniques must be employed. The present manuscript derives a saddlepoint approximation for the Expected Shortfall associated with certain random variables that permit a stochastic representation in terms of some underlying random variables possessing a moment generating function. The new approximation can be evaluated quickly and reliably, and provides excellent accuracy. The doubly noncentral t distribution is considered as an example.