Expected shortfall and beyond

Financial institutions have to allocate so-called economic capital in order to guarantee solvency to their clients and counterparties. Mathematically speaking, any methodology of allocating capital is a risk measure, i.e. a function mapping random variables to the real numbers. Nowadays value-at-risk (VaR), which is defined as a fixed level quantile of the random variable under consideration, is the most popular risk measure. Unfortunately, it fails to reward diversification, as it is not subadditive. In the search for a suitable alternative to VaR, expected shortfall (ES) (or conditional VaR or tail VaR) has been characterized as the smallest coherent and law invariant risk measure to dominate VaR. We discuss these and some other properties of ES as well as its generalization to a class of coherent risk measures which can incorporate higher moment effects. Moreover, we suggest a general method on how to attribute ES risk contributions to portfolio components. (C) 2002 Elsevier Science B.V. All rights reserved.