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, 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 value-at-risk, Expected Shortfall (or conditional value-at-risk or tail value-at-risk) has been characterized as the smallest coherent and law invariant risk measure to dominate value-at-risk. We discuss these and some other properties of Expected Shortfall 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 Expected Shortfall risk contributions to portfolio components.