**Expected Shortfall and Beyond**
by Dirk Tasche of Deutsche Bundesbank October 20, 2002 **Abstract:** 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.
**JEL Classification:** D81, C13.
**Keywords:** Expected Shortfall, Value-at-Risk, Spectral Risk Measure, coherence, risk contribution
**Published in:** *Journal of Banking & Finance*, Vol. 26, No. 7, (July 2002), pp. 1519-1533.
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