Expected Shortfall (ES) in several variants has been proposed as remedy for the deficiencies of Value-at-Risk (VaR) which in general is not a coherent risk measure. In fact, most definitions of ES lead to the same results when applied to continuous loss distributions. Differences may appear when the underlying loss distributions have discontinuities. In this case even the coherence property of ES can get lost unless one took care of the details in its definition. We compare some of the definitions of Expected Shortfall, pointing out that there is one which is robust in the sense of yielding a coherent risk measure regardless of the underlying distributions. Moreover, this Expected Shortfall can be estimated effectively even in cases where the usual estimators for VaR fail.

We discuss the coherence properties of expected shortfall (ES) as a financial risk measure. This statistic arises in a natural way from the estimation of the ‘average of the 100% worst losses’ in a sample of returns to a portfolio. Here p is some fixed confidence level. We also compare several alternative representations of ES which turn out to be more appropriate for certain purposes (J.E.L.: G20, C13, C14).

Expected shortfall (ES) has been widely accepted as a risk measure that is conceptually superior to value-at-risk (VaR). At the same time, however, it has been criticized for issues relating to backtesting. In particular, ES has been found not to be elicitable, which means that backtesting for ES is less straightforward than, for example, backtesting for VaR. Expectiles have been suggested as potentially better alternatives to both ES and VaR. In this paper, we revisit the commonly accepted desirable properties of risk measures such as coherence, comonotonic additivity, robustness and elicitability. We check VaR, ES and expectiles with regard to whether or not they enjoy these properties, with particular emphasis on expectiles. We also consider their impact on capital allocation, an important issue in risk management. We find that, despite the caveats that apply to the estimation and backtesting of ES, it can be considered a good risk measure. As a consequence, there is no sufficient evidence to justify an all-inclusive replacement of ES by expectiles in applications. For backtesting ES, we propose an empirical approach that consists of replacing ES by a set of four quantiles, which should allow us to make use of backtesting methods for VaR.

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.

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.