Executive Summary

Expected shortfall (ES), also known as conditional value at risk (CVaR), is a measure of ﬁnancial risk belonging to the class of spectral measures of risk, which are, in turn, an important subclass of coherent measures of risk.

ES represents a natural evolution of the concept of value at risk (VaR), sharing all its advantages. Notably, ES also possesses the property of subadditivity, the absence of which is the main source of VaR’s drawbacks.

For riskmanagement applications, ES requires exactly the same technology as VaR, in particular for modeling portfolios’ distributions. Any VaR engine (such as VCV, historical, or Montecarlo) works perfectly as an ES engine as well.

ES also probes the risk associated with extreme events, whereas VaR is blind to any tail risk with a probability of occurrence smaller than the chosen confidence level.

The main concept of ES was already known in actuarial science long ago, by several names and with slightly different definitions. Consensus on the modern version was reached only after 2001, when it was shown to be the only one that is, indeed, subadditive, and therefore coherent.
Introduction
In the mid1990s, value at risk (VaR) marked a true revolution in the practice of financial risk management. Fostered by the 1996 Amendment to the Basle Capital Accord, and standardized thanks to the diffusion of risk metrics methods, it soon became (and probably still is) the benchmark risk measure in finance. Why it was so successful is easy to understand. VaR conveys immediate information about a portfolio’s riskiness, showing the potential amount of money lost for some chosen probability on a given time horizon. Moreover, VaR does this job whatever the type of risks to which the portfolio is subject, and condenses their joint effects in a single synthetic number. Compare this simplicity with the cumbersome risk report of a complex portfolio based on more traditional greeks (a countless list of numbers with different meanings and units of measurement that cannot be summed up in a unique synthetic risk measure), and you will soon realize why VaR soon embodied the dreams of any risk or financial officer (remember the legendary J. P. Morgan’s, CEO, Dennis Weatherstone’s “4.15 Risk Report”).
Incidentally, in the same years, mathematical finance laid the foundations of modern risk theory. People started realizing that not all the statistics of a portfolio can be considered a legitimate risk measure. The appearance of (4) represented the first remarkable attempt to single out the fundamental principles associated with financial risk, and the corresponding properties that a coherent risk measure must satisfy. Quite surprisingly, VaR turned out to fail the crucial test of subadditivity, namely the fundamental property associated with the risk diversiﬁcation principle. Subadditivity means that given any coherent measure of risk ρ and any two portfolios X and Y, the risk ρ(X + Y) of the portfolios’ sum should never exceed the sum ρ(X) + ρ(Y) of the portfolios’ risks.
ρ(X + Y) ≤ ρ(X) + ρ(Y) (1)
The difference between the right and lefthand sides is clearly interpreted as the hedging risk benefit coming from the addition of X and Y, and is supposed to be positive in general cases, and zero only when X and Y happen to be perfectly correlated (actually, comonotonic is the right word), thus providing no mutual hedge at all. The problem with VaR is that one can easily find paradoxical situations where VaR(X + Y ) > VaR(X)+ VaR(Y), showing a strictly negative hedging benefit, and, therefore, a clear violation of the diversification principle. Not being subadditive, VaR is not coherent, which means that despite all its practical advantages, it lacks any theoretical foundations (1, 3).
A crucial question is, therefore, whether there exist alternative risk measures that share with VaR all its advantages, and yet lie on a sound theoretical basis in such a way to conform with fundamental risk principles. The answer is yes, and expected shortfall (ES), also known as conditional value at risk (CVaR), is the most popular of such risk measures.
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