Expected shortfall (ES) is a risk measure—a concept used in the field of financial risk measurement to evaluate the market risk or credit risk of a portfolio. The "expected shortfall at q% level" is the expected return on the portfolio in the worst q % {\displaystyle q\%} of cases. ES is an alternative to value at risk that is more sensitive to the shape of the tail of the loss distribution. Expected shortfall is also called conditional value at risk (CVaR), average value at risk (AVaR), tail value at risk (TVaR), conditional tail expectation (CTE), expected tail loss (ETL), and superquantile. These names are often used interchangeably, although several definitions exist in the literature. These definitions coincide in many cases, but may differ for certain types of loss distributions.
Background Risk measures are used both in mathematical finance and in actuarial science, and the value-at-risk and expected shortfall measures are often expressed using different sign conventions and tail conventions in these disciplines. The discussion that follows takes the mathematical finance point of view. In mathematical finance, risk measures arise when considering the profit/loss distribution, i.e., payoff, for a financial portfolio, modeled as a random variable X {\displaystyle X} . This can take positive or negative values, and downside risk corresponds to quantiles with α {\displaystyle \alpha } close to 0. A risk threshold α ∈ [ 0 , 1 ] {\displaystyle \alpha \in [0,1]} is selected, and VaR α ( X ) {\displaystyle \operatorname {VaR} _{\alpha }(X)} is defined to be the absolute value of the α {\displaystyle \alpha } quantile of X {\displaystyle X} (ignoring some technicalities). This is also the 1 − α {\displaystyle 1-\alpha } quantile of − X {\displaystyle -X} . The expected shortfall at level α {\displaystyle \alpha } is then defined as the average value of VaR γ ( X ) {\displaystyle \operatorname {VaR} _{\gamma }(X)} for γ {\displaystyle \gamma } in the interval [ 0 , α ] {\displaystyle [0,\alpha ]} , i.e., it is the average VaR over all levels below α {\displaystyle \alpha } . Expected shortfall is often considered preferable to VaR because it accounts for the severity of the failure, not only the chance of failure. Further, it is a coherent spectral measure of financial portfolio risk, while VaR is not. This is a collection of mathematical properties, one of which ensures that diversification of a portfolio never leads to a higher measure of risk. Viewing the value produced by a risk measure as a capital reserve requirement, ES at level α {\displaystyle \alpha } is always more conservative than VaR at the same level, i.e., ES is always at least as big as VaR at the same level.
Formal definition If X {\displaystyle X} is an integrable random variable representing the payoff of a portfolio at some future time and 0 < α ≤ 1 {\displaystyle 0<\alpha \leq 1} then the expected shortfall of X {\displaystyle X} at level α {\displaystyle \alpha } is
ES α ( X ) = 1 α ∫ 0 α VaR γ ( X ) d γ {\displaystyle \operatorname {ES} _{\alpha }(X)={\frac {1}{\alpha }}\int _{0}^{\alpha }\operatorname {VaR} _{\gamma }(X)\,d\gamma }
where VaR γ {\displaystyle \operatorname {VaR} _{\gamma }} is the value at risk. Several other definitions appear in the literature under the names ES, TVaR, AVaR, CTE, and CVaR. The formulation above as an integral of VaR values is coherent and well-defined in the general case. Other definitions typically coincide under common assumptions such as continuity of the loss distribution, but may differ for distributions with atoms. The above definition is equivalent to
ES α ( X ) = − 1 α ( E [ X 1 { X ≤ x α } ] + x α ( α − P [ X ≤ x α ] ) ) {\displaystyle \operatorname {ES} _{\alpha }(X)=-{\frac {1}{\alpha }}\left(\operatorname {E} [X\ 1_{\{X\leq x_{\alpha }\}}]+x_{\alpha }(\alpha -P[X\leq x_{\alpha }])\right)}
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