Expected Shortfall

From Open Risk Manual


Expected Shortfall (ES) is a Risk Measure used in the context of Quantitative Risk Management of Market Risk or Credit Risk in a Portfolio Management context. The "expected shortfall at level \alpha % " is the Expected Loss experienced in the portfolio in the worst \alpha \% of scenarios.

ES is an alternative to Value at Risk that is addressing some widely criticized attributes of VaR as a risk measure. Expected shortfall is also called conditional value at risk (CVaR), average value at risk (AVaR), and expected tail loss (ETL).


If X is a Random Variable expressing the Risk Distribution of a portfolio at some future time and 0 < \alpha < 1 then we define the expected shortfall as

 \operatorname ES_\alpha = -\frac{1}{\alpha} \int_0^\alpha \operatorname{VaR}_\gamma(X) \, d\gamma

where \operatorname{VaR}_\gamma is the Value at Risk.


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