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Concept
Expected shortfall
Résumé
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 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), expected tail loss (ETL), and superquantile.
ES estimates the risk of an investment in a conservative way, focusing on the less profitable outcomes. For high values of it ignores the most profitable but unlikely possibilities, while for small values of it focuses on the worst losses. On the other hand, unlike the discounted maximum loss, even for lower values of the expected shortfall does not consider only the single most catastrophic outcome. A value of often used in practice is 5%.
Expected shortfall is considered a more useful risk measure than VaR because it is a coherent spectral measure of financial portfolio risk. It is calculated for a given quantile-level and is defined to be the mean loss of portfolio value given that a loss is occurring at or below the -quantile.
If (an Lp) is the payoff of a portfolio at some future time and then we define the expected shortfall as
where is the value at risk. This can be equivalently written as
where is the lower -quantile and is the indicator function. The dual representation is
where is the set of probability measures which are absolutely continuous to the physical measure such that almost surely. Note that is the Radon–Nikodym derivative of with respect to .
Expected shortfall can be generalized to a general class of coherent risk measures on spaces (Lp space) with a corresponding dual characterization in the corresponding dual space. The domain can be extended for more general Orlicz Hearts.
If the underlying distribution for is a continuous distribution then the expected shortfall is equivalent to the tail conditional expectation defined by .
Source officielle
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