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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.
Value at risk is, however, coherent, under the assumption of elliptically distributed losses (e.g. normally distributed) when the portfolio value is a linear function of the asset prices. However, in this case the value at risk becomes equivalent to a mean-variance approach where the risk of a portfolio is measured by the variance of the ...
However, it can be bounded by coherent risk measures like Conditional Value-at-Risk (CVaR) or entropic value at risk (EVaR). CVaR is defined by average of VaR values for confidence levels between 0 and α. However VaR, unlike CVaR, has the property of being a robust statistic. A related class of risk measures is the 'Range Value at Risk' (RVaR ...
Under some other settings, TVaR is the conditional expectation of loss above a given value, whereas the expected shortfall is the product of this value with the probability of it occurring. [3] The former definition may not be a coherent risk measure in general, however it is coherent if the underlying distribution is continuous. [4]
Along the scenario approach, it is also possible to pursue a risk-return trade-off. [7] [8] Moreover, a full-fledged method can be used to apply this approach to control. [9] First constraints are sampled and then the user starts removing some of the constraints in succession. This can be done in different ways, even according to greedy algorithms.
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In financial mathematics and economics, a distortion risk measure is a type of risk measure which is related to the cumulative distribution function of the return of a financial portfolio. Mathematical definition