FRM Part I · FRM Exam Part I · Measures of Financial Risk
Which statement about expected shortfall (ES) and VaR is correct?
Expected shortfall is a coherent risk measure, satisfying all four axioms including subadditivity, while VaR is not coherent in general because it can violate subadditivity for certain loss distributions, such as portfolios with discrete, concentrated default risks.
- AES is coherent, while VaR is not coherent in general because it can violate subadditivityCorrect
- BVaR is coherent for all distributions, while ES violates monotonicity
- CBoth VaR and ES are coherent only if losses are normally distributed and neither is otherwise
- DES violates translation invariance, whereas VaR satisfies it
Explanation
ES satisfies monotonicity, translation invariance, positive homogeneity and subadditivity, so it is coherent. VaR can violate subadditivity for non-elliptical or discrete loss distributions. Under normality VaR happens to be subadditive, but it is not coherent in general.
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