FRM Part I · FRM Exam Part I · Measures of Financial Risk
A risk committee compares VaR and expected shortfall for two portfolios, A and B, whose combined position is the sum of the two. Which statement is correct?
ES is subadditive for any loss distributions, so the combined portfolio's ES never exceeds the sum of the stand-alone ES values. VaR can violate this property, which is a key reason ES is regarded as coherent.
- AES is subadditive, so the ES of the combined portfolio never exceeds the sum of the individual ES valuesCorrect
- BVaR is subadditive, but ES is not, so ES can exceed the sum
- CNeither measure can be subadditive when distributions are discrete
- DES is subadditive only when losses are normally distributed
Explanation
ES is a coherent risk measure and therefore satisfies subadditivity for any loss distributions. VaR can violate subadditivity, for example with discrete or heavy-tailed losses. Therefore, only the first statement is right.
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