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FRM Part I · FRM Exam Part I · Measures of Financial Risk

Which property distinguishes ES from VaR when assessing whether a risk measure is coherent?

ES satisfies subadditivity for any loss distribution, so the ES of a combined portfolio never exceeds the sum of the separate ES values. This makes ES a coherent risk measure, whereas VaR can violate subadditivity and penalise diversification.

  1. AES satisfies subadditivity, so the ES of a combined portfolio is no greater than the sum of the individual ES valuesCorrect
  2. BES fails subadditivity for portfolios of independent assets
  3. CES satisfies subadditivity only when returns are normally distributed
  4. DES is coherent because it ignores losses beyond VaR

Explanation

ES is a coherent measure: it satisfies monotonicity, translation invariance, positive homogeneity and subadditivity for any distribution. VaR can violate subadditivity, particularly with non-elliptical or discrete loss distributions. ES being coherent depends on averaging tail losses, not ignoring them.

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