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

Which statement best explains why Value at Risk fails to be a coherent risk measure while expected shortfall is coherent?

VaR is not coherent because it can fail subadditivity: combining portfolios can produce a VaR larger than the sum of the separate VaRs, contradicting diversification. It does satisfy monotonicity, positive homogeneity and translation invariance. Expected shortfall satisfies all four coherence properties.

  1. AVaR can violate subadditivity, so the VaR of a combined portfolio may exceed the sum of component VaRsCorrect
  2. BVaR violates monotonicity, so a portfolio with larger losses in every state can have a lower VaR
  3. CVaR violates positive homogeneity, so doubling a position does not double its VaR
  4. DVaR violates translation invariance, so adding cash does not change the VaR

Explanation

VaR satisfies monotonicity, positive homogeneity and translation invariance, but it can fail subadditivity, notably for portfolios with discrete, skewed or fat-tailed losses. Expected shortfall satisfies all four properties. The other options misstate properties VaR does satisfy.

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