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FRM Part II · FRM Exam Part II · Estimating Market Risk Measures: An Introduction and Overview

A risk manager is reviewing the axioms of a coherent risk measure (monotonicity, subadditivity, positive homogeneity, translation invariance). Which statement correctly describes the weakness of Value at Risk (VaR) relative to Expected Shortfall (ES)?

VaR can fail subadditivity, meaning the VaR of a combined portfolio may exceed the sum of the individual VaRs. It still satisfies monotonicity, positive homogeneity and translation invariance. Expected Shortfall satisfies all four coherence axioms, which is why it is considered coherent.

  1. AVaR can violate subadditivity, so the VaR of a combined portfolio can exceed the sum of the VaRs of its partsCorrect
  2. BVaR violates monotonicity, so a portfolio with uniformly worse outcomes can show a lower VaR
  3. CVaR violates positive homogeneity, so doubling all positions does not double the VaR
  4. DVaR violates translation invariance, so adding cash does not reduce the VaR by the amount added

Explanation

VaR is monotonic, positively homogeneous and translation invariant, but it is not generally subadditive. Diversification can therefore appear to increase VaR for non-elliptical or discrete loss distributions. ES satisfies all four axioms and is coherent.

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