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

A risk manager reviews four properties of a coherent risk measure: monotonicity, translation invariance, positive homogeneity and subadditivity. Which statement about Value-at-Risk (VaR) is correct?

VaR satisfies monotonicity, translation invariance and positive homogeneity but can fail subadditivity for non-elliptical or discrete loss distributions, so it is not a coherent risk measure. Diversification can then appear to increase measured risk.

  1. AVaR satisfies all four properties for any return distribution
  2. BVaR satisfies monotonicity but can violate subadditivityCorrect
  3. CVaR always satisfies subadditivity but violates monotonicity
  4. DVaR violates translation invariance but satisfies subadditivity

Explanation

VaR is monotonic, translation invariant and positively homogeneous, but it is not subadditive in general, so combining portfolios can show a VaR larger than the sum of the stand-alone VaRs. Hence it is not coherent. The option claiming VaR is always subadditive is wrong because discrete or skewed loss distributions give counterexamples.

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