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

A risk manager reviews four properties that a risk measure may satisfy: monotonicity, translation invariance, homogeneity, and subadditivity. Under the standard definition, a risk measure is coherent only if it satisfies all four. Which of the following statements about Value at Risk (VaR) is correct?

VaR is not a coherent risk measure because it can fail subadditivity for some distributions. It does satisfy monotonicity, translation invariance and homogeneity, but the VaR of a combined portfolio can exceed the sum of the stand-alone VaRs, which contradicts diversification logic.

  1. AVaR is coherent because it satisfies subadditivity for all return distributions
  2. BVaR is not coherent because it can fail subadditivity for some distributionsCorrect
  3. CVaR is not coherent because it fails monotonicity
  4. DVaR is not coherent because it fails translation invariance

Explanation

VaR satisfies monotonicity, translation invariance and homogeneity, but it can violate subadditivity when losses are non-normal, for example with discrete or skewed credit-type payoffs. So the combined VaR of two portfolios can exceed the sum of the individual VaRs. The other options attribute failures to properties VaR does satisfy.

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