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FRM Part I · FRM Exam Part I · Calculating and Applying VaR

A risk manager reviews four properties that a risk measure may satisfy: monotonicity, translation invariance, positive homogeneity and subadditivity. Value at Risk (VaR) is known to fail to be a coherent risk measure in general. Which property does VaR fail to satisfy in general?

VaR fails subadditivity in general. The VaR of a combined portfolio can exceed the sum of stand-alone VaRs, so diversification may appear to increase risk. VaR does satisfy monotonicity, translation invariance and positive homogeneity, which is why it is not coherent while expected shortfall is.

  1. AMonotonicity
  2. BTranslation invariance
  3. CSubadditivityCorrect
  4. DPositive homogeneity

Explanation

VaR satisfies monotonicity, translation invariance and positive homogeneity. It can violate subadditivity, so the VaR of a combined portfolio can exceed the sum of the individual VaRs, which is why VaR is not coherent. Expected shortfall satisfies all four properties.

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