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

Why is VaR generally not regarded as a coherent spectral risk measure?

VaR places all its weight on a single quantile and none on worse losses, so its spectrum is not non-decreasing in loss severity. As a result it is not coherent and can violate subadditivity, unlike expected shortfall or other risk-averse spectral measures.

  1. AIts weighting function puts all weight on one quantile, which is not a non-decreasing risk-averse spectrum and allows subadditivity to failCorrect
  2. BIts weighting function gives larger weights to larger losses, which violates monotonicity
  3. CIt cannot be computed for continuous loss distributions
  4. DIt averages all losses beyond the quantile, so it ignores the confidence level

Explanation

VaR corresponds to a point mass at a single quantile, so the weights are zero beyond it and are not non-decreasing in severity. This can cause subadditivity to fail, so diversification can appear to raise VaR. Increasing weights are the requirement for coherence, not a violation, and averaging beyond the quantile describes expected shortfall.

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