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

A risk manager builds a spectral risk measure as a weighted average of the quantiles of a loss distribution. Which condition on the weighting function (risk spectrum) is needed for the measure to be coherent?

The spectrum must be non-negative, integrate to one, and be non-decreasing in loss severity, so worse outcomes get at least as much weight as milder ones. This risk-averse weighting is what guarantees coherence, including subadditivity, in a spectral risk measure.

  1. AWeights are non-negative, sum (integrate) to one, and do not decrease as loss severity increasesCorrect
  2. BWeights are equal across all quantiles, so every outcome counts the same
  3. CWeights are non-negative and decrease as loss severity increases, so frequent small losses dominate
  4. DWeights are zero everywhere except at a single chosen quantile

Explanation

A spectral measure is coherent when the spectrum is non-negative, integrates to one, and is non-decreasing in loss severity, which reflects risk aversion. Equal weights give the expected loss, which is coherent but does not express risk aversion, so it does not meet the severity-weighting condition. Decreasing weights would understate tail losses, and a single-quantile weight is VaR, which is not coherent.

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