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

A risk manager is comparing risk measures for a portfolio's loss distribution. Which statement about spectral risk measures is correct?

A spectral risk measure is a weighted average of loss quantiles with non-negative weights that integrate to one and rise with the quantile level, so worse losses get larger weights. This risk-aversion condition ensures coherence; decreasing weights or a single-quantile weight such as VaR do not qualify.

  1. AA spectral risk measure is a weighted average of loss quantiles, where the weights must be non-negative and non-decreasing in the quantile level and integrate to oneCorrect
  2. BA spectral risk measure assigns equal weights to all quantiles of the loss distribution, so it always equals the expected loss
  3. CA spectral risk measure is a weighted average of quantiles whose weights must decrease as the quantile moves into the tail
  4. DA spectral risk measure places all weight on a single quantile, so VaR is its only member

Explanation

A spectral measure M = integral of phi(p) q_p dp requires phi non-negative, non-decreasing (greater weight to worse outcomes, reflecting risk aversion) and integrating to one. Decreasing weights would reward tail losses less and violate the risk-aversion condition. Equal weights give expected loss, which is only a special case, and a single-quantile weight (VaR) is not a valid spectral weighting.

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