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FRM Part II · FRM Exam Part II · Estimating Market Risk Measures: An Introduction and Overview

For a spectral risk measure to be coherent, which condition must the risk-aversion (weighting) function satisfy?

The weights must be non-negative, sum to one, and rise with the loss quantile, so larger losses get at least as much weight as smaller ones. This reflects risk aversion and ensures coherence, including subadditivity. Weights that fall with loss size would break the property.

  1. AWeights must be non-negative, sum to one, and be non-decreasing in the loss quantileCorrect
  2. BWeights must be equal across all quantiles
  3. CWeights must be zero below the confidence level and one above it
  4. DWeights must be non-negative and non-increasing as the loss quantile rises

Explanation

Coherence requires non-negative weights summing to one and increasing with the loss quantile, so that the user is more averse to larger losses. Non-increasing weights would imply less concern for worse losses and violate coherence. A step function with zero below the cutoff and equal weights above gives expected shortfall, which is a special case but not the general condition.

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