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

Expected shortfall at the 95% confidence level can be viewed as a spectral risk measure. Which weighting scheme on the ordered loss quantiles represents it?

Expected shortfall at 95% gives equal weight to every loss in the worst 5% tail, equal to one divided by 0.05, and zero weight to all other outcomes. It is therefore a spectral measure with a constant tail weight, unlike exponential weighting that varies smoothly.

  1. AEqual weight of 1/(0.05) on all losses in the worst 5% tail and zero weight on all other outcomesCorrect
  2. BWeight of 0.95 on the 95th percentile loss and 0.05 on the median
  3. CWeights that rise smoothly and exponentially with the loss quantile across the whole distribution
  4. DEqual weights on every outcome in the distribution

Explanation

ES at 95% averages the worst 5% of losses, so each tail quantile gets constant weight 1/0.05 = 20 on the density scale and the rest get zero. Equal weights on all outcomes give the mean. Exponential weights describe an exponential-spectral measure, not ES.

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