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

A bank wants a risk measure that reflects a manager's subjective risk aversion while remaining subadditive. It considers a spectral measure with weights of 0.5 on the worst 1% of outcomes, 0.3 on outcomes from the 1% to 5% worst band, and 0.2 on the 5% to 10% worst band, each applied as an average within its band. Which assessment is correct?

The measure is coherent because the weight density per unit probability, 50, 7.5 and 4, is non-negative and decreases as outcomes become milder, while total weight sums to one. These conditions satisfy the spectral coherence requirement, and equal band weights are not needed.

  1. AIt is coherent because weights per unit probability are higher for worse bands (50, 7.5, 4), integrate to one, and are non-increasing from the tailCorrect
  2. BIt is not coherent because weights over bands are not equal
  3. CIt is not coherent because it uses more than one quantile
  4. DIt is coherent only if the weights per band are equal

Explanation

Per-unit-probability weights are 0.5/0.01 = 50, 0.3/0.04 = 7.5, 0.2/0.05 = 4. They are non-negative, fall as outcomes get milder, and total probability mass of weights is 0.5+0.3+0.2 = 1. That satisfies the conditions for a coherent spectral measure. Equal weights are not required, and using several quantiles is allowed.

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