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

A risk manager reviews four candidate risk measures for a trading desk. Under the axioms of coherence (monotonicity, subadditivity, positive homogeneity, translational invariance), which statement about Value at Risk (VaR) and expected shortfall (ES) is correct?

Expected shortfall is a coherent risk measure, while VaR can violate subadditivity, so a merged portfolio may show a larger VaR than the sum of its parts. Raising the confidence level or the measure's popularity does not make VaR coherent.

  1. AVaR is coherent because it is easy to estimate and widely used in regulation
  2. BES is coherent, while VaR can violate subadditivityCorrect
  3. CBoth VaR and ES violate subadditivity for non-normal distributions
  4. DVaR is coherent only when computed at a 99% confidence level

Explanation

ES satisfies all four coherence axioms. VaR can fail subadditivity, so the VaR of a combined portfolio can exceed the sum of the individual VaRs. Confidence level does not repair this failure.

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