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

Losses of a portfolio are normally distributed with mean zero and a standard deviation of USD 10 million. The 99% standard normal quantile is 2.326 and the density at that quantile is 0.0267. What is the 99% Expected Shortfall (approximately)?

The 99% ES is about USD 26.7 million. For normal losses with zero mean, ES equals sigma times the density at the quantile divided by one minus the confidence level: 10 × 0.0267 / 0.01 = 26.7. This exceeds the 99% VaR of about 23.3 million.

  1. AUSD 26.7 millionCorrect
  2. BUSD 23.3 million
  3. CUSD 2.67 million
  4. DUSD 32.6 million

Explanation

For a normal loss with mean zero, ES = sigma × phi(z)/(1−α) = 10 × 0.0267/0.01 = 26.7. The VaR is 23.26, so ES exceeds it. Omitting the division by 1−α would give 2.67.

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