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FRM Part I · FRM Exam Part I · Measures of Financial Risk

Portfolio returns are normally distributed with mean zero and a daily standard deviation of USD 1.0 million in loss terms. The 99% VaR is 2.326 standard deviations. Using the normal density value φ(2.326) = 0.02665, what is the 99% ES (to two decimals)?

The 99% ES is about USD 2.67 million. For a zero-mean normal distribution, ES equals sigma times the density at the VaR quantile divided by one minus the confidence level: 0.02665 divided by 0.01 equals 2.665 million, which exceeds the 2.33 million VaR.

  1. AUSD 2.67 millionCorrect
  2. BUSD 2.33 million
  3. CUSD 2.65 million
  4. DUSD 3.29 million

Explanation

For a zero-mean normal, ES = σ·φ(z)/(1−c) = 0.02665/0.01 = 2.665, about 2.67 million. Using VaR (2.33) ignores the tail average. The figure 2.65 results from rounding φ incorrectly, and 3.29 is not derived from this formula.

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