FRM Part I · FRM Exam Part I · Calculating and Applying VaR
Losses of a portfolio are normally distributed with mean zero and standard deviation USD 10 million. The 99% VaR is 2.33 standard deviations. The standard normal density at 2.33 is approximately 0.0264. What is the approximate 99% expected shortfall?
The 99% expected shortfall is about USD 26.7 million. For zero-mean normal losses, ES equals sigma times the normal density at the VaR quantile divided by one minus the confidence level, giving roughly 10 × 0.0267 / 0.01. This exceeds the 23.3 million VaR, as ES must.
- AUSD 23.3 million
- BUSD 26.7 millionCorrect
- CUSD 29.9 million
- DUSD 2.67 million
Explanation
For normal losses with mean zero, ES = sigma × φ(z)/(1−α) = 10 × 0.0264/0.01 = 26.4 million, which is closest to 26.7 million given rounding of φ (the exact value φ(2.326)=0.0267 gives 26.7). The 23.3 million figure is the VaR. The 2.67 million figure omits dividing by 0.01.
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