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FRM Part II · FRM Exam Part II · Non-parametric Approaches

A bank's 95% one-day historical simulation VaR from the original sample is USD 4.00 million. A bootstrap with many resamples yields a distribution of VaR estimates with a mean of USD 4.20 million and a standard deviation of USD 0.25 million. Assuming approximate normality of the bootstrap distribution, which is the approximate 95% confidence interval for VaR (using 1.96 standard deviations around the bootstrap mean)?

The interval is the bootstrap mean of 4.20 plus or minus 1.96 times 0.25, which is 0.49. That gives roughly USD 3.71 million to USD 4.69 million. Centering on the original 4.00 estimate would be incorrect here.

  1. AUSD 3.51 million to USD 4.49 million
  2. BUSD 3.71 million to USD 4.69 millionCorrect
  3. CUSD 3.75 million to USD 4.25 million
  4. DUSD 3.20 million to USD 5.20 million

Explanation

Interval = 4.20 ± 1.96 × 0.25 = 4.20 ± 0.49, giving 3.71 to 4.69. The first option centers on the original 4.00 rather than the bootstrap mean. The third uses only one standard deviation and is the wrong width. The fourth uses a wrong multiplier of 4 standard deviations.

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