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FRM Part I · FRM Exam Part I · Calculating and Applying VaR

A bank backtests a 99% one-day VaR model over 250 trading days and observes 7 exceptions. Assuming exceptions are independent, which statement is most accurate? (Expected exceptions at 99% = 2.5; the standard deviation of the number of exceptions is √(250 × 0.01 × 0.99) ≈ 1.57.)

Expected exceptions are 2.5, so seven is 4.5 above expectation, which is about 2.9 standard deviations given a standard deviation near 1.57. That is statistically unlikely under a correct model, indicating the VaR model probably understates risk.

  1. AThe count is about 2.9 standard deviations above expectation, so the model is likely understating riskCorrect
  2. BThe count is about 1.0 standard deviation above expectation, so the model is clearly adequate
  3. CThe count is below expectation, so the model is overstating risk
  4. DThe count is about 4.5 standard deviations above expectation, so the model must be rejected only on regulatory grounds

Explanation

Expected exceptions are 250 × 1% = 2.5. The excess is 7 − 2.5 = 4.5; dividing by 1.57 gives about 2.87 standard deviations. This is statistically significant at conventional levels, suggesting the VaR model underestimates risk. Treating 4.5 itself as the number of standard deviations omits the division by the standard deviation.

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