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FRM Part I · FRM Exam Part I · Hypothesis Testing

A risk manager backtests a 95% one-day VaR over 500 days and records 35 exceptions. Using the normal approximation to the binomial, with a two-sided test at 5% significance (critical value 1.96), which conclusion is correct?

Reject the model. Expected exceptions are 25 with a standard deviation near 4.87, giving a z-statistic of about 2.05, which is above the 1.96 two-sided critical value, so the observed 35 exceptions are statistically too many for a 95% VaR.

  1. AReject the model because z is about 2.06, which exceeds 1.96Correct
  2. BDo not reject the model because z is about 1.29
  3. CReject the model because z is about 4.60
  4. DDo not reject the model because the exception rate of 7% is below 10%

Explanation

Expected exceptions = 500 x 0.05 = 25. Standard deviation = sqrt(500 x 0.05 x 0.95) = sqrt(23.75) = 4.873. z = (35 - 25)/4.873 = 2.05, approximately 2.06 with rounding, above 1.96, so reject. Using the variance 23.75 instead of its square root would give a far smaller z, and the 10% comparison is irrelevant.

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