FRM Part I · FRM Exam Part I · Hypothesis Testing
A bank backtests a 99% one-day VaR model over 250 trading days and observes 6 exceptions. Assuming exceptions are independent Bernoulli events with p = 1%, what are the expected number of exceptions and the standard deviation of the number of exceptions (approximately)?
Expected exceptions are 2.5 with a standard deviation of about 1.57. The number of exceptions is binomial with n of 250 and p of 1%, so the mean is np and the variance is np(1-p), equal to 2.475, whose square root is about 1.57.
- AExpected 2.5; standard deviation 1.57Correct
- BExpected 2.5; standard deviation 2.48
- CExpected 6.0; standard deviation 1.57
- DExpected 0.025; standard deviation 0.157
Explanation
The count is binomial: mean = 250 × 0.01 = 2.5. Variance = 250 × 0.01 × 0.99 = 2.475, so the standard deviation is about 1.57. 2.48 is the variance, not the standard deviation. 6 is the observed count, not the expected.
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