FRM Part II · FRM Exam Part II · Backtesting VaR
A bank's 95% one-day VaR is backtested over 250 days with 20 exceptions. Using the normal approximation to the binomial under the null that the model is correct (p = 0.05), what is the z-statistic, and what is the conclusion at a 5% two-sided level (critical value 1.96)? Use z = (x - pT)/sqrt(p(1-p)T).
The z-statistic is 2.18: expected exceptions are 12.5, the standard deviation is about 3.446, and (20 - 12.5)/3.446 = 2.18. Since this exceeds 1.96, the null that the model is correctly calibrated is rejected at the 5% two-sided level.
- Az = 2.18; reject the model as miscalibratedCorrect
- Bz = 1.45; do not reject the model
- Cz = 2.18; do not reject the model
- Dz = 4.36; reject the model as miscalibrated
Explanation
Expected exceptions = 0.05 x 250 = 12.5. Standard deviation = sqrt(0.05 x 0.95 x 250) = sqrt(11.875) = 3.446. z = (20 - 12.5)/3.446 = 2.18, which exceeds 1.96, so reject. The 4.36 option wrongly divides by half the standard deviation; 1.45 uses a wrong base.
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