FRM Part II · FRM Exam Part II · Backtesting VaR
A bank backtests a 95% one-day VaR over 500 days and observes 40 exceptions. The Kupiec likelihood ratio statistic is LR = -2 ln[(0.95)^460 (0.05)^40] + 2 ln[(0.92)^460 (0.08)^40]. Under the null, LR follows which distribution, and what is the 5% critical value?
The Kupiec LR statistic is asymptotically chi-square with one degree of freedom, so the 5% critical value is 3.84. The 6.63 figure is the 1% level, and two degrees of freedom belong to the joint conditional coverage test.
- AChi-square with 1 degree of freedom, 3.84Correct
- BChi-square with 2 degrees of freedom, 5.99
- CChi-square with 1 degree of freedom, 6.63
- DStandard normal, 1.96
Explanation
The Kupiec POF statistic compares one restricted parameter (the exception probability fixed at 5%) with the unrestricted maximum likelihood estimate (40/500 = 8%), so it is chi-square with 1 degree of freedom. The 5% critical value is 3.84; 6.63 is the 1% value, and 2 df applies to the conditional coverage test.
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