FRM Part II · FRM Exam Part II · Beyond Exceedance-Based Backtesting of Value-at-Risk Models
A risk manager wants to backtest a bank's full predictive VaR distribution rather than only counting exceedances. She converts each day's realized P&L into the probability integral transform (PIT) value using the model's forecast distribution. If the model is correctly specified, what should the PIT series look like?
Under a correctly specified model, the PIT values are independent draws from a uniform distribution on [0,1]. Distribution backtests compare the empirical PIT distribution to this uniform. Only after applying the inverse normal transform do they become standard normal, as used in the Berkowitz test.
- AIndependent draws from a standard uniform distribution on [0,1]Correct
- BIndependent draws from a normal distribution with mean 0 and variance 1 for the raw PIT values
- CA series with an increasing trend as volatility rises
- DDraws that cluster in the tails of [0,1]
Explanation
The PIT of a realized outcome under the true forecast distribution is uniform on [0,1] and independent over time. Distribution tests such as Kolmogorov-Smirnov and Anderson-Darling compare the empirical PIT distribution with this uniform. The normal option applies only after a further inverse-normal transform, as in Berkowitz, not to raw PIT values.
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