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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 its probability integral transform (PIT) value using the model's forecast distribution. If the model is correctly specified, which distribution should these PIT values follow?

The PIT values should be independent and identically distributed uniform on [0,1]. If the forecast distribution is correct, each realized outcome is equally likely to fall at any quantile. The normal result only appears after an additional inverse-normal transform, as in the Berkowitz test.

  1. AStandard normal, independent across days
  2. BUniform on [0,1], independent and identically distributedCorrect
  3. CBinomial with success probability equal to 1 minus the VaR confidence level
  4. DStudent's t with degrees of freedom equal to the number of observations minus one

Explanation

Under a correctly specified forecast distribution, the PIT of the realized outcome is uniform on [0,1] and independent over time. Standard normality only arises after applying the inverse normal transform, which is the Berkowitz step. The binomial describes exceedance counts, not the PIT values.

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