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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 test whether a bank's daily trading P&L is consistent with the full predictive distribution produced by its VaR model, not just the 1% tail. The P&L observations are first converted into probability integral transform (PIT) values using the model's forecast distribution. If the model is correct, which distribution should these PIT values follow?

If the VaR model's forecast distribution is correct, the PIT values should be independent and uniformly distributed on [0,1]. A normal distribution appears only after applying the inverse normal transformation, which is the extra step used in the Berkowitz test.

  1. AStandard normal distribution
  2. BUniform distribution on [0,1] with independent observationsCorrect
  3. CStudent t distribution with 4 degrees of freedom
  4. DBinomial distribution with probability equal to the VaR tail level

Explanation

The probability integral transform of a random variable using its true cumulative distribution function gives a uniform [0,1] variable. With correct dynamic forecasts the PIT series is also independent and identically distributed. Normality arises only after a further inverse-normal transformation, as used in the Berkowitz test.

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