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FRM Part II · FRM Exam Part II · Beyond Exceedance-Based Backtesting of Value-at-Risk Models

A risk analyst evaluates a bank's daily 99% VaR model using the probability integral transform (PIT). For each day, she computes the value of the model's forecast cumulative distribution function evaluated at the realized P&L. If the model's forecast distribution is correct, which distribution should these PIT values follow?

PIT values should follow a uniform distribution on [0, 1] if the forecast distribution is correct, because applying a variable's true CDF to its realizations yields uniform outcomes. A standard normal arises only after a further inverse-normal transformation of the PIT values.

  1. AStandard normal distribution
  2. BUniform distribution on the interval [0, 1]Correct
  3. CBinomial distribution with p = 1%
  4. DChi-square distribution with one degree of freedom

Explanation

The PIT of a random variable through its own true cumulative distribution function is uniform on [0,1]. If the forecast distribution is correct, realized PIT values should be i.i.d. uniform. The normal distribution only appears after a further inverse-normal transformation.

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