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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 VaR model using the probability integral transform (PIT). For each day, the analyst computes the model's predicted cumulative distribution function evaluated at the realized P&L. If the model's predicted distributions are correct, which statement describes the resulting PIT series?

With a correct model, PIT values are independent and identically distributed uniform(0,1) variables. Each is the predicted CDF evaluated at the realized P&L, so every quantile is hit with the right frequency. Normality only appears after an additional inverse-normal transformation.

  1. AThe values are independent and identically distributed uniform(0,1) random variablesCorrect
  2. BThe values are independent standard normal random variables with mean zero
  3. CThe values are all below the VaR confidence level, such as 0.01
  4. DThe values are positively autocorrelated with mean 0.5

Explanation

Under a correctly specified model, applying the predicted CDF to the realized outcome gives a uniform(0,1) variable each day, and independence holds when the forecasts use all available information. Normal values arise only after a further inverse-normal transformation. Values below 0.01 would occur only about 1% of the time.

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