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

A bank's trading desk model forecasts P&L as normal with a standard deviation of 2 million. Over 250 days, a histogram of PIT values shows a pronounced U-shape, with many observations near 0 and near 1 and few near 0.5. What is the most appropriate interpretation?

A U-shaped PIT histogram indicates the model underestimates risk: realized P&L lands in the forecast tails too often, so the forecast distribution is too narrow or too thin-tailed. An overstated risk forecast would instead give a hump-shaped histogram concentrated near the middle.

  1. AThe model overestimates risk, with forecast distributions too wide
  2. BThe model underestimates risk, with forecast tails too thin or the distribution too narrowCorrect
  3. CThe model is well calibrated but biased in the mean only
  4. DThe model is correct because extreme PIT values occur in both tails

Explanation

A U-shaped PIT histogram means realized outcomes fall in the forecast tails more often than expected, so the forecast distribution is too narrow or too light-tailed. An overly wide forecast would produce a hump-shaped histogram with mass concentrated near 0.5. A pure mean bias would produce a skewed histogram, not a symmetric U.

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