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

Using the same loss-positive quantile score S = (1{L > VaR} - 0.01)(L - VaR) at the 99% level, a model forecasts VaR = 5.0 and the realised loss is 2.0 (no exceedance). What is the score, and what does it show about the penalty design?

The score is 0.03. With no exceedance, the indicator is zero, so the score is minus 0.01 times minus 3.0. The small positive penalty shows the scoring function also penalises excessive conservatism, not only breaches, so overstating VaR is not free.

  1. A0.03; a small penalty is incurred even without an exceedance, reflecting an overly conservative forecastCorrect
  2. B0.03 with a sign of zero; no penalty is ever incurred without an exceedance
  3. C2.97; non-exceedances are penalised more than exceedances
  4. D0.00; scores are zero unless VaR is breached

Explanation

No exceedance so the indicator is 0. S = (0 - 0.01)(2.0 - 5.0) = (-0.01)(-3.0) = 0.03, a positive penalty. This small penalty on unused conservatism discourages forecasting VaR far above the true quantile.

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