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

For a 95% VaR forecast, a quantile score is defined as S = (1{L > VaR} - 0.05)... applied as a loss: if realized loss L is at most VaR, penalty = 0.05 x (VaR - L); if L exceeds VaR, penalty = 0.95 x (L - VaR). Model A forecasts VaR = 10 and the realized loss is 14. Model B forecasts VaR = 12 for the same day. What is the difference in penalty (A minus B)?

Model A's penalty is 0.95 times 4, or 3.8, and Model B's is 0.95 times 2, or 1.9. Both exceeded, so the larger weight applies. A minus B equals 1.9, showing the higher VaR forecast is rewarded on a day when the loss was large.

  1. A1.9Correct
  2. B3.8
  3. C-1.9
  4. D1.7

Explanation

Model A: L=14 exceeds 10, penalty = 0.95 x 4 = 3.8. Model B: L=14 exceeds 12, penalty = 0.95 x 2 = 1.9. Difference A minus B = 3.8 - 1.9 = 1.9. Using 0.05 weight wrongly would give 0.2 - 0.1 = 0.1; the sign-reversed option is -1.9.

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