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

A bank backtests its 97.5% one-day VaR and ES. Over a test sample, VaR was exceeded on five days. The realised losses on those days exceeded VaR by: day 1 loss 12 vs VaR 10; day 2 loss 14 vs VaR 10; day 3 loss 11 vs VaR 10; day 4 loss 17 vs VaR 10; day 5 loss 16 vs VaR 10 (all USD millions). The model's average forecast ES on those days was USD 13 million. What is the average realised loss on exceedance days, and how does it compare with forecast ES?

The average realised loss on the five exceedance days is 70/5 = USD 14 million, which is above the forecast ES of USD 13 million. This suggests the model underestimates tail severity, so ES appears understated, which is directly relevant to ES calibration.

  1. AUSD 14 million, above forecast ES, suggesting ES is underestimatedCorrect
  2. BUSD 13 million, equal to forecast ES, suggesting no ES misspecification
  3. CUSD 4 million, below forecast ES, suggesting ES is overestimated
  4. DUSD 14 million, above forecast ES, but this is irrelevant to ES calibration

Explanation

Average loss = (12+14+11+17+16)/5 = 70/5 = 14. This exceeds the forecast ES of 13, which indicates the tail severity is underestimated. USD 4 million is the average excess over VaR (20/5), not the loss itself, and the realised average tail loss is directly relevant to ES calibration.

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