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Select probability mass rather than a row count
At confidence c=90%, select the worst 10% probability mass. In the original model, take all 5% at loss 50 and another 5% from the 25% probability at loss 10. The remaining probability at 10 is outside this selected tail.
Calculate the weighted tail loss
Expected shortfall is (0.05×50+0.05×10)/0.10=30 loss units. Its denominator is the selected 10% probability, not the full model or the number of listed outcomes.
| Loss | Available probability | Included probability | Weighted tail loss |
|---|---|---|---|
| 50 | 5% | 5% | 2.5 |
| 10 | 25% | 5% | 0.5 |
| Total | Not a new distribution | 10% | 3 |
Keep atoms explicit
In a discrete model, averaging every loss at or above VaR can include too much probability. Averaging only losses strictly above VaR can include too little. The exact-tail-mass convention resolves that boundary through fractional inclusion; it does not split the economic outcome into a different loss amount.
Retain the learning scope
This original model explains the arithmetic used by the tool. An FRM or CFA study task should retain its specified sign, confidence, horizon and estimator. It is not a regulatory implementation or an investment forecast.