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Use the original loss probabilities
| Loss units | Probability | Cumulative probability |
|---|---|---|
| −10 | 30% | 30% |
| 0 | 40% | 70% |
| 10 | 25% | 95% |
| 50 | 5% | 100% |
Compute amounts beyond the chosen threshold
At t=10, losses −10, 0 and 10 have zero positive excess. Loss 50 has excess 40 with probability 5%. Expected excess loss is therefore 0.05×40=2, rather than 0.05×50=2.5.
Compare selected thresholds
| Threshold t | Expected positive excess | t + excess / 0.10 |
|---|---|---|
| 0 | 5 | 50 |
| 10 | 2 | 30 |
| 50 | 0 | 50 |
Connect to the exact-tail result
The function is convex and piecewise linear in this finite model. Its slope changes where a threshold passes a loss amount; the minimum occurs at 10 for 90% confidence and equals the previously calculated ES 30. The calculator uses direct tail-mass allocation rather than optimizing thresholds.
Check an exact boundary separately
At 95%, the objective is 50 throughout the threshold interval 10 to 50. The threshold minimizer need not be unique even when the ES value is unique.