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Retain the cumulative probabilities
| Loss units | Probability | Cumulative probability |
|---|---|---|
| −10 | 30% | 30% |
| 0 | 40% | 70% |
| 10 | 25% | 95% |
| 50 | 5% | 100% |
Locate the first value satisfying the rule
At loss 0, cumulative probability is 70%, below 95%. At loss 10 it is 95%, which satisfies F(l)≥0.95. Therefore the lower quantile is 10. Selecting the next row would instead apply a different strict-crossing rule at this boundary.
Calculate the selected 5% average
At 95% confidence the worst-tail mass is 5%. All of it is supplied by loss 50. Expected shortfall is (0.05×50)/0.05=50. Probability strictly above VaR 10 is 5%; probability at or above it is 30%.
Keep quantile and severity separate
VaR 10 and ES 50 are consistent. The threshold describes a cumulative crossing; ES describes the selected worse tail’s average. Neither number is a promised limit on future loss.