On this page
Use the same threshold
In the original loss model, the lower 90% quantile is 10. Loss 50 carries 5% probability and loss 10 carries another 25%. The definition of the event determines whether that boundary mass is included.
Compare the events
| Event | Included outcomes | Probability |
|---|---|---|
| L > 10 | 50 | 5% |
| L ≥ 10 | 10 and 50 | 30% |
| L ≤ 10 | −10, 0 and 10 | 95% |
Pair the correct complements
The complement of L>10 is L≤10. The complement of L≥10 is L<10. Mixing those pairs produces a probability discrepancy equal to the mass at the threshold.
Do not infer ES from the event label alone
A selected exact-tail average may use part of the boundary mass. The tool reports both strict and inclusive probabilities so readers can see why an unqualified exceedance statement would be incomplete.