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Hold outcomes and probabilities unchanged
Losses remain −10, 0, 10 and 50 with probabilities 30%, 40%, 25% and 5%. F(10)=95% and F(50)=100%. The only changed input is confidence.
Compare exact and just-above levels
| Confidence | VaR loss units | ES loss units |
|---|---|---|
| 95% exactly | 10 | 50 |
| 95.0001% | 50 | 50 |
| 97.5% | 50 | 50 |
Explain the constant tail average
At 95% the selected 5% mass is entirely loss 50. Just above 95%, the selected mass becomes slightly smaller and remains inside the same loss-50 atom. Every included unit of probability has the same loss amount, so ES remains 50.
Identify discretisation instead of a new scenario
The VaR jump is a feature of the declared finite distribution and quantile convention. No economic loss outcome changed. Retain confidence precision and the sample rule when comparing reports, rather than describing the jump as a newly observed market event.