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Define the original finite model
Loss is zero with probability 99% and 100 with probability 1%. Its mean is 1 and variance 99, so dispersion is √99. Using the lower cumulative 99% quantile gives VaR zero; exact worst-1% ES is 100.
Specify a separate matched law
A normal loss model with mean 1 and dispersion √99 has the same first two moments but a different shape and unbounded support. Its 99% VaR is approximately 24.146869 and ES approximately 27.518547.
Read the model distinction
| Law | Mean | Variance | 99% VaR | 99% ES |
|---|---|---|---|---|
| Finite 0/100 losses | 1 | 99 | 0 | 100 |
| Normal with matched moments | 1 | 99 | 24.146869 | 27.518547 |
Avoid a universal direction claim
Here the normal threshold exceeds the finite lower quantile while its ES is smaller. The example does not prove normal models always understate or overstate every tail measure. It proves that moments alone do not determine the requested tail calculation.