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State each component’s original distribution
Each component loses 100 with probability 4% and zero with probability 96%. Under the lower 95% quantile rule, each individual VaR is zero because F(0)=96%. Independence is an explicit joint-model assumption.
Construct the joint outcomes
| Event | Combined loss | Joint probability |
|---|---|---|
| Neither loses | 0 | 92.16% |
| Only A loses | 100 | 3.84% |
| Only B loses | 100 | 3.84% |
| Both lose | 200 | 0.16% |
Locate the combined crossing
The combined zero-loss probability is only 92.16%, below 95%. At loss 100, cumulative probability is 99.84%, so combined VaR is 100. It exceeds the sum of individual VaRs, 0+0. This disproves universal VaR subadditivity.
Check the selected-tail measure
Each individual ES is (0.04×100)/0.05=80. The combined worst 5% includes 0.16% at loss 200 and 4.84% at loss 100, yielding (0.0016×200+0.0484×100)/0.05=103.2. That is below the individual ES sum 160.
Keep one counterexample in context
The example does not mean every combination raises VaR or determine a real portfolio allocation. It shows why an aggregation property needs the measure definition and a valid joint distribution. Independence cannot be inferred from two separate marginal loss tables.