What does it mean when people say VaR is not subadditive?

Question

AcadiFi · Accepted Answer

Subadditivity means the risk of a combined portfolio should not exceed the sum of the risks of its parts. A risk measure that satisfies subadditivity rewards diversification in a sensible way. VaR can fail this property for certain non-normal or discontinuous loss distributions. In those cases, combining positions can move a loss into the selected quantile even when each position's stand-alone VaR looks small.

Suppose Harborline Credit Book has two loans. Each loan has a low probability of a large default loss, and each stand-alone 95 percent VaR may be zero because default is outside the 95 percent cutoff. When the loans are combined, the chance that at least one defaults may move inside the 95 percent tail, producing a positive portfolio VaR. The combined VaR can therefore exceed the sum of stand-alone VaRs.

The FRM lesson is not that VaR is useless. It is that VaR can ignore tail severity and can behave poorly for lumpy risks. Expected Shortfall is often preferred in this context because it is coherent under standard conditions and looks beyond the cutoff. When a question mentions diversification and VaR together, watch for subadditivity. On the exam, also state the horizon, confidence level, data source, modeling assumption, and validation step, because those choices usually explain why two reasonable VaR answers differ materially. Practice more in our FRM question bank.

What does it mean when people say VaR is not subadditive?

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