hard · FRM Part 2 Operational Risk
A firm aggregates operational risk capital across two units of measure, each with a 99.9% standalone VaR of $100 million. Risk managers debate the diversification benefit. One argues that because operational losses are heavy-tailed, combining the two units must produce a diversification benefit (sub-additivity) at the 99.9% level, just as for normal risks.
Which statement is correct regarding the additivity of these VaRs?
- VaR is always sub-additive for independent risks by strict mathematical definition, so the combined 99.9% VaR is strictly below $200 million no matter how heavy the underlying loss-severity tails are in this particular case.
- For sufficiently heavy-tailed independent losses with tail index below 1 (infinite mean), VaR can be super-additive, so the combined 99.9% VaR may exceed $200 million; summing standalone VaRs is not guaranteed to be conservative.
- Because the two units of measure are different business lines, perfect-dependence summation always applies by convention, and the combined VaR equals exactly $200 million by rule, leaving no room at all for modeling judgment here.
- Sub-additivity of VaR holds for any elliptically-distributed set of risks, and since operational losses sum many small independent effects they are asymptotically elliptical, guaranteeing a diversification benefit under the classical CLT.
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