hard · FRM Part 2 Operational Risk
A bank aggregates operational-risk capital across seven business-line/event-type cells. Each cell's $99.9% VaR is computed from its own LDA, and the bank wishes to recognize diversification. Two methods are on the table: (A) sum the seven stand-alone VaRs (perfect-dependence assumption), and (B) combine the seven aggregate-loss distributions with a Student-t copula (nu = 6) and read the $99.9% quantile of the pooled distribution. A model reviewer warns that even method (B) can produce a diversified capital number ABOVE the simple sum (A).
Under what condition is this warning technically correct?
- When the cell severities are heavy-tailed enough to be non-sub-additive in the tail and the t-copula's tail dependence concentrates joint extremes, the $99.9% quantile of the sum can exceed the sum of the $99.9% quantiles.
- Only if the t-copula is mis-specified with an invalid negative correlation matrix across the seven cells, which forces an internally inconsistent quantile that mechanically exceeds the comonotonic sum every single time.
- Whenever the degrees-of-freedom parameter nu is below thirty, because low degrees of freedom make the fitted copula density integrate to more than one, inflating the pooled quantile above the simple additive benchmark figure.
- It is never technically correct: the comonotonic sum in method (A) is always the theoretical maximum possible value of VaR, so any copula-based diversified number in (B) is strictly bounded above by (A) by mathematical construction alone.
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