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?

  1. 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.
  2. Only if the t-copula is mis-specified with a negative correlation matrix, which forces an internally inconsistent quantile that mechanically exceeds the comonotonic sum.
  3. Whenever nu < 30, because low degrees of freedom make the copula density integrate to more than one, inflating the pooled quantile above the additive benchmark.
  4. It is never correct: the comonotonic sum (A) is the theoretical maximum of VaR, so any copula-based number (B) is bounded above by (A) by construction.

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