hard · FRM Part 2 Operational Risk
A bank models a single high-severity operational loss cell with a lognormal severity (μ=10, σ=2.5, in log-dollars) and a Poisson frequency with annual intensity λ=20. The risk team computes the 99.9% aggregate capital using the single-loss approximation (SLA): VaR_α ≈ F^-1big(1-(1-α)/(λ)big). A reviewer argues this materially understates the true 99.9% aggregate VaR.
Which statement best identifies the reason the SLA is biased here and the direction of the bias?
- The SLA omits the mean-aggregate offset; because the body of the compound distribution contributes a positive shift, the SLA understates true VaR by approximately the expected aggregate loss λ,e^μ+σ^2/2.
- The SLA is asymptotically exact only as αto1 for subexponential severities, but at finite λ and high σ it should be corrected upward by adding the mean of the aggregate loss, λ,e^μ+σ^2/2, which the first-order SLA drops.
- The SLA overstates VaR because it attributes the entire tail to one loss, whereas diversification across the λ expected events lowers the quantile by a factor of √(λ).
- The SLA is unbiased for any subexponential severity at α=99.9%, so the reviewer is mistaken; the apparent gap is an artifact of using log-dollars rather than dollars in F^-1.
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