hard · FRM Part 2 Operational Risk

Which critique most precisely identifies the resulting bias in the fitted severity tail?

1. Because larger firms report only above higher thresholds, naive pooling left-truncates big-firm data more severely, biasing the fitted scaling exponent and steepening (thinning) the inferred severity tail unless truncation is modeled in the likelihood.
2. The threshold is irrelevant to severity tail estimation because it removes only small losses below the fixed cutoff, which by definition have no influence on the shape or fit of the high-quantile severity tail or its estimated parameters.
3. Scaling losses by revenue with an exponent beta strictly below one guarantees an unbiased tail because sublinear scaling automatically corrects any data-collection truncation that varies systematically across different firm sizes and reporting regimes.
4. The bias inflates the tail: higher reporting thresholds imposed on large firms disproportionately drop their small losses from the pooled sample and mechanically push the fitted tail index toward heavier tails, overstating the required regulatory capital charge.

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