hard · FRM Part 2 Operational Risk
A bank scales external operational-loss data to its own size before fitting severities. It uses a power-law scaling L_scaled = L_ext × (S_bank/S_ext)^β, where S is annual revenue and the estimated scaling exponent is hatβ=0.23. An analyst notes that the external consortium suffers a reporting threshold: only losses above $1 million are collected, and larger firms tend to have higher thresholds.
Which critique most precisely identifies the resulting bias in the fitted severity tail?
- 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.
- The threshold is irrelevant to severity tail estimation because it removes only small losses, which by definition have no influence on a high-quantile tail fit.
- Scaling by revenue with β<1 guarantees an unbiased tail because sublinear scaling automatically corrects any data-collection truncation across firm sizes.
- The bias inflates the tail: higher thresholds for big firms drop small losses and mechanically push the fitted tail index toward heavier tails, overstating capital.
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