Severity Modelling with Policy-Limit Censoring

No - this is not a random deletion. Dropping claims recorded at the limit removes the largest losses by construction. The fitted $18,200 is therefore an estimate of E[Y | Y < 250,000], the conditional mean below the limit, not the unconditional mean the pricing needs.

A quick bound. At least 2.4% of claims are ≥ $250,000, and each contributes at least $250,000 to the mean. So the unconditional mean implied by the truncated fit is at least:

E[Y] ≥ 0.976 × 18,200 + 0.024 × 250,000 = 17,763 + 6,000 = $23,763

That is already within about 3% of the incurred-based $24,600 - and it is a lower bound, because the censored claims actually average well above the limit. Using the analyst's $18,200 as the unlimited mean understates severity by roughly 26% before any tail adjustment.

Conclusion on the “reserving noise” claim. The gap is not noise. It is mechanically implied by censoring: almost all of the $6,400 difference is the excess layer the analyst deleted. The incurred estimate is the one to trust, and it will if anything understate the true unlimited mean until the tail above $250,000 is estimated properly.

A second problem. The covariate coefficients are biased too, not just the intercept. Censoring probability varies by injury type - slip-and-fall claims are less likely to reach $250,000 than product liability or auto-related claims - so deleting the censored claims systematically over-represents the low-severity injury types and distorts the very relativities the filing is built on.

Option 1 - censored likelihood (preferred). Treat the 202 claims as right-censored at $250,000. For uncensored claims the likelihood contribution is f(y); for censored claims it is S(250,000), the survival function at the limit. Fit a parametric severity distribution (lognormal, Weibull or gamma) with covariates on the location parameter and a log link. This uses every observation, estimates the tail shape from the censoring pattern, and yields an unconditional mean directly. The identifying assumption is the parametric tail form, so fit at least two families and compare AIC and the implied means.

Option 2 - Kaplan–Meier / empirical survival. Nonparametric: compute the survival function treating the 202 as censored times. This gives a defensible limited expected value E[min(Y, 250k)] with no distributional assumption, but it cannot estimate the mean beyond the largest observed value, so a parametric tail is still needed for the excess.

Option 3 - two-part model. A logistic model for P(Y ≥ 250k) and a separate model for the excess amount given exceedance, calibrated to umbrella/excess experience. Requires data the analyst does not have.

A subtlety worth flagging. Some of the 202 may have settled at the limit for policy or expense reasons rather than being genuinely unknown above it. If so, they are point masses at $250,000, not censored observations, and the likelihood needs a discrete component at the limit. Check the claim notes before choosing.

Additional data needed: large-loss experience from umbrella or excess policies written over this book, the reinsurer's large-loss bordereau, or industry excess severity curves - any of which pins down the tail that the 2.4% censoring rate leaves weakly identified.

Relativities. A relativity is a ratio of predicted means at two covariate settings, so a common downward bias in the level cancels - if the censoring is non-differential with respect to the covariate. Here it is differential: the censoring probability depends on injury type, which is exactly the covariate being priced. Slip-and-fall claims are less likely to hit the limit, so the truncated sample over-weights low-severity slip-and-fall claims, and exp(0.22) = 1.246 is likely understated - the true relativity is greater than 1.246 because the high-severity slip-and-fall claims were preferentially deleted. The same logic applies to territory. So no, the relativities cannot be filed as they stand; refit with the censored likelihood and compare the coefficients directly. If they move materially, the truncation was not innocuous.

The limit factor. The limit factor is E[min(Y, 250k)]/E[Y]. The truncated fit can estimate the numerator reasonably well - limited severity is exactly what the retained data measure. It cannot estimate the denominator, because the denominator needs the excess layer the analyst deleted. Reproducing the arithmetic: 0.94 × 24,600 = $23,124 of limited severity and $1,476 of excess, which spread over 2.4% of claims implies an average of about $61,500 excess per limited claim, i.e. a mean above-limit severity near $311,000. That is a plausible figure but it is an input, not an output, of the current model - the 0.94 is being carried forward without being tested against this experience at all.

Recommendation for the filing. Re-fit with censoring, then compute the limit factor as LEF(250k)/mean from the same fitted distribution, so numerator and denominator are internally consistent. Present the current 0.94 as unsupported by this analysis, and disclose the censoring rate and treatment in the filing memo.