Increased Limits Factors from Censored CGL Severity Data

The claim contributes its capped value to limited losses at each limit: $100k, $250k, $500k, and $1M. Divided by 620 claims, its contribution to LAS is $161, $403, $806, and $1,613 respectively. Removing it gives LAS of $14,339 at $100k, $19,397 at $250k, $23,194 at $500k, and $25,887 at $1M. Relative to the $100k LAS, the ILFs become 1.000, 1.353, 1.618, and 1.806. So the $1M ILF falls from 1.897 to 1.806 - removing the large claim makes the ILF lower, not higher. The claim is therefore not the reason the company's ILF is below the 2.15 benchmark. The real issue is that the observed severity distribution has a short tail: only one claim above $1.5M in 620 claims. Deleting it would leave the data with essentially no tail information. The benchmark gap is driven by thin data and by policy-limit censoring: policies with $100k limits cannot report losses above $100k, so the empirical distribution is not a random sample of ground-up severity. The direct LAS method also assumes that the mix of risks buying each limit is the same, which is unlikely.

Fit a parametric severity distribution - Pareto, lognormal, or a mixed model - to the uncensored portion of the data, using the $4.2M claim as the only direct observation of the tail. Do not discard it; it is the most informative point for the tail. Instead, control its influence by fitting the tail separately (for example, a Pareto to the excess over $1M) and by showing sensitivity to its value. Because 620 claims are thin, blend the company's fitted ILFs with the industry study using credibility based on claim counts. A simple Bühlmann approach: Z = n/(n+K) with n = 620 and K = 1,500, giving Z ≈ 0.29. The blended ILF at $1M = 0.29 × company fitted + 0.71 × industry. If the company's fitted ILF is 1.85 and the industry is 2.15, the blended value is 2.06. File a value in that range, not the raw 1.897. Also segment by policy limit and adjust for limit selection; if the company's $1M policies are larger risks with different severity, the industry study may be more representative of the ground-up distribution. Document the censoring adjustment and the model fit, for example with Kolmogorov-Smirnov or Anderson-Darling statistics.

The strongest argument is that 1.897 is not the company's own experience in the sense of a representative sample: it is a censored, limit-selected statistic computed from 620 claims with one tail observation. The standard error of the $1M LAS is large. A bootstrap or Bayesian posterior would show a wide interval, likely spanning the benchmark. Filing the point estimate would underprice the $1M limit and create adverse selection: buyers of higher limits would be subsidized by buyers of lower limits, and the company would attract risks that expect large liability losses. The reinsurer's benchmark is also not arbitrary; it reflects a broader pool and a longer tail. File a blended value, for example 2.05–2.10, with a documented credibility calculation and a sensitivity table. If the state allows, file a range or a phased implementation. The chief actuary's own-experience argument would be valid only if the company had thousands of claims and no censoring, which it does not.