Credibility and Large-Loss Treatment for a New Commercial Auto Segment
A specialty carrier has written commercial auto for food-delivery fleets since 2022. Three accident years (2022-2024) are now available: on-level earned premium is $4.20M and reported incurred loss and LAE is $3.90M, a 92.9% ratio. The book averages 310 policies and about 585 ultimate claims over the three years. Two claims dominate the experience: a 2022 intersection accident reserved at $1.20M and a 2023 fatality reserved at $0.90M. The company's credibility standard is full credibility at 1,082 claims (limited fluctuation, 90% confidence, 5% precision). The pricing actuary's prior loss and LAE ratio for this class, before any large-loss load, is 72.0%. An external excess loss study indicates that losses above a $250,000 per-claim threshold add 12.0 percentage points to the loss and LAE ratio for this class. The company reinsures losses above $1.0M per claim. The chief underwriter wants to non-renew the segment, citing the 92.9% ratio.
1. Cap the experience at $250,000 per claim and compute the capped loss and LAE ratio. Explain what capping is intended to accomplish here.
Read the worked answer
Capping. The $1.20M claim becomes $0.25M, removing $0.95M. The $0.90M claim becomes $0.25M, removing $0.65M. Total removed $1.60M.
Capped incurred = 3.90 - 1.60 = $2.30M; capped loss and LAE ratio = 2.30 / 4.20 = 54.8%.
What capping accomplishes. The two claims are not evidence that the expected ratio is 92.9%. They are two draws from a large-loss distribution that is highly skewed, and in a 310-policy book neither their presence nor their absence says much about the mean of that distribution. The external excess study estimates the mean of the layer far more efficiently because it draws on far more exposure. Capping separates the stable, high-frequency component, which the segment's own data can measure, from the volatile component, which should be priced from aggregate data.
Two cautions before using the cap. First, check the reserve adequacy on the two claims themselves; a fatality may still develop, and the $1.20M claim may be understated. Second, the cap at $250,000 does not match the $1.0M reinsurance attachment, so the retained layer between $250,000 and $1.0M sits inside the experience and must be covered consistently by the load, not left as a gap.
2. Credibility-weight the capped experience against the prior and add the large-loss load. Show each step.
Read the worked answer
Credibility. Limited fluctuation on claim counts: Z = sqrt(585 / 1,082) = sqrt(0.5407) = 0.735
Complement. The pricing actuary's 72.0% class ratio. It must be on the same basis as the experience, meaning it must be a capped (before large-loss) ratio; if the 72.0% already includes large losses, it must be restated to a capped basis before weighting. Consistency of basis matters more than the arithmetic.
Weighting.
- 0.735 x 54.8% = 40.3%
- 0.265 x 72.0% = 19.1%
- Credibility-weighted capped ratio = 59.4%
Large-loss load. 59.4% + 12.0 points = 71.4%.
The complement should not be the observed 92.9%: that is the statistic under test, and using it as its own complement gives the experience full credibility by the back door. Nor should the complement be a different segment or a different retention. If the prior is not credible enough to serve as a complement, the correct response is to widen the range and disclose it, not to substitute the raw experience.
3. Given a permissible loss and LAE ratio of 65.0% at current rates, compute the indicated change and respond to the underwriter's non-renewal proposal.
Read the worked answer
Indication. 71.4% / 65.0% - 1 = +9.8%. The unadjusted 92.9% would imply about +43%, so roughly three quarters of the apparent problem is two claims.
Responding to the non-renewal proposal.
- Credibility. 585 claims is only 73.5% credible, and essentially all of the deviation sits in the least credible part of the distribution. A 310-policy book will produce two large claims in some three-year windows and none in others; the ratio will swing violently with no change in the underlying book.
- Maturity. The segment began in 2022, so 2024 and 2025 are not fully developed and part of the indicated deterioration may be reserve immaturity rather than real loss emergence.
- Internal consistency. The $250,000 cap and the $1.0M reinsurance attachment do not match. The retained $250,000 to $1.0M layer is inside the capped experience but outside the external 12-point load. Either raise the cap to the retention and derive the load from a lower layer, or keep the cap and source a load that covers the same retained range.
- Specific claim follow-up. Confirm the fatality reserve and whether the $1.20M claim is expected to breach the $1.0M reinsurance attachment; the recoverable changes the net view.
Recommendation. File the +9.8% indication, disclose the range and the treatment of the two claims, and monitor frequency by fleet size and by delivery radius, since those are the exposures most likely to drive the large-loss frequency. Exiting the segment is a decision about appetite for volatility, not a conclusion supported by this loss ratio.