Experience Rating an Employer's Liability Risk with One Large Claim
The Appointed Actuary is rating a group employer's liability policy for a textile group. The coming year's exposure is 12,000 employee-years and the current annual premium is Rs 28.8 lakh, i.e. Rs 240 per employee-year, unchanged for three years. The insurer's own five-year textile book is 400,000 employee-years with 640 claims and an average claim cost of Rs 62,000. The prospect's own five-year experience is 60,000 employee-years and 96 claims totalling Rs 112 lakh, including one boiler explosion claim of Rs 52 lakh; no other claim exceeded Rs 8 lakh. Industry data indicates a boiler explosion type loss occurs about once in 50,000 employee-years in this industry, with an average cost of about Rs 45 lakh. The class expense ratio is 25% and the board's target underwriting profit margin is 5%.
1. Derive the attritional loss cost for the prospect and for the insurer's book, and decompose the difference between the two into frequency and severity.
Read the worked answer
Prospect, excluding the large claim. 95 claims totalling Rs 60 lakh over 60,000 employee-years.
- Frequency = 95/60,000 = 0.001583 claims per employee-year
- Severity = 60,00,000/95 = Rs 63,158
- Attritional loss cost = 60,00,000/60,000 = Rs 100 per employee-year
Insurer's textile book. 640 claims, average Rs 62,000, over 400,000 employee-years.
- Frequency = 640/400,000 = 0.001600
- Severity = Rs 62,000
- Loss cost = (640 x 62,000)/400,000 = 3.968 crore/400,000 = Rs 99.20 per employee-year
Decomposition. The prospect's attritional loss cost is 100/99.20 = 1.008, i.e. less than 1% above the book. Of that, frequency is 0.001583/0.001600 = 0.990 (1% below book) and severity is 63,158/62,000 = 1.019 (1.9% above book). The two roughly offset. The prospect is not a materially worse risk than the book for attritional losses, and any rate action must therefore come from the large-loss exposure and from the fact that the current Rs 240 rate has not been reviewed for three years - not from a claim that the client's ordinary claims experience is poor.
2. Apply credibility to the attritional loss cost, add an appropriate large-loss load, and derive the indicated premium and rate change.
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Credibility. Full-credibility standard on claim counts, n_full = (1.645/0.05)^2 = 1,082 claims. The prospect has 95 attritional claims, so Z = sqrt(95/1,082) = 0.296. The prospect's own attritional experience gets only 30% weight; the complement is the insurer's textile book at Rs 99.20 per employee-year.
Credible attritional loss cost = 0.296 x 100 + 0.704 x 99.20 = 29.6 + 69.8 = Rs 99.4 per employee-year.
Note how little the credibility weighting changes the answer here - 99.4 against a book rate of 99.2 - because the prospect and the book are so close. Credibility only bites when the segment genuinely diverges; it should still be computed and disclosed, because the next renewal may diverge sharply.
Large-loss load. Do not add the Rs 52 lakh claim into the average; it is one observation and would swamp the average. Load it from frequency and severity: 1/50,000 x Rs 4,500,000 = Rs 90 per employee-year. The observed one claim in 60,000 employee-years is consistent with the assumed 1 in 50,000, so no frequency revision is warranted.
Total expected loss cost = 99.4 + 90 = Rs 189.4 per employee-year.
Indicated premium. Permissible loss ratio = 1 - 0.25 - 0.05 = 0.70. Required premium per employee-year = 189.4/0.70 = Rs 270.6. Annual premium = 270.6 x 12,000 = Rs 32.47 lakh, against Rs 28.8 lakh currently. Indicated change = +12.8%.
3. The client's broker argues the Rs 52 lakh claim was a one-off and should be ignored altogether. Respond, and set out how you would structure the renewal to keep the account while protecting the insurer.
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The broker is half right. Excluding the claim from the attritional average is correct, because a single Rs 52 lakh event inside a Rs 60 lakh attritional total would distort the average of 95 ordinary claims by a factor of nearly two. But excluding it from the price is not: the exposure that produced it still exists. The explosion hazard is a property of the boiler plant, not a random accident, and if the plant is unchanged the hazard is unchanged. The honest answer to the broker is: it has been excluded from the average and separately priced at a loaded cost of Rs 90 per employee-year, which is 47% of the total expected loss cost and is not a rounding item.
Structuring the renewal while keeping the account:
- Cap the rate movement. File +12.8% but offer a staged movement, e.g. +8% now with +5% next year conditional on the risk-improvement actions below, and disclose the uncapped indication.
- Make the large-loss load conditional on risk control. Ask for the boiler inspection certificate, pressure-vessel maintenance records and a documented hot-work permit system. If the insurer's engineer confirms improvement, the load can be reduced in the following year - put that in writing.
- Use a per-risk deductible or self-retention. A Rs 10 lakh deductible on employer's liability removes the frequency layer and gives the client a reason to control small claims, in exchange for a rate credit.
- Consider a per-risk excess of loss for the insurer's own book if this type of exposure is common across the textile account - if a Rs 52 lakh event occurred twice in five years, the load would need to be revised sharply upward, and reinsurance becomes cheaper than the load.
Be explicit in the file note about the risk of winning the account at Rs 240. The 12.8% indication is modest precisely because the credibility weight is low; that low weight cuts both ways.