Catastrophe Model Blending and a Per-Occurrence XoL Purchase Decision
A coastal homeowners writer has $2.4B of total insured value concentrated in one state. Management is buying a per-occurrence catastrophe excess-of-loss program. Two vendor models are available. Model A produces a 1-in-100 occurrence loss of $210M and a 1-in-250 of $360M; Model B produces $150M and $240M. Model A's occurrence exceedance probabilities are 0.070 at $100M, 0.045 at $150M, 0.028 at $200M, 0.018 at $250M, and 0.009 at $350M. Model B's are 0.050, 0.030, 0.018, 0.010, and 0.004 at the same thresholds; both models assume no loss above $500M. The company's own 40-year experience has no loss above $90M, but the record includes a near-miss hurricane that missed the peak exposure by 30 miles. The proposed layer is $100M excess of $100M, quoted at $14M. The CFO wants to know whether to buy it. The state regulator requires the company to demonstrate that its catastrophe load is not double-counting reinsurance. The company's target return on capital implies a 12% cost of capital on retained catastrophe risk.
1. Estimate the expected annual recovery from the $100M xs $100M layer under each model, using the provided exceedance probabilities, and assess the $14M quote.
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Use the layer formula E[recovery] = ∫ from 100 to 200 of S(y) dy. Approximate each segment with the trapezoid of survival probabilities. Model A: 100–150 is 50 × (0.070+0.045)/2 = 2.875; 150–200 is 50 × (0.045+0.028)/2 = 1.825; 200–250 is 50 × (0.028+0.018)/2 = 1.150; 250–350 is 100 × (0.018+0.009)/2 = 1.350; 350–500 is 150 × (0.009+0)/2 = 0.675. Total ≈ 7.9M. Model B: 100–150 is 50 × (0.050+0.030)/2 = 2.00; 150–200 is 50 × (0.030+0.018)/2 = 1.20; 200–250 is 50 × (0.018+0.010)/2 = 0.70; 250–350 is 100 × (0.010+0.004)/2 = 0.70; 350–500 is 150 × (0.004+0)/2 = 0.30. Total ≈ 4.9M. A 50/50 blend is about 6.4M. The $14M premium is about 2.2 times the blended expected recovery, or 1.8 times the higher model. Cat XoL layers often price at 1.3–2.0 times expected loss, so the quote is at the high end but not absurd, especially for a layer with high model uncertainty and a near-miss history. The expected recovery is sensitive to the tail assumption above $350M; if Model A's tail is thicker, expected recovery could be 8–9M, making the quote closer to 1.6 times. Do not evaluate the quote on expected loss alone; include capital relief and the cost of retaining the risk.
2. How should the company blend the two models and reflect the near-miss experience? How should the catastrophe load avoid double counting the reinsurance?
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Blend the models with weights based on out-of-sample validation, not equal weights by default. If both models were calibrated to the same historical storms, use a 50/50 weight or a 60/40 weight toward the model that better reproduces the company's own loss experience and the near-miss footprint. Apply a model-uncertainty load to the blended EP curve - typically 20–30% on the tail probabilities - because the 1-in-250 estimate is not a statistical estimate with a known standard error. The near-miss is not a loss observation and should not be entered as a data point; it is evidence that the tail is plausible and should prevent the company from discounting the model output. For the rate filing, the catastrophe load should be the net cost of risk: expected retained loss after reinsurance, plus the cost of reinsurance (premium minus expected recovery), plus the cost of capital on retained risk. Do not add the gross cat load and then also expense the reinsurance premium; that double counts the transferred loss. Show the regulator a gross and net cat load and reconcile them.
3. The CFO asks whether to buy the $14M layer. Give a recommendation using the capital-relief calculation and the model results.
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Use the 1-in-250 as a capital metric. Without the layer, the retained 1-in-250 loss is $360M under Model A or $240M under Model B; with the layer, it is $260M or $140M. The layer therefore reduces required capital by $100M at that return period, if the regulator and rating agency give full credit. At a 12% cost of capital, the annual benefit is $12M. The net cost of the layer is premium minus expected recovery: $14M − $7.9M = $6.1M under Model A, or $14M − $4.9M = $9.1M under Model B; the blended net cost is about $7.6M. On those numbers the capital relief exceeds the net cost by roughly $4M per year, so buying is justified. Two caveats: first, the capital credit may be less than 100%, especially if the regulator uses a different return period; at 50% credit the benefit is $6M, roughly break-even. Second, the 1-in-250 is a single point; the company should compute the reduction in the full capital distribution, for example TVaR at 99.5%, and include the reinsurer's credit risk. Recommendation: buy, but negotiate the price toward $11–12M or restructure to a $75M xs $100M layer, which would reduce the net cost while preserving most of the capital benefit.