Industry: Insurance and Reinsurance Product: ModelRisk Application: Policy Pricing
The deterministic GLM on a coastal-Florida homeowners book reports an expected annual claim of $1,180 per policy. The Monte Carlo simulation on the same exposure data reports an unconditional mean of $3,311 — and a 99th-percentile annual claim of $96,185, eighty times the GLM mean. The deterministic engine is wrong on the average AND on the tail, and on exactly the cells where being wrong matters most to the carrier's solvency. The tail is what bankrupts insurers, and the tail is what point estimates cannot price.
The chart above is the entire argument in one picture. Only 14% of policy-years on this cell see any claim at all, so the unconditional mean of $3,311 is dwarfed by a conditional mean of $23,384 among the years that do, and the 99th percentile reaches $96,185 with a TVaR-99 of $186,668. A national personal-lines carrier rebuilt its homeowners pricing model in ModelRisk across 1.8 million in-force policies in 12 states. The deterministic GLM-based pricing engine still runs — it is fast and regulator-familiar — but every quoted premium is now overlaid with a Monte Carlo "true-cost" distribution, and any policy where the deterministic premium sits below the 60th percentile of simulated claim cost is flagged for actuarial review before the bind.
The deterministic engine assumed a constant Poisson rate \(\lambda\) per rating cell. The simulation replaces the constant with a Negative Binomial count distribution — equivalent to a Poisson with a Gamma-distributed rate, which correctly captures the over-dispersion seen in 8 years of historical claim counts (variance/mean ratio of 1.8, not 1.0). For a representative coastal cell, the mean annual claim count is 0.21, the P90 is 1, and the P99 is 3 — the carrier's median policy has zero claims most years, but the right tail of frequency is real.
For severity per claim, the model uses a LogNormal body for the bulk of routine claims (water damage, wind, theft) — \(\mu = 8.2\), \(\sigma = 1.35\), giving a body mean of about $9,400 per claim — spliced to a Generalized Pareto tail above $100,000 to capture catastrophic loss events (whole-house fire, total-loss hurricane). The single-LogNormal version the deterministic engine used understated the 99th-percentile claim by 38% on coastal cells — the difference between pricing a policy adequately and writing it at a loss.
For each iteration the engine samples a count \(N\) from the Negative Binomial, then \(N\) severities from the spliced distribution, and sums. The opening chart is the result of 100,000 iterations on a representative cell, and it rewards a second look.
The deterministic GLM reported an expected annual cost of $1,180 and applied a fixed 22% risk load to get a premium of $1,440. The simulation says: unconditional mean = $3,311, P99 = $96,185, TVaR 99 = $186,668. Most policy-years (86%) see zero claims, but among the 14% that do, the conditional mean is $23,384 — and the right tail above $100k is real and frequent enough to matter. The 22% load isn't slightly low; on this cell it would need to roughly double to keep the long-run loss ratio in target range, and even then the tail would still consume capital. The right answer was a structurally different rating plan that priced per-claim severity by sub-zone, not a flat load on a GLM mean.
The deterministic engine — and the original draft of the new probabilistic version — handled claims inflation as geometric Brownian motion. GBM compounds at a constant drift with constant volatility and never switches regime — it is appropriate for asset prices, not for a CPI series that has been through a regime shift. The team replaced GBM with a two-state regime-switching process: a low-inflation state (mean 2.5%, vol 0.7%) and a high state (mean 6.0%, vol 1.5%) with persistence 0.85. The 99th-percentile claim-inflation path under the regime-switching model is materially fatter than under GBM, and the loss-ratio backtest against the post-2021 period is six times more accurate.
Claims cost depends on more than the policy itself. Three external drivers feed every iteration:
Once the cost distribution is in place, the pricing decision becomes a distribution comparison. The current pricing ("22% load") was compared against an actuarially adequate "31% load" and a competitive "16% load":
Over a five-year cumulative view on this cell the differences are stark in the tail. The adequate (+31%) strategy holds a mean loss ratio of 1.03 with a P90 of 4.62; the current (+22%) load runs a mean of 1.07 and a P90 of 4.96; the competitive (+16%) load pushes the mean to 1.09 with a P90 pinned at 5.00. None of the three loads tames the catastrophe-driven right tail on this cell — even the adequate load expects to break even at best — which is precisely the point: a flat percentage load on a GLM mean cannot price a coastal book, and the simulation is what makes that trade-off visible.
Sensitivity ranking on the recommended premium per policy:
The biggest mover is the catastrophe-shock probability, varying it across its plausible 4%–12% range moves the recommended premium by ±$210 per coastal policy. Severity tail-shape \(\xi\) and demand-surge multiplier come next. That ranking redirected the pricing team's external-data spend to a hurricane-frequency research subscription rather than further refinement of the routine-claim severity body.
Premium-setting on a fixed-percentage risk load is pricing the mean; premium-setting on a simulated cost distribution is pricing the shape. On long-tail personal-lines books with catastrophe exposure, the shape is the only thing that distinguishes a profitable book from a market-share book the company cannot afford.