Pareto severity model
/ pa-RAY-toh /
You have probably heard the rough idea that 'the biggest 20% of cases cause 80% of the cost'. That is the spirit of the Pareto distribution, named after the economist Vilfredo Pareto. In insurance it is the classic model for claim severity when a small number of very large losses dominate the total — exactly the heavy-tailed behaviour seen in liability, large property, and reinsurance.
A Pareto severity says the chance that a single loss exceeds an amount x falls off as a power of x rather than dropping sharply: roughly, the survival probability behaves like (theta/(x+theta))^alpha, with a scale parameter theta and a shape parameter alpha that controls tail weight. A smaller alpha means a heavier tail — bigger losses are relatively more likely. A striking feature: if alpha is 1 or below, the theoretical mean is infinite, and if alpha is 2 or below, the variance is infinite. The Pareto also has a neat property for insurance: the distribution of the amount above a threshold (the excess over a deductible or attachment point) is itself Pareto, which makes excess-of-loss pricing tractable.
The Pareto is a workhorse for the upper layers of risk — exactly where reinsurers and capital models live. Because its tail is governed by alpha, estimating that single shape parameter well is enormously important and notoriously hard: the data that pin down the tail are precisely the rare large claims you have few of. Two honest cautions: a Pareto fitted to mostly small-and-medium claims can extrapolate wildly in the tail where it matters most, and a Pareto with low alpha can imply an infinite mean, which is a mathematical signal that 'expected loss' is not a stable, finite quantity for that risk — a sobering thing to confront when setting a price.
A reinsurer prices a layer covering losses above 1,000,000. Fitting a Pareto with shape alpha = 1.5 to large claims, it finds the chance a loss exceeds 5,000,000 is roughly five times less likely than exceeding 1,000,000, not the astronomically smaller figure a thin-tailed model would give. Because the excess over any threshold is again Pareto, the layer's expected cost has a clean closed form.
A power-law tail: doubling the loss size cuts its probability only modestly, not drastically.
If a fitted Pareto has shape alpha at or below 1, its theoretical mean is infinite — a warning that 'average loss' is not a stable target for that risk, and that extrapolating from limited tail data is especially fragile.