Pareto distribution
/ pa-RAY-toh /
The Pareto distribution is the heavy-tailed model actuaries reach for when the rare giant loss dominates everything else. It is the mathematics behind the '80-20 rule' — the observation that a small fraction of events accounts for the bulk of the total. Its tail decays so slowly that enormous outcomes, while individually rare, are common enough to drive the average and the capital you must hold.
A common form has a shape parameter α (alpha) and a scale θ (theta), with tail probability P(X > x) falling off like a power of x rather than exponentially. That power-law decay is the crux: compared with a normal or even a lognormal, the Pareto assigns far more probability to extreme values. Its tail is so heavy that some Paretos have infinite variance, and if α ≤ 1 even the mean is infinite — the giant losses are large enough and frequent enough to overwhelm any average. The smaller α is, the heavier the tail and the more dangerous the risk.
In insurance the Pareto is the canonical severity model for large claims, excess-of-loss reinsurance layers, and catastrophe and liability exposures — anywhere a handful of losses dwarf the rest. It is invaluable precisely because it refuses to let you forget the tail. But it must be handled with respect: estimating α from limited data is notoriously unstable, the very largest losses are sparse by nature, and a model whose mean may not even exist demands humility about how confidently any single number can summarize the risk.
A reinsurer covering the layer above $1m models large fire losses with a Pareto. Because the tail is so heavy, doubling the policy limit can more than double the expected cost of the layer — the giant losses sit out where the Pareto puts most of its weight.
The Pareto's power-law tail is built to model the rare losses that dominate the bill.
A Pareto with shape α ≤ 1 has no finite mean and α ≤ 2 no finite variance — sample averages from such data look stable then lurch when a giant loss arrives, so ordinary intuitions about 'the average' can mislead badly.