claim severity distribution
Now flip from 'how many claims' to 'how big is one claim'. Of all the claims that did happen, most are small — a dented bumper, a cracked window — but every so often there is a huge one, like a house fire or a serious injury lawsuit. If you collect the dollar amount of each claim and look at how those amounts are spread out, you get the claim severity distribution.
It is the probability distribution of X, the size of a single claim, where X is a positive amount (and usually continuous — it can be 1,000 or 1,247.36). Because losses cannot be negative and a few are enormous, severity distributions are typically right-skewed: a long tail stretching out to large values. Common choices are the exponential, gamma, lognormal, Weibull, and Pareto. We summarise severity by its mean (average claim size) but also by its tail, because the rare giant claim drives much of the risk and the capital needed.
Severity is the second half of the frequency-severity story and is where policy features bite. A deductible removes the bottom slice of each loss, a policy limit chops off the top, and coinsurance scales everything down — all of these reshape the severity distribution that the insurer actually pays on. The crucial honest point: the mean alone is dangerously incomplete for severity. Two books can share an average claim of 5,000, yet one has a fat tail that occasionally produces a 5,000,000 loss; that tail, not the mean, determines solvency, reinsurance needs, and price.
A fire insurer records claim sizes of 400, 900, 1,500, 2,200, 3,000 and one outlier of 480,000. The mean is dragged up to about 81,000 by that single huge loss, even though five of the six claims are under 3,000. A lognormal or Pareto fit captures this long right tail far better than a symmetric model.
Most claims are small, a few are enormous — severity is right-skewed with a long tail.
Never describe severity by its average alone. The tail — the chance and size of the rare giant claim — usually drives the real risk, and an average can hide a distribution that occasionally bankrupts an unprepared insurer.