When events happen, how bad are they? Loss severity models
Interactive loss severity explorer comparing Log-Normal, Pareto (heavy-tail) and Gamma distributions. Adjust parameters and see the mean, median, 95th and 99th percentile claim sizes, and the share of total cost from the largest 5% of claims, update in real time.
Severity models describe the size of a loss given that one has occurred. Claim amounts, recovery costs and market drawdowns share two features: they are non-negative and right-skewed, so the mean sits above the median and a small share of large losses accounts for a large share of the total.
Three distributions cover most practical cases. The Log-Normal suits moderate, attritional losses. The Pareto has a power-law tail: as its tail index α approaches 1 the mean becomes unstable and then infinite, which is why liability and catastrophe lines are priced on the tail rather than the average. The Gamma is flexible with a lighter tail, often used for property damage and healthcare costs.
The module shades everything beyond the 95th percentile and reports how much of the expected cost those largest 5% of claims carry: about 20% under the default Log-Normal, 17% under the default Pareto (α = 2.5) and 15% under the default Gamma, rising above 60% for a Pareto with α = 1.2. That gap between the typical claim and the tail is what reinsurance and capital exist for, and the reason this site is called Beyond Averages.
Log-Normal probability density
f(x) = 1 / (x σ √(2π)) · exp(−(ln x − μ)² / 2σ²)
μ and σ are the mean and standard deviation of ln x (defaults 8 and 0.8). The mean claim is exp(μ + σ²/2), about $4,105 at the defaults, and the median exp(μ), about $2,981; the p-th percentile is exp(μ + z_p σ), where z_p is the standard normal quantile (1.645 for the 95th).
Frequently asked questions
What is the difference between Log-Normal and Pareto severity?
Both are right-skewed, but the Pareto tail decays as a power law, so extreme losses are far more likely. Under a Pareto with α ≤ 2 the variance is infinite; with α ≤ 1 even the mean is infinite. The Log-Normal's tail decays faster, so very large losses are rarer and every moment is finite.
Why do actuaries care about the 95th and 99th percentile of claim size?
Pricing uses the mean, but capital, reinsurance attachment points and reserve adequacy depend on the tail. Under the default Log-Normal here the 99th percentile (about $19,000) is nearly five times the mean: a book where a handful of claims can decide the year's result.
Which distribution should I use for insurance claim sizes?
Fit candidates to data and compare tail behaviour, not just goodness of fit at the centre. Common starting points are the Log-Normal for attritional property and motor claims, the Gamma for moderately skewed costs such as healthcare, and a Pareto or spliced model for liability and catastrophe excess layers.
Interactive explorer
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