Fit Claim Frequency and Severity to Data

Distribution	SIC	Mean claim US$ thousand	99th pct US$ thousand	99.9th pct US$ thousand	Used
Lognormal	3,066.9	206.2	1,989	5,748
Gamma	3,077.4	174.8	934	1,449
Pareto (type 2)	3,052.4	174.2	1,287	3,243	✔

The business question

A commercial liability insurer is setting next year's reserves and capital for a small book. It has counted its claims for ten years — about 40 a year — and has the settled amounts of 250 claims: most are below US$200,000, but a few run into millions. The obvious step is to fit distributions to these data and simulate next year.

Two questions decide how good that simulation is. Which distribution describes the claim sizes, especially the rare large ones that drive the 1-in-200 year? And how sure can the insurer be of the fitted parameters, with only ten years and 250 claims?

The model fits three candidate severity distributions, ranks them by SIC, compares their tails, and simulates next year's claims with the fitted parameters fixed and with their uncertainty.

Who it is for: Pricing and capital actuaries, reserving teams, risk modellers fitting distributions to loss data.
Main outputs: SIC ranking of three severity distributions and their tails · fitted claim rate with its uncertainty · next year's total claims with and without parameter uncertainty: mean, 95th and 99.5th percentiles.

How the model works

Three sheets after this one, read from left to right. Each one feeds the next.

Inputs sheet
Ten years of claim counts and 250 claim amounts, one column each (Data sheet). Which severity distribution to use, from a list — the best by SIC, or one chosen by hand.
Fits sheet
A Poisson is fitted to the counts; Lognormal, Gamma and Pareto (type 2) distributions to the amounts, each ranked by SIC. Next year's claims are simulated with VoseAggregateMC twice: with the fitted parameters fixed (Uncertainty = FALSE), and with their uncertainty (Uncertainty = TRUE), which redraws plausible parameters from the data in every iteration.
Results sheet
The candidates' SIC and their 99th and 99.9th percentile claims; the fitted claim rate and its range; next year's total claims with and without parameter uncertainty.

ModelRisk techniques

Fit objects VoseLognormalFitObject
Model selection VoseSIC
Tail percentiles VosePercentile
Parameter uncertainty
Aggregate claims VoseAggregateMC

Find similar models

The results in detail

Next year's claims

Total claims (US$m) with the fitted parameters fixed, and with their uncertainty.

Next year's total claims, with parameter uncertainty

Histogram of With parameter uncertainty from the saved simulation.

Measure	Unit	10th pct	Median	90th pct	Mean
Total claims, parameters fixed	US$m	4.51	6.68	9.78	6.98
Total claims, with parameter uncertainty	US$m	4.42	6.63	9.94	7.03
Claim rate, with parameter uncertainty	a year	37.4	40.0	42.6	40.0

Measure	Unit	Value
1-in-200 total, parameters fixed (99.5th percentile)	US$m	15.97
1-in-200 total, with parameter uncertainty	US$m	17.39
Increase from parameter uncertainty		9%

How to read these results

The three distributions fit the bulk of the claims almost equally well, so the SIC values are close. They part company in the tail, where the data are thinnest: the 1-in-1,000 claim differs several-fold between them. For reserves and capital, the tail is what matters — check it against the largest claims you know of.
The Gamma distribution has a light tail and badly understates large claims. The Pareto (type 2), chosen by SIC here, has a heavy one, as liability claims usually do.
Ten years and 250 claims do not pin the parameters down. With their uncertainty included, the 1-in-200 total is higher than with the fitted parameters treated as exact — reserves and capital set on fixed parameters are too low.
Both 1-in-200 figures rest on only 50 of the 10,000 samples and move by several per cent from run to run: run 100,000 samples before setting capital on them.
Parameter uncertainty shrinks as data accumulate; it never disappears. Always fit with Uncertainty = TRUE for decisions that depend on the tail.

See it move. Open the workbook, change an input on the Inputs sheet and run the simulation: every table and chart on this page updates in Excel.

Download the Excel model Get the free ModelRisk trial

Inputs and assumptions

Every assumption in the model, with its uncertainty and where it comes from. The figures are illustrative.

Severity distribution 1 input

Which distribution describes the claim amounts.

Assumption	Unit	Value
Distribution to useBest by SIC picks the one with the lowest SIC on the Fits sheet.		Best by SIC

Units and conventions

Money: Claim amounts in US$ thousand; totals in US$ million (US$m).
SIC: Schwarz information criterion = k·ln(n) − 2·ln(maximum likelihood), as VoseSIC returns it: the lower, the better the fit, after a penalty for the number of parameters. (ModelRisk's fitting windows show its negative.)
Parameter uncertainty: How far the fitted parameters could be from the true ones, given the amount of data.

Key assumptions and limitations

Claims: Claim sizes are independent and identically distributed; amounts are already inflated to next year.
Frequency: Claim numbers are Poisson with a rate estimated from the ten years.
Data: The claims are invented for this example (drawn from a heavy-tailed distribution).

How to use it

Paste your claim counts and claim amounts into their columns on the Data sheet: any number of years and claims. The model reads every number in each column.
Look at the SIC ranking and the tails on the Fits sheet; choose a distribution on the Inputs sheet if needed.
Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set). It takes a few minutes: every iteration refits the three distributions; 1,000 samples are enough while exploring.
Use the totals with parameter uncertainty for reserves and capital: they include what the data cannot tell.

Simulate this model with ModelRisk’s classic engine: it does not run in Turbo mode yet.

ModelRisk functions used

VosePoissonFitObject, VoseLognormalFitObject, VoseGammaFitObject, VosePareto2FitObject: Distributions fitted to data, without (FALSE) or with (TRUE) parameter uncertainty.
VoseSIC: The Schwarz information criterion of a fit: lower is better.
VosePercentile, VoseMean: Percentiles and the mean of a fitted distribution object.
VoseAggregateMC: Next year's total: a claim number, then one claim size for every claim.
VoseSimPercentile, VoseSimMean: Statistics of the simulation on the Results sheet.

Related example models

All example models

An illustrative example: the figures are invented for teaching and do not describe a real company, project or asset. Last updated 9 October 2026.

Fitting Claim Frequency and Severity to Data

Which distribution fits the claim sizes?