Frequency and Severity Models
Select loss models from the insurance question, observation process, and tail—not from a distribution list.
Frequency and Severity Models
Probability models turn incomplete claim experience into statements about unobserved losses. The right starting point is the insurance process:
Model-selection map
| Question | Data unit | First models | Essential complication |
|---|---|---|---|
| How large is an ordinary positive claim? | one ground-up loss | Gamma, lognormal | deductible, limit, inflation |
| How heavy is the extreme tail? | threshold exceedance or event loss | Pareto, GPD | threshold uncertainty, dependence |
| How many claims occur? | count plus exposure period | Poisson, negative binomial | exposure, heterogeneity, seasonality |
| Which model predicts unseen claims? | training and validation samples | likelihood-based candidates | censoring, truncation, parameter uncertainty |
| Why are there several claim populations? | latent or observed groups | mixture/hierarchical model | identifiability and changing mix |
No distribution is “the insurance distribution.” A body model that predicts typical claims well can fail at reinsurance layers; a tail model can waste information if fitted to every small claim.
The four objects to report
For a loss :
| Object | Definition | Insurance use |
|---|---|---|
| CDF | probability a loss is no larger than | |
| Survival | attachment and exceedance probability | |
| Quantile | VaR and scenario thresholds | |
| Stop-loss mean | expected unlimited layer loss above |
The mean alone is not enough. Two models can share and imply radically different high quantiles and reinsurance costs.
Observation comes before fitting
For every sample, label whether losses are:
- ground-up, insurer-paid, or reinsurer-paid;
- exact, right-censored at a limit, or truncated below a reporting threshold;
- nominal or adjusted to a common price level;
- per claim, occurrence, policy, claimant, or coverage;
- independent observations or claims clustered by event/policyholder.
The severity pages begin with ordinary positive losses, tail models focus on extremes, count models use exposure, fitting respects the observation mechanism, and mixtures represent heterogeneity.
A compact analysis protocol
- Define the target random variable and decision.
- Reconcile and visualise raw and transformed data.
- Choose several plausible models from mechanism and support.
- Fit using the correct likelihood.
- check centre, tail, calibration, and held-out prediction separately.
- quantify parameter and model sensitivity.
- apply policy and reinsurance terms only on the stated loss basis.
In this course, Poisson, Gamma, and Pareto name distributions; frequency, severity, survival, and hazard name concepts. Symbols are defined locally because parameterisations differ across textbooks and software.