Generalised linear model (insurance)

Category: Actuarial fundamentals · Reviewed by Tim Roche, Director · PI & Commercial · Last reviewed

Generalised linear model (insurance)

A generalised linear model (GLM) is a statistical regression framework introduced by Nelder and Wedderburn (1972) that extends ordinary linear regression to response variables with non-normal distributions and non-linear link functions. GLMs became the dominant pricing tool in personal-lines insurance from the late 1990s and remain so today.

Components

1. Random component — a response distribution from the exponential family (Poisson, gamma, Tweedie, binomial, inverse Gaussian).
2. Systematic component — a linear predictor η = β₀ + β₁X₁ + … + βₖXₖ.
3. Link function g — relates the mean to the linear predictor: g(μ) = η.

For motor frequency: Poisson with log link. For severity: gamma with log link. For pure premium: Tweedie with log link.

Why GLMs in pricing?

• Allow simultaneous estimation of multiple rating factors avoiding the bias of one-way analysis.
• Provide statistical inference (standard errors, p-values, deviance comparisons).
• Multiplicative log-link models are intuitive and align with insurer rating books.
• Software (SAS Enterprise Miner, R, Emblem) is mature and widely used.

Practical pricing pipeline

1. Data preparation — exposure, claim count, paid amount, rating factors.
2. One-way analysis — initial inspection.
3. GLM for frequency (Poisson).
4. GLM for severity (gamma) on positive claims.
5. Multiplied to obtain pure premium, or Tweedie GLM directly.
6. Smoothing and credibility weighting of factor levels.
7. Calibration and back-testing.
8. Translation to rate book.

GLMs vs machine learning

GLMs are increasingly supplemented by GBMs (gradient boosted models), random forests and neural networks for pricing. Regulators and rating bureaux generally accept ML outputs only when paired with explainability (SHAP values), bias and fair-presentation reviews, and clear governance.

References

• Nelder, J.A. and Wedderburn, R.W.M. (1972). Generalized Linear Models. JRSS A.
• Ohlsson, E. and Johansson, B. (2010). Non-Life Insurance Pricing with Generalized Linear Models. Springer.
• IFoA. Practical Guide to GI Pricing.

Cross-references

• Frequency-severity model
• Insurance pricing
• Manual rating

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Maintained by Matt Bartlett, Director, Apex Insurance Brokers Limited. FCA FRN 724952. Companies House 07014570.