Actuarial Statistics · Generalised linear models

Applications of GLMs in Actuarial Work: Frequency, Severity and Logistic Models

Updated 11 October 2026 · Fact-checked

A GLM links the mean of a response to rating factors through a link function. For claim counts use Poisson with log link and an offset of log exposure. For claim sizes use gamma with log link. For yes/no outcomes use binomial with logit link. Interpret exp(coefficient) as a multiplier against the base level.

Understand Applications of GLMs in Actuarial Work

Insurers need to price risks using many factors at once, such as age, vehicle type and area. One-way tables mislead because factors overlap. A generalised linear model (GLM) fits all factors together and gives each its own effect.

A GLM has three parts: a response distribution from the exponential family, a linear predictor η = β0 + β1x1 + ..., and a link function g with g(μ) = η. In practice, claims work splits into two models. Frequency is the number of claims per unit of exposure. Severity is the average size of a claim, given that a claim occurred.

For frequency, the usual choice is Poisson with a log link. The log link gives a multiplicative model: μ = exp(β0) × exp(β1x1) × ... Each factor level multiplies the expected claims by a relativity. For severity, the gamma distribution with a log link is standard, because claim sizes are positive and right-skewed, and the variance grows with the square of the mean.

Exposure differs between policies. A policy in force for 6 months should expect fewer claims than one in force for 12 months. The offset handles this. With a log link you write log(μ) = log(exposure) + η. The term log(exposure) has its coefficient fixed at 1, not estimated. Then the model predicts the claim rate per unit of exposure.

For a binary outcome, such as whether a policyholder lapses or makes a claim, use logistic regression: binomial response with logit link, logit(p) = ln(p ÷ (1 − p)) = η. The coefficients are changes in log-odds. exp(β) is an odds ratio. Pricing combines the two models: expected cost per exposure = frequency × severity.

Key rules to remember

GLM structure: g(μ) = η = β0 + β1x1 + ... + βpxp; g is the link function. Y is from the exponential family with mean μ.
Poisson frequency with offset: ln(μ) = ln(e) + β0 + β1x1 + ... so μ = e × exp(β0 + β1x1 + ...); e is exposure. The offset coefficient is fixed at 1, not estimated.
Relativity under log link: relativity = exp(β); Multiplier on the mean for that level against the base level, other factors held fixed.
Logit link: logit(p) = ln(p ÷ (1 − p)) = η, p = exp(η) ÷ (1 + exp(η)); Gives p between 0 and 1.
Odds ratio: odds ratio = exp(β); Multiplier on the odds p ÷ (1 − p) for a one-unit increase in x, or for a level against the base.
Pure premium: expected cost per exposure = expected frequency × expected severity; Valid when the two models are treated as independent given the rating factors.
Gamma severity: Var(Y) = φ μ²; φ is the dispersion parameter. Log link is common.

How to solve Applications of GLMs in Actuarial Work questions

Use this order for any GLM application question, whether it asks you to choose a model, read output or compute a prediction.

1Identify the response: count, claim amount, or binary outcome. This fixes the distribution family: Poisson, gamma or binomial.
2Choose the link. Use log for multiplicative Poisson and gamma models. Use logit for binomial. State why, in one sentence.
3Check exposure. If policies have different periods of cover, include an offset of ln(exposure) for a log-link count model.
4Write the linear predictor for the case in the question, including the base level of each factor. Factor levels use dummy variables and the base level has β = 0.
5Compute η, then invert the link: μ = exp(η) for log, p = exp(η) ÷ (1 + exp(η)) for logit. Add the offset before exponentiating.
6Interpret: exp(β) is a relativity or odds ratio. Say what it multiplies, and note that other factors are held constant.
7Where asked, assess the model using deviance, standard errors or p-values, and comment on fit or overdispersion.

Quickest way: Link, then exponentiate

When to use it: Use this for MCQs and short prediction questions where the model coefficients are given.

Add the intercept and the coefficients for the levels that apply. Skip any base level.
If there is an offset, add ln(exposure) to the sum.
Apply the inverse link: exp for log, exp ÷ (1 + exp) for logit.
For a comparison between two groups, just take exp of the difference in coefficients. The intercept cancels.
Check the result is sensible: a probability must be between 0 and 1, and a Poisson mean must be positive.

Common mistakes in Applications of GLMs in Actuarial Work

Reading a log-link coefficient as an additive change in claims.
Linear regression habits carry over.
Fix: Under a log link the effect is multiplicative. Report exp(β) as a relativity. A β of 0.2 means about 22% more, not 0.2 more claims.
Leaving out the offset, or estimating a coefficient for exposure.
Exposure is treated as just another covariate.
Fix: Put ln(exposure) in as an offset with coefficient fixed at 1. Then the model predicts claims per unit of exposure.
Forgetting to take the log of exposure.
Students add exposure directly to the linear predictor.
Fix: With a log link the offset is ln(exposure). Adding raw exposure breaks the multiplicative structure.
Reading a logistic coefficient as a change in probability.
The coefficient is on the log-odds scale, which feels like the probability scale.
Fix: exp(β) multiplies the odds, not the probability. Convert back with p = exp(η) ÷ (1 + exp(η)) to get probabilities.
Using a normal model for claim severity.
Normal is the default from linear regression.
Fix: Claim sizes are positive and skewed. Use gamma (or lognormal on a log scale) and state that variance rises with the mean.
Including the base level's coefficient or mixing up which level is the base.
R's output omits the base level, so it is easy to lose track.
Fix: Identify the missing level in the output. It has β = 0 and is absorbed in the intercept.

Worked examples

Example 1

A Poisson GLM with log link models motor claim counts. The fitted model is ln(μ) = ln(exposure) − 2.30 + 0.40 (if age under 25) + 0.15 (if urban). A policy has exposure 0.5 years, the driver is under 25 and the car is garaged in an urban area. Find the expected number of claims and the relativity of urban to non-urban.

Show the solution

Linear predictor without offset: η = −2.30 + 0.40 + 0.15 = −1.75.
Add the offset: ln(0.5) = −0.6931. Total = −1.75 − 0.6931 = −2.4431.
μ = exp(−2.4431) = 0.5 × exp(−1.75).
exp(−1.75) = 0.17377, so μ = 0.5 × 0.17377 = 0.0869.
Urban relativity = exp(0.15) = 1.1618.

Answer: Expected claims ≈ 0.087. Urban policies have about 16.2% higher claim frequency than non-urban, other factors fixed.

Example 2

A logistic regression for whether a policyholder lapses gives logit(p) = −1.2 + 0.8 × (premium increase over 10%) − 0.5 × (has multi-policy discount). (a) Find the lapse probability for a policyholder with a premium increase over 10% and no discount. (b) Find the odds ratio for the discount.

Show the solution

(a) η = −1.2 + 0.8 = −0.4.
Odds = exp(−0.4) = 0.6703.
p = odds ÷ (1 + odds) = 0.6703 ÷ 1.6703 = 0.4013.
(b) Odds ratio = exp(−0.5) = 0.6065.
This means the discount multiplies the odds of lapse by 0.6065, holding the premium increase fixed.

Answer: (a) p ≈ 0.401. (b) Odds ratio ≈ 0.607, so the discount cuts the odds of lapse by about 39%.

Exam tips

Always state the distribution, the link and the reason for each. Marks are usually given for the justification, not just the name.
When asked about exposure, say the word offset and give ln(exposure) with coefficient 1.
In R output, find the base level, the estimate column, the standard errors and the deviance lines. Say what each tells you before you interpret it.
Convert coefficients to exp(β) and write the meaning in a plain sentence, such as 'claim frequency is 16% higher, other factors fixed'.
Show full working for predictions. Even if you slip on arithmetic, the correct method earns marks.

Practice questions from Generalised linear models

Applications of GLMs in Actuarial Work: frequently asked questions

What is an offset in a Poisson GLM?

An offset is a term in the linear predictor with a coefficient fixed at 1. For claim counts it is ln(exposure) under a log link. It makes the model predict the claim rate per unit of exposure.

Why use a gamma GLM for claim severity?

Claim amounts are positive and right-skewed, and their spread grows with the mean. A gamma distribution has variance φμ², which matches this. The log link keeps predictions positive.

How do I interpret GLM output in R?

Read the coefficient estimates on the link scale, then use exp() for relativities or odds ratios. Check standard errors and p-values for significance. Use the null and residual deviance to judge how much the model explains.

Why not model pure premium directly?

You can, with a Tweedie distribution. The usual approach is to model frequency and severity separately, which gives more insight into each driver and is simple to explain.