Generalized Linear Models for Insurance Pricing
How GLMs form the backbone of modern insurance pricing and the key considerations for actuaries.
The GLM Framework
Generalized linear models (GLMs) are the workhorse of modern insurance pricing. A GLM extends ordinary linear regression by allowing the response variable to follow any distribution in the exponential family and by connecting the mean of the response to the linear predictor through a link function. This flexibility makes GLMs suitable for the non-normal data that actuaries routinely encounter: claim counts (discrete, non-negative), claim amounts (positive, right-skewed), and loss ratios (bounded or semi-bounded).
A GLM consists of three components. The random component specifies the distribution of the response variable (Poisson, negative binomial, gamma, inverse Gaussian, or Tweedie). The systematic component is a linear predictor: eta = beta_0 + beta_1 x_1 + beta_2 x_2 + ... + beta_p x_p, where the x variables are rating factors. The link function g connects the mean of the response to the linear predictor: g(mu) = eta.
Common GLM Specifications for Insurance
For insurance frequency models (predicting the number of claims per policy), actuaries typically use a Poisson or negative binomial distribution with a log link. The log link produces a multiplicative rating structure: the predicted frequency for a policy with characteristics x_1, x_2, ..., x_p is exp(beta_0) times exp(beta_1 x_1) times exp(beta_2 x_2) and so on. Each factor exp(beta_j x_j) is a relativity that multiplies the base rate, which aligns naturally with how insurance rate manuals are structured.
For severity models (predicting the average claim amount given that a claim occurs), a gamma distribution with a log link is the standard choice. The gamma distribution accommodates the positive, right-skewed nature of claim amounts. A log-normal model (linear regression on log-transformed claims) is an alternative that may be more appropriate when the variance increases more than proportionally with the mean.
The Tweedie distribution, with power parameter between 1 and 2, models aggregate losses (pure premium) directly by combining the frequency and severity components into a single model. This can be advantageous when frequency-severity interactions are suspected or when the modeling objective is pure premium rather than its components.
Exposure and Offset Variables
In frequency models, exposure (measured as earned policy years, earned car years, or similar units) must be incorporated correctly. The standard approach is to include the log of exposure as an offset variable in the linear predictor. An offset is a variable with a fixed coefficient of 1, so the model predicts log(frequency) = log(exposure) + beta_0 + beta_1 x_1 + ..., which is equivalent to modeling the rate (claims per unit of exposure) rather than the raw count.
Failing to include exposure as an offset, or including it as a regular predictor variable, produces incorrect results because the relationship between claim counts and exposure is known to be proportional (a policy in force for half a year should have half the expected claims of a full-year policy), and this proportionality should not be estimated from the data.
Handling Categorical Variables
Insurance rating uses many categorical variables: vehicle make and model, territory, occupation class, and construction type, among others. Categorical variables with many levels present challenges for GLMs. Territory, for instance, may have hundreds or thousands of levels, some with very limited data. Several approaches address this issue:
- Grouping: combine levels with similar experience into broader categories, reducing the number of parameters while preserving meaningful distinctions
- Regularization: apply a penalty to the parameter estimates (ridge regression or LASSO) to shrink estimates for data-sparse levels toward the overall mean
- Spatial smoothing: for geographic variables, impose smoothness constraints so that nearby territories have similar relativities
- Credibility blending: weight each level's estimate between its own experience and a prior (typically the neighboring levels or the overall mean), with the weight depending on data volume
Interaction Effects and Model Selection
Interaction effects between rating variables should be tested systematically. An interaction between age and vehicle type, for example, might indicate that young drivers have disproportionately high claim rates for sports cars compared to sedans. In a multiplicative model, an interaction means that the combined effect of two variables differs from the product of their individual effects.
Model selection involves balancing fit (how well the model explains observed data) with parsimony (avoiding overfitting by using too many parameters). Information criteria such as AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) penalize model complexity, with BIC imposing a stronger penalty. Cross-validation provides a more direct assessment of predictive performance on unseen data. Actuaries typically compare several candidate models using these criteria and select the model that best balances predictive accuracy with interpretability and regulatory acceptability.
Model Validation
Validating a GLM for insurance pricing involves several checks. Compare observed and predicted loss ratios across each rating variable to verify that the model captures the key patterns in the data. Lift charts show how well the model discriminates between high-risk and low-risk policies. Residual plots reveal systematic patterns that the model misses, such as non-linear effects or unmodeled interactions.
Out-of-sample testing (holding out a portion of the data or using a more recent time period) assesses how well the model generalizes beyond the training data. Double lift charts compare the new model against the current rating plan to quantify the improvement in risk discrimination. These validation steps are essential not only for model quality but also for supporting regulatory filings, where actuaries must demonstrate that the proposed rates are actuarially justified.
Regulatory and Practical Considerations
Actuaries must consider regulatory requirements when building pricing GLMs. Some jurisdictions restrict which rating variables may be used (for example, prohibiting gender in auto insurance pricing). Others require specific forms for rate filings or mandate that rates not be unfairly discriminatory. The interpretability of GLMs is a significant advantage in regulatory contexts, as the multiplicative structure and explicit relativities are straightforward to explain and review.
From a practical standpoint, GLMs require clean, well-prepared data. Data quality issues (missing values, coding errors, inconsistent definitions) can significantly affect model results. Actuaries should invest time in data exploration and cleaning before fitting models, and document their data preparation steps thoroughly for regulatory filings and peer review.