Ratemaking: Advanced Techniques and GLM Applications

GLM-Based Ratemaking

Modern ratemaking uses GLMs to model frequency and severity separately. Claim frequency is modeled with Poisson or negative binomial regression using a log link, with exposure as an offset: ln(E[N_i]) = ln(w_i) + x_i' * beta. Claim severity uses Gamma or inverse Gaussian regression with a log link: ln(E[X_i | X_i > 0]) = x_i' * gamma. The pure premium is the product of predicted frequency and predicted severity. This multiplicative structure allows rating factors to be applied independently across rating dimensions.

The log link function is standard for both frequency and severity because it ensures that predictions are always positive and produces multiplicative rating factors. If the coefficient for a binary variable (such as male versus female) is beta_j = 0.15, the rating relativity is exp(0.15) = 1.162, meaning that group has 16.2% higher expected claims than the reference group, all else equal. This multiplicative interpretation aligns naturally with how insurance rates are structured.

Frequency Modeling in Detail

The Poisson GLM is the starting point for frequency modeling. Each policy i has exposure w_i (measured in policy-years or earned car-years) and observed claim count N_i. The model assumes N_i follows a Poisson distribution with mean mu_i = w_i * exp(x_i' * beta). The exposure enters as an offset term ln(w_i) in the linear predictor, so the model estimates the claim rate per unit of exposure rather than the raw count.

When the data exhibits overdispersion (the variance of claim counts exceeds the mean, which is common in insurance data), the Poisson model underestimates standard errors and produces overly significant coefficient estimates. The negative binomial GLM addresses overdispersion by adding a dispersion parameter that allows the variance to exceed the mean. An alternative is the quasi-Poisson model, which adjusts the standard errors by a dispersion factor without changing the point estimates.

Model diagnostics for frequency GLMs include checking the dispersion statistic (sum of squared Pearson residuals divided by degrees of freedom, which should be near 1 for a well-specified Poisson model), examining residual plots for patterns, and comparing observed and predicted claim counts across rating classes.

Severity Modeling in Detail

Severity is modeled among claims that have actually occurred, so the dataset consists only of policies with at least one claim. The Gamma GLM with a log link is the most common choice, as it accommodates the right-skewed, positive-valued nature of claim amounts. The inverse Gaussian distribution is an alternative when severity data is even more right-skewed than the Gamma can handle.

Large claims require special attention in severity modeling. A single catastrophic claim can unduly influence the fitted model if left unmodified. Common approaches include capping individual claim amounts at a threshold (such as the 99th percentile), fitting separate models for attritional and large claims, or using robust estimation methods. The choice of approach depends on the line of business and the purpose of the model (pricing versus reserving versus capital modeling).

Regularization and Variable Selection

Regularization techniques address two common problems in GLM ratemaking: multicollinearity (when rating variables are correlated, making individual coefficient estimates unstable) and model complexity (too many variables or factor levels leading to overfitting). Ridge regression adds an L2 penalty (proportional to the sum of squared coefficients) that shrinks coefficients toward zero without eliminating them. Lasso regression adds an L1 penalty (proportional to the sum of absolute coefficients) that can shrink some coefficients exactly to zero, performing automatic variable selection.

Elastic net combines both penalties, with a mixing parameter alpha controlling the balance between L1 and L2 regularization. When alpha = 1, elastic net reduces to lasso; when alpha = 0, it reduces to ridge. In practice, elastic net with alpha between 0.1 and 0.5 often works well for insurance data, providing some variable selection while maintaining stability when predictors are correlated. The regularization strength (lambda) is chosen by cross-validation.

Modeling Nonlinear Effects and Interactions

Continuous rating variables (such as driver age, vehicle age, or sum insured) often have nonlinear effects on claim frequency or severity. Rather than using arbitrary age bands (which create discontinuities), restricted cubic splines or natural splines model these effects as smooth curves within the GLM framework. A restricted cubic spline with k knots adds k minus 2 basis function terms to the linear predictor, producing a smooth curve that is linear in the tails (beyond the boundary knots).

Interaction terms capture situations where the effect of one rating variable depends on the value of another. For example, the effect of vehicle type on claim frequency might differ between urban and rural territories. In the GLM, this is modeled by including the product of the two variables (or their factor level indicators) in the linear predictor. Adding interactions increases model complexity, so they should be included only when supported by domain knowledge or statistical evidence (such as a significant improvement in AIC or a deviance test).

Implementation Considerations

Implementing a new GLM-based rating plan involves several practical considerations beyond model fitting. Off-balance adjustments ensure that the total premium from the new model matches the total premium from the existing plan, preventing unintended aggregate rate changes. Capping limits the maximum rate change any individual policyholder can experience in a single renewal (for example, no more than 25% increase), smoothing the transition to new rates.

Territorial rating deserves special attention because geographic areas are numerous and individual territories may have sparse data. Spatial smoothing techniques (such as credibility-based territorial adjustments or spatial GLMs with geographic covariates) borrow information from neighboring territories to stabilize estimates. The resulting rate relativities should be checked for geographic consistency and reasonableness before implementation.

Common Pitfalls

A frequent error is including exposure in the severity model. Exposure belongs in the frequency model as an offset. The severity model conditions on a claim having occurred and models the distribution of claim amounts, which does not depend on how long the policy was in force. Another common mistake is comparing AIC values between a Poisson and a negative binomial model directly. Since these models have different likelihoods, AIC comparisons require care, and a deviance test or a test based on the overdispersion parameter is more appropriate.

When extracting rating relativities, remember to exponentiate the GLM coefficients. The raw coefficients are on the log scale and do not directly represent the multiplicative factor applied to the base rate. Also ensure that the base level for each categorical variable is clearly defined and consistent across all model outputs.