Actuarial Notes

Generalized Linear Models for Insurance Rating

By Mark Goldburd, Anand Khare, Dan Tevet and Dmitriy Guller · Casualty Actuarial Society · 2020 · CAS Monograph No. 5

Study guides: Exam MAS-II, Exam PCPA, Exam 8

A comprehensive guide to creating an insurance rating plan using generalized linear models (GLMs), with an emphasis on application over theory. Written for actuaries practising in property/casualty insurance, it presents the technical foundations through insurance examples, then covers the model-building process, data preparation, selection of model form, model refinement and model validation, and briefly discusses extensions to the GLM. It is number 5 in the CAS Monograph Series; the copy CAS now hosts is the second edition's 2025 revision, whose contents and Chapter 7 match the 2020 printing page for page.

On the syllabus

  • Exam MAS-II — objectives C10–C11; Chapter 7.
  • Exam PCPA — objectives B1–B2; Chapter 7.
  • Exam 8 — objectives A1, A4–A8.

1 Introduction

  • The authors know of no single text for the practising actuary that serves as a definitive reference for the use of GLMs in classification ratemaking; the monograph aims to give the reader the tools to build a market-ready classification plan from raw premium and loss data.
  • It assumes familiarity with the earlier CAS exams, including the Actuarial Standards of Practice and Werner and Modlin's Basic Ratemaking (2010).

2 Overview of Technical Foundations

  • 2.1 The Components of the GLM
    • 2.1.1 The Random Component: The Exponential Family
    • 2.1.2 The Systematic Component
    • 2.1.3 An Example
  • 2.2 Exponential Family Variance
  • 2.3 Variable Significance
  • 2.4 Types of Predictor Variables
    • 2.4.1 Treatment of Continuous Variables
    • 2.4.2 Treatment of Categorical Variables
    • 2.4.3 Choose Your Base Level Wisely!
  • 2.5 Weights
  • 2.6 Offsets
  • 2.7 An Inventory of Distributions
    • 2.7.1 Distributions for Severity
    • 2.7.2 Distributions for Frequency
    • 2.7.3 A Distribution for Pure Premium: the Tweedie Distribution
  • 2.8 Logistic Regression
  • 2.9 Correlation Among Predictors, Multicollinearity and Aliasing
  • 2.10 Limitations of GLMs

3 The Model-Building Process

  • 3.1 Setting Objectives and Goals
  • 3.2 Communicating with Key Stakeholders
  • 3.3 Collecting and Processing Data
  • 3.4 Conducting Exploratory Data Analysis
  • 3.5 Specifying Model Form
  • 3.6 Evaluating Model Output
  • 3.7 Validating the Model
  • 3.8 Translating the Model into a Product
  • 3.9 Maintaining and Rebuilding the Model

4 Data Preparation and Considerations

  • 4.1 Combining Policy and Claim Data
  • 4.2 Modifying the Data
  • 4.3 Splitting the Data
    • 4.3.1 Train and Test
    • 4.3.2 Train, Validation and Test
    • 4.3.3 Use Your Data Wisely!
    • 4.3.4 Cross Validation

5 Selection of Model Form

  • 5.1 Choosing the Target Variable
    • 5.1.1 Frequency/Severity versus Pure Premium
    • 5.1.2 Policies with Multiple Coverages and Perils
    • 5.1.3 Transforming the Target Variable
  • 5.2 Choosing the Distribution
  • 5.3 Variable Selection
  • 5.4 Transformation of Variables
    • 5.4.1 Detecting Non-Linearity with Partial Residual Plots
    • 5.4.2 Binning Continuous Predictors
    • 5.4.3 Adding Polynomial Terms
    • 5.4.4 Using Piecewise Linear Functions
    • 5.4.5 Natural Cubic Splines
  • 5.5 Grouping Categorical Variables
  • 5.6 Interactions
    • 5.6.1 Interacting Two Categorical Variables
    • 5.6.2 Interacting a Categorical Variable with a Continuous Variable
    • 5.6.3 Interacting Two Continuous Variables

6 Model Refinement

  • 6.1 Some Measures of Model Fit
    • 6.1.1 Log-Likelihood
    • 6.1.2 Deviance
    • 6.1.3 Limitations on the Use of Log-Likelihood and Deviance
  • 6.2 Comparing Candidate Models
    • 6.2.1 Nested Models and the F-Test
    • 6.2.2 Penalized Measures of Fit
  • 6.3 Residual Analysis
    • 6.3.1 Deviance Residuals
    • 6.3.2 Working Residuals
  • 6.4 Assessing Model Stability

7 Model Validation and Selection

  • The chapter's techniques choose between final candidate models, including proprietary rating plans whose data and form are unavailable: they need only historical observations scored with each model's predictions, and they compare models in a way accessible to the business decision-makers who often choose between them.
  • 7.1 Assessing Fit with Plots of Actual vs. Predicted
    • Plot the actual target against the predicted target for each model, on holdout data; large datasets are often grouped into percentiles (100 buckets of equal model weight) and plotted on a log scale.
  • 7.2 Measuring Lift
    • Lift is a model's economic value: its ability to charge each insured an actuarially fair rate and so prevent adverse selection. It is relative, the lift of one model over another, and is always measured on holdout data.
    • 7.2.1 Simple Quantile Plots
      • Sort by predicted loss cost, bucket into quantiles of equal exposure, and plot actual against predicted pure premium; the winner is judged on predictive accuracy, monotonicity, and the vertical distance between the first and last quantiles.
    • 7.2.2 Double Lift Charts
      • Sort on the ratio of Model A's to Model B's predicted loss cost, so the first and last quantiles hold the risks the two models disagree on most; the winner more closely matches the actual pure premium in each quantile.
    • 7.2.3 Loss Ratio Charts
      • Sort on predicted loss ratio and plot the actual loss ratio of each quantile; they are simple to understand and explain, since loss ratios are the most commonly used measure of profitability.
    • 7.2.4 The Gini Index
      • Sort by predicted loss cost and plot the cumulative percentage of losses against the cumulative percentage of exposures (the Lorenz curve); the Gini index is twice the area between that curve and the line of equality.
      • It quantifies a rating plan's ability to separate the best and worst risks, not its profitability.
  • 7.3 Validation of Logistic Regression Models
    • Quantile plots, Lorenz curves and the Gini index carry over to logistic regression models by sorting on predicted probability.
    • 7.3.1 Receiver Operating Characteristic (ROC) Curves
      • A discrimination threshold turns a predicted probability into a yes/no prediction; the four outcomes on a test set form a confusion matrix, giving the sensitivity (true positive rate) and the specificity.
      • The ROC curve plots the true positive rate against the false positive rate over all thresholds; the area under it, the AUROC, is 0.500 for a model with no predictive power and 1.000 for a perfect one.
      • AUROC = 0.5 \times normalized Gini + 0.5, so the AUROC and the Gini index are not separate validation metrics.

8 Model Documentation

  • 8.1 The Importance of Documenting Your Model
  • 8.2 Check Yourself
  • 8.3 Stakeholder Management
  • 8.4 Code as Documentation

9 Other Topics

  • 9.1 Modeling Coverage Options with GLMs (Why You Probably Shouldn't)
  • 9.2 Territory Modeling
  • 9.3 Ensembling

10 Variations on the Generalized Linear Model

  • 10.1 Generalized Linear Mixed Models (GLMMs)
  • 10.2 GLMs with Dispersion Modeling (DGLMs)
  • 10.3 Generalized Additive Models (GAMs)
  • 10.4 MARS Models
  • 10.5 Elastic Net GLMs

Bibliography

Appendix

Related readings

  • Predictive Modeling Applications in Actuarial Science Volume 1 (Frees, Derrig and Meyers - 2014) — the Bibliography cites four of its chapters (5, 6, 15 and 16) for fuller technical treatment of GLMs and the other models discussed
  • Generalized Linear Models for Insurance Data (De Jong and Heller - 2008) — cited in the Bibliography
  • Basic Ratemaking (Werner - 2016) — Chapter 1 assumes the reader knows its ratemaking material; the monograph cites the 2010 edition

Sources