CAS MONOGRAPH SERIES NUMBER 5 Second Edition GENERALIZED LINEAR MODELS FOR INSURANCE RATING Second Edition Mark Goldburd, FCAS, MAAA Anand Khare, FCAS, FIA, CPCU Dan Tevet, FCAS Dmitriy Guller, FCAS CASUALTY ACTUARIAL SOCIETYCasualty Actuarial Society 1 This monograph is a comprehensive guide to creating an insurance rating plan using generalized linear models (GLMs), with an emphasis on application over theory. It is written for actuaries practicing in the property/casualty insurance industry and assumes the reader is familiar with actuarial terms and methods. The text includes a lengthy section on technical foundations that is presented using examples that are specific to the insurance industry. Other covered topics include the model-building process, data preparation, selection of model form, model refinement, and model validation. Extensions to the GLM are briefly discussed. GENERALIZED LINEAR MODELS FOR INSURANCE RATING Second Edition Mark Goldburd, FCAS, MAAA Anand Khare, FCAS, FIA, CPCU Dan Tevet, FCAS Dmitriy Guller, FCAS Casualty Actuarial Society 4350 North Fairfax Drive, Suite 250 Arlington, Virginia 22203 www.casact.org (703) 276-3100 Generalized Linear Models for Insurance Rating By Mark Goldburd, Anand Khare, Dan Tevet, and Dmitriy Guller Copyright 2020 by the Casualty Actuarial Society All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. For information on obtaining permission for use of the material in this work, please submit a written request to the Casualty Actuarial Society. Library of Congress Cataloging-in-Publication Data Generalized Linear Models for Insurance Rating / Mark Goldburd, Anand Khare, Dan Tevet, and Dmitriy Guller ISBN 978-1-7333294-3-9 (print edition) ISBN 978-1-7333294-4-6 (electronic edition) 1. Actuarial science. 2. Classification ratemaking. 3. Insurance—mathematical models. I. Goldburd, Mark. II. Khare, Anand. III. Tevet, Dan. Copyright 2019, Casualty Actuarial Society Contents 1. Introduction .................................................................................................. 1 2. Overview of Technical Foundations .............................................................. 2 2.1. The Components of the GLM ...............................................................2 2.1.1. The Random Component: The Exponential Family ....................3 2.1.2. The Systematic Component ........................................................4 2.1.3. An Example ................................................................................5 2.2. Exponential Family Variance ..................................................................7 2.3. Variable Significance ..............................................................................8 2.3.1. Standard Error ............................................................................8 2.3.2. p-value ........................................................................................9 2.3.3. Confidence Interval ....................................................................9 2.4. Types of Predictor Variables .................................................................10 2.4.1. Treatment of Continuous Variables ...........................................10 2.4.2. Treatment of Categorical Variables ............................................12 2.4.3. Choose Your Base Level Wisely! ................................................15 2.5. Weights ................................................................................................16 2.6. Offsets .................................................................................................17 2.7. An Inventory of Distributions ..............................................................19 2.7.1. Distributions for Severity ..........................................................19 2.7.2. Distributions for Frequency ......................................................21 2.7.3. A Distribution for Pure Premium: the Tweedie Distribution .....22 2.8. Logistic Regression ...............................................................................25 2.9. Correlation Among Predictors, Multicollinearity and Aliasing .............27 2.10. Limitations of GLMs ...........................................................................28 3. The Model-Building Process ....................................................................... 31 3.1. Setting Objectives and Goals ................................................................31 3.2. Communication with Key Stakeholders ...............................................32 3.3. Collecting and Processing Data ............................................................32 3.4. Conducting Exploratory Data Analysis ................................................32 3.5. Specifying Model Form ........................................................................33 3.6. Evaluating Model Output ....................................................................33 3.7. Validating the Model ...........................................................................33 3.8. Translating the Model into a Product ...................................................33 3.9. Maintaining and Rebuilding the Model ...............................................34 Contents 4. Data Preparation and Considerations ........................................................ 35 4.1. Combining Policy and Claim Data ......................................................35 4.2. Modifying the Data .............................................................................37 4.3. Splitting the Data.................................................................................38 4.3.1. Train and Test ...........................................................................40 4.3.2. Train, Validation and Test .........................................................40 4.3.3. Use Your Data Wisely! ..............................................................40 4.3.4. Cross Validation ........................................................................41 5. Selection of Model Form ............................................................................. 43 5.1. Choosing the Target Variable ...............................................................43 5.1.1. Frequency/Severity versus Pure Premium ..................................43 5.1.2. Policies with Multiple Coverages and Perils ...............................44 5.1.3. Transforming the Target Variable ...............................................45 5.2. Choosing the Distribution ...................................................................46 5.3. Variable Selection .................................................................................47 5.4. Transformation of Variables .................................................................48 5.4.1. Detecting Non-Linearity with Partial Residual Plots .................48 5.4.2. Binning Continuous Predictors .................................................49 5.4.3. Adding Polynomial Terms .........................................................51 5.4.4. Using Piecewise Linear Functions .............................................53 5.4.5. Natural Cubic Splines ...............................................................55 5.5. Grouping Categorical Variables ............................................................55 5.6. Interactions ..........................................................................................55 5.6.1. Interacting Two Categorical Variables........................................56 5.6.2. Interacting a Categorical Variable with a Continuous Variable .......................................................58 5.6.3. Interacting Two Continuous Variables.......................................61 6. Model Refinement ....................................................................................... 62 6.1. Some Measures of Model Fit ................................................................62 6.1.1. Log-Likelihood .........................................................................62 6.1.2. Deviance ...................................................................................63 6.1.3. Limitations on the Use of Log-Likelihood and Deviance...........64 6.2. Comparing Candidate Models .............................................................64 6.2.1. Nested Models and the F-Test ...................................................64 6.2.2. Penalized Measures of Fit ..........................................................66 6.3. Residual Analysis..................................................................................67 6.3.1. Deviance Residuals ...................................................................67 6.3.2. Working Residuals ....................................................................70 6.4. Assessing Model Stability .....................................................................73 7. Model Validation and Selection .................................................................. 75 7.1. Assessing Fit with Plots of Actual vs. Predicted .....................................75 7.2. Measuring Lift .....................................................................................76 7.2.1. Simple Quantile Plots ...............................................................77 7.2.2. Double Lift Charts ....................................................................78