Predictive Modeling Applications in Actuarial Science Volume II: Case Studies in Insurance Predictive modeling uses data to forecast future events. It exploits relationships between explanatory variables and the predicted variables from past occurrences to predict future out- comes. Forecasting fnancial events is a core skill that actuaries routinely apply in insurance and other risk-management applications. Predictive Modeling Applications in Actuarial Sci- ence emphasizes lifelong learning by developing tools in an insurance context, providing the relevant actuarial applications, and introducing advanced statistical techniques that can be used to gain a competitive advantage in situations with complex data. Volume II examines applications of predictive modeling. Where Volume I developed the foundations of predictive modeling, Volume II explores practical uses for techniques, focus- ing especially on property and casualty insurance. Readers are exposed to a variety of tech- niques in concrete, real-life contexts that demonstrate their value, and the overall value of predictive modeling, for seasoned practicing analysts as well as those just starting out. Edward W. (Jed) Frees is the Hickman-Larson Chair of Actuarial Science at the University of Wisconsin–Madison. He received his PhD in mathematical statistics in 1983 from the University of North Carolina at Chapel Hill and is a Fellow of both the Society of Actuaries (SoA) and the American Statistical Association (the only Fellow of both organizations). Regarding his research, Professor Frees has won several awards for the quality of his work, including the Halmstad Prize for best paper published in the actuarial literature (four times). Glenn Meyers, PhD, FCAS, MAAA, CERA, retired from ISO at the end of 2011 after a 37-year career as an actuary. He holds a BS in mathematics and physics from Alma Col- lege, an MA in mathematics from Oakland University, and a PhD in mathematics from SUNY at Albany. A frequent speaker at Casualty Actuarial Society (CAS) meetings, he has served, and continues to serve, the CAS and the International Actuarial Association on vari- ous research and education committees. He has also served on the CAS Board of Directors. He has several published articles in the Proceedings of the Casualty Actuarial Society, Vari- ance, and Actuarial Review. His research contributions have been recognized by the CAS through his being a three-time winner of the Woodward-Fondiller Prize; a two-time win- ner of the Dorweiler Prize; and a winner of the DFA Prize, the Reserves Prize, the Matthew Rodermund Service Award, and theMichelbacher Signifcant Achievement Award. In retire- ment, he still spends some of his time on his continuing passion for actuarial research. Richard A. Derrig is founder and principal of OPAL Consulting LLC, which is a frm that provides research and regulatory support to property casualty insurance clients. Primary areas of expertise include fnancial pricing models, database and data-mining design, fraud detection planning and implementation, and expert testimony for regulation and litigation purposes. INTERNATIONAL SERIES ON ACTUARIAL SCIENCE Editorial Board Christopher Daykin (Independent Consultant and Actuary) Angus Macdonald (Heriot-Watt University) The International Series on Actuarial Science, published by Cambridge University Press in conjunction with the Institute and Faculty of Actuaries, contains textbooks for students tak- ing courses in or related to actuarial science, as well as more advanced works designed for continuing professional development or for describing and synthesizing research. The series is a vehicle for publishing books that refect changes and developments in the curriculum, that encourage the introduction of courses on actuarial science in universities, and that show how actuarial science can be used in all areas where there is long-term fnancial risk. A complete list of books in the series can be found at www.cambridge.org/statistics. Recent titles include the following: Computation and Modelling in Insurance and Finance Erik Bølviken Solutions Manual for Actuarial Mathematics for Life Contingent Risks (2nd Edition) David C.M. Dickson, Mary R. Hardy & Howard R. Waters Actuarial Mathematics for Life Contingent Risks (2nd Edition) David C.M. Dickson, Mary R. Hardy & Howard R. Waters Risk Modelling in General Insurance Roger J. Gray & Susan M. Pitts Financial Enterprise Risk Management Paul Sweeting Regression Modeling with Actuarial and Financial Applications Edward W. Frees Predictive Modeling Applications in Actuarial Science, Volume I: Predictive Modeling Techniques Edited by Edward W. Frees, Richard A. Derrig & Glenn Meyers Nonlife Actuarial Models Yiu-Kuen Tse Generalized Linear Models for Insurance Data Piet De Jong & Gillian Z. Heller PREDICTIVE MODELING APPLICATIONS IN ACTUARIAL SCIENCE Volume II: Case Studies in Insurance EDWARD W. FREES University of Wisconsin–Madison GLENN MEYERS RICHARD A. DERRIG OPAL Consulting LLC 32 Avenue of the Americas, New York, NY 10013 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107029880 © Cambridge University Press 2016 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2016 Printed in the United States of America by Sheridan Books, Inc. A catalogue record for this publication is available from the British Library. ISBN 978-1-107-02988-0 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate. Contents Contributors page xi Preface xv Acknowledgments xix 1 Pure Premium Modeling Using Generalized Linear Models 1 Ernesto Schirmacher 1.1 Introduction 1 1.2 Data Characteristics 3 1.3 Exploratory Data Analysis 6 1.4 Frequency Modeling 12 1.5 Severity Modeling 23 1.6 Pure Premium 30 1.7 Validation 34 1.8 Conclusions 37 References 38 2 Applying Generalized Linear Models to Insurance Data: Frequency/Severity versus Pure Premium Modeling 39 Dan Tevet 2.1 Introduction 39 2.2 Comparing Model Forms 40 2.3 The Dataset and Model Forms 44 2.4 Results 47 Appendix 2.A Proof of Equivalence between Pure Premium Model Forms 55 Conclusion 57 Appendix 2.B The Gini Index 57 References 58 vii viii Contents 3 Generalized Linear Models as Predictive Claim Models 60 Greg Taylor and James Sullivan 3.1 Review of Loss Reserving 60 3.2 Additional Notation 63 3.3 GLM Background 64 3.4 Advantages of GLMs 66 3.5 Diagnostics 68 3.6 Example 73 3.7 Conclusion 97 References 98 4 Frameworks for General Insurance Ratemaking: Beyond the Generalized Linear Model 100 Peng Shi and James Guszcza 4.1 Introduction 100 4.2 Data 102 4.3 Univariate Ratemaking Framework 104 4.4 Multivariate Ratemaking Frameworks 113 4.5 Model Comparisons 122 4.6 Conclusion 123 References 124 5 Using Multilevel Modeling for Group Health Insurance Ratemaking: A Case Study from the Egyptian Market 126 Mona S. A. Hammad and Galal A. H. Harby 5.1 Motivation and Background 126 5.2 Data 130 5.3 Methods and Models 141 5.4 Results 144 5.5 Conclusions 146 Acknowledgments 147 Appendix 147 References 157 6 Clustering in General Insurance Pricing 159 Ji Yao 6.1 Introduction 159 6.2 Overview of Clustering 160 6.3 Dataset for Case Study 161 6.4 Clustering Methods 163 6.5 Exposure-Adjusted Hybrid (EAH) Clusering Method 168 Contents ix 6.6 Results of Case Study 171 6.7 Other Considerations 177 6.8 Conclusions 178 References 179 7 Application of Two Unsupervised Learning Techniques to Questionable Claims: PRIDIT and Random Forest 180 Louise A. Francis 7.1 Introduction 180 7.2 Unsupervised Learning 181 7.3 Simulated Automobile PIP Questionable Claims Data and the Fraud Issue 182 7.4 The Questionable Claims Dependent Variable Problem 185 7.5 The PRIDIT Method 185 7.6 Processing the Questionable Claims Data for PRIDIT Analysis 187 7.7 Computing RIDITS and PRIDITS 187 7.8 PRIDIT Results for Simulated PIP Questionable Claims Data 188 7.9 How Good Is the PRIDIT Score? 189 7.10 Trees and Random Forests 192 7.11 Unsupervised Learning with Random Forest 194 7.12 Software for Random Forest Computation 195 7.13 Some Findings from the Brockett et al. Study 201 7.14 Random Forest Visualization via Multidimensional Scaling 202 7.15 Kohonen Neural Networks 204 7.16 Summary 205 References 206 8 The Predictive Distribution of Loss Reserve Estimates over a Finite Time Horizon 208 Glenn Meyers 8.1 Introduction 208 8.2 The CAS Loss Reserve Database 210 8.3 The Correlated Chain Ladder Model 212 8.4 The Predictive Distribution of Future Estimates 213 8.5 The Implications for Capital Management 216 8.6 Summary and Conclusions 223 References 223