Actuarial Modelling of Claim Counts

Michel Denuit – Professor, Institute of Actuarial Science, UCL, Belgium. Michel Denuit is Professor of Statistics and Actuarial Science at the Université Catholique de Louvain, Belgium. His major fields of research are risk theory and stochastic inequalities. He has (co-)authored numerous articles that have appeared in applied and theoretical journals and served as member of the editorial board for several journals (including Insurance: Mathematics and Economics). He is a section editor on Wiley’s Encyclopedia of Actuarial Science, and is the author of two previous books, one of them with Wiley.Xavier Maréchal – Université Catholique de Louvain, Belgium & CEO of Reacfin, Belgium.Sandra Pitrebois – Université Catholique de Louvain, Belgium & Secura Belgian Re, Brussels.Jean-François Walhin – Université Catholique de Louvain, Belgium & Secura Belgian Re, Brussels

• Foreword xiiiPreface xvNotation xxvPart I Modelling Claim Counts 11 Mixed Poisson Models for Claim Numbers 31.1 Introduction 31.1.1 Poisson Modelling for the Number of Claims 31.1.2 Heterogeneity and Mixed Poisson Model 41.1.3 Maximum Likelihood Estimation 41.1.4 Agenda 51.2 Probabilistic Tools 51.2.1 Experiment and Universe 51.2.2 Random Events 51.2.3 Sigma-Algebra 61.2.4 Probability Measure 61.2.5 Independent Events 71.2.6 Conditional Probability 71.2.7 Random Variables and Random Vectors 81.2.8 Distribution Functions 81.2.9 Independence for Random Variables 91.3 Poisson Distribution 101.3.1 Counting Random Variables 101.3.2 Probability Mass Function 101.3.3 Moments 101.3.4 Probability Generating Function 111.3.5 Convolution Product 121.3.6 From the Binomial to the Poisson Distribution 131.3.7 Poisson Process 171.4 Mixed Poisson Distributions 211.4.1 Expectations of General Random Variables 211.4.2 Heterogeneity and Mixture Models 221.4.3 Mixed Poisson Process 251.4.4 Properties of Mixed Poisson Distributions 261.4.5 Negative Binomial Distribution 281.4.6 Poisson-Inverse Gaussian Distribution 311.4.7 Poisson-LogNormal Distribution 331.5 Statistical Inference for Discrete Distributions 351.5.1 Maximum Likelihood Estimators 351.5.2 Properties of the Maximum Likelihood Estimators 371.5.3 Computing the Maximum Likelihood Estimators with the Newton–Raphson Algorithm 401.5.4 Hypothesis Tests 411.6 Numerical Illustration 441.7 Further Reading and Bibliographic Notes 461.7.1 Mixed Poisson Distributions 461.7.2 Survey of Empirical Studies Devoted to Claim Frequencies 461.7.3 Semiparametric Approach 472 Risk Classification 492.1 Introduction 492.1.1 Risk Classification, Regression Models and Random Effects 492.1.2 Risk Sharing in Segmented Tariffs 502.1.3 Bonus Hunger and Censoring 512.1.4 Agenda 522.2 Descriptive Statistics for Portfolio A 522.2.1 Global Figures 522.2.2 Available Information 522.2.3 Exposure-to-Risk 532.2.4 One-Way Analyses 542.2.5 Interactions 582.2.6 True Versus Apparent Dependence 592.3 Poisson Regression Model 622.3.1 Coding Explanatory Variables 622.3.2 Loglinear Poisson Regression Model 642.3.3 Score 642.3.4 Multiplicative Tariff 652.3.5 Likelihood Equations 662.3.6 Interpretation of the Likelihood Equations 672.3.7 Solving the Likelihood Equations with the Newton–Raphson Algorithm 672.3.8 Wald Confidence Intervals 692.3.9 Testing for Hypothesis on a Single Parameter 692.3.10 Confidence Interval for the Expected Annual Claim Frequency 702.3.11 Deviance 712.3.12 Deviance Residuals 722.3.13 Testing a Hypothesis on a Set of Parameters 722.3.14 Specification Error and Robust Inference 722.3.15 Numerical Illustration 732.4 Overdispersion 792.4.1 Explanation of the Phenomenon 792.4.2 Interpreting Overdispersion 792.4.3 Consequences of Overdispersion 802.4.4 Modelling Overdispersion 802.4.5 Detecting Overdispersion 812.4.6 Testing for Overdispersion 822.5 Negative Binomial Regression Model 832.5.1 Likelihood Equations 832.5.2 Numerical Illustration 852.6 Poisson-Inverse Gaussian Regression Model 862.6.1 Likelihood Equations 862.6.2 Numerical Illustration 862.7 Poisson-LogNormal Regression Model 872.7.1 Likelihood Equations 872.7.2 Numerical Illustration 882.8 Risk Classification for Portfolio A 892.8.1 Comparison of Competing models with the Vuong Test 892.8.2 Resulting Risk Classification for Portfolio A 902.9 Ratemaking using Panel Data 902.9.1 Longitudinal Data 902.9.2 Descriptive Statistics for Portfolio B 922.9.3 Poisson Regression with Serial Independence 942.9.4 Detection of Serial Dependence 972.9.5 Estimation of the Parameters using GEE 1012.9.6 Maximum Likelihood in the Negative Binomial Model for Panel Data 1052.9.7 Maximum Likelihood in the Poisson-Inverse Gaussian Model for Panel Data 1062.9.8 Maximum Likelihood in the Poisson-LogNormal Model for Panel Data 1072.9.9 Vuong Test 1092.9.10 Information Criteria 1102.9.11 Resulting Classification for Portfolio B 1102.10 Further Reading and Bibliographic Notes 1112.10.1 Generalized Linear Models 1112.10.2 Nonlinear Effects 1122.10.3 Zero-Inflated Models 1122.10.4 Fixed Versus Random Effects 1132.10.5 Hurdle Models 1132.10.6 Geographic Ratemaking 1142.10.7 Software 116Part II Basics of Experience Rating 1193 Credibility Models for Claim Counts 1213.1 Introduction 1213.1.1 From Risk Classification to Experience Rating 1213.1.2 Credibility Theory 1213.1.3 Limited Fluctuation Theory 1223.1.4 Greatest Accuracy Credibility 1223.1.5 Linear Credibility 1233.1.6 Financial Equilibrium 1233.1.7 Combining a Priori and a Posteriori Ratemaking 1233.1.8 Loss Function 1243.1.9 Agenda 1243.2 Credibility Models 1243.2.1 A Simple Introductory Example: the Good Driver / Bad Driver Model 1243.2.2 Credibility Models Incorporating a Priori Risk Classification 1263.3 Credibility Formulas with a Quadratic Loss Function 1283.3.1 Optimal Least-Squares Predictor 1283.3.2 Predictive Distribution 1293.3.3 Bayesian Credibility Premium 1303.3.4 Poisson-Gamma Credibility Model 1313.3.5 Predictive Distribution and Bayesian Credibility Premium 1323.3.6 Numerical Illustration 1333.3.7 Discrete Poisson Mixture Credibility Model 1353.3.8 Discrete Approximations for the Heterogeneous Component 1363.3.9 Linear Credibility 1443.4 Credibility Formulas with an Exponential Loss Function 1493.4.1 Optimal Predictor 1493.4.2 Poisson-Gamma Credibility Model 1513.4.3 Linear Credibility 1523.4.4 Numerical Illustration 1523.5 Dependence in the Mixed Poisson Credibility Model 1553.5.1 Intuitive Ideas 1553.5.2 Stochastic Order Relations 1563.5.3 Comparisons of Predictive Distributions 1563.5.4 Positive Dependence Notions 1573.5.5 Dependence Between Annual Claim Numbers 1573.5.6 Increasingness in the Linear Credibility Model 1583.6 Further Reading and Bibliographic Notes 1583.6.1 Credibility Models 1583.6.2 Claim Count Distributions 1593.6.3 Loss Functions 1593.6.4 Credibility and Regression Models 1593.6.5 Credibility and Copulas 1603.6.6 Time Dependent Random Effects 1613.6.7 Credibility and Panel Data Models 1623.6.8 Credibility and Empirical Bayes Methods 1634 Bonus-Malus Scales 1654.1 Introduction 1654.1.1 From Credibility to Bonus-Malus Scales 1654.1.2 The Nature of Bonus-Malus Scales 1664.1.3 Relativities 1664.1.4 Bonus-Malus Scales and Markov Chains 1664.1.5 Financial Equilibrium 1674.1.6 Agenda 1674.2 Modelling Bonus-Malus Systems 1684.2.1 Typical Bonus-Malus Scales 1684.2.2 Characteristics of Bonus-Malus Scales 1694.2.3 Trajectory 1704.2.4 Transition Rules 1714.3 Transition Probabilities 1724.3.1 Definition 1724.3.2 Transition Matrix 1734.3.3 Multi-Step Transition Probabilities 1744.3.4 Ergodicity and Regular Transition Matrix 1764.4 Long-Term Behaviour of Bonus-Malus Systems 1764.4.1 Stationary Distribution 1764.4.2 Rolski–Schmidli–Schmidt–Teugels Formula 1794.4.3 Dufresne Algorithm 1824.4.4 Convergence to the Stationary Distribution 1834.5 Relativities with a Quadratic Loss Function 1844.5.1 Relativities 1844.5.2 Bayesian Relativities 1854.5.3 Interaction between Bonus-Malus Systems and a Priori Ratemaking 1894.5.4 Linear Relativities 1914.5.5 Approximations 1934.6 Relativities with an Exponential Loss Function 1944.6.1 Bayesian Relativities 1944.6.2 Fixing the Value of the Severity Parameter 1964.6.3 Linear Relativities 1964.6.4 Numerical Illustration 1974.7 Special Bonus Rule 2004.7.1 The Former Belgian Compulsory System 2004.7.2 Fictitious Levels 2004.7.3 Determination of the Relativities 2004.7.4 Numerical Illustration 2024.7.5 Linear Relativities for the Belgian Scale 2074.8 Change of Scale 2084.8.1 Migration from One Scale to Another 2084.8.2 Kolmogorov Distance 2084.8.3 Distances between the Random Effects 2094.8.4 Numerical Illustration 2094.9 Dependence in Bonus-Malus Scales 2134.10 Further Reading and Bibliographic Notes 213Part III Advances in Experience Rating 2175 Efficiency and Bonus Hunger 2195.1 Introduction 2195.1.1 Pure Premium 2195.1.2 Statistical Analysis of Claim Costs 2195.1.3 Large Claims and Extreme Value Theory 2205.1.4 Measuring the Efficiency of the Bonus-Malus Scales 2205.1.5 Bonus Hunger and Optimal Retention 2205.1.6 Descriptive Statistics for Portfolio C 2215.2 Modelling Claim Severities 2225.2.1 Claim Severities in Motor Third Party Liability Insurance 2225.2.2 Determining the Large Claims with Extreme Value Theory 2235.2.3 Generalized Pareto Fit to the Costs of Large Claims 2275.2.4 Modelling the Number of Large Claims 2295.2.5 Modelling the Costs of Moderate Claims 2305.2.6 Resulting Price List for Portfolio C 2365.3 Measures of Efficiency for Bonus-Malus Scales 2405.3.1 Loimaranta Efficiency 2405.3.2 De Pril Efficiency 2425.4 Bonus Hunger and Optimal Retention 2465.4.1 Correcting the Estimations for Censoring 2465.4.2 Number of Claims and Number of Accidents 2495.4.3 Lemaire Algorithm for the Determination of Optimal Retention Limits 2515.5 Further Reading and Bibliographic Notes 2555.5.1 Modelling Claim Amounts in Related Coverages 2555.5.2 Tweedie Generalized Linear Model 2555.5.3 Large Claims 2565.5.4 Alternative Approaches to Risk Classification 2575.5.5 Efficiency 2575.5.6 Optimal Retention Limits and Bonus Hunger 2576 Multi-Event Systems 2596.1 Introduction 2596.2 Multi-Event Credibility Models 2606.2.1 Dichotomy 2606.2.2 Multivariate Claim Count Model 2606.2.3 Bayesian Credibility Approach 2616.2.4 Summary of Past Claims Histories 2626.2.5 Variance-Covariance Structure of the Random Effects 2636.2.6 Variance-Covariance Structure of the Annual Claim Numbers 2636.2.7 Estimation of the Variances and Covariances 2646.2.8 Linear Credibility Premiums 2646.2.9 Numerical Illustration for Portfolio A 2686.3 Multi-Event Bonus-Malus Scales 2706.3.1 Types of Claims 2706.3.2 Markov Modelling for the Multi-Event Bonus-Malus Scale 2736.3.3 Determination of the relativities 2746.3.4 Numerical Illustrations 2746.4 Further Reading and Bibliographic Notes 2767 Bonus-Malus Systems with Varying Deductibles 2777.1 Introduction 2777.2 Distribution of the Annual Aggregate Claims 2787.2.1 Modelling Claim Costs 2787.2.2 Discretization 2797.2.3 Panjer Algorithm 2817.3 Introducing a Deductible within a Posteriori Ratemaking 2847.3.1 Annual Deductible 2847.3.2 Per Claim Deductible 2857.3.3 Mixed Case 2857.4 Numerical Illustrations 2867.4.1 Claim Frequencies 2867.4.2 Claim Severities 2867.4.3 Annual Deductible 2877.4.4 Per Claim Deductible 2887.4.5 Annual Deductible in the Mixed Case 2897.4.6 Per Claim Deductible in the Mixed Case 2897.5 Further Reading and Bibliographic Notes 2908 Transient Maximum Accuracy Criterion 2938.1 Introduction 2938.1.1 From Stationary to Transient Distributions 2938.1.2 A Practical Example: Creating a Special Scale for New Entrants 2938.1.3 Agenda 2958.2 Transient Behaviour and Convergence of Bonus-Malus Scales 2958.3 Quadratic Loss Function 2978.3.1 Transient Maximum Accuracy Criterion 2978.3.2 Linear Scales 3028.3.3 Financial Balance 3058.3.4 Choice of an Initial Level 3078.4 Exponential Loss Function 3088.5 Numerical Illustrations 3088.5.1 Scale −1/Top 3088.5.2 −1/+2 Scale 3158.6 Super Bonus Level 3198.6.1 Mechanism 3198.6.2 Initial Distributions 3198.6.3 Transient Relativities 3198.7 Further Reading and Bibliographic Notes 3239 Actuarial Analysis of the French Bonus-Malus System 3259.1 Introduction 3259.2 French Bonus-Malus System 3269.2.1 Modelling Claim Frequencies 3269.2.2 Probability Generating Functions of Random Vectors 3279.2.3 CRM Coefficients 3279.2.4 Computation of the CRMs at Time t 3289.2.5 Global CRM 3299.2.6 Multivariate Panjer and De Pril Recursive Formulas 3309.2.7 Analysis of the Financial Equilibrium of the French Bonus-Malus System 3339.2.8 Numerical Illustration 3359.3 Partial Liability 3389.3.1 Reduced Penalty and Modelling Claim Frequencies 3389.3.2 Computations of the CRMs at Time t 3389.3.3 Financial Equilibrium 3409.3.4 Numerical Illustrations 3419.4 Further Reading and Bibliographic Notes 342Bibliography 345Index 355

"An excellent text on experience rating that picks up the actuarial reader at almost every level. The authors have found an excellent balance between theory and practice." (Journal of the American Statistical Association, June 2009)