Guangyuan Gao Bayesian Claims Reserving Methods in Non-life Insurance with Stan An Introduction Bayesian Claims Reserving Methods in Non-life Insurance with Stan Guangyuan Gao Bayesian Claims Reserving Methods in Non-life Insurance with Stan An Introduction 123 GuangyuanGao Schoolof Statistics Renmin University of China Beijing,China ISBN978-981-13-3608-9 ISBN978-981-13-3609-6 (eBook) https://doi.org/10.1007/978-981-13-3609-6 LibraryofCongressControlNumber:2018963287 ©SpringerNatureSingaporePteLtd.2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd. Theregisteredcompanyaddressis:152BeachRoad,#21-01/04GatewayEast,Singapore189721, Singapore Preface Bayesian models are very popular in non-life claims reserving. This monograph provides a review of Bayesian claims reserving models and their underlying Bayesianinferencetheory.Itinvestigatesthreetypesofclaimsreservingmodelsin Bayesian framework: chain ladder models, basis expansion models involving tail factor, and multivariate copula models. One of the core techniques in Bayesian modeling is inferential methods. This monograph largely relies on Stan, a spe- cializedsoftwareenvironmentwhichappliesHamiltonianMonteCarlomethodand variational Bayes.Thismonographhasthefollowingthree distinguishingfeatures: (cid:129) It has a thorough review of various aspects of Bayesian statistics and relates them to claims reserving problems. (cid:129) It addresses three important points in claims reserving: tail development, stochastic version of payments per claim incurred method, and aggregation of liabilities from correlated portfolios. (cid:129) It provides explicit Stan code for non-life insurance claims reserving. Beijing, China Guangyuan Gao September 2018 v Acknowledgements IamverygratefultoBorekPuza,Richard Cumpston, HanlinShang,TimHiggins, Bronwen Whiting, Steven Roberts, and Xu Shi at the Australian National University.The discussionwith themhelps improve thequality ofthismonograph significantly. I am also very grateful to Chong It Tan and Yanlin Shi at the Macquarie University, and Shengwang Meng at the Renmin University of China. Theyprovideconstructivecomments.IwouldliketothanktheResearchSchoolof Finance, Actuarial Studies and Statistics at the Australian National University for the financial support. I also would like to thank the School of Statistics at the Renmin University of China for the research funds including National Social Science Fund of China (Grant No. 16ZDA052), MOE National Key Research BasesforHumanitiesandSocialSciences(GrantNo.16JJD910001),andthefunds for building world-class universities (disciplines). vii Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Bayesian Inference and MCMC. . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Bayesian Claims Reserving Methods . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Monograph Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 The General Notation Used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Bayesian Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Bayesian Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 The Single-Parameter Case. . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.2 The Multi-parameter Case . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.3 Choice of Prior Distribution . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.4 Asymptotic Normality of the Posterior Distribution . . . . . . 20 2.2 Model Assessment and Selection. . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.1 Posterior Predictive Checking. . . . . . . . . . . . . . . . . . . . . . 21 2.2.2 Residuals, Deviance and Deviance Residuals. . . . . . . . . . . 25 2.2.3 Bayesian Model Selection Methods . . . . . . . . . . . . . . . . . 28 2.2.4 Overfitting in the Bayesian Framework. . . . . . . . . . . . . . . 32 2.3 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3 Advanced Bayesian Computation . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1 Markov Chain Monte Carlo (MCMC) Methods . . . . . . . . . . . . . . 35 3.1.1 Markov Chain and Its Stationary Distribution . . . . . . . . . . 36 3.1.2 Single-Component Metropolis-Hastings (M-H) Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.3 Gibbs Sampler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1.4 Hamiltonian Monte Carlo (HMC). . . . . . . . . . . . . . . . . . . 43 ix x Contents 3.2 Convergence and Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2.2 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 OpenBUGS and Stan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3.1 OpenBUGS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3.2 Stan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4 Modal and Distributional Approximations . . . . . . . . . . . . . . . . . . 58 3.4.1 Laplace Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.4.2 Variational Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.5 A Bayesian Hierarchical Model for Rats Data . . . . . . . . . . . . . . . 60 3.5.1 Classical Regression Models . . . . . . . . . . . . . . . . . . . . . . 60 3.5.2 A Bayesian Bivariate Normal Hierarchical Model . . . . . . . 63 3.5.3 A Bayesian Univariate Normal Hierarchical Model . . . . . . 66 3.5.4 Reparameterization in the Gibbs Sampler . . . . . . . . . . . . . 68 3.6 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4 Bayesian Chain Ladder Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.1 Non-life Insurance Claims Reserving Background . . . . . . . . . . . . 73 4.1.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.1.2 Run-Off Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.1.3 Widely-Used Claims Reserving Methods . . . . . . . . . . . . . 76 4.2 Stochastic Chain Ladder Models . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2.1 Frequentist Chain Ladder Models . . . . . . . . . . . . . . . . . . . 78 4.2.2 A Bayesian Over-Dispersed Poisson (ODP) Model . . . . . . 84 4.3 A Bayesian ODP Model with Tail Factor. . . . . . . . . . . . . . . . . . . 90 4.3.1 Reversible Jump Markov Chain Monte Carlo . . . . . . . . . . 92 4.3.2 RJMCMC for Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.4 Estimation of Claims Liability in WorkSafe VIC . . . . . . . . . . . . . 98 4.4.1 Background of WorkSafe Victoria . . . . . . . . . . . . . . . . . . 98 4.4.2 Estimation of the Weekly Benefit Liability Using Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.4.3 Estimation of the Doctor Benefit Liability Using a Compound Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.6 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5 Bayesian Basis Expansion Models . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.1 Aspects of Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.1.1 Basis Functions of Splines . . . . . . . . . . . . . . . . . . . . . . . . 118 5.1.2 Smoothing Splines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.1.3 Low Rank Thin Plate Splines. . . . . . . . . . . . . . . . . . . . . . 122 5.1.4 Bayesian Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Contents xi 5.2 Two Simulated Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.2.1 A Model with a Trigonometric Mean Function and Normal Errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.2.2 A Gamma Response Variable with a Log-Logistic Growth Curve Mean Function . . . . . . . . . . . . . . . . . . . . . 133 5.3 Application to the Doctor Benefit . . . . . . . . . . . . . . . . . . . . . . . . 143 5.3.1 Claims Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.3.2 PPCI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.3.3 Combining the Ultimate Claims Numbers with the Outstanding PPCI. . . . . . . . . . . . . . . . . . . . . . . . 147 5.3.4 Computing Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.5 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6 Multivariate Modelling Using Copulas . . . . . . . . . . . . . . . . . . . . . . . 153 6.1 Overview of Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.1.1 Sklar’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.1.2 Parametric Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.1.3 Measures of Bivariate Association . . . . . . . . . . . . . . . . . . 157 6.1.4 Inference Methods for Copulas. . . . . . . . . . . . . . . . . . . . . 159 6.2 Copulas in Modelling Risk Dependence. . . . . . . . . . . . . . . . . . . . 167 6.2.1 Structural and Empirical Dependence Between Risks. . . . . 168 6.2.2 The Effects of Empirical Dependence on Risk Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.3 Application to the Doctor and Hospital Benefits. . . . . . . . . . . . . . 170 6.3.1 Preliminary GLM Analysis Using a Gaussian Copula . . . . 171 6.3.2 A Gaussian Copula with Marginal Bayesian Splines . . . . . 174 6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 6.5 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 7.1 The Three Claims Reserving Models. . . . . . . . . . . . . . . . . . . . . . 185 7.1.1 A Compound Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 7.1.2 A Bayesian Natural Cubic Spline Basis Expansion Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 7.1.3 A Copula Model with Bayesian Margins. . . . . . . . . . . . . . 187 7.2 A Suggested Bayesian Modelling Procedure. . . . . . . . . . . . . . . . . 188 7.3 Other Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.3.1 Bayesian Methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.3.2 Actuarial Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 xii Contents Appendix A: Derivations .... ..... .... .... .... .... .... ..... .... 191 Appendix B: Other Sampling Methods.. .... .... .... .... ..... .... 199 Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 203