Bayesian Claims Reserving Methods in Non-life Insurance with Stan: An Introduction

210 Pages·2018·8.495 MB·English

by  Guangyuan Gao

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
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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