Stochastic Claims Reserving _ Methods in Insurance

Transcription

1 Stochastic Claims Reserving _ Methods in Insurance and John Wiley & Sons, Ltd

2 ! Contents Preface Acknowledgement, xiii r xi» J.. '..- 1 Introduction and Notation : : Claims process.:.-.. : Accounting principles and accident years Inflation.:,.., Structural framework to the claims-reserving problem Fundamental properties of the claims reserving process Known and unknown claims Outstanding loss liabilities, classical notation,.,, General remarks., 12 2 Basic Methods Chain-ladder method (distribution-free) Bornhuetter-Ferguson method Number of IBNyR claims, Poisson model ' Poisson derivation of the CL algorithm ' " 27 3 Chain-Ladder Models Mean square error of prediction Chain-ladder method Mack model (distribution-free CL model), Conditional process variance,, Estimation error for single accident years Conditional MSEP, aggregated accident years,,, Bounds in the unconditional approach Results and interpretation,, Aggregation of accident years Proof of Theorems 3.17, 3.18 and 3.20! Analysis of error terms in the CL method ' Classical CL model...,! Enhanced CL model Interpretation,..-...,-' 72

3 3.4.4 CL estimator in the enhanced model Conditional process and parameter prediction errors CL factors and parameter estimation error Parameter estimation 81 Bayesian Models Benktander-Hovinen method and Cape-Cod model Benktander-Hovinen method Cape-Cod model Credible claims reserving methods Minimizing quadratic loss functions Distributional examples to credible claims reserving Log-normal/Log-normal model Exact Bayesian models Overdispersed Poisson model with gamma prior distribution Exponential dispersion family with its associated conjugates Markov chain Monte Carlo methods Btihlmann-Straub credibility model Multidimensional credibility models Hachemeister regression model Other credibility models Kalman filter 160 Distributional Models Log-normal model for cumulative claims Known variances crj Unknown variances Incremental claims (Overdispersed) Poisson model, Negative-Binomial model Log-normal model for incremental claims Gamma model Tweedie's compound Poisson model Wright's model 199 Generalized Linear Models Maximum likelihood estimators Generalized linear models framework Exponential dispersion family Parameter estimation in the EDF MLE for the EDF Fisher's scoring method Mean square error of prediction Other GLM models Bornhuetter-Ferguson method, revisited MSEP in the BF method, single accident year MSEP in the BF method, aggregated accident years 230

4 Bootstrap Methods Introduction Efron's non-parametric bootstrap Parametric bootstrap Log-normal model for cumulative sizes > Generalized linear models Chain-ladder method Approach 1: Unconditional estimation error Approach 3: Conditional estimation error ' Mathematical thoughts about bootstrapping methods Synchronous bootstrapping of seemingly unrelated regressions 253 Multivariate Reserving Methods General multivariate framework Multivariate chain-ladder method Multivariate CL model ' Conditional process variance ' Conditional estimation error for single accident years Conditional MSEP, aggregated accident years Parameter estimation Multivariate additive loss reserving method Multivariate additive loss reserving model Conditional process variance ' Conditional estimation error for single accident years Conditional MSEP, aggregated accident years Parameter estimation Combined Multivariate CL and ALR method Combined CL and ALR method: the model Conditional cross process variance Conditional cross estimation error for single accident years Conditional MSEP, aggregated accident years Parameter estimation 321 Selected Topics I: Chain-Ladder Methods Munich chain-ladder The Munich chain-ladder model Credibility approach to the MCL method MCL Parameter estimation CL Reserving: A Bayesian inference model Prediction of the ultimate claim c Likelihood function and posterior distribution Mean square error of prediction Credibility chain-ladder Examples Markov chain Monte Carlo methods 364'

5 10 Selected Topics II: Individual Claims Development Processes Modelling claims development processes for individual claims Modelling framework Claims reserving categories Separating IBNeR and IBNyR claims Statistical Diagnostics Testing age-to-age factors Model choice Age-to-age factors Homogeneity in time and distributional assumptions Correlations Diagonal effects Non-parametric smoothing 401 Appendix A: Distributions A.I Discrete distributions A.I.1 Binomial distribution A.I.2 Poisson distribution A.I.3 Negative-Binomial distribution A.2 Continuous distributions A.2..1 Uniform distribution A.2..2 Normal distribution A.2,.3 Log-normal distribution A.2,.4 Gamma distribution A.2,,5 Beta distribution Bibliography Index