Preface
Acknowledgement
1. Introduction and Notation
1.1 Claims process
1.1.1 Accounting principles and accident years
1.1.2 Inflation
1.2 Structural framework to the claims-reserving problem
1.2.1 Fundamental properties of the claims reserving process
1.2.2 Known and unknown claims
1.3 Outstanding loss liabilities, classical notation
1.4 General remarks
2 Basic Methods
2.1 Chain-ladder method (distribution-free)
2.2 Bornhuetter-Ferguson method
2.3 Number of IBNyR claims, Poisson model
2.4 Poisson derivation of the CL algorithm
3 Chain-Ladder Models
3.1 Mean square error of prediction
3.2 Chain-ladder method
3.2.1 Mack model (distribution-free CL model)
3.2.2 Conditional process variance
3.2.3 Estimation error for single accident years
3.2.4 Conditional MSEP, aggregated accident years
3.3 Bounds in the unconditional approach
3.3.1 Results and interpretation
3.3.2 Aggregation of accident years
3.3.3 Proof of Theorems 3.17, 3.18 and 3.20
3.4 Analysis of error terms in the CL method
3.4.1 Classical CL model
3.4.2 Enhanced CL model
3.4.3 Interpretation
3.4.4 CL estimator in the enhanced model
3.4.5 Conditional process and parameter prediction errors
3.4.6 CL factors and parameter estimation error
3.4.7 Parameter estimation
4 Bayesina Models
4.1 Benktander-Hovinen method and Cape-Cod model
4.1.1 Benktander-Hovinen method
4.1.2 Cape-Cod model
4.2 Credible claims reserving method
4.2.1 Mimimizing quadratic loss functions
4.2.2 Distributional examples to credible claims reserving
4.2.3 Log-normal/Log-normal model
4.3 Exact Bayesian models
4.3.1 Overdispersed Poisson model with gamma prior distribution
4.3.2 Exponential dispersion family with its associated conjugates
4.4 Markov chain Monte Carlo methods
4.5 Bühlmann-Straub credibility model
4.6 Multidimensional credibility models
4.6.1 Hachemeister regression model
4.6.2 Other credibility models
4.7 Kalman filter
5 Distributional Methods
5.1 Log-normal model for cumulative claims
5.1.1 Known variances
5.1.2 Unknown variances
5.2 Incremental claims
5.2.1 (Overdispersed) Poisson model
5.2.2 Negative-Binomial model
5.2.3 Log-normal model for incremental claims
5.2.4 Gamma model
5.2.5 Tweedie's compound Poisson model
6.2.6 Wright's model
6 Generalized Linear Models
6.1 Maximum likelihood estimators
6.2 Generalized linear models framework
6.3 Exponential dispersion family
6.4 Parameter estimation in the EDF
6.4.1 MLE for the EDF
6.4.2 Fisher's scoring method
6.4.3 Mean square error of prediction
6.5 Other GLM models
6.6 Bornhuetter-Ferguson method, revisited
6.6.1 MSEP in the BF method, single accident year
6.6.2 MSEP in the BF method, aggregated accident years
7 Bootstrap Methods
7.1 Introduction
7.1.1 Efron's non-parametric bootstrap
7.1.2 Parametric bootstrap
7.2 Log-normal model for cumulative sizes
7.3 Generalized linear models
7.4 Chain-ladder method
7.4.1 Approach 1: Unconditional estimation error
7.4.2 Approach 3: Conditional estimation error
7.5 Mathematical thoughts about boootstrapping methods
7.6 Synchronous bootstrapping of seemingly unrelated regressions
8 Multivariate Reserving Methods
8.1 General multivariate framework
8.2 Multivariate chain-ladder method
8.2.1 Multivariate CL model
8.2.2 Conditional process variance
8.2.3 Conditional estimation error for single accident years
8.2.4 Conditional MSEP, aggregated accident years
8.2.5 Parameter estimation
8.3 Multivariate additive loss reserving method
8.3.1 Multivariate additive loss reserving model
8.3.2 Conditional process variance
8.3.3 Conditional estimation error for single accident years
8.3.4 Conditional MSEP, aggregated years
8.3.5 Parameter estimation
8.4 Combined Multivariate CL and ALR method
8.4.1 Combined CL and ALR method: the model
8.4.2 Conditional cross process variance
8.4.3 Conditional cross estimation error for single accident years
8.4.4 Conditional MSEP, aggregated accident years
8.4.5 Parameter estimation
9 Selected Topics I: Chain-Ladder Methods
9.1 Munich chain-ladder
9.1.1 The Munich chian-ladder model
9.1.2 Credibility approach to the MCL method
9.1.3 MCL Parameter estimation
9.2 CL Reserving: A Bayesian inference model
9.2.1 Prediction of the ultimate claim
9.2.2 Likelihood function and posterior distribution
9.2.3 Mean square error of prediction
9.2.4 Credibility chain-ladder
9.2.5 Examples
9.2.6 Markov chain Monte Carlo methods
10 Selected Topics II: Individual Claims Development Processes
10.1 Modelling claims development processes for individual claims
10.1.1 Modelling framework
10.1.2 Claims reserving categories
10.2 Separating IBNer and IBNyR claims
11 Statistical Diagnostics
11.1 Testing age-to-age factors
11.1.1 Model choice
11.1.2 Age-to-age factors
11.1.3 Homogeneity in time and distributional assumptions
11.1.4 Correlations
11.1.5 Diagonal effects
11.2 Non-parametric smoothing
Appendix A: Distributions
A.1 Discrete distributions
A.1.1 Binomial distribution
A.1.2 Poisosn distribution
A.1.3 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
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