1 Introduction and Notation 1
1.1 Claims process 1
1.1.1 Accounting principles and accident years 2
1.1.2 Inflation 3
1.2 Structural framework to the claims-reserving problem 5
1.2.1 Fundamental properties of the claims reserving process 7
1.2.2 Known and unknown claims 9
1.3 Outstanding loss liabilities, classical notation 10
1.4 General remarks 12
2 Basic Methods 15
2.1 Chain-ladder method (distribution-free) 15
2.2 Bornhuetter–Ferguson method 21
2.3 Number of IBNyR claims, Poisson model 25
2.4 Poisson derivation of the CL algorithm 27
3 Chain-Ladder Models 33
3.1 Mean square error of prediction 33
3.2 Chain-ladder method 36
3.2.1 Mack model (distribution-free CL model) 37
3.2.2 Conditional process variance 41
3.2.3 Estimation error for single accident years 44
3.2.4 Conditional MSEP, aggregated accident years 55
3.3 Bounds in the unconditional approach 58
3.3.1 Results and interpretation 58
3.3.2 Aggregation of accident years 63
3.3.3 Proof of Theorems 3.17, 3.18 and 3.20 64
3.4 Analysis of error terms in the CL method 70
3.4.1 Classical CL model 70
3.4.2 Enhanced CL model 71
3.4.3 Interpretation 72
viii Contents
3.4.4 CL estimator in the enhanced model 73
3.4.5 Conditional process and parameter prediction errors 74
3.4.6 CL factors and parameter estimation error 75
3.4.7 Parameter estimation 81
4 Bayesian Models 91
4.1 Benktander–Hovinen method and Cape–Cod model 91
4.1.1 Benktander–Hovinen method 92
4.1.2 Cape–Cod model 95
4.2 Credible claims reserving methods 98
4.2.1 Minimizing quadratic loss functions 98
4.2.2 Distributional examples to credible claims reserving 101
4.2.3 Log-normal/Log-normal model 105
4.3 Exact Bayesian models 113
4.3.1 Overdispersed Poisson model with gamma prior distribution 114
4.3.2 Exponential dispersion family with its associated conjugates 122
4.4 Markov chain Monte Carlo methods 131
4.5 Bühlmann–Straub credibility model 145
4.6 Multidimensional credibility models 154
4.6.1 Hachemeister regression model 155
4.6.2 Other credibility models 159
4.7 Kalman filter 160
5 Distributional Models 167
5.1 Log-normal model for cumulative claims 167
5.1.1 Known variances  2
j 170
5.1.2 Unknown variances 177
5.2 Incremental claims 182
5.2.1 (Overdispersed) Poisson model 182
5.2.2 Negative-Binomial model 183
5.2.3 Log-normal model for incremental claims 185
5.2.4 Gamma model 186
5.2.5 Tweedie’s compound Poisson model 188
5.2.6 Wright’s model 199
6 Generalized Linear Models 201
6.1 Maximum likelihood estimators 201
6.2 Generalized linear models framework 203
6.3 Exponential dispersion family 205
6.4 Parameter estimation in the EDF 208
6.4.1 MLE for the EDF 208
6.4.2 Fisher’s scoring method 210
6.4.3 Mean square error of prediction 214
6.5 Other GLM models 223
6.6 Bornhuetter–Ferguson method, revisited 223
6.6.1 MSEP in the BF method, single accident year 226
6.6.2 MSEP in the BF method, aggregated accident years 230
Contents ix
7 Bootstrap Methods 233
7.1 Introduction 233
7.1.1 Efron’s non-parametric bootstrap 234
7.1.2 Parametric bootstrap 236
7.2 Log-normal model for cumulative sizes 237
7.3 Generalized linear models 242
7.4 Chain-ladder method 244
7.4.1 Approach 1: Unconditional estimation error 246
7.4.2 Approach 3: Conditional estimation error 247
7.5 Mathematical thoughts about bootstrapping methods 248
7.6 Synchronous bootstrapping of seemingly unrelated
regressions 253
8 Multivariate Reserving Methods 257
8.1 General multivariate framework 257
8.2 Multivariate chain-ladder method 259
8.2.1 Multivariate CL model 259
8.2.2 Conditional process variance 264
8.2.3 Conditional estimation error for single accident years 265
8.2.4 Conditional MSEP, aggregated accident years 272
8.2.5 Parameter estimation 274
8.3 Multivariate additive loss reserving method 288
8.3.1 Multivariate additive loss reserving model 288
8.3.2 Conditional process variance 295
8.3.3 Conditional estimation error for single accident
years 295
8.3.4 Conditional MSEP, aggregated accident years 297
8.3.5 Parameter estimation 299
8.4 Combined Multivariate CL and ALR method 308
8.4.1 Combined CL and ALR method: the model 308
8.4.2 Conditional cross process variance 313
8.4.3 Conditional cross estimation error for single accident
years 315
8.4.4 Conditional MSEP, aggregated accident years 319
8.4.5 Parameter estimation 321
9 Selected Topics I: Chain-Ladder Methods 331
9.1 Munich chain-ladder 331
9.1.1 The Munich chain-ladder model 333
9.1.2 Credibility approach to the MCL method 335
9.1.3 MCL Parameter estimation 340
9.2 CL Reserving: A Bayesian inference model 346
9.2.1 Prediction of the ultimate claim 351
9.2.2 Likelihood function and posterior distribution 351
9.2.3 Mean square error of prediction 354
9.2.4 Credibility chain-ladder 359
9.2.5 Examples 361
9.2.6 Markov chain Monte Carlo methods 364
x Contents
10 Selected Topics II: Individual Claims Development Processes 369
10.1 Modelling claims development processes for individual claims 369
10.1.1 Modelling framework 370
10.1.2 Claims reserving categories 376
10.2 Separating IBNeR and IBNyR claims 379
11 Statistical Diagnostics 391
11.1 Testing age-to-age factors 391
11.1.1 Model choice 394
11.1.2 Age-to-age factors 396
11.1.3 Homogeneity in time and distributional assumptions 398
11.1.4 Correlations 399
11.1.5 Diagonal effects 401
11.2 Non-parametric smoothing 401
Appendix A: Distributions 405
A.1 Discrete distributions 405
A.1.1 Binomial distribution 405
A.1.2 Poisson distribution 405
A.1.3 Negative-Binomial distribution 405
A.2 Continuous distributions 406
A.2.1 Uniform distribution 406
A.2.2 Normal distribution 406
A.2.3 Log-normal distribution 407
A.2.4 Gamma distribution 407
A.2.5 Beta distribution 408
Bibliography 409
Index 417