Stochastic Claims Reserving Methods in Insurance (E-Book) von Mario V. Wüthrich

Preface xi

Acknowledgement xiii

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 BornhuetterFerguson 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

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 BenktanderHovinen method and CapeCod model 91

4.1.1 BenktanderHovinen method 92

4.1.2 CapeCod 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 BühlmannStraub 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 _j² 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 Tweedies compound Poisson model 188

5.2.6 Wrights 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 Fishers scoring method 210

6.4.3 Mean square error of prediction 214

6.5 Other GLM models 223

6.6 BornhuetterFerguson 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

7 Bootstrap Methods 233

7.1 Introduction 233

7.1.1 Efrons 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

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

Claims reserving is central to the insurance industry. Insurance liabilities depend on a number of different risk factors which need to be predicted accurately. This prediction of risk factors and outstanding loss liabilities is the core for pricing insurance products, determining the profitability of an insurance company and for considering the financial strength (solvency) of the company.

Following several high-profile company insolvencies, regulatory requirements have moved towards a risk-adjusted basis which has lead to the Solvency II developments. The key focus in the new regime is that financial companies need to analyze adverse developments in their portfolios. Reserving actuaries now have to not only estimate reserves for the outstanding loss liabilities but also to quantify possible shortfalls in these reserves that may lead to potential losses. Such an analysis requires stochastic modeling of loss liability cash flows and it can only be done within a stochastic framework. Therefore stochastic loss liability modeling and quantifying prediction uncertainties has become standard under the new legal framework for the financial industry.

This book covers all the mathematical theory and practical guidance needed in order to adhere to these stochastic techniques. Starting with the basic mathematical methods, working right through to the latest developments relevant for practical applications; readers will find out how to estimate total claims reserves while at the same time predicting errors and uncertainty are quantified. Accompanying datasets demonstrate all the techniques, which are easily implemented in a spreadsheet. A practical and essential guide, this book is a must-read in the light of the new solvency requirements for the whole insurance industry.

Mario V. Wüthrich holds a Ph.D. in mathematics from ETH Zurich (The Swiss Federal Institute of Technology Zurich). He completed his postdoctoral work on statistical physics in 2000 at the University of Nijmegen in The Netherlands. From 2000 to 2005, he held an actuarial position at Winterthur Insurance (Switzerland) where he was responsible for claims reserving in non-life insurance, as well as developing and implementing the Swiss Solvency Test. Since 2005, he has served as senior researcher and lecturer at ETH Zurich with teaching duties in actuarial and financial mathematics. He serves on the board of the Swiss Association of Actuaries (SAA) and is joint editor of the Bulletin SAA.

Michael Merz has been Assistant Professor for Statistics, Risk and Insurance at the University of Tübingen since October 2006. He was awarded the internationally renowned SCOR Actuarial Prize 2004 for his doctoral thesis in risk theory. After completing his doctorate, he worked in the actuarial department of the Baloise insurance company in Basel/Switzerland and gained valuable practical working experience in actuarial science and quantitative risk management. His main research interests are actuarial science and quantitative risk management, with special emphasis on claims reserving and risk theory. He is a referee for many academic journals and has published extensively in leading academic journals, including theASTIN Bulletin and theScandinanvian Actuarial Journal.