Loss Models: From Data to Decisions (Hardcover, 5)

Preface xvii

PART I INTRODUCTION

1 Modeling 3

1.1 The model-based approach 3

1.2 Organization of this book 5

2 Random variables 9

2.1 Introduction 9

2.2 Key functions and four models 11

2.2.1 Exercises 19

3 Basic distributional quantities 21

3.1 Moments 21

3.1.1 Exercises 28

3.2 Percentiles 29

3.2.1 Exercises 30

3.3 Generating functions and sums of random variables 31

3.3.1 Exercises 32

3.4 Tails of distributions 33

3.4.1 Classification based on moments 33

3.4.2 Comparison based on limiting tail behavior 34

3.4.3 Classification based on the hazard rate function 35

3.4.4 Classification based on the mean excess loss function 36

3.4.5 Equilibrium distributions and tail behavior 37

3.4.6 Exercises 38

3.5 Measures of Risk 40

3.5.1 Introduction 40

3.5.2 Risk measures and coherence 40

3.5.3 Value-at-Risk 42

3.5.4 Tail-Value-at-Risk 43

3.5.5 Exercises 47

PART II ACTUARIAL MODELS

4 Characteristics of Actuarial Models 51

4.1 Introduction 51

4.2 The role of parameters 51

4.2.1 Parametric and scale distributions 52

4.2.2 Parametric distribution families 54

4.2.3 Finite mixture distributions 54

4.2.4 Data-dependent distributions 57

4.2.5 Exercises 58

5 Continuous models 61

5.1 Introduction 61

5.2 Creating new distributions 61

5.2.1 Multiplication by a constant 62

5.2.2 Raising to a power 62

5.2.3 Exponentiation 64

5.2.4 Mixing 64

5.2.5 Frailty models 67

5.2.6 Splicing 68

5.2.7 Exercises 70

5.3 Selected distributions and their relationships 74

5.3.1 Introduction 74

5.3.2 Two parametric families 74

5.3.3 Limiting distributions 74

5.3.4 Two heavy-tailed distributions 76

5.3.5 Exercises 77

5.4 The linear exponential family 77

5.4.1 Exercises 80

6 Discrete distributions 81

6.1 Introduction 81

6.1.1 Exercise 82

6.2 The Poisson distribution 82

6.3 The negative binomial distribution 84

6.4 The binomial distribution 87

6.5 The (a, b, 0) class 88

6.5.1 Exercises

6.6 Truncation and modification at zero

6.6.1 Exercises

7 Advanced discrete distributions

7.1 Compound frequency distributions

7.1.1 Exercises

7.2 Further properties of the compound Poisson class

7.2.1 Exercises

7.3 Mixed frequency distributions

7.3.1 General mixed frequency distribution 111

7.3.2 Mixed Poisson distributions 113

7.3.3 Exercises 118

7.4 Effect of exposure on frequency 119

7.5 An inventory of discrete distributions 120

7.5.1 Exercises 120

8 Frequency and severity with coverage modifications 123

8.2.1 Exercises 128

8.3 The loss elimination ratio and the effect of inflation for ordinary deductibles 129

8.3.1 Exercises 131

8.4 Policy limits 132

8.4.1 Exercises 133

8.5 Coinsurance, deductibles, and limits 134

8.5.1 Exercises 135

8.6 The impact of deductibles on claim frequency 137

8.6.1 Exercises 141

9 Aggregate loss models 143

9.1 Introduction 143

9.1.1 Exercises 146

9.2 Model choices 146

9.2.1 Exercises 147

9.3 The compound model for aggregate claims 147

9.5 Computing the aggregate claims distribution 166

9.6 The recursive method 168

9.6.1 Applications to compound frequency models 170

9.6.2 Underflow/overflow problems 172

9.6.3 Numerical stability 173

9.6.4 Continuous severity 173

9.6.5 Constructing arithmetic distributions 174

9.6.6 Exercises 177

9.7 The impact of individual policy modifications on aggregate payments 181

9.7.1 Exercises 183

9.8 The individual risk model 184

9.8.1 The model 184

9.8.2 Parametric approximation 186

9.8.3 Compound Poisson approximation 187

9.8.4 Exercises 190

PART III MATHEMATICAL STATISTICS

10 Introduction to Mathematical Statistics 195

10.1 Introduction and four data sets 195

10.2 Point estimation 197

10.2.1 Introduction 197

10.2.2 Measures of quality 198

10.2.3 Exercises 208

10.3 Interval estimation 210

10.3.1 Exercises 212

10.4 Construction of Parametric Estimators 212

10.4.1 Method of moments and percentile matching 212

10.4.2 Exercises 215

10.5 Tests of hypotheses 218

10.5.1 Exercise 222

11 Maximum likelihood estimation 223

11.1 Introduction 223

11.2 Individual data 225

11.2.1 Exercises 226

11.3 Grouped data 229

11.3.1 Exercises 230

11.4 Truncated or censored data 230

11.4.1 Exercises 235

11.5 Variance and interval estimation for maximum likelihood estimators 236

11.5.1 Exercises 241

11.6 Functions of aymptotically normal estimators 242

11.6.1 Exercises 244

11.7 Nonnormal confidence intervals 244

11.7.1 Exercise 247

12 Frequentist estimation for discrete distributions 249

12.1 Poisson 249

12.2 Negative binomial 252

12.3 Binomial 255

12.4 The (a, b, 1) class 257

12.5 Compound models 261

12.6 Effect of exposure on maximum likelihood estimation 263

12.7 Exercises 264

13 Bayesian estimation 269

13.1 Definitions and Bayes’ Theorem 269

13.2 Inference and prediction 273

13.2.1 Exercises 279

13.3 Conjugate prior distributions and the linear exponential family 284

13.3.1 Exercises 285

13.4 Computational issues 286

PART IV CONSTRUCTION OF MODELS

14 Construction of empirical models 289

14.1 The Empirical Distribution 289

14.2 Empirical distributions for grouped data 293

14.2.1 Exercises 295

14.3 Empirical estimation with right censored data 298

14.3.1 Exercises 309

14.4 Empirical estimation of moments 313

14.4.1 Exercises 319

14.5 Empirical estimation with left truncated data 319

14.5.1 Exercises 324

14.6 Kernel density models 325

14.6.1 Exercises 329

14.7 Approximations for large data sets 330

14.7.1 Introduction 330

14.7.2 Using individual data points 332

14.7.3 Interval-based methods 336

14.7.4 Exercises 339

14.8 Maximum likelihood estimation of decrement probabilities 340

14.8.1 Exercise 343

14.9 Estimation of transition intensities 343

15 Model selection 345

15.1 Introduction 345

15.2 Representations of the data and model 346

15.3 Graphical comparison of the density and distribution functions 347

15.3.1 Exercises 352

15.4 Hypothesis tests 352

15.4.1 Kolmogorov–Smirnov test 352

15.4.2 Anderson–Darling test 355

15.4.3 Chi-square goodness-of-fit test 355

15.4.4 Likelihood ratio test 360

15.4.5 Exercises 361

15.5 Selecting a model 363

15.5.1 Introduction 363

15.5.2 Judgment-based approaches 364

15.5.3 Score-based approaches 365

15.5.4 Exercises 372

PART V CREDIBILITY

16 Introduction and Limited Fluctuation Credibility 381

16.1 Introduction 381

16.2 Limited fluctuation credibility theory 383

16.3 Full credibility 384

16.4 Partial credibility 387

16.5 Problems with the approach 390

16.6 Notes and References 391

16.7 Exercises 391

17 Greatest accuracy credibility 395

17.1 Introduction 395

17.2 Conditional distributions and expectation 397

17.3 The Bayesian methodology 401

17.4 The credibility premium 409

17.5 The Bu¨hlmann model 412

17.6 The Bu¨hlmann–Straub model 416

17.7 Exact credibility 421

17.8 Notes and References 425

17.9 Exercises 426

18 Empirical Bayes parameter estimation 439

18.1 Introduction 439

18.2 Nonparametric estimation 442

18.3 Semiparametric estimation 452

18.4 Notes and References 454

18.5 Exercises 454

19 Simulation 461

19.1 Basics of simulation 461

19.1.1 The simulation approach 462

19.1.2 Exercises 466

19.2 Simulation for specific distributions 466

19.2.1 Discrete mixtures 466

19.2.2 Time or age of death from a life table 467

19.2.3 Simulating from the (a, b, 0) class 468

19.2.4 Normal and lognormal distributions 470

19.2.5 Exercises 471

19.3 Determining the sample size 471

19.3.1 Exercises 473

19.4 Examples of simulation in actuarial modeling 474

19.4.1 Aggregate loss calculations 474

19.4.2 Examples of lack of independence 474

19.4.3 Simulation analysis of the two examples 475

19.4.4 Using simulation to determine risk measures 478

19.4.5 Statistical analyses 478

19.4.6 Exercises 480

A An inventory of continuous distributions 483

A.1 Introduction 483

A.2 Transformed beta family 487

A.2.1 Four-parameter distribution 487

A.2.2 Three-parameter distributions 487

A.2.3 Two-parameter distributions 489

A.3 Transformed gamma family 492

A.3.1 Three-parameter distributions 492

A.3.2 Two-parameter distributions 493

A.3.3 One-parameter distributions 494

A.4 Distributions for large losses 495

A.4.1 Extreme value distributions 495

A.4.2 Generalized Pareto distributions 496

A.5 Other distributions 496

A.6 Distributions with finite support 498

B An inventory of discrete distributions 501

B.4.1 Some compound distributions 506

B.5 A hierarchy of discrete distributions 508

C Frequency and severity relationships 509

D The recursive formula 511

E Discretization of the severity distribution 513

E.1 The method of rounding 513

E.2 Mean preserving 514

E.3 Undiscretization of a discretized distribution 514

References 517

Index 0

1 Introduction 1

2 Chapter 2 solutions 3

2.1 Section 2.2 3

3 Chapter 3 solutions 9

3.1 Section 3.1 9

3.2 Section 3.2 14

3.3 Section 3.3 15

3.4 Section 3.4 15

3.5 Section 3.5 19

4 Chapter 4 solutions 23

4.1 Section 4.2 23

5 Chapter 5 solutions 29

5.1 Section 5.2 29

5.2 Section 5.3 38

5.3 Section 5.4 39

6 Chapter 6 solutions 41

6.1 Section 6.1 41

6.2 Section 6.5 41

6.3 Section 6.6 42

7 Chapter 7 solutions 45

7.1 Section 7.1 45

7.2 Section 7.2 46

7.3 Section 7.3 48

7.4 Section 7.5 52

8 Chapter 8 solutions 57

8.1 Section 8.2 57

8.2 Section 8.3 58

8.3 Section 8.4 61

8.4 Section 8.5 61

8.5 Section 8.6 65

9 Chapter 9 solutions 69

9.1 Section 9.1 69

9.2 Section 9.2 69

9.3 Section 9.3 70

9.4 Section 9.4 79

9.5 Section 9.6 80

9.6 Section 9.7 85

9.7 Section 9.8 87

10 Chapter 10 solutions 93

10.1 Section 10.2 93

10.2 Section 10.3 97

10.3 Section 10.4 98

10.4 Section 10.5 102

11 Chapter 11 solutions 105

11.1 Section 11.2 105

11.2 Section 11.3 110

11.3 Section 11.4 110

11.4 Section 11.5 115

11.5 Section 11.6 120

11.6 Section 11.7 122

12 Chapter 12 solutions 123

12.1 Section 12.7 123

13 Chapter 13 solutions 129

14 Chapter 14 Solutions 141

14.1 Section 14.2 141

14.2 Section 14.3 145

14.3 Section 14.4 150

14.4 Section 14.5 152

14.5 Section 14.6 157

14.6 Section 14.7 159

14.7 Section 14.8 160

15 Chapter 15 solutions 161

15.1 Section 15.3 161

15.2 Section 15.4 161

15.3 Section 15.5 171

16 Chapter 16 solutions

16.1 Section 16.7

17 Chapter 17 Solutions

17.1 Section 17.9

18 Chapter 18 Solutions

18.1 Section 18.5 211

19 Chapter 19 solutions 219

19.1 Section 19.1 219

19.2 Section 19.2 220

19.3 Section 19.3 221

19.4 Section 19.4 221