Loss Models: From Data to Decisions

Praise for the Third Edition

"This book provides in–depth coverage of modelling techniquesused throughout many branches of actuarial science. . . . Theexceptional high standard of this book has made it a pleasure toread." Annals of Actuarial Science

Newly organized to focus exclusively on material tested in theSociety of Actuaries′ Exam C and the Casualty Actuarial Society′sExam 4, Loss Models: From Data to Decisions, Fourth Editioncontinues to supply actuaries with a practical approach to the keyconcepts and techniques needed on the job. With updated materialand extensive examples, the book successfully provides theessential methods for using available data to construct models forthe frequency and severity of future adverse outcomes.

The book continues to equip readers with the tools needed forthe construction and analysis of mathematical models that describethe process by which funds flow into and out of an insurancesystem. Focusing on the loss process, the authors explore keyquantitative techniques including random variables, basicdistributional quantities, and the recursive method, and discusstechniques for classifying and creating distributions. Parametric,non–parametric, and Bayesian estimation methods are thoroughlycovered along with advice for choosing an appropriate model.

New features of this Fourth Edition include:

Expanded discussion of working with large data sets, nowincluding more practical elements of constructing decrementtables Added coverage of methods for simulating several specialsituations An updated presentation of Bayesian estimation, outliningconjugate prior distributions and the linear exponential family aswell as related computational issues

Throughout the book, numerous examples showcase the real–worldapplications of the presented concepts, with an emphasis oncalculations and spreadsheet implementation. A wealth of newexercises taken from previous Exam C/4 exams allows readers to testtheir comprehension of the material, and a related FTP sitefeatures the book′s data sets.

Loss Models, Fourth Edition is an indispensable resourcefor students and aspiring actuaries who are preparing to take theSOA and CAS examinations. The book is also a valuable reference forprofessional actuaries, actuarial students, and anyone who workswith loss and risk models.

Preface xiii

PART I INTRODUCTION

1 Modeling 3

1.1 The model–based approach 3

1.2 Organization of this book 5

2 Random variables 7

2.1 Introduction 7

2.2 Key functions and four models 9

3 Basic distributional quantities 19

3.1 Moments 19

3.2 Percentiles 27

3.3 Generating functions and sums of random variables 29

3.4 Tails of distributions 31

3.5 Measures of Risk 38

PART II ACTUARIAL MODELS

4 Characteristics of Actuarial Models 49

4.1 Introduction 49

4.2 The role of parameters 49

5 Continuous models 59

5.1 Introduction 59

5.2 Creating new distributions 59

5.3 Selected distributions and their relationships 72

5.4 The linear exponential family 75

6 Discrete distributions 79

6.1 Introduction 79

6.2 The Poisson distribution 80

6.3 The negative binomial distribution 83

6.4 The binomial distribution 85

6.5 The (a, b, 0) class 86

6.6 Truncation and modification at zero 89

7 Advanced discrete distributions 95

7.1 Compound frequency distributions 95

7.2 Further properties of the compound Poisson class 101

7.3 Mixed frequency distributions 107

7.4 Effect of exposure on frequency 114

7.5 An inventory of discrete distributions 114

8 Frequency and severity with coverage modifications117

8.1 Introduction 117

8.2 Deductibles 117

8.3 The loss elimination ratio and the effect of inflation forordinary deductibles 122

8.4 Policy limits 125

8.5 Coinsurance, deductibles, and limits 127

8.6 The impact of deductibles on claim frequency 131

9 Aggregate loss models 137

9.1 Introduction 137

9.2 Model choices 140

9.3 The compound model for aggregate claims 141

9.4 Analytic results 155

9.5 Computing the aggregate claims distribution 159

9.6 The recursive method 161

9.7 The impact of individual policy modifications on aggregatepayments 173

9.8 The individual risk model 176

PART III CONSTRUCTION OF EMPIRICAL MODELS

10 Review of mathematical statistics 187

10.1 Introduction 187

10.2 Point estimation 188

10.3 Interval estimation 196

10.4 Tests of hypotheses 198

11 Estimation for complete data 203

11.1 Introduction 203

11.2 The empirical distribution for complete, individual data207

11.3 Empirical distributions for grouped data 211

12 Estimation for modified data 217

12.1 Point estimation 217

12.2 Means, variances, and interval estimation 225

12.3 Kernel density models 236

12.4 Approximations for large data sets 240

PART IV PARAMETRIC STATISTICAL METHODS

13 Frequentist estimation 253

13.1 Method of moments and percentile matching 253

13.2 Maximum likelihood estimation 259

13.3 Variance and interval estimation 272

13.4 Non–normal confidence intervals 280

13.5 Maximum likelihood estimation of decrement probabilities282

14 Frequentist Estimation for discrete distributions285

14.1 Poisson 285

14.2 Negative binomial 289

14.3 Binomial 291

14.4 The (a, b, 1) class 293

14.5 Compound models 297

14.6 Effect of exposure on maximum likelihood estimation 299

14.7 Exercises 300

15 Bayesian estimation 305

15.1 Definitions and Bayes theorem 305

15.2 Inference and prediction 309

15.3 Conjugate prior distributions and the linear exponentialfamily 320

15.4 Computational issues 322

16 Model selection 323

16.1 Introduction 323

16.2 Representations of the data and model 324

16.3 Graphical comparison of the density and distributionfunctions 325

16.4 Hypothesis tests 330

16.5 Selecting a model 342

PART V CREDIBILITY

17 Introduction and Limited Fluctuation Credibility357

17.1 Introduction 357

17.2 Limited fluctuation credibility theory 359

17.3 Full credibility 360

17.4 Partial credibility 363

17.5 Problems with the approach 366

17.6 Notes and References 367

17.7 Exercises 367

18 Greatest accuracy credibility 371

18.1 Introduction 371

18.2 Conditional distributions and expectation 373

18.3 The Bayesian methodology 377

18.4 The credibility premium 385

18.5 The Buhlmann model 388

18.6 The Buhlmann–Straub model 392

18.7 Exact credibility 397

18.8 Notes and References 401

18.9 Exercises 402

19 Empirical Bayes parameter estimation 415

19.1 Introduction 415

19.2 Nonparametric estimation 418

19.3 Semiparametric estimation 428

19.4 Notes and References 430

19.5 Exercises 430

PART VI SIMULATION

20 Simulation 437

20.1 Basics of simulation 437

20.2 Simulation for specific distributions 442

20.3 Determining the sample size 448

20.4 Examples of simulation in actuarial modeling 450

Appendix A: An inventory of continuous distributions459

A.1 Introduction 459

A.2 Transformed beta family 463

A.3 Transformed gamma family 467

A.4 Distributions for large losses 470

A.5 Other distributions 471

A.6 Distributions with finite support 473

Appendix B: An inventory of discrete distributions475

B.1 Introduction 475

B.2 The (a, b, 0) class 476

B.3 The (a, b, 1) class 477

B.4 The compound class 480

B.5 A hierarchy of discrete distributions 482

Appendix C: Frequency and severity relationships 483

Appendix D: The recursive formula 485

Appendix E: Discretization of the severity distribution487

E.1 The method of rounding 487

E.2 Mean preserving 488

E.3 Undiscretization of a discretized distribution 488

Appendix F: Numerical optimization and solution of systems ofequations 491

F.1 Maximization using Solver 491

F.2 The simplex method 495

F.3 Using Excel® to solve equations 496

References 501

STUART A. KLUGMAN, PhD, FSA, CERA, is Staff Fellow(Education) at the Society of Actuaries (SOA) and PrincipalFinancial Group Distinguished Professor Emeritus of ActuarialScience at Drake University. He served as SOA vice–president from2001–2003.

HARRY H. PANJER, PhD, is Distinguished Professor Emeritusin the Department of Statistics and Actuarial Science at theUniversity of Waterloo, Canada. He is past president of theCanadian Institute of Actuaries and the Society of Actuaries.

GORDON E. WILLMOT, PhD, FSA, FCIA, is Munich Re Chair inInsurance and Professor in the Department of Statistics andActuarial Science at the University of Waterloo, Canada. Dr.Willmot currently focuses his research on the analysis of insurancelosses, with an emphasis on the theory and application of aggregateclaims models.