# Loss Models : From Data to Decisions

**by**Klugman, Stuart A.; Panjer, Harry H.; Willmot, Gordon E.

### 9781118315323

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## Summary

## Author Biography

**STUART A. KLUGMAN, PhD, FSA, CERA, **is Staff Fellow (Education) at the Society of Actuaries (SOA) and Principal Financial Group Distinguished Professor Emeritus of Actuarial Science at Drake University. He served as SOA vice-president from 2001-2003.

**HARRY H. PANJER, PhD, **is Distinguished Professor Emeritus in the Department of Statistics and Actuarial Science at the University of Waterloo, Canada. He is past president of the Canadian Institute of Actuaries and the Society of Actuaries.

**GORDON E. WILLMOT, PhD, FSA, FCIA,** is Munich Re Chair in Insurance and Professor in the Department of Statistics and Actuarial Science at the University of Waterloo, Canada. Dr. Willmot currently focuses his research on the analysis of insurance losses, with an emphasis on the theory and application of aggregate claims models.

## Table of Contents

**PART I INTRODUCTION**

**1 Modeling 3**

1.1 The model-based approach 3

1.2 Organization of this book 7

**2 Random variables 11**

2.1 Introduction 11

2.2 Key functions and four models 13

**3 Basic distributional quantities 25**

3.1 Moments 25

3.2 Percentiles 36

3.3 Generating functions and sums of random variables 38

3.4 Tails of distributions 41

3.5 Measures of Risk 50

**PART II ACTUARIAL MODELS**

**4 Characteristics of Actuarial Models 63**

4.1 Introduction 63

4.2 The role of parameters 65

**5 Continuousmodels 77**

5.1 Introduction 77

5.2 Creating new distributions 77

5.3 Selected distributions and their relationships 93

5.4 The linear exponential family 98

**6 Discrete distributions 103**

6.1 Introduction 103

6.2 The Poisson distribution 104

6.3 The negative binomial distribution 108

6.4 The binomial distribution 111

6.5 The (a, b, 0) class 113

6.6 Truncation and modification at zero 117

**7 Advanced discrete distributions 125**

7.1 Compound frequency distributions 125

7.2 Further properties of the compound Poisson class 133

7.3 Mixed frequency distributions 139

7.4 Effect of exposure on frequency 148

7.5 An inventory of discrete distributions 149

**8 Frequency and severity with coverage modifications 153**

8.1 Introduction 153

8.2 Deductibles 155

8.3 The loss elimination ratio and the effect of inflation for ordinary deductibles 161

8.4 Policy limits 164

8.5 Coinsurance, deductibles, and limits 167

8.6 The impact of deductibles on claim frequency 171

**9 Aggregate loss models 179**

9.1 Introduction 179

9.2 Model choices 184

9.3 The compound model for aggregate claims 186

9.4 Analytic results 203

9.5 Computing the aggregate claims distribution 209

9.6 The recursive method 211

9.7 The impact of individual policy modifications on aggregate payments 227

9.8 The individual risk model 232

**PART III CONSTRUCTION OF EMPIRICAL MODELS**

**10 Review of mathematical statistics 245**

10.1 Introduction 245

10.2 Point estimation 246

10.3 Interval estimation 257

10.4 Tests of hypotheses 260

**11 Estimation for complete data 267**

11.1 Introduction 267

11.2 The empirical distribution for complete, individual data 273

11.3 Empirical distributions for grouped data 278

**12 Estimation for modified data 285**

12.1 Point estimation 285

12.3 Kernel density models 308

12.4 Approximations for large data sets 314

**PART IV PARAMETRIC STATISTICAL METHODS**

**13 Frequentist estimation 331**

13.1 Method of moments and percentile matching 332

13.2 Maximum likelihood estimation 339

13.3 Variance and interval estimation 355

13.4 Non-normal confidence intervals 365

13.5 Maximum likelihood estimation of decrement probabilities 369

**14 Frequentist Estimation for discrete distributions 373**

14.1 Poisson 373

14.2 Negative binomial 378

14.3 Binomial 380

14.4 The (a, b, 1) class 384

14.5 Compound models 389

14.6 Effect of exposure on maximum likelihood estimation 391

14.7 Exercises 392

**15 Bayesian estimation 397**

15.1 Definitions and Bayes’ theorem 398

15.2 Inference and prediction 402

15.3 Conjugate prior distributions and the linear exponential family 416

15.4 Computational issues 419

**16 Model selection 421**

16.1 Introduction 421

16.2 Representations of the data and model 422

16.3 Graphical comparison of the density and distribution functions 424

16.4 Hypothesis tests 430

16.5 Selecting a model 445

**PART V CREDIBILITY**

**17 Introduction and Limited Fluctuation Credibility 467**

17.1 Introduction 467

17.2 Limited fluctuation credibility theory 470

17.3 Full credibility 471

17.4 Partial credibility 475

17.5 Problems with the approach 480

17.6 Notes and References 480

17.7 Exercises 480

**18 Greatest accuracy credibility 485**

18.1 Introduction 485

18.2 Conditional distributions and expectation 489

18.3 The Bayesian methodology 494

18.4 The credibility premium 503

18.5 The Buhlmann model 507

18.6 The Buhlmann?Straub model 511

18.7 Exact credibility 518

18.8 Notes and References 522

18.9 Exercises 523

**19 Empirical Bayes parameter estimation 541**

19.1 Introduction 541

19.2 Nonparametric estimation 544

19.3 Semiparametric estimation 557

19.4 Notes and References 559

19.5 Exercises 560

**PART VI SIMULATION**

20 Simulation 567

20.1 Basics of simulation 567

20.2 Simulation for specific distributions 573

20.3 Determining the sample size 580

20.4 Examples of simulation in actuarial modeling 583

**Appendix A: An inventory of continuous distributions 597**

A.1 Introduction 597

A.2 Transformed beta family 601

A.3 Transformed gamma family 606

A.4 Distributions for large losses 609

A.5 Other distributions 611

A.6 Distributions with finite support 613

**Appendix B: An inventory of discrete distributions 615**

B.1 Introduction 615

B.2 The (*a*, *b*, 0) class 616

B.3 The (*a*, *b*, 1) class 618

B.4 The compound class 621

B.5 A hierarchy of discrete distributions 623

**Appendix C: Frequency and severity relationships 625**

**Appendix D: The recursive formula 629**

**Appendix E: Discretization of the severity distribution 631**

E.1 The method of rounding 631

E.2 Mean preserving 632

E.3 Undiscretization of a discretized distribution 633

**Appendix F: Numerical optimization and solution of systems of equations 635**

F.1 Maximization using Solver 636

F.2 The simplex method 640

F.3 Using Excel® to solve equations 641

References 647