# Loss Models: From Data to Decisions, 4th Edition

# Loss Models: From Data to Decisions, 4th Edition

ISBN: 978-1-118-31532-3

Sep 2012

536 pages

## Description

Praise for the *Third Edition*

**"This book provides in-depth coverage of modelling techniques used throughout many branches of actuarial science. . . . The exceptional high standard of this book has made it a pleasure to read." —Annals of Actuarial Science**

Newly organized to focus exclusively on material tested in the Society of Actuaries' Exam C and the Casualty Actuarial Society's Exam 4, *Loss Models: From Data to Decisions, Fourth Edition* continues to supply actuaries with a practical approach to the key concepts and techniques needed on the job. With updated material and extensive examples, the book successfully provides the essential methods for using available data to construct models for the frequency and severity of future adverse outcomes.

The book continues to equip readers with the tools needed for the construction and analysis of mathematical models that describe the process by which funds flow into and out of an insurance system. Focusing on the loss process, the authors explore key quantitative techniques including random variables, basic distributional quantities, and the recursive method, and discuss techniques for classifying and creating distributions. Parametric, non-parametric, and Bayesian estimation methods are thoroughly covered along with advice for choosing an appropriate model.

New features of this *Fourth Edition* include:

- Expanded discussion of working with large data sets, now including more practical elements of constructing decrement tables
- Added coverage of methods for simulating several special situations
- An updated presentation of Bayesian estimation, outlining conjugate prior distributions and the linear exponential family as well as related computational issues

Throughout the book, numerous examples showcase the real-world applications of the presented concepts, with an emphasis on calculations and spreadsheet implementation. A wealth of new exercises taken from previous Exam C/4 exams allows readers to test their comprehension of the material, and a related FTP site features the book's data sets.

*Loss Models, Fourth Edition* is an indispensable resource for students and aspiring actuaries who are preparing to take the SOA and CAS examinations. The book is also a valuable reference for professional actuaries, actuarial students, and anyone who works with loss and risk models.

To explore our additional offerings in actuarial exam preparation visit www.wiley.com/go/c4actuarial .

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**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 modifications 117**

8.1 Introduction 117

8.2 Deductibles 117

8.3 The loss elimination ratio and the effect of inflation for ordinary 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 aggregate payments 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 data 207

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 probabilities 282

**14 Frequentist Estimation for discrete distributions 285**

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 exponential family 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 distribution functions 325

16.4 Hypothesis tests 330

16.5 Selecting a model 342

**PART V CREDIBILITY**

**17 Introduction and Limited Fluctuation Credibility 357**

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 distributions 459**

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 distributions 475**

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 distribution 487**

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 of equations 491**

F.1 Maximization using Solver 491

F.2 The simplex method 495

F.3 Using Excel® to solve equations 496

**References 501**