Loss Models: From Data to Decisions / Edition 4

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An update of one of the most trusted books on constructing and analyzing actuarial models for the C/4 actuarial exam This new, abridged edition has been thoroughly revised and updated to include the essential material related to Exam C of the Society of Actuaries' and Casualty Actuarial Society's accreditation programs. The book maintains an approach to modeling and forecasting that utilizes tools related to risk theory, loss distributions, and survival models. Random variables, basic distributional quantities, the recursive method, and techniques for classifying and creating distributions are also discussed. Both parametric and non-parametric estimation methods are thoroughly covered along with advice for choosing an appropriate model. The book continues to distinguish itself by providing over 400 exercises that have appeared on previous examinations. The emphasis throughout is now placed on calculations and spreadsheet implementation. Additional features of the Fourth Edition include: extended discussions of risk management and risk measures, including Tail-Value-at-Risk; expanded coverage of copula models and their estimation; new sections on extreme value distributions and their estimations, compound frequency class of distributions, and estimation for the compound class; and motivating examples from fields of insurance and business. All data sets are available on an FTP site. An assortment of supplements (both print and electronic) is available. Loss Models, Fourth Edition is an essential resource for students and aspiring actuaries who are preparing to take the SOA and CAS preliminary examinations C/4. It is also a must-have reference for professional actuaries, graduate students in the actuarial field, and anyone who works with loss and risk models in their everyday work.

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Editorial Reviews

Allianz Global Risk Report
...gives the reader a good insight.
This applied textbook for students of actuarial science focuses on the loss process: the outflow of cash due to the payment of benefits. The authors discuss models for the amount of loss for a single claim, a number of payments, and aggregate loss. They also describe credibility theory and long-term models. The text does not assume substantial knowledge of insurance systems, but does assume a solid background in mathematical statistics and calculus. Annotation c. by Book News, Inc., Portland, Or.
From the Publisher
"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 and review." (Annals of Actuarial Science, 2008)

“This book will be necessary for all academic programs in actuarial science. It will also serve as an important reference for practicing actuaries.” (Mathematical Assoc. of America, June 2009) 

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Product Details

  • ISBN-13: 9781118315323
  • Publisher: Wiley
  • Publication date: 9/11/2012
  • Series: Wiley Series in Probability and Statistics Series , #968
  • Edition description: New Edition
  • Edition number: 4
  • Pages: 536
  • Sales rank: 314,634
  • Product dimensions: 10.00 (w) x 7.10 (h) x 1.20 (d)

Meet the Author

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.

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Table of Contents

Preface xiii


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


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


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


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


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


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

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