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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.
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
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