Loss Models Further Topics

An essential resource for constructing and analyzing advanced actuarial models 


Loss Models: Further Topics presents extended coverage of modeling through the use of tools related to risk theory, loss distributions, and survival models. The book uses these methods to construct and evaluate actuarial models in the fields of insurance and business. Providing an advanced study of actuarial methods, the book features extended discussions of risk modeling and risk measures, including Tail-Value-at-Risk. Loss Models: Further Topics contains additional material to accompany the Fourth Edition of Loss Models: From Data to Decisions, such as:

• Extreme value distributions
• Coxian and related distributions
• Mixed Erlang distributions
• Computational and analytical methods for aggregate claim models
• Counting processes
• Compound distributions with time-dependent claim amounts
• Copula models
• Continuous time ruin models
• Interpolation and smoothing

The book is an essential reference for practicing actuaries and actuarial researchers who want to go beyond the material required for actuarial qualification. Loss Models: Further Topics is also an excellent resource for graduate students in the actuarial field.

STUART A. KLUGMAN, PhD, is Staff Fellow (Education) at the Society of Actuaries and Principal Financial Group Distinguished Professor Emeritus of Actuarial Science at Drake University. Dr. Klugman is a two-time recipient of the Society of Actuaries' Presidential Award.

HARRY H. PANJER, PhD, is Distinguished Professor Emeritus in the Department of Statistics and Actuarial Science at the University of Waterloo, Canada. Dr. Panjer was previously president of the Canadian Institute of Actuaries and the Society of Actuaries.

GORDON E. WILLMOT, PhD, is Munich Re Chair in Insurance and Professor in the Department of Statistics and Actuarial Science at the University of Waterloo, Canada. Dr. Willmot has authored more than eighty-five articles in the areas of risk theory, queuing theory, distribution theory, and stochastic modeling in insurance.

Preface xi

1 Introduction 1

2 Coxian and related distributions 3

2.1 Introduction 3

2.2 Combinations of exponentials 4

2.3 Coxian-2 distributions 8

3 Mixed Erlang distributions 11

3.1 Introduction 11

3.2 Members of the mixed Erlang class 13

3.3 Distributional properties 18

3.4 Mixed Erlang claim severity models 22

4 Extreme value distributions 25

4.1 Introduction 25

4.2 Distribution of the maximum 27

4.3 Stability of the maximum of the extreme value distribution 31

4.4 The Fisher—Tippett theorem 33

4.5 Maximum domain of attraction 34

4.6 Generalized Pareto distributions 37

4.7 Stability of excesses of the generalized Pareto 39

4.8 Limiting distributions of excesses 40

4.9 Parameter estimation 42

5 Analytic and related methods for aggregate claim models 55

5.1 Introduction 55

5.2 Elementary approaches 58

5.3 Discrete analogues 63

5.4 Right-tail asymptotics for aggregate losses 67

6 Computational methods for aggregate models 77

6.1 Recursive techniques for compound distributions 77

6.2 Inversion methods 79

6.3 Calculations with approximate distributions 84

6.4 Comparison of methods 90

6.5 The individual risk model 91

7 Counting Processes 101

7.1 Nonhomogeneous birth processes 101

7.2 Mixed Poisson processes 117

8 Discrete Claim Count Models 125

8.1 Unification of the (a, b, 1) and mixed Poisson classes 125

8.2 A class of discrete generalized tail-based distributions 133

8.3 Higher order generalized tail-based distributions 140

8.4 Mixed Poisson properties of generalized tail-based distributions 146

8.5 Compound geometric properties of generalized tail-based distributions 153

9 Compound distributions with time dependent claim amounts 165

9.1 Introduction 165

9.2 A model for inflation 169

9.3 A model for claim payment delays 180

10 Copula models 193

10.1 Introduction 193

10.2 Sklar’s theorem and copulas 194

10.3 Measures of dependency 196

10.4 Tail dependence 197

10.5 Archimedean copulas 198

10.6 Elliptical copulas 203

10.7 Extreme value copulas 206

10.8 Archimax copulas 210

10.9 Estimation of parameters 210

10.10 Simulation from Copula Models 218

11 Continuous-time ruin models 223

11.1 Introduction 223

11.2 The adjustment coefficient and Lundberg’s inequality 225

11.3 An integrodifferential equation 233

11.4 The maximum aggregate loss 238

11.5 Cramer’s asymptotic ruin formula and Tijms’ approximation 242

11.6 The Brownian motion risk process 249

11.7 Brownian motion and the probability of ruin 253

12 Interpolation and smoothing 259

12.1 Introduction 259

12.2 Interpolation with Splines 261

12.3 Extrapolating with splines 268

12.4 Smoothing with Splines 269

Appendix A: An inventory of continuous distributions 277

A.1 Introduction 277

A.2 transformed beta family 281

A.3 transformed gamma family 285

A.4 Distributions for large losses 288

A.5 Other distributions 289

A.6 Distributions with finite support 291

Appendix B: An inventory of discrete distributions 293

B.1 Introduction 293

B.2 The (a, b, 0) class 294

B.3 The (a, b, 1) class 295

B.4 The compound class 298

B.5 A hierarchy of discrete distributions 299

Appendix C: Discretization of the severity distribution 301

C.1 The method of rounding 301

C.2 Mean preserving 302

C.3 Undiscretization of a discretized distribution 302

Appendix D: Solutions to Exercises 305

D.1 Chapter 4 305

D.2 Chapter 5 307

D.3 Chapter 6 308

D.4 Chapter 7 310

D.5 Chapter 8 316

D.6 Chapter 10 321

D.7 Chapter 11 324

D.8 Chapter 12 344

References 349

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