Loss Models

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 xi1 Introduction 12 Coxian and related distributions 32.1 Introduction 32.2 Combinations of exponentials 42.3 Coxian-2 distributions 73 Mixed Erlang distributions 113.1 Introduction 113.2 Members of the mixed Erlang class 123.3 Distributional properties 183.4 Mixed Erlang claim severity models 224 Extreme value distributions 234.1 Introduction 234.2 Distribution of the maximum 254.2.1 From a fixed number of losses 254.2.2 From a random number of losses 274.3 Stability of the maximum of the extreme value distribution 294.4 The Fisher–Tippett theorem 304.5 Maximum domain of attraction 324.6 Generalized Pareto distributions 344.7 Stability of excesses of the generalized Pareto 364.8 Limiting distributions of excesses 374.9 Parameter estimation 394.9.1 Maximum likelihood estimation from the extreme value distribution 394.9.2 Maximum likelihood estimation for the generalized Pareto distribution 424.9.3 Estimating the Pareto shape parameter 444.9.4 Estimating extreme probabilities 474.9.5 Mean excess plots 494.9.6 Further reading 494.9.7 Exercises 495 Analytic and related methods for aggregate claim models 515.1 Introduction 515.2 Elementary approaches 535.3 Discrete analogues 585.4 Right-tail asymptotics for aggregate losses 635.4.1 Exercises 716 Computational methods for aggregate models 736.1 Recursive techniques for compound distributions 736.2 Inversion methods 756.2.1 Fast Fourier transform 756.2.2 Direct numerical inversion 786.3 Calculations with approximate distributions 806.3.1 Arithmetic distributions 806.3.2 Empirical distributions 836.3.3 Piecewise linear cdf 846.3.4 Exercises 856.4 Comparison of methods 866.5 The individual risk model 876.5.1 Definition and notation 876.5.2 Direct calculation 886.5.3 Recursive calculation 897 Counting Processes 977.1 Nonhomogeneous birth processes 977.1.1 Exercises 1127.2 Mixed Poisson processes 1127.2.1 Exercises 1168 Discrete Claim Count Models 1198.1 Unification of the (a, b, 1) and mixed Poisson classes 1198.2 A class of discrete generalized tail-based distributions 1278.3 Higher order generalized tail-based distributions 1348.4 Mixed Poisson properties of generalized tail-based distributions 1398.5 Compound geometric properties of generalized tail-based distributions 1468.5.1 Exercises 1569 Compound distributions with time dependent claim amounts 1599.1 Introduction 1599.2 A model for inflation 1639.3 A model for claim payment delays 17310 Copula models 18710.1 Introduction 18710.2 Sklar’s theorem and copulas 18810.3 Measures of dependency 18910.3.1 Spearman’s rho 19010.3.2 Kendall’s tau 19010.4 Tail dependence 19110.5 Archimedean copulas 19210.5.1 Exercise 19710.6 Elliptical copulas 19710.6.1 Exercise 19910.7 Extreme value copulas 20010.7.1 Exercises 20210.8 Archimax copulas 20310.9 Estimation of parameters 20310.9.1 Introduction 20310.9.2 Maximum likelihood estimation 20410.9.3 Semiparametric estimation 20610.9.4 The role of deductibles 20610.9.5 Goodness-of-fit testing 20810.9.6 An example 20910.9.7 Exercise 21010.10 Simulation from Copula Models 21110.10.1 Simulating from the Gaussian copula 21310.10.2 Simulating from the t copula 21311 Continuous-time ruin models 21511.1 Introduction 21511.1.1 The Poisson process 21511.1.2 The continuous-time problem 21611.2 The adjustment coefficient and Lundberg’s inequality 21711.2.1 The adjustment coefficient 21711.2.2 Lundberg’s inequality 22111.2.3 Exercises 22311.3 An integrodifferential equation 22411.3.1 Exercises 22811.4 The maximum aggregate loss 22911.4.1 Exercises 23811.5 Cramer’s asymptotic ruin formula and Tijms’ approximation 24011.5.1 Exercises 24311.6 The Brownian motion risk process 24511.7 Brownian motion and the probability of ruin 24912 Interpolation and smoothing 25512.1 Introduction 25512.2 Interpolation with splines 25712.2.1 Exercises 26312.3 Extrapolating with splines 26412.3.1 Exercise 26512.4 Smoothing with splines 26512.4.1 Exercise 272A An inventory of continuous distributions 273A.1 Introduction 273A.2 Transformed beta family 277A.2.1 Four-parameter distribution 277A.2.2 Three-parameter distributions 277A.2.3 Two-parameter distributions 279A.3 transformed gamma family 281A.3.1 Three-parameter distributions 281A.3.2 Two-parameter distributions 282A.3.3 One-parameter distributions 283A.4 Distributions for large losses 284A.4.1 Extreme value distributions 284A.4.2 Generalized Pareto distributions 285A.5 Other distributions 285A.6 Distributions with finite support 287B An inventory of discrete distributions 289B.1 Introduction 289B.2 The (a, b, 0) class 290B.3 The (a, b, 1) class 291B.3.1 The zero-truncated subclass 291B.3.2 The zero-modified subclass 293B.4 The compound class 294B.4.1 Some compound distributions 294B.5 A hierarchy of discrete distributions 295C Discretization of the severity distribution 297C.1 The method of rounding 297C.2 Mean preserving 298C.3 Undiscretization of a discretized distribution 298D Solutions to Exercises 301D.1 Chapter 4 301D.2 Chapter 5 303D.3 Chapter 6 304D.4 Chapter 7 305D.5 Chapter 8 312D.6 Chapter 10 316D.7 Chapter 11 319D.8 Chapter 12 333References 339Index 345