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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…mehr

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Produktbeschreibung
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.

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  • Produktdetails
  • Verlag: John Wiley & Sons
  • Seitenzahl: 368
  • Erscheinungstermin: 29.08.2013
  • Englisch
  • ISBN-13: 9781118573747
  • Artikelnr.: 39420418
Autorenporträt
STUART A. KLUGMAN, PhD, is Staff Fellow (Education) atthe Society of Actuaries and Principal Financial GroupDistinguished Professor Emeritus of Actuarial Science at DrakeUniversity. Dr. Klugman is a two-time recipient of the Society ofActuaries' Presidential Award. HARRY H. PANJER, PhD, is Distinguished Professor Emeritusin the Department of Statistics and Actuarial Science at theUniversity of Waterloo, Canada. Dr. Panjer was previously presidentof the Canadian Institute of Actuaries and the Society ofActuaries. GORDON E. WILLMOT, PhD, is Munich Re Chair in Insuranceand Professor in the Department of Statistics and Actuarial Scienceat the University of Waterloo, Canada. Dr. Willmot has authoredmore than eighty-five articles in the areas of risk theory, queuingtheory, distribution theory, and stochastic modeling ininsurance.
Inhaltsangabe
Preface xiii 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(r) to solve equations 496 References 501

Preface xiii About the Companion Website xv Part I Introduction 1 Modeling 3 1.1 The Model-Based Approach 3 1.1.1 The Modeling Process 3 1.1.2 The Modeling Advantage 5 1.2 The Organization of This Book 6 2 Random Variables 9 2.1 Introduction 9 2.2 Key Functions and Four Models 11 2.2.1 Exercises 19 3 Basic Distributional Quantities 21 3.1 Moments 21 3.1.1 Exercises 28 3.2 Percentiles 29 3.2.1 Exercises 31 3.3 Generating Functions and Sums of Random Variables 31 3.3.1 Exercises 33 3.4 Tails of Distributions 33 3.4.1 Classification Based on Moments 33 3.4.2 Comparison Based on Limiting Tail Behavior 34 3.4.3 Classification Based on the Hazard Rate Function 35 3.4.4 Classification Based on the Mean Excess Loss Function 36 3.4.5 Equilibrium Distributions and Tail Behavior 38 3.4.6 Exercises 39 3.5 Measures of Risk 41 3.5.1 Introduction 41 3.5.2 Risk Measures and Coherence 41 3.5.3 Value at Risk 43 3.5.4 Tail Value at Risk 44 3.5.5 Exercises 48 Part II Actuarial Models 4 Characteristics of Actuarial Models 51 4.1 Introduction 51 4.2 The Role of Parameters 51 4.2.1 Parametric and Scale Distributions 52 4.2.2 Parametric Distribution Families 54 4.2.3 Finite Mixture Distributions 54 4.2.4 Data-Dependent Distributions 56 4.2.5 Exercises 59 5 Continuous Models 61 5.1 Introduction 61 5.2 Creating New Distributions 61 5.2.1 Multiplication by a Constant 62 5.2.2 Raising to a Power 62 5.2.3 Exponentiation 64 5.2.4 Mixing 64 5.2.5 Frailty Models 68 5.2.6 Splicing 69 5.2.7 Exercises 70 5.3 Selected Distributions and Their Relationships 74 5.3.1 Introduction 74 5.3.2 Two Parametric Families 74 5.3.3 Limiting Distributions 74 5.3.4 Two Heavy-Tailed Distributions 76 5.3.5 Exercises 77 5.4 The Linear Exponential Family 78 5.4.1 Exercises 80 6 Discrete Distributions 81 6.1 Introduction 81 6.1.1 Exercise 82 6.2 The Poisson Distribution 82 6.3 The Negative Binomial Distribution 85 6.4 The Binomial Distribution 87 6.5 The (a, b, 0) Class 88 6.5.1 Exercises 91 6.6 Truncation and Modification at Zero 92 6.6.1 Exercises 96 7 Advanced Discrete Distributions 99 7.1 Compound Frequency Distributions 99 7.1.1 Exercises 105 7.2 Further Properties of the Compound Poisson Class 105 7.2.1 Exercises 111 7.3 Mixed-Frequency Distributions 111 7.3.1 The General Mixed-Frequency Distribution 111 7.3.2 Mixed Poisson Distributions 113 7.3.3 Exercises 118 7.4 The Effect of Exposure on Frequency 120 7.5 An Inventory of Discrete Distributions 121 7.5.1 Exercises 122 8 Frequency and Severity with Coverage Modifications 125 8.1 Introduction 125 8.2 Deductibles 126 8.2.1 Exercises 131 8.3 The Loss Elimination Ratio and the Effect of Inflation for Ordinary Deductibles 132 8.3.1 Exercises 133 8.4 Policy Limits 134 8.4.1 Exercises 136 8.5 Coinsurance, Deductibles, and Limits 136 8.5.1 Exercises 138 8.6 The Impact of Deductibles on Claim Frequency 140 8.6.1 Exercises 144 9 Aggregate Loss Models 147 9.1 Introduction 147 9.1.1 Exercises 150 9.2 Model Choices 150 9.2.1 Exercises 151 9.3 The Compound Model for Aggregate Claims 151 9.3.1 Probabilities and Moments 152 9.3.2 Stop-Loss Insurance 157 9.3.3 The Tweedie Distribution 159 9.3.4 Exercises 160 9.4 Analytic Results 167 9.4.1 Exercises 170 9.5 Computing the Aggregate Claims Distribution 171 9.6 The Recursive Method 173 9.6.1 Applications to Compound Frequency Models 175 9.6.2 Underflow/Overflow Problems 177 9.6.3 Numerical Stability 178 9.6.4 Continuous Severity 178 9.6.5 Constructing Arithmetic Distributions 179 9.6.6 Exercises 182 9.7 The Impact of Individual Policy Modifications on Aggregate Payments 186 9.7.1 Exercises 189 9.8 The Individual Risk Model 189 9.8.1 The Model 189 9.8.2