Loss Models: From Data to Decisions

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 Parametric Approximation 191 9.8.3 Compound Poisson Approximation 193 9.8.4 Exercises 195 Part III Mathematical Statistics 10 Introduction to Mathematical Statistics 201 10.1 Introduction and Four Data Sets 201 10.2 Point Estimation 203 10.2.1 Introduction 203 10.2.2 Measures of Quality 204 10.2.3 Exercises 214 10.3 Interval Estimation 216 10.3.1 Exercises 218 10.4 The Construction of Parametric Estimators 218 10.4.1 The Method of Moments and Percentile Matching 218 10.4.2 Exercises 221 10.5 Tests of Hypotheses 224 10.5.1 Exercise 228 11 Maximum Likelihood Estimation 229 11.1 Introduction 229 11.2 Individual Data 231 11.2.1 Exercises 232 11.3 Grouped Data 235 11.3.1 Exercises 236 11.4 Truncated or Censored Data 236 11.4.1 Exercises 241 11.5 Variance and Interval Estimation for Maximum Likelihood Estimators 242 11.5.1 Exercises 247 11.6 Functions of Asymptotically Normal Estimators 248 11.6.1 Exercises 250 11.7 Nonnormal Confidence Intervals 251 11.7.1 Exercise 253 12 Frequentist Estimation for Discrete Distributions 255 12.1 The Poisson Distribution 255 12.2 The Negative Binomial Distribution 259 12.3 The Binomial Distribution 261 12.4 The (a, b, 1) Class 264 12.5 Compound Models 268 12.6 The Effect of Exposure on Maximum Likelihood Estimation 269 12.7 Exercises 270 13 Bayesian Estimation 275 13.1 Definitions and Bayes' Theorem 275 13.2 Inference and Prediction 279 13.2.1 Exercises 285 13.3 Conjugate Prior Distributions and the Linear Exponential Family 290 13.3.1 Exercises 291 13.4 Computational Issues 292 Part IV Construction of Models 14 Construction of Empirical Models 295 14.1 The Empirical Distribution 295 14.2 Empirical Distributions for Grouped Data 300 14.2.1 Exercises 301 14.3 Empirical Estimation with Right Censored Data 304 14.3.1 Exercises 316 14.4 Empirical Estimation of Moments 320 14.4.1 Exercises 326 14.5 Empirical Estimation with Left Truncated Data 327 14.5.1 Exercises 331 14.6 Kernel Density Models 332 14.6.1 Exercises 336 14.7 Approximations for Large Data Sets 337 14.7.1 Introduction 337 14.7.2 Using Individual Data Points 339 14.7.3 Interval-Based Methods 342 14.7.4 Exercises 346 14.8 Maximum Likelihood Estimation of Decrement Probabilities 347 14.8.1 Exercise 349 14.9 Estimation of Transition Intensities 350 15 Model Selection 353 15.1 Introduction 353 15.2 Representations of the Data and Model 354 15.3 Graphical Comparison of the Density and Distribution Functions 355 15.3.1 Exercises 360 15.4 Hypothesis Tests 360 15.4.1 The Kolmogorov-Smirnov Test 360 15.4.2 The Anderson-Darling Test 363 15.4.3 The Chi-Square Goodness-of-Fit Test 363 15.4.4 The Likelihood Ratio Test 367 15.4.5 Exercises 369 15.5 Selecting a Model 371 15.5.1 Introduction 371 15.5.2 Judgment-Based Approaches 372 15.5.3 Score-Based Approaches 373 15.5.4 Exercises 381 Part V Credibility 16 Introduction to Limited Fluctuation Credibility 387 16.1 Introduction 387 16.2 Limited Fluctuation Credibility Theory 389 16.3 Full Credibility 390 16.4 Partial Credibility 393 16.5 Problems with the Approach 397 16.6 Notes and References 397 16.7 Exercises 397 17 Greatest Accuracy Credibility 401 17.1 Introduction 401 17.2 Conditional Distributions and Expectation 404 17.3 The Bayesian Methodology 408 17.4 The Credibility Premium 415 17.5 The Bühlmann Model 418 17.6 The Bühlmann-Straub Model 422 17.7 Exact Credibility 427 17.8 Notes and References 431 17.9 Exercises 432 18 Empirical Bayes Parameter Estimation 445 18.1 Introduction 445 18.2 Nonparametric Estimation 448 18.3 Semiparametric Estimation 459 18.4 Notes and References 460 18.5 Exercises 460 Part VI Simulation 19 Simulation 467 19.1 Basics of Simulation 467 19.1.1 The Simulation Approach 468 19.1.2 Exercises 472 19.2 Simulation for Specific Distributions 472 19.2.1 Discrete Mixtures 472 19.2.2 Time or Age of Death from a Life Table 473 19.2.3 Simulating from the (a, b, 0) Class 474 19.2.4 Normal and Lognormal Distributions 476 19.2.5 Exercises 477 19.3 Determining the Sample Size 477 19.3.1 Exercises 479 19.4 Examples of Simulation in Actuarial Modeling 480 19.4.1 Aggregate Loss Calculations 480 19.4.2 Examples of Lack of Independence 480 19.4.3 Simulation Analysis of the Two Examples 481 19.4.4 The Use of Simulation to Determine Risk Measures 484 19.4.5 Statistical Analyses 484 19.4.6 Exercises 486 A An Inventory of Continuous Distributions 489 A.1 Introduction 489 A.2 The Transformed Beta Family 493 A.2.1 The Four-Parameter Distribution 493 A.2.2 Three-Parameter Distributions 493 A.2.3 Two-Parameter Distributions 494 A.3 The Transformed Gamma Family 496 A.3.1 Three-Parameter Distributions 496 A.3.2 Two-Parameter Distributions 497 A.3.3 One-Parameter Distributions 499 A.4 Distributions for Large Losses 499 A.4.1 Extreme Value Distributions 499 A.4.2 Generalized Pareto Distributions 500 A.5 Other Distributions 501 A.6 Distributions with Finite Support 502 B An Inventory of Discrete Distributions 505 B.1 Introduction 505 B.2 The (a, b, 0) Class 506 B.3 The (a, b, 1) Class 507 B.3.1 The Zero-Truncated Subclass 507 B.3.2 The Zero-Modified Subclass 509 B.4 The Compound Class 509 B.4.1 Some Compound Distributions 510 B.5 A Hierarchy of Discrete Distributions 511 C Frequency and Severity Relationships 513 D The Recursive Formula 515 E Discretization of the Severity Distribution 517 E.1 The Method of Rounding 517 E.2 Mean Preserving 518 E.3 Undiscretization of a Discretized Distribution 518 References 521 Index 529