--- title: "Loss Models: From Data to Decisions, Fifth Edition" tags: - authorship/other - exclude-from-word-count - topic/risk - type/media/book author: Gordon E. Willmot & Harry H. Panjer & 3 edition: Fifth publisher: John Wiley & Sons series: Wiley Series in Probability and Statistics subtitle: From Data to Decisions type: book year: 2019 --- # Loss Models: From Data to Decisions, Fifth Edition %% This note, with the exception of comments like this one (reserved for notes on transcription) consists only of content from the text. For commentary see the companion [[loss-models]]. %% ## Part I Introduction ### 1. Modeling #### 1.1 The Model-Based Approach ##### 1.1.1 The Modeling Process The model-based approach should be considered in the context of the objectives of any given problem. Many problems in actuarial science involve the building of a mathematical model that can be used to forecast or predict insurance costs in the future. ##### 1.1.2 The Modeling Advantage #### 1.2 The Organization of This Book ### 2. Random Variables #### 2.1 Introduction #### 2.2 Key Functions and Four Models ##### 2.2.1 Exercises ### 3. Basic Distributional Quantities #### 3.1 Moments ##### 3.1.1 Exercises #### 3.2 Percentiles ##### 3.2.1 Exercises #### 3.3 Generating Functions and Sums of Random Variables ##### 3.3.1 Exercises #### 3.4 Tails of Distributions ##### 3.4.1 Classification Based on Moments ##### 3.4.2 Comparison Based on Limiting Tail ##### 3.4.3 Classification Based on the Hazard Rate Function ##### 3.4.4 Classification Based on the Mean Excess Loss Function The mean excess ##### 3.4.5 Equilibrium Distributions and Tail Behavior ##### 3.4.6 Exercises #### 3.5 Measures of Risk ##### 3.5.1 Introduction ##### 3.5.2 Risk Measures and Coherence ##### 3.5.3 Value at Risk ##### 3.5.4 Tail Value of Risk ##### 3.5.5 Exercises ## Part II Actuarial Models ### 4. Characteristics Of Actuarial Models #### 4.1 Introduction #### 4.2 The Role of Parameters ##### 4.2.1 Parametric and Scale Distributions ##### 4.2.2 Parametric Distribution Families ##### 4.2.3 Finite Mixture Distributions ##### 4.2.4 Data-Dependent Distributions ##### 4.2.5 Exercises ### 5. Continuous Models #### 5.1 Introduction #### 5.2 Creating New Distributions ##### 5.2.1 Multiplication by a Constant ##### 5.2.2 Raising to a Power ##### 5.2.3 Exponentiation ##### 5.2.4 Mixing ##### 5.2.5 Frailty Models ##### 5.2.6 Splicing ##### 5.2.7 Exercises #### 5.3 Selected Distributions and Their Relationships ##### 5.3.1 Introduction ##### 5.3.2 Two Parametric Families ##### 5.3.3 Limiting Distributions ##### 5.3.4 Two Heavy-Tailed Distributions ##### 5.3.5 Exercises #### 5.4 The Linear Exponential Family ##### 5.4.1 Exercises ### 6. Discrete Distributions #### 6.1 Introduction ##### 6.1.1 Exercise #### 6.2 The Poisson Distribution #### 6.3 The Negative Binomial Distribution #### 6.4 The Binomial Distribution #### 6.5 The (a,b) Class ##### 6.5.1 Exercises #### 6.6 Truncation and Modification at zero ##### 6.6.1 Exercises ### 7. Advanced Discrete Distributions #### 7.1 Compound Frequency Distributions ##### 7.1.1 Exercises #### 7.2 Further Properties of the Compound Poisson Class ##### 7.2.1 Exercises #### 7.3 Mixed-Frequency Distributions ##### 7.3.1 The General Mixed-Frequency Distribution ##### 7.3.2 Mixed Poisson Distributions ##### 7.3.3 Exercises #### 7.4 The Effect of Exposure on Frequency #### 7.5 An Inventory of Discrete Distributions ##### 7.5.1 Exercises ### 8. Frequency And Severity With Coverage Modifications #### 8.1 Introduction #### 8.2 Deductibles ##### 8.2.1 Exercises #### 8.3 The Loss Elimination Ratio and the Effect of Inflation for Ordinary Deductibles ##### 8.3.1 Exercise #### 8.4 Policy Limits ##### 8.4.1 Exercises #### 8.5 Coinsurance, Deductibles, and Limits ##### 8.5.1 Exercises #### 8.6 The Impact of Deductibles on Claim Frequency ##### 8.6.1 Exercises ### 9. Aggregate Loss Models #### 9.1 Introduction ##### 9.1.1 Exercises #### 9.2 Model Choices ##### 9.2.1 Exercises #### 9.3 The Compound Model for Aggregate Claims ##### 9.3.1 Probabilities and Moments ##### 9.3.2 Stop-Loss Insurance ##### 9.3.3 The Tweedle Distribution ##### 9.3.4 Exercises #### 9.4 Analytic Results ##### 9.4.1 Exercises #### 9.5 Computing the Aggregate Claims Distribution #### 9.6 The Recursive Method ##### 9.6.1 Applications to Compound Frequency Models ##### 9.6.2 Underflow/Overflow Problems ##### 9.6.3 Numerical Stability ##### 9.6.4 Continuous Severity ##### 9.6.5 Constructing Arithmetic Distributions ##### 9.6.6 Exercises #### 9.7 The Impact of Individual Policy Modifications on Aggregate Payments ##### 9.7.1 Exercises #### 9.8 The individual Risk Model ##### 9.8.1 The Model ##### 9.8.2 Parametric Approximation ##### 9.8.3 Compound Poisson Approximation ##### 9.8.4 Exercises ## Part III Mathematical Statistics ### 10. Introduction To Mathematical Statistics #### 10.1 Introduction and Four Data Sets #### 10.2 Point Estimation ##### 10.2.1 Introduction ##### 10.2.2 Measures of Quality ##### 10.2.3 Exercises #### 10.3 Interval Estimation ##### 10.3.1 Exercises #### 10.4 The Construction of Parametric Estimators ##### 10.4.1 The Method of Moments and Percentile Matching ##### 10.4.2 Exercises #### 10.5 Tests of Hypotheses ##### 10.5.1 Exercise ### 11. Maximum Likelihood Estimation #### 11.1 Introduction #### 11.2 Individual Data ##### 11.2.1 Exercises #### 11.3 Grouped Data ##### 11.3.1 Exercises #### 11.4 Truncated or Censored Data ##### 11.4.1 Exercises #### 11.5 Variance and Interval Estimation for Maximum Likelihood Estimators ##### 11.5.1 Exercises #### 11.6 Functions of Asymptotically Normal Estimators ##### 11.6.1 Exercises #### 11.7 Nonnormal Confidence Intervals ##### 11.7.1 Exercise ### 12. Frequentist Estimation For Discrete Distributions #### 12.1 The Poisson Distribution #### 12.2 The Negative Binomial Distribution #### 12.3 The Binomial Distribution #### 12.4 The (a,b) Class #### 12.5 Compound Models #### 12.6 The Effect of Exposure on Maximum Likelihood Estimation #### 12.7 Exercises ### 13. Bayesian Estimation #### 13.1 Definitions and Bayes' Theorem #### 13.2 Inference and Prediction ##### 13.2.1 Exercises #### 13.3 Conjugate Prior Distributions and the Linear Exponential Family ##### 13.3.1 Exercises #### 13.4 Computational Issues ## Part IV Construction Of Models ### 14. Construction Of Empirical Models #### 14.1 The Empirical Distribution #### 14.2 Empirical Distributions for Grouped Data ##### 14.2.1 Exercises #### 14.3 Empirical Estimation with Right Censored Data ##### 14.3.1 Exercises #### 14.4 Empirical Estimation of Moments ##### 14.4.1 Exercises #### 14.5 Empirical Estimation with Left Truncated Data ##### 14.5.1 Exercises #### 14.6 Kernel Density Models ##### 14.6.1 Exercises #### 14.7 Approximations for large Data Sets ##### 14.7.1 Introduction ##### 14.7.2 Using Individual Data Points ##### 14.7.3 Interval-Based Methods ##### 14.7.4 Exercises #### 14.8 Maximum Likelihood Estimation of Decrement Probabilities ##### 14.8.1 Exercise #### 14.9 Estimation of Transition Intensities ### 15. Model Selection #### 15.1 Introduction #### 15.2 Representations of the Data and Model #### 15.3 Graphical Comparison of the Density and Distribution Functions ##### 15.3.1 Exercises #### 15.4 Hypothesis Tests ##### 15.4.1 The Kolmogorov--Smirnov Test ##### 15.4.2 The Anderson--Darling Test ##### 15.4.3 The Chi-Square Goodness-of-Fit Test ##### 15.4.4 The Likelihood Ratio Test ##### 15.4.5 Exercises #### 15.5 Selecting a Model ##### 15.5.1 Introduction ##### 15.2.2 Judgement-Based Approaches ##### 15.5.3 Score-Based Approaches ##### 15.5.4 Exercises ## Part V Credibility ### 16. Introduction To Limited Fluctuation Credibility #### 16.1 Introduction #### 16.2 Limited Fluctuation Credibility Theory #### 16.3 Full Credibility #### 16.4 Partial Credibility #### 16.5 Problems with the Approach #### 16.6 Notes and References #### 16.7 Exercises ### 17. Greatest Accuracy Credibility #### 17.1 Introduction #### 17.2 Conditional Distributions and Expectation #### 17.3 The Bayesian Methodology #### 17.4 The Credibility Premium #### 17.5 The Bühlmann Model #### 17.6 The Bühlmann--Straub Model #### 17.7 Exact Credibility #### 17.8 Notes and References #### 17.9 Exercises ### 18. Empirical Bayes Parameter Estimation #### 18.1 Introduction #### 18.2 Nonparametric Estimation #### 18.3 Semiparametric Estimation #### 18.4 Notes and References #### 18.5 Exercises ## Part VI Simulation ### 19. Simulation #### 19.1 Basics of Simulation ##### 19.1.1 The Simulation Approach ##### 19.1.2 Exercises #### 19.2 Simulation for Specific Distributions ##### 19.2.1 Discrete Mixtures ##### 19.2.2 Time or Age of Death from a Life Table ##### 19.2.3 Simulating from the (a,b) Class ##### 19.2.4 Normal and Lognormal Distributions ##### 19.2.5 Exercises #### 19.3 Determining the Sample Size ##### 19.3.1 Exercises #### 19.4 Examples of Simulation in Actuarial Modeling ##### 19.4.1 Aggregate Loss Calculations ##### 19.4.2 Examples of Lack of Independence ##### 19.4.3 Simulation Analysis of the Two Examples ##### 19.4.4 The Use of Simulation to Determine Risk Measures ##### 19.4.5 Statistical Analyses ##### 19.4.6 Exercises ## Appendix A An Inventory Of Continuous Distributions ### A.1 Introduction ### A.2 The Transformed Beta Family #### A.2.1 The Four-Parameter Distribution #### A.2.2 Three-Parameter Distributions #### A.2.3 Two-Parameter Distributions ### A.3 The Transformed Gamma Family #### A.3.1 Three-Parameter Distributions #### A.3.2 Two-Parameter Distributions #### A.3.3 One-Parameter Distributions ### A.4 Distributions For Large Losses #### A.4.1 Extreme Value Distributions #### A.4.2 Generalized Pareto Distributions ### A.4.5 Other Distributions ### A.4.6 Distributions With Finite Support ## Appendix B An Inventory Of Discrete Distributions ### B.1 Introduction ### B.2 The (a,b,0) Class ### B.3 The (a,b,1) Class #### B.3.1 The Zero-Truncated Subclass #### B.3.2 The Zero-Modified Subclass ### B.4 The Compound Class #### B.4.1 Some Compound Distributions ### B.5 A Hierarchy Of Discrete Distributions ## Appendix C Frequency And Severity Relationships ## Appendix D The Recursive Formula ## Appendix E Discretization Of The Severity Distribution ### E.1 The Method Of Rounding ### E.2 Mean Preserving ### E.3 Undiscretization Of A Discretized Distribution ## References ## Index