Loss Models

Chapter 1: Modeling

• • 1.1 The Model-Based Approach
• • 1.2 The Organization of This Book

Chapter 2: Random Variables

• • 2.1 Introduction
• • 2.2 Key Functions and Four Models

Chapter 3: Basic Distributional Quantities

• • 3.1 Moments
• • 3.2 Percentiles
• • 3.3 Generating Functions and Sums of Random Variables
• • 3.4 Tails of Distributions
• • 3.5 Measures of Risk

Chapter 4: Characteristics of Actuarial Models

• • 4.1 Introduction
• • 4.2 The Role of Parameters

Chapter 5: Continuous Models

• • 5.1 Introduction
• • 5.2 Creating New Distributions
• • 5.3 Selected Distributions and Their Relationships
• • 5.4 The Linear Exponential Family

Chapter 6: Discrete Distributions

• • 6.1 Introduction
• • 6.2 The Poisson Distribution
• • 6.3 The Negative Binomial Distribution
• • 6.4 The Binomial Distribution
• • 6.5 The (𝑎, 𝑏, 0) Class
• • 6.6 Truncation and Modification at Zero

Chapter 7: Advanced Discrete Distributions

• • 7.1 Compound Frequency Distributions
• • 7.2 Further Properties of the Compound Poisson Class
• • 7.3 Mixed-Frequency Distributions
• • 7.4 The Effect of Exposure on Frequency
• • 7.5 An Inventory of Discrete Distributions

Chapter 8: Frequency and Severity with Coverage Modifications

• • 8.1 Introduction
• • 8.2 Deductibles
• • 8.3 The Loss Elimination Ratio and the Effect of Inflation for Ordinary Deductibles
• • 8.4 Policy Limits
• • 8.5 Coinsurance, Deductibles, and Limits
• • 8.6 The Impact of Deductibles on Claim Frequency

Chapter 9: Aggregate Loss Models

• • 9.1 Introduction
• • 9.2 Model Choices
• • 9.3 The Compound Model for Aggregate Claims
• • 9.4 Analytic Results
• • 9.5 Computing the Aggregate Claims Distribution
• • 9.6 The Recursive Method
• • 9.7 The Impact of Individual Policy Modifications on Aggregate Payments
• • 9.8 The Individual Risk Model

Chapter 10: Introduction to Mathematical Statistics

• • 10.1 Introduction and Four Data Sets
• • 10.2 Point Estimation
• • 10.3 Interval Estimation
• • 10.4 The Construction of Parametric Estimators
• • 10.5 Tests of Hypotheses

Chapter 11: Maximum Likelihood Estimation

• • 11.1 Introduction
• • 11.2 Individual Data
• • 11.3 Grouped Data
• • 11.4 Truncated or Censored Data
• • 11.5 Variance and Interval Estimation for Maximum Likelihood Estimators
• • 11.6 Functions of Asymptotically Normal Estimators
• • 11.7 Nonnormal Confidence Intervals

Chapter 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, 1) Class
• • 12.5 Compound Models
• • 12.6 The Effect of Exposure on Maximum Likelihood Estimation
• • 12.7 Exercises

Chapter 13: Bayesian Estimation

• • 13.1 Definitions and Bayes’ Theorem
• • 13.2 Inference and Prediction
• • 13.3 Conjugate Prior Distributions and the Linear Exponential Family
• • 13.4 Computational Issues

Chapter 14: Construction of Empirical Models

• • 14.1 The Empirical Distribution
• • 14.2 Empirical Distributions for Grouped Data
• • 14.3 Empirical Estimation with Right Censored Data
• • 14.4 Empirical Estimation of Moments
• • 14.5 Empirical Estimation with Left Truncated Data
• • 14.6 Kernel Density Models
• • 14.7 Approximations for Large Data Sets
• • 14.8 Maximum Likelihood Estimation of Decrement Probabilities
• • 14.9 Estimation of Transition Intensities

Chapter 15: Model Selection

• • 15.1 Introduction
• • 15.2 Representations of the Data and Model
• • 15.3 Graphical Comparison of the Density and Distribution Functions
• • 15.4 Hypothesis Tests
• • 15.5 Selecting a Model

Chapter 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

Chapter 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

Chapter 18: Empirical Bayes Parameter Estimation

• • 18.1 Introduction
• • 18.2 Nonparametric Estimation
• • 18.3 Semiparametric Estimation
• • 18.4 Notes and References
• • 18.5 Exercises

Chapter 19: Simulation

• • 19.1 Basics of Simulation
• • 19.2 Simulation for Specific Distributions
• • 19.3 Determining the Sample Size
• • 19.4 Examples of Simulation in Actuarial Modeling

Chapter A: An Inventory of Continuous Distributions

• • A.1 Introduction
• • A.2 The Transformed Beta Family
• • A.3 The Transformed Gamma Family
• • A.4 Distributions for Large Losses
• • A.5 Other Distributions
• • A.6 Distributions with Finite Support

Chapter B: An Inventory of Discrete Distributions

• • B.1 Introduction
• • B.2 The (𝑎, 𝑏, 0) Class
• • B.3 The (𝑎, 𝑏, 1) Class
• • B.4 The Compound Class
• • B.5 A Hierarchy of Discrete Distributions

Chapter C: Frequency and Severity Relationships

Chapter D: The Recursive Formula

Chapter E: Discretization of the Severity Distribution

• • E.1 The Method of Rounding
• • E.2 Mean Preserving
• • E.3 Undiscretization of a Discretized Distribution