Computational Actuarial Science with R 1st Edition by Arthur Charpentier 1466592591 9781466592599

Chapter 1 Introduction

1.1 R for Actuarial Science?

1.1.1 From Actuarial Science to Computational Actuarial Science

1.1.2 The S Language and the R Environment

1.1.3 Vectors and Matrices in Actuarial Computations

1.1.4 R Packages

1.1.5 S3 versus S4 Classes

1.1.6 R Codes and Efficiency

1.2 Importing and Creating Various Objects, and Datasets in R

1.2.1 Simple Objects in R and Workspace

1.2.2 More Complex Objects in R: From Vectors to Lists

1.2.2.1 Vectors in R

1.2.2.2 Matrices and Arrays

1.2.2.3 Lists

1.2.3 Reading csv or txt Files

1.2.4 Importing Excel® Files and SAS® Tables

1.2.5 Characters, Factors and Dates with R

1.2.5.1 Strings and Characters

1.2.5.2 Factors and Categorical Variables

1.2.5.3 Dates in R

1.2.6 Symbolic Expressions in R

1.3 Basics of the R Language

1.3.1 Core Functions

1.3.2 From Control Flow to “Personal” Functions

1.3.2.1 Control Flow: Looping, Repeating and Conditioning

1.3.2.2 Writing Personal Functions

1.3.3 Playing with Functions (in a Life Insurance Context)

1.3.4 Dealing with Errors

1.3.5 Efficient Functions

1.3.6 Numerical Integration

1.3.7 Graphics with R: A Short Introduction

1.3.7.1 Basic Ready-Made Graphs

1.3.7.2 A Simple Graph with Lines and Curves

1.3.7.3 Graphs That Can Be Obtained from Standard Functions

1.3.7.4 Adding Shaded Area to a Graph

1.3.7.5 3D Graphs

1.3.7.6 More Complex Graphs

1.4 More Advanced R

1.4.1 Memory Issues

1.4.2 Parallel R

1.4.3 Interfacing R and C/C++

1.4.4 Integrating R in Excel®

1.4.5 Going Further

1.5 Ending an R Session

1.6 Exercises

Part I Methodology

Chapter 2 Standard Statistical Inference

2.1 Probability Distributions in Actuarial Science

2.1.1 Continuous Distributions

2.1.2 Discrete Distributions

2.1.3 Mixed-Type Distributions

2.1.4 S3 versus S4 Types for Distribution

2.2 Parametric Inference

2.2.1 Maximum Likelihood Estimation

2.2.2 Moment Matching Estimation

2.2.3 Quantile Matching Estimation

2.2.4 Maximum Goodness-of-Fit Estimation

2.3 Measures of Adequacy

2.3.1 Histogram and Empirical Densities

2.3.2 Distribution Function Plot

2.3.3 QQ-Plot, PP-Plot

2.3.4 Goodness-of-Fit Statistics and Tests

2.3.5 Skewness—Kurtosis Graph

2.4 Linear Regression: Introducing Covariates in Statistical Infer­ence

2.4.1 Using Covariates in the Statistical Framework

2.4.2 Linear Regression Model

2.4.3 Inference in a Linear Model

2.5 Aggregate Loss Distribution

2.5.1 Computation of the Aggregate Loss Distribution

2.5.2 Poisson Process

2.5.3 From Poisson Processes to Levy Processes

2.5.4 Ruin Models

2.6 Copulas and Multivariate Distributions

2.6.1 Definition of Copulas

2.6.2 Archimedean Copulas

2.6.3 Elliptical Copulas

2.6.4 Properties and Extreme Copulas

2.6.5 Copula Fitting Methods

2.6.6 Application and Copula Selection

2.7 Exercises

Chapter 3 Bayesian Philosophy

3.1 Introduction

3.1.1 A Formal Introduction

3.1.2 Two Kinds of Probability

3.1.3 Working with Subjective Probabilities in Real Life

3.1.4 Bayesianism for Actuaries

3.2 Bayesian Conjugates

3.2.1 Historical Perspective

3.2.2 Motivation on Small Samples

3.2.3 Black Swans and Bayesian Methodology

3.2.4 Bayesian Models in Portfolio Management and Finance

3.2.5 Relation to Biihlmann Credibility

3.2.6 Noninformative Priors

3.3 Computational Considerations

3.3.1 Curse of Dimensionality

3.3.2 Monte Carlo Integration

3.3.3 Markov Chain Monte Carlo

3.3.4 MCMC Example in R

3.3.5 JAGS and Stan

3.3.6 Computational Conclusion and Specific Packages

3.4 Bayesian Regression

3.4.1 Linear Model from a Bayesian Perspective

3.4.2 Extension to Generalized Linear Models

3.4.3 Extension for Hierarchical Structures

3.5 Interpretation of Bayesianism

3.5.1 Bayesianism and Decision Theory

3.5.2 Context of Discovery versus Context of Justification

3.5.3 Practical Classical versus Bayesian Statistics Revisited

3.6 Conclusion

3.7 Exercises

Chapter 4 Statistical Learning

4.1 Introduction and Motivation

4.1.1 The Dataset

4.1.2 Description of the Data

4.1.3 Scoring Tools

4.1.4 Recoding the Variables

4.1.5 Training and Testing Samples

4.2 Logistic Regression

4.2.1 Inference in the Logistic Model

4.2.2 Logistic Regression on Categorical Variates

4.2.3 Step-by-Step Variable Selection

4.2.3.1 Forward Algorithm

4.2.3.2 Backward Algorithm

4.2.4 Leaps and Bounds

4.2.5 Smoothing Continuous Covariates

4.2.6 Nearest-Neighbor Method

4.3 Penalized Logistic Regression: From Ridge to Lasso

4.3.1 Ridge Model

4.3.2 Lasso Regression

4.4 Classification and Regression Trees

4.4.1 Partitioning

4.4.2 Criteria and Impurity

4.5 From Classification Trees to Random Forests

4.5.1 Bagging

4.5.2 Boosting

4.5.3 Random Forests

Chapter 5 Spatial Analysis

5.1 Introduction

5.1.1 Point Pattern Data

5.1.2 Random Surface Data

5.1.3 Spatial Interaction Data

5.1.4 Areal Data

5.1.5 Focus of This Chapter

5.2 Spatial Analysis and GIS

5.3 Spatial Objects in R

5.3.1 SpatialPoints Subclass

5.3.2 SpatialPointsDataFrame Subclass

5.3.3 SpatialPolygons Subclass

5.3.3.1 First Elementary Example

5.3.3.2 Second Example

5.3.4 SpatialPolygonsDataFrame Subclass

5.4 Maps in R

5.5 Reading Maps and Data in R

5.6 Exploratory Spatial Data Analysis

5.6.1 Mapping a Variable

5.6.2 Selecting Colors

5.6.3 Using the RgoogleMaps Package

5.6.4 Generating KML Files

5.6.4.1 Adding a Legend to a KML File

5.7 Testing for Spatial Correlation

5.7.1 Neighborhood Matrix

5.7.2 Other Neighborhood Options

5.7.3 Moran’s I Index

5.8 Spatial Car Accident Insurance Analysis

5.9 Spatial Car Accident Insurance Shared Analysis

5.10 Conclusion

Chapter 6 Reinsurance and Extremal Events

6.1 Introduction

6.2 Univariate Extremes

6.2.1 Block Maxima

6.2.2 Exceedances above a Threshold

6.2.3 Point Process

6.3 Inference

6.3.1 Visualizing Tails

6.3.2 Estimation

6.3.2.1 Generalized Extreme Value Distribution

6.3.2.2 Poisson-Generalized Pareto Model

6.3.2.3 Point Process

6.3.2.4 Other Tail Index Estimates

6.3.3 Checking for the Asymptotic Regime Assumption

6.3.3.1 Mean Excess Plot

6.3.3.2 Parameter Stability

6.3.4 Quantile Estimation

6.4 Model Checking

6.4.1 Quantile Quantile Plot

6.4.2 Probability—Probability Plot

6.4.3 Return Level Plot

6.5 Reinsurance Pricing

6.5.1 Modeling Occurence and Frequency

6.5.2 Modeling Individual Losses

Part II Life Insurance

Chapter 7 Life Contingencies

7.1 Introduction

7.2 Financial Mathematics Review

7.3 Working with Life Tables

7.4 Pricing Life Insurance

7.5 Reserving Life Insurances

7.6 More Advanced Topics

7.7 Health Insurance and Markov Chains

7.7.1 Markov Chain with R

7.7.2 Valuation of Cash Flows

7.7.3 APV of Benefits and Reserves

7.8 Exercises

7.8.1 Financial Mathematics

7.8.2 Demography

7.8.3 Pricing Life Insurance

7.8.4 Reserving Life Insurances

7.8.5 More Advanced Topics

Chapter 8 Prospective Life Tables

8.1 Introduction

8.2 Smoothing Mortality Data

8.2.1 Weighted Constrained Penalized Regression Splines

8.2.2 Two-Dimensional P-Splines

8.3 Lee—Carter and Related Forecasting Methods

8.3.1 Lee—Carter (LC) Method

8.3.2 Lee—Miller (LM) Method

8.3.3 Booth—Maindonald—Smith (BMS) Method

8.3.4 Hyndman—Ullah (HU) Method

8.3.5 Robust Hyndman-Ullah (HUrob) Method

8.3.6 Weighted Hyndman-Ullah (HUw) Method

8.4 Other Mortality Forecasting Methods

8.5 Coherent Mortality Forecasting

8.6 Life Table Forecasting

8.7 Life Insurance Products

8.8 Exercises

Chapter 9 Prospective Mortality Tables and Portfolio Experience

9.1 Introduction and Motivation

9.2 Notation, Data, and Assumption

9.3 The Methods

9.3.1 Method 1: Approach Involving One Parameter with the SMR

9.3.2 Method 2: Approach Involving Two Parameters with a Semiparametric Relational Model

9.3.3 Method 3: Poisson GLM Including Interactions with Age and Calendar Year

9.3.4 Method 4: Nonparametric Smoothing and Application of the Im­provement Rates

9.3.5 Completion of the Tables: The Approach of Denuit and Goderniaux

9.4 Validation

9.4.1 First Level: Proximity between the Observations and the Model

9.4.2 Second Level: Regularity of the Fit

9.4.3 Third Level: Consistency and Plausibility of the Mortality Trends

9.5 Operational Framework

9.5.1 The Package ELT

9.5.2 Computation of the Observed Statistics and Importation of the Reference

9.5.3 Execution of the Methods

9.5.4 Process of Validation

9.5.5 Completion of the Tables

Chapter 10 Survival Analysis

10.1 Introduction

10.2 Working with Incomplete Data

10.2.1 Data Importation and Some Statistics

10.2.2 Building the Appropriate Database

10.2.3 Some Descriptive Statistics

10.3 Survival Distribution Estimation

10.3.1 Hoem Estimator of the Conditional Rates

10.3.2 Kaplan—Meier Estimator of the Survival Function

10.4 Regularization Techniques

10.4.1 Parametric Adjustment

10.4.2 Semiparametric Adjustment: Brass Relational Model

10.4.3 Nonparametric Techniques: Whittaker—Henderson Smoother

10.4.3.1 Application

10.5 Modeling Heterogeneity

10.5.1 Semiparametric Framework: Cox Model

10.5.2 Additive Models

10.6 Validation of a Survival Model

Part III Finance

Chapter 11 Stock Prices and Time Series

11.1 Introduction

11.2 Financial Time Series

11.2.1 Introduction

11.2.2 Data Used in This Chapter

11.2.3 Stylized Facts

11.3 Heteroskedastic Models

11.3.1 Introduction

11.3.2 Standard GARCH(1,1) Model

11.3.3 Forecasting Heteroskedastic Model

11.3.4 More Efficient Implementation

11.4 Application: Estimation of the VaR Based on the POT and GARCH Model

11.5 Conclusion

Chapter 12 Yield Curves and Interest Rates Models

12.1 A Brief Overview of the Yield Curve and Scenario Simulation

12.2 Yield Curves

12.2.1 Description of the Datasets

12.2.2 Principal Component Analysis

12.3 Nelson—Siegel Model

12.3.1 Estimating the Nelson-Siegel Model with R

12.4 Svensson Model

12.4.1 Estimating the Svensson Model with R

Chapter 13 Portfolio Allocation

13.1 Introduction

13.2 Optimization Problems in R

13.2.1 Introduction

13.2.2 Linear Programming

13.2.3 Quadratic Programming

13.2.4 Nonlinear Programming

13.3 Data Sources

13.4 Portfolio Returns and Cumulative Performance

13.5 Portfolio Optimization in R

13.5.1 Introduction

13.5.2 Mean—Variance Portfolio

13.5.3 Robust Mean-Variance Portfolio

13.5.4 Minimum Variance Portfolio

13.5.5 Conditional Value-at-Risk Portfolio

13.5.6 Minimum Drawdown Portfolio

13.6 Display Results

13.6.1 Efficient Frontier

13.6.2 Weighted Return Plots

13.7 Conclusion

Part IV Non-Life Insurance

Chapter 14 General Insurance Pricing

14.1 Introduction and Motivation

14.1.1 Collective Model in General Insurance

14.1.2 Pure Premium in a Heterogenous Context

14.1.3 Dataset

14.1.4 Structure of the Chapter and References

14.2 Claims Frequency and Log-Poisson Regression

14.2.1 Annualized Claims Frequency

14.2.2 Poisson Regression

14.2.3 Ratemaking with One Categorical Variable

14.2.4 Contingency Tables and Minimal Bias Techniques

14.2.5 Ratemaking with Continuous Variables

14.2.6 A Poisson Regression to Model Yearly Claim Frequency

14.3 From Poisson to Quasi-Poisson

14.3.1 NB1 Variance Form: Negative Binomial Type I

14.3.2 NB2 Variance Form: Negative Binomial Type II

14.3.3 Unstructured Variance Form

14.3.4 Nonparametric Variance Form

14.4 More Advanced Models for Counts

14.4.1 Negative Binomial Regression

14.4.2 Zero-Inflated Models

14.4.3 Hurdle Models

14.5 Individual Claims, Gamma, Log-Normal, and Other Regres­sions

14.5.1 Gamma Regression

14.5.2 The Log-Normal Model

14.5.3 Gamma versus Log-Normal Models

14.5.4 Inverse Gaussian Model

14.6 Large Claims and Ratemaking

14.6.1 Model with Two Kinds of Claims

14.6.2 More General Model

14.7 Modeling Compound Sum with Tweedie Regression

14.8 Exercises

Chapter 15 Longitudinal Data and Experience Rating

15.1 Motivation

15.1.1 A Priori Rating for Cross-Sectional Data

15.1.2 Experience Rating for Panel Data

15.1.3 From Panel to Multilevel Data

15.1.4 Structure of the Chapter

15.2 Linear Models for Longitudinal Data

15.2.1 Data

15.2.2 Fixed Effects Models

15.2.3 Models with Serial Correlation

15.2.4 Models with Random Effects

15.2.5 Prediction

15.3 Generalized Linear Models for Longitudinal Data

15.3.1 Specifying Generalized Linear Models with Random Effects

15.3.2 Case Study: Experience Rating with Bonus—Malus Scales in R

15.3.2.1 Bonus—Malus Scales

15.3.2.2 Transition Rules, Transition Probabilities and Stationary Distribution

15.3.2.3 Relativities

Chapter 16 Claims Reserving and IBNR

16.1 Introduction

16.1.1 Motivation

16.1.2 Outline and Scope

16.2 Development Triangles

16.3 Deterministic Reserving Methods

16.3.1 Chain-Ladder Algorithm

16.3.2 Tail Factors

16.4 Stochastic Reserving Models

16.4.1 Chain-Ladder in the Context of Linear Regression

16.4.2 Mack Model

16.4.3 Poisson Regression Model for Incremental Claims

16.4.4 Bootstrap Chain-Ladder

16.4.5 Reserving Based on Log-Incremental Payments

16.5 Quantifying Reserve Risk

16.5.1 Ultimo Reserve Risk

16.5.2 One-Year Reserve Risk

16.6 Discussion

16.7 Exercises

Chapter 17 Bibliography