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Computational Actuarial Science with R

Name: Computational Actuarial Science with R
Author: Arthur Charpentier
ISBN: 9781498759823

by Arthur Charpentier

August 2014

Intermediate to advanced

656 pages

16h 11m

English

Chapman and Hall/CRC

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Preliminaries
Preface
Contributors
Chapter 1 Introduction
1.1 R for Actuarial Science?1.1.1 From Actuarial Science to Computational Actuarial Science1.1.2 The S Language and the R Environment1.1.3 Vectors and Matrices in Actuarial Computations1.1.4 R Packages1.1.5 S3 versus S4 Classes1.1.6 R Codes and Efficiency1.2 Importing and Creating Various Objects, and Datasets in R1.2.1 Simple Objects in R and Workspace1.2.2 More Complex Objects in R: From Vectors to Lists1.2.2.1 Vectors in R1.2.2.2 Matrices and Arrays1.2.2.3 Lists1.2.3 Reading csv or txt Files1.2.4 Importing Excel® Files and SAS® Tables1.2.5 Characters, Factors and Dates with R1.2.5.1 Strings and Characters1.2.5.2 Factors and Categorical Variables1.2.5.3 Dates in R1.2.6 Symbolic Expressions in R1.3 Basics of the R Language1.3.1 Core Functions1.3.2 From Control Flow to “Personal” Functions1.3.2.1 Control Flow: Looping, Repeating and Conditioning1.3.2.2 Writing Personal Functions1.3.3 Playing with Functions (in a Life Insurance Context)1.3.4 Dealing with Errors1.3.5 Efficient Functions1.3.6 Numerical Integration1.3.7 Graphics with R: A Short Introduction1.3.7.1 Basic Ready-Made Graphs1.3.7.2 A Simple Graph with Lines and Curves1.3.7.3 Graphs That Can Be Obtained from Standard Functions1.3.7.4 Adding Shaded Area to a Graph1.3.7.5 3D Graphs1.3.7.6 More Complex Graphs1.4 More Advanced R1.4.1 Memory Issues1.4.2 Parallel R1.4.3 Interfacing R and C/C++1.4.4 Integrating R in Excel®1.4.5 Going Further1.5 Ending an R Session1.6 ExercisesFigure 1.1Figure 1.2Figure 1.3Figure 1.4Figure 1.5Figure 1.6Figure 1.7Figure 1.8Table 1.1Table 1.2
Part I Methodology
Chapter 2 Standard Statistical Inference
2.1 Probability Distributions in Actuarial Science2.1.1 Continuous Distributions2.1.2 Discrete Distributions2.1.3 Mixed-Type Distributions2.1.4 S3 versus S4 Types for Distribution2.2 Parametric Inference2.2.1 Maximum Likelihood Estimation2.2.2 Moment Matching Estimation2.2.3 Quantile Matching Estimation2.2.4 Maximum Goodness-of-Fit Estimation2.3 Measures of Adequacy2.3.1 Histogram and Empirical Densities2.3.2 Distribution Function Plot2.3.3 QQ-Plot, PP-Plot2.3.4 Goodness-of-Fit Statistics and Tests2.3.5 Skewness—Kurtosis Graph2.4 Linear Regression: Introducing Covariates in Statistical Inference2.4.1 Using Covariates in the Statistical Framework2.4.2 Linear Regression Model2.4.3 Inference in a Linear Model2.5 Aggregate Loss Distribution2.5.1 Computation of the Aggregate Loss Distribution2.5.2 Poisson Process2.5.3 From Poisson Processes to Levy Processes2.5.4 Ruin Models2.6 Copulas and Multivariate Distributions2.6.1 Definition of Copulas2.6.2 Archimedean Copulas2.6.3 Elliptical Copulas2.6.4 Properties and Extreme Copulas2.6.5 Copula Fitting Methods2.6.6 Application and Copula Selection2.7 ExercisesFigure 2.1Figure 2.2Figure 2.3Figure 2.4Figure 2.5Figure 2.6Figure 2.7Figure 2.8Figure 2.9Figure 2.10Figure 2.11Figure 2.12Figure 2.13Figure 2.14Figure 2.15Figure 2.16Figure 2.17Figure 2.18Figure 2.19Figure 2.20Figure 2.21Table 2.1
Chapter 3 Bayesian Philosophy
3.1 Introduction3.1.1 A Formal Introduction3.1.2 Two Kinds of Probability3.1.3 Working with Subjective Probabilities in Real Life3.1.4 Bayesianism for Actuaries3.2 Bayesian Conjugates3.2.1 Historical Perspective3.2.2 Motivation on Small Samples3.2.3 Black Swans and Bayesian Methodology3.2.4 Bayesian Models in Portfolio Management and Finance3.2.5 Relation to Biihlmann Credibility3.2.6 Noninformative Priors3.3 Computational Considerations3.3.1 Curse of Dimensionality3.3.2 Monte Carlo Integration3.3.3 Markov Chain Monte Carlo3.3.4 MCMC Example in R3.3.5 JAGS and Stan3.3.6 Computational Conclusion and Specific Packages3.4 Bayesian Regression3.4.1 Linear Model from a Bayesian Perspective3.4.2 Extension to Generalized Linear Models3.4.3 Extension for Hierarchical Structures3.5 Interpretation of Bayesianism3.5.1 Bayesianism and Decision Theory3.5.2 Context of Discovery versus Context of Justification3.5.3 Practical Classical versus Bayesian Statistics Revisited3.6 Conclusion3.7 ExercisesFigure 3.1Figure 3.2Figure 3.3Figure 3.4Figure 3.5Figure 3.6Figure 3.7Table 3.1Table 3.2
Chapter 4 Statistical Learning
4.1 Introduction and Motivation4.1.1 The Dataset4.1.2 Description of the Data4.1.3 Scoring Tools4.1.4 Recoding the Variables4.1.5 Training and Testing Samples4.2 Logistic Regression4.2.1 Inference in the Logistic Model4.2.2 Logistic Regression on Categorical Variates4.2.3 Step-by-Step Variable Selection4.2.3.1 Forward Algorithm4.2.3.2 Backward Algorithm4.2.4 Leaps and Bounds4.2.5 Smoothing Continuous Covariates4.2.6 Nearest-Neighbor Method4.3 Penalized Logistic Regression: From Ridge to Lasso4.3.1 Ridge Model4.3.2 Lasso Regression4.4 Classification and Regression Trees4.4.1 Partitioning4.4.2 Criteria and Impurity4.5 From Classification Trees to Random Forests4.5.1 Bagging4.5.2 Boosting4.5.3 Random ForestsFigure 4.1Figure 4.2Figure 4.3Figure 4.4Figure 4.5Figure 4.6Figure 4.7Figure 4.8Figure 4.9Figure 4.10Figure 4.11Figure 4.12Figure 4.14Figure 4.13Figure 4.15Figure 4.16Figure 4.18Figure 4.17Figure 4.20Figure 4.19Figure 4.21Figure 4.22Table 4.1Table 4.2
Chapter 5 Spatial Analysis
5.1 Introduction5.1.1 Point Pattern Data5.1.2 Random Surface Data5.1.3 Spatial Interaction Data5.1.4 Areal Data5.1.5 Focus of This Chapter5.2 Spatial Analysis and GIS5.3 Spatial Objects in R5.3.1 SpatialPoints Subclass5.3.2 SpatialPointsDataFrame Subclass5.3.3 SpatialPolygons Subclass5.3.3.1 First Elementary Example5.3.3.2 Second Example5.3.4 SpatialPolygonsDataFrame Subclass5.4 Maps in R5.5 Reading Maps and Data in R5.6 Exploratory Spatial Data Analysis5.6.1 Mapping a Variable5.6.2 Selecting Colors5.6.3 Using the RgoogleMaps Package5.6.4 Generating KML Files5.6.4.1 Adding a Legend to a KML File5.7 Testing for Spatial Correlation5.7.1 Neighborhood Matrix5.7.2 Other Neighborhood Options5.7.3 Moran's I Index5.8 Spatial Car Accident Insurance Analysis5.9 Spatial Car Accident Insurance Shared Analysis5.10 ConclusionFigure 5.1Figure 5.2Figure 5.3Figure 5.4Figure 5.5Figure 5.6Figure 5.7Figure 5.8Figure 5.9Figure 5.10Figure 5.11Figure 5.12Figure 5.13Figure 5.15Figure 5.14Figure 5.16Figure 5.17Figure 5.18Figure 5.19Figure 5.20Figure 5.21Figure 5.22Figure 5.23Table 5.1
Chapter 6 Reinsurance and Extremal Events
6.1 Introduction6.2 Univariate Extremes6.2.1 Block Maxima6.2.2 Exceedances above a Threshold6.2.3 Point Process6.3 Inference6.3.1 Visualizing Tails6.3.2 Estimation6.3.2.1 Generalized Extreme Value Distribution6.3.2.2 Poisson-Generalized Pareto Model6.3.2.3 Point Process6.3.2.4 Other Tail Index Estimates6.3.3 Checking for the Asymptotic Regime Assumption6.3.3.1 Mean Excess Plot6.3.3.2 Parameter Stability6.3.4 Quantile Estimation6.4 Model Checking6.4.1 Quantile Quantile Plot6.4.2 Probability—Probability Plot6.4.3 Return Level Plot6.5 Reinsurance Pricing6.5.1 Modeling Occurence and Frequency6.5.2 Modeling Individual LossesFigure 6.1Figure 6.2Figure 6.3Figure 6.4Figure 6.5Figure 6.6Figure 6.7Figure 6.8Figure 6.9Figure 6.10Figure 6.11Figure 6.12Figure 6.13

Part II Life Insurance
Chapter 7 Life Contingencies
7.1 Introduction7.2 Financial Mathematics Review7.3 Working with Life Tables7.4 Pricing Life Insurance7.5 Reserving Life Insurances7.6 More Advanced Topics7.7 Health Insurance and Markov Chains7.7.1 Markov Chain with R7.7.2 Valuation of Cash Flows7.7.3 APV of Benefits and Reserves7.8 Exercises7.8.1 Financial Mathematics7.8.2 Demography7.8.3 Pricing Life Insurance7.8.4 Reserving Life Insurances7.8.5 More Advanced TopicsFigure 7.1Figure 7.2Figure 7.3
Chapter 8 Prospective Life Tables
8.1 Introduction8.2 Smoothing Mortality Data8.2.1 Weighted Constrained Penalized Regression Splines8.2.2 Two-Dimensional P-Splines8.3 Lee—Carter and Related Forecasting Methods8.3.1 Lee—Carter (LC) Method8.3.2 Lee—Miller (LM) Method8.3.3 Booth—Maindonald—Smith (BMS) Method8.3.4 Hyndman—Ullah (HU) Method8.3.5 Robust Hyndman-Ullah (HUrob) Method8.3.6 Weighted Hyndman-Ullah (HUw) Method8.4 Other Mortality Forecasting Methods8.5 Coherent Mortality Forecasting8.6 Life Table Forecasting8.7 Life Insurance Products8.8 ExercisesFigure 8.1Figure 8.2Figure 8.3Figure 8.4Figure 8.5Figure 8.6Figure 8.7Figure 8.8Figure 8.9Figure 8.10Figure 8.11Figure 8.12Figure 8.13Figure 8.14Figure 8.15Figure 8.16Table 8.1
Chapter 9 Prospective Mortality Tables and Portfolio Experience
9.1 Introduction and Motivation9.2 Notation, Data, and Assumption9.3 The Methods9.3.1 Method 1: Approach Involving One Parameter with the SMR9.3.2 Method 2: Approach Involving Two Parameters with a Semiparametric Relational Model9.3.3 Method 3: Poisson GLM Including Interactions with Age and Calendar Year9.3.4 Method 4: Nonparametric Smoothing and Application of the Improvement Rates9.3.5 Completion of the Tables: The Approach of Denuit and Goderniaux9.4 Validation9.4.1 First Level: Proximity between the Observations and the Model9.4.2 Second Level: Regularity of the Fit9.4.3 Third Level: Consistency and Plausibility of the Mortality Trends9.5 Operational Framework9.5.1 The Package ELT9.5.2 Computation of the Observed Statistics and Importation of the Reference9.5.3 Execution of the Methods9.5.4 Process of Validation9.5.5 Completion of the TablesFigure 9.1Figure 9.2Figure 9.3Figure 9.4Figure 9.5Figure 9.6Figure 9.7Figure 9.8Figure 9.9Figure 9.10Table 9.1Table 9.2Table 9.3Table 9.4
Chapter 10 Survival Analysis
10.1 Introduction10.2 Working with Incomplete Data10.2.1 Data Importation and Some Statistics10.2.2 Building the Appropriate Database10.2.3 Some Descriptive Statistics10.3 Survival Distribution Estimation10.3.1 Hoem Estimator of the Conditional Rates10.3.2 Kaplan—Meier Estimator of the Survival Function10.4 Regularization Techniques10.4.1 Parametric Adjustment10.4.2 Semiparametric Adjustment: Brass Relational Model10.4.3 Nonparametric Techniques: Whittaker—Henderson Smoother10.4.3.1 Application10.5 Modeling Heterogeneity10.5.1 Semiparametric Framework: Cox Model10.5.2 Additive Models10.6 Validation of a Survival ModelFigure 10.1Figure 10.2Figure 10.3Figure 10.4Figure 10.5Figure 10.6Figure 10.7Figure 10.8
Part III Finance
Chapter 11 Stock Prices and Time Series
11.1 Introduction11.2 Financial Time Series11.2.1 Introduction11.2.2 Data Used in This Chapter11.2.3 Stylized Facts11.3 Heteroskedastic Models11.3.1 Introduction11.3.2 Standard GARCH(1,1) Model11.3.3 Forecasting Heteroskedastic Model11.3.4 More Efficient Implementation11.4 Application: Estimation of the VaR Based on the POT and GARCH Model11.5 ConclusionFigure 11.1Figure 11.2Figure 11.3Table 11.1
Chapter 12 Yield Curves and Interest Rates Models
12.1 A Brief Overview of the Yield Curve and Scenario Simulation12.2 Yield Curves12.2.1 Description of the Datasets12.2.2 Principal Component Analysis12.3 Nelson—Siegel Model12.3.1 Estimating the Nelson-Siegel Model with R12.4 Svensson Model12.4.1 Estimating the Svensson Model with RFigure 12.2Figure 12.1Figure 12.3Figure 12.4Figure 12.5Figure 12.6Figure 12.7Figure 12.8
Chapter 13 Portfolio Allocation
13.1 Introduction13.2 Optimization Problems in R13.2.1 Introduction13.2.2 Linear Programming13.2.3 Quadratic Programming13.2.4 Nonlinear Programming13.3 Data Sources13.4 Portfolio Returns and Cumulative Performance13.5 Portfolio Optimization in R13.5.1 Introduction13.5.2 Mean—Variance Portfolio13.5.3 Robust Mean-Variance Portfolio13.5.4 Minimum Variance Portfolio13.5.5 Conditional Value-at-Risk Portfolio13.5.6 Minimum Drawdown Portfolio13.6 Display Results13.6.1 Efficient Frontier13.6.2 Weighted Return Plots13.7 ConclusionFigure 13.1Figure 13.2Figure 13.3Figure 13.4Figure 13.5Figure 13.6Figure 13.7Figure 13.8Table 13.1
Part IV Non-Life Insurance
Chapter 14 General Insurance Pricing
14.1 Introduction and Motivation14.1.1 Collective Model in General Insurance14.1.2 Pure Premium in a Heterogenous Context14.1.3 Dataset14.1.4 Structure of the Chapter and References14.2 Claims Frequency and Log-Poisson Regression14.2.1 Annualized Claims Frequency14.2.2 Poisson Regression14.2.3 Ratemaking with One Categorical Variable14.2.4 Contingency Tables and Minimal Bias Techniques14.2.5 Ratemaking with Continuous Variables14.2.6 A Poisson Regression to Model Yearly Claim Frequency14.3 From Poisson to Quasi-Poisson14.3.1 NB1 Variance Form: Negative Binomial Type I14.3.2 NB2 Variance Form: Negative Binomial Type II14.3.3 Unstructured Variance Form14.3.4 Nonparametric Variance Form14.4 More Advanced Models for Counts14.4.1 Negative Binomial Regression14.4.2 Zero-Inflated Models14.4.3 Hurdle Models14.5 Individual Claims, Gamma, Log-Normal, and Other Regressions14.5.1 Gamma Regression14.5.2 The Log-Normal Model14.5.3 Gamma versus Log-Normal Models14.5.4 Inverse Gaussian Model14.6 Large Claims and Ratemaking14.6.1 Model with Two Kinds of Claims14.6.2 More General Model14.7 Modeling Compound Sum with Tweedie Regression14.8 ExercisesFigure 14.1Figure 14.2Figure 14.3Figure 14.4Figure 14.5Figure 14.6Figure 14.7Figure 14.8Figure 14.9
Chapter 15 Longitudinal Data and Experience Rating
15.1 Motivation15.1.1 A Priori Rating for Cross-Sectional Data15.1.2 Experience Rating for Panel Data15.1.3 From Panel to Multilevel Data15.1.4 Structure of the Chapter15.2 Linear Models for Longitudinal Data15.2.1 Data15.2.2 Fixed Effects Models15.2.3 Models with Serial Correlation15.2.4 Models with Random Effects15.2.5 Prediction15.3 Generalized Linear Models for Longitudinal Data15.3.1 Specifying Generalized Linear Models with Random Effects15.3.2 Case Study: Experience Rating with Bonus—Malus Scales in R15.3.2.1 Bonus—Malus Scales15.3.2.2 Transition Rules, Transition Probabilities and Stationary Distribution15.3.2.3 RelativitiesFigure 15.1Figure 15.2Table 15.1Table 15.2Table 15.3Table 15.4Table 15.5
Chapter 16 Claims Reserving and IBNR
16.1 Introduction16.1.1 Motivation16.1.2 Outline and Scope16.2 Development Triangles16.3 Deterministic Reserving Methods16.3.1 Chain-Ladder Algorithm16.3.2 Tail Factors16.4 Stochastic Reserving Models16.4.1 Chain-Ladder in the Context of Linear Regression16.4.2 Mack Model16.4.3 Poisson Regression Model for Incremental Claims16.4.4 Bootstrap Chain-Ladder16.4.5 Reserving Based on Log-Incremental Payments16.5 Quantifying Reserve Risk16.5.1 Ultimo Reserve Risk16.5.2 One-Year Reserve Risk16.6 Discussion16.7 ExercisesFigure 16.1Figure 16.2Figure 16.3Figure 16.4Figure 16.5Figure 16.6Figure 16.7Figure 16.8Figure 16.9Figure 16.10Figure 16.11Figure 16.12Figure 16.13Figure 16.14Figure 16.15
Chapter 17 Bibliography

Content preview from Computational Actuarial Science with R

Chapter 2

Standard Statistical Inference

Christophe Dutang

Université de Strasbourg and Université du Maine, Le Mans, France Strasbourg, France

2.1 Probability Distributions in Actuarial Science

Let X be our quantity of interest. Actuarial models rely on particular assumptions on the probability distribution of X. When X represents the claim amount or the life length of an individual, one expects X to have a distribution $ℝ_{+}$ , whereas when X represents the claim number, we deal with distribution on ℕ. But, characterizing the support of the random variable X is a necessary but not a sufficient step to characterize our quantity of interest.

In the discrete case, probability distributions are generally characterized by the mass probability function ...

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Computational Actuarial Science with R

by Arthur Charpentier

Standard Statistical Inference

2.1 Probability Distributions in Actuarial Science

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