R Programming for Actuarial Science

Cover
Title Page
Copyright Page
Dedication
Table of Contents
About the Companion Website
Introduction
   1 Main Objectives of This Book
   2 Who Is This Book For?
   3 How to Use This Book
   4 Book Structure
   5 Chapter Style
   6 Examples and Exercises
   7 Verification of Code and Calculations – Best Practice
   8 Website: www.wiley.com/go/rprogramming.com
   9 R or Microsoft Excel?
   10 Caveats
   11 Acknowledgements
1 R : What You Need to Know to Get Started
   1.1 Introduction
   1.2 Getting Started: Installation of R and RStudio
      1.2.1 Installing R
      1.2.2 What Is RStudio?
      1.2.3 Inputting R Commands
   1.3 Assigning Values
   1.4 Help in R
   1.5 Data Objects in R
   1.6 Vectors
      1.6.1 Numeric Vectors
      1.6.2 Logical Vectors
      1.6.3 Character Vectors
      1.6.4 Factor Vectors
   1.7 Matrices
   1.8 Dataframes
   1.9 Lists
   1.10 Simple Plots and Histograms
   1.11 Packages
   1.12 Script Files
   1.13 Workspace, Saving Objects, and Miscellany
   1.14 Setting Your Working Directory
   1.15 Importing and Exporting Data
      1.15.1 Importing Data
      1.15.2 Exporting Data
   1.16 Common Errors Made in Coding
   1.17 Next Steps
   1.18 Recommended Reading
   1.19 Appendix: Coercion
2 Functions in R
   2.1 Introduction
      2.1.1 Objectives
      2.1.2 Core and Package Functions
      2.1.3 User-Defined Functions
   2.2 An Introduction to Applying Core and Package Functions
      2.2.1 Examples of Simple, Common Functions
   2.3 User-Defined Functions
      2.3.1 What does a “udf” consist of?
      2.3.2 Naming Conventions
      2.3.3 Examples and Exercises
   2.4 Using Loops in R – the “for” Function
   2.5 Integral Calculus in R
      2.5.1 The “Integrate” Function
      2.5.2 Numerical Integration
   2.6 Recommended Reading
3 Financial Mathematics (1): Interest Rates and Valuing Cashflows
   3.1 Introduction
   3.2 The Force of Interest
   3.3 Present Value of Future Cashflows
   3.4 Instantaneous Forward Rates and Spot Rates
   3.5 Non-Constant Force of Interest
      3.5.1 Discrete Cashflows
      3.5.2 Cashflows Which Are Continuous
   3.6 Effective and Nominal Rates of Interest
      3.6.1 Effective Rates of Interest
      3.6.2 Why Do We Use Effective Rates?
      3.6.3 Nominal Interest Rates
   3.7 Appendix: Force of Interest – An Analogy with Mortality Rates
   3.8 Recommended Reading
4 Financial Mathematics (2): Miscellaneous Examples
   4.1 Introduction
   4.2 Writing Annuity Functions
      4.2.1 Writing a function for an annuity certain
   4.3 The ‘presentValue’ Function
   4.4 Annuity Function
   4.5 Bonds – Pricing and Yield Calculations
   4.6 Bond Pricing: Non-Constant Interest Rates
   4.7 The Effect of Future Yield Changes on Bond Prices Throughout the Term of the Bond
   4.8 Loan Schedules
      4.8.1 Introduction
      4.8.2 Method 1
      4.8.3 Method 2
   4.9 Recommended Reading
5 Fundamental Statistics: A Selection of Key Topics . Dr A Kume
   5.1 Introduction
   5.2 Basic Distributions in Statistics
   5.3 Some Useful Functions for Descriptive Statistics
      5.3.1 Introduction
      5.3.2 Bivariate or Higher Order Data Structure
   5.4 Statistical Tests
      5.4.1 Exploring for Normality or Any Other Distribution in the Data
      5.4.2 Goodness-of-fit Testing for Fitted Distributions to Data
         5.4.2.1 Continuous distributions
         5.4.2.2 Discrete distributions
      5.4.3 T-tests
         5.4.3.1 One sample test for the mean
         5.4.3.2 Two sample tests for the mean
      5.4.4 F-test for Equal Variances
   5.5 Main Principles of Maximum Likelihood Estimation
      5.5.1 Introduction
      5.5.2 MLE of the Exponential Distribution
         5.5.2.1 Obtaining the MLE numerically using R
         5.5.2.2 Obtaining the MLE analytically
      5.5.3 Large Sample (Asymptotic) Properties of MLE
      5.5.4 Fitting Distributions to Data in R Using MLE
      5.5.5 Likelihood Ratio Test, LRT
   5.6 Regression: Basic Principles
      5.6.1 Simple Linear Regression
      5.6.2 Quantifying Uncertainty on
      5.6.3 Analysis of Variance in Regression
         5.6.3.1 R2 and adjusted R2 Coefficient of Determination
      5.6.4 Some Visual Diagnostics for the Proposed Simple Regression Model
   5.7 Multiple Regression
      5.7.1 Introduction
      5.7.2 Regression and MLE
         5.7.2.1 Multivariate Regression
      5.7.3 Tests
         5.7.3.1 Likelihood Ratio Test in Regression
         5.7.3.2 Akaike Information Criterion: AIC
         5.7.3.3 AIC and Regression model selection
         5.7.3.4 Bayesian Information Criterion: BIC
      5.7.4 Variable Selection, Finding the Most Appropriate Sub-Model
      5.7.5 Backward Elimination
      5.7.6 Forward Selection
      5.7.7 Using AIC/BIC Criteria
      5.7.8 LRT in Model Selection
      5.7.9 Automatic Search Using R-squared Criteria
      5.7.10 Concluding Remarks on Test Data
      5.7.11 Modelling Beyond Linearity
   5.8 Dummy/Indicator Variable Regression
      5.8.1 Introducing Categorical Variables
      5.8.2 Continuous and Indicator Variable Predictors – Including Load in the Model
   5.9 Recommended Reading
6 Multivariate Distributions, and Sums of Random Variables
   6.1 Multivariate Distributions – Examples in Finance
   6.2 Simulating Multivariate Normal Variables
   6.3 The Summation of a Number of Random Variables
   6.4 Conclusion
   6.5 Recommended Reading
7 Benefits of Diversification
   7.1 Introduction
   7.2 Background
   7.3 Key Mathematical Ideas
   7.4 Running Simulations
   7.5 Recommended Reading
8 Modern Portfolio Theory
   8.1 Introduction
   8.2 2-Asset Portfolio
   8.3 3-Asset Portfolio
   8.4 Introduction of a Risk-free Asset to the Portfolio
      8.4.1 Adding a Risk-free Asset
      8.4.2 Capital Market Line and the Sharpe Ratio
      8.4.3 Borrowing to Obtain Higher Returns
   8.5 Appendix: Lagrange Multiplier Method
   8.6 Recommended Reading
9 Duration – A Measure of Interest Rate Sensitivity
   9.1 Introduction
   9.2 Duration – Definitions and Interpretation
   9.3 Duration Function in R
   9.4 Practical Applications of Duration
   9.5 Recommended Reading
10 Asset-Liability Matching: An Introduction
   10.1 Introduction
   10.2 What Interest Rates Do Institutions Use To Measure Their Liabilities?
   10.3 Variance of the Solvency Position
   10.4 Characteristics of Various Asset Classes and Liabilities
   10.5 Our Scenarios
   10.6 Results
   10.7 Simulations
   10.8 Exercise and Discussion – an Insurer With Predominately Short-Term Liabilities
   10.9 Potential Exercise
   10.10 Conclusions
   10.11 Recommended Reading
11 Hedging: Protecting Against a Fall in Equity Markets
   11.1 Introduction
   11.2 Our Example
      11.2.1 Futures Contracts – A Brief Explanation
      11.2.2 Our Task
   11.3 Adopting a Better Hedge
   11.4 Allowance for Contract and Portfolio Sizes
   11.5 Negative Hedge Ratio
   11.6 Parameter and Model Risk
   11.7 A Final Reminder on Hedging
   11.8 Recommended Reading
12 Immunisation – Redington and Beyond
   12.1 Introduction
   12.2 Outline of Redington Theory and Alternatives
   12.3 Redington’s Theory of Immunisation
   12.4 Changes in the Shape of the Yield Curve
   12.5 A More Realistic Example
      12.5.1 Determining a Suitable Bond Allocation
      12.5.2 Change in Yield Curve Shape
      12.5.3 Liquidity Risk
   12.6 Conclusion
   12.7 Recommended Reading
13 Copulas
   13.1 Introduction
   13.2 Copula Theory – The Basics
   13.3 Commonly Used Copulas
      13.3.1 The Independent Copula
      13.3.2 The Gaussian Copula
      13.3.3 Archimedian Copulas
      13.3.4 Clayton Copula
      13.3.5 Gumbel Copula
   13.4 Copula Density Functions
   13.5 Mapping from Copula Space to Data Space
   13.6 Multi-dimensional Data and Copulas
   13.7 Further Insight into the Gaussian Copula: A Non-rigorous View
   13.8 The Real Power of Copulas
   13.9 General Method of Fitting Distributions and Simulations – A Copula Approach
      13.9.1 Fitting the Model
      13.9.2 Simulating Data Using the mvdc and rMvdc Functions
   13.10 How Non-Gaussian Copulas Can Improve Modelling
   13.11 Tail Correlations
   13.12 Exercise (Challenging)
   13.13 Appendix 1 – Copula Properties
   13.14 Appendix 2 – Rank Correlation and Kendall’s Tau, τ
   13.15 Recommended Reading
14 Copulas – A Modelling Exercise
   14.1 Introduction
   14.2 Modelling Future Claims
      14.2.1 Data
      14.2.2 Fitting Appropriate Marginal Distributions
      14.2.3 Fitting The Copula
      14.2.4 Assessing Risk From the Analysis of Simulated Values
      14.2.5 Comparison with the Gaussian Copula Model
      14.2.6 Comparison of the Models with the Data
   14.3 Another Example: Banking Regulator
   14.4 Conclusion
15 Bond Portfolio Valuation: A Simple Credit Risk Model
   15.1 Introduction
   15.2 Our Example Bond Portfolio
      15.2.1 Description
      15.2.2 The Transition Matrix
      15.2.3 Correlation Matrix
      15.2.4 Simulations and Results
      15.2.5 Incorporating Interest Rate Risk – A Simple Adjustment
      15.2.6 Portfolio Consisting of Highly Correlated Bonds
   15.3 Further Development of this Model
   15.4 Recommended Reading
16 The Markov 2-State Mortality Model
   16.1 Introduction
   16.2 Markov 2-State Model
   16.3 Simple Applications of the 2-State Model
   16.4 Estimating Mortality Rates from Data
   16.5 An Example: Calculating Mortality Rates for One Age Band
   16.6 Uncertainty in Our Estimates
   16.7 Next Steps?
   16.8 Appendices
      16.8.1 Informal Discussion of μ
      16.8.2 Intuitive meaning of fx(t)
   16.9 Recommended Reading
17 Approaches to Fitting Mortality Models: The Markov 2-state Model and an Introduction to Splines
   17.1 Introduction
   17.2 Graduation of Mortality Rates
   17.3 Fitting Our Data
      17.3.1 Objective
      17.3.2 Summarised Data
   17.4 Model Fitting with Least Squares
   17.5 Individual Member Data
   17.6 Comparing Life Tables with a Parametric Formula
   17.7 Splines: An Introduction
      17.7.1 Overview
      17.7.2 Data
      17.7.3 Fitting the Model: Spline regression
      17.7.4 Adjusted Dataset
   17.8 Summary
   17.9 Recommended Reading
18 Assessing the Suitability of Mortality Models: Statistical Tests
   18.1 Introduction
   18.2 Theory
   18.3 Our Mortality Data and Various Proposed Mortality Rates
   18.4 Testing the Standard Table Rates – Table 1,
      18.4.1 Data and initial plot
      18.4.2 x2 test
      18.4.3 Signs Test – for Overall Bias
      18.4.4 Serial Correlations Test; Testing for Bias Over Age Ranges
      18.4.5 Analysing the Distribution of Deviances
      18.4.6 logL, AIC Calculations
      18.4.7 Conclusions on Conclusions on
   18.5 Graduation of Mortality Rates by Adjusting a Standard Table
      18.5.1 Testing Table 2,
      18.5.2 Adjusting Table 2
   18.6 Testing Graduated Rates Obtained from a Parametric Formula,
   18.7 Comparing Our Candidate Rates
   18.8 Over-fitting
   18.9 Other Thoughts
   18.10 Appendix – Alternative Calculations of LogL’s
   18.11 Recommended Reading
19 The Lee-Carter Model
   19.1 Introduction
   19.2 Using the L-C Model to Create Data and Fit the Model
      19.2.1 Introducing the Lee-Carter Model
      19.2.2 Calculating the Parameter Values
      19.2.3 Interpretation of ax, bx, and kt
   19.3 Using L-C to Model Actual Mortality Data from HMD
   19.4 Using the lca Function in the Demography Package
   19.5 Constructing Your Own Demogdata Object
   19.6 Forecasting Mortality Rates
   19.7 Case Study: The Impact of the HIV Virus on Mortality Rates
   19.8 Recommended Reading
20 The Kaplan-Meier Estimator
   20.1 Introduction
   20.2 What Is Censoring?
      20.2.1 Non-Informative Censoring
   20.3 Defining the Relevant Event
   20.4 K-M Theory
   20.5 Introductory Example: Monitoring Delays in Making Claim Payments
   20.6 Lung Cancer Example
      20.6.1 Basic Results
      20.6.2 Comparison of Male and Female Rates
      20.6.3 Doctor Assessment Scores – ph.ecog
   20.7 Issues with the Kaplan-Meier Model
   20.8 Recommended reading
21 Cox Proportionate Hazards Regression Model
   21.1 Introduction
   21.2 Cox Model Equation
   21.3 Applications
      21.3.1 Smokers’ Mortality: Small Data Set
      21.3.2 Smokers’ Mortality: Larger Data Set
      21.3.3 Multiple covariates and interactions
   21.4 Comparison of Cox and Kaplan Meier Analyses of Lung Cancer Data
   21.5 Recommended Reading
22 Markov Multiple State Models: Applications to Life Contingencies
   22.1 Introduction
   22.2 The Markov Property
   22.3 Markov Chains and Jump Models
      22.3.1 Examples
      22.3.2 Differences between Markov Chain and Markov Jump Models
   22.4 Markov Chains (Discrete Time)
      22.4.1 Applying Markov Chains to Estimate Future Probabilities
      22.4.2 Markov Chain Model – NCD
      22.4.3 Coding Exercise for Markov Chains
   22.5 Markov Jump Models
      22.5.1 Example – Simple 3-State Model (All Transitions Possible)
      22.5.2 Example – H-S-D Model
   22.6 Non-Constant Rates
   22.7 Premium Calculations
   22.8 Transition Rate Estimation
   22.9 Multiple Decrement Models
      22.9.1 Introduction
      22.9.2 Using a Numerical Approach for the above Fixed Rate Problems
      22.9.3 An Exact Approach
      22.9.4 Age-Dependent Rates
   22.10 Recommended Reading
23 Contingencies I
   23.1 Introduction
   23.2 What is Meant by “Contingencies” in an Actuarial Context?
   23.3 The Life Table
   23.4 Expected Present Values of the Key Contingency Functions
   23.5 Writing Our Own Code – Some Introductory Exercises
   23.6 The Lifecontingencies Package
      23.6.1 The Lifetable and Actuarialtable Objects
      23.6.2 Application to Actual Mortality Tables: AM92 and AF92
      23.6.3 Annuities
      23.6.4 Annuities Paid more Frequently than Annually
      23.6.5 Increasing Annuities
      23.6.6 Reversionary Annuities
      23.6.7 Example: Annuity Company Valuation
      23.6.8 Life Assurance functions
      23.6.9 Assurance Policies with immediate Payment on Death: Ax
   23.7 Simulation of Future Lifetimes
   23.8 Recommended Reading
24 Contingencies II
   24.1 Introduction
   24.2 Mortality Tables: AM92
   24.3 Uncertainty in Present Values: Variance
   24.4 Simulations
      24.4.1 Single Policy
      24.4.2 Portfolios with 100 Policies – Portfolio Claim Distribution from Simulations
   24.5 Simulation of Annuities
   24.6 Premium Calculations
   24.7 Profits – Probability Distributions of Single Policies and Portfolios
   24.8 Progression of expected profits throughout the lifetime of a policy: no reserves held
   24.9 Policy Values
      24.9.1 Calculating Policy Values
      24.9.2 Recursive Formulae – Discrete and Continuous (Thiele)
      24.9.3 Recursive Equation with 3 States – HSD Model
   24.10 Profits from Policies where Reserves Are Held
      24.10.1 Calculating the Profit Vector
      24.10.2 Measures of Profit and Profit Testing
   24.11 Profit Uncertainty: Interest Rate and Mortality Risk
   24.12 Risk Capital and Risk-adjusted Return Measures
   24.13 Unit-linked Policies
      24.13.1 Introduction
      24.13.2 Example with Deterministic and Stochastic Projections
   24.14 Additional Exercises
   24.15 Appendix: Dependent and Independent Rates
   24.16 Recommended Reading
25 Actuarial Risk Theory – An Introduction: Collective and Individual Risk Models
   25.1 Introduction
   25.2 Collective Risk Model
   25.3 Poisson Compound Collective Risk Model
   25.4 Applications of the Model
      25.4.1 Setting Appropriate Reserves and Premium Pricing
      25.4.2 Increasing the Number of Independent Policies
      25.4.3 Adopting a Normal Distribution Approximation
      25.4.4 Return on Capital
      25.4.5 Skewness of the Compound Poisson Model
      25.4.6 Sum of Compound Poisson Distributions
   25.5 Compound Binomial Collective Risk Model
   25.6 Compound Negative Binomial Distribution
   25.7 Panjer’s Recursion Formula
   25.8 Closing Thoughts on Collective Risks Models
   25.9 Individual Risk Model
      25.9.1 Standard Individual Risk Model
      25.9.2 Alternative Model – ‘The Poisson Individual Risk Model’
   25.10 Issues with Heterogeneity
   25.11 Policies Which Are Not Independent
   25.12 Incorporating Parameter Uncertainty in the Models
   25.13 Claim Amount Distributions: Alternatives to the Gamma Distribution
   25.14 Conclusions
   25.15 Recommended Reading
26 Collective Risk Models: Exercise
   26.1 Introduction
   26.2 Analysis of Claims Data
   26.3 Running Simulations
   26.4 Tails of the Distribution
   26.5 Allowing for Parameter Uncertainty
   26.6 Conclusions
   26.7 Recommended Reading
27 Generalised Linear Models: Poisson Regression
   27.1 Introduction
   27.2 Examples/Exercises/Data
   27.3 Brief Recap on Multiple Linear Regression
   27.4 Generalised Linear Models (“GLMs”)
   27.5 Goodness of Fit of GLMs
   27.6 Poisson Regression
      27.6.1 Introduction
      27.6.2 Using Poisson Regression to Model Claim Numbers
   27.7 Data with Varying Exposure Periods
      27.7.1 Claim Rates and the Offset
      27.7.2 Application to Aggregated Data in Section 27.1
   27.8 Categorical and Continuous Variables
      27.8.1 Problem with Continuous Variables
      27.8.2 Categorical Variables
   27.9 Interaction between Variables
   27.10 Over-dispersion
   27.11 Miscellaneous Exercises
   27.12 Further Study / Next Steps
   27.13 Recommended Reading
28 Extreme Value Theory
   28.1 Introduction
   28.2 Why Use EVT?
   28.3 Generalised Pareto Distribution – “GPD”
   28.4 EVT Analysis of Historic Daily Equity Market Returns (S&P 500)
      28.4.1 Basic EVT Analysis
      28.4.2 Will a Normal Distribution (and Other Alternatives) Do Just as Well?
   28.5 Data for Further EVT Analysis
   28.6 Recommended Reading
29 Introduction to Machine Learning: k-Nearest Neighbours (kNN)
   29.1 Introduction
   29.2 Example 1 – Identifying a Fruit Type
      29.2.1 Data
      29.2.2 Overview of the Process
      29.2.3 How does the kNN Algorithm Work?
      29.2.4 Normalising Our Data
      29.2.5 Varying k
      29.2.6 Using Our Model
   29.3 Analysis of Our Model – the Confusion Matrix
   29.4 Example 2 – Cancer Diagnoses
   29.5 Conclusion
   29.6 Recommended Reading
30 Time Series Modelling in R – Dr A Kume
   30.1 Introduction
   30.2 Linear Regression Versus Autoregressive Model
   30.3 Three Components for Time Series Modelling
   30.4 Stationarity
   30.5 Main Tools in R for ARIMA Modelling
      30.5.1 PACF as a Derivation of ACF and Their General Behaviour for ARMA(p,q) Models
      30.5.2 How to Simulate and Obtain the Theoretical Values of ACF and PACF for ARMA Models
   30.6 Identifying a Set Possible Models to the Data Including the Order of Differencing
      30.6.1 Model Fitting to Time Series Data
      30.6.2 Parameter Estimation for Pure Auto-Regressive Models
      30.6.3 Diagnostic Plots
      30.6.4 Forecasting
   30.7 Dealing with Real Data far from Stationary
      30.7.1 Non Parametric Approaches
      30.7.2 Airline Data Modelling Using Multiplicative Seasonal Models
   30.8 Recommended Reading
31 Volatility Models – GARCH
   31.1 Introduction
   31.2 Why Use GARCH Models?
   31.3 Outline of the Chapter
   31.4 Key Theoretical Concepts with GARCH
   31.5 Simulation of Data Using a GARCH Model
   31.6 Fitting a GARCH Model to Data, and Analysis
      31.6.1 Fitting a GARCH Model
      31.6.2 Further Analysis of the Data; Comparison with the Normal Distribution
      31.6.3 Further Analysis of the Data; Volatility Clustering
   31.7 A Note on Correlation and Dependency
   31.8 GARCH Long-Term Variance
   31.9 Exercise: Shocks to Global Equity Markets – The Global Financial Crisis 2008, and COVID-19
   31.10 Extensions to the GARCH Model
   31.11 Appendix – A Mixture of Normal Distributions
   31.12 Recommended Reading
32 Modelling Future Stock Prices Using Geometric Brownian Motion: An Introduction
   32.1 Introduction
      32.1.1 Discrete Gaussian Random Walk
   32.2 Geometric Brownian Motion
   32.3 Applications of GBM, and Simulating Prices
   32.4 Recommended Reading
33 Financial Options: Pricing, Characteristics, and Strategies
   33.1 Introduction
   33.2 What is a Financial Option?
   33.3 What are Financial Options Used for?
   33.4 Black, Scholes and Merton Differential Equation
      33.4.1 Assumptions Underlying B-S-M Formulation
      33.4.2 Solution to B-S-M Equation for European Call Options
      33.4.3 Call Option Price Function
   33.5 Calculating the Option Price Using Simulations
   33.6 Factors Which Affect the Price of a Call Option
      33.6.1 Share Price
      33.6.2 Time to Expiry
      33.6.3 Combined Effect of Share Price and Time to Expiry
      33.6.4 Other Factors
   33.7 Greeks
   33.8 Volatility of Call Option Positions
   33.9 Put Options
   33.10 Delta Hedging
   33.11 Sketch of the B-S-M Derivation
   33.12 Further Tasks
   33.13 Appendix
   33.14 Recommended Reading
Index
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