R Programming for Actuarial Science

R Programming for Actuarial Science
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 YourWorking 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 DoWe 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
	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,