Title Details

1 Introduction 1

1.1 Basic Challenges in Risk Management 1

1.2 Value at Risk 3

1.3 Further Challenges in Risk Management 6

2 Applied Linear Algebra for Risk Managers 11

2.1 Vectors and Matrices 11

2.2 Matrix Algebra in Practice 17

2.3 Eigenvectors and Eigenvalues 21

2.4 Positive Definite Matrices 24

3 Probability Theory for Risk Managers 27

3.1 Univariate Theory 27

3.1.1 Random variables 27

3.1.2 Expectation 31

3.1.3 Variance 32

3.2 Multivariate Theory 33

3.2.1 The joint distribution function 33

3.2.2 The joint and marginal density functions 34

3.2.3 The notion of independence 34

3.2.4 The notion of conditional dependence 35

3.2.5 Covariance and correlation 35

3.2.6 The mean vector and covariance matrix 37

3.2.7 Linear combinations of random variables 38

3.3 The Normal Distribution 39

4 Optimization Tools 43

4.1 Background Calculus 43

4.1.1 Single-variable functions 43

4.1.2 Multivariable functions 44

4.2 Optimizing Functions 47

4.2.1 Unconstrained quadratic functions 48

4.2.2 Constrained quadratic functions 50

4.3 Over-determined Linear Systems 52

4.4 Linear Regression 54

5 Portfolio Theory I 63

5.1 Measuring Returns 63

5.1.1 A comparison of the standard and log returns 64

5.2 Setting Up the Optimal Portfolio Problem 67

5.3 Solving the Optimal Portfolio Problem 70

6 Portfolio Theory II 77

6.1 The Two-Fund Investment Service 77

6.2 A Mathematical Investigation of the Optimal Frontier 78

6.2.1 The minimum variance portfolio 78

6.2.2 Covariance of frontier portfolios 78

6.2.3 Correlation with the minimum variance portfolio 79

6.2.4 The zero-covariance portfolio 79

6.3 A Geometrical Investigation of the Optimal Frontier 80

6.3.1 Equation of a tangent to an efficient portfolio 80

6.3.2 Locating the zero-covariance portfolio 82

6.4 A Further Investigation of Covariance 83

6.5 The Optimal Portfolio Problem Revisited 86

7 The Capital Asset Pricing Model (CAPM) 91

7.1 Connecting the Portfolio Frontiers 91

7.2 The Tangent Portfolio 94

7.2.1 The market’s supply of risky assets 94

7.3 The CAPM 95

7.4 Applications of CAPM 96

7.4.1 Decomposing risk 97

8 Risk Factor Modelling 101

8.1 General Factor Modelling 101

8.2 Theoretical Properties of the Factor Model 102

8.3 Models Based on Principal Component Analysis (PCA) 105

8.3.1 PCA in two dimensions 106

8.3.2 PCA in higher dimensions 112

9 The Value at Risk Concept 117

9.1 A Framework for Value at Risk 117

9.1.1 A motivating example 120

9.1.2 Defining value at risk 121

9.2 Investigating Value at Risk 122

9.2.1 The suitability of value at risk to capital allocation 124

9.3 Tail Value at Risk 126

9.4 Spectral Risk Measures 127

10 Value at Risk under a Normal Distribution 131

10.1 Calculation of Value at Risk 131

10.2 Calculation of Marginal Value at Risk 132

10.3 Calculation of Tail Value at Risk 134

10.4 Sub-additivity of Normal Value at Risk 135

11 Advanced Probability Theory for Risk Managers 137

11.1 Moments of a Random Variable 137

11.2 The Characteristic Function 140

11.2.1 Dealing with the sum of several random variables 142

11.2.2 Dealing with a scaling of a random variable 143

11.2.3 Normally distributed random variables 143

11.3 The Central Limit Theorem 145

11.4 The Moment-Generating Function 147

11.5 The Log-normal Distribution 148

12 A Survey of Useful Distribution Functions 151

12.1 The Gamma Distribution 151

12.2 The Chi-Squared Distribution 154

12.3 The Non-central Chi-Squared Distribution 157

12.4 The F-Distribution 161

12.5 The t-Distribution 164

13 A Crash Course on Financial Derivatives 169

13.1 The Black–Scholes Pricing Formula 169

13.1.1 A model for asset returns 170

13.1.2 A second-order approximation 172

13.1.3 The Black–Scholes formula 174

13.2 Risk-Neutral Pricing 176

13.3 A Sensitivity Analysis 179

13.3.1 Asset price sensitivity: The delta and gamma measures 179

13.3.2 Time decay sensitivity: The theta measure 182

13.3.3 The remaining sensitivity measures 183

14 Non-linear Value at Risk 185

14.1 Linear Value at Risk Revisited 185

14.2 Approximations for Non-linear Portfolios 186

14.2.1 Delta approximation for the portfolio 188

14.2.2 Gamma approximation for the portfolio 189

14.3 Value at Risk for Derivative Portfolios 190

14.3.1 Multi-factor delta approximation 190

14.3.2 Single-factor gamma approximation 191

14.3.3 Multi-factor gamma approximation 192

15 Time Series Analysis 197

15.1 Stationary Processes 197

15.1.1 Purely random processes 198

15.1.2 White noise processes 198

15.1.3 Random walk processes 199

15.2 Moving Average Processes 199

15.3 Auto-regressive Processes 201

15.4 Auto-regressive Moving Average Processes 203

16 Maximum Likelihood Estimation 207

16.1 Sample Mean and Variance 209

16.2 On the Accuracy of Statistical Estimators 211

16.2.1 Sample mean example 211

16.2.2 Sample variance example 212

16.3 The Appeal of the Maximum Likelihood Method 215

17 The Delta Method for Statistical Estimates 217

17.1 Theoretical Framework 217

17.2 Sample Variance 219

17.3 Sample Skewness and Kurtosis 221

17.3.1 Analysis of skewness 222

17.3.2 Analysis of kurtosis 223

18 Hypothesis Testing 227

18.1 The Testing Framework 227

18.1.1 The null and alternative hypotheses 227

18.1.2 Hypotheses: simple vs compound 228

18.1.3 The acceptance and rejection regions 228

18.1.4 Potential errors 229

18.1.5 Controlling the testing errors/defining the acceptance region 229

18.2 Testing Simple Hypotheses 230

18.2.1 Testing the mean when the variance is known 231

18.3 The Test Statistic 233

18.3.1 Example: Testing the mean when the variance is unknown 234

18.3.2 The p-value of a test statistic 236

18.4 Testing Compound Hypotheses 237

19 Statistical Properties of Financial Losses 241

19.1 Analysis of Sample Statistics 244

19.2 The Empirical Density and Q–Q Plots 247

19.3 The Auto-correlation Function 247

19.4 The Volatility Plot 252

19.5 The Stylized Facts 253

20 Modelling Volatility 255

20.1 The RiskMetrics Model 256

20.2 ARCH Models 258

20.2.1 The ARCH(1) volatility model 260

20.3 GARCH Models 264

20.3.1 The GARCH(1, 1) volatility model 265

20.3.2 The RiskMetrics model revisited 268

20.3.3 Summary 269

20.4 Exponential GARCH 269

21 Extreme Value Theory 271

21.1 The Mathematics of Extreme Events 271

21.1.1 A naive attempt 273

21.1.2 Example 1: Exponentially distributed losses 273

21.1.3 Example 2: Normally distributed losses 274

21.1.4 Example 3: Pareto distributed losses 275

21.1.5 Example 4: Uniformly distributed losses 275

21.1.6 Example 5: Cauchy distributed losses 276

21.1.7 The extreme value theorem 277

21.2 Domains of Attraction 278

21.2.1 The Fr´echet domain of attraction 280

21.3 Extreme Value at Risk 283

21.4 Practical Issues 286

21.4.1 Parameter estimation 286

21.4.2 The choice of threshold 287

22 Simulation Models 291

22.1 Estimating the Quantile of a Distribution 291

22.1.1 Asymptotic behaviour 293

22.2 Historical Simulation 296

22.3 Monte Carlo Simulation 299

22.3.1 The Choleski algorithm 300

22.3.2 Generating random numbers 302

23 Alternative Approaches to VaR 309

23.1 The t-Distributed Assumption 309

23.2 Corrections to the Normal Assumption 313

24 Backtesting 319

24.1 Quantifying the Performance of VaR 319

24.2 Testing the Proportion of VaR Exceptions 320

24.3 Testing the Independence of VaR Exceptions 323

References 327

Index 331