Quantitative Portfolio Management (eBook)

Preface 6
Contents 10
1 Returns and the Gaussian Hypothesis 14
1.1 Measure of the Performance 14
1.1.1 Return 14
1.1.2 Rate of Return 15
1.2 Probabilistic and Empirical Definitions 16
1.3 Goodness of Fit Tests 18
1.3.1 Example: Testing the Normality of the Returns of the DAX 30 19
1.4 Further Statistical Results 21
1.4.1 Convergence of the Density Function Estimate 21
1.4.2 Tests Based on Cumulative Distribution Function Estimates 22
1.4.3 Tests Based on Order Statistics 24
1.4.4 Parameter Estimation and Confidence Intervals 25
1.5 Market Data with Python 29
1.5.1 Data Extraction for the DAX 30 29
1.5.2 Statistical Analysis for the DAX 30 29
A Few References 31
2 Utility Functions and the Theory of Choice 32
2.1 Utility Functions and Preferred Investments 32
2.1.1 Risk Appetite and Concavity 33
2.2 Gaussian Laws and Mean-Variance Implications 36
2.3 Efficient Investment Strategies 37
A Few References 38
3 The Markowitz Framework 39
3.1 Investment and Self-Financing Portfolios 39
3.1.1 Notations and Definitions 40
3.1.2 Representations of the Portfolios 40
3.1.3 Return of a Portfolio 42
3.2 Absence of Arbitrage Opportunities 44
3.2.1 Analysis of the Variance-Covariance Matrix 45
3.2.2 The Correlation Matrix 45
3.3 Multidimensional Estimations 47
3.3.1 Wishart, Hotelling's T2 and Fisher–Snedecor Distributions 47
3.3.2 Mean Vector and Variance-Covariance Matrix Estimates 51
3.3.3 Confidence Domain and Statistical Tests 56
3.4 Maket Data with Python 57
A Few References 61
4 Markowitz Without a Risk-Free Asset 63
4.1 The Optimisation Problem 63
4.2 The Geometric Nature of the Set F(?,m) 66
4.3 The Two Fund Theorem 68
4.3.1 Example with Two Assets: Importance of the Correlation 68
4.4 Alternative Parametrisation of F(?, m) and Conclusion 69
A Few References 71
5 Markowitz with a Risk-Free Asset 72
5.1 The Optimisation Problem 73
5.2 Capital Market Line and Limit Cone C(?, m) 74
5.2.1 The Market Portfolio 76
5.2.2 The Tangent Portfolio 79
5.2.3 More Geometric Properties 80
5.3 The Security Market Line 80
5.3.1 The Security Market Line and ``Arbitrage'' Detections 82
5.4 Market Data with Python 84
5.4.1 The Frontier and Capital Market Line for the DAX 30 Components 84
5.4.2 Adding Additional Constraints 88
5.5 Stability of the Solutions 92
5.5.1 Stabilisation by Correlation Adjustment 93
5.6 The Bayesian Approach 94
5.6.1 Jeffrey's Prior ?0 on M and 97
5.6.2 Gaussian Prior ?0 on M 99
5.6.3 The Black–Litterman Model 102
A Few References 104
6 Performance and Diversification Indicators 106
6.1 The Sharpe Ratio 106
6.2 The Jensen Index 107
6.3 The Treynor Index 108
6.4 Other Risk/Return Indicators 109
6.5 The Diversification Ratio 109
A Few References 112
7 Risk Measures and Capital Allocation 114
7.1 Definition of a Risk Measure 114
7.2 Risk Measure in the Markowitz Framework 116
7.2.1 The Markowitz Risk Measure 116
7.2.2 Value at Risk 118
7.2.3 Expected Shortfall 119
7.3 Euler's Formula and Capital Allocation 120
7.3.1 Example of Risk Measure and Capital Allocation 122
7.4 Return on Risk-Adjusted Capital 123
7.4.1 Maximising the RORAC 123
7.4.2 Capital Allocation for a Positive Homogeneous Risk Measure 124
7.4.3 Example: Euler Allocation 127
7.4.4 Example: RORAC for Optimal Portfolios 128
7.4.5 Calculation of a Portfolio VaR, from Observed Asset Prices 130
7.4.6 Example: Boostrap Historical Simulation for a Portfolio VaR 131
A Few References 134
8 Factor Models 135
8.1 Definitions and Notations 135
8.1.1 The Tangent Portfolio as a Factor 137
8.1.2 Endogenous and Exogenous Factors 137
8.1.3 Standard Form for a Factor Model 138
8.2 Identifying the Coefficients When the Factors Are Known 139
8.2.1 Regression on the Factors 141
8.3 Example of a Factor Model 141
8.4 APT Models 143
8.4.1 Example of an APT Model 146
8.4.2 Further Remarks 147
8.4.3 Standard Form for an APT Model 147
8.5 Alternative Definition of an APT Model 147
8.5.1 Estimation of the Risk Premia in an APT Model 148
A Few References 149
9 Identification of the Factors 150
9.1 Total Inertia and Trace of the Variance-Covariance Matrix 150
9.2 Total Inertia of the Projection 151
9.3 Principal Component Analysis and Factors 153
9.3.1 PCA of the Matrix of Variance-Covariance 153
9.3.2 PCA of the Correlation Matrix 156
9.4 Principal Components and Eigenvalues Visualisation 156
9.5 Python: Application to the DAX 30 Components 157
9.5.1 Factors Explaining the Variance for the DAX 30 Components 158
9.5.2 Explanation of the Factors for the DAX 30 Components 159
A Few References 162
10 Exercises and Problems 164
10.1 Midterm Exam, November 2015 164
Master M1: Mido 2nd November 2015 (Midterm Exam: Portfolio Management) 164
10.1.1 Solutions: Midterm Exam, November 2015 166
Master M1: Mido 2015–2016 (Midterm Exam: Portfolio Management) 166
10.2 Exam, January 2016 168
Master M1: Mido 5th January 2016 (Exam: Portfolio Management: Time 1h 30min) 168
10.2.1 Solutions: Exam, January 2016 171
Master M1: Mido 5th January 2016 (Exam: Portfolio Management) 171
10.3 Midterm Exam, November 2016 173
Master M1: Mido 3rd November 2016 (Exam: Portfolio Management: Time 2h) 173
10.3.1 Solutions: Midterm Exam, November 2016 175
10.4 Exam, January 2017 177
Master M1: Mido 11th January 2017 (Exam: Portfolio Management: Time 2h) 177
10.4.1 Solutions: Exam, January 2017 181
10.5 Midterm Exam, November 2017 183
Master M1: Mido 2nd November 2017 (Midterm Exam: Portfolio Management: Time 2h) 183
10.5.1 Solutions: Midterm Exam, November 2017 186
10.6 Exam, January 2018 188
Master M1: Mido 15th January 2018 (Exam: Portfolio Management: Time 2h) 188
10.6.1 Solutions: Exam, January 2018 191
10.7 Midterm Exam, October 2018 193
Master M1: Mido 29th October 2018 (Midterm Exam: Portfolio Management: Time 2h) 193
10.7.1 Solutions: Midterm Exam October 2018 196
A The Lagrangian 199
A.1 Main Results 199
A.1.1 Solution of the Markowitz Problem 201
B Parametrisations 203
B.1 Confidence Domain for an Estimator of M 203
B.2 Confidence Domain for an Observation Ri 204
Bibliography 206
Index 210