Contents Preface xiii Acknowledgements xv About the Authors xvii CHAPTER 1 Introduction 1 1.1 Evolution of Portfolio Optimization 1 1.2 Role of Quantitative Techniques 1 1.3 Organization of the Book 4 Contents Preface xiii Acknowledgements xv About the Authors xvii CHAPTER 1 Introduction 1 1.1 Evolution of Portfolio Optimization 1 1.2 Role of Quantitative Techniques 1 1.3 Organization of the Book 4 CHAPTER 2 History of Portfolio Optimization 7 2.1 Early beginnings 7 2.

2 Harry Markowitz''s Modern Portfolio Theory (1952) 9 2.3 Black-Litterman Model (1990s) 13 2.4 Alternative Methods: Risk Parity, Hierarchical Risk Parity and Machine Learning 19 2.4.1 Risk Parity 19 2.4.2 Hierarchical Risk Parity 26 2.4.

3 Machine Learning 27 2.5 Notes 31 PART ONE Foundations of Portfolio Theory CHAPTER 3 Modern Portfolio Theory 35 3.1 Efficient Frontier and Capital Market Line 35 3.1.1 Case Without Riskless Asset 35 3.1.2 Case With a Riskless Asset 41 3.2 Capital Asset Pricing Model 48 3.

2.1 Case Without Riskless Asset 48 3.2.2 Case With a Riskless Asset 52 3.3 Multifactor Models 54 3.4 Challenges of Modern Portfolio Theory 59 3.4.1 Estimation Techniques in Portfolio Allocation 60 3.

4.2 Non-Elliptical Distributions and Conditional Value-at-Risk (CVaR) 63 3.5 Quantum Annealing in Portfolio Management 65 3.6 Mean-Variance Optimization with CVaR Constraint 67 3.6.1 Problem Formulation 67 3.6.2 Optimization Problem 68 3.

6.3 Clarification of Optimization Classes 68 3.6.4 Numerical Example 69 3.7 Notes 70 CHAPTER 4 Bayesian Methods in Portfolio Optimization 73 4.1 The Prior 75 4.2 The Likelihood 79 4.3 The Posterior 80 4.

4 Filtering 83 4.5 Hierarchical Bayesian Models 87 4.6 Bayesian Optimization 89 4.6.1 Gaussian Processes in a Nutshell 90 4.6.2 Uncertainty Quantification and Bayesian Decision Theory 94 4.7 Applications to Portfolio Optimization 96 4.

7.1 GP Regression for Asset Returns 96 4.7.2 Decision Theory in Portfolio Optimization 96 4.7.3 The Black-Litterman Model 99 4.8 Notes 103 PART TWO Risk Management CHAPTER 5 Risk Models and Measures 107 5.1 Risk Measures 107 5.

2 VaR and CVaR 109 5.2.1 VaR 110 5.2.2 CVaR 112 5.3 Estimation Methods 116 5.3.1 Variance-Covariance Method 116 5.

3.2 Historical Simulation 116 5.3.3 Monte Carlo Simulation 117 5.4 Advanced Risk Measures: Tail Risk and Spectral Measures 118 5.4.1 Tail Risk Measures 118 5.4.

2 Spectral Measures 120 5.5 Notes 123 CHAPTER 6 Factor Models and Factor Investing 125 6.1 Single and Multifactor Models 126 6.1.1 Statistical Models 127 6.1.2 Macroeconomic Models 128 6.1.

3 Cross-sectional Models 130 6.2 Factor Risk and Performance Attribution 135 6.3 Machine Learning in Factor Investing 141 6.4 Notes 144 CHAPTER 7 Market Impact, Transaction Costs, and Liquidity 145 7.1 Market Impact Models 145 7.2 Modeling Transaction Costs 148 7.2.1 Single Asset 151 7.

2.2 Multiple Assets 154 7.3 Optimal Trading Strategies 155 7.3.1 Mei, DeMiguel, and Nogales (2016) 156 7.3.2 Skaf and Boyd (2009) 159 7.4 Liquidity Considerations in Portfolio Optimization 161 7.

4.1 MV and Liquidity 162 7.4.2 CAPM and Liquidity 163 7.4.3 APT and Liquidity 165 7.5 Notes 167 PART THREE Dynamic Models and Control CHAPTER 8 Optimal Control 171 8.1 Dynamic Programming 171 8.

2 Approximate Dynamic Programming 171 8.3 The Hamilton-Jacobi-Bellman Equation 172 8.4 Sufficiently Smooth Problems 174 8.5 Viscosity Solutions 176 8.6 Applications to Portfolio Optimization 180 8.6.1 Classical Merton Problem 180 8.6.

2 Multi-asset Portfolio with Transaction Costs 181 8.6.3 Risk-sensitive Portfolio Optimization 183 8.6.4 Optimal Portfolio Allocation with Transaction Costs 184 8.6.5 American Option Pricing 184 8.6.

6 Portfolio Optimization with Constraints 184 8.6.7 Mean-variance Portfolio Optimization 185 8.6.8 Schödinger Control in Wealth Management 185 8.7 Notes 187 CHAPTER 9 Markov Decision Processes 189 9.1 Fully Observed MDPs 191 9.2 Partially Observed MDPs 192 9.

3 Infinite Horizon Problems 194 9.4 Finite Horizon Problems 198 9.5 The Bellman Equation 200 9.6 Solving the Bellman Equation 203 9.7 Examples in Portfolio Optimization 205 9.7.1 An MDP in Multi-asset Allocation with Transaction Costs 205 9.7.

2 A POMDP for Asset Allocation with Regime Switching 205 9.7.3 An MDP with Continuous State and Action Spaces for Option Hedging with Stochastic Volatility 206 9.8 Notes 207 CHAPTER 10 Reinforcement Learning 209 10.1 Connections to Optimal Control 211 10.1.1 Policy Iteration 212 10.1.

2 Value Iteration 214 10.1.3 Continuous vs. Discrete Formulations 215 10.2 The Enviro.