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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Serie: Wiley Finance
- Sprache: Englisch
Expert guidance on implementing quantitative portfolio optimization techniques
In Quantitative Portfolio Optimization: Theory and Practice, renowned financial practitioner Miquel Noguer, alongside physicists Alberto Bueno Guerrero and Julian Antolin Camarena, who possess excellent knowledge in finance, delve into advanced mathematical techniques for portfolio optimization. The book covers a range of topics including mean-variance optimization, the Black-Litterman Model, risk parity and hierarchical risk parity, factor investing, methods based on moments, and robust optimization as well as machine learning and reinforcement technique. These techniques enable readers to develop a systematic, objective, and repeatable approach to investment decision-making, particularly in complex financial markets.
Readers will gain insights into the associated mathematical models, statistical analyses, and computational algorithms for each method, allowing them to put these techniques into practice and identify the best possible mix of assets to maximize returns while minimizing risk. Topics explored in this book include:
- Specific drivers of return across asset classes
- Personal risk tolerance and it#s impact on ideal asses allocation
- The importance of weekly and monthly variance in the returns of specific securities
Serving as a blueprint for solving portfolio optimization problems, Quantitative Portfolio Optimization: Theory and Practice is an essential resource for finance practitioners and individual investors It helps them stay on the cutting edge of modern portfolio theory and achieve the best returns on investments for themselves, their clients, and their organizations.
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Veröffentlichungsjahr: 2025
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Preface
Acknowledgements
About the Authors
Chapter 1: Introduction
1.1 Evolution of Portfolio Optimization
1.2 Role of Quantitative Techniques
1.3 Organization of the Book
Chapter 2: History of Portfolio Optimization
2.1 Early beginnings
2.2 Harry Markowitz’s Modern Portfolio Theory (1952)
2.3 Black-Litterman Model (1990s)
2.4 Alternative Methods: Risk Parity, Hierarchical Risk Parity and Machine Learning
2.5 Notes
Part One: Foundations of Portfolio Theory
Chapter 3: Modern Portfolio Theory
3.1 Efficient Frontier and Capital Market Line
3.2 Capital Asset Pricing Model
3.3 Multifactor Models
3.4 Challenges of Modern Portfolio Theory
3.5 Quantum Annealing in Portfolio Management
3.7 Notes
Chapter 4: Bayesian Methods in Portfolio Optimization
4.1 The Prior
4.2 The Likelihood
4.3 The Posterior
4.4 Filtering
4.5 Hierarchical Bayesian Models
4.6 Bayesian Optimization
4.7 Applications to Portfolio Optimization
4.8 Notes
Part Two: Risk Management
Chapter 5: Risk Models and Measures
5.1 Risk Measures
5.2 VaR and CVaR
5.3 Estimation Methods
5.4 Advanced Risk Measures: Tail Risk and Spectral Measures
5.5 Notes
Chapter 6: Factor Models and Factor Investing
6.1 Single and MultiFactor Models
6.2 Factor Risk and Performance Attribution
6.3 Machine Learning in Factor Investing
6.4 Notes
Chapter 7: Market Impact, Transaction Costs, and Liquidity
7.1 Market Impact Models
7.2 Modeling Transaction Costs
7.3 Optimal Trading Strategies
7.4 Liquidity Considerations in Portfolio Optimization
7.5 Notes
Part Three: Dynamic Models and Control
Chapter 8: Optimal Control
8.1 Dynamic Programming
8.2 Approximate Dynamic Programming
8.3 The Hamilton-Jacobi-Bellman Equation
8.4 Sufficiently Smooth Problems
8.5 Viscosity Solutions
8.6 Applications to Portfolio Optimization
8.7 Notes
Chapter 9: Markov Decision Processes
9.1 Fully Observed MDPs
9.2 Partially Observed MDPs
9.3 Infinite Horizon Problems
9.4 Finite Horizon Problems
9.5 The Bellman Equation
9.6 Solving the Bellman Equation
9.7 Examples in Portfolio Optimization
9.8 Notes
Chapter 10: Reinforcement Learning
10.1 Connections to Optimal Control
10.2 The Environment and The Reward Function
10.3 Agents Acting in an Environment
10.4 State-Action and Value Functions
10.5 The Policy
10.6 On-Policy Methods
10.7 Off-Policy Methods
10.8 Applications to Portfolio Optimization
10.9 Notes
Part Four: Machine Learning and Deep Learning
Chapter 11: Deep Learning in Portfolio Management
11.1 Neurons and Activation Functions
11.2 Neural Networks and Function Approximation
11.3 Review of Some Important Architectures
11.4 Physics-Informed Neural Networks
11.5 Applications to Portfolio Optimization
11.6 The Case for and Against Deep Learning
11.7 Notes
Chapter 12: Graph-based Portfolios
12.1 Graph Theory-Based Portfolios
12.2 Graph Theory Portfolios: MST and TMFG
12.3 Hierarchical Risk Parity
12.4 Notes
Chapter 13: Sensitivity-based Portfolios
13.1 Modeling Portfolios Dynamics with PDEs
13.2 Optimal Drivers Selection: Causality and Persistence
13.3 AAD Sensitivities Approximation
13.4 Hierarchical Sensitivity Parity
13.5 Implementation
13.6 Conclusion
Part Five: Backtesting
Chapter 14: Backtesting in Portfolio Management
14.1 Introduction
14.2 Data Preparation and Handling
14.3 Implementation of Trading Strategies
14.4 Types of Backtests
14.5 Performance Metrics
14.6 Avoiding Common Pitfalls
14.7 Advanced Techniques
14.8 Case Study: Applying Backtesting to a Real-World Strategy
14.9 Impact of Market Conditions on Backtest Results
14.10 Integration with Portfolio Management
14.11 Tools and Software for Backtesting
14.12 Regulatory Considerations
14.13 Conclusion
Chapter 15: Scenario Generation
15.1 Historical Scenarios
15.2 Bootstrapping Scenarios
15.3 Copula-Based Scenarios
15.4 Risk Factor Model-Based Scenarios
15.5 Time Series Model Scenarios
15.6 Variational Autoencoders
15.7 Generative Adversarial Networks (GANs)
Appendix
A.1 Software and Tools for Portfolio Optimization
Bibliography
Index
End User License Agreement
List of Illustrations
Chapter 2
Figure 2.1 Timeline of portfolio allocation models.
Chapter 3
Figure 3.1 Minimum variance frontier.
Figure 3.2 Mean-variance efficient frontier and capital market line.
Chapter 4
Figure 4.1 Beta distribution as the PDF of the Hurst exponent, .
Figure 4.2 Trace of samples of from the Beta(100,100) prior.
Figure 4.3 Fractional and geometric fractional Brownian motions for different Hurst expon...
Figure 4.4 Independent draws from the Gaussian likelihood in equation (4.13).
Figure 4.5 Top: Estimated density of the MCMC sampled posterior in gray and the ground tr...
Chapter 9
Figure 9.1 MDP Framework: Schematic representation of how the components (i.e., agent, st...
Figure 9.2 Example of POMDP.
Chapter 11
Figure 11.1 Deep Neural Network: Depiction of a DNN. Data is fed into the input layer and ...
Figure 11.2 DNN with weights, biases, and activations: In this representation, the DNN is ...
Chapter 13
Figure 13.1 Embedded space of sensitivities: Average sensitivities of asset , and common...
Figure 13.2 Methodology modules: sensitivities are extracted from multilayer perceptrons (...
Figure 13.3 NAVs for US portfolio top mean-variance methods and 1/N: NAV starting from 01/...
Figure 13.4 NAVs for US portfolio for top mean-variance methods, 1/N, HRP, and HSP for dif...
Figure 13.5 NAVs for US portfolio for top mean-variance methods, 1/N, HRP, and HSP for dif...
Figure 13.6 NAVs for US portfolio long-term investments for top mean-variance methods, HRP...
Figure 13.7 NAVs for US portfolio long-term investments zoom of Figure 13.6: from 06/2015 ...
Figure 13.8 Long-term investments. US portfolio long-term investments performance metrics ...
List of Tables
Chapter 10
Table 10.1 Performance comparison of DRL and MVO approaches as reported in Sood and Balch...
Chapter 12
Table 12.1 Comparison of Portfolio Compositions.
Chapter 13
Table 13.1 US portfolio performance metrics for top mean-variance methods and 1/N: Return...
Table 13.2 US portfolio performance metrics for 1/N, HRP, HSP for diff-erent model hyperp...
Table 13.3 EU portfolio performance metrics for top mean-variance methods, 1/N, HRP, HSP ...
Table 13.4 US portfolio long-term investments performance metrics for top mean-variance m...
Table 13.5 US portfolio long-term investments performance metrics for top mean-variance m...
Guide
Cover
Table of Contents
Title Page
Copyright
Preface
Acknowledgements
About the Authors
Begin Reading
Appendix
Bibliography
Index
End User License Agreement
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Quantitative Portfolio Optimization
Advanced Techniques and Applications
MIQUEL NOGUER ALONSO
JULIÁN ANTOLÍN CAMARENA
ALBERTO BUENO GUERRERO
Copyright © 2025 by Miquel Noguer Alonso, Julián Antolín Camarena, and Alberto Bueno Guerrero. All rights reserved.
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Library of Congress Cataloging-in-Publication Data:
Names: Noguer Alonso, Miquel, author. | Camarena, Julián Antolín, author. | Bueno Guerrero, Alberto, author.
Title: Quantitative portfolio optimization: Advanced techniques and applications / Miquel Noguer Alonso, Julián Antolín Camarena, Alberto Bueno Guerrero.
Description: Hoboken, New Jersey: John Wiley and Sons, Inc, [2025] | Series: Wiley finance | Includes bibliographical references and index. | Summary: “*Quantitative Portfolio Optimization: Advanced Techniques and Applications* offers a comprehensive exploration of portfolio optimization, tracing its evolution from Harry Markowitz’s Modern Portfolio Theory to contemporary techniques. The book combines foundational models like CAPM and Black-Litterman with advanced methods including Bayesian statistics, machine learning, and quantum computing. It bridges theory and practice through detailed explanations, real-world data applications, and case studies. Aimed at finance professionals, researchers, and students, the book provides tools and insights to address future financial challenges and contributes to the field’s ongoing development”– Provided by publisher.
Identifiers: LCCN 2024041151 (print) | LCCN 2024041152 (ebook) | ISBN 9781394281312 (hardback) | ISBN 9781394281336 (pdf) | ISBN 9781394281329 (epub)
Subjects: LCSH: Portfolio management–Mathematical models. | Asset allocation–Mathematical models. | Finance–Mathematical models.
Classification: LCC HG4529.5.N537 2025 (print) | LCC HG4529.5 (ebook) | DDC 332.6–dcundefined
LC record available at https://lccn.loc.gov/2024041151
LC ebook record available at https://lccn.loc.gov/2024041152
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Preface
Quantitative portfolio optimization is a cornerstone of modern financial management, providing a rigorous framework for balancing risk and return in investment portfolios. This book, Quantitative Portfolio Optimization: Advanced Techniques and Applications, aims to serve as both a comprehensive introduction for those new to the field and a deep dive into the latest advancements for experienced practitioners and researchers.
The genesis of portfolio optimization can be traced back to Harry Markowitz’s Modern Portfolio Theory (MPT) in the 1950s, which introduced the now-fundamental concepts of diversification and mean-variance optimization. Since then, the field has evolved significantly, integrating a wide array of quantitative methods, including Bayesian statistics, machine learning algorithms, and advanced optimization techniques. These methods have transformed portfolio management from a discipline grounded in basic statistical principles to one that leverages innovative computational techniques to solve increasingly complex problems.
This book is structured to reflect this evolution, beginning with foundational theories before progressing to advanced applications. We explore not only traditional models such as the Capital Asset Pricing Model (CAPM) and the Black-Litterman model but also the latest developments in areas such as reinforcement learning, deep learning, and graph-based portfolio construction. Additionally, we cover emerging topics like quantum computing’s role in portfolio optimization and the integration of partial differential equations (PDEs) for modeling portfolio dynamics.
Each chapter is meticulously designed to bridge theory with practice, offering detailed explanations of the mathematical underpinnings of each technique, followed by practical applications using real-world data. The mathematical rigor is complemented by code implementations and case studies that demonstrate the practical utility of the methods discussed.
Whether you are a professional, researcher, or student in the field of finance, we hope this book enhances your understanding of quantitative portfolio optimization and equips you with the knowledge to apply these techniques effectively. As the financial markets continue to evolve, so must the methods we use to manage them. We believe that the approaches detailed in this book will be instrumental in addressing the challenges and opportunities of tomorrow’s financial world.
Finally, we extend our deepest gratitude to our colleagues, students, and family members, whose support and encouragement have been invaluable throughout the creation of this book. It is our sincere hope that this work contributes meaningfully to the ongoing development of the field of quantitative portfolio optimization.
Acknowledgements
Miquel I would like to express my gratitude to all those who have contributed to the development and success of this book, especially my co-authors and Alejandro Rodriguez Dominguez. Special thanks to my colleagues in the Quant Community. My mother Maria del Carmen, my sons Jordi and Arnau, and my brother Jordi who always supported me. A lot of people in finance, mathematics, and computer science have inspired me to author this book, to do my best. To Emmanuel, Garud, Peter, Petter, Matthew, Igor, and Gordon. To my dad Jordi, I love you.
Julián I am deeply indebted to my wife, Esther; my parents, Antonio and Cecilia; my brother, Omar; and all of my friends who always support me in whatever I may choose to do. I thank and love you all. And of course, my much beloved pug, Nibbler, who always puts a smile on my face. I love you, kiddo.
I also thank my friends and co-authors Miquel and Alberto, with whom I have had engrossing and fun conversations, and without whom I would not have been a part of this book.
Alberto My first thanks go to my wife Elena, for her continued support and encouragement in everything I do. Also, to my children, Iván, Leo, Elena, and Hugo, for their help, respect, and affection. I could not forget my mother, María del Carmen, who brought me into this world and has always believed in me; and my father, Francisco, who is no longer with us, and who instilled in me his love of knowledge. Special thanks also go to my brother, Francisco José, who has always been an example for me.
About the Authors
Miquel Noguer Alonso is a seasoned financial professional and academic with over 30 years of experience in the industry. He is the Founder of the Artificial Intelligence Finance Institute and Head of Development at Global AI. His career includes roles such as Executive Director at UBS AG and CIO for Andbank. He has served on the European Investment Committee UBS for a decade. He is on the advisory board of FDP Institute and CFA NY.
Mr. Noguer is also an academic, teaching AI, Big Data, and Fintech at institutions like NYU Courant Institute, NYU Tandon, Columbia University, and ESADE. He pioneered the first Fintech and Big Data course at the London Business School in 2017. He is the author of 10 papers on Artificial Intelligence.
He holds an MBA and a Degree in business administration from ESADE, a PhD in quantitative finance from UNED, and other prestigious certifications. His research interests span asset allocation, machine learning, algorithmic trading, and Fintech.
Julián Antolín Camarena holds a bachelor’s, master’s and a PhD in physics. For his master’s, he worked on the foundations of quantum mechanics examining alternative quantization schemes and their application to exotic atoms to discover new physics. His PhD dissertation work was on computational and theoretical optics, electromagnetic scattering from random surfaces, and nonlinear optimization. He then went on to a postdoctoral stint with the US Army Research Laboratory working on inverse reinforcement learning for human-autonomy teaming. He then worked as an Applied AI Researcher at Point72 Asset Management where his research was to develop algorithms for time series analysis using machine learning algorithms and developing deep learning architectures. Currently, works in applying AI to quantitative bioimaging and biophysics at Arizona State University.
Alberto Bueno Guerrero has two bachelor’s degrees in physics and economics, and a PhD in banking and finance. Since he received his doctorate, he has dedicated himself to research in mathematical finance. His work has been presented at various international conferences and published in journals such as Quantitative Finance; Journal of Derivatives, Mathematics; and Chaos, Solitons and Fractals.
Among the topics addressed in his research are the term structure of interest rates, the valuation and hedging of derivatives, the immunization of bond portfolios, and a quantum-mechanical model for interest rate derivatives. His article “Bond Market Completeness Under Stochastic Strings with Distribution-Valued Strategies” has been considered a feature article in Quantitative Finance.
In 2020, he was interviewed for the article “Interviews with Researchers Who Started Their Career in Physics but Moved to Finance,” for the special issue of the Journal of Derivatives titled “Physics and Financial Derivatives.”
He has also functioned as anonymous reviewer for various journals, and regularly reviews articles for Mathematical Reviews of the American Mathematical Society.
Chapter 1Introduction
1.1 Evolution of Portfolio Optimization
Portfolio optimization has undergone significant transformation since its inception. Initially, the focus was on maximizing returns without much regard for risk. This changed with the introduction of Modern Portfolio Theory (MPT) by Harry Markowitz in the 1950s, which introduced the concept of balancing risk and return. Markowitz’s mean-variance optimization laid the groundwork for the systematic assessment of portfolio risk and diversification.
Over the years, portfolio optimization has evolved to incorporate various advanced techniques and models. These include the Capital Asset Pricing Model (CAPM), Arbitrage Pricing Theory (APT), and more sophisticated approaches like the Black-Litterman model, risk parity, and hierarchical risk parity. Recently, machine learning methods have also been integrated into portfolio optimization, providing new ways to manage complex data and uncover hidden patterns in financial markets. Moreover, the integration of reinforcement learning, and graph-based methods has opened new avenues for dynamic and complex portfolio strategies. Sensitivity-based portfolios, which focus on the sensitivity of portfolio returns to changes in underlying factors, have also become an important aspect of modern portfolio management.
1.2 Role of Quantitative Techniques
Quantitative techniques play a crucial role in modern portfolio optimization. These techniques allow for the systematic analysis and management of risk, the development of models to predict asset returns, and the optimization of portfolios to achieve desired outcomes. Key quantitative methods used in portfolio optimization include:
Mean-Variance Optimization
: This foundational technique balances expected return against risk, measured as the variance of returns. It involves calculating the expected returns and covariances of all assets, then solving for the weights that minimize portfolio variance subject to a desired return. The efficient frontier is derived from this process, representing the set of optimal portfolios.
Factor Models
: These models, such as the CAPM and multifactor models, explain asset returns based on various macroeconomic factors or firm-specific factors. The CAPM, for example, relates an asset’s return to the return of the market portfolio, adjusted for the asset’s sensitivity to market movements.
Bayesian Methods
: Bayesian techniques incorporate prior beliefs and observed data to update the estimation of expected returns and risks. The Black-Litterman model is a popular application in portfolio optimization, combining market equilibrium with investor views to produce more stable and diversified portfolios. Bayesian methods are particularly useful for handling parameter uncertainty and incorporating subjective views.
Machine Learning
: Machine learning algorithms are used to identify patterns in large datasets, making them valuable for predictive modeling in portfolio optimization. Techniques like neural networks, decision trees, and generative models can uncover complex relationships between asset returns and various predictors. These methods can enhance the forecasting of returns and risks as well as optimize trading strategies.
Neural Networks:
These are used to model nonlinear relationships between inputs and outputs. In portfolio optimization, they can predict asset returns based on historical data and other variables.
Decision Trees:
These algorithms split the data into subsets based on feature values, creating a tree-like model of decisions. They are useful for capturing the nonlinear relationships in financial data and can be used to identify important variables influencing asset returns.
Generative Models:
These models, such as Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs), are used to generate new data samples that are like the training data. In portfolio optimization, generative models can be used to simulate realistic market scenarios and generate synthetic data for stress testing and risk management.
Reinforcement Learning (RL)
: RL involves training algorithms to make sequences of decisions by rewarding desirable actions and penalizing undesirable ones. In portfolio optimization, RL can dynamically adjust the asset allocation based on market conditions and investment goals. An RL agent learns a policy that maximizes cumulative rewards, which can correspond to returns in a portfolio context. Techniques like Q-learning and policy gradients are commonly used in RL for portfolio management.
Q-learning:
This algorithm learns the value of actions in different states and aims to maximize the expected reward over time. It updates its estimates using the Bellman equation.
Policy Gradients:
These methods optimize the policy directly by computing gradients of the expected reward with respect to the policy parameters.
Graph-based Methods
: These methods use graph theory to represent and analyze the relationships between assets. Graphs can model the dependencies and correlations among assets, aiding in the construction of diversified and robust portfolios.
Graph Theory:
This involves studying graphs, which are mathematical structures used to model pairwise relations between objects. In portfolio optimization, nodes can represent assets, and edges can represent the correlations or co-movements between them.
Hierarchical Risk Parity (HRP):
This approach uses clustering and tree structures to construct portfolios. It aims to distribute risk more evenly across different clusters of assets, improving diversification.
Sensitivity-based Portfolios
: These portfolios focus on the sensitivity of portfolio returns to changes in underlying factors, such as economic variables or market indices. By analyzing how small changes in these factors impact on the portfolio, managers can better understand and manage risk.
Sensitivity Analysis:
This involves examining how the variation in the output of a model can be attributed to different variations in the inputs. In portfolio optimization, sensitivity analysis helps in understanding the impact of changes in asset returns and other factors on the portfolio performance.
Partial Differential Equations (PDEs):
PDEs can be used to model the dynamics of portfolio values over time, considering factors like interest rates and asset prices. Solving these equations provides insights into the optimal portfolio allocation under different market conditions.
Risk Measures and Management
: Techniques like Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are used to quantify the risk of loss in a portfolio. These measures are essential for understanding potential downside risks and making informed decisions about risk mitigation strategies. Advanced risk measures also consider tail risks and the distribution of returns.
Optimization Algorithms
: Several optimization algorithms are employed to solve portfolio optimization problems. These include:
Quadratic Programming:
Used in mean-variance optimization to find the optimal asset weights that minimize portfolio variance for a given return.
Monte Carlo Simulation:
Used to model the probability of different outcomes in a process that cannot easily be predicted due to the intervention of random variables. In portfolio optimization, it is used to simulate the performance of different portfolio strategies under various market conditions.
Genetic Algorithms:
These algorithms mimic natural selection processes to generate high-quality solutions for optimization problems. They are particularly useful in finding optimal portfolios in large, complex investment universes.
Dynamic Programming:
Applied in multi-period portfolio optimization to make decisions that consider the evolution of the portfolio over time.
1.3 Organization of the Book
This book is structured to provide a comprehensive understanding of quantitative portfolio optimization techniques, from foundational theories to advanced applications. The chapters are organized as the list describes:
Chapter 2
: History of Portfolio Optimization:
A review of the key developments in portfolio optimization, from early theories to modern advancements.
Chapter 3
: Modern Portfolio Theory:
A detailed study of mean-variance analysis, the CAPM, and APT, including their applications and limitations. We introduce a new framework Mean Variance with CVAR constraints.
Chapter 4
: Bayesian Methods in Portfolio Optimization:
An exploration of Bayesian techniques and their application to portfolio optimization.
Chapter 5
: Risk Models and Measures:
A discussion on various risk measures, including Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), and their estimation methods.
Chapter 6
: Factor Models and Factor Investing:
Examination of single and multifactor models, factor risk, and performance attribution.
Chapter 7
: Market Impact, Transaction Costs, and Liquidity:
Insights into market impact, transaction costs, and liquidity considerations in portfolio optimization.
Chapter 8
: Optimal Control:
Coverage of dynamic programming, optimal control, and their applications in portfolio optimization.
Chapter 9
: Markov Decision Processes:
Discussion on fully observed and partially observed MDPs, infinite and finite horizon problems, and the Bellman equation.
Chapter 10
: Reinforcement Learning:
Examination of reinforcement learning techniques and their applications in portfolio optimization.
Chapter 11
: Deep Learning in Portfolio Management:
Introduction to deep learning methods and their integration into portfolio management.
Chapter 12
: Graph-based Portfolios:
Exploration of graph theory-based portfolios and their applications.
Chapter 13
: Sensitivity-based Portfolios:
Insights into modeling portfolio dynamics with partial differential equations and sensitivity analysis.
Chapter 14
: Backtesting in Portfolio Management:
Discussion on backtesting methods, trading rules, and transaction costs.
Chapter 15
: Scenario Generation:
Techniques for generating scenarios and their application in portfolio optimization.
This structure ensures a logical progression from basic concepts to advanced techniques, providing readers with the tools and knowledge necessary to optimize portfolios effectively in today’s complex financial markets.
Chapter 2History of Portfolio Optimization
This chapter is dedicated to a non-exhaustive review of the main contributions to portfolio optimization, from Markowitz’s precursors to the modern machine learning methods, including Markowitz’s mean-variance approach, the Black-Litterman model, Risk Parity, and Hierarchical Risk Parity. Only the Black-Litterman model and the Risk Parity approach will be addressed in depth. An alternative derivation of the Black-Litterman model, based on Bayesian methods, will be presented in Chapter 4. A more detailed treatment of the Markowitz model, the Hierarchical Risk Parity algorithm and some machine learning methods will be postponed to subsequent chapters.
2.1 Early beginnings
Harry Markowitz is unanimously recognized as the father of Modern Portfolio Theory (MPT). His seminal works, Markowitz (1952, 1959), settled the foundations of the mean-variance analysis upon which other aspects of MPT were built.
The groundbreaking nature of Markowitz (1952) can be inferred from its scarce set of references: Uspensky (1937), Williams (1938) and Hicks (1939). The first one is a text on Mathematical Probability, to which the reader was referred. Of the two remaining references, only Williams (1938) played a role in the development of the 1952 paper, as we will see soon. Therefore, it is not surprising that Mark Rubinstein stated the following: “What has always impressed me most about Markowitz’s 1952 paper is that it seemed to come out of nowhere” (Rubinstein, 2002). Nevertheless, the key ideas of mean-variance analysis (i.e., diversification, mean returns, and a risk measure as variables) were present in previous literature. The rest of the section will be dedicated to analyzing some of these early contributions.
According to Markowitz (1999), written references to the concept of diversification can be traced back to Shakespeare’s The Merchant of Venice, in which the following passage can be found (Act I, Scene I):
My ventures are not in one bottom trusted,
Nor to one place; nor is my whole state
Upon the fortune of this present year
Therefore my merchandise makes me no sad.
In the academic field, Daniel Bernouilli, in his 1738 paper on the Saint Petersburg paradox (Bernouilli, 1954), offered this diversification advice for risk-averse investors: “... it is advisable to divide goods which are exposed to some small danger into several portions rather than to risk them all together.” In the twentieth century, the first reference related to Portfolio Theory appears in The Nature of Capital and Income (Fisher, 1906), in which variance is suggested as a measure of economic risk. Hicks, in his treatise Theory of Money (Hicks, 1935), introduces a qualitative analysis regarding the probabilities associated with the risk of an investment, concluding that they can be represented by a mean value and an appropriate measure of risk (although he does not mention any specific measure).
Marschak (1938) takes a step forward specifying the nature of the parameter measuring risk: “It is sufficiently realistic, however, to confine ourselves, for each yield, to two parameters only: the mathematical expectation (‘lucrativity’) and the coefficient of variation (‘risk’).” Although Marschak was the advisor of Markowitz’s thesis, we cannot state (as Markowitz himself acknowledges) that Marschak had an influence on Markowitz’s work. In fact, Marschak did not inform to Markowitz of the existence of Marschak (1938).
The only previous work that had a clear impact on Markowitz is Williams (1938). According to Markowitz (1991): “The basic principles of portfolio theory came to me one day while I was reading John Burr Williams The Theory of Investment Value. Williams was remarkably prescient. He provided the first derivation of the ‘Gordon growth formula,’ the Modigliani-Miller Capital Structure Irrelevancy Theorem, and strongly advocated the dividend discount model. But Williams had very little to say about the effects of risk on valuation (pp. 67–70), because he believed that all risk could be diversified away.” In the opinion of Mark Rubinstein: “Markowitz had the brilliant insight that, while diversification would reduce risk, it would not generally eliminate it” (Rubinstein, 2002).
We cannot conclude this section dedicated to the predecessors of MPT without considering Roy (1952), of which Markowitz wrote the following: “On the basis of Markowitz (1952), I am often called the father of modern portfolio theory (MPT), but Roy can claim an equal share of this honor” (Markowitz, 1999). In fact, Roy (1952) presents an analysis very similar to that of Markowitz (1952). Specifically, Roy includes correlations between asset prices in the analysis and, like Markowitz, realizes that “the principle of maximising expected return does not explain the well-known phenomenon of the diversification of resources among a wide range of assets.” Moreover, Roy also considers the expected value of returns, ; and the standard deviation of returns, , as the only parameters on which investment decisions are based. However, instead of minimizing the standard deviation, as Markowitz did, Roy maximizes , where is a level of returns that can be considered a disaster. This maximization procedure leads Roy to obtain the efficient frontier as a hyperbola in the space. According to Markowitz (1999), the main differences between Roy’s paper and his 1952 paper are that Markowitz (1952) worked only with long positions and allowed the investors to select one portfolio from the efficient frontier, whereas Roy (1952) worked also with short positions and recommended one specific portfolio.
Given that Roy arrived at the same results as Markowitz independently and using similar methods, it is reasonable to wonder why Roy did not also receive the Nobel Prize. Markowitz thought the reason was his greater visibility: Roy’s paper was his last (and only) publication in finance, whereas Markowitz wrote two books and a collection of papers related to this subject (Markowitz, 1999).
2.2 Harry Markowitz’s Modern Portfolio Theory (1952)
Before delving into the analysis of Markowitz’s paper, we introduce some of the definitions and the notation that we will use. We consider portfolios composed of risky assets, with returns ; which are random variables with expectations and covariances , . We will usually work in matrix form, with the vector of expected returns and the covariance matrix with . The variances of asset returns are given by the diagonal elements of , . As the assets in the portfolio are risky assets, we have , , and then, is a positive definite matrix, that is, for any nonzero -vector . In addition, we will assume that is nonsingular, , meaning that none of the asset returns is perfectly correlated with the return of the portfolio composed of the remaining assets.
Each portfolio is determined by a vector in which is the proportion of investor’s wealth allocated to the -th asset. By this definition we have , or in matrix notation, , where is the -vector given by . The expected return of the portfolio is given by , and the variance of its return is .
In his 1952 paper, Markowitz begins by rejecting the maximization of expected returns as a guiding principle for investment behavior. This decision is rooted in the often-understated principle of diversification, a key aspect of Markowitz’s work. In his own words: “[...] a rule of behavior which does not imply the superiority of diversification must be rejected both as a hypothesis and as a maxim.”
Diversification, as a rule leading to the reduction of the risk of a portfolio, measured by the variance of its return, finds support in theoretical arguments. As the following simple result shows, under certain assumptions, perfectly diversified portfolios become asymptotically risk-free. In other words, the variance of the portfolio return diminishes as the number of component assets increases.
Proposition 2.1
Consider an equally weighted portfolio whose asset returns are independent random variables with the same variance. Then, the variance of the portfolio return, , satisfies
Proof
As the portfolio is equally weighted, we have , and by the independence assumption, , where is the identity matrix. Then, the portfolio variance is
from where we obtain the desired result.
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While the previous result is intellectually appealing, its assumptions are hardly satisfied in actual markets. First, there is empirical evidence suggesting that returns follow stable Paretian distributions, characterized by infinite variances (Mandelbrot, 1963; Fama, 1965a).1 Second, it is difficult, if not impossible, to find a substantial number of assets with totally independent returns and equal variance. Therefore, as Markowitz himself pointed out: “The returns from securities are too intercorrelated. Diversification cannot eliminate all variance.”
Once the diversification principle is accepted, Markowitz turns his attention to the maximization of expected returns, for the case of no short sales allowed. We will formalize Markowitz’s reasoning in the following result, which also includes short sales.
Proposition 2.2
If all the available assets have different expected returns, the portfolio with maximum expected return consists of only one asset with the highest expected return. This result holds true whether short sales are allowed or not.
Proof
Consider first the case of short sales allowed. The optimization problem can be stated as
with variable . The corresponding Lagrangian is
and the first-order conditions are
From equation (2.1) we obtain , and all the assets in the portfolio must have the same expected return . As by hypothesis all the expected returns are different, the portfolio must be composed of only one asset, with the highest expected return.
If short sales are not allowed, we must include the additional constraint , i.e., , . The Karush-Kuhn-Tucker first-order conditions for the problem are
where is a Lagrange multiplier, is a vector of Lagrange multipliers, and “” stands for the Hadamard (component-wise) product. If the -th asset is included in the optimal portfolio, then , and by (2.3), . Replacing this value in equation (2.1), we obtain , and thus, all the assets in the portfolio must have the same expected return. Therefore, the optimal portfolio is composed only of the asset with the highest return.
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Remark 2.1
In the unlikely case in which there are several assets with the same expected return, an optimal portfolio would be composed of all these assets in any proportion.
Considering the previous result, and diversification as a first principle, Markowitz rejects the maximization of portfolio return as a maxim. The next step involves considering the expected return-variance of return rule, which leads to the so-called mean-variance (MV) analysis.
The assumptions of this approach, not explicitly stated by Markowitz, are the following (Constantinides and Malliaris, 1995).
The investor considers only the mean and the variance of returns to form her optimal portfolios.
As pointed out in Meucci (2005), considering only the first two moments of the distribution of returns can be considered as an approximation to the broader problem of dealing with all the moments. This approximation is exact if portfolio returns follow a normal distribution, although this is not an assumption of the MV analysis.
Given the expected return of the portfolio, the investor will choose the portfolio with the lowest variance of returns.
Markowitz (1952) assumes that the knowledge of expected returns is available to the investor, without addressing the problem of its estimation. On the other hand, he considers variance as a measure of portfolio risk, and then, the investor wants to reduce its value, while maintaining expected returns as high as possible. In his words, the investor “considers expected return a desirable thing and variance of returns an undesirable thing.”
The investment horizon is one period.
This is a restrictive assumption of the model, which will be relaxed in subsequent chapters of this book, where we consider the dynamic nature of portfolio management. Nevertheless, as pointed out in Kolm et al. (2014b), practitioners typically use one-period models to rebalance portfolios from one period to another.
The investor’s individual decisions do not affect market prices.
This assumption is equivalent to excluding market impact (i.e., changes in prices caused by the trade itself) from the analysis. As market impact depends on the ratio of the trade size to the average trade volume, Markowitz is assuming, implicitly, that the investor rebalances her portfolio by trading in relatively tiny amounts.
Fractional shares may be purchased.
This is a mathematical requirement not satisfied in actual markets, although it does not impose a severe restriction because portfolios are defined by their weights in each asset, and not by absolute amounts.
Transaction costs and taxes do not exist.
This assumption separates the model from the actual functioning of markets. We will see in subsequent chapters how to include transaction costs in the study.
Investors can sell assets short.
In fact, Markowitz (1952) does not allow short sales to simplify the analysis. We will present the MV approach with short sales allowed and refer to the situation in which they are forbidden.
Once stated the MV optimization problem, Markowitz (1952) intentionally avoids the mathematical formulation and instead presents a geometrical analysis for the case of only three assets. We avoid here these geometrical considerations and refer the reader to the original paper.
With the preceding considerations, we can now state the optimization program associated with the MV analysis as
with variable .
The problem in equation (2.4) is a convex problem with strictly convex objective function, and then the optimal set contains at most one point. We will find this optimal solution in Chapter 3.
2.3 Black-Litterman Model (1990s)
In 1989, while at Goldman Sachs, Fisher Black and Robert Litterman were assigned the task of developing an asset allocation model with the objective of diversifying their clients’ global bond portfolios. The result of their work is known today as the Black-Litterman model and was published in the Journal of Fixed Income (Black and Litterman, 1991a).
The authors begin the presentation of their model by considering a portfolio manager who maximizes total return for any given level of risk, namely, who follows the MV approach. Nevertheless, they propose a model that goes beyond the standard framework, to address the main problem that portfolio managers face when applying the Markowitz model: the weights in optimal portfolios are very sensitive to small changes in expected returns, giving rise to extremely unbalanced portfolios (Green and Hollifield, 1992). Furthermore, this problem involves considering the degree of uncertainty about the expected returns, which has no place in MV analysis.
Black and Litterman rule out some of the solutions provided to these problems, such as restricting the weights in the portfolio or including transaction costs. Instead, they propose incorporating two key ingredients into the analysis. On the one hand, taking the values of expected returns, which are inputs to the Markowitz model (see Chapter 3), from an equilibrium model, specifically, the International CAPM (ICAPM).2 According to the authors themselves, the use of these equilibrium values “significantly ameliorates the usual tendency of mean-variance models to map seemingly reasonable views into what appears to be extremely unbalanced portfolios.”
On the other hand, allowing portfolio managers to modify the equilibrium values according to their own views, specifying relative strengths for each view. In the words of Black and Litterman: “The simple idea that expected returns ought to be consistent with market equilibrium, except to the extent that the investor explicitly states otherwise, turns out to be of critical importance in making practical use of the model.”
The assumptions of the Black-Litterman model can be stated as follows:
The expected return of each asset is consistent with the equilibrium, with a given confidence level determined by the portfolio manager.
In this assumption, the term “consistent” means that the difference between equilibrium expected returns and “true” expected returns is a random variable with zero mean. The confidence level of the equilibrium expected returns is included as a parameter in the covariance matrix of the random variable (see
equation [2.7]
).
The portfolio manager includes her views on each asset, with a given confidence level about each view.
These views can be expressed in absolute or relative terms with respect to the equilibrium values of expected returns.
The errors corresponding to the equilibrium values and to the manager views follow a zero-mean Gaussian distribution and are independent.
This is a simplifying assumption that will allow us to obtain the distribution of the estimator of expected returns.
In order to formalize the previous assumptions, we will consider, as usual, that the equilibrium model that generates the expected returns is the CAPM of Sharpe (1964), Lintner (1965) and Mossin (1966) (see Chapter 3). Therefore, the equilibrium expected return of the -th asset, , is given by (see equation [2.5])
where is the return of the riskless asset, is the beta of the -th asset, , and , are, respectively, the variance and the expected return of the market portfolio.3
From equation (2.5) we have
where is the equilibrium expected excess return of the -th asset, and is the weight of the -th asset in the market portfolio. This expression can be written in matrix form as
where , and is the market portfolio.
According to assumptions i) and iii), we can write
where is the vector of true expected excess returns, and is the parameter expressing the confidence on the equilibrium estimates. Values of close to zero would imply a high confidence in the estimates.
On the other hand, assumptions ii) and iii) lead to
where is a -vector representing the manager views, is a matrix expressing the modifications of equilibrium values, and a invertible matrix accounting for the manager’s confidence in her own views on the assets. The matrix is usually taken as a diagonal matrix whose elements represent the variances of the errors in the views for each asset. Values close to zero would correspond to high confidence in the views.
For example, in a portfolio with three assets (), if the manager has two views (one absolute and the other relative):
The first asset will have a return of 1.2%,
The second asset will outperform the third one by 3%,
and she assigns a standard deviation of 0.5% to the first view and a 1.7% to the second view, we will have
The main result of the model is an estimator of the true expected excess return, and it is presented in the following theorem.
Theorem 2.1
The best linear unbiased estimator (BLUE) of the expected excess return in the Black-Litterman model is given by
and satisfies
Proof
Stacking equations (2.7) and (2.8), we have the linear model
where , and , with the identity matrix. Equation (2.11) is a general linear regression model. As is known, by the Gauss-Markov Theorem (Kariya and Kurata, 2004; Theorem 2.1), the estimator
is the BLUE estimator of , with
Replacing the expressions of , and in (2.12) and (2.13), we have
and
Finally, by assumption iii), has a Gaussian distribution, and (2.10) holds.
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Once the theorem is stated and proved, some remarks are worth making.
Remark 2.2
The existence of the Black-Litterman estimatorof equation (2.9) is guaranteed by the Gauss-Markov Theorem, as long as the matrixhas full rank. Moreover, this theorem does not require any distributional assumption, thus equation (2.9) is valid even in the non-Gaussian case.
Remark 2.3
Theorem 2.1is not valid for the case in which, i.e., when the manager is 100% confident in the equilibrium values. In this case, from equation (2.7) we have, , and the Black-Litterman approach is not relevant. Another situation in whichTheorem2.1 does not hold is when the manager has views different from the equilibrium estimates () with a100% confidence (). Then, does not exist and we cannot apply equation (2.9).
The following result proves the consistency of Theorem 2.1 with the CAPM when the manager does not incorporate any views.
Corollary 2.1
If the manager has no views modifying equilibrium estimates ( ), then, and she will end up holding the market portfolio.
Proof
The equality is immediate from equation (2.9). On the other hand, the minimum variance portfolio is given by (see equation [3.18]) , with a Lagrange multiplier. Replacing equation (2.6) in this expression, we get
As and are portfolios, we have
and then, .
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Corollary 2.2
The Black-Litterman estimator of equation (2.9) can be written as
where, and the weightsandare matrices given by
that satisfy, whereis the identity matrix of order.
Proof
It suffices to consider that and replace by in equation (2.9).
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When the manager is 100% confident in her views, , and from equation (2.8) we have and . Then, is the estimate of the expected returns in that case. Therefore, Corollary 2.2 has the interpretation that the Black-Litterman estimator is a weighted linear combination of equilibrium and investor estimates.
According to Fabozzi et al. (2007), the most important contribution of the Black-Litterman model is that it adjusts the entire vector of equilibrium expected excess returns with the manager’s views. Due to the correlation between returns, views on only a few assets imply changes in the expected excess returns of all assets. Mathematically, the contribution of the manager’s views to the estimator in equation (2.9) is determined by the term . Since is a matrix and is a -vector, the effect of the views will end up propagating to the components of the estimator, even when . Intuitively, estimation errors spread to all assets, making less sensitive to errors in asset-specific views, mitigating the problem initially exposed by Black and Litterman.
2.4 Alternative Methods: Risk Parity, Hierarchical Risk Parity and Machine Learning
2.4.1 Risk Parity
The risk parity approach was born in 1996 with the launch of the All Weather Fund by Bridgewater Associates, but the concept of Risk Parity Portfolios first appeared in Qian (2005), where they are defined as “a family of efficient beta portfolios that allocate market risk equally across asset classes, including stocks, bonds and commodities.”
The reasoning behind the risk parity approach is to realize that the traditional practice of diversification based on balancing a portfolio in terms of capital allocation can lead to a high concentration of risk in certain types of assets. To illustrate this effect, Qian (2005) presents as an example, a 60/40 portfolio with 60% stocks and 40% bonds, apparently balanced, in which stocks contribute 93% of risk, while bonds only account for the remaining 7%.
Based on these considerations, if we want to limit the impact of large losses coming from one of the components of the portfolio, we need the expected contribution to risk being (approximately) the same for all asset types in the portfolio, which leads to the risk parity approach.
To formalize these ideas, we define the risk contribution of the -th asset in the portfolio, , as
where is the volatility of the portfolio, used as a measure of its risk. The following result states that the risk contribution is well-defined.
Proposition 2.3
The risk of the portfolio is equal to the sum of the risk contributions of its components:
Proof
It is not difficult to show that for any , and then is a homogeneous function of degree one. By Euler’s Theorem, we obtain
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As we have seen, the original risk parity approach considered the equalization of the risk contributions of all assets. As this concept has evolved to include different extensions, we will give the portfolios obtained under this approach the name of Equal Risk Contributions (ERC) portfolios. Thus, following Maillard et al. (2010), we define an ERC portfolio as the portfolio that satisfies
As the following result states, it is possible to find expressions for the weights, , of ERC portfolios in some specific cases.
Proposition 2.4
a) If the correlations between assets are equal, then
b) If the volatilities of all assets are equal, then
c) In the general case of unrestricted volatilities and correlations
whereis the beta of the-th asset (see Chapter 3).
Proof
a) Using equation (2.14) and that for , we have
and therefore, the condition for ERC portfolios is satisfied if . From this expression we get
from where we obtain equation (2.17).
b) Following a reasoning similar to that of the case a), we have
and then, implies
Imposing the condition , we find equation (2.18).
c) Considering that , we have . Replacing this result in equation (2.14), we get . Following a procedure analogous to the one used previously, we obtain
Applying that for ERC portfolios , , we find that , from where the last equality in (2.19) follows.
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It is important to note that of the three expression that appear in Proposition 2.4, only equation (2.17) can be considered as an explicit solution for the weights. Expressions in equations (2.18) and (2.19) provide an intuitive interpretation of the solution, but do not allow it to be obtained explicitly, since the right-hand side also depends on the weights. Thus, in the general case without restrictions on correlations and volatilities, it is necessary to resort to numerical algorithms.
Maillard et al. (2010) present three algorithms for ERC portfolios. The first one is the sequential quadratic programming (SQP) algorithm given by
This first algorithm is the one preferred by the authors, since it is the easiest to solve as it does not incorporate nonlinear inequality constraints. Nevertheless, they are cases in which numerical optimization is tricky (see Maillard et al. [2010] for details and solutions).
The second algorithm is
with an arbitrary constant, and the ERC portfolio is given by . This formulation has the advantage of having a unique solution for any value of , with (Maillard et al., 2010; Appendix A.2).
The third algorithm is a modification of the second one, given by
As the constant in the second algorithm, the constant can be interpreted as the minimum level of diversification in order to obtain an ERC portfolio. It is not difficult to verify that an ERC portfolio obtained solving equation (2.20) with a value can be obtained from equation (2.21) with the value
In the extreme case with , the first constraint disappears and the solution is the global minimum variance (GMV) portfolio, that is, the portfolio with the smallest possible variance for any given expected return (see Chapter 3). Another extreme case appears when , which corresponds to the equally weighted (EW) portfolio.4 This last algorithm allows us to show that the risk of ERC portfolios is placed between that of GMV and EW portfolios.
Proposition 2.5
The volatility, , of an ERC portfolio satisfies
whereandare, respectively, the volatilities of the GMV and the EW portfolios.
Proof
Taking logarithms in the arithmetic mean-geometric mean inequality
we have
and using , we obtain
Thus, the maximum value of in the program in equation (2.21) is , and therefore, .
On the other hand, , since the first constraint in equation (2.21) is less restrictive with than with . Therefore, the solution satisfies
for every . We know that for any value of , the solution of program (2.20) determines an ERC portfolio with . Replacing this value in the inequality (2.23) we obtain
where in the last equality we have used that the solution of (2.20) satisfies . Using equation (2.25) in (2.22), we have that , and then . Thus, we can write (2.24) as
which concludes the proof.
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The following result states that ERC portfolios are MV efficient if the Sharpe ratios of the assets are equal and the correlations between them also coincide.
Theorem 2.2
Consider the expected excess return of the-th asset, , whereis the return of the riskless asset, and assume thatforandfor, with. Then, the ERC portfolio is a MV-efficient portfolio.
Proof
MV-efficient portfolios are given by (see Chapter 3, equations [3.14]–[3.15])
where and . Premultiplying by we obtain
On the other hand, by equation (2.14), we can write the vector of risk contributions, , as
where is the matrix with and for . Replacing equations (2.26) in (2.27), we get
and thus, the individual risk contribution of a MV-efficient portfolio is . Applying the ERC condition , we obtain . By assumption, we have , , and then . Using we arrive at
By a) of Proposition 2.4, we know that, with equal pairwise correlations, expression in equation (2.28) is an ERC portfolio, and the proof is complete.
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Kaya and Lee (2012) have empirically studied whether the conditions for optimality of ERC portfolios are met, with negative results, at least for short horizons. However, for longer-term allocations, the portfolios are close to optimal.
Among the virtues of the risk parity approach mentioned in the literature, we can highlight the following:
Risk Parity Portfolios provide better downside protection (Qian, 2005, 2006)
. This is because expected percentage contributions to loss bear close relationship to the percentage contributions to risk.
It is not necessary to formulate expected return assumptions (Chaves et al., 2011)
. The only input is the covariance matrix, which can be estimated more accurately than expected returns (Merton, 1980).
Risk Parity Portfolios show a higher Sharpe ratio than competing approaches such as GMV or MV (Chaves et al., 2011)
. Moreover, the Sharpe ratios of Risk Parity Portfolios are less volatile over time.
Like any other approach, risk parity also has deficiencies, among them we can point out the following:
The expected returns of Risk Parity Portfolios may be too low to meet an investor’s desired return (Hurst et al., 2010; Qian, 2011)
. To avoid this problem, managers usually resort to leverage to raise the expected returns.
Risk Parity Portfolios are sensitive to the inclusion decision for assets (Chaves et al., 2011)
. The methodology does not provide information about how many asset classes and what asset classes to include.
