## 1.3 Outline of the book

This book is organized into two main parts, comprising a total of 16 chapters, along with two appendices at the end. The content of each of the chapters is outlined next.

Part I. *Financial Data*: This part focuses on financial data modeling, which is a necessary component before the portfolio design.

Chapter 2 discusses

*stylized facts*in financial data. These unique characteristics differentiate financial data from other types of data. Some of these characteristics include lack of stationarity, volatility clustering, heavy-tailed distributions, and strong asset correlation. This chapter provides a concise and visual overview of these stylized facts to help readers better understand and analyze financial data.Chapter 3 focuses on

*i.i.d. modeling*in financial data. Although the i.i.d. model is a simplistic approximation, it is still widely used in practice. However, challenges arise due to non-Gaussian distributions and noise, which are often ignored in financial literature. To address these challenges, robust and heavy-tailed estimators for the mean vector and the covariance matrix are necessary, and this chapter provides detailed explanations for these estimators. Furthermore, incorporating prior information through techniques such as shrinkage, factor modeling, and Black-Litterman fusion can significantly improve the accuracy of estimates. Due to the breadth of topics covered in this chapter, it is rather long, but it provides readers with a comprehensive understanding of i.i.d. modeling for financial data.Chapter 4 explores the application of

*time series models*to financial data to capture temporal dependencies for both mean modeling and variance modeling. While mean models provide debatable improvement over the i.i.d. approach, variance models, including GARCH-related models and stochastic volatility models, have shown to be effective in capturing the volatility of financial data (the latter showing improved results but at a higher computational cost). This chapter presents a unified modeling approach through state space modeling with special emphasis on the use of the efficient Kalman filtering, which notably allows to approximate stochastic volatily models with low computational cost.Chapter 5 focuses on

*financial graphs*and their applications in financial data analysis. While graphical modeling of financial data originated in 1999, many methods have since been proposed. Among these methods, sparse Gaussian models are suitable for providing basic insights, low-rank formulations can be used to cluster assets, and heavy-tailed models are appropriate for accounting for non-Gaussian data. Graph-based techniques can provide valuable visual and analytical tools for financial data analysis. This chapter provides an overview of cutting-edge techniques on graph modeling of financial assets, allowing readers to gain a deeper understanding of the applications and benefits of financial graphs in data analysis.

Part II. *Portfolio Optimization*: This part contains a wide range of chapters covering various portfolio formulations with corresponding algorithms and examples.

Chapter 6 provides a comprehensive introduction to

*portfolio basics*. The chapter covers fundamental topics such as portfolio notation, cumulative return calculation, transaction costs, portfolio rebalancing, practical portfolio constraints, portfolio measures of performance, simple heuristic portfolios, and basic risk-based portfolios. While the chapter covers the basics, it also includes an interesting nugget on the interpretation of the heuristic quintile portfolio, widely used by practitioners, as a formally derived robust portfolio. This chapter serves as an excellent starting point for readers new to portfolio management, providing them with the foundational knowledge necessary to understand and build portfolios.Chapter 7 delves into the topic of

*modern portfolio theory*, which is the main focus of the majority of textbooks on portfolio design. In this book, this chapter serves as a starting point for exploring a wide range of different portfolio formulations. The chapter begins with an introduction to the basic mean–variance portfolio and then moves on to the often-ignored maximum Sharpe ratio portfolio, for which several practical numerical algorithms are presented in detail (such as bisection, Dinkelbach, and Schaible transform-based methods). The Kelly portfolio and utility-based portfolios are also overviewed. The chapter concludes with a discussion of a recently proposed universal algorithm that can be utilized to solve portfolios based on any trade-off between the mean and variance. Overall, this chapter provides readers with a comprehensive understanding of modern portfolio theory and its practical applications.Chapter 8 focuses on

*portfolio backtesting*, which is essential in strategy evaluation. Many biases, such as overfitting, can invalidate backtesting results, making it a challenging task. As a consequence, published backtests should not be trusted blindly. This chapter delves into the common pitfalls and dangers of backtesting, which are often ignored in textbooks, and puts forward the approach of multiple randomized backtests to help mitigate risks. The chapter also discusses the benefits of stress testing with resampled data to complement the backtesting results. By providing readers with a comprehensive understanding of the challenges of backtesting and suggesting practical solutions to overcome them, this chapter serves as an essential guide for portfolio assessment.Chapter 9 explores

*high-order portfolios*, which introduce high-order moments in the mean–variance formulation. This idea dates back to the beginning of modern portfolio theory, but until recently, it was impractical due to difficulties in parameter estimation, excessive memory requirements, and complexity of optimization methods for a realistic number of assets. This chapter covers all the basics of high-order portfolios and introduces recent advances that make this approach practical.Chapter 10 considers

*portfolios with alternative measures of risk*. While variance is the most commonly used measure of risk in portfolio optimization, many advanced risk measures, such as downside risk, semivariance, CVaR, and drawdown, can also be incorporated. These measures can be formulated in convex form, allowing for the use of efficient algorithms. This chapter provides an overview of these sophisticated alternatives to Markowitz’s seminal mean–variance formulation.Chapter 11 presents

*risk parity portfolios*, which aim to diversify risk allocation beyond equal capital allocation. These portfolios were proposed by practitioners and rely on using granular asset risk contributions rather than overall portfolio risk. This chapter presents risk parity portfolios progressively, starting from a naive diagonal formulation and progressing to sophisticated convex and nonconvex formulations. It also covers a wide range of numerical algorithms, including newly proposed techniques.Chapter 12 overviews

*graph-based portfolios*, which utilize graphical representations of asset relationships learned from data to improve the portfolio design. Graph-based portfolios enable hierarchical clustering and novel formulations that account for asset interconnectivity, enhancing portfolio construction. This chapter provides a comprehensive overview of all existing graph-based portfolios, presenting a unified view of the different approaches.Chapter 13 covers

*index tracking portfolios*, which are designed to mimic an index under the assumption that the market is efficient and cannot be beaten. Sparse index tracking further improves this approach by using few assets, posing a sparse regression problem. This chapter provides a state-of-the-art overview of the existing methodologies and introduces new formulations for index tracking portfolios. It also includes a cutting-edge algorithm that automatically selects the right level of sparsity, making index tracking more efficient and effective.Chapter 14 gives an overview of

*robust portfolios*, which aim to address the inevitable parameter estimation errors that can lead to meaningless or catastrophic results if ignored. While optimal portfolio solutions may seem ideal in theory, practical implementation requires techniques like robust optimization and resampling methods. This chapter covers these standard techniques, providing readers with a comprehensive understanding of robust portfolios and how to optimize them.Chapter 15 explores

*pairs trading or statistical arbitrage portfolios*, which are market-neutral strategies designed to be orthogonal to the market trend. These strategies trade on the oscillations among different assets, making them a popular technique in advanced portfolio management. This chapter provides an overview of the basics of pairs trading and statistical arbitrage, as well as exploring the more sophisticated use of Kalman filtering.Chapter 16 presents the concept of

*deep learning portfolios*, which utilize deep learning techniques to analyze financial time series data and optimize portfolios. While deep learning has revolutionized fields like natural language processing and computer vision, its potential in finance remains uncertain due to challenges such as limited availability of nonstationary data and the weakness of the signal buried in noise. This chapter provides a standalone account of deep learning and the current efforts in the financial arena, acknowledging the risk of becoming quickly obsolete but still providing a good starting point.

Appendices A and B. *Preliminaries on Optimization*: This final part provides an overview of basic concepts in optimization theory (Appendix A) and a concise account of practical algorithms (Appendix B) used throughout the book.