Introduction
This book originates from notes I wrote for two university courses. The first is ORIE5256 - Topics in Risk Management and Portfolio Construction, a course offered in the program for M.S. in Financial Engineering at Cornell University. The second is MATH-GA 2708.001 - Algorithmic Trading & Quantitative Strategies, offered in the Mathematics Department at New York University. When I set out to write this book, my objective was to write the quantitative introduction I had wanted to read at the beginning of my journey in finance. Given the scope and goals of quantitative investing, it is only possible to cover a small fraction of it in a course, or even in a book. To address this problem, I made three choices.
First and most important, I aim for synthesis. A book is, first of all, a knowledge filter. In the preface to his classic (Kelley, 1955), Kelley wrote that he wanted to title his book "What Every Young Analyst Should Know"; that book was barely three hundred pages long. It still feels fresh and necessary today. In order to keep my book of manageable length, my working principle has been to focus on real-world problems and then use the simplest techniques that allow me to address the problem at hand. A recurrent theme in the book is that almost everything in it is either linear or quadratic. In the process of writing, I have ruthlessly eliminated topics of secondary importance, material that was too hard for the payoff that it gave the readers, and also topics or ideas that are not sufficiently well-formed, or too experimental. Even if you choose not to read my book, I implore you to internalize the following lesson, learned by practitioners through sweat and tears: theory is cheap. There are thousands and thousands of theory papers, in love with technical virtuosity but oblivious of reality. Do not fall into temptation; by applications be driven.1
Second, I consider risk management and portfolio management as intrinsically connected. Asset return modeling, volatility estimation, portfolio optimization, ex-ante and ex-post performance analytics are all linked. For example, hedging belongs to risk and portfolio optimization, and analysis of performance feeds back into portfolio construction. I have avoided redundancy as much as possible. Sections often refer to earlier ones or are linked to later ones. As I was revising my book draft, whenever I found I had introduced some topic (often because I was infatuated with it) and then never used it, I exiled it to a long file made out of "rejected sections" that lives in my laptop. That is the sad part. The happy part: it's surprising how tall a tree can grow with a bit of pruning. Out of metaphor, there is a lot of material in this book, and it gets challenging at times.
Third, I occasionally integrate some standard financial results approaches with tools from the field of statistical learning. The former is applied in fundamental factor modeling, portfolio optimization, and performance attribution. I use the latter for the estimation of statistical models and backtesting. My hope is that the integration of these different approaches is seamless.
The questions that I address in this book are:
- How do I model returns in a way that allows me to generate risk and return forecasts?
- What are excess returns?
- How do I model multivariate returns?
- How do I describe and forecast risk?
- How do I test risk forecasts and return forecasts?
- How do I backtest alphas?
- How do I monetize these signals?
- How do I optimize a portfolio?
- What is the impact of risk and alpha errors on performance?
- How do I account for transaction costs in portfolio management?
- How do I hedge a portfolio?
- How do I improve?
- How do I allocate risk over time?
- How do I distinguish skill from luck?
The style of the book is also, I hope, a bit different. I have kept in mind the six values that Italo Calvino (Calvino, 1999) hoped to preserve in the current millennium: Lightness, Quickness, Exactitude, Visibility, Multiplicity, and Consistency. My aversion to advanced mathematics notwithstanding, I must warn the reader that the book is not easy. During my lectures, I have induced more than one student into a comatose stupor. Afterwards, a few students left finance altogether and successfully pursued careers in entertainment. Another student keeps sending me postcards from Ibiza. A handful have become portfolio managers at hedge funds and risk managers. Yet, it is the easiest book I could write for the task at hand, and it is written in the friendliest style I am capable of. Also, I would be lying-and conveying the wrong message-if I claimed that the problems I present are now settled, and that the book is the last word on the subject. On the contrary, you and I are in this book together, and together we shall keep a beginner's mind (Suzuki, 1970): a spirit of openness and curiosity, even when facing advanced topics. I will point out the limits of my theories and the open problems for you to work on. If you are old enough to have lived in the seventies and liked punk music, you may remember a cyclostyled zine (Figure 1). On its second page it showed three open chords; below them, a command: "NOW FORM A BAND." May this book be your field guide to being a punk quantitative researcher. It will be a life well lived.
Figure 1 Punk fanzine Sideburn #1, page 2 (1977).
Source: Flickr.com/Dunk.
Prerequisites
The book should be accessible to a beginning graduate or advanced undergraduate student in Physics, Mathematics, Statistics, or Engineering. This means having a working relationship, and if possible a romantic one, with advanced linear algebra, probability theory, and statistics. Even more important is to have a deep interest in quantitative modeling of real-life phenomena. Many readers will be either members of a systematic trading team, or work as quantitative researchers in the central team of a hedge fund or a quantitative asset manager.
The book's material is organized in such a way that you do not need to go through mathematical proofs. You can rely only on informal statements of mathematical results in the main body of the chapters, and that will suffice to understand the main points. The appendices at the end of the chapters contain more rigorous statements, proofs, and background material. If you plan on actively doing research, you should study them, eventually.
Even if you read only the main body, you should be used to thinking in mathematical models. The Book of Nature is written in a mathematical language.2 Be comfortable with:
- Working with linear algebra, at least at the level of Strang (2019) and Trefethen and Bau (1997).
- Some applied probability, at the level of Ross (2023). Exposure to some simple control theory and state-space models helps. You can come to this from econometrics (Harvey, 1990; Shumway and Stoffer, 2011), control theory (Simon2006), or statistics (Hyndman et al., 2008).
- Some optimization modeling is a plus. The first few chapters of Boyd and Vandenberghe (2004) would be ideal. However, I will cover the basic theory in an appendix.
Organization
Like Caesar's Gaul, the book is broadly divided into three parts. The first part focuses on returns modeling. I cover the basics of GARCH early on because they are needed for factor modeling, and then I cover factor models because they are necessary for everything. I have separate chapters for fundamental and statistical models. These topics are covered in depth, and both the treatment and some of the modeling approaches are novel. Finally, I cover data snooping/backtesting as a separate chapter, since it is a central element of the investment process.
The second part is devoted to portfolio construction and performance analysis, both ex ante and ex post. The focus is on mean-variance optimization (MVO). I emphasize the geometric intuition behind much of mean-variance optimization. Rotations, projections, and angles are prevalent throughout. This allows for a synthetic, elegant characterization of performance and for concise proofs. The statistics of the Sharpe Ratio are covered in some detail. The decomposition of payoffs into timing components, factor and idiosyncratic Profit and Loss (PnL), and stock selection versus sizing of positions is rigorously demonstrated. Model error plays an important role in this part. If an optimization problem is Othello, then model error must be Iago: it can drive the optimization insane. Unlike in Shakespeare's tragedies, we can try to rewrite the endings and turn them into comedies.
The third part is the shortest. It contains results about intertemporal volatility allocation and performance attribution. These are essential components of the investment process and belong in a book with the word "Elements" in its title.
Each chapter is organized like an onion. The first sections convey the essential ideas using simple quantitative methods and are aimed at a broad audience. Sections marked with a star "" are more advanced and can be skipped on a first reading. Proofs of new results or basic technical material are relegated to the appendices at the end of the chapters. The goal is not to disrupt the flow of learning. As mentioned at the beginning of this preface, the content of this book was taught first and written later. It is meant to be read aloud and discussed, but it...
