Advanced Portfolio Management. Giuseppe A. Paleologo

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simple heuristics to advanced optimizations.

       When will you need this: Always. This will become your second nature. You will break the ice at cocktail parties mentioning how much risk decomposition helped you in your life.

3.1 Introduction

On July 3, 1884, the Customer's Afternoon Letter (owned by Dow Jones & Co.) began publishing the first stock index: a simple price average of nine transportation companies and two industrial ones. In 1886 it published the first Dow Jones Industrial Average. In 1889, the newspaper became The Wall Street Journal, and over time more indices were created. Indices provided a benchmark against which to compare one's investment; and they are a summary of the overall behavior of the market or of a specific sector. A typical benchmarking exercise: if we hold a stock, on any given day we first look at the overall market return, as provided by the index, and then we compute the out- or underperformance of the stock compared to the market. When we look at indices as market summaries, we implicitly know that they describe most, or at least some, of the stock returns for that market segment. In a very real way, having an index gives us a way to describe performance and variation of stock returns. Factor models capture these two intuitive facts, make it rigorous, and extend them in many directions.

A first extension aims to offer more flexibility in the relationship between stock and benchmark. For example, a cyclical stock in the financial sector like Synchrony Financial (ticker: SYF) moves more in sync (lame pun) with the market than, say, Walmart (ticker: WMT), a large, stable, defensive stock. Figure 3.1 bears this out. We take the daily returns of Synchrony and Wal-Mart and regress them against the daily returns of SPY.¹ The regression coefficient is denoted “beta”. If the market returns an incremental 1%, the stock returns an incremental (beta) * (1%), everything else being equal. It is a measure of market sensitivity that differs from stock to stock. Figure 3.1 shows the regression of SYF daily returns against SP500 futures returns, for the period January 2, 2018, to December 31, 2019. Let us go with the assumption that we can estimate the true beta of a stock to the market; i.e., the estimation error of the beta doesn't really matter. Then a simple decomposition of market return + stock-specific return gives us a great deal of information. On the benchmarking side, we now know what fraction of the return is attributable to the market. It may seem that SYF is outperforming the market in a bull market and Wal-Mart is underperforming. However, after decomposing returns, it may be the other way around: SYF is just a leveraged bet on the market, and after we remove the market contribution, SYF has underperformed, and WMT has outperformed. Another benefit from the linear relationship between market and stock returns is that it establishes a relationship among all stock returns. The market is the common link. For the overwhelming majority of stocks, the beta to the market is positive, but it can take values in a wide interval; several stocks exhibit betas higher than two. This relationship has implications for expected returns and risk as well. We will delve deeper into both later in this chapter. But before we proceed, we introduce a new term, “alpha”. In your daily job, it's alpha that will pay your salary. Beta, on the other hand, can get you fired. This explains why so many portfolio managers have the symbol

tattooed on their bodies, while no one ever thought of getting a tattoo with the symbol . Not even risk managers.

Figure 3.1 Linear regression of Synchrony's (SYF) and Wal-Mart's (WMT) daily returns against the SPY daily returns, using daily returns for the years 2018 and 2019.

3.2 Alpha and Beta

Consider the regression line for SYF again. The complete formula for a linear regression includes an intercept and an error term, or residual. We write this explicitly. For a given stock,

(3.1)

Visually, this relationship is shown in Figure 3.2. We already introduced the term . This market component of the stock return is also called the systematic return of a stock. The first term on the left is the intercept ; it is a constant. When the market return is zero, the daily stock return is in expectation equal to alpha. No one observes expected returns, however. Even in the absence of market returns, the realized return of the stock would be . The last term is the “noise” around the stock return; it is also called idiosyncratic return of the stock. The terms specific and residual are also common, and we use all of them interchangeably. We would like to believe that this nuisance term is specific to the company: every commonality among the stocks comes the beta and the market return. The simple model of Equation (3.1), with a single systematic source of return for all the stocks, is called a single-factor model.

Figure 3.2 Flowchart illustrating the relationships between market, idio and asset returns, mediated by betas and offset by alphas. Market return times beta is added to alpha and idiosyncratic return to yield the total return.

      The term is only dependent on the company and nothing else. If that is the case, then the nuisance of many stocks diversifies away, a phenomenon that we will revisit many times in the following chapters. Alpha is the expected value of idiosyncratic return, and is the noise masking it. So, if we take the expectation of a stock return, what do we get? Alpha and Beta are constant, epsilon has expectation equal to zero, and the market return has a non-zero return.

      There

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