Regression, Beta & Alpha

The last lesson said beta measures an asset's systematic risk. This lesson shows where beta actually comes from: linear regression, one of the most used tools in quantitative finance. Regression is how the abstract idea of a relationship between two variables becomes a concrete, measured number. Along the way you get beta and alpha, two of the key quantities in investing.

What regression does

Linear regression fits a straight line through a cloud of data points to describe how one variable changes as another changes. You have a dependent variable, the thing you are trying to explain, and an independent variable, the thing you use to explain it. The fitted line has two parts. The slope tells you how much the dependent variable changes for each one-unit change in the independent variable, and it is the heart of the relationship. The intercept is where the line crosses the vertical axis, the value when the independent variable is zero. The line is chosen to be as close as possible to all the points, by making the total squared distance from the points to the line as small as it can be.

Regression gives us beta

Now the finance payoff. The most famous application is the market model: you regress a stock's returns on the overall market's returns. Here the stock's return is the dependent variable and the market's return is the independent variable. The slope of that regression line is the stock's beta. Beta measures how sensitive the stock is to movements in the whole market, which is exactly the systematic risk from the last lesson. This is the concrete origin of beta. It is simply the slope you get when you fit a line relating a stock's returns to the market's returns.

Interpreting beta

A beta of 1.5 means that when the market rises 1 percent, the stock has tended to rise about 1.5 percent, and when the market falls 1 percent, the stock has tended to fall about 1.5 percent. High-beta stocks amplify market moves, and low-beta stocks dampen them. Because beta captures only the systematic, market-related risk, it is the risk measure at the center of the Capital Asset Pricing Model, which says expected return should rise with beta.

Alpha: return the market does not explain

The intercept of that same regression is closely related to alpha. Alpha is the part of a stock's return that is not explained by the market's movements. If beta captures how much of a stock's return comes from riding the market, alpha is what is left over, often interpreted as the value a manager or strategy adds beyond simply taking on market risk. A positive alpha means the investment did better than its beta and the market would predict, and a negative alpha means it did worse. Generating persistent, genuine positive alpha is the goal of active investing, and it is famously hard, because most apparent alpha turns out to be either hidden risk, luck, or the kind of overfitting this unit keeps warning about.

Beta is the return you earned just for showing up to the market. Alpha is what you can claim was skill, and most of it, on close inspection, was not.

The cautions

• Regression measures association, not causation, the same principle that applied to correlation. A fitted relationship does not prove one variable drives the other.
• Regression assumes a linear relationship, so a curved true relationship will be described poorly by a straight line.
• Beta is estimated from historical data and can change over time, so a past beta is not a guaranteed guide to the future.
• Apparent alpha is often illusory, arising from luck, unmeasured risks, or overfitting rather than genuine skill.

You can now measure an asset's systematic risk with beta and its excess return with alpha. The final building block combines return and risk into a single score for judging performance: the Sharpe ratio.