6. The CAPM: beta as the slope of a regression

Level 4: Correlation, regression and the myth of causality

Look up the beta of the same share at two different data providers and they will probably give you two different numbers.

Neither of them is wrong. Beta is not a physical property of the company: it is the slope of a line, and it depends on how you draw it.

Beta is a slope

In the market theory course you met beta intuitively, as a measure of how much a share amplifies or dampens the market's movements.

Now you can formalize it precisely. A share's beta is, literally, the slope of a simple linear regression in which:

• The dependent variable is the return of the share.
• The independent variable is the return of the market.

It is exactly the same procedure as in module 3, applied with the market as the sole factor.

The CAPM formula

The capital asset pricing model, known as the CAPM, uses that beta to build a formula for expected return. In words:

expected return of the asset = risk-free rate + beta multiplied by (expected market return minus the risk-free rate).

The term in brackets is called the market risk premium, or equity risk premium. It is the additional compensation that the market as a whole offers over and above the risk-free asset.

How to read the value of beta

Interpreting the value of beta is straightforward:

• A beta of 1 means the share tends to move, on average, one for one with the market.
• A beta of 1.5, typical of cyclical or growth sectors, amplifies market moves by an extra 50% in both directions.
• A beta of 0.6, more characteristic of defensive sectors such as utilities or consumer staples, dampens those moves.
• A negative beta, uncommon but possible in certain assets and periods, implies that the asset tends to move in the opposite direction to the market.

Alpha is the intercept

The intercept of that same regression also has a name of its own in this context: it is alpha.

It is the part of the share's return that is not explained by its exposure to the market. And it is precisely the number an active manager is chasing when he claims to beat the market.

That said, there is a condition for taking it seriously. The computed alpha may turn out to be statistically indistinguishable from zero, given the margin of error that comes with the sample size and the volatility of returns.

If that is the case, there is no solid evidence of genuine skill beyond the plain exposure to market risk that beta already captures.

A share and an index fund

The R² of this regression of a single share against the market is usually modest, typically between 20% and 40%.

That confirms something you already saw in module 4: much of the daily movement of an individual share responds to its own idiosyncratic risk —company news, results, management decisions— and not to the market as a whole.

Now apply that same regression to a highly diversified index fund, with hundreds of positions. The R² usually exceeds 90%.

The reason is that the idiosyncratic risk of each individual company tends to cancel out within the portfolio. What remains is systematic risk, tied to the market, and it explains almost all the remaining variation.

Beta moves

An important practical nuance: beta is not a fixed physical constant of each share, but an estimate. It depends on three decisions:

• The time window used.
• The frequency of the data: daily, weekly, monthly.
• The index chosen as the benchmark for the "market".

That is why it is perfectly normal for two different data providers to publish different betas for the same share, without either being wrong. They simply use different settings.

Computing beta over a rolling window, updated periodically, also reveals something else: a company's risk profile evolves over time, for example when its business model changes.

An incomplete reference point

The CAPM, with its single market factor, is a useful but incomplete simplification. The empirical evidence shows that it does not fully explain why some shares systematically return more than others.

That shortfall is exactly the gap that the multifactor models of the previous module try to close, by adding further factors to market beta.

Even so, the CAPM beta remains the industry's standard reference point for adjusting returns for risk.