Is the CAPM beta equivalent to the coefficient estimate of an OLS regression?

Question

The $\beta_i$ of an asset or portfolio is defined as its covariance with the market (which itself therefore has a beta of $\beta_m = 1$). The CAPM looks a lot like a simple linear regression model. Is its $\beta$ equivalent to the coefficient estimate obtained from an ordinary least squares regression on the CAPM equation?

Answer 1

They are actually exactly the same thing. CAPM say that expected risk premia are “explained" by the risk premium on the mean variance efficient (MVE) portfolio $$ R^i_{t+1} - R^f = \delta (R^{MVE}_{t+1}-R^f) + \varepsilon_{t+1} $$ De facto, you are saying that the systematic risk is just the projection of risk premia on MVE risk premium, and OLS are exactly the same thing as a linear projection. Indeed it must be the case that $$\delta= \frac{Cov(R^{MVE}_{t+1},R^i_{t+1})}{Var(R^{MVE}_{t+1})}=\beta^{OLS}$$

And by the way, when investors price by means of discounted expected payoffs, i.e.when $$ p_t^i=\mathbb{E}_t \frac{d_{t+1}}{1+R^i_{t+1}} $$ expected returns always admit a beta representation, i.e. we can always rewrite $$ \mathbb{E}_t R^i_{t+1} =\alpha_0 + \beta\lambda $$

Answer 2

No. The CAPM is an equilibrium model. It describes relationships between expectations. With OLS you typically estimate a distribution for realized returns in the future (or sometimes even only in the past...).

$R^i_{t+1} - R^f = \delta (R^{MVE}_{t+1}-R^f) + \varepsilon_{t+1}$

is not an equation from the CAPM.

The CAPM SML is: $r+E(r_i - r) = r+\beta_i E(r_M-r)$

There is no index $t$, and no random term, as the CAPM describes only expectations, and only for one period.

an equation like this:

$\mathbb{E}_t R^i_{t+1} =\alpha_0 + \beta\lambda$

can always be estimated for any pair of dependent and independent variable and any data set, no matter how investors price anything.

It's sad that so many use historical data only to estimate betas - as it reads on any retail mutual fund fact sheet, that past performance is no guarantee for future performance... the same holds for distributions... of course.

But then, let's say you're a publicly traded corporation, how you estimate beta to get a discount rate for a future risky cash flow, should depend on how you believe the market estimates beta.

In principle, though, any approach you find convincing, could be justified, as Sharpe pointed out:

“Much confusion has arisen concerning the relationship between the equilibrium results of the CAPM and the underlying relationships among security returns. As can be seen, the CAPM makes no assumptions about the “return generating process”. Hence, its results are completely consistent with any such process.” William F. Sharpe, Capital Asset Prices with and without Negative Holdings, Nobel lecture, Dec. 7th 1990, Economic Sciences 1990, p. 320

Answer 3

I'm not sure I'd use the word "equivalent" but you have the right idea. Instead, I would say OLS regression is a very common way of estimating CAPM beta.

CAPM is a linear model to calculate the an appropriate return of an asset given its non-diversifiable risk. OLS is really a method for solving certain types of linear models. So OLS can be used on the CAPM model but it is used on many, many other models as well.

Also, there are other methods used for estimating CAPM betas other than OLS. Some may even have better predictive power, but OLS is by far the most common.