# How to compute a Beta - regression

Dear all,

I asked the question to countless professors and professionals, and no one seems able to confidently prove and explain how a Beta is computed: the question is, should we use Re-Rf and Rm-Rf in our regression and the intercept will be our “Alpha”, or use Re and Rm?

I wrote my master thesis about “The quest for a less imperfect pricing model”, based on the work of Professors Fama and French, which computed its 3- 4- and 5-factor regressions using Re-Rf and Rm-Rf.

If Re-Rf and Rm-Rf shall be used, then how come the Beta can also be computed using Cov(Re,Rm)/Var(Rm) and not Cov(Re-Rf,Rm-Rf)/Var(Rm-Rf)?

Best,

Adding a constant to X or Y (or both) doesn’t change their covariance, nor their variances; i.e.,

Cov(X, Y) = Cov(X + c, Y) = Cov(X, Y + k) = Cov(X + c, Y + k)

Var(X) = Var(X + c)

Var(Y) = Var(Y + k)

You are totally right, Magician, from a mathematical perspective; however, Rf is NOT a constant: over long time periods, Rf can actually change significantly.

Does it still mean that the two methods (subtracting or not subtracting Rf) should give the same result?

I believe it should not: Let assume that Re is constantly equal to Rf over our regression period, our Beta will be 0 since Re-Rf will be always 0, which is not necessary the case when NOT deducting Rf.

Re Rm Rf Re-Rf Rm-Rf 0.1 0.2 0.1 0 0.1 0.5 0.3 0.2 0.3 0.1 -0.2 -0.3 0 -0.2 -0.3 -0.6 0 0.1 -0.7 -0.1 0 0.1 0.2 -0.2 -0.1 0.2 0.4 0 0.2 0.4 0.1 0 0.1 0 -0.1 Re & Rm Re-Rf & Rm-Rf COVAR.S : 0.050 0.043 VAR.S : 0.053 0.050 BETA : 0.938 0.867

You’re correct: if the risk-free rate changes over the period, then there’s no reason that they should give the same result.

1. should we use covariance.p & var.p or covariance.s &var.s on Excel? In other words, are covariance and variance computed assuming population or sample? I believe sample makes more sense, but some websites and professors’ slides use population.

2. if Rf changes over time and the 2 betas are thus different, which one should be used?

1. I’d use sample variance and covariance, but know that you’ll get the same beta either way.

2. I’d use Re vs. Rm; beta is supposed to measure the influence of changes in the market return on changes in the stock’s return. If you use Re − Rf vs. Rm − Rf, then you’re introducing the influence of changes in the risk-free rate on changes in Re, which isn’t what beta’s supposed to measure.

Perfect! You explain very well why one should use Re vs. Rm with the CAPM.

Professor Damodaran also uses this method and states that: Intercept - (Riskfree Rate / n) x (1- Beta) = Jensen’s Alpha (http://people.stern.nyu.edu/adamodar/podcasts/cfspr17/session7.pdf)

When using a multifactor regression, however, like in the Fama and French models (that add SMB and HML as premiums), would you use, in this case, Re − Rf vs. Rm − Rf like they did in their work?

So . . . you’re asking me if I’m going to tell a Nobel laureate that he got it wrong?

Great! So, just to summarize:

Both can be used technically in the CAPM, but Re vs. Rm makes more sense since we want to study the impact of absolute market movement on our stock.

In multifactor models, same thing, but the fact that Rm − Rf is a factor like SMB and HML explains why Fama and French probably used Rm − Rf and Re − Rf

PS: It’s thank to you that I passed all three exams in 3 years with amazing grades, you are better at explaining and finance than most professors I had; finance is more of an art than a pure science, it is therefore you right to tell a Nobel laureate that an alternative way to compute Beta may be more appropriate

Even if finance were more science than art, it would still be a right to tell an authority there may exist reasonable alternatives. The most frequently suffered logical fallacy, in my opinion, is that of an appeal to authority; each day we fall prey to the “Oh, [authority figure] said such and such-- must be true and can’t be questioned.” Nobel laureates are people too and there exists a great deal of politics (especially nowadays).

Anyhow, don’t use Excel-- get R. Use R commander if you are uncomfortable with programming (although it would be hugely beneficial to learn some statistical programming). Excel is not meant for data analysis and many papers were written exposing why this is dangerous and makes your work more prone to substantial errors and risk of irreproducible results.

I have to say, I thought CAPM was the Ri = rf + bi(Rm-rf)

which really shows a company return, Ri, as a function of the market risk premium. So a CAPM beta is different from regressing Ri on Rm which is the “beta” for seeing how changes in Ri relate to the market return.

Fama and French uses differences because it asked questions about “premia” or “excesses” for being in any of those categories, which makes sense in the CAPM context.

Again, the “classic” beta of movement in Ri relative to changes in Rm, would be better calculated by regressing Ri on Rm.

Keep in mind that any of these calculations implicitly assume that nothing else impacts Ri; if Rm is not the only explanatory variable in reality, your estimator of beta will be statistically biased. Good estimation should be a higher goal than sticking with old but useful theory.