# French and Fama Three Factor Model - What is the correct formula?

I hope you can help me with the following question. What is the correct way to write the formula for the French and Fama Three Factor Model. I have currently found three versions of this formula, these being: The question I have, is there are difference between these or are all the same? Primarily I am interested in the explanation for the position of the Alpha and the risk free rate, I also notice that the error term is sometimes noted sometimes not. Is it a mere question of preference or are there deeper implications as to how you write it? I thank you for any feedback!

• The one with the error term is the theoretical form, the one with the alpha constant is what gets estimated by regression. The first one uses a strange inconsistent notation with some b's and some beta's. But really, there is not much difference here at all. – noob2 Oct 12 '15 at 15:19
• I would suggest to stick with the last one. That's cleaner and the one usually found in the literature. – CharlesM Oct 12 '15 at 18:13

There are issues with all three of these. @noob2 has pointed out the inconsistency in the first one. It also uses no subscripts. The second one is probably least wrong. However, there is no index to the error term and no intercept. The third one uses too many indices. It has an index on the intercept and indices on the slopes. Granted, it is possible to estimate a time-varying version of Fama-French that would require indices here. However, that is not what normally people who estimate Fama-French models are doing.

One other quibble is that they each use $i$ as an index. Typically $i$ is used for cross-sectional indices and $t$ is used for time indices. It's not really a significant issue, but if you start mixing in panel models, then it becomes important.

So here's how I would write it: $$r_{t}-r_{t,f}=\alpha+\beta_{mkt}\left(r_{t,mkt}-r_{t,f}\right)+\beta_{SMB}r_{t,SMB}+\beta_{HML}r_{t,HML} + \varepsilon_{t}$$ I like making clear that they are all different types of returns. I have a tendency to not work with the excess returns, but it is common in academic finance.

• Thank you for the elaboration John, may I ask why you decided not to use the error term at the end of the suggested formula? – Noir Oct 13 '15 at 8:22
• Because I forgot! Now fixed. – John Oct 13 '15 at 13:46

It's just a different notation. If you need factors already constructed you can get them on the web, for Example on French's site.

• Hi user42469, welcome to Quant.SE! Thanks for the answer, I added the link to French's site and made it a bit more concise. – Bob Jansen Oct 12 '15 at 18:26

I have been thinking about this issue a lot recently after reading Cochrane's excellent textbook. After some struggles, I think the key to understand all this is to pay attention to the subscripts, because they are important. Here is my take on your question.

I will use the form as so: $$r^e_{i,t} = \alpha_i+\beta_{i,MKT}(r_{MKT,t}-r_{f,t})+\beta_{i,SMB} r_{SMB,t}+\beta_{i,HML}r_{HML,t}+\epsilon_{i,t}$$

This is population regression. You can think of projecting excess return on to the space of the factors and a constant. If you read the theory part seriously, you can see this is how we define the $\beta$s in the expected return-beta form of any linear factor model.To think about this relationship, you can fix i (or the vectorized form $r^e_t=[r^e_1,\cdots,r^e_N]'$), then we can think of the regression above as a time-series regression as

$$r^e_{t} = \alpha+\beta_{MKT}(r_{MKT,t}-r_{f,t})+\beta_{SMB} r_{SMB,t}+\beta_{HML}r_{HML,t}+\epsilon_{t}$$

This is the usual linear regression you see in any textbook. By construction, the $E[\epsilon_t|factors]=0$.

The more interesting thing is to think about what we are doing when we look at the intercept to see if the model does a good job pricing our left hand side portfolio.

First of all, when you talk about a factor model, the main implication of it is actually on the expected return not on return. $$E[r^e_i] = \beta_i \lambda$$. where $\lambda$ is the risk premium associated with your factor. There are two things we need to pay attention to.

• The subscript doesn't involve t, because we are dealing with an unconditional model which we think of the realization of returns and beta at each time as drawn from a distribution of $r^e_i$ and $\beta_i$ for each firm.
• You may wonder where is the factor in this model. The factor only enters into the model through its risk premium. It's only in the special case when your factors are excess returns, the risk premium $\lambda=E[f]$.

Now with these concepts clear up, we can proceed to understand Fama French 3-factor model.So what they propose is that the 3 factors are three long-short portfolio returns, namely MKT, SMB and HML. Given what I have talked about above, the model (APT, ICAPM) essentially says: $$E[r^e_i] = \beta_{i,MKT}E[r_{MKT}-r_f]+\beta_{i,SMB} E[r_{SMB}]+\beta_{i,HML} E[r_{HML}]$$

Remember, these relationship are predicted by the model to hold in the population. What we observe is only the historical realization of these returns, so we need to estimate this relationship.

Finally, let's relate back to the time-series regression of Fama French at the beginning. By taking the unconditional expectation of both sides of the first equation and compared to the theoretical model above. You can see the implication for those MKT, SMB and HML being factors that perfectly price assets is that $\alpha_i =0$ $\forall i$ jointly. To test this, we can use the GRS(1989) test statistics.