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I am wondering about the interpretation of the loading of the SMB factor. Some papers (e.g., here) state that $\beta_{SMB}>0.5$ implies a portfolio is weighted more towards small caps. In other work (e.g., here), it says that it is popular believe that a portfolio is small caps weighted if $\beta_{SMB}>0$. I find the latter to be more intuitive: If the smb excess returns increases, and our portfolio return increases (by whatever small amount) than our portfolio is weighted more towards small caps.

Now, what of both is true? And where do different interpretations of this coefficient come from?

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  • $\begingroup$ Upvoting because I am interested in a discussion on this question. $\endgroup$
    – KaiSqDist
    Commented Apr 3 at 13:27

2 Answers 2

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Well this is actually a very simple question.

Suppose of run a Fama-French 3-factor model regression on a portfolio $i$:

$$ r_{i,t} - r_f = \alpha_i + \beta_{i,mkt} (r_{mkt} - r_f) + \beta_{i,HML}HML_t + \beta_{i,SMB} SMB_t + \epsilon_{i,t}$$

You got some coefficients $\beta$. Now suppose you want to replicate return as close as possible the return of portfolio $i$ using only the factors.

To replicate the portfolio $p$ return, you need to hold a portfolio that has the following weights:

  • Risk-free: $1-\beta_{i,mkt}$
  • Market: $\beta_{i,mkt}$
  • HML: $\beta_{i,HML}$
  • SMB: $\beta_{i,SMB}$

This is the best you can do to replicate the return of portfolio $p$. In particular your replicating portfolio will have a correlation with portfolio $p$ of $\sqrt{R^2}$ where $R^2$ is the R-square of the regression.

In other words:

  1. You have a weight of $\beta_{i,mkt}$ on the market portfolio.
  2. A weight of $\beta_{i,SMB}$ in small caps and a weight of $-\beta_{i,SMB}$ in large-caps.
  3. A weight of $\beta_{i,HML}$ in value firms and weight of $-\beta_{i,HML}$ in growth firms.

As you can see if $\beta_{i,SMB}>0$ you are long small caps and short large-caps. If $\beta_{i,SMB}<0$ you are short small caps and long large caps.

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    $\begingroup$ This actually makes a lot of sense! One thing: "A weight of $β_{i,SMB}$ in small caps and $β_{i,SMB}$ in large-caps.", but wouldn't that mean I am equally weighted in small caps and large caps irrespective of the beta coefficient for SMB? Or do you mean $β_{i,SMB}$ in small caps and $-1*β_{i,SMB}$ in large caps? $\endgroup$
    – n_arch
    Commented Apr 3 at 13:48
  • $\begingroup$ Sorry you are right. It's $-1 \beta$. I fixed it. $\endgroup$
    – phdstudent
    Commented Apr 3 at 19:06
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This is not necessarily true in general; it only means that the portfolio varies with the SMB factor.

The average size of a portfolio and the HML beta do not align perfectly (although they are correlated).

Hence, you can construct a portfolio of only large stocks with a big HML beta.

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  • $\begingroup$ This is simply wrong. $\endgroup$
    – phdstudent
    Commented Apr 7 at 5:13
  • $\begingroup$ How? Characteristics are not equal to betas? $\endgroup$
    – alejandroll10
    Commented Apr 8 at 12:58

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