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I have the following well-known formula for Fama and French's three-factor model

$ R_{it}-R_{ft}=\alpha_i+\beta_{i1}(R_{m,t}-R_{f,t})+\beta_{i2}SMB_t+\beta_{i3}HML_t+\epsilon_{it}$.

My question is: Why do we not take the size (market capitalization) of the firm itself instead of constructing a risk factor based on the sizes of other firms and why are the correlation between the firm's excess return and the size factor (which concerns the other firms) an explanation for the size?

Most explanations concerning the size factor are about the size of the firm itself. For example, small firms are more riskier because their small size makes them more susceptible to outside influences. However, it is not the size of the firm that is featured in the excess return equation, but the return on a portfolio of other firms based on size. So what explanation could you give why the correlation with this portfolio yields a higher return.

In this question I have focused on the size factor but my question also holds for the value factor. Could someone help me out? Thanks in advance!

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    $\begingroup$ Fama French use the 'risk factor approach' you mentioned, but other researchers use the 'caracteristics approach' which as you describe consists of looking at the size of the firm itself. There is no agreement which is better (and it may depend what your goal is). A writeup about the issue is here alphaarchitect.com/2017/10/31/… $\endgroup$ – noob2 Feb 27 '20 at 20:52
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You can do exactly what you are saying, look at Fama Macbeth type regression analysis, with cross sectional regressions as opposed to time series regression.

In a cross sectional approach you first get an exposure of the stock to each factor in the approach you have seen in Fama-French and then for each time point apply a cross sectional regression of the returns of each asset and the exposure to a specific factor and you can find the “factor return” or risk premium.

You can regress in step 1 using basically anything you like, number of light bulbs sold on a Wednesday morning with return to get some kind of beta to that factor. For market cap and other fundamentals you can use a standardised value for the exposure values instead of doing a time series regression. Take a look at the following link for a replication of the Bloomberg factor model, it includes a size factor:

https://run.unl.pt/bitstream/10362/16787/1/Costa.R_2016.pdf

This method is the one used mostly in practice by the big guys (Barra etc), it means you can have all kinds of factors whereas the standard time series regression method you speak about you must generate portfolios for each factor like you say.

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  • $\begingroup$ That link is gold! $\endgroup$ – Validus Oculus Jun 26 at 23:31
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Fama and French are trying to capture risk factors across a broad cross-section of assets. The way that they go about doing this is by sorting a bunch of stocks into a portfolio by their size to proxy for this "factor". It would not really make sense to use specifically one firms market cap as the size factor. They are trying to see if size itself is priced as a risk factor, not the size of a specific firm. So correlation with the size portfolio implies that the specific asset you are analyzing is exposed to the size factor (usually because it is a very small/very large market cap stock) and thus experiences the excess returns associated with taking on this risk. If you used just the one specific firms market cap on the right hand side and used it to explain its return it probably wouldn't really make sense. The firms market cap would remain largely constant (at least relatively) over time; that is if you classify it as "small" today it would probably still be classified as "small" months from now. Yet it would likely experience an excess return. You suspect that part of this excess return is due to its market cap but its market cap will not have varied over the period and therefore you will have no relationship really to observe in that type of analysis. Thats kind of why they use the factor mimicking portfolios.

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