# Fama-French 3-factor model: factors implying risk

The Fama-French three-factor risk model is given by $$r=R_f+\beta_m(K_m-R_f) + \beta_s\cdot\mathit{SMB}+\beta_v\cdot\mathit{HML}+\alpha$$ where $r$ is the return, $R_f$ is the risk-free rate, $K_m$ is the market return, $SMB$ is the Small Minus Big factor and $HML$ is the High Minus Low factor. $\alpha$ and the $\beta$'s are estimated from the data and $\alpha$ can be interpreted as excess return not explained by the model.

Both $\beta_s$ and $\beta_v$ are highly significant and thus the CAPM (where $\beta_s = \beta_v = 0$) does not hold. This result can be explained in the two ways: the first is that this is a stock pricing anomaly and $\alpha$ can be obtained. This does not seem to be the case as the anomaly has persisted.

The second is that investing in small cap or value stocks carries extra risk and that the FF 3-factor model just explains risk better than the CAPM does. This interpretation is in line with semi-strong EMH but does not satisfy me. Why should small cap or value stocks be more risky? Economists can come up with all kinds of interesting narratives but have these narratives been tested quantitatively? And what were the results?

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This question is part of this weeks topic challenge, see meta.quant.stackexchange.com/q/1369/35?cb=1 for more info. –  Bob Jansen Oct 20 '13 at 11:33
I think small cap indices get raped the most during crises. –  user2763361 Oct 23 '13 at 13:31
That would be quite easy to show statistically and I guess that result would be well known in that case. –  Bob Jansen Oct 23 '13 at 19:01
This holds for Australia (just knowledge from my work); but would be interesting to see for other countries. –  user2763361 Oct 24 '13 at 1:07

There is no definitive answer to this question and there are infinite papers out there. I personally think they are better explained as mispricings. Several points:

1) Persistence of HML does not imply it has to be a risk factor. If there are idiosyncratic mispricings in individual stocks, then by construction, the ones that look cheap are going to be actually cheap.

This is a related (though somewhat involved) criticism of the risk-factor interpretation: http://finance.wharton.upenn.edu/~rlwctr/papers/9315.PDF

2) Here is some nice evidence that those factors aren't risk-factors and are more likely mispricings. At least they seem better explained by characteristics than factors: http://ww.andreisimonov.com/Microstr_PhD/MSU_09/DanielTitman97.pdf

3) There are attempts to find theoretical explanations for those factors. For instance people hypothesized (if my memory works) that the value factor is correlated with human capital, and therefore despite value stocks have higher returns, growth stocks are better hedges for investors if you think they are optimizing overall (taking into account the variability of their human capital).

4) Aside: yes those factors work quite consistently across time and markets: http://www.master272.com/finance/longshort/ff_growthvalue.pdf

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Thanks, that is a lot to digest. It would be nice to have some papers that argue the other side. I've some ideas where to look but no time at the moment. More later. –  Bob Jansen Oct 23 '13 at 19:03
@BobJansen You may want to look at this paper. rady.ucsd.edu/faculty/directory/valkanov/pub/classes/mfe/docs/… –  J Li Oct 24 '13 at 12:19