How does the Fama and French 3-factor model explain stock covariance?

Question

Does it at all? If so, how?

It is understood that size and value play a role in determining returns and there are proposed explanation those these, but what about covariance?

Answer 1

There are two ways I could think of to answer this question. First, and I think this is what you are asking, "what is the covariance structure of two assets under the Fama-French 3-factor model?" Consider two assets $i$ and $j$.

Let's start with their returns under the Fama-French factor model: $$ r_{i,t} = \alpha_{i} + \beta_{1,i} MKT_{t} + \beta_{2,i} HML_{t}+ \beta_{3,i}SMB_{t} + \epsilon_{i,t}$$ $$ r_{j,t} = \alpha_{j} + \beta_{1,j} MKT_{t} + \beta_{2,j} HML_{t}+ \beta_{3,j}SMB_{t} + \epsilon_{j,t}$$ Under the assumptions of the factor model setup:

1. $Cov(\epsilon_{j,t}, \epsilon_{i,t}) = 0$
2. $E[\epsilon_{j,t}]=E[\epsilon_{i,t}]=0$
3. $Var(\epsilon_{k,t})=\sigma^2_{\epsilon_{k}}, \forall k $
4. $Var(MKT_{t})=\sigma^2_{M}$
5. $Var(HML_{t})=\sigma^2_{H}$
6. $Var(SMB_{t})=\sigma^2_{S}$
7. The covariance of two factors is $Cov(F_{k},F_{\ell})=\sigma_{k,\ell}$ for $k\neq\ell$
8. All alphas and betas are constants

Now let's estimate the covariance of the returns when $i\neq j$:

$$ Cov(r_{i,t}, r_{j,t}) = Cov(\alpha_{i} + \beta_{1,i} MKT_{t} + \beta_{2,i} HML_{t}+ \beta_{3,i}SMB_{t}, \\ \alpha_{j} + \beta_{1,j} MKT_{t} + \beta_{2,j} HML_{t}+ \beta_{3,j}SMB_{t} + \epsilon_{j,t})$$ Recall that $ Cov(aX+bY,cW+dZ) = \\ ac \cdot Cov(X,W) + bc \cdot Cov(Y,W)\\ + ad \cdot Cov(X,Z) + bd \cdot Cov(Y,Z)$

These assumptions imply that $$ Cov(r_{i,t}, r_{j,t}) = \beta_{i,1} \beta_{j,1} \sigma^2_{M} + \\ \beta_{i,1} \beta_{j,2} \sigma_{M,H} + \\ \beta_{i,1} \beta_{j,3} \sigma_{M,S} + \\ \beta_{i,2} \beta_{j,1} \sigma_{M,H} + \\ \beta_{i,2} \beta_{j,2} \sigma^2_{H} + \\ \beta_{i,2} \beta_{j,3} \sigma_{H,S} + \\ \beta_{i,3} \beta_{j,1} \sigma_{M,S} + \\ \beta_{i,3} \beta_{j,2} \sigma_{H,S} + \\ \beta_{i,3} \beta_{j,3} \sigma^2_{S} \\ = \beta_{i,1} \beta_{j,1} \sigma^2_{M} + \beta_{i,2} \beta_{j,2} \sigma^2_{H} + \beta_{i,3} \beta_{j,3} \sigma^2_{S} + (\beta_{i,1} \beta_{j,2} + \beta_{i,2} \beta_{j,1}) \sigma_{M,H} + (\beta_{i,1} \beta_{j,3} + \beta_{i,3} \beta_{j,1}) \sigma_{M,S} + (\beta_{i,2} \beta_{j,3} + \beta_{i,3} \beta_{j,2} ) \sigma_{H,S} $$ And a $Var(r_{k}) = $ $$ \beta_{k,1}^2 \sigma^2_{M} + \beta_{k,2}^2\sigma^2_{H} + \beta_{k,3}^2 \sigma^2_{S} + 2\beta_{k,1} \beta_{k,2} \sigma_{M,H} + 2\beta_{k,1} \beta_{k,3} \sigma_{M,S} + 2\beta_{k,2} \beta_{k,3} \sigma_{H,S} + \sigma^2_{\epsilon_{k}}$$

The alternative question might be "why would assets follow the data generating process that is found in Fama-French". I'm not sure that there is a good answer to that. Factor models are nice and convenient, and Fama and French developed theirs to match the observation that historically small caps and stocks with a high book-to-market ratio have out performed. That wasn't really a theory argument, although it is probably possible to write down a theory model where asset prices follow a Fama French 3-factor process, I don't know of any. One paper says:

Fama & French (1993) contend that stock returns can be described by three factors, viz, market, size and book-to-market equity. However, the model lacks any well-built academic justification as to why size and book-to-market describe the cross-sectional differences in predicted returns of stocks.

AN EMPIRICAL EXAMINATION OF THE FAMA-FRENCH MODEL IN INDIA

Answer 2

It is not clear to what covariance this question refers to.

As in the CAPM model the covariance among assets is reduced to $\beta$, through the covariance with the market index.

$\beta_i=\sigma_{i,M}/\sigma_M^2$

Then, if CAPM is correct, we could explain the covariance relating the $\beta$s

$\sigma_{i,j}=\beta_i*\beta_j*\sigma_M^2$

In the context of multifactor models something similar occurs, since, in order to use those models, it is necessary to estimate $\beta$s, or sensitivities, related to each proposed factors.