Why in Fama-French factor model relative market capitalization and book-to-market aren't used directly for predicting return rate?

Question

Fama and French use the following formula for predicting stock returns

\begin{align*} r=r_{riskfree} + \beta_1(r_{market}-r_{riskfree})+\beta_2(SMB)+\beta_3(HML) \end{align*}

which basically means that in order to find expected return for a certain stock, you should regress its excess returns against market excess returns, small-minus-big portfolio returns and value-minus-growth portfolio returns.

This feels extremely counterintuitive because if we believe that there is a premium for small-cap companies, why isn't information about capitalization used directly (e.g. relative capitalization to market capitalization or percentile)? Is it just because it works better or there is some intuition behind this?

By not doing so, Fama-French model make it possible for a large-cap stock actually to earn small-cap stock premium provided it for whatever reason has $\beta_{2}>0$, which would make no sense in terms of interpretability.

There is also no intuition behind how SMB and HML portfolio are formed (why, for instance, not 40% smallest cap stocks minus 30% largest cap stocks instead of 50/50? No specific justification here for any division)

In case it is not clear what I mean, consider the following model instead: \begin{align*} r - (\beta_1(r_{market}-r_{riskfree})+r_{riskfree}) = (cap\:premium)*(cap\:percentile) + (value\:premium)*(book-to-market) + const \end{align*} By running a regression through different stocks, capitalization premium (expected to be negative) and value premium can be found and then used to find what excess return (over CAPM's prediction) a stock can generate. Here relative market capitalization and book-to-market are included directly, so here it isn't possible for a larger company with same systematic risk and book-to-market ratio to have higher expected return what makes more sense

Answer 1

So, let me begin with a disclaimer. Elsewhere, I make three mathematical arguments that would impinge on this question in a way you didn't ask, but I feel I should disclose.

The first is that since returns are a ratio of prices times a ratio of quantities plus dividends adjusted for bankruptcy and mergers, returns cannot have a population mean, even if stationary. That would imply that Fama-French cannot be valid. It is non-sensical, mathematically. That is also true for the CAPM it attempts to falsify.

The second is that since Frequentist probabilities are incoherent in the de Finetti sense of the word, any model built on Frequentist axioms will force arbitrage opportunities created by the calculations into existence, defeating an absence of arbitrage assumption.

The third argument is that the calculus the CAPM is built on assumes that all parameters are known and that nobody makes estimates. For some scientific models, the difference between the two is negligible. For finance, the assumption is catastrophic because of a proof in 1958 that models like the CAPM or Fama-French cannot have a solution within the axioms.

Let us assume that none of the above are true. Let us also be concerned with the validity of the CAPM. Let us also assume that we have to make estimates for all parameter values.

The CAPM is built on Frequentist axioms so we will use Frequentist decision theory to test it.

In Frequentist decision theory, we assert a null hypothesis is true and choose a cut-off value for the null such as $p<.05$ or some other value. While I assume you know that, it also impinges on the discussion so I am making that explicit.

The only real prediction in the CAPM is that any intercept will be zero. Unfortunately, for a variety of statistical reasons, that isn't really a testable hypothesis.

Implicitly, however, if all information is contained in $\beta$, then the effect of all other possible factors should be zero, regardless of how you construct them. The null of interest is that $\beta_{SML}=0$ and $\beta_{HML}=0$. If the null is falsified then the CAPM is falsified. So far, we are just inside standard inferential methods, but we are going to wander into decision theory.

If both $\hat{\beta}$ are in the acceptance region, then Frequentist decision theory requires you to behave as if the null is true. In that case, the Capital Asset Pricing Model should be treated as if true. Note that Frequentist decision theory does not assert the truth or falsehood of the null. It says you should behave as if true.

If either $\hat{\beta}$ are in the rejection region, then you are to behave as if the CAPM is false. If you stopped there, then you would have inference the way Ronald Fisher understood it. Falsifying the null has no information in it other than that the null is probably false. Nothing else. Fisher did not have an alternative hypothesis as an idea.

However, because you are controlling the frequencies with the null, setting a cut-off, and presumably controlling for power, you can make a stronger decision. You can decide to reject the null and behave as if the alternative that you specified is true.

Fama and French merely rejected the null. The CAPM is falsified. Nonetheless, the field took the next step and took up the alternative as if true.

Your question has caught on to one of the problems of Fama French. It has no economic reason within mean-variance finance for it to be true. It can create counter-intuitive results because of the independence assumption of the factors in ordinary least squares.

It also has no holiness to its structure. Why create the quantile cuts where they were? The model did not come down from the mountain and was not inscribed in stone? It was made up of choices that were practical. There is no reason to believe it is associated with the data generation function.

The weakness in Frequentist decision theory is that when your alternative lacks a grounding in theory, it is the "as of now" best model, but holds no intrinsic validity.

The difficulty is created by the absence of a binary choice. The Fama-French model is not the only possible way to specify an alternative to the CAPM. If it were, then it would be the one and only alternative and all would be good.

To see a simple example of that, consider a loaded dice that either comes up even 2/5ths, 3/5ths, or 4/5ths of the time. Imagine you made 13 rolls of the dice. If you choose 2/5ths as your null and it is rejected, the maximum likelihood estimator cannot map to either of the alternatives as 5 does not go into 13. The null is rejected at zero as well. Of course, if you observe a zero, the most likely loading is 2/5ths. That loading is excluded as it is rejected.

Null hypothesis methods have problems when you cannot think in terms of a binary choice. There is nothing special about $H_0:\theta=2/5$ as you could have chosen $H_0:\theta=3/5$ or $H_0:\theta=4/5$ as equally valid. Not 2/5ths doesn't allow you to distinguish between the validity of 3/5 or 4/5.

Fama-French was just one of an infinite number of possible alternative constructions.

Answer 2

why isn't information about capitalization used directly

One of the reasons is dimensions of both sides of the equation: the original specification regresses returns on returns, while your proposed one - on dollars or ratios. This has an important consequence for asset pricing predictions of the model (after all, pricing assets is what the model is supposed to do!): in the original specification the alpha must be indistinguishable from 0, since for any asset it is always possible to hedge the risk factors using the betas, hence whatever is left would earn no risk premium.

Another prediction lost in your specification is (to see, take the expectation of both sides of the Fama-French equation) that the risk premia are equal to the expected value of the regressors. Since we know that risk premia is measured in the units of returns, your specification loses this trait.

Then, the original specification allows for an easier construction of portfolios which is essential to reduce problem dimensions (from thousands of stocks to dozens of portfolios) and reduce much noise: a return of a portfolio of assets is a much more reasonable thing to have than a ratio.

Finally, @Kevin's comment to your question conveys another (not less) important reason, namely, that it is not what the stock is that matters but how it behaves.