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I am trying to understand the Fama-French factor model, or any kind of CAPM extensions really. What is really puzzling me is the use of mimicking portfolios. Fama and French create mimicking portfolios (indices) of the SMB and HML factors, so similarly as the market risk-factor (index). My question is: why not simply use the size and the book-to-market ratio as a dependent variable. What is the need to create a mimicking portfolio?

I am trying to find answers in previous posts, here and here, but it is not really answering my question.

The only explanation that I can personally come up with is that we are not actually interested in predicting returns, where you would have a dependent variable specific to the firm (say, $S_{i}$ being size of firm $i$), but we are rather interested in explaining the returns (in that case, $S^{*}_{i}$ are the average returns of all firms contained in bucket $i$). Is this intuition maybe correct?

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The innovation of Fama and French's Three Factor Model wasn't in finding book to market ratios forecast returns but in reconciling that empirical regularity with the standard framework of macro-finance theory which views higher expected returns as compensation for macroeconomic risk of hedging concern to investors.

Predicting returns using a stock's book to market ratio is arguably acceptable for pure forecasting, but it wouldn't sit well in the academic, macro-finance research program of trying to explain expected returns using macroeconomic risk. The guiding intuition of that research program is that a security can have higher expected returns if its like selling insurance, if it tends to deliver its cashflows in good states of the world rather than the bad states.

How I'd summarize the logic of the Fama French Three Factor Model:

  • We know from previous papers etc... that value stocks with high book to market ratios tend to have higher average returns.
  • Leap of faith step: Let's construct a portfolio long value stocks (i.e. high book to market firms) and short growth stocks and call the portfolio return HML.
    • Note: Why HML would be a variable of hedging concern to investors is debatable and not clear at all? This IMHO is the big theoretical weakness or mystery of this whole exercise.
  • Super cool observation! Stocks with high book to market ratios tend to covary positively with the HML portfolio. We can explain the variation in expected returns of value stocks using the variation in their covariance with some macroeconomic variable (the return of the HML portfolio). The market has some kind of factor structure.

Further Explanation: why explain with covariance rather than characteristics?

Let $x_i$ be a characteristic of firm $i$ (eg. the firm's book to market ratio), let $R_i$ be a random variable denoting the return of firm $i$, and let $c_i = \operatorname{Cov}(R_i, S)$ where $S$ is the excess return of a particular portfolio of hedging concern to investors. The heart of your question is why do typical academic asset pricing models take the form of equation (1), equating expected returns as linear in covariance with some $S$:

$$ \operatorname{E}[R_i] - r_f = c_i \gamma \tag{1}$$ rather than something like the more flexible equation (2): $$ \operatorname{E}[R_i] - r_f = f(x_i) \tag{2}$$

The short answer is that (1) is inherently consistent with the "Law of One Price" and financial-economic theory while form (2) is far less constrained and can allow all kinds of craziness.

Example for intuition: do two securities that deliver the same stochastic cashflow tomorrow necessarily have the same price today?

Imagine we live in a two period world you have two securities $i$ and $j$ that will deliver the same cashflow $Z$ next period. Imagine the two securities sell for prices $p_i$ and $p_j$ respectively hence $R_i = \frac{Z}{p_i}$ and $R_j = \frac{Z}{p_j}$.

If equation (1) is true, Law of One Price holds:

Equation (1) implies:

$$ \operatorname{E}\left[ \frac{Z}{p_i} \right] - r_f = \operatorname{Cov}\left(\frac{Z}{p_i}, S \right) \gamma$$

Solving for the price:

$$ p_i = \frac{\operatorname{E}\left[ Z \right] - \operatorname{Cov}\left(Z, S \right) \gamma}{ r_f}$$

$$ p_j = \frac{\operatorname{E}\left[ Z \right] - \operatorname{Cov}\left(Z, S \right) \gamma}{ r_f}$$

You thus have the immensely logical $p_i = p_j$ and $R_i = R_j$.

I haven't proved anything here, but it turns out that the Law of One Price implies being able to write an equation of form (1).

If expected returns follow equation (2)... bizarre things are allowed?

It's not hard to construct a hypothetical example where:

  • Two firms deliver the same stochastic cashflow $Z$ next period.
  • The firms have different accounting values $x_i \neq x_j$ (e.g. due to different firm histories)

If equation (2) is true, then securities $i$ and $j$ can't have the same price which is bizarre.

Practical reality (a lot of trading firms take approach 2):

In standard financial economic theory, it is covariances that should matter rather than firm characteristics. Expected return models should take form (1) rather than form (2).

But it turns a practical reality is that if you're not particularly concerned about offending the priests of rational economic theory, a well constructed statistical model of form (2) will arguably outperform the more theoretically sound structure of form (1).

Why might you be better off predicting returns with characteristics rather than estimated covariances? Some (incomplete) list of possibilities:

  • Characteristics are better predictors of the true covariance rather than historical returns.
    • Covariances change over time and perhaps something is true like $c_{it} = g( x_{it})$ (i.e. covariances are some function of a characteristic)?
  • Behavioral finance stuff is important and enforcing consistency with Law of One Price (LOOP) is just empirically wrong.

If covariances vs. characteristics interests you, you might one to read Daniel and Titman (1997), "Evidence on the Characteristics of Cross Sectional Variation in Stock Returns."

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    $\begingroup$ These are the kind of answers I wish I could upvote twice :) $\endgroup$
    – Kevin
    Mar 22, 2023 at 15:09
  • $\begingroup$ 1. How do I reconcile your first paragraph with your first note This IMHO is the big theoretical weakness or mystery of this whole exercise? 2. Regarding Super cool observation! Stocks with high book to market ratios tend to covary positively with the HLM portfolio. Is that tongue in cheek? Since the positive correlation arises by construction. 3. Regarding the heart of the question, you emphasize the functional form while I think what matters is looking at a certain characteristic vs. covariance with a portfolio. These are two very different angles. $\endgroup$ Mar 23, 2023 at 11:48
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    $\begingroup$ 5. The reference Daniel and Titman seems to be from 1997, not 2012. 6. Thank you for a thought provoking answer! :) $\endgroup$ Mar 23, 2023 at 12:04
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    $\begingroup$ @RichardHardy On the other hand, I'm not aware of a good story for why HML should be a risk factor. It's not obvious? Is this really what's going on? The CAPM at least had clean, believable stories for why the market return should be a risk factor. $\endgroup$ Mar 23, 2023 at 16:19
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    $\begingroup$ I think @MatthewGunn is raising a good point. "I'm not aware of a good story for why HML should be a risk factor." There're a great many theories that explain the value premium but are compatible with a conditional CAPM in which value stocks simply have higher market betas. $\endgroup$
    – Kevin
    Mar 23, 2023 at 16:31
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Even if your concern is prediction, the mimicking portfolio might be of interest.

This is because using individual firm characteristics as risk factors is noisy and does not reflect the true underlying risk factors that drive expected returns. By constructing mimicking portfolios, the systematic risk factors are isolated. If you want to predict via the factor and not some noisy instantiation of it, you may want to go with the mimicking portfolio.

Nonetheless, the original motivation of Fama-French is as you describe. In [1] it is noted that

A mimicking portfolio may represent an unobservable factor when the stock sensitivities to the factor are believed to be disclosed by some firm characteristics. A well-known example of this case is in the Fama and French (1993), where the characteristics such as firm size or book-to-market ratio are supposed to reveal the loadings of the firms to some latent distress factors.

[1] Asgharian, Hossein. 2004. "A Comparative Analysis of Ability of Mimicking Portfolios in Representing the Background Factors," Working Papers 2004:10, Lund University, Department of Economics.

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