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This is a general question that applies to the CAPM and any version of the APT (e.g. the Fama & French three factor model). Speaking in terms of the APT:

Assuming a simple one-index version of the APT I have:

\begin{equation} R_i = \alpha_i + \beta_{1,i}f_1 + \epsilon_i, \end{equation}

where, for each asset $i$, $R$ denotes the return, $\alpha$ denotes a constant, $\beta_1$ denotes the factor loading on factor $f_1$ and $\epsilon$ denotes the idiosyncratic error. It is well known and easy to proof that this implies:

\begin{equation} E(R_i) = r_f + \beta_{1,i}\lambda_1, \end{equation}

where $\lambda_i$ denotes the risk premium associated with the corresponding factor.

Now, this clearly states that I can predict the expected value of an asset return cross-sectionally, that is in the same period of my factor, as well as factor loading realization. There is no subscript $t$! Nonetheless, models such as the APT are commonly used to predict the next periods returns, i.e.:

\begin{equation} E(R_{i,t+1}) = r_{f,t} + \beta_{1,i,t}\lambda_{1,t}. \end{equation}

My question: Why can I predict returns in $t+1$ with the model - the original APT does relate to expectation within a cross-section? Going from formula 2 to formula 3 necessarily implies that one assumes the factor loadings are constant across $t$ to $t+1$. This is not a reasonable assumption in my opinion.

The only explanation I can come up with:

Usually the $\beta_{1,i}$ is estimated via time-series regressions. When sticking to formula 2, this necessarly implies that I use $R_{i,t}$ when estimating $\beta_{1,i}$ and when estimating $\lambda_i$. Put differently, my LHS variable in step one is implicitly part of my RHS variable in step two (as it is estimated based on it) - that makes limited sense, probably. When using the expected future return relation in the third formula, I only use $R_{i,t}$ when estimating $\beta_{1,i}$. Hence, formulating it like this empirically is cleaner.

EDIT: To add to my point: Consider the Cochrane 2011 JF Presidential Adress. On page 1059 he mentions the FF model, relating expected returns in $t$ to factors in $t$. On page 1062 he then goes on to say "More generally, “time-series” forecasting regressions, “cross-sectional” regressions, and portfolio mean returns are really the same thing. All we are ever really doing is understanding a big panel-data forecasting regression,

\begin{equation} R^{ei}_{t+1}=a+b'C_{it}+\epsilon^i_{t+1}. \end{equation}

This is exactly what I am finding confusing: How is the cross sectional regression he explicitly formulates earlier, the same as this prediction regression? It is one thing to talk about expected returns in $t$, as the theory on cross-sectional variation does, and another thing to talk about expected returns in $t+1$.

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    $\begingroup$ Hopefully someone who's a practitioner with these AP models will provide a thorough answer. My short answer is that in my opinion these models are outdated and should be treated as an intellectual curiosity: times have moved on and I don't personally know anyone who uses for example the CAPM for anything practical. Your remarks are completely reasonable. I would personally never use a model that is based on a cross-section of assets at a fixed point in time to predict anything about the future. $\endgroup$ Apr 17, 2023 at 13:07
  • $\begingroup$ PS: I am aware that people still use variants of the Famma-French model in asset management: but these would be much more sophisticated versions of the model. I don't think the old models from the 80s or even 70s are used in their original form for anything practical nowadays, except for academic discussions. $\endgroup$ Apr 17, 2023 at 13:08
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    $\begingroup$ @JanStuller, check out survey studies quoted in section 13.8 of Berk & DeMarzo "Corporate Finance: The Core" 4th edition (2016), and you will find that the CAPM is the most widely used model among CFOs for calculating the cost of capital. It can also be inferred (indirectly) that the CAPM is also the most popular model among investors in mutual funds. The first of the surveys is somewhat dated, but I suspect the picture in the corporate world has not changed drastically since then (that is a guess). When it comes to money managers and such, the story may be very different, as you indicate. $\endgroup$ Apr 17, 2023 at 13:30
  • $\begingroup$ @RichardHardy: ok, that's a fair point. $\endgroup$ Apr 17, 2023 at 13:44
  • $\begingroup$ Thanks for your remarks. I can tell you from my own experience that for example in many simple small/mid cap M&A processes, the CAPM is still used for cost of equity. However, my question also pertains to academics who use the models to predict returns, not only professionals. $\endgroup$
    – shenflow
    Apr 17, 2023 at 14:48

1 Answer 1

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Short answer: Yes and no.

Long answer: Yes, as you correctly point out with the Cochrane reference, you can use a factor model to predict stock market returns. How good is that prediction, will depend on how well you are estimating means/variances and covariances. Let's proceed in steps, and let me work as the CAPM as the factor model, but everything below can be extended to a factor model:

  1. The conditional CAPM implies: $$ E_t[r_{i}] - r_f = \beta_{i,t} E_t[r_m - r_f] $$

  2. Covariances and consequently betas are usually stable in short-horizons so I can write: $$ E_t[r_{i}] - r_f = \beta_{i} E_t[r_m - r_f] $$

  3. The equation above is valid for one period ahead: $$ E_{t+1}[r_{i}] - r_f = \beta_{i} E_{t+1}[r_m - r_f] $$

So if you want to predict the stock return of any asset (assuming for now that covariances are stable), you only need to predict the stock market return. Now that's where things get tricky.

The best two references to understand this are:

  1. Cochrane (2008) - The dog that did not bark
  2. Goyal and Welch (2007)

The first tells you what economists mean by equity premium being predictable. It basically implies that some variable (or state variable) predicts the equity premium. Cochrane argues that mathematically either dividend growth or returns must be predictable. He shows that the latter is true. Take a look at table (1):

enter image description here

The dividend-price ratio predicts the equity premium. When D/P is high the returns are high. But these are low-frequency in-sample estimates.

The second reference (Goyal), shows that equity premium is predictable in-sample but not out-of-sample. So you cannot trade on this predictability - which basically implies that you cannot forecast the ex-post return (at a monthly frequency). Ex-ante we know that equity premium moves with some state variables in the economy (i.e. expected returns are high in recessions) but in practice this cannot be exploited economically.

So can you predict the return of a stock long-term? Yes - if their beta is stable. Can you predict it next month? No, because you can't predict the market risk-premium.

Let me expand a bit on predictability of the market risk-premium vs predictability of variances/covariances (or betas):

Consider total US stock market between 1928-2022:

  1. $T_{years} = 95$;
  2. Average excess return of stocks over $r_f$: $\bar{r}_{annual} = 0.082$
  3. With a standard deviation of return of: $\sigma_{annual} = 0.197$

What is the confidence interval for the mean (which can be time-varying) and the standard deviation?

  1. SE error for the mean: $2.02\%$
  2. Standard error for the volatility $\approx 1.43\%$ (assuming normality)

So confidence interval for the mean: $8.2\% \pm 1.96 \times 2.02\% = [4.23\% - 12.16\%]$

And confidence interval for the volatility: $[0.17 \text{ to } 0.22]$

So means are much harder to estimate than volatilities. And that is the issue: how to forecast the mean return of the market!

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    $\begingroup$ Okay, I understand. A question, though: There is a vast amount of literature that uses e.g. the CAPM Beta or other factors/characteristics as a RHS variable in a simple prediction regression. I.e. C=e.g. CAPM Beta in the formula by Cochrane that I am referring to. E.g. Bali et al. (2017; Empirical Asset Pricing) explicitly state that "we would expect to find a positive cross-sectional relation ebtween beta and future excess returns (p. 131)". Why would I expect that? Is this logic based on some simple mental extrapolation of the (in this case) CAPM? $\endgroup$
    – shenflow
    Apr 18, 2023 at 7:31
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    $\begingroup$ For consistency, I would replace $E_t[r_{i}] - r_f = \beta_{i,t} E_t[r_m - r_f]$ with $E_t[r_{i}] - r_f = \beta_{i,t} (E_t[r_m] - r_f)$ or $E_t[r_{i} - r_f] = \beta_{i,t} E_t[r_m - r_f]$. Also, if we are dealing with multiple time periods, should the returns perhaps have time indices? $\endgroup$ Apr 18, 2023 at 7:33
  • $\begingroup$ Thanks for making those edits @RichardHardy. shenflow, yes you would expect that. No extrapolation if betas move slowly . $\endgroup$
    – phdstudent
    Apr 18, 2023 at 13:47
  • $\begingroup$ No problem. I did not touch the things I commented about (you may or may not wish to address them), I only fixed a repeated typo. $\endgroup$ Apr 18, 2023 at 13:52
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    $\begingroup$ Actually, your argument still holds - the underlying factor is still the same for all assets. Got it - thanks! Will accept your answer if there are no other (possible more thorough) answers to the question. $\endgroup$
    – shenflow
    Apr 20, 2023 at 8:14

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