Factor Models, Conditional Returns, and the Limits of Forecasting

# Why can I use equilibrium asset pricing models to predict future returns?


# Why can I use equilibrium asset pricing models to predict future returns?












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 $t+1$.

## Answer by phdstudent (score 4)

https://quant.stackexchange.com/a/75250

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:

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

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

- 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:

- Cochrane (2008) - The dog that did not bark

- 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):

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:

- $T_{years} = 95$;

- Average excess return of stocks over $r_f$: $\bar{r}_{annual} = 0.082$

- 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?



- 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!