Factor Models, Little Green Men, And Machine Learning

Economists use machine learning (ML) to study asset prices in two different ways. Approach #1: use these techniques to predict the cross-section of expected returns—i.e., to predict which stocks are most likely to have high or low future returns. e.g., see here, here, or here. Approach #2: use them to try to uncover the “true asset-pricing model”—a.k.a., the “set of priced risk factors”.

Many economists dismiss approach #1, arguing that predicting future stock returns is a job for traders not academics. Instead, it’s much more common for researchers to adopt approach #2. The conventional wisdom is that we, as researchers, will learn something deep and fundamental about how financial markets work if one of these new ML techniques uncovers a factor model that perfectly explains the cross-section of expected returns. There’s a widely held view that doing empirical asset-pricing research means attributing differences in expected returns to some risk-return tradeoff with an intuitive story attached to it.

But… not so fast. There’s actually something paradoxical about the logic of approach #2. There’s a problem with the conventional wisdom. And, the goal of this post is to explain what that special something is.

Factor Models

But first: factor models. What are economists talking about when they say they’re trying to find the “true asset-pricing model” or the “set of priced risk factors”? To get a handle on this terminology, consider regressing the returns of each stock, $R_{n,t}$ , on lagged values of some predictive variable, $X_{n,t-1}$ :

$\begin{equation*} R_{n,t} = \hat{a} + \hat{b} \cdot X_{n,t-1} + \hat{e}_{n,t} \end{equation*}$

The results of a predictive regression like this one can be interpreted as trading-strategy returns. You can read the estimated $\hat{b}$ as the return to a zero-cost portfolio that’s long high- $X$ stocks and short low- $X$ stocks:

$\begin{equation*} \hat{b} \propto {\textstyle \frac{1}{N} \cdot \sum_n} \, (R_{n,t} - \bar{R}_t) \cdot (X_{n,t-1} - \bar{X}_{t-1}) \end{equation*}$

Thus, $\hat{b} > 0$ implies both that stocks with high predictor values yesterday, $(X_{n,t-1} - \bar{X}_{t-1}) > 0$ , tended to have high excess returns today, $(R_{n,t} - \bar{R}_t) > 0$ , and also that it would have been profitable to trade on $X$ today.

It could be that an estimated $\hat{b} > 0$ represents arbitrage profits. But, maybe trading on $X$ is only profitable because it requires investors to bear lots of non-diversifiable risk? Imagine that investors are all really worried about not having enough money during future market crashes, $R_{\mathit{Mkt},t} \ll 0$ . Then, if the predictive variable $X_{n,t-1}$ turned out to be capturing exposure to market risk,

$\begin{equation*} X_{n,t-1} = {\textstyle \frac{\mathrm{Cov}[R_{n,t}, \, R_{\mathit{Mkt},t}]}{\mathrm{Var}[R_{\mathit{Mkt},t}]}} \end{equation*}$

the profits earned by trading on $X$ would represent compensation for holding a portfolio that will deliver terrible returns during market crashes—i.e., at the worst possible time as far as investors are concerned. And, when economists think this is what’s going on, they typically write the predictive variable as $\beta_{n,t-1}^{(\mathit{Mkt})}$ rather than $X_{n,t-1}$ . This is what they’re talking about when they speak of “market beta”.

So far so good. Now, for the final step. Notice that this compensation-for-risk logic doesn’t just apply when the risk factor is market returns. You can replace $R_{\mathit{Mkt},t}$ with any variable so long as the variable defines some sort of bad aggregate outcome in investors’ eyes. e.g., think about something like a drop in market liquidity. So, looking for the “true asset-pricing model” or the “set of priced risk factors” means looking for a collection of $K \geq 1$ variables $\{R_{1,t}, \ldots,\,R_{K,t} \}$ such that, if we assume investors are worried about not having enough money when these risk factors are negative, then every difference in expected returns is perfectly explained by differences exposure to these $K$ priced risk factors:

$\begin{equation*} \mathrm{E}_{t-1}[R_{n,t}] = {\textstyle \sum_{k=1}^K} \, \lambda_t^{(k)} \cdot \beta_{n,t-1}^{(k)} \end{equation*}$

Above, each $\lambda_t^{(k)} > 0$ is a market-wide constant called the price of risk associated with the $k$ th factor.

I really want to emphasize the logic here. When an economist says a factor model explains the cross-section of expected returns, he’s saying that investors all have the same $K \geq 1$ risk factors in mind when making their respective portfolio choices. If one of these risk factors were to go negative, investors would consider it a bad state of the world; if all of them were to go negative, it’d be apocalyptic. The clain is that investors are all really worried about having enough money when these various kinds of bad outcomes occur. So, as a result, they’re willing to pay extra for assets whose returns are less correlated with these $K$ risk factors—i.e., for assets that are more likely to have positive returns when risk factors are negative. Therefore, in equilibrium, these assets will have higher prices today and thus lower expected future returns.

Little Green Men

By now, researchers have proposed lots of different candidate factor models. Some might even say there’s a “factor zoo”. Each model makes its own claim about a specific set of risk factors that all investors are worried about. And yet, there’s no general consensus among researchers (let alone investors) about which is correct. This disagreement should already give you pause, but now ask yourself this: If you have to use an ML algorithm to identify the correct “set of priced risk factors” in investors’ “true asset-pricing model”, how did investors find these variables in the first place? A few investors certainly understand the ML toolkit today, but most certainly do not. And, no one was aware of these ideas twenty something years ago.

As a thought experiment, suppose that tomorrow while doing other research you encounter an ML algorithm, which was first discovered in 2010, that always outputs a factor model which perfectly explains the cross-section of expected returns. Does it make sense to claim that this ML algorithm is able to find the “true asset-pricing model” at work in, say, 1985? By assumption, when you feed data from 1985 into the algorithm, the output will be a “set of priced risk factors” that perfectly explains the cross-section of expected returns in 1985. But, could these risk factors possibly reflect how Madonna-loving 1985 investors were thinking about risk and return? No. Of course not. If the algorithm wasn’t discovered until 2010, could 1985 investors have known about this “set of priced risk factors”?

Let’s make the thought experiment even more extreme. Suppose that little green men come to earth tomorrow and secretly give you an alien computer that operates based on principles never before seen by humans. There’s absolutely nothing like it here on earth. And, this advanced computer comes pre-programmed with correspondingly advanced ML algorithms. And, imagine that one of these algorithms works like the algorithm described above. It always outputs a set of $K \geq 1$ risk factors that perfectly explain the cross-section of expected returns. Do these risk factors tell us anything about how human investors view risk in earthly markets? Again: No. Of course not. To discover them you had to use an advanced alien technology with absolutely no analog here on earth. So, how could this risk factors be capturing earthly investors’ views about risk and return? The algorithm simply produces an excellent set of predictive variables that take the form of partial correlations with each asset’s returns—i.e., that take the form of $\beta_{n,t-1}^{(k)}$ s.

Machine Learning

I’m quite bullish about the prospects of ML in asset pricing. I think researchers have barely scratched the surface. I just don’t think that approach #2—i.e., searching for the “true asset-pricing model”/”set of priced risk factors”—is a sensible way to apply the ML toolkit. Although academics tend to poo poo approach #1 as lacking in economic content, it’s simply not true. There are lots of situations where we’re perfectly happy to have good return predictions at the price of not understanding where this fit comes from. Traders are obviously OK with this Faustian bargain. But, so too are researchers. It’s not like the Fama-French 3-factor model is popular because we have an economic understanding of what the size and value factors represent.

Financial economists like to think about the market and its investors as something separate. But, it’s just not so. We are the investors in our asset-pricing models. There’s no separation. And, this fact should be reflected in our models. For me, this is the most interesting economic insight that comes with applying ML algorithms to study asset prices. If the tools that we use to find predictors change, then the predictors that our theoretical investors find should change, too. In his AFA presidential address, John Cochrane writes that, “to address these questions in the zoo of new variables, I suspect we will have to use different methods… For one variable, portfolio sorts and regressions both work. But we cannot chop portfolios $27$ ways… so, I do not see how to do it by a high-dimensional portfolio sort.” Whatever those different methods end up being (ML or otherwise), we’d better not be modeling asset-pricing equilibria the same way after they get introduced.

Factor Models

Little Green Men

Machine Learning

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