Replacing linear factors with a non-linear, characteristic approach in quant equity

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All right, we're back with another clip from the archives, this time it's season 4, episode 9 with Vivek Viswanathan. For three decades, equity quants have largely lived under the authoritative rule of the Fama French three-factor model and other linear sorts. In this episode, Vivek provides a cogent alternative to that orthodoxy. Specifically, he explains why an unconstrained, characteristic-driven portfolio will be able to more efficiently capture behavioral-based market anomalies. I think it's an absolute master class for alternative thinking in quant equity. I want to add, it was really tough to clip particular parts of this episode. I decided I wanted to go for one continuous section in the middle, but Vivek's comments both at the beginning and at the end about Chinese markets, I think provide a tremendous example and insight into how people can think about finding alpha in alternative markets, and the episode is certainly worth a listen just for those sections themselves.

Alright, let's dive in. I hope you enjoy.

So you spent a good part of your career, I think it was nine years at research affiliates, which was a firm that was a real innovator in the smart beta space. I know since joining Raelian, you sort of shed off that Factor-oriented view of the world, and I'm curious as to what led to that change. This one's gonna be a long and technical one. So forgive the soliloquy I'm about to give here.

So let me clear out some potential semantic misunderstanding here. When I say I don't have a factor view of the world, I mean anomalies as linear factors. I believe that the equity market factor is an extremely useful tool. I believe industries and countries drive the covariance matrix. So, not having a factor view of the world is specifically not viewing anomalies as risk factors or linear, univariate mappings to return, and I hope to make that claim that when you think about anomalies, you need not think about risk factors and about long-short portfolios built on linear characteristics sorts.

Those are useful for publishing papers and for heuristic understanding. But they are not useful for portfolio construction or for mapping how characteristics relate to expected returns. There are a few high-level ideas.

I want to touch on the first is where the idea of factors come from. Factors are an idea that stems from arbitrage pricing theory, or APT, that the expected return of an asset is a linear combination of priced risks. And you might argue that even if you don't believe in APT with respect to all these anomaly factors, you can still believe that factors are a useful tool. I'll address that later. But for now, I want to talk about why the characteristics that we see that predict return should not be considered priced risk factors.

There are a few things we would expect to see if we believe things like value, growth, profitability, cash flow, accruals, post-earnings announcement drift, news sentiment, momentum, whether those were price risks. So first, one that few people talk about is that we would expect to see products that sold high valuation, low profitability, negative earnings, surprising every other negative factor tilt you can imagine. Because those would be great hedges for whatever latent risk investors are trying to avoid if these anomalies are risk. It stands to reason that investors would want to hedge these risks, would happily pay their counterparties to hedge these risks. But those products largely don't exist.

There's no hedge or latent risk factor product that earns a negative expected return in exchange for risk hedging. No one is selling these supposed insurance products to investors; moreover, no one has figured out what risks this multitude of factors is meant to price. What risk is profitability a proxy for? What risk is conservative accounting a proxy for? What about earnings growth? What about momentum? No one really seems to know.

Now, to certainly the case of value and size embodies some aspect of risk. Imagine you have two firms with exactly the same expected cash flows, but one is riskier than the other; then the riskier one will have a higher discount rate and will be smaller cap and deeper value. So, if there are latent risks, they will probably show up at least in part in value and size. But it's really difficult to make the case for the multitude of other signals.

And of course, if factors were price risks, we would expect to see covariances driving returns, not characteristics, but they don't. That was found by Daniel and Titman in 1996 with respect to value. But it's also been shown about most other anomalies as well. And in fact, my thesis shows this with regard to the '97 McLean and Ponta factors at least. In 80% of cases, and compute globally, it's characteristics that drive return.

It's true that there have been attempts to recover this idea that if you squint hard enough, covariances are driving returns. So in 2019, Gukelly and Shows used a variant of an autoencoder to account for the predictability of returns in that encoder, and say you can compress the information in characteristics and outperform other linear asset pricing models. And of course you can, because, as we'll discuss, the relationship between characteristics and returns is nonlinear. So if you put a nonlinear model against a few linear ones, you're probably gonna win out. But that doesn't mean you found price risk.

It just means that auto encoders are good at compressing information, and nonlinear models outperform linear ones. The real test would be putting it up against a neural network that didn't have a bottleneck layer in it. In other words, does the act of compressing your factor space improve or hurt your predictability, and from all of our analysis?

Compressing the factor space hurts you really want that a hundred plus signals helping your prediction. Yet another piece of evidence against the risk factor approach to understanding anomalies is how we construct covariance matrices. When we build covariance matrices, you might account for market, industries, countries, principal components of return, and admittedly size and value, but you wouldn't account for post earnings announcement drift or operating profitability factors. They are not meaningful parts of the covariance structure now. Why does this matter? Who cares whether anyone thinks factors are price risk? It matters. Because if you believed that factors were price risks, you would sit on one side of the trade forever. Expecting to get paid for taking that risk. You would expect that someone would happily pay you a premium to take on this factor risk, but it's a behavioral anomaly. Then you're keenly aware that the market might become efficient to that signal.

But before I talk about why factors are not even particularly useful as tools, I want to talk about the number of factors. So there's this idea of a factor zoo brought up by John Cochran that there are too many factors and there's a mystery here. And indeed he's right. There is a mystery but the way much of the literature has tried to explain it is not correct. Way back when I read Harvey, Leo, and Zhu's and the cross-section of expected return, and later Ho, Shui, and Jang's replicating anomalies, and if you read those papers, you might think that most cross-sectional anomalies are data-stripped and that markets are more efficient than the anomaly literature suggests, or that most anomalies can be collapsed into other anomalies so that they're only a handful of anomalies out there. And by the way, I briefly believed all of those things, right? Very briefly though, like maybe a few months. Some pretty quick empirical test can refute them. So as it turns out, you really want those 100 plus signals in your expected return model.

The first bit of evidence for this is Jacobs and Mahler's 2019 paper anomalies across the globe. They found that if you create cap-weighted long-short portfolios out of the 241 anomalies that have been found in the literature and equally weight them—that is, equally weight the factors—then you will earn significant excess returns in 38 out of 39 markets, and that's significant at a 0.01 level. The odd one out happens to be Turkey. Now let's think about this for a second. These anomalies were discovered in the US. If they were the result of data snooping, then they would not produce excess returns in other markets. But they deliver excess returns, on average, in 38 out of 39 markets.

Now you might argue that the returns might be driven by only a handful of these 241 anomalies, maybe five out of the 241. Let's say that's where my thesis comes in. I didn't look at 241 anomalies, but I did look at the 86 out of 97 McLean and Ponto anomalies that can be calculated globally from 1995 to 2018, and 44 out of 86 of those earn a significant Fama French three-factor alpha, and that's at a point of five level. Two of those 86 are size and value themselves. So they can't produce a Fama French three-factor alpha. So, 44 out of 84 anomalies deliver a significant Fama French three-factor alpha globally. That is over half of the anomalies found in the literature. That's too many significant anomalies for the result to be driven purely by luck.

Now you might ask, why does the Fama French three-factor alpha matter? Why not just look at whether the returns are significant in their own right without looking at the Fama French three-factor alpha? The answer is that almost every signal is a quality signal and quality signals as a whole tend to be negatively correlated with size and value.

AQR showed this with respect to size and their size matters. The ideal holds for a variety of quality factors, even ones that come from alternative data sources. So, that is why the FF three-factor alpha is so important, and why you will see so many anomalies with insignificant return but significant FF three-factor alphas.

Now, I want to briefly talk about Ho-Shui and Jiang's paper replicating anomalies because there's a particular issue with it that some of these other papers that try to collapse anomalies have as well. They first look at factors that earn significant returns in their own right, and then they try to explain only those factors. The insignificant factors are never tested against a factor model. But this is the issue with that approach: if you regress insignificant factors on a set of other factors, those insignificant factors might have a positively significant alpha, and that is now a new anomaly. You don't get to ignore those guys. If your asset pricing anomaly generates more significant alphas than there were significant factor returns in the first place, that is not a successful model because there is now more significant factors than before; a linear combination of factors is still a factor.

So, I hit on two things already: one is that factors are behavioral anomalies. The second is that there are many anomalies now. We have these a hundred or more potential anomalies, and we don't believe they're risk factors. So what are they? They are information about long horizon expected return of the firm, and this is information that can interact and be nonlinear, and it can be incorporated into price in one period or another, and this is exactly how we would view the world if we never learned about factors. We look at a company and say, it has this gross profitability, this operating profitability, these accruals, I read this negative piece of news and this positive thing from their annual report, they're being sued by a competitor for anti-competitive practices. And you take all this information and you get a value for the firm, and you compare that to the price and you say this stock is overvalued or undervalued; you want your model to effectively do the same thing practically.

That means you need a machine learning model, at the very least you need linear Ridge so you can prevent yourself from overfitting to those a hundred-plus signals. But if you want interactions and nonlinearities, you need gradient boosting and random forests, if not feed-forward neural networks. You don't have to predict the value of the firm, though. Yeah, practically you're predicting return, but you want to include all of those signals and throw in price and valuation as additional features.

Now, once you have your model, how do you improve your product? Instead of trying to build new cross-sectional factors, you're trying to provide information to your model about the value of this firm so it can learn more about expected return. So, you're still engineering features, you're just not building factors.

In summary, these anomalies aren't risk factors; they're characteristics that predict return, and they can do so through interactions and nonlinearities. There are many, many characteristics, and you want to search for this data everywhere you can and use the latest tools to try to encode information about stock expected returns.

So, I think for a lot of quants who got their education in the last 15 years, it can be very difficult to sort of mentally unchain themselves from the traditional factor framework, and often it sort of requires seeing a non-factor framework in practice to sort of break that mental box.

So I'm curious whether you could go into Maybe what that non-factor framework does look like in practice versus the more traditional linear models. Absolutely, first you need an expected return model and assuming you're predicting cross-sectional equity returns, that model should utilize.

Some subset of things that fall under machinery, if you aren't trading in individual stocks, Then what you do is dependent on the amount of data that you have in cross-sectional equities. You generally have a lot, so your models can be far more sophisticated.

If you're looking at commodity futures on a monthly horizon, I might use linear Ridge. But the cross-section is probably not big enough for anything more complicated.

If you're looking at asset allocation on the other hand, It's even fine to use static weights there, because it's hard to build conditional expected returns in that space that are better than unconditional expected returns.

When you're predicting cross-sectional equity returns with a hundred signals, You do not want to use OLS because you will overfit and get no excess returns in your auto sample back test. I can't emphasize enough how different your experience will be using OLS.

Versus using a model with some form of regularization built-in that prevents your parameters from overfitting. But if you're predicting on an annual horizon, You can rely on linear Ridge and maybe random force.

If you're predicting on a quarterly or monthly horizon, You want linear Ridge, random force, and gradient boosting; you might be able to use neural networks on monthly returns, But we don't.

If you're predicting on a fine horizon like daily or intraday, Then you definitely want to use neural networks. Probably only neural networks. Now, the next and most important step is you predict returns in each time step.

Let's say every month, using models that could only be fit in prior months. If you're predicting a return in July 2010, you can only use models that fit data up until June 2010, in other words, You can only use models that were fit to prior period data to predict the next period's return.

That is how real life works, and you want your model to behave similar. These are what are called pseudo out-of-sample or quasi out-of-sample back test. We tend to just call them out-of-sample back test, because the word back test already implies that they can't literally be out of sample.

Now, let's go back to our predicting returns in 2010 example. Let's say your data set starts in 1995, in 2010, You have 15 years of data to fit your returns, in 2020, You have 25 years of data to fit your returns. So your back test assumes you have less information than you do now.

That does mean your back test will somewhat understate your ability to earn returns, But that's probably more than offset by the fact that markets get more efficient over time.

Now once you have your expected returns, you need a covariance matrix. There are many different ways to do this. Well, you probably want to account for structural sources of covariance like industry, country, size, and assume the remaining variance is residual.

But there are other ways, for example using some number of principal components and then assuming the remaining variances residual works fine, too. You also need to account for various decay horizons and you need to shrink the loadings on sources of covariance.

I'm gonna skip over that because it's probably its own conversation and it's just not good podcast conversation.

You need a whiteboard or something now. You have your machine learning expected returns and you have this covariance matrix. You want to do mean tracking error optimization with respect to your benchmark if you're a relative return investor. If you're trying to max my sharp ratio, you do mean variance optimization. You also want to have constraints on industry weights, country weights, and individual stock weights.

It's difficult to communicate your model that you're uncertain about the covariance matrix. And constraints are a clumsy but effective way to say, look, I did my best on the covariance, but take it with a grain of salt. Let's make sure I not to get too crazy here with the offsetting bets.

I mentioned earlier that an equally weighted basket of China factors has something like a 1.5 information ratio from 2010 until now, using all these methods on global and China specific signals in a walk-forward out-of-sample way. An optimized portfolio in China large cap will produce an IR of about 2.8 after transaction costs since 2010. A walk-forward optimized portfolio in China small produces IR of about 3.5 after transaction costs. It will do that partially by building good expect returns. But also by drastically reducing tracking. At that information ratio, your tracking error will be between four and six percent. So, your back-tested expect a return will be 12 to 24 percent excess returns.

I'm sorry, given such a low tracking error, you might get classified as enhanced indexing despite your expected return. That's something to be wary of. To deal with this, you can relax your constraints and reduce your risk aversion, but your information ratio will fall. You cannot maintain very high information ratios while taking high tracking error because of the zero lower bound on weights.

Now if you have a long-short portfolio, you can kind of go to town here. Now, there's some things worth realizing about expected return loss. If the model doesn't think you can perform well on a market, it's not gonna bullshit you. So, if you run the same walk-forward optimized model in US large cap, you would earn an excess return of 2% and an information ratio of 0.6 after transaction costs since 2010. In Hong Kong, you would get an information ratio of zero after transaction costs. You wouldn't earn any excess returns on the market. If you run an EM large, you get about 1.6 after transaction costs. If you run in China large, it's about 2.8.

If you're trying to do walk-forward prediction in an efficient market, you're gonna have mediocre results as you would in real life. If 90% of the active managers in the US underperform over any 10-year period, it would be strange if we found a model that did incredibly well.

Now another useful insight is that if you build a solid covariance matrix, your optimized model may show no significant factor tilts using traditional factor attribution. And we actually had a discussion about this at our firm just last week. We fed one of our newly launched products into some factor attribution software and lo and behold, we had virtually no factor tilts relative to our benchmark. The one factor tilt we did have, momentum, did not perform well over the past three months or so, and yet our active return over this benchmark that we had no significant factor tilts against was a positive 5%. And this is because factor tilts don't really matter. They don't tell you they expect a return of the portfolio. They don't even do a good job classifying the risk of your portfolio.

A profitable retail stock in China A shares

It's not going to be any more correlated with a profitable tech stock in China H shares than an unprofitable tech stock in China H Shares in the end, even though you're using value signals and profitability signals and many other signals those very well may not show up as linear factor tilts since your models nonlinear and accounts for covariance risk and

Now naturally clients might have the question if you can't use factor attribution

How do you know if you're taking the same risk across two portfolios and cross two managers?

You would take the correlation of the active return of the two portfolios now if you're worried that this is too backward looking you can build an expected covariance matrix of

The underlying securities and calculate the expected correlation of the two portfolios

That's hard

But at least it's accurate

if you're using factor loadings to compare portfolios

You might as well use astrology for all the good. It's going to do you

I

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