The Fama-Miller Center at Chicago Booth jointly with EDHEC and the Review of Financial Studies will host a conference on September 27–28, 2018 in Chicago, on the theme “New Methods for the Cross Section of Returns.” Conference announcement here and call for papers here.

Papers are invited for submission on this broad theme, including:

- Which characteristics provide incremental information for expected returns?
- How can we tame the factor zoo?
- What are the key factors explaining cross-sectional variation in expected returns?
- How many factors do we need to explain the cross section?
- How can we distinguish between competing factor models?
- Do anomaly returns correspond to new factors?

Why a blog post for this among the hundreds of interesting conferences? Naked self-interest. I agreed to give the keynote talk, so the better the conference papers, the more fun I have!

This is, I think, a hot topic, and lots of people are making good progress on it. It's a great time for a conference, and I look forward to catching up and trying to integrate what has been done and were we have to go.

My sense of the topic and the challenge: (Some of this reprises points in "Discount rates," but not all)

Both expected returns and covariances seem to be stable functions of characteristics, like size and book/market ratio. The expected return and covariance of an individual stock seems to vary a lot over time. So we need to build ER(characteristics) and then see if it lines up with covariance(R, factors | characteristics), where factors are also portfolios formed on the basis of characteristics.

But which characteristics? Already there is a zoo of somewhat overlapping characteristics. There is the data snooping and out of sample success question, and the multiple regression question. I think index construction rather than multiple regression is the way to think about it though. Of 55 accounting variables that seem to forecast returns, each one adds some information. The question is not, can we get by with just 5 of them, but which 5 linear combinations of all of them? Yet a vastly overfit multiple regression is not the answer.

The main problem is interaction terms. Is the book to market effect stronger among small firms? In

Does d matter? Yes it does! But with 300 right hand variables, and now you want nonlinearities, the scope of interaction effects explodes to far greater than the number of data points.

"Big data" techniques may help. In the end "machine learning" amounts to huge nonlinear regressions, and the various algorithms amount to ways to impose useful structure on interaction effects. But they are designed for problems with a lot more data and a lot better R2 than we have, and since they don't tell us what the question is that they answer, it's not obvious that structure that helps to predict which cat video you will choose also helps to model our interaction effects.

Even here, it seems we should start by thinking what these characteristics mean. One overarching theme for me is the present value relation. Price/Dividend or Book/Market = a long present value of future returns and a long present value of future dividend growth or cash flow growth. Thus, a variable that helps to forecast cashflows give P/D or B/M must also forecast returns. If you can forecast cashflows are higher but that doesn't raise the price, you must also forecast returns are lower. (Old point, but we're here to make progress on old points.) It makes sense to me in this context that many accounting variables help to forecast returns -- because they help to forecast cashflows, i.e. to clean up B/M of its cashflow component, or to clean B up.

Similarly, the

This suggests in the end that we're really not interested in which 5 accounting variables forecast returns. We're not really even interested in which linear combinations of 55 accounting variables forecast returns. We need to take a third step, past firm name as key characteristic, past accounting variables as characteristic, to the underlying quantity like expected cashflow growth as the key characteristic.

But not all characteristics have an interpretable meaning like this. What does momentum mean anyway? Eventually, even characteristics like this should have some underlying economic unity, like trading volume or dealer leverage or something similar.

Beyond the obvious data mining, there is a phenomenon worth pursuing, suggested by Rob Arnott Noah Beck, and Vitali Kalesnik: If you look for characteristics or strategies that earn high returns, it is likely that you find ones that started at low valuations and ended up at high valuations in your sample. At a minimum, searches for characteristics that forecast returns should control for such in-sample rises in valuations.

Different characteristics also ought to matter for different horizons of expected returns. Once we learn returns are not iid, it is quite possible that there are some signals that work for short horizons, and others at longer horizons. Some signals are much more persistent than others. This means that the expected return (characteristics) may look quite different at different horizons -- and hopefully a lot simpler for the long-run investor. Price pressure signals may matter a lot for day and week returns, and dissappear in months and years. But of course a long run investor in a highly managed portfolio can unwind horizons. Still many investors do buy and hold...

*Update:*In response to the good nonlinear comment. Yes! What is the functional form by which a characteristic is related to returns? To what extent are we finding results in extreme 1-10 equal weighted portfolios that are just dusty corners of the market? The trouble is that nonlinearities chew up scarce data points quickly. Worse, what are the interaction terms? ER(c1) is bad enough to estimate nonparametrically (expected returns in characteristic sorted portfolios are just an inefficient box-weighted nonparametric regression), but ER(c1, c2) explodes, and ER(c1, ... c300) is truly hopeless.The main problem is interaction terms. Is the book to market effect stronger among small firms? In

R_t+1 = a + b*size_t + c*B/M_t + d*(size_t x B/M_t) + error_t+1

Does d matter? Yes it does! But with 300 right hand variables, and now you want nonlinearities, the scope of interaction effects explodes to far greater than the number of data points.

"Big data" techniques may help. In the end "machine learning" amounts to huge nonlinear regressions, and the various algorithms amount to ways to impose useful structure on interaction effects. But they are designed for problems with a lot more data and a lot better R2 than we have, and since they don't tell us what the question is that they answer, it's not obvious that structure that helps to predict which cat video you will choose also helps to model our interaction effects.

*End update.*Even here, it seems we should start by thinking what these characteristics mean. One overarching theme for me is the present value relation. Price/Dividend or Book/Market = a long present value of future returns and a long present value of future dividend growth or cash flow growth. Thus, a variable that helps to forecast cashflows give P/D or B/M must also forecast returns. If you can forecast cashflows are higher but that doesn't raise the price, you must also forecast returns are lower. (Old point, but we're here to make progress on old points.) It makes sense to me in this context that many accounting variables help to forecast returns -- because they help to forecast cashflows, i.e. to clean up B/M of its cashflow component, or to clean B up.

Similarly, the

*term structure*of risk premiums matters. If I can forecast returns in 10 years given price will be higher, then I must forecast that returns in one year will be lower.This suggests in the end that we're really not interested in which 5 accounting variables forecast returns. We're not really even interested in which linear combinations of 55 accounting variables forecast returns. We need to take a third step, past firm name as key characteristic, past accounting variables as characteristic, to the underlying quantity like expected cashflow growth as the key characteristic.

But not all characteristics have an interpretable meaning like this. What does momentum mean anyway? Eventually, even characteristics like this should have some underlying economic unity, like trading volume or dealer leverage or something similar.

Beyond the obvious data mining, there is a phenomenon worth pursuing, suggested by Rob Arnott Noah Beck, and Vitali Kalesnik: If you look for characteristics or strategies that earn high returns, it is likely that you find ones that started at low valuations and ended up at high valuations in your sample. At a minimum, searches for characteristics that forecast returns should control for such in-sample rises in valuations.

Different characteristics also ought to matter for different horizons of expected returns. Once we learn returns are not iid, it is quite possible that there are some signals that work for short horizons, and others at longer horizons. Some signals are much more persistent than others. This means that the expected return (characteristics) may look quite different at different horizons -- and hopefully a lot simpler for the long-run investor. Price pressure signals may matter a lot for day and week returns, and dissappear in months and years. But of course a long run investor in a highly managed portfolio can unwind horizons. Still many investors do buy and hold...

The mapping between characteristics and factors remains a puzzle. Each characteristic that forecasts returns seems to correspond to a new factor. The 1-10 portfolio sorted on the characteristic earns a positive return, but the 12, ... 9,10 portfolio returns line up with their betas on a high minus low factor. And the new factor seems uncorrelated with the old factors. That's nice in revealing APT logic at work, and when it holds it's nice confirmation that the expected return is not just fishing bias. A fished expected return need not correspond to a covariance. But are there really dozens if not more priced factors? (On both issues, I'm still a fan of Charles Clarke's synthesis. Expected return itself is the summary characteristic, and level slope and curvature of expected returns is the summary set of priced factors. This seems to escape the troubles of 5+ factor models that all look just about alike.)

Well, maybe. Who said there should only be one priced factor? Of the assumptions behind the CAPM, the one that says we have shared all risks and all hold the same total wealth portfolio is perhaps the fishiest. Most investors have substantial outside income, which should drive them away from the common portfolio. Of course they should be avoiding stocks correlated with their business income, not doubling or tripling down on it. (Most of my neighbors' jobs, houses, and portfolios are all loaded into tech!) But in that world there could well be dozens if not hundreds of risks on sale. Anything a mass of investors wants to avoid will generate a premium to induce the other mass of investors to over-weight it. So there is no presumption other than esthetic really that we do not have as many risk factors as there are brands of toothpaste.

Here too I'm slipping in to the common presumption that betas are cashflow betas. Once we learn about the large amount of price variation that comes from discount rates, then betas are discount rate betas -- correlations between the expected return of one stock rising and the expected return of another group of stocks rising. The economic foundations of multidimensional discount rate betas are at best confusing to me. Lots to do. That also leaves open the trading and institutional finance aspects. Maybe short-run discount rate betas are basic supply and demand sorts of things, limited risk absorption by the existing traders.

The existing traders at least can spread the price pressure out across correlated securities, leading to factor APT behavior, but they cannot eliminate it. Of course new traders should come in and wipe out such transitory opportunities, but arbitrage does not wipe out risk premiums.

*All*of us must trade and share in a new risk before the premium goes away. Risk premiums depend on the

*average*investor.

Here too, it seems there is not

*enough*intermediation. Short-lasting expected returns mean temporary price components. The buy and hold investor can ignore temporary price movements, as he or she can ignore price fluctuations of long-term bonds. Yes, it's better still to market time them if they exist. But the average investor needs an intermediary to do that. And, there still is the embarrassment of all of these temporary price pressure stories that intermediaries on the whole seem amazingly incapable of delivering alpha.

Well, I should not have gone on so long. I'm not on the program committee. I'm a bit aware of a lot of good work on these issues, and a lot more to go. I hope to learn a lot in September.

Great to know! The cross section of expected returns is one of the most fascinating and important topics in asset pricing.

ReplyDelete"which 5 linear combinations of all of them"

ReplyDeleteAll that work, by so many people, and it seems everyone is using least squares fits of linear models trying to predict non-linear effects.

Good point. See update.

DeleteHi John,

ReplyDeleteThanks for the shout out. My journey through "Discount Rates" continues. Matthew Linn and I posted a new paper, "Characteristics and the Cross-section of Covariances," exploring the idea (your idea) that covariances could be modeled as functions of characteristics.

[First draft here: https://ssrn.com/abstract=3141622]

There's a lot still to do, but we think it's pretty interesting. The conference sounds great, we'll be submitting. We'd love to get comments at this stage.

A little shy to ask because I am sure that I am missing something obvious (hence the "anonymous" choice...), but it is a serious question :)

ReplyDelete"Who said there should only be one priced factor?"

Given that there is always a single beta representation, shouldn't (absolutely) all risk premiums (perfectly) covary with each other?

Even if the SDF is made up of several factors, shouldn't it always be possible to find the one on the MVF that prices everything, and then all premiums should covary with this factor (and therefore with each other)?