Monday, December 15, 2014

Level, Slope and Curve Factor Model - What's this all about?

John Cochrane recently blogged about my paper.  My co-blogger asked me to explain it a little more, and I realized that a lot is lost if you aren't deep into the finance jargon.  Additionally, what's the deeper significance of the whole endeavor.  Here is an attempt to explain my goal and interpretation in a broader context.

First, let me define a "factor." A factor is just meant to describe common movements across stocks. All technology stocks go up together when there is some good news about the tech industry. All energy stocks go down together on some bad energy news (maybe lower world energy demand). The goal is to stop describing a stock as "Ford" and start describing it as a combination of factors.

The factors we really care about aren't things like industries. We want to understand the "priced" risk factors. These are the factors that represent risk to investors. I can diversify my exposure to ups and downs in tech or industry or automotives by just buying lots stocks. So we don't expect any higher returns to holding tech stocks.

Additionally, it turns out there are lots of things that predict stock returns.  For instance, certain characteristics of companies: small companies have higher returns than large companies, "cheap" companies have higher returns than expensive companies. Each of these are associated with factors. Small stocks go up and down together and to some degree move opposite large stocks. Cheap stocks go up and down and to some degree move opposite expensive stocks.

The literature now is postulating more and more possible risk factors.  In contrast,  I am arguing a lot of those factors are manifestations of three main risk factors level, slope and curve. Here's what I do. I use all of these characteristics and I sort stocks into "high return" stocks and "low return" stocks. I sort into 25 portfolios. The 25th portfolio we expect to have really high returns. The 1st we expect to have really low returns. The factors describe how those portfolios move relative to each other.

Level describes that stocks go up and down together. This has been known for 50 years or more. On good economic news all stocks go up, on bad economic news all stocks go down. The most common model of risk, the CAPM, basically describes all stocks risk as how much they go up and down with the overall market.

The "slope factor" describes that high return stocks and low return stocks often move opposite of one another. (I think) It means that whatever risk high return stocks are exposed to changes over time. Sometimes investors need to be compensated a lot to hold risky high return stocks (with high expected returns), sometimes not so much.

The "curve factor" describes that sometimes high return stocks and low return stocks go up or down together, while medium return stocks don't even move. The curve factor is the hardest to interpret. It may be related to the volatility of the slope factor above, but that's kind of speculative.

So now I can describe every stock as some manifestation of level, slope and curve. Yesterday, I needed theories to describe hundreds of priced factors. Now I only need theories to describe THREE!!! That's pretty good. And since we've been explaining level for years, we really just have two. So, I don't know why high return stocks and low return stocks move opposite each other, maybe some risk factor, as in these companies will do badly in bad times, maybe these stocks are susceptible to fits of irrational exuberance. But if I can explain level, slope and curve, I can explain why some stocks have higher returns than others.


  1. I am hoping you can help me understand why I shouldn't expect this result. As I mentioned on John Cochrane's blog, in factor models, like Fama-French, we have a hypothesis about latent factors causing expected returns, while a PCA reduces correlated observations into the most important linearly independent composites. Factor analysis and PCA are not the same thing, but we often expect that the factors will pick up some linear combination of the most relevant principal components.

    To me the result is a basis shift - project returns on factors with direct economic interpretation, like the Fama-French three factor model, or project onto a parsimonious description rooted on the principal components (giving level, slope, and curvature). If a factor model is good at explaining returns, it must capture most of the information in the largest principal components, hence it will capture level, slope, and curvature information. Is this not a re-statement of the same information of any successful factor model? This what I don't understand. Stated the other way, is it even possible for a successful factor model to not get level, slope, and curvature right? I do like the result, but is the significance more examining how well other factor models capture level, slope, and curvature information? Thanks for any comments, I am just trying to understand, not criticize.


  2. Thanks for your comment. "What is the significance?" and "should we have expected it?" are different questions.

    On the former, the short answer is we don't live in the FF 3 factor world anymore. There are 13 factors in my paper, including the Fama and French five factor model! There are many, many more in the literature.

    The standard procedure looks like this. Find a spread in returns on some firm characteristic. Ex. Illiquid stocks earn higher returns than liquid stocks. Next, build a factor by buying illiquid stocks and shorting liquid stocks. Finally, see if that factor prices other assets. If so, that factor represents priced risk.

    The world looks quite complicated from this perspective. There are potentially hundreds of factors. Are they all priced factors? Some certainly are, but even those, are all portions of those factors priced? What if sometimes value stocks move up and down together, because lots of energy stocks happen to be value stocks in one year and the factor movement is really an industry movement?

    My idea is to sort on all the things we've learned about expected returns first. Then ask, how do high return stocks move relative to low return stocks? The goal is to average out all the unpriced factors like industry and focus in on the priced factors.

    In the end, we need to write theory to explain the priced factors. How do we model why stocks behave this way. It would be very hard to do that for hundreds of factors, but the world looks simpler now. Maybe this is more manageable.

    I started from John's perspective in Discount Rates. His call to arms was how can we organize the cross-section? How can we make sense of it, so we can go back to writing theory? The paper attempts to organize the cross-section and provide a way forward.

    Should we have expected it? Maybe, but in some sense asking "what would we expect to see if we ran principal components on portfolios sorted by expected return?" misses the point. The big idea in my paper is why you should want to do this. I find the people that really like this paper are the ones that don't think the answer is surprising. As John says in his post, "And he comes up with level, slope, and curvature, which is always the answer and thus beautiful. We just had to know which question to ask."

    As John alludes in his post, the LSC structure is elegant. We see it in many places. To those, used to seeing it. They are more likely to think this is the right way of looking at the data. It just seems right. To the rest, I think the curvature factor is a bit of a hang up. If you ask most people what they'd expect to get, I think they'd say level and slope. The literature is filled with slope factors. Any time we sort in one dimension (value, momentum. size, investments...) we get a slope factor. To not get one would be a big violation of the APT. Curvature factors are pretty rare in the stock literature though.

    Hope that answers you question.

  3. Thank you for your reply. The more I think about your paper, the more I really like the result. I always felt that large factor models mostly represented sophisticated fishing expeditions with lots of unpriced factor contamination. Your result helps focus the problem by reducing everything to the returns themselves. Now factor models need to account for only three things, regardless of their complicated structure. It's a great way to approach the problem and helps us all focus on the economics driving returns, as opposed to getting lost in the weeds of the latest factor model.

    A beautifully simple, fantastic result. Congratulations!