Thursday, December 11, 2014

Level, Slope and Curve for Stocks

"The Level, Slope and Curve Factor Model for Stocks" is an interesting and important empirical finance paper by Charles Clarke at the  University of Connecticut.

Charles uses the Fama-French (2008) variables to forecast stock returns, i. e.,  size, book to market, momentum, net issues, accruals, investment, and profitability. \[ Ret_{i,t+1} = \beta_0 + \beta_1 Size_{i,t} + \beta_2 BtM_{i,t} + \beta_3 Mom_{i,t} + \beta_4 zeroNS_{i,t} + \beta_5 NS_{i,t} + \beta_6 negACC_{i,t} + \] \[ + \beta_7 posACC_{i,t} + \beta_8 dAtA_{i,t} + \beta_9 posROE_{i,t} + \beta_{10} negROE_{i,t} + e_{i,t+1} \] He forms 25 portfolios based on the predicted average return from this regression, from high to low expected returns.  Then, he finds the principal components of these 25 portfolio returns.

Source: Charles Clarke

And the result is... hold your breath... Level, Slope and Curvature! The picture on the left plots the weights and loadings of the first three factors. The x axis are the 25 portfolios, ranked from the one with low average returns to 25 with high average return. The graph represents the weights -- how you combine each portfolio to form each factor in turn -- and also the loadings -- how much each portfolio return moves when the corresponding factor moves by one.

No surprise, the 3 factors explain almost all the variance of the 25 portfolios returns, and the three factors provide a factor pricing model with very low alphas; the APT works.

Now, why am I so excited about this paper?

There are now dozens -- above 300 in the literature (see  Green, Hand, and Zhang and Harvey, Liu and Zhou) -- of variables that supposedly forecast stock returns in the cross section. The first, hard, question is which of these really matter, in a multiple regression sense, and how much data mining is there in the whole business?

The next, harder, and less examined, question is, how do these patterns in mean returns correspond to covariances?  Each variable seems also to be a factor in the variance sense -- assets sorted by variables that forecast returns turn out to move together ex-post. But how many such factors do we really need? To explain the cross-section of average returns, do we need growth and profitability factors in the presence of value? Look at Fama and French and  Robert Novy-Marx wrestling with one factor vs. another.  Discount Rates wrestled with this question, suggesting that we need to model the covariance matrix as a function of characteristics, essentially running regressions of the product \( R_{i,t+1}R_{j,t+1} \) on the same right hand variables, somehow factor analyze that, somehow sort through the same multiple regression/fishing problem to see which characteristics are really important to second moments, and then see if the first moment function of characteristics is linearly proportional to covariance as a function of characteristics. Ugh.

Charles cuts through the latter huge multiple-regression chaos. His big idea is,  look at the only characteristic that matters, the expected return itself!  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. The fishing problem in expected returns remains, but relating the expected returns to factors is much simpler.

More deeply, I think Charles is leading us down a second step of how we think about asset pricing models. First, we thought of expected return and betas of individual companies. But those are unstable over time, so on average all companies look about the same. Then, we thought of expected return and betas as functions of characteristics like size and book to market, ignoring the company name. That worked well with one or two characteristics, but it's falling apart with hundreds of characteristics. By using expected return itself as the only characteristic for second moments, Charles dramatically simplifies the task.

Lustig, Roussanov and Verdehlan  did something quite similar for the carry trade. Sorting countries by expected return, they found a stable structure, and level slope and curvature factors; they found the slope factor accounted for expected returns.  But that was still basically using only one signal, so I didn't see the big point. In Charles' paper, the level slope and curvature factors of the expected-return portfolios allow you to  avoid the whole highly multivariate modeling of the covariance matrix.


(Students: factor analysis is really easy. [Q,L] = eig(cov(rx)) in matlab, where rx is the T x N vector of returns. The columns of Q are then the weights and loadings of the principal components. Detailed explanation starting p. 551 here. )


  1. This post made me think of a 2005 paper "Level-Slope-Curvature - Fact or
    Artefact?" by Lord and Pessler.

    Still interesting, but it makes me wonder if the pretty results you see are due to the nature of the PCA. I need to read the Clarke paper though.

  2. I'm am completely unknowledgeable on this topic so not expecting a reply. That said could level of returns, slope of returns and curvature of returns mean level of risk, sensitivity to risk and sensitivity to sensitivity of risk..

  3. Small correction: factor analysis \neq principal components.

  4. By doing this, how we can separate the style of the stocks? I mean we known that there are some common factors but we cannot say that this stock is like value or momentum etc. It seems like the Roll theorem that there is always a MV portfolio which can be used as the only factor pricing but we cannot characterize the individual stocks (like value, mom….)

  5. By doing this, how we can separate the style of the stocks? I mean we known that there are some common factors but we cannot say that this stock is like value or momentum etc. It seems like the Roll theorem that there is always a MV portfolio which can be used as the only factor pricing but we cannot characterize the individual stocks (like value, mom….)

  6. Isn't it a mathematical tautology that if you try to explain data ordinally sorted, you'll get three factors that 'look like' level, slope, and curvature? Apply PCA to the yield curve, you'll get the same thing. I bet you get the level, slope and curvature applied to country GDP growth arrayed from Haiti to Singapore.

    1. I can't see the tautology. Do you have a reference?

    2. I wasn't able to get level, slope, and curvative when basically doing this to simulated factor models (taking an approach similar to RMT but with independent uncorrelated factors and some simple assumptions about the distributions of coefficients).

      However, the paper I reference above suggests that there are some conditions under which you will get eigenvectors that look like that. Perhaps more realistic assumptions about factor models will let me generate that result in an RMT-like fashion.

    3. Think about a monotonic curve (eg, the yield curve, the 25 expected return stock portfolios). The pcs that will best explain that will be 1) level, 2) slope and 3) curvature.

    4. I do appreciate the comments on the paper. I've tried to join the discussion myself, but my comments didn't make it through. Here is one more try.

      It isn't generally true that all that is required is a monotonic sort to produce level, slope and curve. It is true that this relationship shows up in the yield curve (in yields and returns). This is well known and cited in the paper. John even covers this in his online PhD course.

      For instance, if you do a simulation where the CAPM holds and sort on beta, you should get just a level factor. I don't know the bond literature as well, but I think if you simulate a Vasicek model and run PCA on the term structure, you'll just get a level factor. I'm open to some mechanical explanation, but I don't think it's just monotonic sorting.

      I do think it's worth pondering, whether the analogy to the yield curve is worth pondering. After all, we know what a yield curve looks like. We know that the level factor in bonds means the yield curve goes up and down sometimes (say persistent inflation news). We know the slope factor tells us the yield curve steepens or flattens. And the curvature factor tells us the yield curve bends. This is describing yields, but also returns.

      We aren't used to thinking about an equivalent yield curve for stocks. But I wonder, is there an equivalent expected return curve that we don't really observe. Across stocks, expected returns rise as a whole (level), steepens as high return stocks get even higher risk premiums (slope), and even bends (curve).

    5. My view: Level slope and curve show up a lot because, the covariance matrices of lots of things often have this structure. There's nothing mechanical about it. If you factor analyze [ 1 rho ; rho 1] you get level and slope. That's another way of saying positive correlation. If you factor analyze [ 1 0; 0 1] you don't. We're seeing a common structure in the world not a mechanical artifact.

    6. Along the lines of Charlie Clarke's last paragraph, the level factor here is just regular equity market beta: the average return across the portfolios is the average return across all stocks.

      The slope factor suggests that high expected return stocks are more correlated to each other than they are to low expected return stocks, which may have arisen due to the use of Fama-French factors, many of which have the same property. It would be interesting to see the correlation of the slope factor to, say, shocks in volatility, i.e., do high-return stocks have high returns because they perform poorly during crises? I believe that Lustig, Roussanov and Verdehlan found such correlation between the currency slope factor and global equity volatility.

      I'm not sure how to interpret the curvature factor. If it's long average-return stocks and short both low-return and high-return stocks, then does it have an expected return close to zero? If so, then curvature would seem to represent an unpriced risk factor.

  7. Dr. Cochrane,
    Two members in the blogosphere mention you.
    First is Scott Sumner, in a praiseworthy light

    Second is Noah Smith on IS-LM models (In a more combative light)

    It seems that you are making waves!

  8. Forgive my ignorance. But isn't the model trying to explain next period returns using current period attributes? To the extent the model has a very high R-square wouldn't that fly in the face of efficient market theory? All the attributes are known and if they are known, they should be priced in and if they are priced in, the attributes should have no predictive power as it relates to future returns...isn't that how it is supposed to be?

    1. There are two regressions to distinguish here. The first regression is the forecasting regression, the one in my post. Indeed these have small R2 values at monthly horizons. The high R2 is the second regression, which is about explaining variance not mean. Here you regress R_it = alpha_i + beta_i'*factor_t + error_it where i is each of the 25 portfolios. Notice it's t on t. The regression is showing how much of the variance of Ri you can explain by variance of the factors, both moving at the same time. Here a high R2 is fine. Efficient markets is silent on the factor structure of returns, and the world is a lot simpler if a few factors can describe large fractions of variance.

  9. I find the result interesting but I don't understand the significance. I'm hoping you can help. 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. Of course 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.

    I wonder if it is helpful to think about the one factor HJM model of the forward curve or a one factor model of the short rate. Since it's a one factor model, any fit to empirical data, which we can do through PCA, will only give you level shifts. Across the curve, a shock at a given time is transmitted through all maturities equally. More factors give more fidelity to the dynamics, allowing for a more realistic evolution. For stocks, the CAPM is too simple, a bit like a one factor model of the forward curve.

    Given the success of the Fama-French three-factor model, should we not expect this result to show up in a PCA in the form of the level, slope, and curvature? What am I missing?

  10. Some of my friends outside of finance asked me what the paper was about. I also speculate on the broader significance. So if you don't understand what the paper is about or you aren't sure what "factor" means in this context, you might want to check out this blog post:

  11. I have done a similar set of calculations on Australian equity data. As Charlie Clarke and Brian Chen say, the flat line factor is simply the market beta effect. If one does the calculations in terms of stock return minus index return, then the first factor disappears.
    Australian data show the same phenomenon as Chen suggests, namely that the higher predicted alpha portfolios are relatively more correlated with each other than with the low predicted alpha portfolios. Indeed the correlation with the between high and low predicted alpha portfolios is generally negative, and generally the correlation is more negative the further apart the portfolios are in terms of expected return. Hence the PCA second factor produces a larger variance by assigning opposite signs to the factor weights between the two groups. As Chen suggests, this form of anti-diversification is what produces the slope on the factor weights.
    The question remains as to whether the slope should be positive or negative. I observe for Australia (and I expect it is true of US data) that the sample variance of the low predicted alpha stocks is larger than that for the high predicted alpha stocks, an observation sometimes offered as an explanation for the Fama-French findings on market efficiency. Consequently, the PCA second factor produces a larger variance by assigning a larger weight to the low predicted alpha portfolio, and a smaller weight to the high predicted alpha portfolios.
    The conclusion: the second factor weights should exhibit a slope, and the factor weights should slope down going from low predicted alpha portfolios to high predicted alpha portfolios.
    In the case of Australian equity data, I find no evidence of a defined shape in the third or subsequent factors.

  12. Hi Professor Cochrane,

    If lots of things share the same type of covariance structure like the Level, Slope and Curvature structure, does it somewhat explain the "puzzle" that there are just a zoo of "factors" that have explanatory power of cross sectional of expected returns, not because they are real proxy of underlying state variables, but just because they have the similar covariance structure as the stock returns? If that's true, that the stock returns are just like, for instance, weather data in Manhattan in terms of their covariance structure, what value does "finding this covariance structure" might add? If the next step for asset pricing is to find any theoretical story to explain Level, Slope and Curvature in stock returns, shouldn't that true underlying story be more universal and must also be applicable to lots of other things with similar covariance structure, instead of just a story related to economics or finance?

  13. I wonder if someone has looked at portfolio returns of stocks formed with high loadings to PC1 , PC2 and PC3 relative to portfolios formed using a linear combination of the forecast variable ranks.


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