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.

Bravo.

(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. )