This article is one in a long series which I hope will help explain the ins and outs of DFA – Dimensional Fund Advisors. NOTE: This is my interpretation and explanation only. For the final word, please refer to the DFA Canada Website.
Introduction To The Fama-French Three Factor Model
First, as a refresher let’s look at the CAPM. Remember, this asset pricing model (the “single” factor model) says that an investor’s expected return is based on their exposure to the market factor (or Beta). Here is the formula again:
E(Rp) = Rf + β(Rm – Rf)
As I alluded to in the last part of this series, the Fama-French Three Factor Model (FFTFM henceforth) uses three factors to explain expected return: 1) Market Factor, 2) Size Factor, and 3) Value Factor. Just as CAPM includes a measure of the amount of exposure to the market factor, the FFTFM includes measures of the exposure to each of the three factors.
The Fama-French Three Factor Model:
E(Rp) = Rf + β(Rm – Rf) + s(Small – Big) + h(High BtM – Low BtM)
(Small – Big) and (High BtM – Low BtM) are analogous to (Rm – Rf). Remember that (Rm – Rf) is the equity premium or market factor – it is just the excess return of stocks over t-bills. Similarly (Small – Big) is the excess return of Small stocks versus Big stocks, and (High BtM – Low BtM) is the excess return of Value stocks over Growth stocks. You can think of each as having their own Beta, except it’s not called Beta for each – it’s only called Beta for measuring the sensitivity to the market factor. There is not really a name for the “other” Betas, they are just represented as “s” and “h” – each is just a measure of the amount of exposure to the other two factors (size and value).
In fact, the way the formula is normally written uses SMB to represent “Small Minus Big” and HML to represent “High BtM Minus Low BtM”. Therefore you will see the formula written as follows (but it means the same as above):
E(Rp) = Rf + β(Rm – Rf) + s(SMB) + h(HML)
And, just to bash you over the head again with the same old thing, in plain english this is saying that the investor’s expected return is equal to the risk-free rate PLUS their exposure to the market factor PLUS their exposure to the size factor PLUS their exposure to the value factor.
Okay Preet, So What?
Well I suppose the best way to explain why this three factor model has garnered so much attention is to look at how Fama and French looked at it. They took the total stock market and they chopped it up into a 5 x 5 matrix (Book to Market quintiles by Size quintiles) as follows:
It is also important to know what R² is – according to Wikipedia: “the coefficient of determination, R2, is the proportion of variability in a data set that is accounted for by a statistical model… R2 is a statistic that will give some information about the goodness of fit of a model. In regression, the R2 coefficient of determination is a statistical measure of how well the regression line approximates the real data points. An R2 of 1.0 indicates that the regression line perfectly fits the data.” Basically the closer to 1.0 the R² is, the more “explanatory power” of a model.
If you look at the CAPM regressions for “monthly value-weight portfolio returns from July 1963 to December 2007” for a total of 534 months of data (information sourced from Eugene Fama and DFA) then the R² values for this 5×5 cross-section of the total market looks as follows (remember, this is for the CAPM):
So what is being said here is that for each slice of the 25 slices of the market based on differing size and value factors, the CAPM really only does a half decent job for larger stocks with low Book to Market values – coincidentally, the S&P 500 has many large cap growth stocks. But once we stray away from this, the CAPM does an increasingly poorer job of explaining the variation in returns. Of course, this begs the question: what about the Three Factor regressions?
Here are the “Three Factor Regressions for Monthly Value-Weight Portfolio Returns from July 1963 to December 2007”, again a total of 534 months of data and this was sourced from Eugene Fama and DFA, and again looking at the R² values:
I’ll stop there for today, but Part XI will address some of the early criticisms of this model. Part XII will look at how widespread this model has become, even though you many not realize it.