First there was the Fama-French three-factor model, then four factors. How about a fifth?

** **

In 1993, the Fama-French three-factor (beta, size and value) model replaced the single-factor capital asset pricing model (CAPM) and became the standard model in finance, explaining more than 90 percent of the variation of returns of diversified portfolios.

While the model was a big improvement over the CAPM, it couldn’t explain some major anomalies. In 1997, Mark Carhart augmented the three-factor model with a fourth factor: momentum. By addressing one of the biggest anomalies, the momentum factor made a large contribution to the explanatory power of the factor model.

The four-factor model has been the workhorse model since.

But like all models, even the four-factor model had problems—there were many anomalies that it couldn’t explain. Kewei Hou, Chen Xue and Lu Zhang, authors of the September 2012 study, “Digesting Anomalies: An Investment Approach,” proposed a new four-factor model that goes a long way toward explaining many of the anomalies that neither the Fama-French three-factor nor the four-factor models explain. They called it the *q*-factor model. The four factors are:

- The market excess return (beta)
- The difference between the return on a portfolio of small-cap stocks and the return on a portfolio of large-cap stocks
- The difference between the return on a portfolio of low-investment stocks and the return on a portfolio of high-investment stocks. Note that the investment factor is highly correlated with the value premium, suggesting that this factor plays a similar role to that of the value factor.
- The difference between the return on a portfolio of high return on equity stocks and the return on a portfolio of low return on equity stocks. Note that the profitability factor has a high correlation with the momentum factor, meaning it would play a similar role to the momentum factor in analyzing performance.

Among their important findings was that the investment and profitability (return on equity) factors are almost totally uncorrelated, meaning that they are independent, or unique, factors.

Professors Fama and French, in a June 2013 paper, “A Five-Factor Asset Pricing Model,” took a close look at the new model, to see if these new factors—investment and profitability—added explanatory power. In other words, if they knew in 1993 what they know today, which model would they have chosen? The following is a summary of their findings:

- While a five-factor (beta, size, value, profitability and investment) doesn’t fully explain the cross section of returns (there are still anomalies), it provides a good description of average returns.
- The model’s main problem is its failure to explain the low average returns on small stocks that invest a lot despite low profitability. (Note that the Fama-French three-factor model has a problem explaining the poor performance of small growth stocks.)
- The performance of the model is not sensitive to the specifics of the way its factors are defined.
- A four-factor model that excludes the value factor (what is referred to as “HML,” or the return of high book-to-market stocks minus the return of low book-to-market stocks) captures average returns as well as any other four-factor model considered.
- A five-factor model (including HML) doesn’t improve the description of average returns from the four-factor model. The reason is that average HML return is captured by the exposures of HML to other factors. Thus, in the five-factor model, HML is redundant for explaining average returns, but may have value in other ways.

Fama and French did note that, “While the five-factor model doesn’t improve the description of average returns of the four-factor model that drops HML**,** the five-factor model may be a better choice in applications. For example, though captured by exposures to other factors, there is a large value premium in average returns that is often targeted by money managers.”

Thus, “in evaluating how investment performance relates to known premiums, we probably want to know the tilts of the portfolios toward each of the factors.”

They added that “for explaining average returns, nothing is lost in using a redundant factor.”

Importantly, they also found that the five-factor model does perform well.

“Unexplained average returns for individual portfolios are almost all close to zero,” Fama and French noted.

Among their other interesting findings were that “controlling for investment, value stocks behave like stocks with robust profitability, even though unconditionally value stocks tend to be less profitable.”

Additionally, Fama and French found that the value, profitability and investment factors are negatively correlated with both the market and the size factor, providing important information regarding potential benefits from portfolios that diversify exposures across factors.

What’s more, they found that “the lethal combination for microcaps is low profitability and high investment; low profitability alone doesn’t appear to be a problem.”

However, they found that this problem doesn’t hold for large stocks with low profitability and high investment.

Another interesting finding is that with this new model, just as the well-known value factor isn’t needed to explain the variation in returns of diversified portfolios, momentum isn’t needed either.

The authors of “Digesting Anomalies” found that their profitability factor has a high correlation (0.50) with the momentum factor, meaning that it plays a similar role to the momentum factor in analyzing performance.

Returning to the question—knowing what we know today, which model would they choose?— Fama and French concluded that since HML seems to be a redundant factor in the sense that its high-average return is fully captured by its exposure to other factors, the four-factor model may well be adequate in some cases, though not all.

“In applications where the sole interest is abnormal returns, our tests suggest that a four-factor model that drops HML performs as well as (no better and no worse than) the five-factor model. But if one is also interested in measuring portfolio tilts toward value, profitability and investment, the five-factor model is the choice.”

Only time will tell if the new model replaces the Fama-French one. But with their endorsement, it does seem like a good possibility.

Larry Swedroe is director of Research for the BAM Alliance, a community of more than 130 independent registered investment advisors throughout the country.

* *

** **