*HML*is redundant?

**EFF/KRF:** There is some confusion about the interpretation of the evidence in Fama and French (2014, “A Five-Factor Model of Expected Returns”) that *HML* is redundant for explaining average U.S. stock returns for 1963-2013.

It doesn’t imply that there is no value premium. When *HML* is defined in the usual way (2x3 sorts on *Size* and *B/M*), its average value for 1963-2013 is a hefty 5.21% per year with a standard deviation of 13.70% and a *t*-statistic of 2.72. This is similar to the market premium in excess of the bill rate, 6.74% per year with a standard deviation of 17.97% and a t-statistic of 2.68. The t-statistics tell us that the underlying expected premiums are reliably greater than zero.

In practical terms, since there is indeed a value premium, it is still quite legitimate for investors to target it in portfolio decisions. The expected benefits from doing so have not changed.

When we say that *HML* is redundant, what we mean is that its average return is fully captured by its exposures to the other factors of the five-factor model. This means *HML* has no information about average returns that is not in other factors, so we do not need *HML* to explain average returns.

Filling in the details, suppose we run a regression of *HML* monthly returns on the other four factors of the five-factor model, specifically, the market factor *R _{M}-R_{F}*, the size factor

*SMB*, the profitability factor

*RMW*(robust minus weak) and the investment factor

*CMA*(conservative minus aggressive),

(1) *HML _{t} = a + b(R_{Mt}-R_{Ft}) + sSMB_{t} + rRMW_{t} + cCMA_{t} + e_{t}*.

(The subscript *t* indicates a particular month of data and the sample of months used in the paper to estimate the intercept *a* and the factor exposures *b*, *s*, *r*, and *c* covers 1963-2013.)

One question answered by this regression is: how much of the average *HML* return can we explain with the exposures of *HML* to the other four factors of the five-factor model, along with the average returns of the factors? To see this, let’s average the variables in the regression over the months in the sample. Using the symbol *A* to denote an average, we have,

(2) *A(HML) = a + bA(R _{M}-R_{F}) + sA(SMB) + rA(RMW) + cA(CMA)*.

(The residual *e _{t}* disappears because the regression sets it average value to zero.)

This equation tells us that the regression intercept, *a*, is the part of the average *HML* return left unexplained by the exposures of *HML* to other factors, in other words, *a* is the alpha of *HML* with respect to other factors. In our tests this alpha is quite close to zero. For example, using the regression and the average monthly 1963-2013 returns for the factors from the standard 2x3 sorts, the equation becomes,

0.37 = -0.04 + 0.01*A(RM-RF)* + 0.02*A(SMB)* + 0.23*A(RMW)* + 1.04*A(CMA)*

= -0.04 + 0.01(0.50) + 0.02(0.29) + 0.23(0.25) + 1.04(0.33)

Thus, of the 37 basis point per month average *HML* monthly return only three basis points is left unexplained by the average returns of the other four factors and the exposures of *HML* to the other factors. This means that at least in U.S. data for 1963-2013, *HML* has little or no information about average returns not captured by the other factors of the four factor model. Standard asset pricing theory then tells us that *HML* is redundant for explaining average returns. If we drop *HML* from the five-factor model the exposures to the other four factors change in such a way that the resulting four-factor model produces the same estimates of expected returns as the five-factor model.

The five-factor model doesn’t improve the description of average returns of the four-factor model that drops *HML*, but the five-factor model is probably a better choice in applications. Thus, though captured by exposures to other factors, there is a large value premium in average returns that is often legitimately targeted by money managers. In evaluating how investment performance relates to known premiums, we probably want to know the tilts of portfolios toward the *Size*, *B/M*, *OP*, and *Inv* factors. And for explaining average returns, nothing is lost in using a redundant factor.

**What are the takeaways?**

(1) There is a value premium in average returns, captured by the average value of *HML*.

(2) In 1963-2013 U.S. data, the average *HML* return is fully captured by the exposures of *HML* to the market, *Size*, profitability, and investment factors of the five-factor model. As a result, a four-factor model that drops *HML* captures average returns as well as the five-factor model. But the five-factor model is still quite useful for measuring the style tilts of managed portfolios.

We emphasize that the redundancy of *HML* may be specific to the period and country we examine. Stay tuned for out-of-sample evidence.

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Eugene Fama and Ken French are members of the Board of Directors of the general partner of, and provide consulting services to Dimensional Fund Advisors LP.