## New Angles on the Size Premium

Many investors and advisors who implement multifactor portfolios tend to focus on capturing the value premium over the size premium, often for the simple reason that, historically, the value premium has been larger.

Others have even challenged the size premium’s very existence, citing a weak and varying historical record. In both situations, it may be that the size premium—and specifically the construction of the size factor—is not fully understood.

My colleague Sean Grover, a member of the investment strategy team at Buckingham and The BAM Alliance, put together the following analysis of size factor construction to help address and clarify this issue. He begins with some basic definitions.

**Defining The Size Factor**

The size factor, as defined by Eugene Fama and Kenneth French, is constructed by sorting all stocks by market capitalization, as determined by market capitalizations of NYSE stocks, into deciles and then taking the weighted average annual return of deciles 6-10 (small stocks) minus the weighted average annual return of deciles 1-5 (large stocks). In other words, it’s the top 50% of stocks ranked by size minus the bottom 50%.

Contrast this with the value factor, which is constructed by sorting stocks on book-to-market ratio and then taking the weighted average annual return from deciles 1-3 (value stocks) minus the weighted average annual return from deciles 8-10 (growth stocks). In other words, it’s the top 30% of stocks ranked by valuation metric minus the bottom 30%.

In this construct, deciles 4-7 are considered core stocks. The 30/30 construction is also used for other established risk factors, such as momentum, profitability, quality and low beta/low volatility. The size factor is the only exception.

Using data from the University of Chicago’s Center for Research in Securities Prices (CRSP), Figure 1 below presents historical returns for grouped market-capitalization deciles.

As illustrated with these market-capitalization deciles, if there’s a premium in some category of stocks over another, then that premium will be larger, the more strictly that category is defined.

As we can see with the more narrow size grouping, as stocks get smaller, their returns get higher.

In addition, the more strictly the category is defined, the more difficult it should be to capture. So in trying to understand the size premium, we ask: How does the construction of the size factor affect the amount of the premium that a portfolio can capture?

**Different Perspectives On Size**

Kenneth French’s data library provides returns for a variety of percentile sorts on market capitalization. Using these, we are able to build several different versions of the size factor in an attempt to answer our question.

The standard size factor—defined above as the weighted average return of the smallest 50% of stocks less that of the largest 50%—will be designated 50/50. We then construct size factors using the smallest (largest) 30%, 20% and 10% of stocks, which we’ll call 30/30, 20/20 and 10/10, respectively.

Figure 2 shows the historical annual premium for these various definitions of the size factor.

As expected, the more narrowly we define “small cap” versus “large cap,” the larger the annual premium. The standard 50/50 size factor has the smallest annual premium and, notably, the premium resulting from the 30/30 construction is actually larger than the value premium (which is 4.83%). Furthermore, all the annual premiums are statistically significant (meaning they have a t-statistic of greater than two).

To measure how much of a given premium a portfolio is able to capture, multifactor investors turn to estimated factor loadings from a factor model regression. For three small-cap indexes, Figure 3 shows the estimated size factor loadings from a four-factor model for our various size factor constructions.

Again, we find that the more narrowly we define the construction of the size factor, the more difficult it is to capture the premium. For each index, the more narrowly the size factor is defined, the smaller the estimated loading. Note that all the estimates shown in Figure 3 are statistically significant.

**Real-World Examples**

We can perform the same exercise for live funds as well. Consider Figure 4, which shows the results for the same exercise as Figure 3 but using three small-cap funds. (In the interest of full disclosure, my firm, Buckingham, recommends DFA funds in constructing client portfolios.)

Again, we find that the more narrowly the size factor is defined, the smaller the estimated loading. In this case, consider that even though there is significant variation between funds for a given size factor construction, each fund still shows a decrease in estimated loadings with stricter size factor definitions.

We have now fleshed out our hypotheses concerning the effect the construction of the size factor has on the magnitude of the size premium, and on a portfolio’s ability to capture it.

Using two funds, the DFA U.S. Micro Cap Fund from Figure 4 and the DFA Large Cap Value Fund, we can take a look at how the differences in construction between the size factor and the value factor affects the amount of the premium a portfolio is able to capture. This is measured simply by multiplying a portfolio’s estimated factor loading by the respective annual premium. Figure 5 shows this calculation for the two funds.

Despite the value premium being larger than the size premium, the portfolio targeting the size factor, the DFA U.S. Micro Cap Fund, actually captures more factor premium. This is due to the higher factor loading, which, we have shown, results directly from the less strict construction of the size factor.

Although brief, this example pretty clearly shows that the magnitude of factor premia should not be the sole consideration for multifactor investors choosing their allocation to risk factors. The important takeaway is that size factor construction—and thus portfolios’ increased ability to capture the size premium—needs to be considered when evaluating the size premium. The investor who ignores the size premium may very well be missing out on sizable factor premia.

If you’re interested in further discussion about the size premium, an article I wrote last year on a paper by Cliff Asness, Andrea Frazzini, Ronen Israel, Tobias Moskowitz and Lasse Pedersen, “Size Matters, If You Control Your Junk,” examines the problem of a disappearing size premium by controlling for the quality factor.

*This commentary originally appeared May 20 on ETF.com*

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