Efficient Frontier
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William J. Bernstein

The Small-Growth Indexing Anomaly

Efficient Frontier regulars should be well acquainted by now with the futility of active management in almost all asset-class categories. As Jack Bogle points out, it’s not higher math, but simple arithmetic: The gross return of the average fund manager must of necessity be the market return, since these folks are the market for all practical purposes. So the net return of the average manager must be the market return minus the average expenses. Because a low-expense index fund has about a 1% expense advantage over the average actively managed fund, it must of necessity beat it by about 1%. When transactional expenses (spreads and impact costs) are taken into account, the gap is even higher. Finally, since there is no evidence of persistence among the prior best performers, it is hopeless to seek managers who can beat the indexes in the long term.

Active-fund proponents argue that certain areas are less efficient than others, particularly small- and mid-cap stocks. Let’s take a look at some raw data. For the five years ending March 31, 2001, I calculated the percentile rankings of the appropriate Vanguard index fund or S&P/Barra index in its Morningstar category; note that 1 is the top percentile and 100 the worst.

Index Fund or S&P/Barra Index


Vanguard Large-Cap Growth


Vanguard 500 Index Fund (Large Cap Blend)


Vanguard Large-Cap Value Fund


Barra Mid-Cap Growth Index


S&P 400 Mid-Cap Index (Mid-Cap Blend)


Barra Mid-Cap Value Index


Vanguard Small-Cap Growth Fund


S&P 600 Small-Cap Index (Small-Cap Blend)


Vanguard Small-Cap Value Fund


As you can see, these data do tend to support this notion, at least in the small-growth area, where 72% of actively managed funds beat the index. However, indexing retains its advantage in the small-value area. Focusing on the small-growth asset class, I’ve plotted the wealth of $1 invested in the Morningstar universe of small-growth funds versus the Barra Small Growth Index:

This is not an isolated bit of data mining—it shows up in multiple data samples using multiple techniques. There is no question that indexing small-growth stocks is a bad idea. As a practical matter, this is not of much importance to the individual investor, since small-growth stocks have the lowest long-term returns of any asset class, as can be seen from the following plot of the growth of $1 invested in each of the six Fama-French asset classes in the past several decades:

However, Mark Carhart, in his impressive study of fund persistence, found no evidence of persistence or superior performance even in the small-growth area. Of interest is the fact that he used four-factor analysis—the extra factor being momentum. Another Fama-French student, James Davis, in an unpublished study, found much the same thing but with an interesting wrinkle. He used only the traditional three factors and found that active growth managers seemed to do better than value managers, with growth/value monthly alphas of 0.20%/-0.11% for large cap, 0.12%/-0.10% for mid cap, and 0.03%/-0.20% for small cap.

What’s going on here? Why does small-growth active-manager outperformance show up in the raw data, growth active-manager outperformance in the three-factor analysis, but not in Carhart’s four-factor analysis? Survivorship bias is one possible answer—Malkiel demonstrated that this was about 1.5% per year in general equity funds as a whole; it’s certainly much larger with small-growth funds. But since January 1994, small-growth funds have beaten the index by 5.44% per year—surely the survivorship bias is not that large.

For some time, I’ve suspected that the answer to this riddle was momentum. Colleague Steve Dunn years ago pointed out to me the blistering performance of John Bogle (Junior!) at N/I Numeric Investors (he has since moved onto his own shop). At the time, Bogle fils was clearly a small-growth momentum investor. So I decided to look at how small-growth outperformance relates to the momentum factor. This factor is available, along with the other three, at Ken French’s wonderful Web site. Here is a look at the cumulative returns of all four factors:

As you can see, the momentum factor is a powerful one, with returns about the same as the value factor. What’s downright spooky about it, however, is how consistent it is after the great depression.

All this is prologue to the point of my whole exercise—the regression of the excess monthly returns of the active managers over the Barra Small Growth Index versus the momentum factor:

What this graph shows is that in months with high momentum returns (x-axis), the active managers tend to beat the index (y-axis). The data are quite clear, with a t stat of 5.67. What the small-growth managers are doing is holding onto their winners—"letting their profits ride."

There are a lot of unanswered questions about momentum as a returns factor. With its near zero SD, it seems to be almost a riskless play. But not a free one. These are monthly data points, formed by going long the best-performing 30% of stocks over the past 11 months and shorting the worst-performing 30% of stocks. So it’s a very expensive strategy that cannot be completely captured in the real world. On the other hand, it is telling you that it’s a bad idea to sell your winners in a small-cap portfolio. The slope of the regression plot is 0.25, suggesting that you get a quarter of the magnitude of the factor by playing it in a reasonable manner.

This is, of course, what a tax-managed small-cap strategy does. The above considerations would predict that a tax-managed index fund should beat a plain-vanilla one. And indeed this is the case: Since January 1999, DFA’s Tax-Managed U.S. 6-10 Small Company Fund has beaten its older U.S. 6-10 Small Company Fund by almost 2% per year, and Vanguard’s Tax-Managed Small Cap Fund has beaten the S&P 600 Index (on which it is based) by 0.69% per year (after expenses no less) over the past two years. This raises the interesting prospect that it may be worthwhile to hold a tax-managed fund in a retirement account.

Whether or not this turns out to be an historical curiosity is anybody’s guess. Riskless excess-returns strategies tend to get arbitraged out of existence, and the momentum story is not exactly a secret. But if any investment behavior is hardwired into human nature, it is trend following. Students of the efficient-market-versus-behavioralist debate should be following future returns of the momentum factor with more than a little interest.

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