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Rethinking Small Caps

Financial Advisor

By Gary A. Miller and Scott A. MacKillop | September 2, 2011

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New research that puts small cap stocks in proper perspective.

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“The results show that, in the 1936-1975 period, the common stock of small firms had, on average, higher risk-adjusted returns than the common stock of larger firms.” That one sentence, which appeared in a paper by Rolf Banz published in the Journal of Financial Economics in 1981, is the foundation for the investment truism, “Small stocks beat large stocks.” As it turns out, Banz was wrong.

Banz’s groundbreaking research appeared originally in his paper, “The Relationship Between Return and Market Value of Common Stocks.” His conclusion was later substantiated by Eugene Fama and Kenneth French in one of the most famous of all investment research papers, “The Cross-Section of Expected Stock Returns,” published in the Journal of Finance in 1992. It has been accepted as true ever since.

For three decades, this belief has had a profound impact on the way financial advisors build portfolios. It also led to the launch in December 1981 of the Dimensional Fund Advisors U.S. Micro Cap Portfolio. (It was originally called the DFA U.S. 9-10 Portfolio, but was recently renamed.) The fund was designed to allow advisors to efficiently take advantage of the “small-beats-large” anomaly. Today, many investment firms emphasize the advantage of “going small.”

However, our research does not support the idea but instead shows it’s easier to argue the opposite-that large beats small.

Our Discovery

We believe that often an anomaly disappears or is diminished once its existence is widely known. The original purpose of our research was to explore what happened to the small-beats-large anomaly after it was revealed by Banz in 1981.

We started looking for our answer in the “post-Banz” period from 1982 through 2010. We analyzed the performance of small stocks relative to large and also compared the performance of both to the performance of the DFA U.S. Micro Cap fund to see if it had successfully capitalized on the idea that small beats large.

What we found was consistent with our original hypothesis-that the anomalies fade after they are discovered. We found that, on a risk-adjusted basis, large stocks performed better than both small stocks and the DFA U.S. Micro Cap Portfolio. No surprises there.

Then we looked at two different periods-the 56 years before Banz’s paper came out (from 1926 to 1981) and the 29 years afterward-and tried to decide which of those two time frames most influenced investment history. We discovered large stocks prevailed in the post-Banz time period and we assumed, based on the earlier research, that small stocks prevailed in the pre-Banz time period. But we wanted to see what happened when you looked at these periods together. So we analyzed what we call the “full historical period”-1926 through 2010.

What we found was puzzling. We found that large stocks beat small stocks on a risk-adjusted basis over the full historical period. Though that in itself did not baffle us, what did was the extent to which large stocks beat small. We did not expect them to so greatly outperform.
To solve this mystery, we revisited the conclusions of the earlier researchers whose findings had been viewed as hallowed truth for so long. We analyzed the pre-Banz period and, for good measure, the exact period examined by Banz himself: 1936 to 1975.

What we found shocked us. Large stocks beat small in both periods. The long-held belief that small stocks beat large on a risk-adjusted basis is simply not supported by the facts.

How We Reached Our Conclusion

We analyzed the returns of two well-established indexes. To represent small stocks, we used the Center for Research in Security Prices (CRSP) 9-10 index. To represent large stocks, we used the S&P 500 index.

We chose these indexes for good reason. The CRSP 9-10 index is the premier small stock index and goes back to 1926. The S&P 500 also goes back to 1926 and is commonly used to represent the large stock universe. Ibbotson’s Stocks, Bonds, Bills and Inflation uses the same indexes for these universes.

For each time period, we calculated the monthly “excess returns” of the CRSP 9-10 index and the S&P 500 index. We defined “excess return” as the return in excess of the risk-free rate. We used three-month T-bills to represent the risk-free rate.

We determined risk-adjusted returns using two measures. First, we determined the “alpha” of small stocks relative to large. A positive alpha indicates a higher risk-adjusted return for small stocks. A negative alpha indicates a higher risk-adjusted return for large stocks.
We also determined the Sharpe ratios for both indexes. A higher Sharpe ratio for one index relative to the other would indicate a higher risk-adjusted return for that index.

We also compared the “lag-adjusted monthly returns” of the CRSP 9-10 index and the S&P 500 index. Fama and French identified a problem that arises when you analyze the monthly returns of small stocks: Their prices can be “stale” because of the lack of trading during the month. Even though the prices “catch up” in the following month, this “lag” understates their true volatility, and thus their risk.
Fama and French addressed that problem this way: They compared small stock prices for a given month to large stock prices for both the current and the previous month. We did the same thing.

Because of the need to “look back” one month at large stock prices when calculating “lag-adjusted” returns, our analysis of periods that begin in 1926 starts in February 1926. This allows us to “look back” at the large stock prices in January 1926. This did not have a material impact on our results.

We also compared the annualized returns of the two indexes. This differs from the approaches taken by Banz and by Fama and French. They relied on comparisons of average monthly returns, not annualized ones. We compared monthly returns as well, but find fault with relying solely on those comparisons.

Investors are more interested in annualized returns than average monthly returns. Investors rarely see monthly return data. Instead, they typically see either quarterly returns or annualized returns. If they see monthly returns at all, it is for a single month, not monthly returns averaged over a long period.

Moreover, if you compare the average monthly returns of two asset classes rather than annualizing their returns, the more volatile asset class will appear to have better performance relative to the less volatile. Thus, Banz and Fama and French exaggerated the advantage of small stocks.

Here is an example of the problem. You have two asset classes: A and B. Asset class A, the more volatile of the two, has a -38% return in one month followed by a +42% return the next month. Asset class B has a -6% return the first month followed by a +10% return the next month. Average the monthly returns of asset class A and you get a return of 2% (-38% + 42% ÷ 2 = 2%). You get the same return if you average the monthly returns for asset class B (-6% + 10% ÷ 2 = 2%). Do they really have the same return?

In fact, asset class B has a far better return than asset class A. If we invest $100 in asset class A, we have $62 after the first month (-38%). Then we gain 42% on our $62 portfolio so now we have just over $88. We experienced a loss of almost 12%. If, instead, we invest $100 in asset class B, we have $94 after the first month and a little over $103 after the second. We gained a little over 3% by investing in the less volatile asset class-quite a difference. When we annualize returns, this problem is eliminated.

A Review of the Results

Let’s begin by examining the post-Banz period (1982-2010). The results are shown in Figure 1.

The absolute returns of the CRSP 9-10 index and the DFA U.S. Micro Cap fund are higher than the returns for the S&P 500 index, although not by a wide margin. This is true whether you look at them on a monthly, a lag-adjusted monthly or an annualized basis. That is what we would expect to find.

The returns of small stocks are generally higher than the returns of large stocks over long periods because of the additional risk associated with investing in them. The question that Banz and Fama and French were attempting to answer was: “Are investors adequately compensated for this additional risk?” They concluded that investors were more than adequately compensated.

Our research suggests this is not the case. Looking at the monthly returns, you see that small stocks and the DFA fund have a slight edge over large stocks if you use alpha as the measure of risk-adjusted return, but large stocks have a higher Sharpe ratio, which means they have better risk-adjusted returns.

Our analysis of lag-adjusted monthly returns gives the advantage to large stocks whether you look at alpha or the Sharpe ratio. That is because the lag-adjustment better captures the true volatility of small stocks. The DFA fund still has a slight edge over large stocks if you use alpha as the measure of risk-adjusted return, but not if you use the Sharpe ratio.

In annualized returns, large stocks have the advantage over both small stocks and the DFA fund whether you look at alpha or the Sharpe ratio.

This suggested that our original hypothesis was correct. Whatever advantage small-cap stocks had in the pre-Banz period, it disappeared in the post-Banz period.

This becomes even clearer if we look at the post-Banz period in greater detail. First, on an absolute-return basis, the CRSP 9-10 and the DFA fund actually had lower returns than the S&P 500 for the first 20 years of the period. Both caught up, but just barely.

Second, DFA’s effort to capitalize on the small-beats-large anomaly was not successful. The DFA fund outperformed the CRSP 9-10, but did not outperform the S&P 500 on a risk-adjusted basis.

Finally, even DFA’s original success in implementing its small-cap strategy faded in recent years. For the 20 years ending 2010, the DFA fund returned 13.48% annualized, while the CRSP 9-10 returned 14.52%. The gap was even wider for the ten years ending 2010. The fund returned 9.63% and the index 12.58%.

Next let’s look at the full historical period (1926 to 2010). The results are shown in Figure 2.

The pattern is similar to the pattern for the post-Banz period. The CRSP 9-10 has an absolute return advantage over the S&P 500.
On a risk-adjusted basis, the monthly returns show the CRSP 9-10 has an advantage over the S&P 500 if you use alpha as the measuring stick, but gives the advantage to the S&P 500 if you use the Sharpe ratio. With lag-adjusted monthly returns, the alpha advantage for the CRSP 9-10 gets smaller and the Sharpe ratio advantage for the S&P 500 gets larger. With annualized returns, the advantage goes to the S&P 500 whether you use alpha or the Sharpe ratio as a measure of risk-adjusted return.

Now let’s examine the pre-Banz period (1926 to 1981). The results are shown in Figure 3.

Again, we find the CRSP 9-10 index has higher absolute returns than the S&P 500 index regardless of how you look at the data-monthly, lag-adjusted monthly or annualized.

Looking at monthly returns, the CRSP 9-10 also has a higher risk-adjusted return than the S&P 500 when using either alpha or the Sharpe ratio as the measure. When looking at lag-adjusted monthly returns, the alpha advantage for small stocks is reduced and the Sharpe ratio advantage shifts to large stocks. Looking at the annualized data, the alpha advantage for small stocks almost disappears and large stocks continue to hold a slight advantage in the Sharpe ratio analysis.

Finally, let’s examine the period from 1936 through 1975. This is the time period originally analyzed by Rolf Banz. The results are set forth in Figure 4.

Once again, we find the absolute returns for the CRSP 9-10 are higher than for the S&P 500, reflecting the greater risk of small stocks. When Banz originally examined this period, he concluded that investors were more than fairly compensated for this additional risk.
That conclusion appears questionable. Looking at monthly returns, the CRSP 9-10 shows a positive alpha relative to the S&P 500, giving small stocks the edge in risk-adjusted return. But the Sharpe ratio analysis suggests that large stocks may actually hold the advantage.
Looking at lag-adjusted monthly returns, the alpha advantage for small stocks drops dramatically and the Sharpe ratio advantage for large stocks increases. Looking at the annualized data, large stocks are superior on a risk-adjusted basis whether you use alpha or the Sharpe ratio as your measuring stick.

Conclusions

We set out to learn what had happened to the small-beats-large anomaly in the days since it was first revealed by Rolf Banz and discovered that it may never have existed at all. If anything, it appears that large stocks beat small. Our findings are counter to those of earlier researchers.

Four factors that may explain the difference in our findings: First, we used the CRSP 9-10 index to represent small stocks and the S&P 500 index to represent large stocks. The earlier researchers constructed their own indexes using individual securities data from CRSP.
Another factor is that the CRSP data is constantly revised. We assume it is revised to make it more accurate, so if this accounts for the differences, we would expect our findings to be more accurate too.

Third, we used annualized returns in our analysis. Again, we believe this is a better approach because it is more relevant to investors and it avoids overstating the benefits of the more volatile asset class.

Finally, our conclusions rely, in part, on our use of “lag-adjusted” monthly returns. Since Fama and French also used them, such returns should not account for the difference in our findings and theirs. However, Banz used raw monthly returns, and this may explain differences in our findings and his.

If the small-beats-large anomaly relies for its existence on the use of certain indexes or the use of monthly instead of annualized returns, it is a highly illusory concept, at best. An advantage either exists or it doesn’t. Investors should not put their money at risk using a mere mathematical chimera.

Our research suggests that it is time to rethink the idea that small stocks outperform large stocks on a risk-adjusted basis. On the contrary, it appears there may actually be a large company stock advantage.