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?