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.