bills as reported by Ibbotson Associates. Our analysis of what past investors would have earned assumes that investors owned broadly diversified portfolios whose returns were equal to the reported market returns. An important statistical caveat is in order. We will use this data to analyze holding periods of as long as 25 years. Using monthly starting points, we have over 600 different 25-year holding periods in our 76 years of data. However, they overlap. There are only three independent 25-year holding periods. Past results are not necessarily indicative of future results although they can serve as a useful reality check against either extreme optimism or pessimism. The purpose of our analysis of historical data is not to forecast either future absolute returns or relative returns among asset classes. It is to demonstrate (1) the enormous differences between nominal and real, after-tax return, (2) the uneven impact of taxes on different asset classes, (3) the impact of time on the perception of risk, and (4) the impact of fixed real spending requirements. The average annual returns since 1926 (calculated as geometric means) are shown in Table 30.1. An investor trying to decide on the allocation between stocks and bonds might conclude that the long-term expected return of stocks is about double that of bonds. This fact would then be weighed against the greater volatility of stocks in determining the trade-off that best suited the investor's objectives. However, during this period, inflation averaged 3.06 percent per year. An adjustment for inflation produces the results shown in Table 30.2. Inflation changes the situation a great deal. Real returns are much lower. The historic real return of stocks is nearly four times that of bonds. The adjustment for inflation might provide greater motivation for ownership of stocks versus bonds. The next step is to adjust returns for the impact of taxes. This is a subjective process. Tax rates have varied a great deal over the past 76 years. At times they have been higher than current rates, but at other times they have been lower. TABLE 30.1 Average Nominal Returns 1926 through 2001 Nominal Stocks 10.68% Bonds 5.33 Cash 3.81 Source: Ibbotson Associates. TABLE 30.2 Impact of Inflation     Nominal Real Stocks 10.68% 7.41% Bonds 5.33 2.21 Cash 3.81 0.73 Source: Ibbotson Associates.