rate of change in the Consumer Price Index. At the bottom of each column are four descriptive statistics. The first is the arithmetic mean or average holding period return. For bills, it is 3.82%; for long-term government bonds, 5.36%; and for large stocks, 13.11%. The numbers in that row imply a positive av- erage excess return suggesting a risk premium of, for example, 1.54% per year on long- term government bonds and 9.29% on large stocks (the average excess return is the average HPR less the average risk-free rate of 3.82%). The second statistic at the bottom of Table 5.2 is the standard deviation. The higher the standard deviation, the higher the variability of the HPR. This standard deviation is based on historical data rather than forecasts of future scenarios as in equation 5.2. The formula for historical variance, however, is similar to equation 5.2:   2 n n Ar r-B 2   t 1 Here, each years outcome (rt) is taken as a possible scenario. Deviations are taken from the historical average, r-, instead of the expected value, E(r). Each historical outcome is taken as equally likely and given a "probability" of 1/n. [We multiply by n/(n - 1) to eliminate statistical bias in the estimate of variance.] Figure 5.3 gives a graphic representation of the relative variabilities of the annual HPR for the three different asset classes. We have plotted the three time series on the same set of axes, each in a different color. The graph shows very clearly that the annual HPR on stocks is the most variable series. The standard deviation of large-stock returns has been 20.21% (and that of small stocks larger still) compared to 8.12% for long-term government bonds and 3.29% for bills. Here is evidence of the risk-return trade-off that characterizes security markets: The markets with the highest average returns also are the most volatile. The other summary measures at the end of Table 5.2 show the highest and lowest annual HPR (the range) for each asset over the 74-year period. The extent of this range is another measure of the relative riskiness of each asset class. It, too, confirms the ranking of stocks as the riskiest and bills as the least risky of the three asset classes. An all-stock portfolio with a standard deviation of 20.21% would represent a very volatile investment. For example, if stock returns are normally distributed with a standard deviation of 20.21% and an expected rate of return of 13.11% (the historical average), in roughly one year out of three, returns will be less than 7.10% (13.11 - 20.21) or greater than 33.32% (13.11 20.21). Figure 5.4 is a graph of the normal curve with mean 13.11% and standard deviation 20.21%. The graph shows the theoretical probability of rates of return within various ranges given these parameters.         2 The importance of the coupon rate when comparing returns is discussed in Part III.