26.4 28.2 27.8 28.2 28.1 28.2 70th centile 38.7 49.7 33.8 35.7 31.6 32.9 30.0 30.0 95th centile 96.3 95.6 54.3 51.8 40.9 39.9 34.1 33.8 Maximum 442.6 NA 136.7 NA 73.7 NA 43.1 NA Mean 28.2 28.2 28.2 28.2 28.2 28.2 28.2 28.2 Standard deviation 41.0 41.0 14.4 14.4 7.1 7.1 3.4 3.4 Skewness (M3) 255.4 0.0 88.7 0.0 44.5 0.0 17.7 0.0 Sample size 1,227 - 131,072 - 32,768 - 16,384 -   Source: Lawrence Fisher and James H. Lorie, "Some Studies of Variability of Returns on Investments in Common Stocks," Journal of Business 43 (April 1970).     Normal and Lognormal Distributions   Modern portfolio theory, for the most part, assumes that asset returns are normally distrib- uted. This is a convenient assumption because the normal distribution can be described completely by its mean and variance, consistent with mean-variance analysis. The argu- ment has been that even if individual asset returns are not exactly normal, the distribution of returns of a large portfolio will resemble a normal distribution quite closely. The data support this argument. Table 6A.1 shows summaries of the results of one-year investments in many portfolios selected randomly from NYSE stocks. The portfolios are listed in order of increasing degrees of diversification; that is, the numbers of stocks in each portfolio sample are 1, 8, 32, and 128. The percentiles of the distribution of returns for each portfolio are compared to what one would have expected from portfolios identical in mean and variance but drawn from a normal distribution. Looking first at the single-stock portfolio (n 1), the departure of the return distribu- tion from normality is significant. The mean of the sample is 28.2%, and the standard de- viation is 41.0%. In the case of normal distribution with the same mean and standard deviation, we would expect the fifth percentile stock to lose 39.2%, but the fifth percentile stock actually lost 14.4%. In addition, although the normal distributions mean coincides with its median, the actual sample median of the single stock was 19.6%, far below the sample mean of 28.2%. In contrast, the returns of the 128-stock portfolio are virtually identical in distribution to the hypothetical normally distributed portfolio. The normal distribution therefore is a pretty good working assumption for well-diversified portfolios. How large a portfolio must be for this result to take hold depends on how far the distribution of the individual stocks is from normality. It appears from the table that a portfolio typically must include at least 32 stocks for the