Normal and Lognormal Distributions

Modern portfolio theory, for the most part, assumes that asset returns are normally distributed. This is a convenient assumption because the normal distribution can be described completely by its mean and variance, consistent with mean-variance analysis. The argument 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 distribution from normality is significant. The mean of the sample is 28.2%, and the standard deviation 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 distribution's 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 one-year return to be close to normally distributed.

There remain theoretical objections to the assumption that individual stock returns are normally distributed. Given that a stock price cannot be negative, the normal distribution cannot be truly representative of the behavior of a holding-period rate of return because it allows for any outcome, including the whole range of negative prices. Specifically, rates of

Figure 6A.2 The lognormal distribution for three values of a.

Pr (X )

Ac =30%


f\ JL? =70% \








Source: J. Atchison and J. A. C. Brown, The Lognormal Distribution (New York: Cambridge University Press, 1976).

Source: J. Atchison and J. A. C. Brown, The Lognormal Distribution (New York: Cambridge University Press, 1976).

return lower than -100% are theoretically impossible because they imply the possibility of negative security prices. The failure of the normal distribution to rule out such outcomes must be viewed as a shortcoming.

An alternative assumption is that the continuously compounded annual rate of return is normally distributed. If we call this rate r and we call the effective annual rate re, then re = er — 1, and because er can never be negative, the smallest possible value for re is — 1, or -100%. Thus this assumption nicely rules out the troublesome possibility of negative prices while still conveying the advantages of working with normal distributions. Under this assumption the distribution of re will be lognormal. This distribution is depicted in Figure 6A.2.

Call re(t) the effective rate over an investment period of length t. For short holding periods, that is, where t is small, the approximation of re(t) = ert — 1 by rt is quite accurate and the normal distribution provides a good approximation to the lognormal. With rt normally distributed, the effective annual return over short time periods may be taken as approximately normally distributed.

For short holding periods, therefore, the mean and variance of the effective holding-period returns are proportional to the mean and variance of the annual, continuously compounded rate of return on the stock and to the time interval.

Therefore, if the standard deviation of the annual, continuously compounded rate of return on a stock is 40% (ct = .40 and ct2 = .16), then the variance of the holding-period return for one month, for example, is for all practical purposes a2(monthly)


CHAPTER 6 Risk and Risk Aversion

and the monthly standard deviation is V.0133 = .1155.

To illustrate this principle, suppose that the Dow Jones Industrial Average went up one day by 50 points from 10,000 to 10,050. Is this a "large" move? Looking at annual, continuously compounded rates on the Dow Jones portfolio, we find that the annual standard deviation in postwar years has averaged about 16%. Under the assumption that the return on the Dow Jones portfolio is lognormally distributed and that returns between successive subperiods are uncorrelated, the one-day distribution has a standard deviation (based on 250 trading days per year) of a(day) = a (year) Vl/250

.16 V250

Applying this to the opening level of the Dow Jones on the trading day, 10,000, we find that the daily standard deviation of the Dow Jones index is 10,000 X .0101 = 101 points per day.

If the daily rate on the Dow Jones portfolio is approximately normal, we know that in one day out of three, the Dow Jones will move by more than 1% either way. Thus a move of 50 points would hardly be an unusual event.


Look again at Table 6A.1. Are you surprised that the minimum rates of return are less negative for more diversified portfolios? Is your explanation consistent with the behavior of the sample's maximum rates of return?


1. The probability distribution of the rate of return can be characterized by its moments. The reward from taking the risk is measured by the first moment, which is the mean of the return distribution. Higher moments characterize the risk. Even moments provide information on the likelihood of extreme values, and odd moments provide information on the asymmetry of the distribution.

2. Investors' risk preferences can be characterized by their preferences for the various moments of the distribution. The fundamental approximation theorem shows that when portfolios are revised often enough, and prices are continuous, the desirability of a portfolio can be measured by its mean and variance alone.

3. The rates of return on well-diversified portfolios for holding periods that are not too long can be approximated by a normal distribution. For short holding periods (e.g., up to one month), the normal distribution is a good approximation for the lognormal.


1. The Smartstock investment consulting group prepared the following scenario analysis for the end-of-year dividend and stock price of Klink Inc., which is selling now at $12 per share:




Dividend ($)

Price ($)





















Compute the rate of return for each scenario and a. The mean, median, and mode.

b. The standard deviation and mean absolute deviation.

c. The first moment, and the second and third moments around the mean. Is the probability distribution of Klink stock positively skewed?


A.1. Investors appear to be more sensitive to extreme outcomes relative to moderate outcomes than variance and higher even moments can explain. Casual evidence suggests that investors are eager to insure extreme losses and express great enthusiasm for highly positively skewed lotteries. This hypothesis is, however, extremely difficult to prove with properly controlled experiments.

A.2. The better diversified the portfolio, the smaller is its standard deviation, as the sample standard deviations of Table 6A.1 confirm. When we draw from distributions with smaller standard deviations, the probability of extreme values shrinks. Thus the expected smallest and largest values from a sample get closer to the mean value as the standard deviation gets smaller. This expectation is confirmed by the samples of Table 6A.1 for both the sample maximum and minimum annual rate.


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