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 as- sumption 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 pe- riods, 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 nor- mally distributed, the effective annual return over short time periods may be taken as ap- proximately 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 com- pounded 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% ( .40 and 2 .16), then the variance of the holding-period return for one month, for example, is for all practical purposes   2(monthly) .16 .0133 12 12 II. Portfolio Theory 6. Risk and Risk Aversion The McGraw−Hill Companies, 2001