The lognormal property of stock prices can be used to provide information on the probability distribution of the continuously compounded rate of return earned on a stock between times 0 and T. If we define the continuously compounded rate of return per annum realized between times 0 and T as x, then
From equation (15.2), it follows that
Thus, the continuously compounded rate of return per annum is normally distributed with mean and standard deviation . As T increases, the standard deviation of x declines. To understand the reason for this, consider two cases: . We are more certain about the average return per year ...