The expected return, R, is
where Rj is the return for the ith possibility, P; is the probability of that return occurring, and n is the total number of possibilities. Thus the expected return is simply a weighted average of the possible returns, with the weights being the probabilities of occurrence. For the distribution of possible returns shown in Table 5.1, the expected return is shown to be 9 percent.
To complete the two-parameter description of our return distribution, we need a measure of the dispersion, or variability, around our expected return. The conventional measure of dispersion is the standard deviation. The greater the standard deviation of returns, the greater the variability of returns, and the greater the risk of the investment. The standard deviation, G, can be expressed mathematically as
where V represents the square root. The square of the standard deviation, cr, is known as the variance of the distribution. Operationally, we generally first calculate a distribution's variance, or the weighted average of squared deviations of possible occurrences from the mean value of the distribution, with the weights being the probabilities of occurrence. Then the square root of this figure provides us with the standard deviation. Table 5.1 reveals our example distribution's variance to be 0.00703. Taking the square root of this value, we find that the distribution's standard deviation is 8.38 percent.
Use of Standard Deviation Information. So far we have been working with a discrete (noncontinuous) probability distribution, one where a random variable, like return, can take on only certain values within an interval. In such cases we do not have to calculate the standard deviation in order to determine the probability of specific outcomes. To determine the probability of the actual return in our example being less than zero, we look at the shaded section of Table 5.1 and see that the probability is 0.05 + 0.10 = 15%. The procedure is slightly more complex when we deal with a continuous distribution, one where a random variable can take on any value within an interval. And, for common stock returns, a continuous distribution is a more realistic assumption, as any number of possible outcomes ranging from a large loss to a large gain are possible.