Assume that we are facing a normal (continuous) probability distribution of returns. It is symmetrical and bell-shaped, and 68 percent of the distribution falls within one standard deviation (right or left) of the expected return; 95 percent falls within two standard deviations; and over 99 percent falls within three standard deviations. By expressing differences from the expected return in terms of standard deviations, we are able to determine the probability that the actual return will be greater or less than any particular amount.
We can illustrate this process with a numerical example. Suppose that our return distribution had been approximately normal with an expected return equal to 9 percent and a standard deviation of 8.38 percent. Let's say that we wish to find the probability that the actual future return will be less than zero. We first determine how many standard deviations 0 percent is from the mean (9 percent). To do this we take the difference between these two values, which happens to be -9 percent, and divide it by the standard deviation. In this case the result is -0.09/0.0838 = -1.07 standard deviations. (The negative sign reminds us that we are looking to the left of the mean.) In general, we can make use of the formula
where R is the return range limit of interest and where Z (the Z-score) tells us how many standard deviations R is from the mean.
Table V in the Appendix at the back of the book can be used to determine the proportion of the area under the normal curve that is Z standard deviations to the left or right of the mean. This proportion corresponds to the probability that our return outcome would be Z standard deviations away from the mean.
Turning to (Appendix) Table V, we find that there is approximately a 14 percent probability that the actual future return will be zero or less. The probability distribution is illustrated in Figure 5.1. The shaded area is located 1.07 standard deviations left of the mean, and, as indicated, this area represents approximately 14 percent of the total distribution.
As we have just seen, a return distribution's standard deviation turns out to be a rather versatile risk measure. It can serve as an absolute measure of return variability - the higher the standard deviation, the greater the uncertainty concerning the actual outcome. In addition, we can use it to determine the likelihood that an actual outcome will be greater or less than a particular amount. However, there are those who suggest that our concern should be with "downside" risk - occurrences less than expected - rather than with variability both above and below the mean. Those people have a good point. But, as long as the return distribution is relatively symmetric - a mirror image above and below the mean - standard deviation still works. The greater the standard deviation, the greater the possibility for large disappointments.