A Brief History of Risk and Return 1-8
Like most statistical concepts, students will struggle remembering the concept of
a normal distribution. In our experience, this is mostly because they are unsure of
their understanding—not that they have not “seen” the material before.
For many different random events in nature, a particular frequency distribution,
the normal distribution (or bell curve) is useful for describing the probability of
ending up in a given range. For example, the idea behind “grading on a curve”
comes from the fact that exam scores often resemble a bell curve.
Figure 1.10 illustrates a normal distribution and its distinctive bell shape. As you
can see, this distribution has a much cleaner appearance than the actual return
distributions illustrated in Figure 1.8. Even so, like the normal distribution, the
actual distributions do appear to be at least roughly mound shaped and
symmetric. When this is true, the normal distribution is often a very good
Also, you will have to remind students that the distributions in Figure 1.9 are
based on only 90 yearly observations, while Figure 1.10 is, in principle, based on
an infinite number. So, if we had been able to observe returns for, say, 1,000
years, we might have filled in a lot of the irregularities and ended up with a much
smoother picture. For our purposes, it is enough to observe that the returns are
at least roughly normally distributed.
The usefulness of the normal distribution stems from the fact that it is completely
described by the average and the standard deviation. If you have these two
numbers, then there is nothing else to know. For example, with a normal
distribution, the probability that we end up within one standard deviation of the
average is about 2/3. The probability that we end up within two standard
deviations is about 95 percent. Finally, the probability of being more than three
standard deviations away from the average is less than 1 percent.
E. The Second Lesson
Observing that there is variability in returns from year-to-year, we see that there
is a significant chance of a large change in value in the returns. So the second
lesson is: The greater the potential reward, the greater the risk.
1.5 More on Average Returns
A. Arithmetic versus Geometric Averages
The geometric average return answers the question: “What was your average
compound return per year over a particular period?”
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