events and treat them as our best guesses (probabilities) of
the likelihood of future business conditions. We could then
say that an investor’s expected return is 10 percent. One-
third of the time the investor gets 30 percent, another one-
third 10 percent, and the rest of the time she suffers a 10
percent loss. This means that, on average, her yearly return
will turn out to be 10 percent.
Expected return =
1
/
3
(0.30) +
1
/
3
(0.10) +
1
/
3
(–0.10) = 0.10.
The yearly
returns will be quite variable, however, ranging
from a 30 percent gain to a 10 percent loss. The “variance” is
a measure of the dispersion of returns. It is defined as the
average squared deviation of each possible return from its
average (or expected) value, which we just saw was 10
percent.
Variance =
1
/
3
(0.30–0.10)
2
+
1
/
3
(0.10–0.10)
2
+
1
/
3
(–0.10–
0.10)
2
=
1
/
3
(0.20)
2
+ 1/3(0.00)
2
+
1
/
3
(–0.20)
2
= 0.0267.
The square root of the variance
is known as the standard
deviation. In this example, the standard deviation equals
0.1634.
Dispersion measures of risk such as variance and standard
deviation have failed to satisfy everyone. “Surely riskiness is
not related to variance itself,” the critics say. “If the
dispersion results from happy surprises—that is, from
outcomes turning out better than expected—no
investors in
their right minds would call that risk.”
It is, of course, quite true that only the possibility of
downward disappointments constitutes risk. Nevertheless, as
a practical matter, as long as the
distribution of returns is
symmetric—that is, as long as the chances of extraordinary
gain are roughly the same as the probabilities for
disappointing returns and losses—a dispersion or variance
measure will suffice as a risk measure.
The greater the
dispersion or variance, the greater the possibilities for
disappointment.
Although the pattern of historical returns from individual
securities has not usually been symmetric,
the returns from
well-diversified portfolios of stocks are at least roughly
symmetrical. The following chart shows the distribution of
monthly security returns for a portfolio invested in the S&P
500-Stock Index over seventy years. It was constructed by
dividing the range of returns into equal intervals (of
approximately 1¼ percent) and
then noting the frequency
(the number of months) with which the returns fell within
each interval. On average, the portfolio returned close to 1
percent per month or about 11 percent per year. In periods
when the market declined sharply, however,
the portfolio
also plunged, losing more than 20 percent in a single month.
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