Although expected return
is the best estimate available of future returns, the actual return is
not likely to equal the expected return. For this reason, investors and
managers would like to have an idea of how precise their estimate might
be. To help quantify the precision of their estimates, you use two
concepts: variance and its square root, the standard deviation.
In a literal sense, the standard deviation is a measure of how far
from the expected value the actual outcome might be. Two stocks may
have the same expected return, but have different levels of risk,
as measured by variance and standard deviation. Just as the calculation
of expected return was based on assumptions, it would be false to
assume that variance and standard deviation encompass all aspects of
risk. These tools merely give you more information based on past
results, or expected future results. For example, consider the
following two stocks, A and B, each of which has an expected return of
5 percent, as noted at the top of each table.
Stock A would have a return of 7 percent from the "Good" outcome,
and only a 3 percent return from a "Bad" outcome. Using the formula
from the previous section on expected return, you can easily see that
the expected return is 5 percent for stock A (half, or .5, of 7
percent, plus half of 3 percent is 5 percent).
Stock B would have a much better return of 15 percent in a "Good"
outcome, but lose 5 percent in a "Bad" outcome. The expected return is
half of 15 percent, plus half of 5 percent, or 5 percent, the same as
that for stock A.
Both stocks have the same expected return. But even in the worst
case, stock A still earns the investor 3 percent. On the other hand,
stock B causes the investor to lose 5 percent in the worst case. This
is a demonstration of why investors consider the variability of
expected returns. Based on the individual's risk tolerance, a 3 percent
return might be acceptable, but the risk of a 5 percent loss might be
regarded as unacceptable.
You can make assumptions regarding the risk tolerance of the
individuals in the challenge problems. In Challenge B, despite the fact
that the investor owned only two stocks, the standard deviation
associated with the returns of Conglomo stock was relatively low. Even
the Bilco stock that the investor considered was assigned a relatively
low standard deviation based on expected returns. By contrast, in
Challenge C, although the investor diversified through mutual funds,
the standard deviation of returns associated with those funds was high.
As a result, each of those respective funds, based on standard
deviation alone, would be considered more risky than the stock
of Conglomo. As a result, you might come to the conclusion that the
investor in Challenge C actually had a higher risk tolerance than the
investor in Challenge B. In fact, the Challenge A investor recognized
the risks associated with his NASDAQ portfolio and sought to reduce his
perceived exposure to risk by investing in the NYSE, which had a lower
standard deviation.
Variance
Most people are risk averse, in that they wish to minimize
the amount of risk they must endure to earn a certain level of expected
return. If investors were indifferent to risk, they would not be
influenced by the differences between stock A and stock B above,
whereas the riskaverse investor would clearly prefer stock A.
Therefore, most people want to know the range, or dispersion, of
possible outcomes, as well as the likelihood of certain outcomes
occurring.
This course will review two methods to calculate the variance of an
expected return. The one you use depends upon the information you have
at your disposal. One method uses the same type of forecast that we
used to calculate expected return. This method can be used when you
have no historical data available to analyze past performance. This can
occur when evaluating the stock of a new technology company or perhaps
when deciding whether or not to invest in a new project. (This method
was used in Challenge B to estimate returns for Bilco stock.)
Alternatively, when you have historical data, you can examine how
the investment has performed in the past and use that as an indication
of how it will perform in the future. By using historical returns on a
firm's stock, you can calculate the variance of past returns. Keep in
mind that even this method does not provide an absolute basis for
determining the riskiness of the stock. Past returns are not always
reliable indicators of future results. Nonetheless, calculating
variance and standard deviation based on historical returns is often
the preferred method, because it relies on historical fact, as opposed
to unquantified speculation regarding the future.
Calculating variance based on historical returns
Variance is calculated using historical data by following the steps below.

Calculate the average historical return.


Calculate the difference between each observed return and the average return.


Square each difference.


Sum the squared differences.


Divide the sum of squared differences by the number of observations minus 1.

The result of Step 5 is the sample variance.
You might want to refer to your statistics definitions to
differentiate between calculating a sample variance and a population
variance. The following example will demonstrate this process.
Example You are considering adding a new stock to your
portfolio that has an expected return of 4 percent. Before you buy this
stock you would like to know its variance. Using historical data,
calculate the variance of this stock's returns.

The mean of these returns is simply the average. So you calculate
the mean by summing all the returns and dividing by the number of
returns.

The calculations below combine steps 2, 3, and 4.

Calculate the difference between each observed return and the average return.


Square each difference.


Sum the squared differences.

The sum of the squared differences (.0024) is listed at the bottom of the calculations.

The next step is to divide the sum of the squared differences by the number of observations minus 1.
Sigma squared is often used as a symbol to represent variance.

You have measured the difference between actual returns and the mean
return. Those differences are then squared, which means that variance
will always be a positive number. If you ever get a negative variance,
check your arithmetic. These squared differences are then averaged,
giving you a measure of the square of the average error in the expected
return.
