A large part of finance deals with the tradeoff between risk
and return. Return, as used here, refers to the percentage increase (or
decrease) in an investment over time. The time period is usually a
month, a quarter, or a year, but could be a day, or any other time
interval. Regardless of the interval, the return is usually given as an
annual percentage unless otherwise stated. (It is very common in
economics and finance to use a term without modifiers, since a return
could be a dollar amount as easily as a percentage return.) Risk is
defined in the next topic, Variance and Standard Deviation.
An expected return
is just that—what do you expect the return to be? This course reviews
methods used to compute the expected return. One is based on historical
evidence; the other uses probabilities assigned to possible outcomes.
Consider the method using historical information first.
Expected Return Using Historical Information
An expectation is another term for weighted average in statistics.
The two concepts are both used to represent a single measure of the
likelihood of something. One common method used to develop an estimate
of expected return on an investment is to simply average the historical
returns. A financial analyst might look at the percentage return on a
stock for the last 10 years and see what the average return has been.
Mathematically, the average is computed as
That is, the historical average of stock i, read "Rbar sub i," equals one Tth (1 divided by T) times the sum of the returns for security i for the time period t. If you have 10 years of historical returns for security A, this formula could be written as
Suppose security A had percentage returns the last 10 years of 12
percent, 14 percent, 8 percent, 4 percent, 10 percent, 18 percent, 10
percent, 12 percent, 15 percent, and 14 percent. The historical average
return would be
On average, stock A returned 9.7 percent over the last 10 years. You
can use this as the expected return for next year. This is the simplest
form of statistics that you can use to compute an expected return for
the next year based on historical data. There are other statistical
techniques to project the expected return based on historical data, but
they are beyond the scope of this course.
Expected Return Using Possible Outcomes with Probabilities
A second method used to provide a measure of expected return
involves developing scenarios (for example, best, worst, and most
likely), estimating what the return would be under each scenario, and
assigning probabilities to each scenario. (The estimation could be as
simple as a set of guesses or as complicated as a full simulation model
of the economy and its impact on a company's revenues.) This technique
has the advantage of using more information than historical information
alone. The problem with this approach is that the probabilities can be
subjective and the result is only as good as the information used and
the model itself. The most useful aspect of this approach is that it
forces the analyst to consider several possibilities and their
consequences.
For this course, you simply want to focus on how to use the outcomes
and their probabilities to derive the expected return. Consider a
simple example that uses only two possible outcomes.
To calculate expected return, first list the possible future
outcomes that will alter the expected cash flows from the investment.
For example, consider the manager of new product development at a
consumer products company that has develped a new product. The firm
must decide whether to accept the cost of producing the product and the
cost of introducing it to the market. The product introduction will be
followed by one of two possible alternative outcomes: There could be
"high" acceptance by the market or "low" acceptance. The possible
future outcomes are listed below.
At this point, the analysis must assign probabilities to each of the
possible outcomes. One set of rules that must always be followed in
calculating expected return is that every outcome must have a
probability assigned that might be 0. Every individual probability is
between 0 and 1. And, all the probabilities must add up to 1. So in the
example, the manager needs to decide the likelihood of each of these
events. Using focus group results, the manager decides that there is a
60 percent chance of high acceptance, and since there are only two
alternatives, there is a 40 percent chance of low acceptance. You now
know you have covered all possible alternatives, since .4 + .6 adds up
to 1.
To calculate the expected return of the new product introduction, an
estimate of the return is needed for each possible outcome. For this
example, assume that there will be a return of 20 percent if there is
high acceptance, and a return of only 6 percent if there is low
acceptance. With this information, you can calculate the expected
return.
Compare these estimates to the estimates in the three challenge
problems. In this case, you were to assign a probability to each of the
possible outcomes, and you would assign expected returns associated
with each of those outcomes. You would not rely on any past data. In
Challenges A and C, the average past performance of the NYSE and the
Japanese fund, respectively, were indicators of expected future
outcomes.
In other words, if you possess historical results, and you believe
that those results are a good indicator of future expectations, you can
derive future expected return by extrapolating those past results to
the future. However, if you do not have historical results, or you do
not believe that history is a good indicator of future performance,
expected return is derived by the weighted average of all possible
returns, weighted by their own probabilities. More formally, it can be
written as
This formula for expected return of security i says to sum the products of the return of security i for each outcome n times the probability of outcome n.
Completing the example is the following information.
The sum of the weighted returns is
As illustrated in the challenges, the same principle applies to
stock returns. Suppose an investor is considering the purchase of a new
firm's stock. As a new stock, there is little historical data to make
any predictions regarding future returns. But the investor knows that
there are four possible states of the economy over the next year: boom,
steady, slow, and bust. Each state of the economy is equally likely to
occur, with a 25 percent chance of each. If the economy booms, then the
stock will return 30 percent, if it is steady, the investor expects to
earn 16 percent. If the economy slows, the investor expects to earn
only 8 percent, and if the economy goes bust, the stock will lose 10
percent. So the investor calculates the expected return as
Here again, the expected return under each state of the economy is
assigned based on a prediction. One way to help formulate that
prediction is to look for historical data on similar investments. For
instance, in Challenge C, although you did not have reliable historical
data on the new technology fund, you elected to develop a set of
possible outcomes based on the past performance of similar technology
funds.
Expected Return of a Portfolio of Stocks
As a next step, you may need to calculate the expected return on a
portfolio of more than one investment. To calculate the expected return
of a portfolio simply compute the weighted average of the expected
returns on all the assets in your portfolio, with each asset's return
weighted by the proportion of the total portfolio each asset represents.
Where
The animation below shows you how to compute the expected return of
a portfolio. (Note: The formula notation in the animation is slightly
different from the notation shown above, but it represents the same
process.)
Would you like another example? Consider an investor who owns three
stocks. Stock A has an expected return of 18 percent, and the investor
owns $25,000 worth of the stock. She also owns $15,000 of stock B,
which has an expected return of 12 percent. The expected return on her
$10,000 worth of stock C is 10 percent. What is the expected return on
her portfolio?
Begin by calculating the relative weight of each stock.
Use E(r_{i}) as the expected return on stock i, then use these weights to calculate the portfolio's overall expected return.
How many decimal places should you use?
1. A friend just gave you his winning stock picks for the future. The stocks and their expected returns, E(r), appear in the following table.
If you invest an equal amount in each, what is your portfolio's expected return?
Solution 1
2. It is your job as an analyst to forecast next year's stock return
for Global Gadgets Inc. Through lots of research and hard work, you
have developed the following set of possible outcomes and associated
probabilities.
What is your expected return for the stock of Global Gadgets Inc. next year?
Solution 2
3. The marketing department has come up with two new products that
it expects to launch next month. The expected return for product X, a
new shampoo, is 15 percent, and it will cost $15 million to launch. The
expected return for product Y, a new fragrant bar soap, is 22 percent,
and it will cost $6 million to launch. What is the expected return to
your company for this portfolio of products?
Solution 3
4. Vanmartin Industries Limited (VIL) and Lightspeed Computer Technologies (LCT) offer the following possible returns.
If you invest $37,000 in Vanmartin Industries Limited and $28,500 in
Lightspeed Computer Technologies, what is your portfolio's expected
return?
Solution 4
_{SourcesDigital Imagery © copyright 2000 PhotoDisc, Inc.}
