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Essential Concepts
Time Value of Money
Evaluating Cash Flows
Risk and Return
Expected Return
Variance and Standard Deviation

PreMBA Analytical Methods
Risk and Return: Expected Return

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 "R-bar sub i," equals one T-th (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.


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.)

Expected Returns
View animation

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(ri) 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

Digital Imagery copyright 2000 PhotoDisc, Inc.

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