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

PreMBA Analytical Methods
Risk and Return: Variance and Standard Deviation

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.


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 risk-averse 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.

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.

How does a business strategy consultant use variance?
Low bandwith

High bandwith

Calculating variance based on expected returns
The second method of calculating variance uses expected returns instead of mean historical returns. The procedure to compute this form of variance is as follows.

Calculate the expected return, E(r), as described earlier, by summing the products of each outcome multiplied by its probability.

Calculate the difference between each possible future return and the expected return.

Square each difference.

Multiply (weight) each squared difference by its associated probability.

Sum the product of the squared differences times their respective probabilities.

The result of Step 5 is the variance of the expected return. The following example will demonstrate this method.

Variance of a Probability Distribution

Example Consider the possible future returns and their associated probabilities for stock B.

Standard Deviation

First, verify that your probabilities sum to 1, then repeat Step 1 by calculating the expected return, E(r), of the possible future returns.

Expected Value of a Probability Distribution

Therefore, stock B's expected return is 4.36 percent. Now calculate the difference between each possible future return and the expected return, and then square the differences. Multiply each of the squared differences by their respective probabilities. The following table depicts each of these steps.

Sum the final column in the above table to derive the variance of the expected return.

Standard Deviation

Since a variance is the (weighted) average of the errors squared, you need to take its square root to express in the same units as the underlying variable, in this case the percentage return. The square root of the variance is called the standard deviation. When dealing with normally distributed variables, approximately 68 percent of all observed values will be within one standard deviation of the mean, and approximately 95 percent of the observed values will be within two standard deviations of the mean. Stock price returns, while not technically normally distributed, may be modeled assuming normality.

Are stock returns normally distributed?

The animation below compares differences in standard deviation across stocks, and their possible implications for an investment's return.

Even though all three stocks in the animation had the same expected return, the range of actual returns differed across the stocks. Variance and standard deviation quantifies how widely dispersed actual returns are relative to the expected return.

Consider the following graphs for Conglomo, Inc. and Bilco, Inc. These graphs show the theoretical frequency distributions of the monthly returns for each firm's common stock as though the returns were normally distributed.

Conglomo's distribution of returns is more concentrated than Bilco's, as illustrated by Conglomo's relatively wider bell curve. A more concentrated distribution is defined as having a smaller standard deviation. The distribution curve appears higher, steeper, and narrower because more observations are occurring close to the expected return. Bilco's distribution is rather flat, reflecting that its returns are less concentrated, or more dispersed, than those of Conglomo Inc.

You can use Excel to compute variance and standard deviation.
Download Variance and Standard Deviation Excel Tutorial

The standard deviation, denoted by Greek letter s, is calculated by taking the square root of the variance.

Or, if variance was calculated using forecasted data,

Standard Deviation of a Probability Distribution

Using the variance from the earlier example, which was calculated based on historical returns, stock A has a mean return of 4 percent with a variance of

These values can be used to calculate the standard deviation of the returns.

Imagine that you are now considering whether to buy stock A. You have an estimate of its expected return, and you know how widely dispersed its previous returns have been. If you are willing to assume that the future will be like the past, you can use the standard deviation calculations to predict the likelihood that any one return will be within a specific range. For example, you know that approximately 68 percent of the time, stock A's monthly return will be within one standard deviation of the mean. Therefore, you can find that range if you assume the standard deviation remains the same in the future.

Thus, 68 percent of the time the stock's return will be between -8 percent and 32 percent. Similarly, you can calculate that 95 percent of the time, the stock's return will be between -28 percent and 52 percent.

Assuming the future will be like the past, this analysis indicates that stock A is likely to generate positive returns.

Standard deviation measures return variability due to factors both specific to the firm, and to factors outside the firm's control. For example, suppose a company becomes embroiled in a major lawsuit, discovers a new technology, or loses a key employee. These events will cause changes in returns. These events are called "firm-specific risks," or "diversifiable risks," reflecting the fact that all firms do not necessarily face the same risks. Macroeconomic factors, such as a rapid rise in inflation, affect all industries, and the performance of the market in general. Therefore, this is called "market," "systematic" or "non-diversifiable" risk, which represents risks created by macroeconomic conditions that all firms face. The next section shows how "firm-specific" risk can be minimized through diversification, leaving the investor exposed only to market risk.

How many decimal places should you use?

1. Compute the variance and standard deviation of historical returns for Nepenthe Hotels.

Solution 1

2. Use the following information to compute the variance and standard deviation of the historical returns of PosterCo.

Solution 2

3. The following table contains the historical returns for TX Drugs, a nationwide chain of retail drug stores. Use Excel to determine the variance and standard deviation of the company's returns.

Solution 3

4. XYZ Inc. has hired you to analyze the variability of their returns. Specifically, XYZ's managers would like you to calculate measures of risk that they can report at an upcoming shareholders' meeting. The company has been in business for five years and has provided the following table of historical returns.

XYZ has also given you a set of possible returns for the coming year. The possible returns and their matching probabilities are presented in the table below.

XYZ Inc. wants to give its shareholders a very complete picture of the company's risk. To do so, its management wants you to first provide two alternative measures of variance and standard deviation. The first is based on historical data, while the second is based on the company's estimates of future returns and associated probabilities. Once you have calculated the separate measures of standard deviation, XYZ Inc. would like you to average the two measures, giving each equal representation. What are the variance and the standard deviation using each method, and what is the average measure of standard deviation using equal weighting?

Solution 4

5. Your boss has asked you to evaluate whether or not to add a stock to your company's investment portfolio. She wants the stock to have an expected return of at least 12 percent, and she wants to be at least 95 percent certain that the return will be higher than 10 percent. Use the historical returns provided below to determine whether the proposed stock meets her criteria.

Solution 5

You can use Excel to compute variance and standard deviation.
Download Variance and Standard Deviation Excel Tutorial

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