Investors want to minimize the risk associated with a given expected return. Diversification can play a role by minimizing firm-specific risks that add to the overall uncertainty of returns. When you add new assets to your portfolio, you are adding the possibility that they will do well during the times your existing assets perform poorly, and vice versa. So without changing your expected return, you are able to lower the variability of returns. In fact, through careful selection of assets to add to your portfolio, you can eliminate most firm-specific risk, which is sometimes called non-systematic risk, and bear only market, or systematic, risk to your portfolio.

Your ability to reduce firm-specific risk in a portfolio depends on the relative correlation of the assets held in the portfolio. Correlation describes the degree to which returns on investments move together. In Challenge B, the investor was able to increase the expected return of his portfolio (from 12.20 percent to 12.59 percent) and actually reduce the risk as measured by standard deviation (from 9.88 percent to 9.52 percent) by adding an additional stock to the portfolio. The primary reason for this simultaneous reduction in risk and increase in expected return was a low correlation of the returns of Conglomo and Bilco (.3).

The following simplified example illustrates the fundamental benefit of diversification.

A simple portfolio
Suppose that you have $2 to invest. You can purchase shares in firm Alpha today at a cost of $1 each. There is a 50 percent probability that after one year the stock's price will rise to $5 (yielding a 400 percent return). There is also a 50 percent probability that the stock price will still be $1 one year from now (yielding, of course, a 0 percent return).

Alternatively, you could purchase shares in firm Beta today at a price of $1 each. There is also a 50 percent probability that Beta's stock price will rise to $5, and a 50 percent probability that Beta's stock price will remain at $1.

Because the payoffs and the probabilities are identical, both investments have the same expected return (200% = 400%(.5) + 0%(.5)) and the same risk (the standard deviation is 200 percent in both cases). So you might be tempted to ask: Is there a benefit to purchasing one share of each stock (or diversifying) rather than purchasing two shares of one? The answer depends upon the correlation of the returns for the two stocks.

Scenario one Alpha's and Beta's stock prices are perfectly negatively correlated.

Consider these conditions. If average temperatures during the year are above normal, then Alpha's stock will rise to $5, but Beta's will remain at $1. If temperatures are below normal, Beta's will rise to $5, but Alpha's will remain at $1. Assume that the probability of each temperature outcome is 50 percent. Your total return from different investment options would look something like this:

Notice that the expected total value of your portfolio is $6 in all three cases ((.5 &multiplication;$2) + (.5 &multiplication; $10) = 1 + 5 = $6). But the returns in the third case have a standard deviation of zero—you are guaranteed $6, no matter what. Even though the two individual investments are risky, there is absolutely no risk associated with the diversified portfolio. However, this is not the case if the two stock prices are related in a different manner.

Scenario two Alpha's and Beta's stock prices are perfectly positively correlated.

Suppose that Alpha and Beta's stock prices always move together. That is, if temperatures are above normal, both stock prices rise to $5. If temperatures are below normal, both prices remain at $1.

Your total returns from the different investment options would look something like this:

Unlike scenario one, the diversified portfolio in this case is no less risky than either of the two individual investment possibilities. The problem is that the stock prices of the two companies are perfectly positively correlated. A perfect positive correlation means that the value of two assets moves in the same direction, by the same percentage, at the same time.

An important principle in finance is that risk reduction cannot be achieved through diversification if the returns on two or more assets are perfectly positively correlated. However, diversification provides benefit if the returns are not perfectly positively correlated. Furthermore, an investor can eliminate all "non-systematic" risk if the two assets are perfectly negatively correlated, as you saw in scenario one. Risk can seldom be eliminated (reduced to zero) through diversification, as in scenario 1, but it can be reduced.

Correlation Coefficient

To fully understand how the benefits of diversification are evaluated, start by considering the correlation coefficient.

The correlation coefficient measures the degree to which two variables (such as two stock prices; indicated here by subscripts to rho, i and j) move together. The correlation coefficient must always be between +1 and -1. (+1 indicates the extreme of perfect positive correlation, and -1 indicates the other extreme of perfect negative correlation.) Perfect positive correlation indicates that whenever stock A goes up, stock B goes up by the same percentage.

In the interactive graphic below, drag and drop the purple line onto the graph to match the type of correlation.

In the first graph, the returns of both stocks are perfectly positively correlated similar to what occurred in scenario two above. You can see that when the first stock increases in value, the second stock increases also, by the same amount. This means that no diversification is possible. Graph two shows two stocks that are perfectly negatively correlated as in scenario one above. When the first stock increases in value, the second decreases in value by the same magnitude, meaning that you can eliminate all risk. The final graph shows stocks that have absolutely no correlation. You can see that there is no pattern at all, and that when the first stock increases in value, the second stock sometimes goes up but is just as likely to go down. Typically, the correlation between stocks will track more closely to this pattern than to either of the others.

The formula for calculating the correlation coefficient for two returns, i and j, is

You should already be familiar with the terms in the denominator, which are the standard deviations of the individual returns. As you recall, standard deviation measures the dispersion of observed returns from the mean. The term in the numerator is called the covariance of the two returns and measures how the observed returns move together, or covary.

Covariance and correlation describe the degree to which two variables move together. You can use correlation and covariance to describe whether two stock prices move in the same direction, whether they move in opposite directions, or whether they move completely independently of one another. Covariance and correlation are also used to ascertain whether a stock price moves in tandem with a given economic variable, such as the stock market.

Covariance indicates the degree to which two variables move in unison. A positive covariance means that the variables tend to move together, and a negative covariance means that they tend to vary inversely. Correlation is a normalized measure that describes both the direction and the degree to which two variables tend to move together. To measure the covariance between historical data points, you would examine deviations from the mean for each variable.

Covariance between X and Y = average of (X - X^*) &multiplication; (Y - Y^*) where:

If you were using probability estimates, you would examine the differences between each variable and the weighted expected return of the data. You may have performed these calculations while solving for the solution of Challenge B.

Riskiness of a Diversified Portfolio

Remember that the standard deviation of the return from an asset gives one measure of the riskiness of the asset. As such, two standard deviation calculations, one for stock A and one for stock B, only tell you the variability of returns for stock A and the variability of returns for stock B. You need to know the correlation coefficient for the returns of two assets, so that you can calculate the standard deviation of a portfolio that contains both assets. You can then compare the riskiness of the portfolio with two assets to the riskiness of the individual assets. To calculate the standard deviation of a portfolio investment with two assets, i and j, use this formula.

Example
When the correlation between two assets is +1.

Suppose you want to invest your wealth in the stocks of ABC, Inc. and XYZ, Inc. The expected returns and standard deviations of the returns for the two corporations are

Now assume that the covariance of the returns from the two companies is

What are the expected return and standard deviation of your portfolio if you invested 50 percent in ABC, Inc. and 50 percent in XYZ, Inc., and their covariance is .0416?

The expected return, you may recall, is the weighted average of the two individual returns.

Calculating the standard deviation of the portfolio requires first finding the correlation coefficient.

The standard deviation of the portfolio is

In this case, there is actually no risk-reducing benefit from diversification, because the two asset returns are perfectly positively correlated. That is, the correlation coefficient is +1. So what happens when you diversify? If you invest 100 percent of your wealth in XYZ, then your portfolio's expected return and standard deviation would be 10 percent and 16 percent, respectively. By diversifying, you increase your expected return to 15 percent, but you also increase your risk to 21 percent. In this example of perfect positive correlation, both the risk and expected return of the portfolio increased as a result of diversification.

Notice that because the correlation between ABC and XYZ is +1, the formula for the portfolio's standard deviation equals to the weighted average of the individual standard deviations.

However, unlike the expected return of a portfolio, the risk of a portfolio is generally not the weighted average of the standard deviation of the individual asset returns. Only in the very rare case in which the returns of the individual assets are perfectly positively related (correlated) is a portfolio's standard deviation the weighted average of the standard deviations of its individual assets.

Compare the previous results to another example in which there are risk-reducing gains from diversification.

Example
When the correlation between two assets is 0.

What if you change your assumption about the covariance between ABC and XYZ to 0, but keep everything else the same?

You can verify that the expected return will not change.

However, with a covariance of 0, the correlation coefficient would be as shown below.

Regardless of the fact that the individual standard deviations do not change, a covariance of 0 causes the correlation coefficient to be 0.

The portfolio standard deviation becomes

Changing the correlation between ABC and XYZ from +1 to 0 causes the portfolio standard deviation to decline from .21 to .1526, while not affecting the portfolio's expected return. This demonstrates how diversification across assets with less than a +1 of correlation can reduce risk while holding the expected return constant. Although these examples use only two assets, the same diversification properties apply to portfolios that contain many assets.

So you have now seen that it is possible to reduce firm-specific risk in your portfolio by diversifying your portfolio among various stocks with weak correlations.

The standard deviation measures total variability or "total risk" in the portfolio. Covariance between individual security returns within the portfolio occurs because macroeconomic forces are going to affect all securities in the portfolio to different degrees. Economic factors such as general inflation and business cycles will affect almost all firms. This risk can be classified as "systematic risk," or "market risk," and cannot be diversified away. Thus there are two classifications of risk encompassed in the standard deviation measure—firm-specific risk and market risk. Expected returns for investors can be increased proportionately to the amount of market risk undertaken. However, the market will not reward investors who take on firm-specific risk with increased returns.

Beta: Measuring Market Risk

Do all portfolios have the same level of market risk? Certainly not. So we need a measure of the market risk carried by a portfolio. This measure of market risk is called "beta" and has applications throughout finance. Investors use beta to evaluate what level of return they should be earning relative to how much market risk is present in their portfolios. Corporations use this value to estimate their cost of capital, which in turn is used to decide whether to start potential projects.

Calculating beta using standard deviation
Fortunately, it is easy to calcuate beta. Beta measures the relation between an individual security and what is called the market portfolio of all possible assets. Watch the animation below to help you better understand what beta is.

Beta Animation View animation

In reality, it is impossible to calculate the return for a portfolio of all assets, such as stocks, bonds, real estate etc. It is common to use the return on the S&P 500 Index as a proxy for a market portfolio. As with individual securities, the standard deviation of the market index can be calculated, as well as the correlation between individual stocks and the market portfolio. Armed with this information, you can calculate the beta of an asset with this formula:

where the subscript m refers to the market index. This formula reads: the beta for security i equals the standard deviation for security i times the correlation between the returns for security i and the market index, m, divided by the standard deviation of the returns for the market index.

Beta describes how much an individual stock's returns fluctuate relative to changes in the market index. Suppose that you calculate a stock's beta to be 2. This would indicate that when the market returns one percent, the stock itself would expect to return two percent. If the market loses three percent, the stock would be expected to lose six percent. Theoretically, when a stock's beta is high, its returns are high when the market is doing well. If a stock has a low beta, such as .5, then its returns will be expected to fluctuate less than the returns of the market overall. You can think of beta as a measure of a stock's sensitivity to the overall market. In this sense it is a good measure of the market risk faced by the firm.

Calculating beta using regression
There is another way to calculate a firm's beta, and that is by using linear regression. Simple linear regression is a method used to demonstrate the relationship between two random variables, such as the return on the stock index and the return on a security. One of these random variables is called the "dependent variable," meaning that its value depends on what happens with the second variable. The second variable is called the "independent variable." Regression calculates how the degree to which the dependent variable reacts to changes in the independent variable. This can be put in an equation form.

Here, beta defines the relation between the two variables. In the context of measuring market risk, the independent variable would be the market index return, while the security return is the dependent variable. Beta measures the relation between the two. The way this is solved with regression is by finding the line that best fits the data points. Beta will be the slope of this line. Consider the following graphs.

The first graph shows all the observations plotted. The regression equation will fit a line through those observations that minimize the squared sum of the errors (depicted above as the vertical measure between each point and the regression line) for every observation and the line itself. (This technique leads to the name "ordinary least squares.") Calculating the best-fit line is commonly done today using spreadsheet functions or statistics software. For example, see the solution to practice problem five.

With a firm understanding of these risk and return concepts, you will be well prepared to venture into more advanced aspects of financial theory and practice.

How many decimal places should you use?

1. Compute the portfolio standard deviation of an equally weighted portfolio consisting of Fransco Corp. and Librell, Inc.; assume a covariance of 0.

Solution 1

2. Compute the portfolio standard deviation based on the following table of information and a correlation coefficient of -.25.

Solution 2

3. Compute the portfolio standard deviation based on the following table of information and a correlation coefficient of .75.

Solution 3

4. You are deciding which of three stocks to add to your portfolio. Using the following information, which of these stocks has the highest total risk? Also calculate each stock's systematic risk. Which stock has more non-diversifiable risk?

Solution 4

5. Using Excel and the data in the spreadsheet below, calculate stock A's beta.

Solution 5