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.

What does a retail banker think about diversification? Low bandwith

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.

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:

X = value of first variable
X^{*} = average of first variable
Y = value of second variable
Y^{*} = average of second variable

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.

This Wall Street Journal article explains how diversification, along with rebalancing, increases portfolio performance. Surprise Benefit from Diversification

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.

w_{i} = the weight, or fraction, of the investment that you have allocated to each asset

= the correlation coefficient

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.

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.

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?