Chapter 4
Introduction
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This chapter will discuss the concept of risk and
how it is measured.
Furthermore, this chapter will discuss:
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Risk aversion
Mean return
Variance and standard deviation of return
Systematic and unsystematic risk
Capital asset pricing model (CAPM)
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What is Risk?

Risk is the uncertainty that an outcome will
vary from our expectations.
 For an investment, it is the notion that cash
flows or percentage returns will be different
than our expectations.
 This includes the “upside” potential as well
as the “downside.”
 As the potential outcomes widen, so does
the risk.
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Declining Marginal Utility for Money

“Who Wants to be a Millionaire” example
Why do most contestants stop and not “take
a chance” for the $1 million question?
 Because the potential utility gain is not as
much as the utility lost from an incorrect
answer (and loss of a major portion of their
winnings up to that point)

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Risk Aversion
and Risk Averse Behavior
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Does this mean we never take risks?
Risk aversion means that we must be
compensated adequately for bearing risk.
This applies to returns for individual investors
investing in stocks or bonds.
It also applies to companies deciding on
investing in new projects for their
shareholders.
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Returns and Distributions
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An example of a distribution is a potential
distribution of dividends.
Each potential dividend is an outcome.
Each outcome has a probability of occurrence
associated with it.
Expected return = average return

Simple average vs. weighted average
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What is the Expected Dividend for the
Following Distribution?
Dividend
Probability
$10
10%
$7
20%
$5
40%
$3
20%
$1
10%
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Expected Value Calculation

($10 + $7 + $5 + $3 + $1)/5 = $5.20
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This is incorrect!
($10 x .10) + ($7 x .20) + ($5 x .40) +
($3 x .20) + ($1 x .10) = $5.10
We have an expected outcome (mean) and a
number of outcomes around the mean.
This is called a distribution.
 A normal “bell-shaped” curve has half the
outcomes to the right of the mean.

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What Can a Distribution Tell Us?
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In this example, the shape of the distribution tells us
about the risk of the investment.
Variance is a measure of risk.
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The variance examines the differences between each
outcome and the expected value.
Variance is a positive number.
In general form, variance is the sum of:
((Outcome 1 – expected value)2 x probability of outcome 1)) + ((Outcome 2
– expected value)2 x probability of outcome 2))+…
((Outcome n – expected value)2 x probability of outcome n))
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Variance and Standard Deviation
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The variance for the previously shown
dividend distribution is $5.68.
In order to make the interpretation more
useful, we take the square root of the
variance.
This is called the standard deviation.
The standard deviation in this example is
$2.38.
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The Normal Distribution
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For a distribution to be normal, 68.3% of the
observations must be between one standard
deviation subtracted from the mean and one
standard deviation added to the mean.
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In this example, one standard deviation from the
mean ranges from $2.72 and $7.48.
Thus, on average, we would expect to receive
a dividend from between $2.72 and $7.48
approximately 2/3 of the time.
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Standard Deviation and Risk Aversion
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Standard deviation is an indication of the risk
of the investment.
Given that most people are risk averse, what
can we say about investments and their
standard deviations?
If investment A and investment B have the
same expected return, but investment B has a
higher standard deviation, which investment
would you choose? Why?
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Coefficient of Variation
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Suppose two investments have different
expected returns and different standard
deviations.
How do we know which one to choose?
Coefficient of variation = standard
deviation / expected return.
The risk averse investor will choose the
lowest risk for the greater return and thus,
the lower ratio.
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Diversification

The concept of a portfolio:
A financial portfolio is a collection of two or
more assets.
 Why do investors hold more than one asset?
 “Don’t put all your eggs in one basket.”
 Investors hold more than one asset in order
to be diversified.
 Investors diversify to improve or hold returns
constant and reduce overall risk.

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Diversification (continued)

If we have two assets that are very similar,
they could both increase or both decrease.
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What does risk aversion tell us about these
alternatives?
If we have two assets that are different, we
can maintain a return by holding them
simultaneously—if one decreases, the other
increases.
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Correlation
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How do we know if the returns of two assets move
in the same direction (or not)?
Correlation coefficient
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Abbreviated  (lowercase Greek or “rho”)
A statistical measure of the relationship between two
variables
What kind of relationship would you expect to find
between the amount of rainfall and umbrella sales?

In this case, would be positive
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Correlation (continued)
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The correlation coefficient ranges from –1.0 to
+1.0.
 If two assets have returns that move together in
perfect lockstep, we can say their returns have
a rho of +1.0.
 If they move in exactly opposite directions, then
the rho of their returns is –1.0.
Given what you know about portfolios, the ideal
pair of assets would have a rho of _____.
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Correlation and Risk
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If the correlation coefficient of the returns of
two assets is +1.0, then the standard
deviation (risk) of the portfolio is simply the
weighted average of the standard
deviations of the two assets.
Thus, there is no risk reduction in this case.
 There would be no benefit from holding
these two assets in portfolio.

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Correlation and Risk (continued)
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On the other hand, what happens if we find two
assets with a correlation of –1.0?
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What is the likelihood of finding two assets with
perfect negative correlation?
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Risk would be completely eliminated.
The standard deviation of the portfolio would be 0.
It is rare if not impossible.
However, all we have to find is two assets with
correlation of less than +1.0 to achieve some benefits
of risk reduction.
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The Market Portfolio
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If we are very risk averse, can we hold an
asset that has no risk?

Risk-free asset has a guaranteed return.
 An
example would be a security issued by the
U.S. Government, such as a treasury bill.

If the return is guaranteed, what is the
standard deviation of the return for the risk-free
asset?
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The Market Portfolio (continued)
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The market portfolio is a theoretical portfolio
comprising all assets in the appropriate proportion.
It is the most “efficient.”
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It provides the most return for a given level of risk.
Since the market portfolio is the best portfolio in
terms of risk and return, we must assume investors
will own it.
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The Capital Market Line
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Depending on risk preferences, investors can
invest part of their funds in the risk-free asset
and part in the market portfolio.
They could also increase their return (but
also the risk) by holding the market portfolio
and borrowing funds.
The line that extends from the risk-free asset
through the market portfolio is called the
capital market line.
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Risk and Its Components
Systematic Risk + Unsystematic Risk = Total Risk
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Systematic risk relates to those factors that affect
all assets in the market.
Unsystematic risk relates to those factors that are
specific to a particular asset.
The market portfolio is so diversified that all
unsystematic risk is removed as assets are added
to it.
Therefore, the only risk in the market portfolio is
systematic.
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How Can We Hold the Market Portfolio?
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Do we need to hold all the assets in the world to
obtain the benefits of the market portfolio?
Research indicates that if we have approximately 30
assets in a portfolio we will have obtained the
maximum benefit from diversification.
Investors hold a “proxy” for the market portfolio—a
mutual fund such as the S&P 500 index fund.
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What is the Relevant Risk
of New Assets?
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We assume investors are diversified and
hold a proxy for the market portfolio.
Therefore, the only risk component relevant
to them is the systematic risk because the
unsystematic risk of an investment will be
diversified away.
Total risk (standard deviation) includes both
types of risk.
Is there a measure of systematic risk?
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Beta
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Beta is the measure of an asset’s systematic
risk relative to the market portfolio.
Beta = xmx /  m
It is found by multiplying the correlation
coefficient of any asset (asset x) and the
market portfolio by the standard deviation of
asset x. This product is divided by the
standard deviation of the market portfolio.
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Beta (continued)
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Betas are compared to the overall market.
The market portfolio has a beta of 1.
If the stock of a company has a beta of 2, it is
twice as risky as the market.
Where can I find betas?
Use linear regression
 Yahoo! Finance website
 Various brokerage firm websites

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Beta as a Predictive Tool
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The capital market line examines return
versus total risk.
The security market line (SML) measures
return of a security against beta.
The SML represents a minimum expected
return given the relevant risk of a security.
Expected Return = Rf + [(Rm – Rf) x ]
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The Capital Asset Pricing Model
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The SML is an equation for a straight line.
Beta is the slope of the line.
This is also known as the Capital Asset
Pricing Model (CAPM).
If a project generates a return higher than
the required rate of return as shown by the
SML, value is created and the project is
accepted. If not, then value is lost and the
project should be rejected.
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Limitations of CAPM

The CAPM cannot always predict the returns of
assets accurately and it has limitations.
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The market portfolio is a theoretical concept; no
consensus on which proxy for the market portfolio is
best.
Betas are calculated based upon historical returns and
then used to predict future returns.
Despite the limitations, CAPM is useful in getting
investors to understand a fundamental relationship
between risk and return.
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©2005 Pearson Education, Inc.
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