Chapter 5 - Risk and Return

CHAPTER
5 RISK
AND RETURN
L E A R N I N G G O A L S
Understand the meaning and fundamentals of risk, Review the two types of risk and the derivation
LG1 LG5
return, and risk preferences. and role of beta in measuring the relevant risk of
both an individual security and a portfolio.
Describe procedures for assessing and measur-
LG2
ing the risk of a single asset. Explain the capital asset pricing model (CAPM),
LG6
its relationship to the security market line (SML),
Discuss the measurement of return and standard
LG3 and shifts in the SML caused by changes in infla-
deviation for a portfolio and the various types tionary expectations and risk aversion.
of correlation that can exist between series of
numbers.
Understand the risk and return characteristics of
LG4
a portfolio in terms of correlation and diversifica-
tion, and the impact of international assets on a
portfolio.
Across the Disciplines WHY THIS CHAPTER MATTERS TO YO U

Accounting: You need to understand the relationship between translate those data into decisions that increase the value of
risk and return because of the effect that riskier projects will the firm.
have on the firm’s annual net income and on your efforts to
Marketing: You need to understand that although higher-risk
stabilize net income.
projects may produce higher returns, they may not be the best
Information systems: You need to understand how to do sensi- choice for the firm if they produce an erratic earnings pattern
tivity and correlation analyses in order to build decision pack- and do not optimize the value of the firm.
ages that help management analyze the risk and return of vari-
Operations: You need to understand how investments in plant
ous business opportunities.
assets and purchases of supplies will be measured by the firm
Management: You need to understand the relationship and to recognize that decisions about such investments will be
between risk and return, and how to measure that relationship made by evaluating the effects of both risk and return on the
in order to evaluate data that come from finance personnel and value of the firm.
212
CITIGROUP
CITIGROUP TAKES ON
NEW ASSOCIATES
A s they chased after hot new financial

services businesses that boosted
earnings quickly, many banks ignored a
key principle of risk management: Diver-
sification reduces risk. They expanded
into risky areas such as investment
banking, stock brokerage, wealth management, and equity investment, and they moved away
from their traditional services such as mortgage banking, auto financing, and credit cards.
Although adding new business lines is a way to diversify, the benefits of diversification come from
balancing low-risk and high-risk activities. As the economy changed, banks ran into problems
with these new, higher-risk services. Banks that had “hedged their bets” by continuing to offer a
variety of services spread across the risk spectrum earned higher returns.
Citigroup is a case study for the benefits of diversification. The company, created in 1998 by
the merger of Citicorp and Travelers Group, provides a broad range of financial products and ser-
vices to 100 million consumers, corporations, governments, and institutions in over 100 countries.
These offerings include consumer banking and credit, corporate and investment banking, com-
mercial finance, leasing, insurance, securities brokerage, and asset management. Under the
leadership of Citigroup CEO Sandy Weill, the company made acquisitions that reduced its depen-
dence on corporate and investment banking. In September 2000, Citigroup bought Associates
First Capital Corp for $31 billion.
With the acquisition of Associates, Citigroup shifted the balance of its business more toward
consumers than toward institutions. Associates’s target market is the lower-middle economic
class. Although these customers are riskier than the traditional bank customer, the rewards are
greater too, because Associates can charge higher interest rates and fees to compensate itself
for taking on the additional risk. The existing consumer finance businesses of both Associates and
Citigroup know how to handle this type of lending and earn solid returns in the process.
A more diversified group of businesses with greater emphasis on the consumer side should
reduce Citigroup’s earnings volatility and improve shareholder value. Commenting in spring 2001
on the corporation’s ability to weather the current economic downturn, Weill said, “The strength
and diversity of our earnings by business, geography, and customer helped to deliver a strong
bottom line in a period of market uncertainty.” Citigroup’s return on equity (ROE) for the first quar-
ter 2001 was 22.5 percent, just above fiscal year 2000’s 22.4 percent and better than its average
ROE of 19 percent for the period 1998 to 2000.
Citigroup and its consumer business units demonstrate several key fundamental financial
concepts: Risk and return are linked, return should increase if risk increases, and diversification
reduces risk. As this chapter will show, firms can use various tools and techniques to quantify
and assess the risk and return for individual assets and for groups of assets.
213
214 PART 2 Important Financial Concepts
LG1 5.1 Risk and Return Fundamentals

To maximize share price, the financial manager must learn to assess two key
determinants: risk and return.1 Each financial decision presents certain risk and
return characteristics, and the unique combination of these characteristics has an
impact on share price. Risk can be viewed as it is related either to a single asset or
portfolio to a portfolio—a collection, or group, of assets. We will look at both, beginning
A collection, or group, of assets. with the risk of a single asset. First, though, it is important to introduce some fun-
damental ideas about risk, return, and risk preferences.
Risk Defined
risk In the most basic sense, risk is the chance of financial loss. Assets having greater
The chance of financial loss or, chances of loss are viewed as more risky than those with lesser chances of loss.
more formally, the variability of More formally, the term risk is used interchangeably with uncertainty to refer to
returns associated with a given
asset.
the variability of returns associated with a given asset. A $1,000 government
bond that guarantees its holder $100 interest after 30 days has no risk, because
there is no variability associated with the return. A $1,000 investment in a firm’s
common stock, which over the same period may earn anywhere from $0 to $200,
is very risky because of the high variability of its return. The more nearly certain
the return from an asset, the less variability and therefore the less risk.
Some risks directly affect both financial managers and shareholders. Table 5.1
briefly describes the common sources of risk that affect both firms and their share-
holders. As you can see, business risk and financial risk are more firm-specific and
therefore are of greatest interest to financial managers. Interest rate, liquidity, and
market risks are more shareholder-specific and therefore are of greatest interest to
stockholders. Event, exchange rate, purchasing-power, and tax risk directly affect
both firms and shareholders. The nearby box focuses on another risk that affects
both firms and shareholders—moral risk. A number of these risks are discussed in
more detail later in this text. Clearly, both financial managers and shareholders
must assess these and other risks as they make investment decisions.
Return Defined
return
The total gain or loss experi- Obviously, if we are going to assess risk on the basis of variability of return, we
enced on an investment over a need to be certain we know what return is and how to measure it. The return is
given period of time; calculated the total gain or loss experienced on an investment over a given period of time. It
by dividing the asset’s cash
is commonly measured as cash distributions during the period plus the change in
distributions during the period,
plus change in value, by its value, expressed as a percentage of the beginning-of-period investment value. The
beginning-of-period investment expression for calculating the rate of return earned on any asset over period t, kt,
value. is commonly defined as
Ct Pt Pt1
kt (5.1)
Pt1
1. Two important points should be recognized here: (1) Although for convenience the publicly traded corporation is
being discussed, the risk and return concepts presented apply to all firms; and (2) concern centers only on the wealth
of common stockholders, because they are the “residual owners” whose returns are in no way specified in advance.
CHAPTER 5 Risk and Return 215
TABLE 5.1 Popular Sources of Risk Affecting Financial Managers

and Shareholders
Source of risk Description
Firm-Specific Risks
Business risk The chance that the firm will be unable to cover its operating costs. Level is driven by the firm’s
revenue stability and the structure of its operating costs (fixed vs. variable).
Financial risk The chance that the firm will be unable to cover its financial obligations. Level is driven by the
predictability of the firm’s operating cash flows and its fixed-cost financial obligations.
Shareholder-Specific Risks
Interest rate risk The chance that changes in interest rates will adversely affect the value of an investment. Most
investments lose value when the interest rate rises and increase in value when it falls.
Liquidity risk The chance that an investment cannot be easily liquidated at a reasonable price. Liquidity is signif-
icantly affected by the size and depth of the market in which an investment is customarily traded.
Market risk The chance that the value of an investment will decline because of market factors that are inde-
pendent of the investment (such as economic, political, and social events). In general, the more a
given investment’s value responds to the market, the greater its risk; and the less it responds, the
smaller its risk.
Firm and Shareholder Risks
Event risk The chance that a totally unexpected event will have a significant effect on the value of the firm
or a specific investment. These infrequent events, such as government-mandated withdrawal of a
popular prescription drug, typically affect only a small group of firms or investments.
Exchange rate risk The exposure of future expected cash flows to fluctuations in the currency exchange rate. The
greater the chance of undesirable exchange rate fluctuations, the greater the risk of the cash flows
and therefore the lower the value of the firm or investment.
Purchasing-power risk The chance that changing price levels caused by inflation or deflation in the economy will
adversely affect the firm’s or investment’s cash flows and value. Typically, firms or investments
with cash flows that move with general price levels have a low purchasing-power risk, and those
with cash flows that do not move with general price levels have high purchasing-power risk.
Tax risk The chance that unfavorable changes in tax laws will occur. Firms and investments with values
that are sensitive to tax law changes are more risky.
where
kt actual,expected, or required rate of return2 during period t
Ct cash (flow) received from the asset investment in the time period
t 1 to t
Pt price (value) of asset at time t
Pt1 price (value) of asset at time t 1
2. The terms expected return and required return are used interchangeably throughout this text, because in an effi-
cient market (discussed later) they would be expected to be equal. The actual return is an ex post value, whereas
expected and required returns are ex ante values. Therefore, the actual return may be greater than, equal to, or less
than the expected/required return.
In Practice
FOCUS ON ETHICS What About Moral Risk?
The poster boy for “moral risk,” holder wealth maximization has to & Johnson); promoting openness
the devastating effects of unethi- be ethically constrained. for employees with concerns;
cal behavior for a company’s What can companies do to weeding out employees who do
investors, has to be Nick Leeson. instill and maintain ethical corpo- not share the company’s ethics
This 28-year-old trader violated rate practices? They can start by values before those employees
his bank’s investing rules while building awareness through a can harm the company’s reputa-
secretly placing huge bets on the code of ethics. Nearly all Fortune tion or culture; assigning an indi-
direction of the Japanese stock 500 companies and about half of vidual the role of ethics director;
market. When those bets proved all companies have an ethics code and evaluating leaders’ ethics
to be wrong, the $1.24-billion spelling out general principles of in performance reviews (as at
losses resulted in the demise of right and wrong conduct. Compa- Merck & Co.).
the centuries-old Barings Bank. nies such as Halliburton and The Leeson saga under-
More than any other single Texas Instruments have gone into scores the difficulty of dealing
episode in world financial history, specifics, because ethical codes with the “moral hazard” problem,
Leeson’s misdeeds underscored are often faulted for being too when the consequences of an
the importance of character in vague and abstract. individual’s actions are largely
the financial industry. Forty-one Ethical organizations also borne by others. John Boatright
percent of surveyed CFOs admit reveal their commitments through argues in his book Ethics in
ethical problems in their organiza- the following activities: talking Finance that the best antidote is to
tions (self-reported percents are about ethical values periodically; attract loyal, hardworking employ-
probably low), and 48 percent of including ethics in required train- ees. Ethicists Rae and Wong tell
surveyed employees admit to ing for mid-level managers (as at us that debating issues is fruitless
engaging in unethical practices Procter & Gamble); modeling if we continue to ignore the char-
such as cheating on expense ethics throughout top management acter traits that empower people
accounts and forging signatures. and the board (termed “tone at the for moral behavior.
We are reminded again that share- top,” especially notable at Johnson
The return, kt, reflects the combined effect of cash flow, Ct, and changes in value,
Pt Pt1, over period t.3
Equation 5.1 is used to determine the rate of return over a time period as
short as 1 day or as long as 10 years or more. However, in most cases, t is 1 year,
and k therefore represents an annual rate of return.
EXAMPLE Robin’s Gameroom, a high-traffic video arcade, wishes to determine the return on
two of its video machines, Conqueror and Demolition. Conqueror was purchased
1 year ago for $20,000 and currently has a market value of $21,500. During the
year, it generated $800 of after-tax cash receipts. Demolition was purchased
4 years ago; its value in the year just completed declined from $12,000 to
$11,800. During the year, it generated $1,700 of after-tax cash receipts. Substi-
3. The beginning-of-period value, Pt1, and the end-of-period value, Pt, are not necessarily realized values. They are
often unrealized, which means that although the asset was not actually purchased at time t 1 and sold at time t,
values Pt1 and Pt could have been realized had those transactions been made.
tuting into Equation 5.1, we can calculate the annual rate of return, k, for each
video machine.
Conqueror (C):
$800 $21,500 $20,000 $2,300
kC 11.5%
$20,000
$20,000

Demolition (D):
$1,700 $11,800 $12,000 $1,500
kD 12.5%
$12,000 $12,000

Although the market value of Demolition declined during the year, its cash flow
caused it to earn a higher rate of return than Conqueror earned during the same
period. Clearly, the combined impact of cash flow and changes in value, mea-
sured by the rate of return, is important.
Historical Returns
Investment returns vary both over time and between different types of invest-
ments. By averaging historical returns over a long period of time, it is possible to
eliminate the impact of market and other types of risk. This enables the financial
decision maker to focus on the differences in return that are attributable primar-
ily to the types of investment. Table 5.2 shows the average annual rates of return
for a number of popular security investments (and inflation) over the 75-year
period January 1, 1926, through December 31, 2000. Each rate represents the
average annual rate of return an investor would have realized had he or she pur-
chased the investment on January 1, 1926, and sold it on December 31, 2000.
You can see that significant differences exist between the average annual rates of
return realized on the various types of stocks, bonds, and bills shown. Later in
this chapter, we will see how these differences in return can be linked to differ-
ences in the risk of each of these investments.
TABLE 5.2 Historical Returns for

Selected Security
Investments (1926–2000)
Investment Average annual return
Large-company stocks 13.0%

Small-company stocks 17.3
Long-term corporate bonds 6.0
Long-term government bonds 5.7
U.S. Treasury bills 3.9
Inflation 3.2%
Source: Stocks, Bonds, Bills, and Inflation, 2001 Yearbook
(Chicago: Ibbotson Associates, Inc., 2001).
FIGURE 5.1
Required (or Expected) Return

Risk Preferences
Risk-Averse
Risk preference behaviors
Averse
Indifferent Risk-Indifferent
Seeking
Risk-Seeking
0 x1 x2
Risk
Risk Preferences
Feelings about risk differ among managers (and firms).4 Thus it is important to
specify a generally acceptable level of risk. The three basic risk preference behav-
iors—risk-averse, risk-indifferent, and risk-seeking—are depicted graphically in
risk-indifferent
Figure 5.1.
The attitude toward risk in which
no change in return would be • For the risk-indifferent manager, the required return does not change as risk
required for an increase in risk.
goes from x1 to x2. In essence, no change in return would be required for the
risk-averse increase in risk. Clearly, this attitude is nonsensical in almost any business
The attitude toward risk in which context.
an increased return would be • For the risk-averse manager, the required return increases for an increase in
required for an increase in risk.
risk. Because they shy away from risk, these managers require higher
risk-seeking expected returns to compensate them for taking greater risk.
The attitude toward risk in which • For the risk-seeking manager, the required return decreases for an increase in
a decreased return would be
risk. Theoretically, because they enjoy risk, these managers are willing to give
accepted for an increase in risk.
up some return to take more risk. However, such behavior would not be
Hint Remember that likely to benefit the firm.
most shareholders are also
risk-averse. Like risk-averse Most managers are risk-averse; for a given increase in risk, they require an
managers, for a given increase increase in return. They generally tend to be conservative rather than aggressive
in risk, they also require an
increase in return on their when accepting risk for their firm. Accordingly, a risk-averse financial manager
investment in that firm. requiring higher returns for greater risk is assumed throughout this text.
Review Questions
5–1 What is risk in the context of financial decision making?
5–2 Define return, and describe how to find the rate of return on an investment.
4. The risk preferences of the managers should in theory be consistent with the risk preferences of the firm. Although
the agency problem suggests that in practice managers may not behave in a manner consistent with the firm’s risk
preferences, it is assumed here that they do. Therefore, the managers’ risk preferences and those of the firm are
assumed to be identical.
5–3 Compare the following risk preferences: (a) risk-averse, (b) risk-indifferent,
and (c) risk-seeking. Which is most common among financial managers?
LG2 5.2 Risk of a Single Asset

The concept of risk can be developed by first considering a single asset held in
isolation. We can look at expected-return behaviors to assess risk, and statistics
can be used to measure it.
Risk Assessment
Sensitivity analysis and probability distributions can be used to assess the general
level of risk embodied in a given asset.
sensitivity analysis
An approach for assessing risk Sensitivity Analysis
that uses several possible-return
estimates to obtain a sense of the Sensitivity analysis uses several possible-return estimates to obtain a sense of the
variability among outcomes. variability among outcomes.5 One common method involves making pessimistic
(worst), most likely (expected), and optimistic (best) estimates of the returns
range associated with a given asset. In this case, the asset’s risk can be measured by the
A measure of an asset’s risk,
which is found by subtracting the
range of returns. The range is found by subtracting the pessimistic outcome from
pessimistic (worst) outcome from the optimistic outcome. The greater the range, the more variability, or risk, the
the optimistic (best) outcome. asset is said to have.
EXAMPLE Norman Company, a custom golf equipment manufacturer, wants to choose the
better of two investments, A and B. Each requires an initial outlay of $10,000,
and each has a most likely annual rate of return of 15%. Management has made
pessimistic and optimistic estimates of the returns associated with each. The three
estimates for each asset, along with its range, are given in Table 5.3. Asset A
appears to be less risky than asset B; its range of 4% (17% 13%) is less than
the range of 16% (23% 7%) for asset B. The risk-averse decision maker would
prefer asset A over asset B, because A offers the same most likely return as B
(15%) with lower risk (smaller range).
TABLE 5.3 Assets A and B
Asset A Asset B
Initial investment $10,000 $10,000

Annual rate of return
Pessimistic 13% 7%
Most likely 15% 15%
Optimistic 17% 23%
Range 4% 16%
5. The term sensitivity analysis is intentionally used in a general rather than a technically correct fashion here to sim-
plify this discussion. A more technical and precise definition and discussion of this technique and of “scenario analy-
sis” are presented in Chapter 10.
Although the use of sensitivity analysis and the range is rather crude, it does
give the decision maker a feel for the behavior of returns, which can be used to
estimate the risk involved.
Probability Distributions
Probability distributions provide a more quantitative insight into an asset’s risk.
probability The probability of a given outcome is its chance of occurring. An outcome with
The chance that a given outcome an 80 percent probability of occurrence would be expected to occur 8 out of 10
will occur. times. An outcome with a probability of 100 percent is certain to occur. Out-
comes with a probability of zero will never occur.
EXAMPLE Norman Company’s past estimates indicate that the probabilities of the pes-
simistic, most likely, and optimistic outcomes are 25%, 50%, and 25%, respec-
tively. Note that the sum of these probabilities must equal 100%; that is, they
must be based on all the alternatives considered.
probability distribution
A model that relates probabilities A probability distribution is a model that relates probabilities to the associ-
to the associated outcomes. ated outcomes. The simplest type of probability distribution is the bar chart,
bar chart which shows only a limited number of outcome–probability coordinates. The bar
The simplest type of probability charts for Norman Company’s assets A and B are shown in Figure 5.2. Although
distribution; shows only a limited both assets have the same most likely return, the range of return is much greater,
number of outcomes and associ- or more dispersed, for asset B than for asset A—16 percent versus 4 percent.
ated probabilities for a given
event.
If we knew all the possible outcomes and associated probabilities, we could
develop a continuous probability distribution. This type of distribution can be
continuous probability thought of as a bar chart for a very large number of outcomes.6 Figure 5.3 pre-
distribution sents continuous probability distributions for assets A and B.7 Note that although
A probability distribution
showing all the possible
assets A and B have the same most likely return (15 percent), the distribution of
outcomes and associated
probabilities for a given event.
FIGURE 5.2
Probability of Occurrence
Probability of Occurrence
Asset A Asset B
Bar Charts
Bar charts for asset A’s and .60 .60
asset B’s returns .50 .50
.40 .40
.30 .30
.20 .20
.10 .10
0 5 9 13 17 21 25 0 5 9 13 17 21 25
Return (%) Return (%)
6. To develop a continuous probability distribution, one must have data on a large number of historical occurrences
for a given event. Then, by developing a frequency distribution indicating how many times each outcome has
occurred over the given time horizon, one can convert these data into a probability distribution. Probability distri-
butions for risky events can also be developed by using simulation—a process discussed briefly in Chapter 10.
7. The continuous distribution’s probabilities change because of the large number of additional outcomes consid-
ered. The area under each of the curves is equal to 1, which means that 100% of the outcomes, or all the possible
outcomes, are considered.
FIGURE 5.3
Probability Density
Continuous Probability
Distributions
Asset A
Continuous probability
distributions for asset A’s
and asset B’s returns
Asset B
0 5 7 9 11 13 15 17 19 21 23 25
Return (%)
returns for asset B has much greater dispersion than the distribution for asset A.
Clearly, asset B is more risky than asset A.
Risk Measurement
In addition to considering its range, the risk of an asset can be measured quanti-
tatively by using statistics. Here we consider two statistics—the standard devia-
tion and the coefficient of variation—that can be used to measure the variability
of asset returns.
Standard Deviation
standard deviation (k) The most common statistical indicator of an asset’s risk is the standard deviation,
The most common statistical k, which measures the dispersion around the expected value.8 The expected value
indicator of an asset’s risk; it , is the most likely return on an asset. It is calculated as follows:9
of a return, k
measures the dispersion around
the expected value. n
kj Prj
k (5.2)
expected value of a return (k) j1
The most likely return on a given
asset. where
kj return for the jth outcome

Prj probability of occurrence of the jth outcome
n number of outcomes considered
8. Although risk is typically viewed as determined by the dispersion of outcomes around an expected value, many
people believe that risk exists only when outcomes are below the expected value, because only returns below the
expected value are considered bad. Nevertheless, the common approach is to view risk as determined by the vari-
ability on either side of the expected value, because the greater this variability, the less confident one can be of the
outcomes associated with an investment.
9. The formula for finding the expected value of return, k, when all of the outcomes, kj, are known and their related
probabilities are assumed to be equal, is a simple arithmetic average:
n
kj
j1
(5.2a)

k n
where n is the number of observations. Equation 5.2 is emphasized in this chapter because returns and related prob-
abilities are often available.
TABLE 5.4 Expected Values of Returns for

Assets A and B
Weighted value
Possible Probability Returns [(1) (2)]
outcomes (1) (2) (3)
Asset A
Pessimistic .25 13% 3.25%
Most likely .50 15 7.50
Optimistic .2
5 17 4.25

Total 1 .00 Expected return 15.00%

Asset B
Pessimistic .25 7% 1.75%

Most likely .50 15 7.50
Optimistic .2
5 23 5.75

Total 1
.0
0 Expected return 15.00%

EXAMPLE The expected values of returns for Norman Company’s assets A and B are pre-
sented in Table 5.4. Column 1 gives the Prj’s and column 2 gives the kj’s. In each
case n equals 3. The expected value for each asset’s return is 15%.
The expression for the standard deviation of returns, k, is10
j1
(kj j
n
k )2 Pr
k (5.3)
In general, the higher the standard deviation, the greater the risk.
EXAMPLE Table 5.5 presents the standard deviations for Norman Company’s assets A and
B, based on the earlier data. The standard deviation for asset A is 1.41%, and the
standard deviation for asset B is 5.66%. The higher risk of asset B is clearly
reflected in its higher standard deviation.
Historical Returns and Risk We can now use the standard deviation as a
measure of risk to assess the historical (1926–2000) investment return data in
Table 5.2. Table 5.6 repeats the historical returns and shows the standard devia-
tions associated with each of them. A close relationship can be seen between the
investment returns and the standard deviations: Investments with higher returns
have higher standard deviations. Because higher standard deviations are associ-
ated with greater risk, the historical data confirm the existence of a positive rela-
10. The formula that is commonly used to find the standard deviation of returns, k, in a situation in which all out-
comes are known and their related probabilities are assumed equal, is

n
(kj k)2
k j1
(5.3a)
n1
where n is the number of observations. Equation 5.3 is emphasized in this chapter because returns and related prob-
abilities are often available.
TABLE 5.5 The Calculation of the Standard Deviation

of the Returns for Assets A and Ba
i kj
k kj k
(kj k
)2 Prj (kj k
)2 Prj
Asset A
1 13% 15% 2% 4% .25 1%

2 15 15 0 0 .50 0
3 17 15 2 4 .25 1

3
(kj k)2 Prj

j1
2%

3
k (kj k
)2 Prj 2%
1.41%
A j1

Asset B
1 7% 15% 8% 64% .25 16%

2 15 15 0 0 .50 0
3 23 15 8 64 .25 1
6

3
(kj k)2 Prj 32%

j1

3
kB (kj k)2 Prj 32
j1
% 5.66%

aCalculations in this table are made in percentage form rather than decimal form—e.g., 13%
rather than 0.13. As a result, some of the intermediate computations may appear to be incon-
sistent with those that would result from using decimal form. Regardless, the resulting stan-
dard deviations are correct and identical to those that would result from using decimal rather
than percentage form.
tionship between risk and return. That relationship reflects risk aversion by mar-
ket participants, who require higher returns as compensation for greater risk. The
historical data in Table 5.6 clearly show that during the 1926–2000 period,
investors were rewarded with higher returns on higher-risk investments.
TABLE 5.6 Historical Returns and Standard

Deviations for Selected Security
Investments (1926–2000)
Investment Average annual return Standard deviation
Large-company stocks 13.0% 20.2%

Small-company stocks 17.3 33.4
Long-term corporate bonds 6.0 8.7
Long-term government bonds 5.7 9.4
U.S. Treasury bills 3.9 3.2
Inflation 3.2% 4.4%
Source: Stocks, Bonds, Bills, and Inflation, 2001 Yearbook (Chicago: Ibbotson Associates,
Inc., 2001).
FIGURE 5.4
Probability Density
Bell-Shaped Curve
Normal probability distribu-
tion, with ranges
68%
95%
99%
0 –3σk –2σk –1σk k +1σk +2σk +3σk

Return (%)
Normal Distribution A normal probability distribution, depicted in Figure

normal probability distribution 5.4, always resembles a “bell-shaped” curve. It is symmetrical: From the peak of
A symmetrical probability distri- the graph, the curve’s extensions are mirror images (reflections) of each other.
bution whose shape resembles a The symmetry of the curve means that half the probability is associated with the
“bell-shaped” curve.
values to the left of the peak and half with the values to the right. As noted on the
figure, for normal probability distributions, 68 percent of the possible outcomes
will lie between 1 standard deviation from the expected value, 95 percent of all
outcomes will lie between 2 standard deviations from the expected value, and
99 percent of all outcomes will lie between 3 standard deviations from the
expected value.11
EXAMPLE If we assume that the probability distribution of returns for the Norman Company
is normal, 68% of the possible outcomes would have a return ranging between
13.59 and 16.41% for asset A and between 9.34 and 20.66% for asset B; 95% of
the possible return outcomes would range between 12.18 and 17.82% for asset A
and between 3.68 and 26.32% for asset B; and 99% of the possible return outcomes
would range between 10.77 and 19.23% for asset A and between 1.98 and
31.98% for asset B. The greater risk of asset B is clearly reflected in its much wider
range of possible returns for each level of confidence (68%, 95%, etc.).
Coefficient of Variation
coefficient of variation (CV ) The coefficient of variation, CV, is a measure of relative dispersion that is useful
A measure of relative dispersion in comparing the risks of assets with differing expected returns. Equation 5.4
that is useful in comparing the gives the expression for the coefficient of variation:
risks of assets with differing
expected returns. k
CV (5.4)

k
The higher the coefficient of variation, the greater the risk.
11. Tables of values indicating the probabilities associated with various deviations from the expected value of a nor-
mal distribution can be found in any basic statistics text. These values can be used to establish confidence limits and
make inferences about possible outcomes. Such applications can be found in most basic statistics and upper-level
managerial finance textbooks.
EXAMPLE When the standard deviations (from Table 5.5) and the expected returns (from
Table 5.4) for assets A and B are substituted into Equation 5.4, the coefficients of
variation for A and B are 0.094 (1.41% 15%) and 0.377 (5.66% 15%),
respectively. Asset B has the higher coefficient of variation and is therefore more
risky than asset A—which we already know from the standard deviation.
(Because both assets have the same expected return, the coefficient of variation
has not provided any new information.)
The real utility of the coefficient of variation comes in comparing the risks of
assets that have different expected returns.
EXAMPLE A firm wants to select the less risky of two alternative assets—X and Y. The
expected return, standard deviation, and coefficient of variation for each of these
assets’ returns are
Statistics Asset X Asset Y
(1) Expected return 12% 20%

(2) Standard deviation 9%a 10%
(3) Coefficient of variation [(2) (1)] 0.75 0.50a
aPreferred asset using the given risk measure.
Judging solely on the basis of their standard deviations, the firm would prefer
asset X, which has a lower standard deviation than asset Y (9% versus 10%).
However, management would be making a serious error in choosing asset X over
asset Y, because the dispersion—the risk—of the asset, as reflected in the coeffi-
cient of variation, is lower for Y (0.50) than for X (0.75). Clearly, using the coef-
ficient of variation to compare asset risk is effective because it also considers the
relative size, or expected return, of the assets.
Review Questions
5–4 Explain how the range is used in sensitivity analysis.
5–5 What does a plot of the probability distribution of outcomes show a deci-
sion maker about an asset’s risk?
5–6 What relationship exists between the size of the standard deviation and
the degree of asset risk?
5–7 When is the coefficient of variation preferred over the standard deviation
for comparing asset risk?
LG3 LG4 5.3 Risk of a Portfolio

In real-world situations, the risk of any single investment would not be viewed
independently of other assets. (We did so for teaching purposes.) New invest-
ments must be considered in light of their impact on the risk and return of the
efficient portfolio portfolio of assets.12 The financial manager’s goal is to create an efficient portfo-
A portfolio that maximizes return lio, one that maximizes return for a given level of risk or minimizes risk for a
for a given level of risk or given level of return. We therefore need a way to measure the return and the stan-
minimizes risk for a given level
of return.
dard deviation of a portfolio of assets. Once we can do that, we will look at the
statistical concept of correlation, which underlies the process of diversification
that is used to develop an efficient portfolio.
Portfolio Return and Standard Deviation

The return on a portfolio is a weighted average of the returns on the individual
assets from which it is formed. We can use Equation 5.5 to find the portfolio
return, kp:
n
kp (w1 k1) (w2 k2) . . . (wn kn) wj kj (5.5)
j1
where
wj proportion of the portfolio’s total dollar value represented by asset j

kj return on asset j
n
Of course,
j=1 wj 1, which means that 100 percent of the portfolio’s assets
must be included in this computation.
The standard deviation of a portfolio’s returns is found by applying the for-
mula for the standard deviation of a single asset. Specifically, Equation 5.3 is
used when the probabilities of the returns are known, and Equation 5.3a (from
footnote 10) is applied when the outcomes are known and their related probabil-
ities of occurrence are assumed to be equal.
EXAMPLE Assume that we wish to determine the expected value and standard deviation of
returns for portfolio XY, created by combining equal portions (50%) of assets X
and Y. The forecasted returns of assets X and Y for each of the next 5 years
(2004–2008) are given in columns 1 and 2, respectively, in part A of Table 5.7. In
column 3, the weights of 50% for both assets X and Y along with their respective
returns from columns 1 and 2 are substituted into Equation 5.5. Column 4 shows
the results of the calculation—an expected portfolio return of 12% for each year,
2004 to 2008.
Furthermore, as shown in part B of Table 5.7, the expected value of these
portfolio returns over the 5-year period is also 12% (calculated by using Equa-
tion 5.2a, in footnote 9). In part C of Table 5.7, portfolio XY’s standard devia-
tion is calculated to be 0% (using Equation 5.3a, in footnote 10). This value
should not be surprising because the expected return each year is the same—
12%. No variability is exhibited in the expected returns from year to year.
12. The portfolio of a firm, which would consist of its total assets, is not differentiated from the portfolio of an
owner, which would probably contain a variety of different investment vehicles (i.e., assets). The differing character-
istics of these two types of portfolios should become clear upon completion of Chapter 10.
TABLE 5.7 Expected Return, Expected Value, and Standard

Deviation of Returns for Portfolio XY
A. Expected portfolio returns
Forecasted return
Expected portfolio
Asset X Asset Y Portfolio return calculationa return, kp
Year (1) (2) (3) (4)
2004 8% 16% (.50 8%) (.50 16%) 12%

2005 10 14 (.50 10%) (.50 14%) 12
2006 12 12 (.50 12%) (.50 12%) 12
2007 14 10 (.50 14%) (.50 10%) 12
2008 16 8 (.50 16%) (.50 8%) 12
B. Expected value of portfolio returns, 2004–2008b
12% 12% 12% 12% 12% 60%

p 12%
k
5 5

C. Standard deviation of expected portfolio returnsc
kp
(12% 12%)2 (12% 12%)2 (12% 12%)2 (12% 12%)2 (12% 12%)2

51

0% 0% 0% 0% 0%

4

0
% 0%
4

aUsing Equation 5.5.

bUsing Equation 5.2a found in footnote 9.
cUsing Equation 5.3a found in footnote 10.
correlation
A statistical measure of the
relationship between any two
series of numbers representing
data of any kind.
positively correlated
Describes two series that move
Correlation
in the same direction. Correlation is a statistical measure of the relationship between any two series of
negatively correlated
numbers. The numbers may represent data of any kind, from returns to test
Describes two series that move scores. If two series move in the same direction, they are positively correlated. If
in opposite directions. the series move in opposite directions, they are negatively correlated.13
13. The general long-term trends of two series could be the same (both increasing or both decreasing) or different
(one increasing, the other decreasing), and the correlation of their short-term (point-to-point) movements in both
situations could be either positive or negative. In other words, the pattern of movement around the trends could be
correlated independent of the actual relationship between the trends. Further clarification of this seemingly inconsis-
tent behavior can be found in most basic statistics texts.
FIGURE 5.5 Perfectly Positively Correlated Perfectly Negatively Correlated

Correlations
N
The correlation between
series M and series N
Return
Return
N
M M
Time Time
correlation coefficient The degree of correlation is measured by the correlation coefficient, which
A measure of the degree of ranges from 1 for perfectly positively correlated series to 1 for perfectly nega-
correlation between two series.
tively correlated series. These two extremes are depicted for series M and N in
perfectly positively correlated Figure 5.5. The perfectly positively correlated series move exactly together; the per-
Describes two positively fectly negatively correlated series move in exactly opposite directions.
correlated series that have a
correlation coefficient of 1.
perfectly negatively correlated

Diversification
Describes two negatively The concept of correlation is essential to developing an efficient portfolio. To
correlated series that have a
reduce overall risk, it is best to combine, or add to the portfolio, assets that have
correlation coefficient of 1.
a negative (or a low positive) correlation. Combining negatively correlated assets
can reduce the overall variability of returns. Figure 5.6 shows that a portfolio
containing the negatively correlated assets F and G, both of which have the same
expected return, k, also has that same return k but has less risk (variability) than
either of the individual assets. Even if assets are not negatively correlated, the
lower the positive correlation between them, the lower the resulting risk.
uncorrelated Some assets are uncorrelated—that is, there is no interaction between their
Describes two series that lack returns. Combining uncorrelated assets can reduce risk, not so effectively as com-
any interaction and therefore bining negatively correlated assets, but more effectively than combining positively
have a correlation coefficient
close to zero.
correlated assets. The correlation coefficient for uncorrelated assets is close to zero
and acts as the midpoint between perfect positive and perfect negative correlation.
FIGURE 5.6 Portfolio of

Diversification Asset F Asset G Assets F and G
Combining negatively Return Return Return
correlated assets to diversify
risk
k k
Time Time Time

The creation of a portfolio that combines two assets with perfectly positively
correlated returns results in overall portfolio risk that at minimum equals that of
the least risky asset and at maximum equals that of the most risky asset. How-
ever, a portfolio combining two assets with less than perfectly positive correla-
tion can reduce total risk to a level below that of either of the components, which
in certain situations may be zero. For example, assume that you manufacture
machine tools. The business is very cyclical, with high sales when the economy is
expanding and low sales during a recession. If you acquired another machine-
tool company, with sales positively correlated with those of your firm, the com-
bined sales would still be cyclical and risk would remain the same. Alternatively,
however, you could acquire a sewing machine manufacturer, whose sales are
countercyclical. It typically has low sales during economic expansion and high
sales during recession (when consumers are more likely to make their own
clothes). Combination with the sewing machine manufacturer, which has nega-
tively correlated sales, should reduce risk.
EXAMPLE Table 5.8 presents the forecasted returns from three different assets—X, Y, and
Z—over the next 5 years, along with their expected values and standard devia-
tions. Each of the assets has an expected value of return of 12% and a standard
TABLE 5.8 Forecasted Returns, Expected Values, and Standard Deviations for
Assets X, Y, and Z and Portfolios XY and XZ
Assets Portfolios
XYa XZb
Year X Y Z (50%X 50%Y) (50%X 50%Z)
2004 8% 16% 8% 12% 8%

2005 10 14 10 12 10
2006 12 12 12 12 12
2007 14 10 14 12 14
2008 16 8 16 12 16
c
Statistics:
Expected value 12% 12% 12% 12% 12%
Standard deviationd 3.16% 3.16% 3.16% 0% 3.16%
aPortfolioXY, which consists of 50% of asset X and 50% of asset Y, illustrates perfect negative correlation because these two return streams
behave in completely opposite fashion over the 5-year period. Its return values shown here were calculated in part A of Table 5.7.
bPortfolio XZ, which consists of 50% of asset X and 50% of asset Z, illustrates perfect positive correlation because these two return streams
behave identically over the 5-year period. Its return values were calculated by using the same method demonstrated for portfolio XY in part A of
Table 5.7.
cBecause the probabilities associated with the returns are not given, the general equations, Equation 5.2a in footnote 9 and Equation 5.3a in foot-
note 10, were used to calculate expected values and standard deviations, respectively. Calculation of the expected value and standard deviation for
portfolio XY is demonstrated in parts B and C, respectively, of Table 5.7.
d The portfolio standard deviations can be directly calculated from the standard deviations of the component assets with the following formula:
kp
w12 12 w22 22 2w1w2r1,212
where w1 and w2 are the proportions of component assets 1 and 2, 1 and 2 are the standard deviations of component assets 1 and 2, and r1,2 is
the correlation coefficient between the returns of component assets 1 and 2.
deviation of 3.16%. The assets therefore have equal return and equal risk. The
return patterns of assets X and Y are perfectly negatively correlated. They move
in exactly opposite directions over time. The returns of assets X and Z are per-
fectly positively correlated. They move in precisely the same direction. (Note: The
returns for X and Z are identical.)14
Portfolio XY Portfolio XY (shown in Table 5.8) is created by combining equal

portions of assets X and Y, the perfectly negatively correlated assets.15 (Calcula-
tion of portfolio XY’s annual expected returns, their expected value, and the
standard deviation of expected portfolio returns was demonstrated in Table 5.7.)
The risk in this portfolio, as reflected by its standard deviation, is reduced to 0%,
whereas the expected return remains at 12%. Thus the combination results in the
complete elimination of risk. Whenever assets are perfectly negatively correlated,
an optimal combination (similar to the 50–50 mix in the case of assets X and Y)
exists for which the resulting standard deviation will equal 0.
Portfolio XZ Portfolio XZ (shown in Table 5.8) is created by combining equal

portions of assets X and Z, the perfectly positively correlated assets. The risk in
this portfolio, as reflected by its standard deviation, is unaffected by this combi-
nation. Risk remains at 3.16%, and the expected return value remains at 12%.
Because assets X and Z have the same standard deviation, the minimum and
maximum standard deviations are the same (3.16%).
Correlation, Diversification,
Risk, and Return
Hint Remember, low In general, the lower the correlation between asset returns, the greater the poten-
correlation between two series tial diversification of risk. (This should be clear from the behaviors illustrated in
of numbers is less positive and
more negative—indicating Table 5.8.) For each pair of assets, there is a combination that will result in the
greater dissimilarity of behavior lowest risk (standard deviation) possible. How much risk can be reduced by this
of the two series. combination depends on the degree of correlation. Many potential combinations
(assuming divisibility) could be made, but only one combination of the infinite
number of possibilities will minimize risk.
Three possible correlations—perfect positive, uncorrelated, and perfect nega-
tive—illustrate the effect of correlation on the diversification of risk and return.
Table 5.9 summarizes the impact of correlation on the range of return and risk
for various two-asset portfolio combinations. The table shows that as we move
from perfect positive correlation to uncorrelated assets to perfect negative corre-
lation, the ability to reduce risk is improved. Note that in no case will a portfolio
of assets be riskier than the riskiest asset included in the portfolio.
14. Identical return streams are used in this example to permit clear illustration of the concepts, but it is not neces-
sary for return streams to be identical for them to be perfectly positively correlated. Any return streams that move
(i.e., vary) exactly together—regardless of the relative magnitude of the returns—are perfectly positively correlated.
15. For illustrative purposes it has been assumed that each of the assets—X, Y, and Z—can be divided up and com-
bined with other assets to create portfolios. This assumption is made only to permit clear illustration of the concepts.
The assets are not actually divisible.
TABLE 5.9 Correlation, Return, and Risk for Various

Two-Asset Portfolio Combinations
Correlation
coefficient Range of return Range of risk
1 (perfect positive) Between returns of two assets Between risk of two assets held
held in isolation in isolation
0 (uncorrelated) Between returns of two assets Between risk of most risky asset
held in isolation and an amount less than risk
of least risky asset but greater
than 0
1 (perfect negative) Between returns of two assets Between risk of most risky asset
held in isolation and 0
EXAMPLE A firm has calculated the expected return and the risk for each of two assets—R
and S.
Asset
Expected return, k Risk (standard deviation),
R 6% 3%
S 8 8
Clearly, asset R is a lower-return, lower-risk asset than asset S.

To evaluate possible combinations, the firm considered three possible corre-
lations—perfect positive, uncorrelated, and perfect negative. The results of the
analysis are shown in Figure 5.7, using the ranges of return and risk noted above.
In all cases, the return will range between the 6% return of R and the 8% return
of S. The risk, on the other hand, ranges between the individual risks of R and S
(from 3% to 8%) in the case of perfect positive correlation, from below 3% (the
risk of R) and greater than 0% to 8% (the risk of S) in the uncorrelated case, and
between 0% and 8% (the risk of S) in the perfectly negatively correlated case.
FIGURE 5.7 Correlation

Possible Correlations Coefficient Ranges of Return Ranges of Risk
Range of portfolio return (kp) +1 (Perfect Positive) +1
and risk (kp) for combina-
tions of assets R and
S for various correlation 0 (Uncorrelated) 0
coefficients
–1 (Perfect Negative) –1
0 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9
kR kS σk σk
R S
Portfolio Return (%) Portfolio Risk (%)

(kp) (σkp)
Note that only in the case of perfect negative correlation can the risk be
reduced to 0. Also note that as the correlation becomes less positive and more
negative (moving from the top of the figure down), the ability to reduce risk
improves. The amount of risk reduction achieved depends on the proportions in
which the assets are combined. Although determining the risk-minimizing combi-
nation is beyond the scope of this discussion, it is an important issue in develop-
ing portfolios of assets.
International Diversification
The ultimate example of portfolio diversification involves including foreign assets
in a portfolio. The inclusion of assets from countries with business cycles that are
not highly correlated with the U.S. business cycle reduces the portfolio’s respon-
siveness to market movements and to foreign currency fluctuations.
Returns from International Diversification

Over long periods, returns from internationally diversified portfolios tend to be
superior to those of purely domestic ones. This is particularly so if the U.S. econ-
omy is performing relatively poorly and the dollar is depreciating in value against
most foreign currencies. At such times, the dollar returns to U.S. investors on a
portfolio of foreign assets can be very attractive. However, over any single short
or intermediate period, international diversification can yield subpar returns, par-
ticularly during periods when the dollar is appreciating in value relative to other
currencies. When the U.S. currency gains in value, the dollar value of a foreign-
currency-denominated portfolio of assets declines. Even if this portfolio yields a
satisfactory return in local currency, the return to U.S. investors will be reduced
when translated into dollars. Subpar local currency portfolio returns, coupled
with an appreciating dollar, can yield truly dismal dollar returns to U.S. investors.
Overall, though, the logic of international portfolio diversification assumes
that these fluctuations in currency values and relative performance will average
out over long periods. Compared to similar, purely domestic portfolios, an inter-
nationally diversified portfolio will tend to yield a comparable return at a lower
level of risk.
Risks of International Diversification

U.S. investors should also be aware of the potential dangers of international
investing. In addition to the risk induced by currency fluctuations, several other
political risk financial risks are unique to international investing. Most important is political
Risk that arises from the risk, which arises from the possibility that a host government will take actions
possibility that a host govern-
harmful to foreign investors or that political turmoil in a country will endanger
ment will take actions harmful to
foreign investors or that political investments there. Political risks are particularly acute in developing countries,
turmoil in a country will where unstable or ideologically motivated governments may attempt to block
endanger investments there. return of profits by foreign investors or even seize (nationalize) their assets in the
host country. An example of political risk was the heightened concern after
Desert Storm in the early 1990s that Saudi Arabian fundamentalists would take
over and nationalize the U.S. oil facilities located there.
Even where governments do not impose exchange controls or seize assets,

international investors may suffer if a shortage of hard currency prevents payment
of dividends or interest to foreigners. When governments are forced to allocate
scarce foreign exchange, they rarely give top priority to the interests of foreign
investors. Instead, hard-currency reserves are typically used to pay for necessary
imports such as food, medicine, and industrial materials and to pay interest on the
government’s debt. Because most of the debt of developing countries is held by
banks rather than individuals, foreign investors are often badly harmed when a
country experiences political or economic problems.
Review Questions
5–8 What is an efficient portfolio? How can the return and standard deviation
of a portfolio be determined?
5–9 Why is the correlation between asset returns important? How does diver-
sification allow risky assets to be combined so that the risk of the portfolio
is less than the risk of the individual assets in it?
5–10 How does international diversification enhance risk reduction? When
might international diversification result in subpar returns? What are
political risks, and how do they affect international diversification?
LG5 LG6 5.4 Risk and Return: The Capital Asset

Pricing Model (CAPM)
The most important aspect of risk is the overall risk of the firm as viewed by
investors in the marketplace. Overall risk significantly affects investment oppor-
tunities and—even more important—the owners’ wealth. The basic theory that
capital asset pricing model
(CAPM)
links risk and return for all assets is the capital asset pricing model (CAPM).16 We
The basic theory that links risk will use CAPM to understand the basic risk–return tradeoffs involved in all types
and return for all assets. of financial decisions.
Types of Risk
To understand the basic types of risk, consider what happens to the risk of a
portfolio consisting of a single security (asset), to which we add securities ran-
domly selected from, say, the population of all actively traded securities. Using
16. The initial development of this theory is generally attributed to William F. Sharpe, “Capital Asset Prices: A The-
ory of Market Equilibrium Under Conditions of Risk,” Journal of Finance 19 (September 1964), pp. 425–442, and
John Lintner, “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital
Budgets,” Review of Economics and Statistics 47 (February 1965), pp 13–37. A number of authors subsequently
advanced, refined, and tested this now widely accepted theory.
FIGURE 5.8
Risk Reduction
Portfolio risk and
Portfolio Risk, σkP

Diversifiable Risk
diversification
Total Risk
Nondiversifiable Risk
1 5 10 15 20 25
Number of Securities (Assets) in Portfolio
the standard deviation of return, kp, to measure the total portfolio risk, Figure
5.8 depicts the behavior of the total portfolio risk (y axis) as more securities are
added (x axis). With the addition of securities, the total portfolio risk declines, as
a result of the effects of diversification, and tends to approach a lower limit.
Research has shown that, on average, most of the risk-reduction benefits of diver-
total risk
The combination of a security’s
sification can be gained by forming portfolios containing 15 to 20 randomly
nondiversifiable and diversifi- selected securities.17
able risk. The total risk of a security can be viewed as consisting of two parts:
Total security risk Nondiversifiable risk Diversifiable risk (5.6)
diversifiable risk Diversifiable risk (sometimes called unsystematic risk) represents the portion of
The portion of an asset’s risk that an asset’s risk that is associated with random causes that can be eliminated
is attributable to firm-specific, through diversification. It is attributable to firm-specific events, such as strikes,
random causes; can be elimi-
nated through diversification. lawsuits, regulatory actions, and loss of a key account. Nondiversifiable risk (also
Also called unsystematic risk. called systematic risk) is attributable to market factors that affect all firms; it can-
not be eliminated through diversification. (It is the shareholder-specific market
nondiversifiable risk risk described in Table 5.1.) Factors such as war, inflation, international inci-
The relevant portion of an asset’s dents, and political events account for nondiversifiable risk.
risk attributable to market
factors that affect all firms;
Because any investor can create a portfolio of assets that will eliminate virtu-
cannot be eliminated through ally all diversifiable risk, the only relevant risk is nondiversifiable risk. Any
diversification. Also called investor or firm therefore must be concerned solely with nondiversifiable risk.
systematic risk. The measurement of nondiversifiable risk is thus of primary importance in select-
ing assets with the most desired risk–return characteristics.
17. See, for example, W. H. Wagner and S. C. Lau, “The Effect of Diversification on Risk,” Financial Analysts Jour-
nal 26 (November–December 1971), pp. 48–53, and Jack Evans and Stephen H. Archer, “Diversification and the
Reduction of Dispersion: An Empirical Analysis,” Journal of Finance 23 (December 1968), pp. 761–767. A more
recent study, Gerald D. Newbould and Percy S. Poon, “The Minimum Number of Stocks Needed for Diversifica-
tion,” Financial Practice and Education (Fall 1993), pp. 85–87, shows that because an investor holds but one of a
large number of possible x-security portfolios, it is unlikely that he or she will experience the average outcome. As a
consequence, the study suggests that a minimum of 40 stocks is needed to diversify a portfolio fully. This study tends
to support the widespread popularity of mutual fund investments.
The Model: CAPM

The capital asset pricing model (CAPM) links nondiversifiable risk and return
for all assets. We will discuss the model in five sections. The first deals with
the beta coefficient, which is a measure of nondiversifiable risk. The second
section presents an equation of the model itself, and the third graphically
describes the relationship between risk and return. The fourth section discusses
the effects of changes in inflationary expectations and risk aversion on the rela-
tionship between risk and return. The final section offers some comments on
the CAPM.
Beta Coefficient
beta coefficient (b) The beta coefficient, b, is a relative measure of nondiversifiable risk. It is an
A relative measure of nondiversi- index of the degree of movement of an asset’s return in response to a change in
fiable risk. An index of the
the market return. An asset’s historical returns are used in finding the asset’s beta
degree of movement of an asset’s
return in response to a change in coefficient. The market return is the return on the market portfolio of all traded
the market return. securities. The Standard & Poor’s 500 Stock Composite Index or some similar
stock index is commonly used as the market return. Betas for actively traded
market return stocks can be obtained from a variety of sources, but you should understand how
The return on the market portfo-
they are derived and interpreted and how they are applied to portfolios.
lio of all traded securities.
Deriving Beta from Return Data An asset’s historical returns are used in
finding the asset’s beta coefficient. Figure 5.9 plots the relationship between the
returns of two assets—R and S—and the market return. Note that the horizontal
(x) axis measures the historical market returns and that the vertical (y) axis mea-
sures the individual asset’s historical returns. The first step in deriving beta
involves plotting the coordinates for the market return and asset returns from
various points in time. Such annual “market return–asset return” coordinates are
shown for asset S only for the years 1996 through 2003. For example, in 2003,
asset S’s return was 20 percent when the market return was 10 percent. By use of
statistical techniques, the “characteristic line” that best explains the relationship
between the asset return and the market return coordinates is fit to the data
points.18 The slope of this line is beta. The beta for asset R is about .80 and that
18. The empirical measurement of beta is approached by using least-squares regression analysis to find the regres-
sion coefficient (bj) in the equation for the “characteristic line”:
kj aj bj km ej
where
kj return on asset j
aj intercept
Cov (kj, km)
bj beta coefficient, which equals
m2
where
Cov (kj, km) covariance of the return on asset j, kj, and the return on the market portfolio, km
m2 variance of the return on the market portfolio
km required rate of return on the market portfolio of securities
ej random error term, which reflects the diversifiable, or unsystematic, risk of asset j
The calculations involved in finding betas are somewhat rigorous. If you want to know more about these calcula-
tions, consult an advanced managerial finance or investments text.
FIGURE 5.9
Asset Return (%) Asset S
Beta Derivationa (1997)
Graphical derivation of beta 35
for assets R and S
30 (2002) (2001)
25
(2003) bS = slope = 1.30
20
(2000) Asset R
15
(1998) 10
(1996)
5 bR = slope = .80
Market
0 Return (%)
–20 –10 10 15 20 25 30 35
–5
(1999) –10
Characteristic Line S –15
Characteristic Line R –20

–25
–30
a All data points shown are associated with asset S. No data points are shown for asset R.
for asset S is about 1.30. Asset S’s higher beta (steeper characteristic line slope)
indicates that its return is more responsive to changing market returns. Therefore
asset S is more risky than asset R.19
Hint Remember that Interpreting Betas The beta coefficient for the market is considered to be
published betas are calculated equal to 1.0. All other betas are viewed in relation to this value. Asset betas may
using historical data. When
investors use beta for decision be positive or negative, but positive betas are the norm. The majority of beta
making, they should recognize coefficients fall between .5 and 2.0. The return of a stock that is half as respon-
that past performance relative sive as the market (b .5) is expected to change by 1/2 percent for each 1 percent
to the market average may not
accurately predict future change in the return of the market portfolio. A stock that is twice as responsive as
performance. the market (b 2.0) is expected to experience a 2 percent change in its return for
each 1 percent change in the return of the market portfolio. Table 5.10 provides
various beta values and their interpretations. Beta coefficients for actively traded
stocks can be obtained from published sources such as Value Line Investment
Survey, via the Internet, or through brokerage firms. Betas for some selected
stocks are given in Table 5.11.
Portfolio Betas The beta of a portfolio can be easily estimated by using the
betas of the individual assets it includes. Letting wj represent the proportion of
19. The values of beta also depend on the time interval used for return calculations and on the number of returns
used in the regression analysis. In other words, betas calculated using monthly returns would not necessarily be com-
parable to those calculated using a similar number of daily returns.
TABLE 5.10 Selected Beta Coefficients and

Their Interpretations
Beta Comment Interpretation
2.0 Move in same Twice as responsive as the market

1.0 direction as Same response as the market
.5 market Only half as responsive as the market
0 Unaffected by market movement
.5 Move in opposite Only half as responsive as the market
1.0 direction to Same response as the market
2.0 market Twice as responsive as the market
TABLE 5.11 Beta Coefficients for Selected Stocks

(March 8, 2002)
Stock Beta Stock Beta
Amazon.com 1.95 Int’l Business Machines 1.05

Anheuser-Busch .60 Merrill Lynch & Co. 1.85
Bank One Corp. 1.25 Microsoft 1.20
Daimler Chrysler AG 1.25 NIKE, Inc. .90
Disney 1.05 PepsiCo, Inc. .70
eBay 2.20 Qualcomm 1.30
Exxon Mobil Corp. .80 Sempra Energy .60
Gap (The), Inc. 1.60 Wal-Mart Stores 1.15
General Electric 1.30 Xerox 1.25
Intel 1.30 Yahoo! Inc. 2.00
Source: Value Line Investment Survey (New York: Value Line Publishing, March 8, 2002).
the portfolio’s total dollar value represented by asset j, and letting bj equal the
beta of asset j, we can use Equation 5.7 to find the portfolio beta, bp:
n
bp (w1 b1) (w2 b2) . . . (wn bn) wj bj (5.7)
j1
n
Of course,
j=1 wj 1, which means that 100 percent of the portfolio’s assets
must be included in this computation.
Portfolio betas are interpreted in the same way as the betas of individual
Hint Mutual fund assets. They indicate the degree of responsiveness of the portfolio’s return to
managers are key users of the changes in the market return. For example, when the market return increases by
portfolio beta and return
concepts. They are continually 10 percent, a portfolio with a beta of .75 will experience a 7.5 percent increase in
evaluating what would happen its return (.75 10%); a portfolio with a beta of 1.25 will experience a 12.5 per-
to the fund’s beta and return if cent increase in its return (1.25 10%). Clearly, a portfolio containing mostly
the securities of a particular
firm were added to or deleted low-beta assets will have a low beta, and one containing mostly high-beta assets
from the fund’s portfolio. will have a high beta.
TABLE 5.12 Austin Fund’s Portfolios

V and W
Portfolio V Portfolio W
Asset Proportion Beta Proportion Beta
1 .10 1.65 .10 .80

2 .30 1.00 .10 1.00
3 .20 1.30 .20 .65
4 .20 1.10 .10 .75
5 .20 1.25 .50 1.05

Totals 1.00 1.00

EXAMPLE The Austin Fund, a large investment company, wishes to assess the risk of two
portfolios it is considering assembling—V and W. Both portfolios contain five
assets, with the proportions and betas shown in Table 5.12. The betas for the two
portfolios, bv and bw, can be calculated by substituting data from the table into
Equation 5.7:
bv (.10 1.65) (.30 1.00) (.20 1.30) (.20 1.10) (.20 1.25)
.165 .300 .260 .220 .250 1.195 1.20

bw (.10 .80) (.10 1.00) (.20 .65) (.10 .75) (.50 1.05)
.080 .100 .130 .075 .525 .91

Portfolio V’s beta is 1.20, and portfolio W’s is .91. These values make sense,
because portfolio V contains relatively high-beta assets, and portfolio W contains
relatively low-beta assets. Clearly, portfolio V’s returns are more responsive to
changes in market returns and are therefore more risky than portfolio W’s.
The Equation
Using the beta coefficient to measure nondiversifiable risk, the capital asset pric-
ing model (CAPM) is given in Equation 5.8:
kj RF [bj (km RF)] (5.8)
where
kj required return on asset j
RF risk-free rate of return, commonly measured by the
return on a U.S. Treasury bill
risk-free rate of interest, RF bj beta coefficient or index of nondiversifiable risk for asset j
The required return on a risk-free km market return; return on the market portfolio of assets
asset, typically a 3-month U.S.
Treasury bill. The CAPM can be divided into two parts: (1) risk-free of interest, RF , which
is the required return on a risk-free asset, typically a 3-month U.S. Treasury bill
U.S. Treasury bills (T-bills)
Short-term IOUs issued by the
(T-bill), a short-term IOU issued by the U.S. Treasury, and (2) the risk premium.
U.S. Treasury; considered the These are, respectively, the two elements on either side of the plus sign in Equa-
risk-free asset. tion 5.8. The (km RF) portion of the risk premium is called the market risk pre-
mium, because it represents the premium the investor must receive for taking the
average amount of risk associated with holding the market portfolio of assets.20
Historical Risk Premiums Using the historical return data for selected secu-
rity investments for the 1926–2000 period shown in Table 5.2, we can calculate
the risk premiums for each investment category. The calculation (consistent with
Equation 5.8) involves merely subtracting the historical U.S. Treasury bill’s aver-
age return from the historical average return for a given investment:
Investment Risk premiuma
Large-company stocks 13.0% 3.9% 9.1%

Small company stocks 17.3 3.9 13.4
Long-term corporate bonds 6.0 3.9 2.1
Long-term government bonds 5.7 3.9 1.8
U.S. Treasury bills 3.9 3.9 0.0
aReturn values obtained from Table 5.2.
Reviewing the risk premiums calculated above, we can see that the risk pre-
mium is highest for small-company stocks, followed by large-company stocks,
long-term corporate bonds, and long-term government bonds. This outcome
makes sense intuitively because small-company stocks are riskier than large-
company stocks, which are riskier than long-term corporate bonds (equity is
riskier than debt investment). Long-term corporate bonds are riskier than long-
term government bonds (because the government is less likely to renege on debt).
And of course, U.S. Treasury bills, because of their lack of default risk and their
very short maturity, are virtually risk-free, as indicated by their lack of any risk
premium.
EXAMPLE Benjamin Corporation, a growing computer software developer, wishes to deter-

mine the required return on an asset Z, which has a beta of 1.5. The risk-free rate
of return is 7%; the return on the market portfolio of assets is 11%. Substituting
bz 1.5, RF 7%, and km 11% into the capital asset pricing model given in
Equation 5.8 yields a required return of
kz 7% [1.5 (11% 7%)] 7% 6% 13
%
The market risk premium of 4% (11% 7%), when adjusted for the asset’s
index of risk (beta) of 1.5, results in a risk premium of 6% (1.5 4%). That risk
premium, when added to the 7% risk-free rate, results in a 13% required return.
Other things being equal, the higher the beta, the higher the required return,
and the lower the beta, the lower the required return.
20. Although CAPM has been widely accepted, a broader theory, arbitrage pricing theory (APT), first described by
Stephen A. Ross, “The Arbitrage Theory of Capital Asset Pricing,” Journal of Economic Theory (December 1976),
pp. 341–360, has received a great deal of attention in the financial literature. The theory suggests that the risk pre-
mium on securities may be better explained by a number of factors underlying and in place of the market return used
in CAPM. The CAPM in effect can be viewed as being derived from APT. Although testing of APT theory confirms
the importance of the market return, it has thus far failed to identify other risk factors clearly. As a result of this fail-
ure, as well as APT’s lack of practical acceptance and usage, we concentrate our attention here on CAPM.
The Graph: The Security Market Line (SML)

security market line (SML) When the capital asset pricing model (Equation 5.8) is depicted graphically, it is
The depiction of the capital called the security market line (SML). The SML will, in fact, be a straight line. It
asset pricing model (CAPM ) as a reflects the required return in the marketplace for each level of nondiversifiable
graph that reflects the required
return in the marketplace for
risk (beta). In the graph, risk as measured by beta, b, is plotted on the x axis, and
each level of nondiversifiable required returns, k, are plotted on the y axis. The risk–return tradeoff is clearly
risk (beta). represented by the SML.
EXAMPLE In the preceding example for Benjamin Corporation, the risk-free rate, RF, was
7%, and the market return, km, was 11%. The SML can be plotted by using the
two sets of coordinates for the betas associated with RF and km, bRF and bm (that
is, bRF 0,21 RF 7%; and bm 1.0, km 11%). Figure 5.10 presents the result-
ing security market line. As traditionally shown, the security market line in Figure
5.10 presents the required return associated with all positive betas. The market
risk premium of 4% (km of 11% RF of 7%) has been highlighted. For a beta for
asset Z, bz, of 1.5, its corresponding required return, kz, is 13%. Also shown in
the figure is asset Z’s risk premium of 6% (kz of 13% RF of 7%). It should be
clear that for assets with betas greater than 1, the risk premium is greater than
that for the market; for assets with betas less than 1, the risk premium is less than
that for the market.
FIGURE 5.10
Security Market Line 17
Security market line (SML) 16
15 SML
with Benjamin Corporation’s
asset Z data shown 14
kz = 13
Required Return, k (%)
12 Asset Z’s
km = 11 Risk
Market
10 Premium
Risk
9 Premium (6%)
8 (4%)
RF = 7
6
5
4
3
2
1
0 .5 1.0 1.5 2.0

bR bm bz
F
Nondiversifiable Risk, b
21. Because RF is the rate of return on a risk-free asset, the beta associated with the risk-free asset, bRF, would equal
0. The 0 beta on the risk-free asset reflects not only its absence of risk but also that the asset’s return is unaffected by
movements in the market return.
Shifts in the Security Market Line

The security market line is not stable over time, and shifts in the security market
line can result in a change in required return. The position and slope of the SML
are affected by two major forces—inflationary expectations and risk aversion—
which are analyzed next.22
Changes in Inflationary Expectations Changes in inflationary expectations

affect the risk-free rate of return, RF. The equation for the risk-free rate of
return is
RF k* IP (5.9)
This equation shows that, assuming a constant real rate of interest, k*, changes in
inflationary expectations, reflected in an inflation premium, IP, will result in corre-
sponding changes in the risk-free rate. Therefore, a change in inflationary expecta-
tions that results from events such as international trade embargoes or major
changes in Federal Reserve policy will result in a shift in the SML. Because the risk-
free rate is a basic component of all rates of return, any change in RF will be
reflected in all required rates of return.
Changes in inflationary expectations result in parallel shifts in the SML in
direct response to the magnitude and direction of the change. This effect can best
be illustrated by an example.
EXAMPLE In the preceding example, using CAPM, the required return for asset Z, kZ, was
found to be 13%. Assuming that the risk-free rate of 7% includes a 2% real
rate of interest, k*, and a 5% inflation premium, IP, then Equation 5.9 con-
firms that
RF 2% 5% 7%
Now assume that recent economic events have resulted in an increase of 3%

in inflationary expectations, raising the inflation premium to 8% (IP1). As a
result, all returns likewise rise by 3%. In this case, the new returns (noted by sub-
script 1) are
RF1 10% (rises from 7% to 10%)

km1 14% (rises from 11% to 14%)
Substituting these values, along with asset Z’s beta (bZ) of 1.5, into the CAPM
(Equation 5.8), we find that asset Z’s new required return (kZ1) can be calculated:
kZ1 10% [1.5 (14% 10%)] 10% 6% 16%

Comparing kZ1 of 16% to kZ of 13%, we see that the change of 3% in asset Z’s
required return exactly equals the change in the inflation premium. The same 3%
increase results for all assets.
22. A firm’s beta can change over time as a result of changes in the firm’s asset mix, in its financing mix, or in exter-
nal factors not within management’s control, such as earthquakes, toxic spills, and so on. The impacts of changes in
beta on value are discussed in Chapter 7.
FIGURE 5.11
SML1
Inflation Shifts SML 17
Impact of increased inflation- kz = 16
1
15 SML
ary expectations on the SML
km = 14
1

kz = 13
12
km = 11
RF = 10
1 Inc.
9 in IP
8
RF = 7
6 IP1
5
IP
4
3
2
1 k*
0 .5 1.0 1.5 2.0

bR bm bz
F
Figure 5.11 depicts the situation just described. It shows that the 3% increase
in inflationary expectations results in a parallel shift upward of 3% in the SML.
Clearly, the required returns on all assets rise by 3%. Note that the rise in the
inflation premium from 5% to 8% (IP to IP1) causes the risk-free rate to rise from
7% to 10% (RF to RF1) and the market return to increase from 11% to 14% (km
to km1). The security market line therefore shifts upward by 3% (SML to SML1),
causing the required return on all risky assets, such as asset Z, to rise by 3%. It
should now be clear that a given change in inflationary expectations will be fully
reflected in a corresponding change in the returns of all assets, as reflected graph-
ically in a parallel shift of the SML.
Changes in Risk Aversion The slope of the security market line reflects the
general risk preferences of investors in the marketplace. As discussed earlier and
shown in Figure 5.1, most investors are risk-averse—they require increased
returns for increased risk. This positive relationship between risk and return is
graphically represented by the SML, which depicts the relationship between non-
diversifiable risk as measured by beta (x axis) and the required return (y axis).
The slope of the SML reflects the degree of risk aversion: the steeper its slope, the
greater the degree of risk aversion, because a higher level of return will be
required for each level of risk as measured by beta. In other words, risk premiums
increase with increasing risk avoidance.
Changes in risk aversion, and therefore shifts in the SML, result from chang-
ing preferences of investors, which generally result from economic, political, and
social events. Examples of events that increase risk aversion include a stock mar-
ket crash, assassination of a key political leader, and the outbreak of war. In gen-
eral, widely accepted expectations of hard times ahead tend to cause investors to
become more risk-averse, requiring higher returns as compensation for accepting
a given level of risk. The impact of increased risk aversion on the SML can best be
demonstrated by an example.
EXAMPLE In the preceding examples, the SML in Figure 5.10 reflected a risk-free rate (RF) of
7%, a market return (km) of 11%, a market risk premium (km RF) of 4%, and a
required return on asset Z (kZ) of 13% with a beta (bZ) of 1.5. Assume that recent
economic events have made investors more risk-averse, causing a new higher mar-
ket return (km1) of 14%. Graphically, this change would cause the SML to shift
upward as shown in Figure 5.12, causing a new market risk premium (km1 RF)
of 7%. As a result, the required return on all risky assets will increase. For asset Z,
with a beta of 1.5, the new required return (kZ1) can be calculated by using
CAPM (Equation 5.8):
kZ1 7% [1.5 (14% 7%)] 7% 10.5% 17.5%

This value can be seen on the new security market line (SML1) in Figure 5.12.
Note that although asset Z’s risk, as measured by beta, did not change, its
required return has increased because of the increased risk aversion reflected in
FIGURE 5.12
22
Risk Aversion Shifts SML 21 SML1
Impact of increased risk 20
aversion on the SML 19
18
kZ = 17.5
1 17
16
15 SML
km = 14
1
kZ = 13
12
km = 11 New Market Risk Premium
10 km – RF = 7%
1
9
8
RF = 7
6
Initial Market
5
Risk Premium
4 km – RF = 4%
3
2
1
0 .5 1.0 1.5 2.0

bR bm bZ
F
In Practice
FOCUS ON PRACTICE What’s at Risk? VAR Has the Answer
Financial managers, always on the would represent an amount, let’s value at risk of that portfolio. If it
lookout for new ways to measure call it D dollars, where the chance was riskier than previously
and manage risk, have added of losing more than D dollars is, thought, traders could take cor-
value-at -risk (VAR ) techniques to say, 1 in 50 over some future time rective action—selling a particular
their repertoire. VAR is a statistical interval, perhaps a week. type of security, for example—to
measure of risk exposure that re- VAR shows companies reduce risk.
flects the potential loss from an whether they are properly diversi- Like any quantitative model,
unlikely, adverse event in a normal, fied and also whether they have VAR has its limitations. Perhaps its
everyday market environment. It sufficient capital. Among its biggest drawback is its reliance on
predicts the drop in a company’s other benefits, it tells managers historical patterns that may not
value that will occur if things go whether their actions are too hold true in the future.
wrong by calculating the financial cautious, identifies risk trouble
Sources: Steve Bergsman, “Delivering the
risk in the future market value of a spots that might not be caught, Risk Management Goods,” Treasury & Risk
portfolio of assets, liabilities, and and provides a way to compare Management, downloaded from www.
equity. business units that measure per- treasuryandrisk.com/trmtechguide/
article13.cgi; Peter Coy, “Taking the Angst
First used by banks and bro- formance differently for internal Out of Taking a Gamble,” Business Week
kerage firms to measure the risk of reporting. (July 14, 1997), pp. 52–53; and Paul Hom and
Ron Tonuzi, “Value-at-Risk: Safety Net or
market movements, VAR now has For example, a bank could Abyss?” Treasury & Risk Management
proponents among nonfinancial take a diverse portfolio of financial (November/December 1998), downloaded
companies such as Xerox, General assets and calculate price swings from www.cfonet.com; Barry Schachter, “An
Irreverent Guide to Value at Risk,” All About
Motors, and GTE. Unlike other risk by measuring performance on Value-at-Risk (Web site), downloaded from
tools that measure risk using stan- specific days in the past. Plotting www.gloriamundi.com.
dard deviation, VAR is stated in the percentage gain or loss for
currency units: for example, VAR hundreds of days would reveal the
the market risk premium. It should now be clear that greater risk aversion results
in higher required returns for each level of risk. Similarly, a reduction in risk
aversion causes the required return for each level of risk to decline.
Some Comments on CAPM

The capital asset pricing model generally relies on historical data. The betas may
or may not actually reflect the future variability of returns. Therefore, the
required returns specified by the model can be viewed only as rough approxima-
tions. Users of betas commonly make subjective adjustments to the historically
determined betas to reflect their expectations of the future.
efficient market The CAPM was developed to explain the behavior of security prices and pro-
A market with the following vide a mechanism whereby investors could assess the impact of a proposed secu-
characteristics: many small
investors, all having the same
rity investment on their portfolio’s overall risk and return. It is based on an
information and expectations assumed efficient market with the following characteristics: many small investors,
with respect to securities; no all having the same information and expectations with respect to securities; no
restrictions on investment, no restrictions on investment, no taxes, and no transaction costs; and rational
taxes, and no transaction costs; investors, who view securities similarly and are risk-averse, preferring higher
and rational investors, who view
securities similarly and are risk-
returns and lower risk.
averse, preferring higher returns Although the perfect world of the efficient market appears to be unrealistic,
and lower risk. studies have provided support for the existence of the expectational relationship
described by CAPM in active markets such as the New York Stock Exchange.23
In the case of real corporate assets, such as plant and equipment, research thus far
has failed to prove the general applicability of CAPM because of indivisibility,
relatively large size, limited number of transactions, and absence of an efficient
market for such assets.
Despite the limitations of CAPM, it provides a useful conceptual framework
for evaluating and linking risk and return. An awareness of this tradeoff and an
attempt to consider risk as well as return in financial decision making should help
financial managers achieve their goals.
Review Questions
5–11 How are total risk, nondiversifiable risk, and diversifiable risk related?
Why is nondiversifiable risk the only relevant risk?
5–12 What risk does beta measure? How can you find the beta of a portfolio?
5–13 Explain the meaning of each variable in the capital asset pricing model
(CAPM) equation. What is the security market line (SML)?
5–14 What impact would the following changes have on the security market
line and therefore on the required return for a given level of risk? (a) An
increase in inflationary expectations. (b) Investors become less risk-averse.
5–15 Why do financial managers have some difficulty applying CAPM in finan-
cial decision making? Generally, what benefit does CAPM provide them?
S U M M A RY
FOCUS ON VALUE
A firm’s risk and expected return directly affect its share price. As we shall see in Chapter 7,
risk and return are the two key determinants of the firm’s value. It is therefore the financial
manager’s responsibility to assess carefully the risk and return of all major decisions in
order to make sure that the expected returns justify the level of risk being introduced.
The way the financial manager can expect to achieve the firm’s goal of increasing its
share price (and thereby benefiting its owners) is to take only those actions that earn returns
at least commensurate with their risk. Clearly, financial managers need to recognize, mea-
sure, and evaluate risk–return tradeoffs in order to ensure that their decisions contribute to
the creation of value for owners.
23. A study by Eugene F. Fama and Kenneth R. French, “The Cross-Section of Expected Stock Returns,” Journal of
Finance 47 (June 1992), pp. 427–465, raised serious questions about the validity of CAPM. The study failed to find
a significant relationship between the historical betas and historical returns on over 2,000 stocks during 1963–1990.
In other words, it found that the magnitude of a stock’s historical beta had no relationship to the level of its histori-
cal return. Although Fama and French’s study continues to receive attention, CAPM has not been abandoned
because its rejection as a historical model fails to discredit its validity as an expectational model. Therefore, in spite
of this challenge, CAPM continues to be viewed as a logical and useful framework—both conceptually and opera-
tionally—for linking expected nondiversifiable risk and return.
REVIEW OF LEARNING GOALS

Understand the meaning and fundamentals of Understand the risk and return characteristics
LG1 LG4
risk, return, and risk preferences. Risk is the of a portfolio in terms of correlation and diver-
chance of loss or, more formally, the variability of sification, and the impact of international assets on
returns. A number of sources of firm-specific and a portfolio. Diversification involves combining
shareholder-specific risks exists. Return is any cash assets with low (less positive and more negative)
distributions plus the change in value expressed as a correlation to reduce the risk of the portfolio.
percentage of the initial value. Investment returns Although the return on a two-asset portfolio will lie
vary both over time and between different types of between the returns of the two assets held in isola-
investments. The equation for the rate of return is tion, the range of risk depends on the correlation
given in Table 5.13. The three basic risk preference between the two assets. If they are perfectly posi-
behaviors are risk-averse, risk-indifferent, and risk- tively correlated, the portfolio’s risk will be
seeking. Most financial decision makers are risk- between the individual asset’s risks. If they are
averse. They generally prefer less risky alternatives, uncorrelated, the portfolio’s risk will be between
and they require higher expected returns as com- the risk of the most risky asset and an amount less
pensation for taking greater risk. than the risk of the least risky asset but greater than
zero. If they are negatively correlated, the portfo-
Describe procedures for assessing and measur- lio’s risk will be between the risk of the most risky
LG2
ing the risk of a single asset. The risk of a sin- asset and zero. International diversification can be
gle asset is measured in much the same way as the used to reduce a portfolio’s risk further. With for-
risk of a portfolio, or collection, of assets. Sensitiv- eign assets come the risk of currency fluctuation
ity analysis and probability distributions can be and political risks.
used to assess risk. In addition to the range, the
standard deviation and the coefficient of variation Review the two types of risk and the deriva-
LG5
are statistics that can be used to measure risk quan- tion and role of beta in measuring the relevant
titatively. The key equations for the expected value risk of both an individual security and a portfolio.
of a return, the standard deviation of a return, and The total risk of a security consists of nondiversifi-
the coefficient of variation are summarized in able and diversifiable risk. Nondiversifiable risk is
Table 5.13. the only relevant risk; diversifiable risk can be
eliminated through diversification. Nondiversifiable
Discuss the measurement of return and stan- risk is measured by the beta coefficient, which is a
LG3
dard deviation for a portfolio and the various relative measure of the relationship between an as-
types of correlation that can exist between series of set’s return and the market return. Beta is derived
numbers. The return of a portfolio is calculated as by finding the slope of the “characteristic line”
the weighted average of returns on the individual that best explains the historical relationship be-
assets from which it is formed. The equation for tween the asset’s return and the market return. The
portfolio return is given in Table 5.13. The portfo- beta of a portfolio is a weighted average of the be-
lio standard deviation is found by using the for- tas of the individual assets that it includes. The
mula for the standard deviation of a single asset. equations for total risk and the portfolio beta are
Correlation—the statistical relationship between given in Table 5.13.
any two series of numbers—can be positive (the
series move in the same direction), negative (the Explain the capital asset pricing model
LG6
series move in opposite directions), or uncorrelated (CAPM), its relationship to the security market
(the series exhibit no discernible relationship). At line (SML), and shifts in the SML caused by changes
the extremes, the series can be perfectly positively in inflationary expectations and risk aversion. The
correlated (have a correlation coefficient of 1) or capital asset pricing model (CAPM) uses beta to re-
perfectly negatively correlated (have a correlation late an asset’s risk relative to the market to the as-
coefficient of 1). set’s required return. The equation for CAPM is
TABLE 5.13 Summary of Key Definitions and Formulas for Risk and Return
Definitions of variables
bj beta coefficient or index of nondiversifiable risk for asset j

bp portfolio beta
Ct cash received from the asset investment in the time period t 1 to t
CV coefficient of variation
expected value of a return
k
kj return for the jth outcome; return on asset j; required return on asset j
km market return; the return on the market portfolio of assets
kp portfolio return
kt actual, expected, or required rate of return during period t
n number of outcomes considered
Pt price (value) of asset at time t
Pt1 price (value) of asset at time t 1
Prj probability of occurrence of the jth outcome
RF risk-free rate of return
k standard deviation of returns
wj proportion of total portfolio dollar value represented by asset j
Risk and return formulas
Rate of return during period t:
Ct Pt Pt1
kt [Eq. 5.1] Coefficient of variation:
Pt1
k
Expected value of a return: CV [Eq. 5.4]

k
for probabilistic data:
Portfolio return:
n
kj Prj
k [Eq. 5.2] n
kp wj kj
j1
[Eq. 5.5]
j1
general formula:
Total security risk Nondiversifiable risk
n
kj
j1
Diversifiable risk [Eq. 5.6]
n
k [Eq. 5.2a] Portfolio beta:
Standard deviation of return: n

bp wj bj [Eq. 5.7]
for probabilistic data: j1

n
Capital asset pricing model
k (k k
) Pr
j
2
j [Eq. 5.3]
j1 (CAPM):
general formula: kj RF [bj (km RF)] [Eq. 5.8]

n
(kj k)2
j1
k [Eq. 5.3a]
n1
given in Table 5.13. The graphical depiction of magnitude and direction of change. Increasing risk
CAPM is the security market line (SML), which aversion results in a steepening in the slope of the
shifts over time in response to changing inflationary SML, and decreasing risk aversion reduces the slope
expectations and/or changes in investor risk aver- of the SML. Although it has some shortcomings,
sion. Changes in inflationary expectations result in CAPM provides a useful conceptual framework for
parallel shifts in the SML in direct response to the evaluating and linking risk and return.
SELF-TEST PROBLEMS (Solutions in Appendix B)

LG3 LG4 ST 5–1 Portfolio analysis You have been asked for your advice in selecting a portfolio
of assets and have been given the following data:
Expected return
Year Asset A Asset B Asset C
2004 12% 16% 12%

2005 14 14 14
2006 16 12 16
No probabilities have been supplied. You have been told that you can create two
portfolios—one consisting of assets A and B and the other consisting of assets A
and C—by investing equal proportions (50%) in each of the two component
assets.
a. What is the expected return for each asset over the 3-year period?
b. What is the standard deviation for each asset’s return?
c. What is the expected return for each of the two portfolios?
d. How would you characterize the correlations of returns of the two assets
making up each of the two portfolios identified in part c?
e. What is the standard deviation for each portfolio?
f. Which portfolio do you recommend? Why?
LG5 LG6 ST 5–2 Beta and CAPM Currently under consideration is a project with a beta, b, of
1.50. At this time, the risk-free rate of return, RF, is 7%, and the return on the
market portfolio of assets, km, is 10%. The project is actually expected to earn
an annual rate of return of 11%.
a. If the return on the market portfolio were to increase by 10%, what would
you expect to happen to the project’s required return? What if the market
return were to decline by 10%?
b. Use the capital asset pricing model (CAPM) to find the required return on
this investment.
c. On the basis of your calculation in part b, would you recommend this invest-
ment? Why or why not?
d. Assume that as a result of investors becoming less risk-averse, the market
return drops by 1% to 9%. What impact would this change have on your
responses in parts b and c?
PROBLEMS
LG1 5–1 Rate of return Douglas Keel, a financial analyst for Orange Industries, wishes to
estimate the rate of return for two similar-risk investments, X and Y. Keel’s
research indicates that the immediate past returns will serve as reasonable esti-
mates of future returns. A year earlier, investment X had a market value of
$20,000, investment Y of $55,000. During the year, investment X generated cash
flow of $1,500 and investment Y generated cash flow of $6,800. The current mar-
ket values of investments X and Y are $21,000 and $55,000, respectively.
a. Calculate the expected rate of return on investments X and Y using the most
recent year’s data.
b. Assuming that the two investments are equally risky, which one should Keel
recommend? Why?
LG1 5–2 Return calculations For each of the investments shown in the following table,
calculate the rate of return earned over the unspecified time period.
Cash flow Beginning-of- End-of-

Investment during period period value period value
A $ 100 $ 800 $ 1,100

B 15,000 120,000 118,000
C 7,000 45,000 48,000
D 80 600 500
E 1,500 12,500 12,400
LG1 5–3 Risk preferences Sharon Smith, the financial manager for Barnett Corporation,
wishes to evaluate three prospective investments: X, Y, and Z. Currently, the
firm earns 12% on its investments, which have a risk index of 6%. The expected
return and expected risk of the investments are as follows:
Expected Expected
Investment return risk index
X 14% 7%
Y 12 8
Z 10 9
a. If Sharon Smith were risk-indifferent, which investments would she select?

Explain why.
b. If she were risk-averse, which investments would she select? Why?
c. If she were risk-seeking, which investments would she select? Why?
d. Given the traditional risk preference behavior exhibited by financial man-
agers, which investment would be preferred? Why?
LG2 5–4 Risk analysis Solar Designs is considering an investment in an expanded prod-
uct line. Two possible types of expansion are being considered. After investigating
the possible outcomes, the company made the estimates shown in the following
table
Expansion A Expansion B
Initial investment $12,000 $12,000

Pessimistic 16% 10%
Most likely 20% 20%
Optimistic 24% 30%
a. Determine the range of the rates of return for each of the two projects.
b. Which project is less risky? Why?
c. If you were making the investment decision, which one would you choose?
Why? What does this imply about your feelings toward risk?
d. Assume that expansion B’s most likely outcome is 21% per year and that
all other facts remain the same. Does this change your answer to part c?
Why?
LG2 5–5 Risk and probability Micro-Pub, Inc., is considering the purchase of one of
two microfilm cameras, R and S. Both should provide benefits over a 10-year
period, and each requires an initial investment of $4,000. Management has con-
structed the following table of estimates of rates of return and probabilities for
pessimistic, most likely, and optimistic results:
Camera R Camera S
Amount Probability Amount Probability
Initial investment $4,000 1.00 $4,000 1.00

Pessimistic 20% .25 15% .20
Most likely 25% .50 25% .55
Optimistic 30% .25 35% .25
a. Determine the range for the rate of return for each of the two cameras.
b. Determine the expected value of return for each camera.
c. Purchase of which camera is riskier? Why?
LG2 5–6 Bar charts and risk Swan’s Sportswear is considering bringing out a line of
designer jeans. Currently, it is negotiating with two different well-known design-
ers. Because of the highly competitive nature of the industry, the two lines of
jeans have been given code names. After market research, the firm has estab-
lished the expectations shown in the following table about the annual rates
of return

Market acceptance Probability Line J Line K
Very poor .05 .0075 .010

Poor .15 .0125 .025
Average .60 .0850 .080
Good .15 .1475 .135
Excellent .05 .1625 .150
Use the table to:

a. Construct a bar chart for each line’s annual rate of return.
b. Calculate the expected value of return for each line.
c. Evaluate the relative riskiness for each jean line’s rate of return using the bar
charts.
LG2 5–7 Coefficient of variation Metal Manufacturing has isolated four alternatives for
meeting its need for increased production capacity. The data gathered relative to
each of these alternatives is summarized in the following table.
Expected Standard
Alternative return deviation of return
A 20% 7.0%
B 22 9.5
C 19 6.0
D 16 5.5
a. Calculate the coefficient of variation for each alternative.

b. If the firm wishes to minimize risk, which alternative do you recommend?
Why?
LG2 5–8 Standard deviation versus coefficient of variation as measures of risk Greengage,
Inc., a successful nursery, is considering several expansion projects. All of the
alternatives promise to produce an acceptable return. The owners are extremely
risk-averse; therefore, they will choose the least risky of the alternatives. Data on
four possible projects follow.
Project Expected return Range Standard deviation
A 12.0% .040 .029

B 12.5 .050 .032
C 13.0 .060 .035
D 12.8 .045 .030
a. Which project is least risky, judging on the basis of range?

b. Which project has the lowest standard deviation? Explain why standard devi-
ation is not an appropriate measure of risk for purposes of this comparison.
c. Calculate the coefficient of variation for each project. Which project will
Greengage’s owners choose? Explain why this may be the best measure of
risk for comparing this set of opportunities.
LG2 5–9 Assessing return and risk Swift Manufacturing must choose between two asset
purchases. The annual rate of return and the related probabilities given in the
following table summarize the firm’s analysis to this point.
Project 257 Project 432

Rate of return Probability Rate of return Probability
10% .01 10% .05

10 .04 15 .10
20 .05 20 .10
30 .10 25 .15
40 .15 30 .20
45 .30 35 .15
50 .15 40 .10
60 .10 45 .10
70 .05 50 .05
80 .04
100 .01
a. For each project, compute:

(1) The range of possible rates of return.
(2) The expected value of return.
(3) The standard deviation of the returns.
(4) The coefficient of variation of the returns.
b. Construct a bar chart of each distribution of rates of return.
c. Which project would you consider less risky? Why?
LG2 5–10 Integrative—Expected return, standard deviation, and coefficient of variation

Three assets—F, G, and H—are currently being considered by Perth Industries.
The probability distributions of expected returns for these assets are shown in
the following table.
Asset F Asset G Asset H

j Prj Return, kj Prj Return, kj Prj Return, kj
1 .10 40% .40 35% .10 40%

2 .20 10 .30 10 .20 20
3 .40 0 .30 20 .40 10
4 .20 5 .20 0
5 .10 10 .10 20
–
a. Calculate the expected value of return, k , for each of the three assets. Which
provides the largest expected return?
b. Calculate the standard deviation, k, for each of the three assets’ returns.
Which appears to have the greatest risk?
c. Calculate the coefficient of variation, CV, for each of the three assets’
returns. Which appears to have the greatest relative risk?
LG2 5–11 Normal probability distribution Assuming that the rates of return associated
with a given asset investment are normally distributed and that the expected
return,
k, is 18.9% and the coefficient of variation, CV, is .75, answer the fol-
lowing questions.
a. Find the standard deviation of returns, k.
b. Calculate the range of expected return outcomes associated with the follow-
ing probabilities of occurrence: (1) 68%, (2) 95%, (3) 99%.
c. Draw the probability distribution associated with your findings in parts a
and b.
LG3 5–12 Portfolio return and standard deviation Jamie Wong is considering building a
portfolio containing two assets, L and M. Asset L will represent 40% of the
dollar value of the portfolio, and asset M will account for the other 60%. The
expected returns over the next 6 years, 2004–2009, for each of these assets, are
shown in the following table.
Expected return
Year Asset L Asset M
2004 14% 20%

2005 14 18
2006 16 16
2007 17 14
2008 17 12
2009 19 10
a. Calculate the expected portfolio return, kp, for each of the 6 years.
p, over the 6-year period.
b. Calculate the expected value of portfolio returns, k
c. Calculate the standard deviation of expected portfolio returns, k , over the
p
6-year period.
d. How would you characterize the correlation of returns of the two assets L
and M?
e. Discuss any benefits of diversification achieved through creation of the
portfolio.
LG3 5–13 Portfolio analysis You have been given the return data shown in the first table
on three assets—F, G, and H—over the period 2004–2007.
Expected return
Year Asset F Asset G Asset H
2004 16% 17% 14%

2005 17 16 15
2006 18 15 16
2007 19 14 17
Using these assets, you have isolated the three investment alternatives shown in
the following table:
Alternative Investment
1 100% of asset F
2 50% of asset F and 50% of asset G
3 50% of asset F and 50% of asset H
a. Calculate the expected return over the 4-year period for each of the three
alternatives.
b. Calculate the standard deviation of returns over the 4-year period for each of
the three alternatives.
c. Use your findings in parts a and b to calculate the coefficient of variation for
each of the three alternatives.
d. On the basis of your findings, which of the three investment alternatives do
you recommend? Why?
LG4 5–14 Correlation, risk, and return Matt Peters wishes to evaluate the risk and return
behaviors associated with various combinations of assets V and W under three
assumed degrees of correlation: perfect positive, uncorrelated, and perfect nega-
tive. The expected return and risk values calculated for each of the assets are
shown in the following table.
Expected Risk (standard

Asset
return, k deviation), k
V 8% 5%
W 13 10
a. If the returns of assets V and W are perfectly positively correlated (correla-

tion coefficient 1), describe the range of (1) expected return and (2) risk
associated with all possible portfolio combinations.
b. If the returns of assets V and W are uncorrelated (correlation coefficient 0),
describe the approximate range of (1) expected return and (2) risk associated
with all possible portfolio combinations.
c. If the returns of assets V and W are perfectly negatively correlated (correla-
tion coefficient 1), describe the range of (1) expected return and (2) risk
associated with all possible portfolio combinations.
LG1 LG4 5–15 International investment returns Joe Martinez, a U.S. citizen living in
Brownsville, Texas, invested in the common stock of Telmex, a Mexican corpo-
ration. He purchased 1,000 shares at 20.50 pesos per share. Twelve months
later, he sold them at 24.75 pesos per share. He received no dividends during
that time.
a. What was Joe’s investment return (in percentage terms) for the year, on the
basis of the peso value of the shares?
b. The exchange rate for pesos was 9.21 pesos per $US1.00 at the time of the
purchase. At the time of the sale, the exchange rate was 9.85 pesos per
$US1.00. Translate the purchase and sale prices into $US.
c. Calculate Joe’s investment return on the basis of the $US value of the shares.
d. Explain why the two returns are different. Which one is more important to
Joe? Why?
LG5 5–16 Total, nondiversifiable, and diversifiable risk David Talbot randomly selected
securities from all those listed on the New York Stock Exchange for his portfo-
lio. He began with a single security and added securities one by one until a total
of 20 securities were held in the portfolio. After each security was added, David
calculated the portfolio standard deviation, k . The calculated values are shown
p
in the following table.
Number of Portfolio Number of Portfolio

securities risk, kp securities risk, kp
1 14.50% 11 7.00%
2 13.30 12 6.80
3 12.20 13 6.70
4 11.20 14 6.65
5 10.30 15 6.60
6 9.50 16 6.56
7 8.80 17 6.52
8 8.20 18 6.50
9 7.70 19 6.48
10 7.30 20 6.47
a. On a set of “number of securities in portfolio (x axis)–portfolio risk (y axis)”

axes, plot the portfolio risk data given in the preceding table.
b. Divide the total portfolio risk in the graph into its nondiversifiable and diver-
sifiable risk components and label each of these on the graph.
c. Describe which of the two risk components is the relevant risk, and explain
why it is relevant. How much of this risk exists in David Talbot’s portfolio?
LG5 5–17 Graphical derivation of beta A firm wishes to estimate graphically the betas
for two assets, A and B. It has gathered the return data shown in the following
table for the market portfolio and for both assets over the last ten years,
1994–2003.
Actual return
Year Market portfolio Asset A Asset B
1994 6% 11% 16%

1995 2 8 11
1996 13 4 10
1997 4 3 3
1998 8 0 3
1999 16 19 30
2000 10 14 22
2001 15 18 29
2002 8 12 19
2003 13 17 26
a. On a set of “market return (x axis)–asset return (y axis)” axes, use the data
given to draw the characteristic line for asset A and for asset B.
b. Use the characteristic lines from part a to estimate the betas for assets A and B.
c. Use the betas found in part b to comment on the relative risks of assets A and B.
LG5 5–18 Interpreting beta A firm wishes to assess the impact of changes in the market
return on an asset that has a beta of 1.20.
a. If the market return increased by 15%, what impact would this change be
expected to have on the asset’s return?
b. If the market return decreased by 8%, what impact would this change be
expected to have on the asset’s return?
c. If the market return did not change, what impact, if any, would be expected
on the asset’s return?
d. Would this asset be considered more or less risky than the market? Explain.
LG5 5–19 Betas Answer the following questions for assets A to D shown in the following
table.
Asset Beta
A .50
B 1.60
C .20
D .90
a. What impact would a 10% increase in the market return be expected to have
on each asset’s return?
b. What impact would a 10% decrease in the market return be expected to have
on each asset’s return?
c. If you were certain that the market return would increase in the near future,
which asset would you prefer? Why?
d. If you were certain that the market return would decrease in the near future,
which asset would you prefer? Why?
LG5 5–20 Betas and risk rankings Stock A has a beta of .80, stock B has a beta of 1.40,
and stock C has a beta of .30.
a. Rank these stocks from the most risky to the least risky.
b. If the return on the market portfolio increased by 12%, what change would
you expect in the return for each of the stocks?
c. If the return on the market portfolio decreased by 5%, what change would
you expect in the return for each of the stocks?
d. If you felt that the stock market was just ready to experience a significant
decline, which stock would you probably add to your portfolio? Why?
e. If you anticipated a major stock market rally, which stock would you add to
your portfolio? Why?
LG5 5–21 Portfolio betas Rose Berry is attempting to evaluate two possible portfolios,
which consist of the same five assets held in different proportions. She is particu-
larly interested in using beta to compare the risks of the portfolios, so she has
gathered the data shown in the following table.
Portfolio weights
Asset Asset beta Portfolio A Portfolio B
1 1.30 10% 30%

2 .70 30 10
3 1.25 10 20
4 1.10 10 20
5 .90 40 20

Totals 100% 100%

a. Calculate the betas for portfolios A and B.

b. Compare the risks of these portfolios to the market as well as to each other.
Which portfolio is more risky?
LG6 5–22 Capital asset pricing model (CAPM) For each of the cases shown in the follow-
ing table, use the capital asset pricing model to find the required return.
Risk-free Market
Case rate, RF return, km Beta, b
A 5% 8% 1.30
B 8 13 .90
C 9 12 .20
D 10 15 1.00
E 6 10 .60
LG5 LG6 5–23 Beta coefficients and the capital asset pricing model Katherine Wilson is won-
dering how much risk she must undertake in order to generate an acceptable
return on her portfolio. The risk-free return currently is 5%. The return on the
average stock (market return) is 16%. Use the CAPM to calculate the beta coef-
ficient associated with each of the following portfolio returns.
a. 10%
b. 15%
c. 18%
d. 20%
e. Katherine is risk-averse. What is the highest return she can expect if she is
unwilling to take more than an average risk?
LG6 5–24 Manipulating CAPM Use the basic equation for the capital asset pricing model
(CAPM) to work each of the following problems.
a. Find the required return for an asset with a beta of .90 when the risk-free rate
and market return are 8% and 12%, respectively.
b. Find the risk-free rate for a firm with a required return of 15% and a beta of
1.25 when the market return is 14%.
c. Find the market return for an asset with a required return of 16% and a beta
of 1.10 when the risk-free rate is 9%.
d. Find the beta for an asset with a required return of 15% when the risk-free
rate and market return are 10% and 12.5%, respectively.
LG1 LG3 LG5 LG6 5–25 Portfolio return and beta Jamie Peters invested $100,000 to set up the follow-
ing portfolio one year ago:
Asset Cost Beta at purchase Yearly income Value today
A $20,000 .80 $1,600 $20,000

B 35,000 .95 1,400 36,000
C 30,000 1.50 — 34,500
D 15,000 1.25 375 16,500
a. Calculate the portfolio beta on the basis of the original cost figures.
b. Calculate the percentage return of each asset in the portfolio for the year.
c. Calculate the percentage return of the portfolio on the basis of original cost,
using income and gains during the year.
d. At the time Jamie made his investments, investors were estimating that the
market return for the coming year would be 10%. The estimate of the risk-
free rate of return averaged 4% for the coming year. Calculate an expected
rate of return for each stock on the basis of its beta and the expectations of
market and risk-free returns.
e. On the basis of the actual results, explain how each stock in the portfolio
performed relative to those CAPM-generated expectations of performance.
What factors could explain these differences?
LG6 5–26 Security market line, SML Assume that the risk-free rate, RF, is currently 9%
and that the market return, km, is currently 13%.
a. Draw the security market line (SML) on a set of “nondiversifiable risk
(x axis)–required return (y axis)” axes.
b. Calculate and label the market risk premium on the axes in part a.
c. Given the previous data, calculate the required return on asset A having a
beta of .80 and asset B having a beta of 1.30.
d. Draw in the betas and required returns from part c for assets A and B on the
axes in part a. Label the risk premium associated with each of these assets,
and discuss them.
LG6 5–27 Shifts in the security market line Assume that the risk-free rate, RF, is currently
8%, the market return, km, is 12%, and asset A has a beta, bA, of 1.10.
a. Draw the security market line (SML) on a set of “nondiversifiable risk
(x axis)–required return (y axis)” axes.
b. Use the CAPM to calculate the required return, kA, on asset A, and depict
asset A’s beta and required return on the SML drawn in part a.
c. Assume that as a result of recent economic events, inflationary expectations
have declined by 2%, lowering RF and km to 6% and 10%, respectively.
Draw the new SML on the axes in part a, and calculate and show the new
required return for asset A.
d. Assume that as a result of recent events, investors have become more risk-
averse, causing the market return to rise by 1%, to 13%. Ignoring the shift in
part c, draw the new SML on the same set of axes that you used before, and
calculate and show the new required return for asset A.
e. From the previous changes, what conclusions can be drawn about the impact
of (1) decreased inflationary expectations and (2) increased risk aversion on
the required returns of risky assets?
LG6 5–28 Integrative—Risk, return, and CAPM Wolff Enterprises must consider several
investment projects, A through E, using the capital asset pricing model (CAPM)
and its graphical representation, the security market line (SML). Relevant infor-
mation is presented in the following table.
Item Rate of return Beta, b
Risk-free asset 9% 0
Market portfolio 14 1.00
Project A — 1.50
Project B — .75
Project C — 2.00
Project D — 0
Project E — .50
a. Calculate the required rate of return and risk premium for each project, given
its level of nondiversifiable risk.
b. Use your findings in part a to draw the security market line (required return
relative to nondiversifiable risk).
c. Discuss the relative nondiversifiable risk of projects A through E.
d. Assume that recent economic events have caused investors to become less
risk-averse, causing the market return to decline by 2%, to 12%. Calculate
the new required returns for assets A through E, and draw the new security
market line on the same set of axes that you used in part b.
e. Compare your findings in parts a and b with those in part d. What conclu-
sion can you draw about the impact of a decline in investor risk aversion on
the required returns of risky assets?
CHAPTER 5 CASE Analyzing Risk and Return on Chargers Products’ Investments
J unior Sayou, a financial analyst for Chargers Products, a manufacturer of sta-

dium benches, must evaluate the risk and return of two assets, X and Y. The
firm is considering adding these assets to its diversified asset portfolio. To assess
the return and risk of each asset, Junior gathered data on the annual cash flow
and beginning- and end-of-year values of each asset over the immediately pre-
ceding 10 years, 1994–2003. These data are summarized in the accompanying
table. Junior’s investigation suggests that both assets, on average, will tend to
Return Data for Assets X and Y, 1994–2003
Asset X Asset Y
Value Value
Year Cash flow Beginning Ending Cash flow Beginning Ending
1994 $1,000 $20,000 $22,000 $1,500 $20,000 $20,000

1995 1,500 22,000 21,000 1,600 20,000 20,000
1996 1,400 21,000 24,000 1,700 20,000 21,000
1997 1,700 24,000 22,000 1,800 21,000 21,000
1998 1,900 22,000 23,000 1,900 21,000 22,000
1999 1,600 23,000 26,000 2,000 22,000 23,000
2000 1,700 26,000 25,000 2,100 23,000 23,000
2001 2,000 25,000 24,000 2,200 23,000 24,000
2002 2,100 24,000 27,000 2,300 24,000 25,000
2003 2,200 27,000 30,000 2,400 25,000 25,000
perform in the future just as they have during the past 10 years. He therefore
believes that the expected annual return can be estimated by finding the average
annual return for each asset over the past 10 years.
Junior believes that each asset’s risk can be assessed in two ways: in isolation
and as part of the firm’s diversified portfolio of assets. The risk of the assets in
isolation can be found by using the standard deviation and coefficient of varia-
tion of returns over the past 10 years. The capital asset pricing model (CAPM)
can be used to assess the asset’s risk as part of the firm’s portfolio of assets.
Applying some sophisticated quantitative techniques, Junior estimated betas for
assets X and Y of 1.60 and 1.10, respectively. In addition, he found that the risk-
free rate is currently 7% and that the market return is 10%.
Required
a. Calculate the annual rate of return for each asset in each of the 10 preceding
years, and use those values to find the average annual return for each asset
over the 10-year period.
b. Use the returns calculated in part a to find (1) the standard deviation and (2)
the coefficient of variation of the returns for each asset over the 10-year
period 1994–2003.
c. Use your findings in parts a and b to evaluate and discuss the return and risk
associated with each asset. Which asset appears to be preferable? Explain.
d. Use the CAPM to find the required return for each asset. Compare this value
with the average annual returns calculated in part a.
e. Compare and contrast your findings in parts c and d. What recommendations
would you give Junior with regard to investing in either of the two assets?
Explain to Junior why he is better off using beta rather than the standard
deviation and coefficient of variation to assess the risk of each asset.
f. Rework parts d and e under each of the following circumstances:

(1) A rise of 1% in inflationary expectations causes the risk-free rate to rise
to 8% and the market return to rise to 11%.
(2) As a result of favorable political events, investors suddenly become less
risk-averse, causing the market return to drop by 1%, to 9%.
WEB EXERCISE Go to the RiskGrades Web site, www.riskgrades.com. This site, from
WW RiskMetrics Group, provides another way to assess the riskiness of stocks and
W
mutual funds. RiskGrades provide a way to compare investment risk across all
asset classes, regions, and currencies. They vary over time to reflect asset-specific
information (such as the price of a stock reacting to an earnings release) and gen-
eral market conditions. RiskGrades operate differently from traditional risk mea-
sures, such as standard deviation and beta.
1. First, learn more about RiskGrades by clicking on RiskGrades Help Center

and reviewing the material. How are RiskGrades calculated? What differ-
ences can you identify when you compare them to standard deviation and
beta techniques? What are the advantages and disadvantages of this mea-
sure, in your opinion?
2. Get RiskGrades for the following stocks using the Get RiskGrade pull-down
menu at the site’s upper right corner. You can enter multiple symbols sepa-
rated by commas. Select all dates to get a historical view.
Company Symbol
Citigroup C
Intel INTC
Microsoft MSFT
Washington Mutual WM
What do the results tell you?
3. Select one of the foregoing stocks and find other stocks with similar risk
grades. Click on By RiskGrade to pull up a list.
4. How much risk can you tolerate? Use a hypothetical portfolio to find out.
Click on Grade Yourself, take a short quiz, and get your personal
RiskGrade measure. Did the results surprise you?
Remember to check the book’s Web site at

www.aw.com/gitman
for additional resources, including additional Web exercises.

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CHAPTER

Across the Disciplines WHY THIS CHAPTER MATTERS TO YO U

A s they chased after hot new financial

LG1 5.1 Risk and Return Fundamentals

TABLE 5.1 Popular Sources of Risk Affecting Financial Managers

Source of risk Description

Firm and Shareholder Risks

TABLE 5.2 Historical Returns for

Investment Average annual return

Large-company stocks 13.0%

Required (or Expected) Return

LG2 5.2 Risk of a Single Asset

TABLE 5.3 Assets A and B

Initial investment $10,000 $10,000

kj return for the jth outcome

TABLE 5.4 Expected Values of Returns for

Pessimistic .25 7% 1.75%

The expression for the standard deviation of returns, k, is10

TABLE 5.5 The Calculation of the Standard Deviation

1 13% 15% 2% 4% .25 1%

(kj k)2  Prj

1 7% 15% 8% 64% .25 16%

(kj k)2  Prj 32%

TABLE 5.6 Historical Returns and Standard

Investment Average annual return Standard deviation

Large-company stocks 13.0% 20.2%

Inflation 3.2% 4.4%

0 –3σk –2σk –1σk k +1σk +2σk +3σk

Normal Distribution A normal probability distribution, depicted in Figure

Statistics Asset X Asset Y

(1) Expected return 12% 20%

LG3 LG4 5.3 Risk of a Portfolio

Portfolio Return and Standard Deviation

wj proportion of the portfolio’s total dollar value represented by asset j

TABLE 5.7 Expected Return, Expected Value, and Standard

A. Expected portfolio returns

2004 8% 16% (.50  8%) (.50  16%) 12%

B. Expected value of portfolio returns, 2004–2008b

12% 12% 12% 12% 12% 60%

C. Standard deviation of expected portfolio returnsc

aUsing Equation 5.5.

FIGURE 5.5 Perfectly Positively Correlated Perfectly Negatively Correlated

perfectly negatively correlated

FIGURE 5.6 Portfolio of

Time Time Time

2004 8% 16% 8% 12% 8%

Portfolio XY Portfolio XY (shown in Table 5.8) is created by combining equal

Portfolio XZ Portfolio XZ (shown in Table 5.8) is created by combining equal

TABLE 5.9 Correlation, Return, and Risk for Various

Clearly, asset R is a lower-return, lower-risk asset than asset S.

FIGURE 5.7 Correlation

Portfolio Return (%) Portfolio Risk (%)

Returns from International Diversification

Risks of International Diversification

Even where governments do not impose exchange controls or seize assets,

LG5 LG6 5.4 Risk and Return: The Capital Asset

Portfolio Risk, σkP

Total security risk Nondiversifiable risk Diversifiable risk (5.6)

The Model: CAPM

The expression for the standard deviation of returns, k, is10

(kj k)2 Prj

(kj k)2 Prj 32%

2004 8% 16% (.50 8%) (.50 16%) 12%

kZ1 10% [1.5 (14% 10%)] 10% 6% 16%

general formula: kj RF [bj (km RF)] [Eq. 5.8]