Across the Disciplines

5

Why This Chapter Matters To You

Accounting: You need to understand the

relationship between risk and return

because of the effect that riskier projects

will have on the firm™s annual net income

and on your efforts to stabilize net income.

Risk

Information systems: You need to under-

stand how to do sensitivity and correlation

analyses in order to build decision pack-

and Return

ages that help management analyze the

risk and return of various business oppor-

tunities.

Management: You need to understand

the relationship between risk and return,

and how to measure that relationship in

order to evaluate data that come from

finance personnel and translate those

data into decisions that increase the value

of the firm.

Marketing: You need to understand that

LEARNING GOALS

although higher-risk projects may produce

higher returns, they may not be the best

Understand the meaning and

LG1

choice for the firm if they produce an fundamentals of risk, return, and risk

erratic earnings pattern and do not opti- aversion.

mize the value of the firm.

Describe procedures for assessing

LG2

Operations: You need to understand how and measuring the risk of a single

investments in plant assets and purchases asset.

of supplies will be measured by the firm

Discuss risk measurement for a single

LG3

and to recognize that decisions about

asset using the standard deviation

such investments will be made by evaluat-

and coefficient of variation.

ing the effects of both risk and return on

Understand the risk and return

the value of the firm. LG4

characteristics of a portfolio in terms

of correlation and diversification, and

the impact of international assets on a

portfolio.

Review the two types of risk and

LG5

the derivation and role of beta in

measuring the relevant risk of both an

individual security and a portfolio.

Explain the capital asset pricing

LG6

model (CAPM) and its relationship to

the security market line (SML).

189

190 PART 2 Important Financial Concepts

T he concept that return should increase if risk increases is fundamental to

modern management and finance. This relationship is regularly observed in

the financial markets, and important clarification of it has led to Nobel prizes. In

this chapter we discuss these two key factors in finance”risk and return”and

introduce some quantitative tools and techniques used to measure risk and return

for individual assets and for groups of assets.

Risk and Return Fundamentals

LG1

To maximize share price, the financial manager must learn to assess two key

determinants: risk and return. 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

to a portfolio”a collection, or group, of assets. We will look at both, beginning

portfolio

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 aversion.

Risk Defined

In the most basic sense, risk is the chance of financial loss. Assets having

risk

The chance of financial loss or, greater chances of loss are viewed as more risky than those with lesser chances

more formally, the variability of of loss. More formally, the term risk is used interchangeably with uncertainty

returns associated with a given

to refer to the variability of returns associated with a given asset. A $1,000 gov-

asset.

ernment 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 box on page 193 focuses on another risk that

affects both firms and shareholders”moral risk.

return

The total gain or loss experi-

Return Defined

enced on an investment over a

given period of time; calculated Obviously, if we are going to assess risk on the basis of variability of return, we

by dividing the asset™s cash

need to be certain we know what return is and how to measure it. The return is

distributions during the period,

the total gain or loss experienced on an investment over a given period of time. It

plus change in value, by its

is commonly measured as cash distributions during the period plus the change in

beginning-of-period investment

value, expressed as a percentage of the beginning-of-period investment value. The

value.

191

CHAPTER 5 Risk and Return

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.

expression for calculating the rate of return earned on any asset over period t, kt,

is commonly defined as

Ct Pt Pt 1

kt (5.1)

Pt 1

where

kt actual,expected, or required rate of return 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

Pt price (value) of asset at time t 1

1

192 PART 2 Important Financial Concepts

The return, kt, reflects the combined effect of cash flow, Ct, and changes in value,

Pt Pt 1, over period t.

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.

Robin™s Gameroom, a high-traffic video arcade, wishes to determine the return on

EXAMPLE

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. Substituting into

Equation 5.1, we can calculate the annual rate of return, k, for each video machine.

$800 $21,500 $20,000 $2,300

Conqueror (C): kC 11.5%

$20,000 $20,000

$1,700 $11,800 $12,000 $1,500

Demolition (D): 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

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).

193

CHAPTER 5 Risk and Return

In Practice

FOCUS ON ETHICS What About Moral Risk?

The poster boy for “moral risk,” holder wealth maximization has to for employees with concerns;

the devastating effects of unethi- be ethically constrained. weeding out employees who do

cal behavior for a company™s What can companies do to not share the company™s ethics

investors, has to be Nick Leeson. instill and maintain ethical corpo- values before those employees

This 28-year-old trader violated rate practices? They can start by can harm the company™s reputa-

his bank™s investing rules while building awareness through a tion or culture; assigning an indi-

secretly placing huge bets on code of ethics. Nearly all Fortune vidual the role of ethics director;

the direction of the Japanese 500 companies and about half of and evaluating leaders™ ethics

stock market. When those all companies have an ethics code in performance reviews (as at

bets proved to be wrong, the spelling out general principles of Merck & Co.).

$1.24-billion losses resulted in right and wrong conduct. Compa- The Leeson saga under-

the demise of the centuries-old nies such as Halliburton and scores the difficulty of dealing

Barings Bank. Texas Instruments have gone into with the “moral hazard” problem,

More than any other single specifics, because ethical codes when the consequences of an

episode in world financial history, are often faulted for being too individual™s actions are largely

Leeson™s misdeeds underscored vague and abstract. borne by others. John Boatright

the importance of character in Ethical organizations also argues in his book Ethics in

the financial industry. Forty-one reveal their commitments through Finance that the best antidote is to

percent of surveyed CFOs admit the following activities: talking attract loyal, hardworking employ-

ethical problems in their organiza- about ethical values periodically; ees. Ethicists Rae and Wong tell

tions (self-reported percents are including ethics in required train- us that debating issues is fruitless

probably low), and 48 percent of ing for mid-level managers (as at if we continue to ignore the char-

surveyed employees admit to Procter & Gamble); modeling acter traits that empower people

engaging in unethical practices ethics throughout top management for moral behavior.

such as cheating on expense and the board (termed “tone at the

accounts and forging signatures. top,” especially notable at Johnson

We are reminded again that share- & Johnson); promoting openness

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.

Risk Aversion

Financial managers generally seek to avoid risk. Most managers are risk-averse”

risk-averse

The attitude toward risk in which for a given increase in risk they require an increase in return. This attitude is

an increased return is required

believed consistent with that of the owners for whom the firm is being managed.

for an increase in risk.

Managers generally tend to be conservative rather than aggressive when accepting

risk. Accordingly, a risk-averse financial manager requiring higher return for

greater risk is assumed throughout this text.

194 PART 2 Important Financial Concepts

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.

5“3 Describe the attitude toward risk of a risk-averse financial manager.

Risk of a Single Asset

LG2 LG3

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

Sensitivity Analysis

An approach for assessing risk

that uses several possible-return Sensitivity analysis uses several possible-return estimates to obtain a sense of the

estimates to obtain a sense of the

variability among outcomes. One common method involves making pessimistic

variability among outcomes.

(worst), most likely (expected), and optimistic (best) estimates of the returns asso-

range ciated with a given asset. In this case, the asset™s risk can be measured by the range

A measure of an asset™s risk,

of returns. The range is found by subtracting the pessimistic outcome from the

which is found by subtracting the

optimistic outcome. The greater the range, the more variability, or risk, the asset

pessimistic (worst) outcome from

is said to have.

the optimistic (best) outcome.

Norman Company, a custom golf equipment manufacturer, wants to choose the

EXAMPLE

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).

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.

195

CHAPTER 5 Risk and Return

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%

times. An outcome with a probability of 100 percent is certain to occur. Out-

comes with a probability of zero will never occur.

Norman Company™s past estimates indicate that the probabilities of the pes-

EXAMPLE

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

probability distribution

must be based on all the alternatives considered.

A model that relates prob-

abilities to the associated

A probability distribution is a model that relates probabilities to the associ-

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

The simplest type of probability

bar charts for Norman Company™s assets A and B are shown in Figure 5.1.

distribution; shows only a limited

number of outcomes and associ- Although both assets have the same most likely return, the range of return is

ated probabilities for a given

much greater, or more dispersed, for asset B than for asset A”16 percent versus

event.

4 percent.

continuous probability If we knew all the possible outcomes and associated probabilities, we could

distribution develop a continuous probability distribution. This type of distribution can be

A probability distribution show-

thought of as a bar chart for a very large number of outcomes. Figure 5.2 presents

ing all the possible outcomes and

continuous probability distributions for assets A and B. Note that although assets

associated probabilities for a

A and B have the same most likely return (15 percent), the distribution of returns

given event.

FIGURE 5.1

Probability of Occurrence

Probability of Occurrence

Asset A Asset B

Bar Charts

.60 .60

Bar charts for asset A™s and

.50 .50

asset B™s returns

.40 .40

.30 .30

.20 .20

.10 .10

05 9 13 17 21 25 05 9 13 17 21 25

Return (%) Return (%)

196 PART 2 Important Financial Concepts

FIGURE 5.2

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 (%)

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

The most common statistical indicator of an asset™s risk is the standard deviation,

standard deviation ( k)

The most common statistical

k, which measures the dispersion around the expected value. The expected value

indicator of an asset™s risk; it

of a return, k, is the most likely return on an asset. It is calculated as follows:1

measures the dispersion around

the expected value. n

k kj Prj (5.2)

expected value of a return (k ) j1

The most likely return on a given

where

asset.

kj return for the jth outcome

Prj probability of occurrence of the jth outcome

n number of outcomes considered

1. 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 (5.2a)

j1

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.

197

CHAPTER 5 Risk and Return

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 .25 17 4.25

Total 1.00 Expected return 15.00%

Asset B

Pessimistic .25 7% 1.75%

Most likely .50 15 7.50

Optimistic .25 23 5.75

Total 1.00 Expected return 15.00%

The expected values of returns for Norman Company™s assets A and B are pre-

EXAMPLE

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, is2

n

k)2

(kj Prj (5.3)

k

j1

In general, the higher the standard deviation, the greater the risk.

Table 5.5 presents the standard deviations for Norman Company™s assets A and

EXAMPLE

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-

2. 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

k)2

(kj

j1

(5.3a)

k

n 1

where n is the number of observations. Equation 5.3 is emphasized in this chapter because returns and related prob-

abilities are often available.

198 PART 2 Important Financial Concepts

TABLE 5.5 The Calculation of the Standard Deviation

of the Returns for Assets A and Ba

(kj k)2 k)2

i kj k kj k Prj (kj 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

k)2

(kj Prj 2%

j1

3

k)2

(kj Prj 2% 1.41%

k

A j1

Asset B

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

2 15 15 0 0 .50 0

3 23 15 8 64 .25 16

3

k)2

(kj Prj 32%

j1

3

k)2

(kj Prj 32% 5.66%

kB

j1

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.

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-

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.

Coefficient of Variation

The coefficient of variation, CV, is a measure of relative dispersion that is useful

coefficient of variation (CV )

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.

199

CHAPTER 5 Risk and Return

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).

When the standard deviations (from Table 5.5) and the expected returns (from

EXAMPLE

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.

A firm wants to select the less risky of two alternative assets”X and Y. The

EXAMPLE

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%

9%a

(2) Standard deviation 10%

0.50a

(3) Coefficient of variation [(2) (1)] 0.75

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.

200 PART 2 Important Financial Concepts

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?

Risk of a Portfolio

LG4

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

portfolio of assets. The financial manager™s goal is to create an efficient portfolio,

one that maximizes return for a given level of risk or minimizes risk for a given

level of return. The statistical concept of correlation underlies the process of

efficient portfolio

diversification that is used to develop an efficient portfolio.

A portfolio that maximizes return

for a given level of risk or

minimizes risk for a given level

Correlation

of return.

correlation

Correlation is a statistical measure of the relationship between any two series of

A statistical measure of the

numbers. The numbers may represent data of any kind, from returns to test

relationship between any two

scores. If two series move in the same direction, they are positively correlated. If

series of numbers representing

the series move in opposite directions, they are negatively correlated.

data of any kind.

The degree of correlation is measured by the correlation coefficient, which

positively correlated

ranges from 1 for perfectly positively correlated series to 1 for perfectly nega-

Describes two series that move

tively correlated series. These two extremes are depicted for series M and N in

in the same direction.

Figure 5.3. The perfectly positively correlated series move exactly together; the per-

negatively correlated

fectly negatively correlated series move in exactly opposite directions.

Describes two series that move

in opposite directions.

correlation coefficient

Diversification

A measure of the degree of

correlation between two series.

The concept of correlation is essential to developing an efficient portfolio. To

reduce overall risk, it is best to combine, or add to the portfolio, assets that

perfectly positively correlated

Describes two positively have a negative (or a low positive) correlation. Combining negatively correlated

correlated series that have a

assets can reduce the overall variability of returns. Figure 5.4 shows that a port-

correlation coefficient of 1.

folio containing the negatively correlated assets F and G, both of which have

perfectly negatively correlated the same expected return, k, also has that same return k but has less risk (vari-

Describes two negatively

ability) than either of the individual assets. Even if assets are not negatively

correlated series that have a

correlated, the lower the positive correlation between them, the lower the

correlation coefficient of 1.

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 posi-

have a correlation coefficient

tively correlated assets. The correlation coefficient for uncorrelated assets is close

close to zero.

201

CHAPTER 5 Risk and Return

FIGURE 5.3 Perfectly Positively Correlated Perfectly Negatively Correlated

Correlations

N

The correlation between

series M and series N

Return

Return

N

M

M

Time

Time

to zero and acts as the midpoint between perfect positive and perfect negative

correlation.

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.

Table 5.7 presents the forecasted returns from three different assets”X, Y, and

EXAMPLE

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

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

FIGURE 5.4 Portfolio of

Assets F and G

Asset F Asset G

Diversification

Return Return Return

Combining negatively

correlated assets to diversify

risk

k k

Time Time Time

202 PART 2 Important Financial Concepts

TABLE 5.7 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

Statistics:c

Expected value 12% 12% 12% 12% 12%

d

Standard deviation 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 are calculated

as shown in the following table.

Forecasted return

Asset X Asset Y Portfolio return calculation Expected portfolio 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

bPortfolioXZ, 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 are calculated using the same

method demonstrated in note a above for portfolio XY.

cBecause the probabilities associated with the returns are not given, the general equation, Equation 5.2a in footnote 1,

is used to calculate expected values as demonstrated below for portfolio XY.

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

kxy 12%

5 5

The same formula is applied to find the expected value of return for assets X, Y, and Z, and portfolio XZ.

dBecause the probabilities associated with the returns are not given, the general equation, Equation 5.3a in footnote 2,

is used to calculate the standard deviations as demonstrated below for portfolio XY.

12%)2 12%)2 (12% 12%)2 12%)2 12%)2

(12% (12% (12% (12%

kxy 51

0% 0% 0% 0% 0% 0

% 0%

4 4

The same formula is applied to find the standard deviation of returns for assets X, Y, and Z, and portfolio XZ.

203

CHAPTER 5 Risk and Return

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.)3

Portfolio XY Portfolio XY (shown in Table 5.7) is created by combining equal

portions of assets X and Y, the perfectly negatively correlated assets.4 The risk in

this portfolio, as reflected by its standard deviation, is reduced to 0%, and the

expected return value remains at 12%. Because both assets have the same

expected return values, are combined in equal parts, and are perfectly negatively

correlated, 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 5.7) 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%.

Whenever perfectly positively correlated assets such as X and Y are combined,

the standard deviation of the resulting portfolio cannot be reduced below that of

the least risky asset; the maximum portfolio standard deviation will be that of the

riskiest asset. Because assets X and Z have the same standard deviation (3.16%),

the minimum and maximum standard deviations are the same (3.16%), which is

the only value that could be taken on by a combination of these assets. This result

can be attributed to the unlikely situation that X and Z are identical assets.

Correlation, Diversification, Risk, and Return

In general, the lower the correlation between asset returns, the greater the poten-

tial diversification of risk. (This should be clear from the behaviors illustrated in

Table 5.7.) For each pair of assets, there is a combination that will result in the

lowest risk (standard deviation) possible. How much risk can be reduced by this

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 5.8 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 correlation,

the ability to reduce risk is improved. Note that in no case will a portfolio of

WW assets be riskier than the riskiest asset included in the portfolio. Further discussion

W

of these relationships is included at the text™s Web site (www.aw.com/gitman).

3. Identical return streams are used in this example to permit clear illustration of the concepts, but it is not necessary

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.

4. 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.

204 PART 2 Important Financial Concepts

TABLE 5.8 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

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.

political risk

Risks of International Diversification

Risk that arises from the

possibility that a host govern-

U.S. investors should also be aware of the potential dangers of international

ment will take actions harmful to

investing. In addition to the risk induced by currency fluctuations, several other

foreign investors or that political

financial risks are unique to international investing. Most important is political

turmoil in a country will

risk, which arises from the possibility that a host government will take actions

endanger investments there.

205

CHAPTER 5 Risk and Return

harmful to foreign investors or that political turmoil in a country will endanger

investments there. Political risks are particularly acute in developing countries,

where unstable or ideologically motivated governments may attempt to block

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 Why must assets be evaluated in a portfolio context? What is an efficient

portfolio?

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?

Risk and Return: The Capital Asset

LG5 LG6

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

links risk and return for all assets is the capital asset pricing model (CAPM).5 We

(CAPM)

will use CAPM to understand the basic risk“return tradeoffs involved in all types

The basic theory that links risk

of financial decisions.

and return for all assets.

5. 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.

206 PART 2 Important Financial Concepts

FIGURE 5.5

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

Types of Risk

To understand the basic types of risk, consider what happens to the risk of a port-

folio consisting of a single security (asset), to which we add securities randomly

selected from, say, the population of all actively traded securities. Using the stan-

dard deviation of return, kp, to measure the total portfolio risk, Figure 5.5

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 diversification can

be gained by forming portfolios containing 15 to 20 randomly selected securities.

The total risk of a security can be viewed as consisting of two parts:

total risk

Total security risk Nondiversifiable risk Diversifiable risk (5.5)

The combination of a security™s

nondiversifiable risk and

Diversifiable risk (sometimes called unsystematic risk) represents the portion of

diversifiable risk.

an asset™s risk that is associated with random causes that can be eliminated

diversifiable risk

through diversification. It is attributable to firm-specific events, such as strikes,

The portion of an asset™s risk that

lawsuits, regulatory actions, and loss of a key account. Nondiversifiable risk (also

is attributable to firm-specific,

called systematic risk) is attributable to market factors that affect all firms; it can-

random causes; can be elimi-

nated through diversification. not be eliminated through diversification. (It is the shareholder-specific market

Also called unsystematic risk.

risk described in Table 5.1.) Factors such as war, inflation, international inci-

dents, and political events account for nondiversifiable risk.

nondiversifiable risk

Because any investor can create a portfolio of assets that will eliminate virtu-

The relevant portion of an asset™s

risk attributable to market ally all diversifiable risk, the only relevant risk is nondiversifiable risk. Any

factors that affect all firms;

investor or firm therefore must be concerned solely with nondiversifiable risk.

cannot be eliminated through

The measurement of nondiversifiable risk is thus of primary importance in select-

diversification. Also called

ing assets with the most desired risk“return characteristics.

systematic risk.

The Model: CAPM

The capital asset pricing model (CAPM) links nondiversifiable risk and return

for all assets. We will discuss the model in four 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

207

CHAPTER 5 Risk and Return

describes the relationship between risk and return. The final section offers

some comments on the CAPM.

Beta Coefficient

The beta coefficient, b, is a relative measure of nondiversifiable risk. It is an

beta coefficient (b)

index of the degree of movement of an asset™s return in response to a change in

A relative measure of nondiversi-

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

coefficient. The market return is the return on the market portfolio of all traded

return in response to a change in

securities. The Standard & Poor™s 500 Stock Composite Index or some similar

the market return.

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 5.6 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

FIGURE 5.6

Asset S

Asset Return (%)

(1997)

Beta Derivationa

Graphical derivation of beta 35

for assets R and S

30 (2002) (2001)

25

bS = slope = 1.30

(2003)

20

Asset R

(2000)

15

(1998) 10

(1996)

5 bR = slope = .80

Market

Return (%)

0 10 15 20 25 30 35

“20 “10

“5

(1999) “10

“15

Characteristic Line S

“20

Characteristic Line R

“25

“30

a All data points shown are associated with asset S. No data points are shown for asset R.

208 PART 2 Important Financial Concepts

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. The slope of this line is beta. The beta for asset R is about .80 and that for

asset S is about 1.30. Asset S™s higher beta (steeper characteristic line slope) indi-

cates that its return is more responsive to changing market returns. Therefore

asset S is more risky than asset R.

Interpreting Betas The beta coefficient for the market is considered to be

equal to 1.0. All other betas are viewed in relation to this value. Asset betas may

be positive or negative, but positive betas are the norm. The majority of beta

coefficients fall between .5 and 2.0. The return of a stock that is half as respon-

sive as the market (b .5) is expected to change by 1/2 percent for each 1 percent

change in the return of the market portfolio. A stock that is twice as responsive as

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 5.9 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 5.10.

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

the portfolio™s total dollar value represented by asset j, and letting bj equal the

beta of asset j, we can use Equation 5.6 to find the portfolio beta, bp:

n

...

bp (w1 b1) (w2 b2) (wn bn) wj bj (5.6)

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

assets. They indicate the degree of responsiveness of the portfolio™s return to

changes in the market return. For example, when the market return increases by

10 percent, a portfolio with a beta of .75 will experience a 7.5 percent increase in

its return (.75 10%); a portfolio with a beta of 1.25 will experience a 12.5 per-

cent increase in its return (1.25 10%). Clearly, a portfolio containing mostly

low-beta assets will have a low beta, and one containing mostly high-beta assets

will have a high beta.

TABLE 5.9 Selected Beta Coefficients and

Their Interpretations

Beta Comment Interpretation

2.0 Twice as responsive as the market

Move in same

1.0 direction as Same response as the market

market

.5 Only half as responsive as the market

0 Unaffected by market movement

.5 Only half as responsive as the market

Move in opposite

1.0 direction to Same response as the market

market

2.0 Twice as responsive as the market

209

CHAPTER 5 Risk and Return

TABLE 5.10 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 Austin Fund, a large investment company, wishes to assess the risk of two

EXAMPLE

portfolios it is considering assembling”V and W. Both portfolios contain five

assets, with the proportions and betas shown in Table 5.11. The betas for the two

portfolios, bv and bw, can be calculated by substituting data from the table into

Equation 5.6:

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.

TABLE 5.11 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

210 PART 2 Important Financial Concepts

The Equation

Using the beta coefficient to measure nondiversifiable risk, the capital asset pric-

ing model (CAPM) is given in Equation 5.7:

kj RF [bj (km RF)] (5.7)

where

kj required return on asset j

RF risk-free rate of return, commonly measured by the

return on a U.S. Treasury bill

bj beta coefficient or index of nondiversifiable risk for asset j

km market return; return on the market portfolio of assets

The CAPM can be divided into two parts: (1) risk-free of interest, RF, which

risk-free rate of interest, RF

The required return on a risk-free is the required return on a risk-free asset, typically a 3-month U.S. Treasury bill

asset, typically a 3-month U.S.

(T-bill), a short-term IOU issued by the U.S. Treasury, and (2) the risk premium.

Treasury bill.

These are, respectively, the two elements on either side of the plus sign in Equa-

tion 5.7. The (km RF) portion of the risk premium is called the market risk pre-

U.S. Treasury bills (T-bills)

Short-term IOUs issued by the mium, because it represents the premium the investor must receive for taking the

U.S. Treasury; considered the

average amount of risk associated with holding the market portfolio of assets.6

risk-free asset.

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 5.7) involves merely subtracting the historical U.S. Treasury bill™s aver-

age return from the historical average return for a given investment:

Risk premiuma

Investment