Unformatted text preview: Risk
and Return
“Believe me! The secret of reaping the
greatest fruitfulness and the greatest
enjoyment from life is to live dangerously!”
—Friedrich Wilhelm Nietzsche Are You the “GoforIt” Type?
The financial crisis has people buzzing about “systematic risk.” This term
means different things in different contexts. Traditionally, systematic risk has
referred to the nondiversifiable risk that comes from the impact the overall
market has on individual investments. This risk is also known as “market
risk” according to the Capital Asset Pricing Model (CAPM) described in
this chapter.
With the financial crisis, however, people have been using the term
systematic risk in a somewhat different way. Many companies, especially
financial firms, are connected to each other in significant ways. With a
financial instrument known as a swap, for example, one company may
have a contract with another company that calls for large payments to be
made by one to the other according to specified terms. If the company that
is obligated to pay does not, then the company that was supposed to receive
the funds might fail. If that company that was supposed to receive the funds
fails, then other companies that it owed money to according to other swaps
might also fail. This chain reaction of default, failure, default, failure could
affect a large number of firms. The larger the firm, the more such relationships
it is likely to have and the greater the chain reaction failures that may occur.
This is the systematic risk the government is concerned about now.
Systematic risk is why the prestigious Wall Street firm Lehman Brothers
declared bankruptcy. The same is true about mortgage giants Fannie Mae
and Freddie Mac. Systematic risk led to Wall Street icon Merrill Lynch selling
Source: The Google Company website (www.google.com/intl/en/options) 160 © Jordan Lewy (http://www.fotolia.com/p/55426) itself at a fire sale price to Bank of America. The fear of systematic risk is
why the federal government pumped billions of dollars into the insurance
giant AIG.
This kind of systematic risk clearly has the potential to do great harm.
This is part of the overall examination of risk conducted in this chapter. Chapter Overview Learning Objectives
After reading this chapter,
you should be able to:
1. Define risk, risk aversion,
and the riskreturn
relationship.
2. Measure risk using the
standard deviation and
coefficient of variation. Business firms face risk in nearly everything they do. Assessing risk is one of the
most important tasks financial managers perform. The events of recent years have
made this painfully obvious. In this chapter we will discuss risk, risk aversion, and
the riskreturn relationship. We will measure risk using standard deviation and
the coefficient of variation. We will identify types of risk and examine ways to
reduce risk exposure or compensate for risk. Finally, we will see how the capital
asset pricing model (CAPM) explains the riskreturn relationship. 4. Explain methods of risk
reduction. Risk 5. Describe how firms
compensate for assuming
risk. The world is a risky place. For instance, if you get out of bed in the morning and
go to class, you run the risk of getting hit by a bus. If you stay in bed to minimize
the chance of getting run over by a bus, you run the risk of getting coronary artery
disease because of a lack of exercise. In everything we do—or don’t do—there is
a chance that something will happen that we didn’t expect. Risk is the potential
for unexpected events to occur.
161 3. Identify the types of
risk that business firms
encounter. 6. Discuss the capital asset
pricing model (CAPM). 162 Part II Essential Concepts in Finance Risk Aversion
Most people try to avoid risks if possible. Risk aversion doesn’t mean that some people
don’t enjoy risky activities, such as skydiving, rock climbing, or automobile racing. In
a financial setting, however, evidence shows that most people avoid risk when possible,
unless there is a higher expected rate of return to compensate for the risk. Faced with
financial alternatives that are equal except for their degree of risk, most people will
choose the less risky alternative.
Risk aversion is the tendency to avoid additional risk. Riskaverse people will avoid
risk if they can, unless they receive additional compensation for assuming that risk. In
finance, the added compensation is a higher expected rate of return. The RiskReturn Relationship
The relationship between risk and required rate of return is known as the riskreturn
relationship. It is a positive relationship because the more risk assumed, the higher
the required rate of return most people will demand. It takes compensation to convince
people to suffer.
Suppose, for instance, that you were offered a job in the Sahara Desert, working
long hours for a boss everyone describes as a tyrant. You would surely be averse to the
idea of taking such a job. But think about it: Is there any way you would take this job?
What if you were told that your salary would be $1 million per year? This compensation
might cause you to sign up immediately. Even though there is a high probability you
would hate the job, you’d take that risk because of the high compensation.1
Not everyone is risk averse, and among those who are, not all are equally risk
averse. Some people would demand $2 million before taking the job in the Sahara
Desert, whereas others would do it for a more modest salary.
People sometimes engage in very risky financial activities, such as buying lottery
tickets or gambling in casinos. This suggests that they like risk and will pay to experience
it. Most people, however, view these activities as entertainment rather than financial
investing. The entertainment value may be the excitement of being in a casino with all
sorts of people, or being able to fantasize about spending the multimilliondollar lotto
jackpot. But in the financial markets, where people invest for the future, they almost
always seek to avoid risk unless they are adequately compensated.
Risk aversion explains the positive riskreturn relationship. It explains why risky
junk bonds carry a higher market interest rate than essentially riskfree U.S. Treasury
bonds. Hardly anyone would invest $5,000 in a risky junk bond if the interest rate on
the bond were lower than that of a U.S. Treasury bond having the same maturity. This is not to suggest that people can be coaxed into doing anything they are averse to doing merely by offering them a
sufficient amount of compensation. If people are asked to do something that offends their values, there may be no amount of
compensation that can entice them. 1 Chapter 7 Risk and Return Measuring Risk
We can never avoid risk entirely. That’s why businesses must make sure that the
anticipated return is sufficient to justify the degree of risk assumed. To do that, however,
firms must first determine how much risk is present in a given financial situation. In
other words, they must be able to answer the question, “How risky is it?”
Measuring risk quantitatively is a rather tall order. We all know when something
feels risky, but we don’t often quantify it. In business, risk measurement focuses on
the degree of uncertainty present in a situation—the chance, or probability, of an
unexpected outcome. The greater the probability of an unexpected outcome, the greater
the degree of risk. Using Standard Deviation to Measure Risk
In statistics, distributions are used to describe the many values variables may have. A
company’s sales in future years, for example, is a variable with many possible values.
So the sales forecast may be described by a distribution of the possible sales values
with different probabilities attached to each value. If this distribution is symmetric, its
mean—the average of a set of values—would be the expected sales value. Similarly,
possible returns on any investment can be described by a probability distribution—
usually a graph, table, or formula that specifies the probability associated with each
possible return the investment may generate. The mean of the distribution is the most
likely, or expected, rate of return.
The graph in Figure 71 shows the distributions of forecast sales for two companies,
Company Calm and Company Bold. Note how the distribution for Company Calm’s
possible sales values is clustered closely to the mean and how the distribution of
Company Bold’s possible sales values is spread far above and far below the mean.2
The narrowness or wideness of a distribution reflects the degree of uncertainty about
the expected value of the variable in question (sales, in our example). The distributions
in Figure 71 show, for instance, that although the most probable value of sales for both
companies is $1,000, sales for Company Calm could vary between $600 and $1,400,
whereas sales for Company Bold could vary between $200 and $1,800. Company Bold’s
relatively wide variations show that there is more uncertainty about its sales forecast
than about Company Calm’s sales forecast.
One way to measure risk is to compute the standard deviation of a variable’s
distribution of possible values. The standard deviation is a numerical indicator of how
widely dispersed the possible values are around a mean. The more widely dispersed
a distribution is, the larger the standard deviation, and the greater the probability that
the value of a variable will be significantly above or below the expected value. The
standard deviation, then, indicates the likelihood that an outcome different from what
is expected will occur.
Let’s calculate the standard deviations of the sales forecast distributions for
Companies Calm and Bold to illustrate how the standard deviation can measure the
degree of uncertainty, or risk, that is present. These two distributions are discrete. If sales could take on any value within a given range, the distribution would be continuous
and would be depicted by a curved line. 2 163 164 Part II Essential Concepts in Finance
Company Calm Sales Distribution Probability of Ocurrence 70%
60%
50%
40%
30%
20%
10%
0 $0 $200 $400 $600 $800 $1,000 $1,200 $1,400 $1,600 $1,800 $2,000 Possible Sales Values Company Bold Sales Distribution Figure 71 Sales
Forecast Distributions for
Companies Calm and Bold
Possible future sales distribution
for two companies. Calm has a
relatively “tight” distribution, and
Bold has a relatively “wide”
distribution. Note that sales for
Company Bold has many more
possible values than the sales
for Company Calm. Probability of Ocurrence 70%
60%
50%
40%
30%
20%
10%
0 $0 $200 $400 $600 $800 $1,000 $1,200 $1,400 $1,600 $1,800 $2,000 Possible Sales Values Calculating the Standard Deviation To calculate the standard deviation of the
distribution of Company Calm’s possible sales, we must first find the expected value,
or mean, of the distribution using the following formula:
Formula for Expected Value, or Mean (µ)
µ = ∑ (V × P) (71) where: µ = the expected value, or mean
∑ = the sum of
V = the possible value for some variable
P = the probability of the value V occurring Applying Equation 71, we can calculate the expected value, or mean, of Company
Calm’s forecasted sales. The following values for V and P are taken from Figure 71: Chapter 7 Risk and Return Calculating the Mean (µ) of Company Calm’s
Possible Future Sales Distribution
Possible
Sales Value (V) Probability
of Occurrence (P) VxP $ 600 .05 30 $ 800 .10 80 $ 1,000 .70 700 $ 1,200 .10 120 $ 1,400 .05 70 Σ = 1.00 Σ = 1,000 = µ Each possible sales value is multiplied by its respective probability. The probability
values, taken from Figure 71, may be based on trends, industry ratios, experience,
or other information sources. We add together the products of each value times its
probability to find the mean of the possible sales distribution.
We now know that the mean of Company Calm’s sales forecast distribution is
$1,000. We are ready to calculate the standard deviation of the distribution using the
following formula:
The Standard Deviation (σ) Formula σ= ∑ P(V − µ ) 2 (72) where: σ = the standard deviation
∑ = the sum of
P = the probability of the value V occurring
V = the possible value for a variable
µ = the expected value To calculate the standard deviation of Company Calm’s sales distribution, we
subtract the mean from each possible sales value, square that difference, and then
multiply by the probability of that sales outcome. These differences squared, times their
respective probabilities, are then added together, and the square root of this number is
taken. The result is the standard deviation of the distribution of possible sales values.
Calculating the Standard Deviation(s) of Company Calm’s
Possible Future Sales Distribution
Possible Sales
Value (V) Probability of
Occurrence (P) V–µ (V – µ)2 P(V – µ)2 $ 600 .05 –400 160,000 8,000 $ 800 .10 –200 40,000 4,000 $1,000 .70 0 0 0 $1,200 .10 200 40,000 4,000 $1,400 .05 400 160,000 8,000
Σ = 24,000 24 , 000 = 155 = σ 165 166 Part II Essential Concepts in Finance This standard deviation result, 155, serves as the measure of the degree of risk
present in Company Calm’s sales forecast distribution.
Let’s calculate the standard deviation of Company Bold’s sales forecast distribution.
Mean (µ) and Standard Deviation (σ) of Company Bold’s
Possible Future Sales Distribution
Possible Sales
Value (V)
$ Probability of
Occurrence (P) 200 .04 Mean Calculation
V×P
8 V–µ (V – µ)2 P(V – µ)2 –800 640,000 25,600 $ 400 .07 28 –600 360,000 25,200 $ 600 .10 60 –400 160,000 16,000 $ 800 .18 144 –200 40,000 7,200 $ 1,000 .22 220 0 0 0 $ 1,200 .18 216 200 40,000 7,200 $ 1,400 .10 140 400 160,000 16,000 $ 1,600 .07 112 600 360,000 25,200 $ 1,800 .04 72 800 640,000 Σ = 1,000 = µ 25,600
Σ = 148,000 148, 000 = 385 = σ Take Note
In the preceding
procedure, we combine
two steps: (1 finding the
mean of the distribution
with Equation 71;
and (2) calculating
the standard deviation
with Equation 72. As you can see, Company Bold’s standard deviation of 385 is over twice that of
Company Calm. This reflects the greater degree of risk in Company Bold’s sales forecast.
Interpreting the Standard Deviation Estimates of a company’s possible sales, or a
proposed project’s future possible cash flows, can generally be thought of in terms of a
normal probability distribution. The normal distribution is a special type of distribution.
It allows us to make statements about how likely it is that the variable in question will
be within a certain range of the distribution.
Figure 72 shows a normal distribution of possible returns on an asset. The vertical
axis measures probability density for this continuous distribution so that the area under
the curve always sums to one. Statistics tells us that when a distribution is normal,
there is about a 67 percent chance that the observed value will be within one standard
deviation of the mean. In the case of Company Calm, that means that, if sales were
normally distributed, there would be a 67 percent probability that the actual sales will
be $1,000 plus or minus $155 (between $845 and $1,155). For Company Bold it means,
if sales were normally distributed, there would be a 67 percent probability that sales
will be $1,000 plus or minus $385 (between $615 and $1,385).
Another characteristic of the normal distribution is that approximately 95 percent
of the time, values observed will be within two standard deviations of the mean. For
Company Calm this means that there would be a 95 percent probability that sales will
be $1,000 plus or minus $155 × 2, or $310 (between $690 and $1,310). For Company
Bold it means that sales will be $1,000 plus or minus $385 × 2, or $770 (between $230
and $1,770). These relationships are shown graphically in Figure 73.
The greater the standard deviation value, the greater the uncertainty about what the
actual value of the variable in question will be. The greater the value of the standard
deviation, the greater the possible deviations from the mean. Chapter 7 167 Risk and Return Company Calm Normal Distribution 0.30% Probability Density 0.25% 0.20% 0.15% 0.10% 0.05% Figure 72 Normal Distribution
0.00% $0 $200 $400 $600 $800 $1,000 $1,200 $1,400 $1,600 Possible Sales Values Using the Coefficient of Variation to Measure Risk
Whenever we want to compare the risk of investments that have different means,
we use the coefficient of variation. We were safe in using the standard deviation to
compare the riskiness of Company Calm’s possible future sales distribution with that
of Company Bold because the mean of the two distributions was the same ($1,000).
Imagine, however, that Company Calm’s sales were 10 times that of Company Bold. If
that were the case and all other factors remained the same, then the standard deviation
of Company Calm’s possible future sales distribution would increase by a factor of 10,
to $1,550. Company Calm’s sales would appear to be much more risky than Company
Bold’s, whose standard deviation was only $385.
To compare the degree of risk among distributions of different sizes, we should use
a statistic that measures relative riskiness. The coefficient of variation (CV) measures
relative risk by relating the standard deviation to the mean. The formula follows:
Coefficient of Variation (CV)
CV = Standard Deviation
Mean (73) The coefficient of variation represents the standard deviation’s percentage of the
mean. It provides a standardized measure of the degree of risk that can be used to
compare alternatives.
To illustrate the use of the coefficient of variation, let’s compare the relative risk
depicted in Company Calm’s and Company Bold’s possible sales distributions. When
we plug the figures into Equation 73, we see: This normal probability
distribution of possible returns
has a mean, the expected value
of $1,000. Interactive Module
Go to www.textbookmedia.
com and find the free
companion material for this
book. Follow the instructions
there. Change the mean,
standard deviation, weight,
and correlation values and
watch the graph. 168 Part II Essential Concepts in Finance $845 Figure 73 The Degree
of Risk Present in Company
Calm’s and Company
Bold’s Possible Future Sales
Values as Measured by
Standard Deviation
The standard deviation shows
there is much more risk present
in Company Bold’s sales
probability distribution than
in Company Calm’s. If the
distributions are normal, then
there is a 67% probability that
Company Calm’s sales will be
between $845 and $1,155,
and a 95% probability sales
will be between $690 and
$1,310. For Company Bold
there is a 67% probability that
sales will be between $615
and $1,385, and a 95%
probability that sales will be
between $230 and $1,770. Company Calm $1,155 –$155 $1,000 +$155 67% Probability
Company Bold $615 –$385 $1,000 +$385 $1,385 67% Probability $690 Company Calm
–$310 $1,000 +$310 $1,310 95% Probability $230 Company Bold
–$770 $1,000 +$770 $1,770 95% Probability Company Calm CVsales = Standard Deviation
155
=
= .155, or 15.5%
Mean
1, 000 Company Bold CVsales = Standard Deviation
385
=
= .385, or 38.5%
Mean
1, 000 Company Bold’s coefficient of variation of possible sales (38.5 percent) is more
than twice that of Company Calm (15.5 percent). Furthermore, even if Company Calm
were 10 times the size of Company Bold—with a mean of its possible future sales of
$10,000 and with a standard deviation of $1,550—it would not change the coefficient
of variation. This would remain 1,550/10,000 = .155, or 15.5 percent. We use the
coefficient of variation instead of the standard deviation to compare distributions that
have means with different values because the CV adjusts for the difference, whereas
the standard deviation does not. The Types of Risks Firms Encounter
Risk refers to uncertainty—the chance that what you expect to happen won’t happen.
The forms of risk that businesses most often encounter are business risk, financial risk,
and portfolio risk. Chapter 7 Risk and Return Business Risk
Business risk refers to the uncertainty a company has with regard to its operating income
(also known as earnings before interest and taxes, or EBIT). The more uncertainty about
a company’s expected operating income, the more business risk the company has. For
example, if we assume that grocery prices remain constant, the only grocery store in a
small town probably has little business risk—the store owners can reliably predict how
much their customers will buy each month. In contrast, a gold mining firm in Wyoming
has a lot of business risk. Because the owners have no idea when, where, or how much
gold they will strike, they can’t predict how much they will earn in any period with any
degree of certainty.
Measuring Business Risk The degree of uncertainty about operating income (and,
therefore, the degree of business risk in the firm) depends on the volatility of operating
income. If operating income is relatively constant, as in the grocery store example, then
there is relatively little uncertainty associated with it. If operating income can take on
many different values, as is the case with the gold mining firm, then there is a lot of
uncertainty about it.
We can measure the variability of a company’s operating income by calculating
the standard deviation of the operating income forecast. A small standard deviation
indicates little variability and, therefore, little uncertainty. A large standard deviation
indicates a lot of variability and great uncertainty.
Some companies are large and others small. So to make comparisons among different
firms, we must measure the risk by calculating the coefficient of variation of possible
operating income values. The higher the coefficient of variation of possible operating
income values, the greater the business risk of the firm.
Table 71 shows the expected value (µ), standard deviation (σ), and coefficient of
variation (CV) of operating income for Company Calm and Company Bold, assuming
that the expenses of both companies vary directly with sales (i.e., neither company has
any fixed expenses).
The Influence of Sales Volatility Sales volatility affects business risk—the more
volatile a company’s sales, the more business risk the firm has. Indeed, when no fixed
costs are present—as in the case of Company Calm and Company Bold—sales volatility
is equivalent to operating income volatility. Table 71 shows that the coefficients of
variation of Company Calm’s and Company Bold’s operating income are 15.5 percent
and 38.5 percent, respectively. Note that these coefficient numbers are exactly the same
numbers as the two companies’ coefficients of variation of expected sales.
The Influence of Fixed Operating Costs In Table 71 we assumed that all of Company
Calm’s and Company Bold’s expenses varied proportionately with sales. We did this
to illustrate how sales volatility affects operating income volatility. In the real world,
of course, most companies have some fixed expenses as well, such as rent, insurance
premiums, and the like. It turns out that fixed expenses magnify the effect of sales
volatility on operating income volatility. In effect, fixed expenses magnify business risk.
The tendency of fixed expenses to magnify business risk is called operating leverage. To
see how this works, refer to Table 72, in which we assume that all of Company Calm’s
and Company Bold’s expenses are fixed. 169 170 Part II Essential Concepts in Finance Table 71 Expected Value (µ), Standard Deviation (σ), and Coefficient of Variation (CV) of Possible Operating Income
Values for Company Calm and Bold, Assuming All Expenses Are Variable
Company Calm
Probability of Occurrence
5% 10% 70% 10% 5% Sales $ 600 $ 800 $ 1,000 $ 1,200 $ 1,400 Variable Expenses $ 516 $ 688 $ 860 $ 1,032 $ 1,204 Operating Income (EBIT) $ 84 $ 112 $ 140 $ 168 $ 196 µ of possible operating income values per equation 71: $140
σ of possible operating income values per equation 72: $21.69
CV of possible operating income values per equation 73: 15.5% Company Bold
Probability of Occurrence
4% 7% 10% Sales $ 200 $ 400 $ Variable Expenses $ 172 $ 344 $ Operating Income (EBIT) $ 28 $ 56 $ 18% 22% 18% 10% 7% 4% 600 $ 800 $ 1,000 $ 1,200 $ 1,400 $1,600 $ 1,800 516 $ 688 $ 860 $ 1,032 $ 1,204 $1,376 $ 1,548 84 $ 112 $ 140 $ 168 $ 196 $ 224 $ 252 µ of possible operating income values per equation 71: $140
σ of possible operating income values per equation 72: $53.86
CV of possible operating income values per equation 73: Take Note
One group of businesses
that is exposed to an
extreme amount of
financial risk because they
operate almost entirely
on borrowed money:
banks and other financial
institutions. Banks get
almost all the money
they use for loans from
deposits—and deposits
are liabilities on the bank’s
balance sheet. Banks must
be careful to keep their
revenues stable. Otherwise,
fluctuations in revenues
would cause losses that
would drive the banks
out of business. Now you
know why the government
regulates financial
institutions so closely! 38.5% As Table 72 shows, the effect of replacing each company’s variable expenses with
fixed expenses increased the volatility of operating income considerably. The coefficient
of variation of Company Calm’s operating income jumped from 15.49 percent when
all expenses were variable to over 110 percent when all expenses were fixed. When all
expenses are fixed, a 15.49 percent variation in sales is magnified to a 110.66 percent
variation in operating income. A similar situation exists for Company Bold.
The greater the fixed expenses, the greater the change in operating income for a
given change in sales. Capitalintensive companies, such as electric generating firms,
have high fixed expenses. Service companies, such as consulting firms, often have
relatively low fixed expenses. Financial Risk
When companies borrow money, they incur interest charges that appear as fixed expenses
on their income statements. (For business loans, the entire amount borrowed normally
remains outstanding until the end of the term of the loan. Interest on the unpaid balance,
then, is a fixed amount that is paid each year until the loan matures.) Fixed interest
charges act on a firm’s net income in the same way that fixed operating expenses act
on operating income—they increase volatility. The additional volatility of a firm’s net
income caused by the fixed interest expense is called financial risk. The phenomenon
whereby a given change in operating income causes net income to change by a larger
percentage is called financial leverage. Chapter 7 171 Risk and Return Table 72 Expected Value (µ), Standard Deviation (σ), and Coefficient of Variation (CV) of Possible
Operating Income Values for Company Calm and Bold, Assuming All Expenses Are Fixed
Company Calm
Probability of Occurrence
5% 10% 70% 10% 5% Sales $ 600 $ 800 $ 1,000 $ 1,200 $ 1,400 Fixed Expenses $ 860 $ 860 $ 860 $ 860 $ 860 Operating Income (EBIT) ($260) ($ 60) $ 140 $ 340 $ 540 µ of possible operating income values per equation 71: $140
σ of possible operating income values per equation 72: $154.92
CV of possible operating income values per equation 73: 110.7%
Company Bold
Probability of Occurrence
4% 7% 10% 18% 22% 18% 10% 7% Sales $ 200 $ 400 $ Fixed Expenses $ 860 $ 860 $ Operating Income (EBIT) ($660) ($460) ($ 260) 600 $ 800 $ 1,000 $ 1,200 $ 1,400 $1,600 $ 1,800 860 $ 860 $ 860 $ 860 $ 860 $ 860 $ 860 ($ $ 140 $ 340 $ 540 $ 740 $ 940 60) µ of possible operating income values per equation 71: $140
σ of possible operating income values per equation 72: $384.71
CV of possible operating income values per equation 73: 274.8% Measuring Financial Risk Financial risk is the additional volatility of net income
caused by the presence of interest expense. We measure financial risk by noting the
difference between the volatility of net income when there is no interest expense and
when there is interest expense. To measure financial risk, we subtract the coefficient
of variation of net income without interest expense from the coefficient of variation of
net income with interest expense. The coefficient of variation of net income is the same
as the coefficient of variation of operating income when no interest expense is present.
Table 73 shows the calculation for Company Calm and Company Bold assuming (1)
all variable operating expenses and (2) $40 in interest expense.
Financial risk, which comes from borrowing money, compounds the effect of
business risk and intensifies the volatility of net income. Fixed operating expenses
increase the volatility of operating income and magnify business risk. In the same way,
fixed financial expenses (such as interest on debt or a noncancellable lease expense)
increase the volatility of net income and magnify financial risk.
Firms that have only equity financing have no financial risk because they have no
debt on which to make fixed interest payments. Conversely, firms that operate primarily
on borrowed money are exposed to a high degree of financial risk. Portfolio Risk
A portfolio is any collection of assets managed as a group. Most large firms employ their
assets in a number of different investments. Together, these make up the firm’s portfolio of 4% 172 Part II Essential Concepts in Finance Table 73 Expected Value (µ), Standard Deviation (σ), and Coefficient of Variation (CV) of Possible Net
Income Values for Company Calm and Bold
Company Calm
Probability of Occurrence
5% 10% 70% 10% 5% Sales $ 600 $ 800 $ 1,000 $ 1,200 $ 1,400 Variable Expenses $ 516 $ 688 $ 860 $ 1,032 $ 1,204 Operating Income (EBIT) $ 84 $ 112 $ 140 $ 168 $ 196 Interest Expense $ 40 $ 40 $ 40 $ $ Net Income $ 44 $ 72 $ 100 40 $ 128 40 $ 156 µ of possible net income values per equation 71: $100
σ of possible net income values per equation 72: $21.69
CV of possible net income values per equation 73: 21.7% Summary:
CV of possible net income values when interest expense is present: 21.7% CV of possible net income values when interest expense is not present (from Table 71): 15.5% Difference (financial risk) 6.2% Company Bold
Probability of Occurrence
4% 7% 10% 18% 22% 18% 10% 7% 4%
$ 1,800 Sales $ 200 $ 400 $ 600 $ 800 $ 1,000 $ 1,200 $ 1,400 $1,600 Variable Expenses $ 172 $ 344 $ 516 $ 688 $ 860 $ 1,032 $ 1,204 $1,376 $ 1,548 Operating Income (EBIT) $ 28 $ 56 $ 84 $ 112 $ 140 $ 168 $ 196 $ 224 $ Interest Expense $ 40 $ 40 $ 40 $ 40 $ Net Income ($ 12) $ 16 $ 44 $ 72 $ 100 40 $ 40 $ 128 $ 40 $ 156 $ 252 40 $ 40 $ 184 $ 212 µ of possible net income values per equation 71: $100
σ of possible net income values per equation 72: $53.86
CV of possible net income values per equation 73: 53.9% Summary:
CV of possible net income values when interest expense is present:
CV of possible net income values when interest expense is not present (from Table 71):
Difference (financial risk) Take Note
For simplicity, Table 73
assumes that neither firm
pays any income taxes.
Income tax is not a fixed
expense, so its presence
would not change the
volatility of net income. 53.9%
38.5%
15.4% assets. Individual investors also have portfolios containing many different stocks or other
investments. Firms (and individuals for that matter) are interested in portfolio returns and
the uncertainty associated with them. Investors want to know how much they can expect to
get back from their portfolio compared with how much they invest (the portfolio’s expected
return) and what the chances are that they won’t get that return (the portfolio’s risk).
We can easily find the expected return of a portfolio, but calculating the standard
deviation of the portfolio’s possible returns is a little more difficult. For example, suppose
Company Cool has a portfolio that is equally divided between two assets, Asset A and Chapter 7 Risk and Return Asset B. The expected returns and standard deviations of possible returns of Asset A
and Asset B are as follows:
Asset A
Expected Return E(R)
Standard Deviation (σ) Asset B 10% 12% 2% 4% Finding the expected return of Company Cool’s portfolio is easy. We simply
calculate the weighted average expected return, E(Rp), of the twoasset portfolio using
the following formula:
Expected Rate of Return of a Portfolio, E(Rp)
Comprised of Two Assets, A and B
E(Rp) = (wa × E(Ra)) + (wb × E(Rb)) (74) where: E(Rp) = the expected rate of return of the portfolio composed of Asset A
and Asset B
wa = the weight of Asset A in the portfolio
E(Ra) = the expected rate of return of Asset A
wb = the weight of Asset B in the portfolio
E(Rb) = the expected rate of return of Asset B According to Equation 74, the expected rate of return of a portfolio containing
50 percent Asset A and 50 percent Asset B is
E(Rp) = (.50 × .10) + (.50 × .12)
= .05 + .06
= .11, or 11% Now let’s turn to the standard deviation of possible returns of Company Cool’s
portfolio. Determining the standard deviation of a portfolio’s possible returns requires
special procedures. Why? Because gains from one asset in the portfolio may offset
losses from another, lessening the overall degree of risk in the portfolio. Figure 74
shows how this works.
Figure 74 shows that even though the returns of each asset vary, the timing of the
variations is such that when one asset’s returns increase, the other’s decrease. Therefore,
the net change in the Company Cool portfolio returns is very small—nearly zero. The
weighted average of the standard deviations of returns of the two individual assets, then,
does not result in the standard deviation of the portfolio containing both assets. The
reduction in the fluctuations of the returns of Company Cool (the combination of assets
A and B) is called the diversification effect.
Correlation How successfully diversification reduces risk depends on the degree of
correlation between the two variables in question. Correlation indicates the degree
to which one variable is linearly related to another. Correlation is measured by the
correlation coefficient, represented by the letter r. The correlation coefficient can take
on values between +1.0 (perfect positive correlation) to 1.0 (perfect negative correlation). 173 174 Part II Essential Concepts in Finance Figure 74 The
Variation in Returns Over
Time for Asset A, Asset B,
and the Combined
Company Cool Portfolio
Figure 74 shows how the
returns of Asset A and Asset B
might vary over time. Notice
that the fluctuations of each
curve are such that gains in
one almost completely offset
losses in the other. The risk of
the Company Cool portfolio
is small due to the offsetting
effects. Percent Return on Investment 12%
11%
10%
9%
8%
Return from Asset A
Return from Asset B
Resulting return from Company Cool portfolio 7%
6% 2009 2010 2011 2012 2013 2014 2015 If two variables are perfectly positively correlated, it means they move together—that
is, they change values proportionately in the same direction at the same time. If two
variables are perfectly negatively correlated, it means that every positive change in one
value is matched by a proportionate corresponding negative change in the other. In the
case of Assets A and B in Figure 74, the assets are negatively correlated.
The closer r is to +1.0, the more the two variables will tend to move with each other
at the same time. The closer r is to 1.0, the more the two variables will tend to move
opposite each other at the same time. An r value of zero indicates that the variables’
values aren’t related at all. This is known as statistical independence.
In Figure 74, Asset A and Asset B had perfect negative correlation (r = 1.0). So
the risk associated with each asset was nearly eliminated by combining the two assets
into one portfolio. The risk would have been completely eliminated had the standard
deviations of the two assets been equal. Take Note
Any time the correlation
coefficient of the returns
of two assets is less than
+1.0, then the standard
deviation of the portfolio
consisting of those
assets will be less than
the weighted average
of the individual assets’
standard deviations. Calculating the Correlation Coefficient Determining the precise value of r between
two variables can be extremely difficult. The process requires estimating the possible
values that each variable could take and their respective probabilities, simultaneously.
We can make a rough estimate of the degree of correlation between two variables by
examining the nature of the assets involved. If one asset is, for instance, a firm’s existing
portfolio, and the other asset is a replacement piece of equipment, then the correlation
between the returns of the two assets is probably close to +1.0. Why? Because there
is no influence that would cause the returns of one asset to vary any differently than
those of the other. A CocaCola® Bottling company expanding its capacity would be
an example of a correlation of about +1.0.
What if a company planned to introduce a completely new product in a new market?
In that case we might suspect that the correlation between the returns of the existing
portfolio and the new product would be something significantly less than +1.0. Why?
Because the cash flows of each asset would be due to different, and probably unrelated,
factors. An example would be Disney buying the Anaheim Ducks National Hockey
League team. Chapter 7 Risk and Return Calculating the Standard Deviation of a TwoAsset Portfolio To calculate the
standard deviation of a portfolio, we must use a special formula that takes the
diversification effect into account. Here is the formula for a portfolio containing two
assets.3 For convenience, they are labeled Asset A and Asset B:
Standard Deviation of a TwoAsset Portfolio σp = 2 2 2 2 wa σ a + w bσ b + 2wa wb ra,bσ aσ b (75) 2
where: σp = the standard (.50 2 )(0.02 2 ) the (.50 2 )(0.04the + (2)(.50)(.50)(−1.0)(.02)(.04)
= deviation of + returns of ) combined portfolio
containing Asset A and Asset B
= Asset A in the (.25)(.0016) − .0004
w = the weight of(.25)(.0004) + twoasset portfolio
a σa = the standard deviation .0004 returns of Asset A
= .0001 + of the − .0004
wb = the weight of Asset B in the twoasset portfolio
= .0001
σb = the standard deviation of the returns of Asset B
= .01, or 1%
ra,b = the correlation coefficient of the returns of Asset A and Asset B The formula may look scary, but don’t panic. Once we know the values for each
factor, we can solve the formula rather easily with a calculator. Let’s use the formula to
find the standard deviation of a portfolio composed of equal amounts invested in Asset
A and Asset B (i.e., Company Cool).
To calculate the standard deviation of possible returns of the portfolio of Company Cool,
we need to know that Company Cool’s portfolio is composed of 50 percent Asset A (wa =
.5) and 50 percent Asset B (wb = .5). The standard deviation of Asset A’s expected returns
is 2 percent (σa = .02), and the standard deviation of Asset B’s expected returns is 4 percent
(σb = .04). To begin, assume the correlation coefficient (r) is 1.0, as shown in Figure 74.
Now we’re ready to use Equation 75 to calculate the standard deviation of
Company Cool’s returns.
σp = 2 2 2 2 2 2 wa σ a + w bσ b + 2wa wb ra,bσ aσ b
2 2 = (.50 )(0.02 ) + (.50 )(0.04 ) + (2)(.50)(.50)(−1.0)(.02)(.04) = (.25)(.0004) + (.25)(.0016) − .0004 = .0001 + .0004 − .0004 = .0001 = .01, or 1% The diversification effect results in risk reduction. Why? Because we are combining
two assets that have returns that are negatively correlated (r = 1.0). The standard deviation
of the combined portfolio is much lower than that of either of the two individual assets (1
percent for Company Cool compared with 2 percent for Asset A and 4 percent for Asset B).
You can adapt the formula to calculate the standard deviations of the returns of portfolios containing more than two assets, but
doing so is complicated and usually unnecessary. Most of the time, you can view a firm’s existing portfolio as one asset and a
proposed addition to the portfolio as the second asset. 3 175 Figure 75 The
Relationship between the
Number of Assets in a
Portfolio and the Riskiness
of the Portfolio
The graph shows that as
each new asset is added to
a portfolio, the diversification
effect causes the standard
deviation of the portfolio to
decrease. After 20 assets have
been added, however, the
effect of adding further assets is
slight. The remaining degree of
risk is nondiversifiable risk. Part II Essential Concepts in Finance Standard Deviation of Portfolio Expected Returns 176 12%
11%
10%
9%
8%
7%
6%
5% Diversifiable Risk 4%
3% Nondiversifiable Risk 2%
1%
0
0 5 10 15 20 25 30 35 Number of Assets in the Portfolio Nondiversifiable Risk Unless the returns of onehalf the assets in a portfolio are
perfectly negatively correlated with the other half—which is extremely unlikely—some
risk will remain after assets are combined into a portfolio. The degree of risk that remains
is nondiversifiable risk, the part of a portfolio’s total risk that can’t be eliminated by
diversifying.
Nondiversifiable risk is one of the characteristics of market risk because it is
produced by factors that are shared, to a greater or lesser degree, by most assets in the
market. These factors might include inflation and real gross domestic product changes.
Figure 75 illustrates nondiversifiable risk.
In Figure 75 we assumed that the portfolio begins with one asset with possible
returns having a probability distribution with a standard deviation of 10 percent. However,
if the portfolio is divided equally between two assets, each with possible returns having
a probability distribution with a standard deviation of 10 percent, and the correlation of
the returns of the two assets is, say +.25, then the standard deviation of the returns of
the portfolio drops to about 8 percent. If the portfolio is divided among greater numbers
of stocks, the standard deviation of the portfolio will continue to fall—as long as the
newly added stocks have returns that are less than perfectly positively correlated with
those of the existing portfolio.
Note in Figure 75, however, that after about 20 assets have been included in the
portfolio, adding more has little effect on the portfolio’s standard deviation. Almost all
the risk that can be eliminated by diversifying is gone. The remainder, about 5 percent
in this example, represents the portfolio’s nondiversifiable risk.
It is clear that the Financial Crisis was caused, at least in part, by the fact that
business leaders and portfolio managers underestimated the extent to which assets could
be correlated. With the chain reaction systematic risk described in the opener to this
chapter, many assets moved up and down (mostly down) with each other at the same Portfolio Returns Chapter 7
20%
19%
18%
17%
16%
15%
14%
13%
12%
11%
10%
9%
8%
7%
6%
5%
4%
3%
2%
1%
0 Risk and Return 177 Portfolio High Risk
Beta = 1.5
Portfolio Average Risk
Beta = Market = 1.0
Portfolio Low Risk
Beta = 0.5 1 2 3 4 Figure 76 Portfolio
Fluctuations and Beta 5 6 7 8 9 10 Periods time. The extent to which these assets moved together had not been seen like this since
The Great Depression. If you estimate that the correlation coefficient of returns of two
assets is .4 and it turns out to be .9, you will be underestimating the risk you are assuming.
Measuring Nondiversifiable Risk Nondiversifiable risk is measured by a term called
beta (β). The ultimate group of diversified assets, the market, has a beta of 1.0. The betas
of portfolios, and individual assets, relate their returns to those of the overall stock market.
Portfolios with betas higher than 1.0 are relatively more risky than the market. Portfolios
with betas less than 1.0 are relatively less risky than the market. (Riskfree portfolios
have a beta of zero.) The more the return of the portfolio in question fluctuates relative to
the return of the overall market, the higher the beta, as shown graphically in Figure 76.
Figure 76 shows that returns of the overall market fluctuated between about 8
percent and 12 percent during the 10 periods that were measured. By definition, the
market’s beta is 1.0. The returns of the averagerisk portfolio fluctuated exactly the
same amount, so the beta of the averagerisk portfolio is also 1.0. Returns of the lowrisk portfolio fluctuated between 6 percent and 8 percent, half as much as the market.
So the lowrisk portfolio’s beta is 0.5, only half that of the market. In contrast, returns
of the highrisk portfolio fluctuated between 10 percent and 16 percent, one and a half
times as much as the market. As a result, the highrisk portfolio’s beta is 1.5, half again
as high as the market.
Companies in lowrisk, stable industries like public utilities will typically have
low beta values because returns of their stock tend to be relatively stable. (When the
economy goes into a recession, people generally continue to turn on their lights and
use their refrigerators; and when the economy is booming, people do not splurge on
additional electricity consumption.) Recreational boat companies, on the other hand, The relative fluctuation in returns
for portfolios of different betas.
The higher the beta, the more
the portfolio’s returns fluctuate
relative to the overall market.
The market itself has a beta
of 1.0. 178 Part II Essential Concepts in Finance tend to have high beta values. That’s because demand for recreational boats is volatile.
(When times are tough, people postpone the purchase of recreational boats. During
good economic times, when people have extra cash in their pockets, sales of these
boats take off.) Dealing with Risk
Once companies determine the degree of risk present, what do they do about it? Suppose,
for example, a firm determined that if a particular project were adopted, the standard
deviation of possible returns of the firm’s portfolio of assets would double. So what?
How should a firm deal with the situation?
There are two broad classes of alternatives for dealing with risk. First, you might
take some action to reduce the degree of risk present in the situation. Second (if the
degree of risk can’t be reduced), you may compensate for the degree of risk you are
about to assume. We’ll discuss these two classes of alternatives in the following sections. RiskReduction Methods
One way companies can avoid risk is simply to avoid risky situations entirely. Most of
the time, however, refusing to get involved is an unsatisfactory business decision. Carried
to its logical conclusion, this would mean that everyone would invest in riskfree assets
only, and no products or services would be produced. Bill Gates, founder and CEO
of Microsoft, didn’t get rich by avoiding risks. To succeed, businesses must take risks.
If we assume that firms (and individuals) are willing to take some risk to achieve the
higher expected returns that accompany that risk, then the task is to reduce the degree
of risk as much as possible. The following three methods help to reduce risk: reducing
sales volatility and fixed costs, insurance, and diversification.
Reducing Sales Volatility and Fixed Costs Earlier in the chapter, we discussed how
sales volatility and fixed operating costs contribute to a firm’s business risk. Firms in
volatile industries whose sales fluctuate widely are exposed to a high degree of business
risk. That business risk is intensified even further if they have large amounts of fixed
operating costs. Reducing the volatility of sales, and the amount of fixed operating costs
a firm must pay, then, will reduce risk.
Reducing Sales Volatility If a firm could smooth out its sales over time, then the
fluctuation of its operating income (business risk) would also be reduced. Businesses
try to stabilize sales in many ways. For example, retail ski equipment stores sell tennis
equipment in the summer, summer vacation resorts offer winter specials, and movie
theaters offer reduced prices for early shows to encourage more patronage during slow
periods.
Insurance Insurance is a timehonored way to spread risk among many participants
and thus reduce the degree of risk borne by any one participant. Business firms insure
themselves against many risks, such as flood, fire, and liability. However, one important
risk—the risk that an investment might fail—is uninsurable. To reduce the risk of
losing everything in one investment, firms turn to another riskreduction technique,
diversification. Chapter 7 Diversification Review Figure 74 and the discussion following the figure. We showed
in that discussion how the standard deviation of returns of Asset A (2 percent) and
Asset B (4 percent) could be reduced to 1 percent by combining the two assets into
one portfolio. The diversification effect occurred because the returns of the two assets
were not perfectly positively correlated. Any time firms invest in ventures whose returns
are not perfectly positively correlated with the returns of their existing portfolios, they
will experience diversification benefits. Compensating for the Presence of Risk
In most cases it’s not possible to avoid risk completely. Some risk usually remains
even after firms use riskreduction techniques. When firms assume risk to achieve an
objective, they also take measures to receive compensation for assuming that risk. In
the sections that follow, we discuss these compensation measures.
Adjusting the Required Rate of Return Most owners and financial managers are
generally risk averse. So for a given expected rate of return, less risky investment projects
are more desirable than more risky investment projects. The higher the expected rate
of return, the more desirable the risky venture will appear. As we noted earlier in the
chapter, the riskreturn relationship is positive. That is, because of risk aversion, people
demand a higher rate of return for taking on a higherrisk project.
Although we know that the riskreturn relationship is positive, an especially difficult
question remains: How much return is appropriate for a given degree of risk? Say, for
example, that a firm has all assets invested in a chain of convenience stores that provides
a stable return on investment of about 6 percent a year. How much more return should
the firm require for investing some assets in a baseball team that may not provide steady
returns4—8 percent? 10 percent? 25 percent? Unfortunately, no one knows for sure, but
financial experts have researched the subject extensively.
One wellknown model used to calculate the required rate of return of an investment
is the capital asset pricing model (CAPM). We discuss CAPM next. Relating Return and Risk: The Capital Asset
Pricing Model
Financial theorists William F. Sharpe, John Lintner, and Jan Mossin worked on the
riskreturn relationship and developed the capital asset pricing model, or CAPM. We
can use this model to calculate the appropriate required rate of return for an investment
project given its degree of risk as measured by beta (β).5 The formula for CAPM is
presented in Equation 76.
CAPM Formula
kp = krf + (km – krf) × β (76) Some major league baseball teams lose money and others make a great deal. Television revenues differ greatly from team to
team, as do ticket sales and salary expenses. 4 See William Sharpe, “Capital Asset Prices: A Theory of Market Equilibrium,” Journal of Finance (September 1964); John
Lintner, “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets,”
Review of Economics and Statistics (February 1965); and Jan Mossin, “Equilibrium in a Capital Asset Market,” Econometrica
(October 1966). 5 179 Risk and Return Take Note
Diversification is a hotly
debated issue among
financial theorists.
Specifically, theorists
question whether a
firm provides value
to its stockholders if
it diversifies its asset
portfolio to stabilize the
firm’s income. Many
claim that individual
stockholders can achieve
diversification benefits
more easily and cheaply
than a firm, so firms
that diversify actually
do a disservice to their
stockholders. What do
you think? Take Note
In capital budgeting a
rate of return to reflect
risk is called a riskadjusted discount rate.
See Chapter 10. 180 Part II Essential Concepts in Finance Table 74 Using the CAPM to Calculate Required Rates of Return for
Investment Projects
Given:
The riskfree rate, krf = 4% The required rate of return on the market, km = 12% Project Low Risk’s beta = 0.5 Project Average Risk’s beta = 1.0
Project High Risk’s beta = 1.5
Required rates of return on the project’s per the CAPM:
Project Low Risk: kp = .04 + (.12  .04) × 0.5
= .04 + .04 = .08, or 8%
Project Average Risk: kp = .04 + (.12  .04) × 1.0
= .04 + .08 = .12, or 12%
Project High Risk: kp = .04 + (.12  .04) × 1.5
= .04 + .12 = .16, or 16%
Given that the riskfree rate of return is 4 percent and the required rate of return on the
market is 12 percent, the CAPM indicates the appropriate required rate of return for a
lowrisk investment project with a beta of .5 is 8 percent. The appropriate required rate of
return for an averagerisk project is the same as that for the market, 12 percent, and the
appropriate rate for a highrisk project with a beta of 1.5 is 16 percent. where: kp = the required rate of return appropriate for the investment project
krf = the riskfree rate of return
km = the required rate of return on the overall market
β = the project’s beta The three components of the CAPM include the riskfree rate of return (krf), the
market risk premium (km – krf), and the project’s beta (β). The riskfree rate of return
(krf) is the rate of return that investors demand from a project that contains no risk.
Riskaverse managers and owners will always demand at least this rate of return from
any investment project.
The required rate of return on the overall market minus the riskfree rate (km –
krf) represents the additional return demanded by investors for taking on the risk of
investing in the market itself. The term is sometimes called the market risk premium.
In the CAPM, the term for the market risk premium, (km – krf), can be viewed as the
additional return over the riskfree rate that investors demand from an “average stock”
or an “averagerisk” investment project. The S&P 500 stock market index is often used
as a proxy for the market.
As discussed earlier, a project’s beta (β) represents a project’s degree of risk relative
to the overall stock market. In the CAPM, when the beta term is multiplied by the market
risk premium term, (km – krf), the result is the additional return over the riskfree rate Chapter 7 Risk and Return 181 Required Rate of Return kp (the measure of reward) 20% Project High Risk
Beta = 1.5
Return = 16% 15%
Project Average Risk
Beta = 1.0
Return = 12% 10%
Project Low Risk
Beta = 0.5
Return = 8% Figure 77 CAPM and
the RiskReturn Relationship 5% 0%
0 0.5 1.0
Beta (the measure of risk) 1.5 This graph illustrates the
increasing return required for
increasing risk as indicated by
the CAPM beta. This graphical
depiction of the riskreturn
relationship according to the
2.0 CAPM is called the security
market line. that investors demand from that individual project. Beta is the relevant risk measure
according to the CAPM. Highrisk (highbeta) projects have high required rates of return,
and lowrisk (lowbeta) projects have low required rates of return.
Table 74 shows three examples of how the CAPM is used to determine the
appropriate required rate of return for projects of different degrees of risk.
As we can see in Table 74, Project High Risk, with its beta of 1.5, has a required rate
of return that is twice that of Project Low Risk, with its beta of 0.5. After all, shouldn’t
we ask for a higher rate of return if the risk is higher? Note also that Project Average
Risk, which has the same beta as the market, 1.0, also has the same required rate of
return as the market (12 percent). The riskreturn relationship for these three projects
is shown in Figure 77.
Remember that the beta term in the CAPM reflects only the nondiversifiable risk
of an asset, not its diversifiable risk. Diversifiable risk is irrelevant because the diversity
of each investor’s portfolio essentially eliminates (or should eliminate) that risk. (After
all, most investors are well diversified. They will not demand extra return for adding a
security to their portfolios that contains diversifiable risk.) The return that welldiversified
investors demand when they buy a security, as measured by the CAPM and beta, relates
to the degree of nondiversifiable risk in the security. What’s Next
In this chapter we examined the riskreturn relationship, types of risk, risk measurements,
riskreduction techniques, and the CAPM. In the next chapter, we will discuss the time
value of money. 182 Part II Essential Concepts in Finance Summary
1. Define risk, risk aversion, and the riskreturn relationship.
In everything you do, or don’t do, there is a chance that something will happen that you
didn’t count on. Risk is the potential for unexpected events to occur.
Given two financial alternatives that are equal except for their degree of risk, most
people will choose the less risky alternative because they are risk averse. Risk aversion
is a common trait among almost all investors. Most investors avoid risk if they can,
unless they are compensated for accepting risk. In an investment context, the additional
compensation is a higher expected rate of return.
The riskreturn relationship refers to the positive relationship between risk and
the required rate of return. Due to risk aversion, the higher the risk, the more return
investors expect.
2. Measure risk using the standard deviation and the coefficient of variation.
Risk is the chance, or probability, that outcomes other than what is expected will occur.
This probability is reflected in the narrowness or width of the distribution of the possible
values of the financial variable. In a distribution of variable values, the standard deviation
is a number that indicates how widely dispersed the possible values are around the
expected value. The more widely dispersed a distribution is, the larger the standard
deviation, and the greater the probability that an actual value will be different than the
expected value. The standard deviation, then, can be used to measure the likelihood that
some outcome substantially different than what is expected will occur.
When the degrees of risk in distributions of different sizes are compared, the
coefficient of variation is a statistic used to measure relative riskiness. The coefficient
of variation measures the standard deviation’s percentage of the expected value. It
relates the standard deviation to its mean to give a risk measure that is independent of
the magnitude of the possible returns.
3. Identify the types of risk that business firms encounter.
Business risk is the risk that a company’s operating income will differ from what is
expected. The more volatile a company’s operating income, the more business risk the
firm contains. Business risk is a result of sales volatility, which translates into operating
income volatility. Business risk is increased by the presence of fixed costs, which magnify
the effect on operating income of changes in sales.
Financial risk occurs when companies borrow money and incur interest charges
that show up as fixed expenses on their income statements. Fixed interest charges act on
a firm’s net income the same way fixed operating expenses act on operating income—
they increase volatility. The additional volatility of a firm’s net income caused by the
presence of fixed interest expense is called financial risk.
Portfolio risk is the chance that investors won’t get the return they expect from a
portfolio. Portfolio risk can be measured by the standard deviation of possible returns
of a portfolio. It is affected by the correlation of returns of the assets making up the
portfolio. The less correlated these returns are, the more gains on some assets offset losses
on others, resulting in a reduction of the portfolio’s risk. This phenomenon is known as
the diversification effect. Nondiversifiable risk is risk that remains in a portfolio after Chapter 7 Risk and Return all diversification benefits have been achieved. Nondiversifiable risk is measured by a
term called beta (β). The market has a beta of 1.0. Portfolios with betas greater than
1.0 contain more nondiversifiable risk than the market, and portfolios with betas less
than 1.0 contain less nondiversifiable risk than the market.
4. Explain methods of risk reduction.
Firms can reduce the degree of risk by taking steps to reduce the volatility of sales or
their fixed costs. Firms also obtain insurance policies to protect against many risks, and
they diversify their asset portfolios to reduce the risk of income loss.
5. Describe how firms compensate for assuming risk.
Firms almost always demand a higher rate of return to compensate for assuming risk.
The more risky a project, the higher the return firms demand.
6. Explain how the capital asset pricing model (CAPM) relates risk and return.
When investors adjust their required rates of return to compensate for risk, the question
arises as to how much return is appropriate for a given degree of risk. The capital asset
pricing model (CAPM) is a model that measures the required rate of return for an
investment or project, given its degree of nondiversifiable risk as measured by beta (β). Equations Introduced in This Chapter
Equation 71. The Expected Value, or Mean (µ), of a Probability Distribution:
µ = ∑ (V × P) where: µ = the expected value, or mean
∑ = the sum of
V = the possible value for some variable
P = the probability of the value V occurring Equation 72. The Standard Deviation:
σ= where: ∑ P(V − µ ) 2 σ = the standard deviation ∑ = the sum of
P = the probability of the value V occurring
V = the possible value for a variable
µ = the expected value 183 184 Part II Essential Concepts in Finance Equation 73. The Coefficient of Variation of a Probability Distribution:
CV = Equation 74. Standard Deviation
Mean The Expected Rate of Return, E(Rp), of a portfolio comprised of
Two Assets, A and B: E(Rp) = (wa × E(Ra)) + (wb × E(Rb)) where: E(Rp) = the expected rate of return of the portfolio composed of Asset A
and Asset B
wa = the weight of Asset A in the portfolio
E(Ra) = the expected rate of return of Asset A
wb = the weight of Asset B in the portfolio
E(Rb) = the expected rate of return of Asset B Equation 75. The Standard Deviation of a TwoAsset Portfolio:
σp = where: 2 2 2 2 w a σ a + w bσ b + 2w a w b ra,bσ aσ b σp = the standard deviation of the returns of the combined portfolio
containing Asset A and Asset B
wa = the weight of Asset A in the twoasset portfolio
σa = the standard deviation of the returns of Asset A
wb = the weight of Asset B in the twoasset portfolio
σb = the standard deviation of the returns of Asset B
ra,b = the correlation coefficient of the returns of Asset A and Asset B Equation 76. The Capital Asset Pricing Model (CAPM):
kp = krf + (km – krf) × β where: kp = the required rate of return appropriate for the investment
project
krf = the riskfree rate of return
km = the required rate of return on the overall market
β = the project’s beta Chapter 7 Risk and Return 185 SelfTest
ST1. For Bryan Corporation, the mean of the
distribution of next year’s possible sales
is $5 million. The standard deviation of
this distribution is $400,000. Calculate the
coefficient of variation (CV) for this distribution
of possible sales. ST2. Investors in Hoeven Industries common stock
have a .2 probability of earning a return of 4
percent, a .6 probability of earning a return
of 10 percent, and a .2 probability of earning
a return of 20 percent. What is the mean of
this probability distribution (the expected rate
of return)? ST3. The standard deviation of the possible returns of
Boris Company common stock is .08, whereas
the standard deviation of possible returns
of Natasha Company common stock is .12.
Calculate the standard deviation of a portfolio
comprised of 40 percent Boris Company stock
and 60 percent Natasha Company stock. The
correlation coefficient of the returns of Boris
Company stock relative to the returns of
Natasha Company stock is –.2. The mean of the normal probability
distribution of possible returns of Gidney and
Cloyd Corporation common stock is 18 percent.
The standard deviation is 3 percent. What is the
range of possible values that you would be 95
percent sure would capture the return that will
actually be earned on this stock? ST6. Dobie’s Bagle Corporation common stock
has a beta of 1.2. The market risk premium
is 6 percent and the riskfree rate is 4 percent.
What is the required rate of return on this stock
according to the CAPM? ST7. Using the information provided in ST6, what
is the required rate of return on the common
stock of Zack’s Salt Corporation? This stock
has a beta of .4. ST8. A portfolio of three stocks has an expected
value of 14 percent. Stock A has an expected
return of 6 percent and a weight of .25 in the
portfolio. Stock B has an expected return of
10 percent and a weight of .5 in the portfolio.
Stock C is the third stock in this portfolio.
What is the expected rate of return of Stock C? What is the standard deviation for the Hoeven
Industries common stock return probability
distribution described in ST2? ST4. ST5. 186 Part II Essential Concepts in Finance Review Questions
1. What is risk aversion? If common stockholders are
risk averse, how do you explain the fact that they
often invest in very risky companies?
2. Explain the riskreturn relationship.
3. Why is the coefficient of variation often a better
risk measure when comparing different projects
than the standard deviation?
4. What is the difference between business risk and
financial risk?
5. Why does the riskiness of portfolios have to
be looked at differently than the riskiness of
individual assets?
6. What happens to the riskiness of a portfolio if
assets with very low correlations (even negative
correlations) are combined? 7. What does it mean when we say that the
correlation coefficient for two variables is +1?
What does it mean if this value is zero? What does
it mean if it is +1?
8. What is nondiversifiable risk? How is it measured?
9. Compare diversifiable and nondiversifiable risk.
Which do you think is more important to financial
managers in business firms?
10. How do riskaverse investors compensate for risk
when they take on investment projects?
11. Given that riskaverse investors demand more
return for taking on more risk when they invest,
how much more return is appropriate for, say, a
share of common stock, than for a Treasury bill?
12. Discuss risk from the perspective of the capital
asset pricing model (CAPM). Build Your Communication Skills
CS1. Go to the library, use business magazines,
computer databases, the Internet, or other
sources that have financial information about
businesses. (See Chapter 5 for a list of specific
resources.) Find three companies and compare
their approaches to risk. Do the firms take
a conservative or an aggressive approach?
Write a one to twopage report, citing specific
evidence of the risktaking approach of each
of the three companies you researched. CS2. Research three to five specific mutual funds.
Then form small groups of four to six. Discuss
whether the mutual funds each group member
researched will help investors diversify the
risk of their portfolio. Are some mutual funds
better than others for an investor seeking
good diversification? Prepare a list of mutual
funds the group would select to diversify its
risk and explain your choices. Present your
recommendations to the class. Chapter 7 187 Risk and Return Problems
71. Manager Paul Smith believes an investment project will have the following
yearly cash flows with the associated probabilities throughout its life of five
years. Calculate the standard deviation and coefficient of variation of the cash
flows.
Probability
of Occurrence Cash Flows($)
$10,000 .05 13,000 .10 16,000 .20 19,000 .30 22,000 .20 25,000 .10 28,000 .05 72. MilkU, an agricultural consulting firm, has developed the following income
statement forecast:
MilkU Income Forecast (in 000’s)
2%
Sales Probability of Occurrence
8%
80%
8% 2% $500 $700 $1,200 $1,700 $1,900 Variable Expenses 250 350 600 850 950 Fixed Operating Expenses 250 250 250 250 250 0 100 350 600 700 Operating Income a.
b.
c.
d. Calculate the expected value of MilkU’s operating income.
Calculate the standard deviation of MilkU’s operating income.
Calculate the coefficient of variation of MilkU’s operating income.
Recalculate the expected value, standard deviation, and coefficient of
variation of MilkU’s operating income if the company’s sales forecast
changed as follows:
10% Sales Standard
Deviation and
Coefficient of
Variation Probability of Occurrence
15%
15%
50% $500 $700 $1,200 $1,700 10%
$1,900 e. Comment on how MilkU’s degree of business risk changed as a result of the
new sales forecast in part d. Measuring Risk 188 Part II Essential Concepts in Finance
Standard
Deviation
and Mean 73. The following data apply to Henshaw Corp. Calculate the mean and the
standard deviation, using the following table.
Probability
of Occurrence Possible Sales $ 1,000
$ 5,000
$ 10,000
$ 15,000
$ 20,000
Measuring Risk .10
.20
.45
.15
.10
∑ = 1.00 74. As a new loan officer in the Bulwark Bank, you are comparing the financial
riskiness of two firms. Selected information from pro forma statements for each
firm follows.
Equity Eddie’s Company Net Income Forecast (in 000’s)
Probability of Occurrence
70%
10% 5% 5%
Operating Income 10% $100 $200 $400 $600 0 0 0 0 0 100 200 400 600 700 Interest Expense
BeforeTax Income $700 Taxes (28%) 28 56 112 168 196 Net Income 72 144 288 432 504 Barry Borrower’s Company Net Income Forecast (in 000’s)
5%
Operating Income $110.0 Probability of Occurrence
70%
10%
10%
$220.0 $440.0 $660.0 5%
$770.0 Interest Expense 40 40 40 40 40 BeforeTax Income 70 180 400 620 730 Taxes (28%) 19.6 50.4 112 173.6 204.4 Net Income 50.4 129.6 288 446.4 525.6 a. Calculate the expected values of Equity Eddie’s and Barry Borrower’s net
incomes.
b. Calculate the standard deviations of Equity Eddie’s and Barry Borrower’s net
incomes.
c. Calculate the coefficients of variation of Equity Eddie’s and Barry
Borrower’s net incomes.
d. Compare Equity Eddie’s and Barry Borrower’s degrees of financial risk. Chapter 7 189 Risk and Return 75. George Taylor, owner of a toy manufacturing company, is considering the
addition of a new product line. Marketing research shows that gorilla action
figures will be the next fad for the six to tenyearold age group. This new
product line of gorillalike action figures and their hightech vehicles will
be called GoRilla. George estimates that the most likely yearly incremental
cash flow will be $26,000. There is some uncertainty about this value because
George’s company has never before made a product similar to the GoRilla. He
has estimated the potential cash flows for the new product line along with their
associated probabilities of occurrence. His estimates follow. Standard
Deviation and
Coefficient of
Variation GoRilla Project
Cash Flows Probability
of Occurrence $20,000 1% $22,000 12% $24,000 23% $26,000 28% $28,000 23% $30,000 12% $32,000 1% a. Calculate the standard deviation of the estimated cash flows.
b. Calculate the coefficient of variation.
c. If George’s other product lines have an average coefficient of variation of 12
percent, what can you say about the risk of the GoRilla Project relative to
the average risk of the other product lines?
76. Assume that a company has an existing portfolio A with an expected return of
9 percent and a standard deviation of 3 percent. The company is considering
adding an asset B to its portfolio. Asset B’s expected return is 12 percent with
a standard deviation of 4 percent. Also assume that the amount invested in A
is $700,000 and the amount to be invested in B is $200,000. If the degree of
correlation between returns from portfolio A and project B is zero, calculate:
a. The standard deviation of the new combined portfolio and compare it with
that of the existing portfolio.
b. The coefficient of variation of the new combined portfolio and compare it
with that of the existing portfolio. Portfolio Risk 77. Zazzle Company has a standard deviation of 288 and a mean of 1,200. What is
its coefficient of variation (CV)? Coefficient of
Variation 78. What is the expected rate of return on a portfolio that has $4,000 invested in
Stock A and $6,000 invested in Stock B? The expected rates of return on these
two stocks are 13 percent and 9 percent, respectively. Expected Rate
of Return 190 Part II Essential Concepts in Finance
Standard
Deviation 79. A twostock portfolio has 30 percent in Stock A, with an expected return of 21
percent and a standard deviation of 5 percent, and the remainder in Stock B,
with an 18 percent expected return and a standard deviation of 2 percent. The
correlation coefficient is 0.6. Determine the standard deviation for this portfolio. Measuring Risk 710. A firm has an existing portfolio of projects with an expected return of 11
percent a year. The standard deviation of these returns is 4 percent. The existing
portfolio’s value is $820,000. As financial manager, you are considering the
addition of a new project, PROJ1. PROJ1’s expected return is 13 percent with
a standard deviation of 5 percent. The initial cash outlay for PROJ1 is expected
to be $194,000.
a. Calculate the coefficient of variation for the existing portfolio.
b. Calculate the coefficient of variation for PROJ1.
c. If PROJ1 is added to the existing portfolio, calculate the weight (proportion)
of the existing portfolio in the combined portfolio.
d. Calculate the weight (proportion) of PROJ1 in the combined portfolio.
e. Assume the correlation coefficient of the cash flows of the existing portfolio
and PROJ1 is zero. Calculate the standard deviation of the combined
portfolio. Is the standard deviation of the combined portfolio higher or lower
than the standard deviation of the existing portfolio?
f. Calculate the coefficient of variation of the combined portfolio.
g. If PROJ1 is added to the existing portfolio, will the firm’s risk increase or
decrease? CAPM 711. Assume the riskfree rate is 5 percent, the expected rate of return on the market
is 15 percent, and the beta of your firm is 1.2. Given these conditions, what is
the required rate of return on your company’s stock per the capital asset pricing
model? CAPM 712. Calculate the expected rates of return for the low, average, and highrisk
stocks:
a. Riskfree rate = 4.5 percent
b. Market risk premium = 12.5 percent
c. Lowrisk beta = .5
d. Averagerisk beta = 1.0
e. Highrisk beta= 1.6 Challenge Problem 713. Your firm has a beta of 1.5 and you are considering an investment project
with a beta of 0.8. Answer the following questions, assuming that shortterm
Treasury bills are currently yielding 5 percent and the expected return on the
market is 15 percent. Chapter 7 191 Risk and Return a. What is the appropriate required rate of return for your company per the
capital asset pricing model?
b. What is the appropriate required rate of return for the investment project per
the capital asset pricing model?
c. If your firm invests 20 percent of its assets in the new investment project,
what will be the beta of your firm after the project is adopted? (Hint:
Compute the weighted average beta of the firm with the new asset, using
Equation 74.)
The following problems (714 to 718) relate to the expected business of Power Software
Company (PSC) (000’s of dollars):
Power Software Company Forecasts
Probability of Occurrence
20%
40% 2%
$800 Sales 8% $1,000 $1,400 $2,000 20% 8% 2% $2,600 $3,000 $3,200 714. Calculate the expected value, standard deviation, and coefficient of variation of
sales revenue of PSC. Measuring Risk 715. Assume that PSC has no fixed expense but has a variable expense that is 60
percent of sales as follows: Business Risk Power Software Company Forecasts
Probability of Occurrence
40%
20% 2%
Variable Expenses 20% 8% 2% $800 Sales 8%
$1,000 $1,400 $2,000
$2,600 $3,000 $3,200 480 600 840 1,200 1,560 1,800 1,920 Calculate PSC’s business risk (coefficient of variation of operating income).
716. Now assume that PSC has a fixed operating expense of $400,000, in addition to
the variable expense of 60 percent of sales, shown as follows: Business Risk Power Software Company Forecasts
2%
Sales 8% 20% Probability of Occurrence
40%
20% 8% 2% $800 $1,000 $1,400 $2,000 $2,600 $3,000 $3,200 Variable Expenses 480 600 840 1,200 1,560 1,800 1,920 Fixed Expenses 400 400 400 400 400 400 400 Recalculate PSC’s business risk (coefficient of variation of operating income). How does this figure compare
with the business risk calculated with variable cost only? 192 Part II Essential Concepts in Finance Various Statistics
and Financial Risk 717. Assume that PSC has a fixed interest expense of $60,000 on borrowed funds.
Also assume that the applicable tax rate is 30 percent. What are the expected
value, standard deviation, and coefficient of variation of PSC’s net income?
What is PSC’s financial risk? Business and
Financial Risk 718. To reduce the various risks, PSC is planning to take suitable steps to reduce
volatility of operating and net income. It has projected that fixed expenses and
interest expenses can be reduced. The revised figures follow:
Power Software Company Forecasts
Probability of Occurrence
60%
13% 1% 6% 13% 6% 1% $800 $1,000 $1,400 $2,000 $2,600 $3,000 $3,200 Variable Expenses 480 600 840 1,200 1,560 1,800 1,92 Fixed Expenses 250 250 250 250 250 250 250 Interest Expense 40 40 40 40 40 40 40 Sales Recalculate PSC’s business and financial risks and compare these figures with
those calculated in problems 716 and 717. The tax rate is 30 percent. Answers to SelfTest
ST1. CV = σ ÷ µ = $400,000 ÷ $5,000,000 =
.08 = 8% ST2. µ = (.2 × .04) + (.6 × .10) + (.2 × .20) = .108 = 10.8% ST3. σ = ([.2 × (.04 – .108)2] + [.6 × (.10 – .108)2] + [.2 × (.20 – .108)2]).5
= [(.2 × .004624) + (.6 × .000064) + (.2 × .008464)].5
= .002656.5 = .0515 = 5.15% ST4. [
σp = (.42 × .082) + (.62 + .122) + (2 × .4 × .6 × (–.2) × .08 × .12)].5
= [(.16 × .0064) + (.36 × .0144) + (–.0009216)].5
= .0052864.5 = .0727 = 7.27% ST5. .18 + (2 × .03) = .24 = 24%
.18 – (2 × .03) = .12 = 12% Chapter 7 Risk and Return Therefore, we are 95% confident that the actual return will be between 12%
and 24%.
ST6. ks = .04 + (.06 × 1.2) = .112 = 11.2% ST7. ks = .04 + (.06 × .4) = .064 = 6.4% ST8. WT of Stock C must be .25 for the total of the weights to equal 1:
.14 = (.06 × .25) + (.10 × .5) + [E(RC) × .25]
.14 = .065 + [E(RC) × .25]
.075 = E(RC) × .25
E(RC) = .30 = 30% 193 ...
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This note was uploaded on 11/17/2011 for the course BUS 330 taught by Professor Nugent during the Fall '08 term at SUNY Stony Brook.
 Fall '08
 Nugent

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