<<

. 11
( 21)



>>

72 can be used to determine the interest rate required to double the money in
an account in a given number of years. To illustrate, an interest rate of 72 / 5
= 14.4 percent is required to double the value of an account in ¬ve years. The
calculator solution in this case is 14.9 percent, so the Rule of 72 again gives a
reasonable approximation of the precise answer.
7. Some ¬nancial institutions even pay interest on accounts that is compounded
continuously. However, continuous compounding is not relevant to healthcare
¬nance, so it will not be discussed here.
8. Most ¬nancial calculators are programmed to calculate the EAR or, given the
EAR, to ¬nd the stated rate. This is called interest rate conversion. Enter the
stated rate and the number of compounding periods per year, and then press the
EFF percent key to ¬nd the EAR.
9. The annual percentage rate (APR) and annual percentage yield (APY) are
terms de¬ned in Truth in Lending and Truth in Savings laws. APR is de¬ned
as Periodic rate — Number of compounding periods per year, so it ignores the
consequences of compounding. Although the APR on a credit card with interest
charges of 1.5 percent per month is 1.5% — 12 = 18.0%, the true effective annual
rate is 19.6 percent.


References
The owner™s manual for your calculator.
The reference manual for your spreadsheet software, or any of the after-market spread-
sheet manuals.
CHAP TER



10
FINANCIAL RISK AND
REQUIRED RETURN

Learning Objectives
After studying this chapter, readers will be able to:

• Explain in general terms the concept of ¬nancial risk.
• De¬ne and differentiate between stand-alone risk and portfolio risk.
• De¬ne and differentiate between corporate risk and market risk.
• Explain the CAPM relationship between risk and required rate of
return.
• Use the CAPM to determine required returns.


Introduction
Two of the most important concepts in healthcare ¬nancial management are
¬nancial risk and required return. What is ¬nancial risk, how is it measured,
and what effect, if any, does it have on required return and hence on manage-
rial decisions? Because so much of ¬nancial decision making involves risk and
return, it is impossible to gain a good understanding of healthcare ¬nancial
management without having a solid appreciation of risk and return concepts.
If investors”both individuals and businesses”viewed risk as a benign
fact of life, it would have little impact on decision making. However, decision
makers for the most part believe that if a risk must be taken, there must be
a reward for doing so. Thus, an investment of higher risk, whether it be an
individual investor™s security investment or a radiology group™s investment
in diagnostic equipment, must offer a higher return to make it ¬nancially
attractive.
In this chapter, basic risk concepts are presented from the perspective
of both individual investors and businesses. Health services managers must be
familiar with both contexts because investors supply the capital that businesses
need to function. In addition, the chapter discusses the relationship between
risk and required rate of return. To be truly useful in ¬nancial decision making,
it is necessary to know the impact of risk on investors™ views of investment
acceptability.


287
288 Healthcare Finance



The Many Faces of Financial Risk
Unfortunately, a full discussion of ¬nancial risk would take many chapters,
perhaps even an entire book, because ¬nancial risk is a very complicated sub-
ject. First of all, the nature of ¬nancial risk depends on whether the investor
is an individual or a business. Then, if the investor is an individual, it depends
on the investment horizon, or the amount of time until the investment pro-
ceeds are needed. To make the situation even more complex, it may even be
dif¬cult to de¬ne, measure, or translate ¬nancial risk into something usable
for decision making. For example, the risk that individual investors face when
saving for retirement is the risk that the amount of funds accumulated will not
be suf¬cient to fund the lifestyle expected during the full term of retirement.
Needless to say, translating such a de¬nition of risk into investment goals is not
easy. The good news is that our primary interest here concerns the ¬nancial
risk inherent in businesses. Thus, our discussion can focus on the fundamental
factors that in¬‚uence the riskiness of real-asset investments and the securities
that businesses sell to raise the capital needed to make the invesments.
Still, two factors come into play that complicate our discussion of ¬-
nancial risk. The ¬rst complicating factor is that ¬nancial risk is seen both by
businesses and the investors in businesses. There is some risk inherent in the
business itself that depends on the type of business. For example, pharma-
ceutical ¬rms are generally acknowledged to face a great deal of risk, while
healthcare providers typically have less risk. Then, investors (i.e., stockholders
and creditors) bear the riskiness inherent in the business, but as modi¬ed by
the nature of the securities they hold. For example, the stock of Beverly
Enterprises is more risky than its debt, although the risk of both securities
depends on the inherent risk of a business that operates in the long-term care
industry. The risk differential arises because of contractual differences between
equity and debt: Debtholders have a ¬xed claim against the cash ¬‚ows and
assets of the business, while stockholders have a residual claim, or a claim to
what is left after all other claimants have been paid.
The second complicating factor results from the fact that the riskiness
of an investment depends on the context in which it is held. For example, a
stock held alone is riskier than the same stock held as part of a large portfolio
of stocks. Similarly, a magnetic resonance imaging (MRI) system operated
independently is riskier than the same system operated as part of a large,
geographically diversi¬ed business that owns and operates numerous types
of diagnostic equipment.



Self-Test 1. What are the two complications that arise when dealing with ¬nancial
Question risk in a business setting?
289
Chapter 10: Financial Risk and Required Return



Introduction to Financial Risk
Generically, risk is de¬ned as “a hazard; a peril; exposure to loss or injury.”
Thus, risk refers to the chance that an unfavorable event will occur. If a
person engages in skydiving, he or she is taking a chance with injury or death;
skydiving is risky. If a person gambles at roulette, he or she is not risking injury
or death but is taking a ¬nancial risk. Even when a person invests in stocks or
bonds, he or she is taking a risk in the hope of earning a positive rate of return.
Similarly, when a healthcare business invests in new assets such as diagnostic
equipment or new hospital beds or a new managed care plan, it is taking a
¬nancial risk.
To illustrate ¬nancial risk, consider two potential personal investments.
The ¬rst investment consists of a one-year, $1,000 face value U.S. Treasury
bill that is bought for $950. Treasury bills are short-term federal debt that are
sold at a discount (i.e., less than face value) and return face, or par, value at
maturity. The investor expects to receive $1,000 at maturity in one year, so
the anticipated rate of return on the T-bill investment is ($1,000 ’ $950) /
$950 = $50 / $950 = 0.053, or 5.3%. Using a ¬nancial calculator:

’950
Inputs 1 1000



= 5.3
Output

The $1,000 payment is ¬xed by contract (the T-bill promises to pay this
amount), and the U.S. government is certain to make the payment, except for
national disaster”a very unlikely event. Thus, there is virtually a 100 percent
probability that the investment will actually earn the 5.3 percent rate of return
that is expected. In this situation, the investment is de¬ned as being riskless,
or risk free.
Now, assume that the $950 is invested in a biotechnology partnership
that will be terminated in one year. If the partnership develops a new commer-
cially valuable product, its rights will be sold and $2,000 will be received from
the partnership for a rate of return of ($2,000 ’ $950) / $950 = $1,050 /
$950 = 1.1053 = 110.53%:
’950
Inputs 1 2000



= 110.53
Output

But if nothing worthwhile is developed, the partnership would be worthless,
no money would be received, and the rate of return would be ($0 ’ $950) /
$950 = ’1.00 = ’100%:
290 Healthcare Finance


’950
Inputs 1 0



= ’100.00
Output

(Most ¬nancial calculators give no solution when the future value is zero, but
if a very small number, for example, 0.0001, is entered for the future value,
the solution for interest rate is ’100.00.)
Now, assume that there is a 50 percent chance that a valuable product
will be developed. In this admittedly unrealistic situation, the expected rate
of return, a statistical concept that will be discussed shortly, is the same 5.3
percent as on the T-bill investment: (0.50 — 110.53%) + (0.50 — [-100%])
= 5.3%. However, the biotechnology partnership is a far cry from being risk-
less. If things go poorly, the realized rate of return will be ’100 percent,
which means that the entire $950 investment will be lost. Because there is a
signi¬cant chance of actually earning a return that is far less than expected,
the partnership investment is described as being very risky.
Thus, ¬nancial risk is related to the probability of earning a return
less than expected. The greater the chance of earning a return far below that
expected, the greater the amount of ¬nancial risk.1


Self-Test 1. What is a generic de¬nition of risk?
Questions 2. Explain the general concept of ¬nancial risk.


Risk Aversion
Why is it so important to de¬ne and measure ¬nancial risk? The reason is that,
for the most part, both individual and business investors dislike risk. Suppose
that a person was given the choice between a sure $1 million and the ¬‚ip
of a coin for either zero or $2 million. In the statistical sense, the expected
dollar return on the coin ¬‚ip is $1 million, the same amount as the sure thing.
Thus, from a statistical standpoint, the return on both choices is the same.
However, just about everyone confronted with this choice would take the
sure $1 million. A person that takes the sure thing is said to be risk averse; a
person who is indifferent between the two alternatives is risk neutral ; and an
individual who prefers the gamble to the sure thing is a risk seeker.
Of course, people and businesses do gamble and take other ¬nancial
chances, so all of us at some time typically exhibit risk-seeking behavior.
However, most individuals would never put a sizable portion of their wealth
at risk, and most health services managers would never “bet the business.”
Most people are risk averse when it really matters.
What are the implications of risk aversion for ¬nancial decision making?
First, given two investments with similar returns but different risk, investors
291
Chapter 10: Financial Risk and Required Return



will favor the lower-risk alternative. Second, investors will require higher re-
turns on higher-risk investments. These behavioral outcomes of risk aversion
have a signi¬cant impact on many facets of ¬nancial decision making and hence
will appear over and over in this book.


Self-Test
1. What does the term risk aversion mean?
Questions
2. What are the implications of risk aversion for ¬nancial decision making?


Probability Distributions
The chance that an event will occur is called probability of occurrence, or just
probability; for example, when rolling a single die, the probability of rolling a
two is one out of six, or 1 / 6 = 0.1667 = 16.67%. If all possible outcomes
related to a particular event are listed, and a probability is assigned to each
outcome, the result is a probability distribution. In the example of the role of
a die, the probability distribution looks like this:

Outcome Probability
0.1667 = 16.67%
1
0.1667 = 16.67%
2
0.1667 = 16.67%
3
0.1667 = 16.67%
4
0.1667 = 16.67%
5
0.1667 = 16.67%
6
1.0000 = 100.00%

The possible outcomes (i.e., the number of dots showing after the die roll)
are listed in the left column, while the probability of each outcome is listed
as both decimals and percentages in the right column. For a complete proba-
bility distribution, which must include all possible outcomes for an event, the
probabilities must sum to 1.0, or 100 percent.
Probabilities can also be assigned to possible outcomes”in this case,
returns”on both personal and business investments. If a person buys stock,
the return will usually come in the form of dividends and capital gains ( selling
the stock for more than the person paid for it) or losses ( selling the stock for
less the person paid for it). Because all stock returns are uncertain, there is
some chance that the dividends will not be as high as expected and that the
stock price will not increase as much as expected or that it will even decrease.
The higher the probabilities of dividends and stock price well below those
expected, the higher the probability that the return will be signi¬cantly less
than expected and hence the greater the risk.
To illustrate the concept using a business investment, consider a hos-
pital evaluating the purchase of a new MRI system. The cost of the system
is an investment, and the net cash in¬‚ows that stem from patient utilization
292 Healthcare Finance



provide the return. The net cash in¬‚ows, in turn, depend on the number of
procedures, charge per procedure, payer discounts, operating costs, and so on.
These values typically are not known with certainty but depend on factors such
as patient demographics, physician acceptance, local market conditions, labor
costs, and so on. Thus, the hospital actually faces a probability distribution
of returns rather than a single return known with certainty. The greater the
probability of returns well below the return anticipated, the greater the risk
of the MRI investment.


Self-Test 1. What is a probability distribution?
Questions 2. How are probability distributions used in ¬nancial decision making?


Expected and Realized Rates of Return
To be most useful, the concept of ¬nancial risk must be de¬ned more precisely
than just the chances of a return well below that anticipated. Table 10.1
contains the estimated return distributions developed by the ¬nancial staff of
Norwalk Community Hospital for two proposed projects: an MRI system and
a walk-in clinic. Here, each economic state re¬‚ects a combination of factors
that dictate each project™s pro¬tability. For example, for the MRI project, the
very poor economic state signi¬es a very competitive market and hence very
low utilization, very high discounts on reimbursements, very high operating
costs, and so on. Conversely, the very good economic state assumes very
high utilization and reimbursement, very low operating costs, and so on. The
economic states are de¬ned in a similar fashion for the walk-in clinic.
The expected rate of return, de¬ned in the statistical sense, is the weight-
ed average of the return distribution, where the weights are the probabilities
of occurrence. For example, the expected rate of return on the MRI system,
E(RMRI), is 10 percent:

E(RMRI ) = Probability of Return 1 — Return 1
+ Probability of Return 2 — Return 2
+ Probability of Return 3 — Return 3 and so on
= (0.10 — [’10%]) + (0.20 — 0%) + (0.40 — 10%)
+ (0.20 — 20%) + (0.10 — 30%)
= 10.0%.
Calculated in a similar manner, the expected rate of return on the walk-in
clinic is 15 percent.
The expected rate of return is the average return that would result,
given the return distribution, if the investment were randomly repeated many
times. In this illustration, if 1,000 clinics were built in different areas, each of
293
Chapter 10: Financial Risk and Required Return


TABLE 10.1
Rate of Return if State Occurs
Probability Norwalk
Economic State of Occurrence MRI Clinic Community
Hospital:
’10% ’20% Estimated
Very poor 0.10
Poor 0.20 0 0 Returns for Two
Average 0.40 10 15
Proposed
Good 0.20 20 30
Projects
Very good 0.10 30 50
1.00



which faced the return distribution given in Table 10.1, the average return on
the 1,000 investments would be 15 percent, assuming the returns in each area
are independent of one another (random). However, only one clinic would
actually be built, and the realized rate of return may be less than the expected
15 percent. Therefore, the clinic investment (as well as the MRI investment)
is risky.
Expected rate of return expresses expectations for the future. When
the managers at Norwalk Community Hospital analyzed the MRI investment,
they expected it to earn 10 percent. However, assume that economic condi-
tions take a turn for the worse and the very poor economic scenario actually
occurs. In this case, the realized rate of return, which is the rate of return
that the investment actually produced as measured at termination, would be a
negative 10 percent. It is the potential of realizing a minus 10 percent return
on an investment that has an expected return of plus 10 percent that produces
risk.
Note that in many situations, especially those arising in classroom illus-
trations, the expected rate of return is not even achievable. For example, an
investment that has a 50 percent chance of a 5 percent return and a 50 percent
chance of a 15 percent return has an expected rate of return of 10 percent. Yet,
assuming the given distribution truly re¬‚ects the complete return potential of
the investment, there is zero probability of actually realizing the 10 percent
expected rate of return.


Self-Test
1. How is the expected rate of return calculated?
Questions
2. What is the economic interpretation of the expected rate of return?
3. What is the difference between the expected rate of return and the
realized rate of return?


Stand-Alone Risk
We can look at the two distributions in Table 10.1 and intuitively conclude
that the clinic is more risky than the MRI system because the clinic has a chance
294 Healthcare Finance



of a 20 percent loss, while the worst possible loss on the MRI system is 10
percent. This intuitive risk assessment is based on the stand-alone risk of the
two investments; that is, we are focusing on the riskiness of each investment
under the assumption that it would be the business™s only asset (operated in
isolation). In the next section, portfolio effects will be introduced, but for
now, let us continue our discussion of stand-alone risk.
Stand-alone risk depends on the “tightness” of an investment™s return
distribution. If an investment has a “tight” return distribution, with returns
falling close to the expected return, it has relatively low stand-alone risk.
Conversely, an investment with a return distribution that is “loose,” and hence
has values well below the expected return, is relatively risky in the stand-
alone sense.
It is important to recognize that risk and return are separate attributes
of an investment. An investment may have a very “tight” distribution of
returns, and hence very low stand-alone risk, but its expected rate of return
might be only 2 percent. In this situation, the investment probably would not
be ¬nancially attractive, in spite of its low risk. Similarly, a high-risk investment
with a suf¬ciently high expected rate of return would be attractive.
To be truly useful, any de¬nition of risk must have some measure, or
numerical value, so we need some way to specify the “degree of tightness” of
an investment™s return distribution. One such measure is standard deviation,
which is often given the symbol “σ ” (Greek lowercase sigma). Standard devi-
ation is a common statistical measure of the dispersion of a distribution about
its mean”the smaller the standard deviation, the “tighter” the distribution
and hence the lower the riskiness of the investment. To illustrate the calcula-
tion of standard deviation, consider the MRI investment™s estimated returns
listed in Table 10.1. Here are the steps:

1. The expected rate of return on the MRI, E(RMRI), is 10 percent.
2. The variance of the return distribution is determined as follows:

Variance = (Probability of Return 1 — [Rate of Return 1 ’ E(RMRI )]2 )
+ (Probability of Return 2 — [Rate of Return 2 ’ E(RMRI )]2 )
and so on
= (0.10 — [’10% ’ 10%]2 ) + (0.20 — [0% ’ 10%]2 )
+ (0.40 — [10% ’ 10%]2 ) + (0.20 — [20% ’ 10%]2 )
+ (0.10 — [30% ’ 10%]2 )
= 120.00.
Variance, like standard deviation, is a measure of the dispersion of a
distribution about its expected value, but it is less useful than standard
deviation because its measurement unit is percent (or dollars) squared,
which has no economic meaning.
295
Chapter 10: Financial Risk and Required Return



3. The standard deviation is de¬ned as the square root of the variance:

Standard deviation (σ ) = Variance

= 120.00 = 10.95% ≈ 11.0%.

Using the same procedure, the clinic investment listed in Table 10.1 was
found to have a standard deviation of returns of about 18 percent.2 Because
the clinic investment™s standard deviation of returns is larger than that of the
MRI investment, the clinic investment has more stand-alone risk than the
MRI investment.
As a general rule, investments with higher expected rates of return have
larger standard deviations than investments with smaller expected returns.
This situation occurs in our MRI and clinic example. In situations where
expected rates of return on investments differ substantially, standard deviation
may not give a good picture of one investment™s stand-alone risk relative to
another. The coef¬cient of variation (CV), which is de¬ned as the standard
deviation of returns divided by the expected return, measures the risk per
unit of return and hence standardizes the measurement of stand-alone risk.
To illustrate, here are the CVs for the MRI and clinic investments:
σ
Coef¬cient of variation = .
E(R )
CVMRI = 11.0%/10.0% = 1.10.
CVClinic = 18.0%/15.0% = 1.20.
In this situation, the clinic investment has slightly more risk per unit of return,
so it is riskier than the MRI as measured by both standard deviation and
coef¬cient of variation. However, note that the clinic™s stand-alone risk as
measured by the coef¬cient of variation is not as great relative to the MRI
as it is when measured by standard deviation. This difference in relative risk
occurs because the clinic has a higher expected rate of return. Finally, note
that coef¬cient of variation has no units; it is just a raw number.


Self-Test
1. What is stand-alone risk?
Questions
2. De¬ne and explain two measures of stand-alone risk?
3. Is one measure better than another?


Portfolio Risk and Return
The preceding section developed a risk measure”standard deviation”that
applies to investments held in isolation. (We also introduced the coef¬cient of
variation, but in most situations the standard deviation will suf¬ce.) However,
most investments are not held in isolation but are held as part of a collection,
or portfolio, of investments. Individual investors typically hold portfolios of
296 Healthcare Finance



securities (i.e., stocks and bonds), while businesses generally hold portfolios
of projects (i.e., product or service lines). When investments are held in
portfolios, the primary concern of investors is not the realized rate of return
on an individual investment but rather the realized rate of return on the entire
portfolio. Similarly, the stand-alone risk of each individual asset in the portfolio
is not important to the investor; what matters is the aggregate riskiness of the
portfolio. Thus, the whole nature of risk and how it is de¬ned and measured
changes when one recognizes that investments are not held in isolation but as
parts of portfolios.

Portfolio Returns
Consider the returns estimated for the seven investment alternatives listed
in Table 10.2. The individual investment alternatives”Investments A, B, C,
and D”could be projects under consideration by South West Clinics, Inc.,
or they could be stocks that are being evaluated as personal investments by
Bruce Duncan. The remaining three alternatives in Table 10.2 are portfolios.
Portfolio AB consists of 50 percent invested in Investment A and 50 percent
in Investment B (e.g., $10,000 invested in A and $10,000 invested in B);
Portfolio AC is an equal-weighted portfolio of Investments A and C; and
Portfolio AD is an equal-weighted portfolio of Investments A and D. As
shown in the bottom of the table, Investments A and B have 10 percent
expected rates of return, while the expected rates of return for Investments C
and D are 15 percent and 12 percent, respectively. Investments A and B have
identical stand-alone risk as measured by standard deviation, 11.0 percent,
while Investments C and D have greater stand-alone risk than A and B.
The expected rate of return on a portfolio, E(Rp), is the weighted average
of the expected returns on the assets that make up the portfolio, with the
weights being the proportion of the total portfolio invested in each asset:

E(Rp ) = (w1 — E[R1 ]) + (w2 — E[R2 ]) + (w3 — E[R3 ]) and so on.
In this case, w1 is the proportion of Investment 1 in the overall portfolio and

TABLE 10.2
Rate of Return if State Occurs
Economic Probability
Estimated
State of Occurrence A B C D AB AC AD
Returns for Four
Individual
’10% ’25% 10% ’17.5%
Very poor 0.10 30% 15% 2.5%
Investments and ’5 ’2.5
Poor 0.20 0 20 10 10 5.0
Average 0.40 10 10 15 0 10 12.5 5.0
Three Portfolios
Good 0.20 20 0 35 25 10 27.5 22.5
’10
Very good 0.10 30 55 35 10 42.5 32.5
1.00

Expected rate of return 10.0% 10.0% 15.0% 12.0% 10.0% 12.5% 11.0%
Standard deviation 11.0% 11.0% 21.9% 12.1% 0.0% 16.4% 10.1%
297
Chapter 10: Financial Risk and Required Return



E(R1) is the expected rate of return on Investment 1, and so on. Thus, the
expected rate of return on Portfolio AB is 10 percent:

E(RAB ) = (0.5 — 10%) + (0.5 — 10%) = 5% + 5% = 10%,
while the expected rate of return on Portfolio AC is 12.5 percent and on AD
is 11.0 percent.
Alternatively, the expected rate of return on a portfolio can be calcu-
lated by looking at the portfolio™s return distribution. To illustrate, consider
the return distribution for Portfolio AC contained in Table 10.2. The port-
folio return in each economic state is the weighted average of the returns on
Investments A and C in that state. For example, the return on Portfolio AC
in the very poor state is (0.5 — [’10 %]) + (0.5 — [’25 %]) = ’17.5%.
Portfolio AC™s return in each other state is calculated similarly. Portfolio AC™s
return distribution now can be used to calculate its expected rate of return:

E(RAC ) = (0.10 — [’17.5%]) + (0.20 — [’2.5%]) + (0.40 — 12.5%)
+ (0.20 — 27.5%) + (0.10 — 42.5%)
= 12.5%.
This is the same value as calculated from the expected rates of return of the
two portfolio components:

(0.5 — 10%) + (0.5 — 15%) = 12.5%.
After the fact, the actual, or realized, returns on Investments A and C
will probably be different from their expected values, and hence the realized
rate of return on Portfolio AC will likely be different from its 12.5 percent
expected return.

Portfolio Risk: Two Assets
When an investor holds a portfolio of assets, the portfolio is in effect a stand-
alone investment, so the riskiness of the portfolio is measured by the stan-
dard deviation of portfolio returns, the previously discussed measure of
stand-alone risk. How does the riskiness of the individual investments in a
portfolio combine to create the overall riskiness of the portfolio? Although
the rate of return on a portfolio is the weighted average of the returns on
the component investments, a portfolio™s standard deviation (i.e., riskiness) is
generally not the weighted average of the standard deviations of the individ-
ual components. The portfolio™s riskiness may be smaller than the weighted
average of each component™s riskiness. Indeed, the riskiness of a portfolio may
be less than the least risky portfolio component and, under certain conditions,
a portfolio of risky assets may be even riskless.
A simple example can be used to illustrate this concept. Suppose that
an individual is given the following opportunity: Flip a coin once; if it comes
298 Healthcare Finance



up heads, he or she wins $10,000, but if it comes up tails, the individual loses
$8,000. This is a reasonable gamble in that the expected dollar return is (0.5 —
$10,000) + (0.5 — [’$8,000]) = $1,000. However, it is highly risky because
the individual has a 50 percent chance of losing $8,000. Thus, risk aversion
would cause most individuals to refuse the gamble, especially if the $8,000
potential loss would result in ¬nancial hardship.
Alternatively, suppose that the individual is given the opportunity to
¬‚ip the coin 100 times, and he or she would win $100 for each head but lose
$80 for each tail. It is possible, although extremely unlikely, that the individual
would ¬‚ip all heads and win $10,000. It is also possible, and also extremely
unlikely, that he or she would ¬‚ip all tails and lose $8,000. But the chances are
very high that the individual would actually ¬‚ip close to 50 heads and 50 tails
and net about $1,000. Even if he or she ¬‚ipped a few more tails than heads,
the individual would still make money on the gamble.
Although each ¬‚ip is very risky in the stand-alone sense, taken collec-
tively the ¬‚ips are not very risky at all. In effect, the multiple ¬‚ipping has
created a portfolio of investments; each ¬‚ip of the coin can be thought of as
one investment, so the individual now has a 100-investment portfolio. Fur-
thermore, the return on each investment is independent of the returns on the
other investments: The individual has a 50 percent chance of winning on each
¬‚ip of the coin regardless of the results of the previous ¬‚ips. By combining the
¬‚ips into a single gamble (i.e., into an investment portfolio), the risk associated
with each ¬‚ip of the coin is reduced. In fact, if the gamble consisted of a very
large number of ¬‚ips, almost all risk would be eliminated: The probability of a
near-equal number of heads and tails would be extremely high, and the result
would be a sure pro¬t. The key to the risk reduction inherent in the portfolio
is that the negative consequences of tossing a tail can be offset by the positive
consequences of tossing a head.
To examine portfolio effects in more depth, consider Portfolio AB in
Table 10.2. Each investment (A or B) has a standard deviation of returns of 10
percent, and hence is quite risky when held in isolation. However, a portfolio
of the two investments has a rate of return of 10 percent in every possible
state of the economy, so it offers a riskless 10 percent return. This result is
veri¬ed by the value of zero for Portfolio AB™s standard deviation of return.
The reason Investments A and B can be combined to form a riskless portfolio
is that their returns move exactly opposite to one another. Thus, in economic
states when A™s returns are relatively low, those of B are relatively high, and
vice versa, so the gains on one investment in the portfolio more than offset
losses on the other.
The movement relationship of two variables (i.e., their tendency to
move either together or in opposition) is called correlation. The correlation
coef¬cient, r, measures this relationship. Investments A and B can be combined
to form a riskless portfolio because the returns on A and B are perfectly
negatively correlated, which is designated by r = ’1.0. In every state where
299
Chapter 10: Financial Risk and Required Return



Investment A has a return higher than its expected return, Investment B has
a return lower than its expected return, and vice versa.
The opposite of perfect negative correlation is perfect positive correla-
tion, with r = +1.0. Returns on two perfectly positively correlated investments
move up and down together as the economic state changes. When the returns
on two investments are perfectly positively correlated, combining the invest-
ments into a portfolio will not lower risk because the standard deviation of
the portfolio is merely the weighted average of the standard deviations of the
two components.
To illustrate the impact of perfect positive correlation, consider Port-
folio AC in Table 10.2. Its expected rate of return, E(RAC), is 12.5 percent,
while its standard deviation is 16.4 percent. Because of perfect positive corre-
lation between the returns on A and C, Portfolio AC™s standard deviation is
the weighted average standard deviation of its components:

σAC = (0.5 — 11.0%) + (0.50 — 21.95%)
= 16.4%.
There is no risk reduction in this situation. The risk of the portfolio is less than
the risk of Investment C, but it is more than the risk of Investment A. Forming
a portfolio does not reduce risk when the returns on the two components are
perfectly positively correlated; the portfolio merely averages the risk of the
two investments.
What happens when a portfolio is created with two investments that
have positive, but not perfectly positive, correlation? Combining the two in-
vestments can eliminate some, but not all, risk. To illustrate, consider Portfolio
AD in Table 10.2. This portfolio has a standard deviation of returns of 10.1
percent, so it is risky. However, Portfolio AD™s standard deviation is not only
less than the weighted average of its components™ standard deviations, (0.5 —
11%) + (0.5 — 12.1%) = 11.6%, it also is less than the standard deviation of
each component. The correlation coef¬cient between the return distributions
for A and D is 0.53, which indicates that the two investments are positively
correlated, but they are not perfectly correlated because the coef¬cient is less
than +1.0. Thus, combining two investments that are positively correlated,
but not perfectly so, lowers risk but does not eliminate it.
Because correlation is the factor that drives risk reduction, a logical
question arises: What is the correlation among the returns on “real-world” in-
vestments? Generalizing about the correlations among real-world investment
alternatives is dif¬cult. However, it is safe to say that the return distributions
of two randomly selected investments”whether they are real assets in a hos-
pital™s portfolio of service lines or ¬nancial assets in an individual™s investment
portfolio”are virtually never perfectly correlated, and hence correlation co-
ef¬cients are never ’1.0 or +1.0. In fact, it is almost impossible to ¬nd ac-
tual investment opportunities with returns that are negatively correlated with
300 Healthcare Finance



one another, or even to ¬nd investments with returns that are uncorrelated
(r = 0). Because all investment returns are affected to a greater or lesser de-
gree by general economic conditions, investment returns tend to be positively
correlated with one another. However, because investment returns are not af-
fected identically by general economic conditions, returns on most real-world
investments are not perfectly positively correlated.
The correlation coef¬cient between the returns of two randomly cho-
sen investments will usually fall in the range of +0.4 to +0.8. Returns on
investments that are similar in nature, such as two inpatient projects in a hos-
pital or two stocks in the same industry, will typically have correlations at the
upper end of this range. Conversely, returns on dissimilar projects or securities
will tend to have correlations at the lower end of the range.

Portfolio Risk: Many Assets
Businesses are not restricted to two projects, and individual investors are not
restricted to holding two-security portfolios. Most companies have tens, or
even hundreds, of individual projects (i.e., product or service lines), and most
individual investors hold many different securities or mutual funds that may
be composed of hundreds of individual securities. Thus, what is most rele-
vant to ¬nancial decision making is not what happens when two investments
are combined into portfolios, but what happens when many investments are
combined.
To illustrate the risk impact of creating large portfolios, consider Fig-
ure 10.1. The ¬gure illustrates the riskiness inherent in holding randomly

FIGURE 10.1
Portfolio Size
and Risk
301
Chapter 10: Financial Risk and Required Return



selected portfolios of one asset, two assets, three assets, four assets, and so
on, considering the correlations that occur among real-world investments.
The plot is based on historical annual returns on common stocks traded on
the New York Stock Exchange (NYSE), but the conclusions reached are ap-
plicable to portfolios made up of any type of investment, including healthcare
providers that offer many different types of services. The plot shows the ave-
rage standard deviation of all one-asset (one-stock) portfolios, the average
standard deviation of all possible two-asset portfolios, the average standard
deviation of all possible three-asset portfolios, and so on.
The riskiness inherent in holding an average one-asset portfolio is rela-
tively high, as measured by its standard deviation of annual returns. The aver-
age two-asset portfolio has a lower standard deviation, so holding an average
two-asset portfolio is less risky than holding a single asset of average risk. The
average three-asset portfolio has an even lower standard deviation of returns,
so an average three-asset portfolio is even less risky than an average two-asset
portfolio. As more assets are randomly added to create larger portfolios, the
average riskiness of the portfolio decreases. However, as more and more as-
sets are added, the incremental risk reduction of adding even more assets de-
creases, and regardless of how many assets are added, some risk always remains
in the portfolio”even with a portfolio of thousands of assets, substantial risk
remains.
The reason that all risk cannot be eliminated by creating a very large
portfolio is that the returns on the component investments, although not
perfectly so, are still positively correlated with one another. In other words, all
investments, both real and ¬nancial, are affected to a lesser or greater degree
by general economic conditions. If the economy booms, all investments tend
to do well, while in a recession all investments tend to do poorly. It is the
positive correlation among real-world asset returns that prevents investors
from creating riskless portfolios.

Diversi¬able Risk Versus Portfolio Risk
Figure 10.1 shows what happens as investors create ever larger portfolios.
As the size of a randomly created portfolio increases, the riskiness of the
portfolio decreases, so a large proportion of the stand-alone risk inherent in an
individual investment can be eliminated if it is held as part of a large portfolio.
For example, recent studies have found that a large stock portfolio has only
about one-half the standard deviation of an average stock.
Thus, if a stock investor wanted to eliminate the maximum amount
of stand-alone risk inherent in owning NYSE stocks, he or she would have
to, at least in theory, own over 3,000 stocks. Such a portfolio is called the
market portfolio because it consists of the entire stock market (or at least one
entire segment of the stock market). However, it is not necessary for individual
investors to own a very large number of stocks to gain the risk-reducing bene¬t
inherent in holding large portfolios. As illustrated in Figure 10.1, most of the
302 Healthcare Finance



bene¬t of diversi¬cation can be obtained by holding a well-diversi¬ed portfolio
of about 50 randomly selected stocks. (A portfolio of 50 healthcare stocks,
for example, is not well diversi¬ed because the stocks are in the same sector
of the economy and hence are not randomly chosen.)
That part of the stand-alone riskiness of an individual investment that
can be eliminated by diversi¬cation (i.e., by holding it as part of a well-
diversi¬ed portfolio) is called diversi¬able risk. That part of the riskiness of
an individual investment that cannot be eliminated by diversi¬cation is called
portfolio risk. Thus, every investment, whether it be the stock of Beverly
Enterprises held by an individual investor or an MRI system operated by a
hospital, has some diversi¬able risk that can be eliminated and some portfolio
risk that cannot be diversi¬ed away. Not all investments bene¬t to the same
degree from portfolio risk-reducing effects, and some portfolios are not truly
well diversi¬ed. In general, however, any investment will have some of its
stand-alone risk eliminated when it is held as part of a portfolio.
Diversi¬able risk, as seen by individuals who invest in stocks, is caused
by events that are unique to a single business, such as new product or ser-
vice introductions, strikes, and lawsuits. Because these events are essentially
random, their effects can be eliminated by diversi¬cation. When one stock in
a portfolio does worse than expected because of a negative event unique to
that ¬rm, another stock in the portfolio will do better than expected because
of a ¬rm-unique positive event. On average, bad events in some companies
will be offset by good events in others, so lower-than-expected returns will
be offset by higher-than-expected returns, leaving the investor with an overall
portfolio return closer to that expected than would be the case if only a single
stock were held.
The same logic can be applied to a ¬rm with a portfolio of projects. Per-
haps hospital returns generated from inpatient surgery are less than expected
because of the trend toward outpatient procedures, but this may be offset by
returns that are greater than expected on state-of-the-art diagnostic services.
(If the hospital offered both inpatient and outpatient surgery, it would be
hedging itself against the trend toward more outpatient procedures because
reduced demand for inpatient surgery would be offset by increased demand
for outpatient surgery.)
The point to be made here is that the negative impact of random events
that are unique to a particular business, or to a particular product or service
within a ¬rm, can be offset by positive events in other businesses or in other
products or services. Thus, the risk caused by random, unique events can
be eliminated by portfolio diversi¬cation. Individual investors can diversify
by holding many securities, and businesses can diversify by operating many
projects.
Unfortunately, not all risk can be diversi¬ed away. Portfolio risk”the
risk that remains even when well-diversi¬ed portfolios are created”stems
from factors that systematically affect all stocks in a portfolio, such as wars,
in¬‚ation, recessions, and high interest rates or all products or services offered
303
Chapter 10: Financial Risk and Required Return



by a business. For example, the increasing power of managed care organiza-
tions could lower reimbursement levels for all services offered by a hospital.
No amount of diversi¬cation by the hospital (except, perhaps, moving into
managed care) could eliminate this risk. Because portfolio risk cannot be elim-
inated, even well-diversi¬ed investors, whether they are individuals with large
securities portfolios or diversi¬ed healthcare companies with many different
service lines, face this type of risk.

Implications for Investors
The ability to eliminate a portion of the stand-alone riskiness inherent in
individual investments has two signi¬cant implications for investors, whether
the investor is an individual who holds securities or a business that offers
products or services.

• Holding a single investment is not rational. Holding a portfolio can
eliminate much of the stand-alone riskiness inherent in individual
investments. Investors who are risk averse should seek to eliminate all
diversi¬able risk. Individual investors can easily diversify their personal
investment portfolios by buying many individual securities or mutual
funds that hold diversi¬ed portfolios. Businesses cannot diversify their
investments as easily as individuals, but businesses that offer a diverse line
of products or services are less risky than businesses that rely on a single
product or service.
• Because an asset held in a portfolio has less risk than when held in
isolation, the traditional stand-alone risk measure of standard deviation is
no longer appropriate for individual assets. Thus, it is necessary to rethink
the de¬nition and measurement of ¬nancial risk for such assets. (Note,
though, that standard deviation remains the correct measure for the
riskiness of an investor™s portfolio because the portfolio is, in effect, a
single asset held in isolation.)



Self-Test
1. What is a portfolio of assets?
Questions
2. What is a well-diversi¬ed portfolio?
3. What happens to the risk of a single asset when it is held as part of a
portfolio of assets?
4. Explain the differences between stand-alone risk, diversi¬able risk, and
portfolio risk.
5. Why should all investors hold portfolios rather than individual assets?
6. Is standard deviation the appropriate risk measure for an individual
asset?
7. Is standard deviation the appropriate risk measure for an investor™s
portfolio of assets?
304 Healthcare Finance



Measuring the Risk of Investments Held in Portfolios
The stand-alone risk of individual investments is reduced when they are held as
part of a portfolio, so standard deviation is no longer the relevant risk measure
for such investments. Because investors are concerned with the overall riski-
ness of the portfolio, which is measured by standard deviation, the appropriate
measure of risk for individual investments in the portfolio is the contribution
of each one to the overall riskiness (standard deviation) of the portfolio. In
this section, we describe how the portfolio risk of individual investments can
be measured.

The Beta Coef¬cient
The most widely used measure of risk for investments held in portfolios
is called the beta coef¬cient, or just beta. It measures the volatility of an
investment™s returns relative to the volatility of the portfolio. To illustrate
the concept of beta, consider Table 10.3, which contains historical annual
returns for three individual investments, H, M, and L, and for a large portfolio,
P. Five years of annual returns are displayed in the table, but three years
of monthly returns, or some other combination, could have been chosen.
Because the returns are historical (i.e., realized) rather than projected, the
probability of occurrence of each return is the same: for ¬ve years of returns,
each return has a probability of 100% / 5 = 20%. For now, the context does not
matter; H, M, and L could be stocks that an individual investor is considering
as an addition to Portfolio P, a stock portfolio, or they could be projects that
are being evaluated by a hospital and hence would be added to the hospital™s
portfolio of services, Portfolio P.
Figure 10.2 plots the historical annual returns on the three individual
investments on the Y-axis versus returns on the portfolio on the X-axis. In-
vestment M has the same volatility as the portfolio. (In fact, Investment M has
the same historical returns as does the portfolio.) However, Investment H is
more volatile than the portfolio: its returns ranged from ’18 to 50 percent.
Conversely, Investment L is less volatile than the portfolio: its returns ranged
only from 2 to 19 percent.
Investments that are more volatile than the portfolio increase the risk-


TABLE 10.3 Rate of Return
Beta Coef¬cient
Year H M L Portfolio (P)
Illustration
1 35% 15% 8% 15%
2 5 5 5 5
’18 ’5 ’5
3 2
4 40 25 18 25
5 50 35 19 35
305
Chapter 10: Financial Risk and Required Return


FIGURE 10.2
Relative
Volatility of
Assets H, M,
and L




iness of the portfolio when they are added, while investments that are less
volatile decrease the riskiness of the portfolio. Thus, the amount of volatility
of an investment, relative to the portfolio, measures the contribution of the
investment to the overall riskiness of the portfolio, and hence relative volatility
measures an individual investment™s portfolio risk. Remember that an indi-
vidual investment™s stand-alone risk is de¬ned as volatility about its mean
(expected) return, so a completely different concept is being used to assess
the portfolio risk of an individual investment.
How should the risks of Investments H, M, and L be measured, con-
sidering that they would be held as part of Portfolio P? In fact, the lines that
are plotted on Figure 10.2 are regression lines in the statistical sense, and the
slope of each line measures the volatility of that investment relative to the
volatility of the portfolio. (The slope of a regression line is a measure of its
steepness and is de¬ned as rise over run.) The regression line for Investment
M has a slope, or beta coef¬cient, of 1.0, which shows that M has the same
volatility as the portfolio and hence has average risk, where average is de¬ned
as the riskiness of the portfolio.3 Investment H has a beta of 1.5, and hence it
is 1.5 times as risky as the portfolio, while L, with a beta of 0.5, is only half as
risky as Portfolio P.
306 Healthcare Finance



A U.S. Treasury security that had a 5 percent realized return in each
year over the same ¬ve-year period as shown in Table 10.3 would have a
horizontal regression line and hence a slope of zero. Such an investment would
have a beta of zero, which signi¬es no portfolio risk. In fact, such a Treasury
security would have no stand-alone risk because a 5 percent rate of return in
each year would result in a standard deviation of zero.
Theoretically, an investment could have a negative beta; the regression
line for such an investment would slope downward. In this case, the invest-
ment™s return would move opposite to the portfolio™s return: in years when
the portfolio™s return was high, the investment™s return would be low, and vice
versa. Such an investment with returns that are negatively correlated with the
portfolio™s returns, has negative portfolio risk, which means that it would have
a signi¬cant risk-reducing impact on the portfolio. Although it is possible to
¬nd investments in the real world that have negative betas based on historical
returns, it is much more dif¬cult to ¬nd an investment that is expected to have
a negative beta on the basis of future returns. The reason is that the returns
on all assets in a portfolio typically are affected in a similar manner by external
economic forces.
Note that, in Figure 10.1, the investment returns (the dots) do not all
fall on the regression line. As previously discussed, the slope of the regression
line (beta) measures the portfolio risk of the investment. The distance of the
points, on average, from the regression line measures the diversi¬able risk of
the investment. The further the points plot from the line, the greater the
amount of risk that is diversi¬ed away when the investment is held as part of a
portfolio. In effect, adding the investment to a portfolio forces the points to
the regression line, because returns on other investments in the portfolio will
drag the points above the line down to the line and pull the points below the
line up to it.

Corporate Risk: Risk Within Business Portfolios
Businesses typically offer a myriad of products or services and thus can be
thought of as having a large number (hundreds or more) of individual activi-
ties, or projects. For example, most managed care organizations offer numer-
ous healthcare plans to diverse groups of enrollees in numerous service areas.
And many hospitals and hospital systems offer a large number of inpatient,
outpatient, and even home health services that cover a wide geographical area
and treat a wide range of illnesses and injuries. Thus, healthcare businesses,
except for the very smallest, actually consist of a portfolio of projects.
What is the riskiness of an individual project to a business with many
projects? Because the project is part of the business™s portfolio of assets, its
stand-alone risk is not relevant”the project is not held in isolation. The
relevant risk of any project to a business with many projects is its contribution
to the business™s overall risk, or the impact of the project on the variability
of the ¬rm™s overall rate of return. Some of the stand-alone riskiness of the
307
Chapter 10: Financial Risk and Required Return



project will be diversi¬ed away by combining the project with the ¬rm™s other
projects. The remaining risk, which is the portfolio risk in a business context,
is called corporate risk.
The quantitative measure of corporate risk is a project™s corporate beta,
or corporate b, which is the slope of the regression line that results when the
project™s returns are plotted on the Y-axis and the overall returns on the ¬rm
are plotted on the X-axis. If Table 10.3 and Figure 10.2 represented returns on
three projects and the overall returns for AtlantiCare, a not-for-pro¬t HMO,
the betas for H, M, and L would be corporate betas. They would measure the
contribution of each project to AtlantiCare™s overall risk.
A project™s corporate beta measures the volatility of returns on the
project relative to the business as a whole, which has a corporate beta of 1.0.
If a project™s corporate beta is 1.5, such as for Project H, its returns are 1.5
times as volatile as the ¬rm™s returns. Such a project increases the volatility of
AtlantiCare™s overall returns and hence increases the riskiness of the business.
A corporate beta of 1.0, such as for Project M, indicates that the project™s
returns have the same volatility as the ¬rm. Hence, the project has the same
risk as AtlantiCare™s average project, which is a hypothetical project with risk
identical to the aggregate business (the portfolio). A corporate beta of 0.5,
such as for Project L, indicates that the project™s returns are less volatile than
the ¬rm™s returns, so the project reduces AtlantiCare™s overall risk.
In closing the discussion of corporate risk (the risk of individual projects
within a business with many projects), it must be noted that we have glossed
over the dif¬culties inherent in implementing the concept in practice. That
discussion will take place in Chapter 15. However, the concept of corporate
risk remains important to health services managers in spite of its implementa-
tion problems.

Market Risk: Risk Within Stock Portfolios
The previous section discussed the riskiness of business projects to an or-
ganization. This section discusses the riskiness of business projects to own-
ers, which for investor-owned corporations are common stockholders. Why
should health services managers be concerned about how stock investors view
risk? The answer is simple: Stock investors are the suppliers of equity capital to
investor-owned businesses, so they set the rates of return that such businesses
must pay to raise that capital. In turn, these rates set the minimum pro¬tabil-
ity that investor-owned businesses must earn on the equity portion of their
real-asset investments. Even managers of not-for-pro¬t ¬rms should have an
understanding of how stock investors view risk because market-set required
rates of return can in¬‚uence the opportunity cost rates used in making real-
asset investments within not-for-pro¬t businesses. Chapter 13 discusses this
topic in detail.
Because stock investors hold well-diversi¬ed portfolios of stocks, the
relevant riskiness of an individual project undertaken by a company whose
308 Healthcare Finance



stock is held in the portfolio is the project™s contribution to the overall riskiness
of that stock portfolio. Some of the stand-alone riskiness of the project will be
diversi¬ed away by combining the project with all the other projects in the
stock portfolio. The remaining portfolio risk is called market risk, which is
de¬ned as the contribution of the project to the riskiness of a well-diversi¬ed
stock portfolio.
How should a project™s market risk be measured? A project™s market
beta, or market b, measures the volatility of the project™s returns relative to
the returns on a well-diversi¬ed portfolio of stocks, which represents a very
large portfolio of individual projects. If Table 10.3 and Figure 10.2 repre-
sented returns on three projects and the overall returns on a well-diversi¬ed
stock portfolio, the betas for H, M, and L would be market betas, and they
would measure the market risk of the projects. The only difference between
the discussion of market risk and the previous discussion of corporate risk is
what de¬nes the relevant portfolio. When focusing on corporate risk, the rel-
evant portfolio is the business™s overall portfolio of projects. When discussing
market risk, the relevant portfolio is a well-diversi¬ed stock portfolio (the
market portfolio).
A project with a market beta of 1.5, such as for Project H, has returns
that are 1.5 times as volatile as the returns on the market and hence increases
the riskiness of a well-diversi¬ed stock portfolio. A market beta of 1.0 indicates
that the project™s returns have the same volatility as the market”such a project
has the same market risk as the market portfolio. A market beta of 0.5 indicates
that the project™s returns are half as volatile as the returns on the market”such
a project reduces the riskiness of a well-diversi¬ed stock portfolio.4
The two types of portfolio risk”corporate and market”are identical in
concept. Both types of risk are de¬ned as the contribution of the investment”
in this case, a business project”to the overall riskiness of the portfolio. Also,
both types of risk are measured by the volatility of the investment™s returns rel-
ative to the volatility of the portfolio. The only difference between corporate
and market risk is what de¬nes the portfolio. In corporate risk, the portfolio is
de¬ned as the collection of projects held within a business; in market risk, the
portfolio is de¬ned as the collection of projects held within a well-diversi¬ed
stock portfolio.
Finally, even though an individual investor™s stock portfolio can be
thought of as a portfolio of many separate projects, the portfolio actually con-
sists of the stocks of individual ¬rms. Individual investors, therefore, are most
concerned with the aggregate risk and return characteristics of the compa-
nies they own rather than the risk and return characteristics of each com-
pany™s projects. This logic leads investors to be more concerned with the
company™s (stock™s) market beta than they are with the market betas of in-
dividual projects.
A stock™s market beta is the slope of the line formed by regressing the
individual ¬rm™s stock returns against the aggregate returns on the market. For
309
Chapter 10: Financial Risk and Required Return



example, the market beta of General Healthcare (GH) was recently reported
to be 0.80. This means that an equity investment in GH is somewhat less risky
to well-diversi¬ed stock investors than an average stock, which has a beta of
1.0. GH™s corporate beta, like all other company™s corporate betas, is 1.0 by
de¬nition. What is relevant to stock investors, because they hold portfolios of
common stocks, is the ¬rm™s market beta, not its corporate beta.


Self-Test
1. What is the de¬nition of portfolio risk?
Questions
2. How is portfolio risk measured?
3. What is a corporate beta, and how is it estimated?
4. What is a market beta, and how is it estimated?
5. Brie¬‚y, what is the difference between corporate risk and market risk?
6. What is the difference between a project™s market beta and the ¬rm™s
market beta?


Portfolio Betas
Individual investors hold portfolios of stocks, each with its own market risk
as measured by the stock™s market beta, while businesses hold portfolios of
projects, each with its own corporate and market betas. What impact does
the beta of a portfolio component have on the overall portfolio™s beta? The
beta of any portfolio of investments, bp, is simply the weighted average of the
individual component betas:

bp = (w1 — b1 ) + (w2 — b2 ) and so on.
Here, bp is the beta of the portfolio, which measures the volatility of the entire
portfolio; w1 is the fraction of the portfolio in Investment 1; b1 is the beta
coef¬cient of Investment 1; and so on.
To illustrate, the stock of GH has a market beta of 0.8, indicating that
its returns to stockholders are somewhat less volatile than the returns on a
well-diversi¬ed stock portfolio (or average stock with a beta of 1.0). Each
project within GH has its own market risk, however, as measured by its market
beta. Some projects may have very high market betas (e.g., over 1.5), while
other projects may have very low market betas (e.g., under 0.5). When all the
projects are combined, the overall market beta of the company is 0.8.
For ease of discussion, assume that GH has only the following three
projects:
Project Market Beta Dollar Investment Proportion
A 1.3 $ 15,000 15.0%
B 1.1 30,000 30.0
C 0.5 55,000 55.0
$100,000 100.0%
310 Healthcare Finance



The weighted average of the project market betas, which is the ¬rm™s market
beta, is 0.8:

Market bGH = (w1 — b1 ) + (w2 — b2 ) + (w3 — b3 )
= (0.15 — 1.3) + (0.30 — 1.1) + (0.55 — 0.5)
= 0.8.
Each project™s market beta re¬‚ects its volatility relative to the market
portfolio. Note that each of GH™s ¬ctitious projects also has a corporate beta
that measures the volatility of the project™s returns relative to that of the
corporation as a whole. The weighted average of these project corporate betas
must equal 1.0, which is the corporate beta of any business.


Self-Test 1. How is the beta of a portfolio related to the betas of the components?
Questions 2. What is the value of the weighted average of all project corporate betas
within a business?


Relevance of the Risk Measures
Thus far, the chapter has discussed in some detail three measures of ¬nancial
risk”stand-alone, corporate, and market”but it is still unclear which risk
is the most relevant in ¬nancial decision making. It turns out that the risk
that is relevant to any ¬nancial decision depends on the particular situation
at hand. When the decision involves a single investment that will be held
in isolation, stand-alone risk is the relevant risk. Here, the risk and return
on the portfolio is the same as the risk and return on the single asset in the
portfolio. In this situation, the riskiness faced by the investor, whether it be
an individual considering a stock purchase or a business considering a MRI
system investment, is de¬ned in terms of returns less than expected, and the
appropriate measure is the standard deviation (or coef¬cient of variation) of
the return distribution.
In most decisions, however, the investment under consideration will
not be held in isolation but will be held as part of an investment portfolio.
Individual investors normally hold portfolios of stocks, while businesses nor-
mally hold portfolios of real-asset investments (projects). Thus, it is clear that
portfolio risk is more relevant to real-world decisions than is stand-alone risk.
However, there are three distinct ownership situations that affect the relevancy
of portfolio risk.

Large Investor-Owned Businesses
For large investor-owned businesses, the primary ¬nancial goal is shareholder
wealth maximization. This means that managerial decisions should focus on
311
Chapter 10: Financial Risk and Required Return



risk and return as seen by the business™s stockholders. Because stockholders
tend to hold large portfolios of securities, and hence a very large portfolio of
individual projects, the most relevant risk of a project under consideration
by a large for-pro¬t ¬rm is the project™s contribution to a well-diversi¬ed
stock portfolio (the market portfolio). Of course, this is the project™s market
risk. Many would argue, and we agree, that corporate and stand-alone risk
cannot be disregarded in all situations. For example, corporate risk, which best
measures the impact of the project on the ¬nancial condition of the business,
clearly is relevant to the business™s nonowner stakeholders, such as managers,
employees, creditors, and suppliers, who should not be totally ignored. Also,
the failure of a project that is large, relative to the business, could bring down
the entire ¬rm. Under such circumstances, the project clearly has high risk to
stockholders even if its market risk is low. The bottom line here is that market
risk should be of primary importance in large investor-owned businesses, but
corporate and stand-alone risk should not be ignored.

Small Investor-Owned Businesses
For small investor-owned businesses, the situation is more complicated. Take,
for example, a three-physician group practice. Here, there is no separation
between management and ownership, and the equity investment position is
complicated by the fact that the business is also the owners™ employer. In this
situation, the primary goal of the business is more likely to be maximization
of owners™ overall well-being than strict shareholder wealth maximization.
For example, owner/managers may value leisure time, as exempli¬ed by three
afternoons of golf, as being more important than additional wealth creation.
To complicate the situation even more, shareholder wealth now consists of
both the value of the ownership position and the professional fees (salaries)
derived from the business.
Thus, in small for-pro¬t businesses, corporate risk is probably more
relevant than market risk. The owner/managers would not want to place the
viability of the business in jeopardy just to increase their expected ownership
value by a small amount. Put another way, the owner/managers are not well
diversi¬ed in regards to the business because a large proportion of their wealth
comes from expected future employment earnings. Because of this, market
risk loses relevance and corporate risk becomes most important. However,
the potential relevancy of stand-alone risk as described in the previous section
also applies.

Not-for-Pro¬t Businesses
Not-for-pro¬t businesses do not have owners, and their goals stem from a
mission statement that generally involves service to society. In this situation,
market risk clearly is not relevant; the concern to managers is the impact of
the project on the riskiness of the business, which is measured by a project™s
312 Healthcare Finance



corporate risk. Thus, the risk measure most relevant here is corporate risk.
Again, however, the stand-alone risk of large projects that could sink the
business is relevant.


Self-Test 1. Explain the situations in which each of the risk types”stand-alone,
Question corporate, and market risk”are relevant.


Interpretation of the Risk Measures
It is important to recognize that none of the risk measures discussed can
be interpreted without some standard of reference. For example, suppose
that Investors™ Healthcare, a for-pro¬t company, is evaluating a project that
has a 0.7 coef¬cient of variation of returns. Does the project have high,
low, or moderate stand-alone risk? We don™t know the answer without more
information. However, knowing that Investor™s Healthcare has an average
coef¬cient of variation of returns of 0.4 on all of its projects enables us to
state that the project has more stand-alone risk than does the average project.
Similarly, suppose the project under consideration has a corporate beta
of 1.4. We know that it has above-average corporate risk because any business
in the aggregate, including Investors™ Healthcare, has a corporate beta of 1.0.
This same project might have a market beta of 1.2. This indicates that the
project is riskier than the average project held in a large stock portfolio, but
how does the project™s market risk compare to the market risk of Investors™
Healthcare? If the business (i.e., its stock) has a market beta of 0.9, the project
also has above-average market risk as compared to the business as a whole.
The point here is that risk is always interpreted against some standard because
without a standard it is impossible to make judgments.
Which risk judgment is most relevant to Investors™ Healthcare? As dis-
cussed in the previous section, market risk is most relevant because the busi-
ness taking on the project is investor owned, and hence managers should be
most concerned about the impact of a new project on stockholders™ risk. How-
ever, as explained in the previous section, corporate and stand-alone risk might
also have some relevance. The good news here is that the proposed project
has above-average risk, when compared to the business™s average project, re-
gardless of which risk measure is used.


Self-Test 1. How are risk measures interpreted?
Question

The Relationship Between Risk and Required Return
This chapter contains a great deal of discussion that focuses on de¬ning and
measuring ¬nancial risk. However, being able to de¬ne and measure ¬nancial
313
Chapter 10: Financial Risk and Required Return



risk is of no value in ¬nancial decision making unless risk can be related to
return; that is, the answer to this question is needed: How much return is
required to compensate investors for assuming a given level of risk? In this
section, the focus is on setting required rates of return on stock investments,
but in other chapters the focus is on setting required rates of return on
individual projects within businesses.

The Capital Asset Pricing Model (CAPM)
The relationship between the market risk of a stock, as measured by its market
beta, and its required rate of return is given by the Capital Asset Pricing Model
(CAPM).
To begin, some basic de¬nitions are needed:

• E(Re) = Expected rate of return on a stock. Because an investment in
stocks is called an equity investment, we use the subscript “e” (as opposed
to “s”) to designate the return.
• R(Re) = Required rate of return on a stock. If E(Re) is less than R(Re), the
stock would not be purchased or it would be sold if it was owned. If E(Re)
was greater than R(Re), the stock should be bought, and a person would
be indifferent about the purchase if E(Re) = R(Re).
• RF = Risk-free rate of return. In a CAPM context, RF is generally
measured by the return on long-term U.S. Treasury bonds.
• b = Market beta coef¬cient of the stock. The market beta of an
average-risk stock is 1.0.
• R(RM) = Required rate of return on a portfolio that consists of all stocks,
which is called the market portfolio. R(RM) is also the required rate of
return on an average-risk (b = 1.0) stock. Note that in practice the market
portfolio is proxied by some stock index such as the NYSE Index or the
S&P 500 Index.
• RPM = Market risk premium = R(RM) ’ RF. This is the additional return
over the risk-free rate required to compensate investors for assuming
average (b = 1.0) risk.
• RPe = Risk premium on the stock = [R(RM) ’ RF] — b = RPM — b. A
stock™s risk premium is less than, equal to, or greater than the premium on
an average stock, depending on whether its beta is less than, equal to, or
greater than 1.0. If b = 1.0, then RPe = RPM.

Using these de¬nitions, the CAPM relationship between risk and required
rate of return is given by the following equation, which is called the Security
Market Line (SML):

R (Re ) = RF + (R[RM ] ’ RF) — b
= RF + (RPM — b).
314 Healthcare Finance



In words, the SML tells us that the required rate of return on a stock is
equal to the risk-free rate plus a premium for bearing the risk inherent in that
stock investment. Furthermore, the risk premium consists of the premium
required for bearing average (beta = 1.0) risk, RPM = (R[RM) ’ RF), multi-
plied by the beta coef¬cient of the stock in question. In effect, the market risk
premium is adjusted up or down on the basis of the riskiness of the individual
stock relative to that of the market (or an average stock).
To illustrate use of the SML, assume that the risk-free rate, RF, is 6
percent; the required rate of return on the market, R[RM], is 12 percent; and
the market beta, b, of General Healthcare (GH) stock is 0.8. According to the
SML, a stock investment in GH has a required rate of return of 10.8 percent:

R (RGH ) = 6% + (12% ’ 6%) — 0.8
= 6% + (6% — 0.8)
= 6% + 4.8% = 10.8%.
If the expected rate of return, E(RGH), were 15 percent, investors should buy
the stock because E(RGH) is greater than R(RGH). Conversely, if E(RGH) =
8%, investors should sell the stock because E(RGH) is less than R(RGH).
A stock with a beta of 1.5, one that is riskier than GH, would have a
required rate of return of 15 percent:

R (Rb=1.5 ) = 6% + (6% — 1.5)
= 6% + 9% = 15%.
An average stock, with b = 1.0, would have a required return of 12 percent,
which is the same as the market return:

R (Rb=1.0 ) = 6% + (6% — 1.0)
= 6% + 6% = 12% = R (RM ).
Finally, a stock with below-average risk, with b = 0.5, would have a required
rate of return of 9 percent.

R (Rb=0.5 ) = 6% + (6% — 0.5)
= 6% + 3% = 9%.
The market risk premium, RPM, depends on the degree of aversion that
investors in the aggregate have to risk. In this example, T-bonds yielded RF
= 6% and an average risk stock had a required rate of return of R(RM) = 12%,
so RPM = 6 percentage points. If investors™ degree of risk aversion increased,
R(RM) might increase to 14 percent, which would cause RPM to increase to 8
percentage points. Thus, the greater the overall degree of risk aversion in the
economy, the higher the required rate on the market and hence the higher
the required rates of return on all stocks.
315
Chapter 10: Financial Risk and Required Return



Also, values for the risk-free rate, RF, and the required rate of return
on the market, R(RM), are in¬‚uenced by in¬‚ation expectations. The higher
investor expectations regarding in¬‚ation, the greater these values and hence
the greater the required rates of return on all stocks.
The SML is often expressed in graphical form, as in Figure 10.3, which
shows the SML when RF = 6% and R(RM) = 12%. Here are the relevant points
concerning Figure 10.3:

• Required rates of return are shown on the vertical axis, while risk as
measured by market beta is shown on the horizontal axis.
• Riskless securities have b = 0; therefore, RF is the vertical axis intercept.
• The slope of the SML re¬‚ects the degree of risk aversion in the economy.
The greater the average investor™s aversion to risk the steeper the slope of
the SML, the greater the risk premium for any stock, and the higher the
required rate of return on all stocks.
• The intercept on the Y (vertical) axis re¬‚ects the level of expected
in¬‚ation. The higher in¬‚ation expectations, the greater both RF and
R(RM). Thus, the higher the SML plots on the graph.
• The values previously calculated for the required rates of return on stocks
with b = 0.5, 1.0, and 1.5 agree with the values shown on the graph.

Both the SML and a stock™s position on it change over time because of
changes in interest rates, investors™ risk aversion, and the individual company™s
(stock™s) beta. Thus, the SML, as well as a stock™s risk, must be evaluated on


FIGURE 10.3
Required Rate The Security
of Return (%) Market Line
18

<<

. 11
( 21)



>>