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