KristinLReiterP_2021_Chapter10FinancialRis_GapenskisHealthcareFi (1)

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CHAPTER 367 10 FINANCIAL RISK AND REQUIRED RETURN Learning Objectives After studying this chapter, readers will be able to • Explain in general terms the concept of financial risk. • Define and differentiate between stand-alone risk and portfolio risk. • Define market risk. • Explain the capital asset pricing model (CAPM) relationship between risk and required rate of return. • Use the CAPM to determine required returns. Introduction Two of the most important concepts in investment analysis are financial risk and required return. What is financial risk, how is it measured, and what effect, if any, does it have on required return (opportunity cost rates) and managerial decisions? Because so much of financial decision-making involves risk and return, it is impossible to understand healthcare finance without hav- ing a solid appreciation of risk and return concepts. If investors—both individuals and organizations—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 is an individual investor’s security investment or a radiology group’s invest- ment in diagnostic equipment, must offer a higher return to make it finan- cially 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 they must understand how investors make decisions regarding the supply of capital as well as how businesses make decisions regarding their use of that capital. In addition, the chapter discusses Copyright 2021. AUPHA/HAP Book. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law. EBSCO Publishing : eBook Collection (EBSCOhost) - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS AN: 2558485 ; Kristin L. Reiter, Paula H. Song.; Gapenski's Healthcare Finance: An Introduction to Accounting and Financial Management, Seventh Edition Account: s4264928.main.eds

Gapenski’s Healthcare Finance 368 the relationship between risk and required rate of return. To be truly useful in financial decision-making, it is necessary to know the impact of risk on investors’ views of investment acceptability. The Many Faces of Financial Risk A full discussion of financial risk would take many chapters—perhaps even an entire book—because financial risk is a complicated subject. First of all, the nature of financial 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 proceeds are needed. To make the situation even more complex, it may be difficult to define, measure, or translate financial 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 sufficient to fund the lifestyle expected during the full term of retirement. Needless to say, trans- lating such a definition of risk into investment goals is not easy. The good news is that our primary interest in this book is the financial risk inherent in businesses. Thus, our discussion focuses on the fundamental factors that influ- ence the riskiness of real-asset investments (e.g., land, buildings, equipment). Still, two factors complicate our discussion of financial risk. The first complicating factor is that financial risk is seen both by businesses and by the investors in those businesses. There is some risk inherent in the business that depends on the type of business. For example, biotechnology firms are gen- erally acknowledged to face a great deal of risk, while hospitals typically have less risk. Then, investors, primarily stockholders and debtholders (creditors), bear the riskiness inherent in the business, but as modified by the nature of the securities they hold . For example, the stock of Magellan Health (www. magellanhealth.com), a for-profit behavioral health company, is more risky than its debt, although the risk of both securities depends on the inherent risk of a business that operates in the behavioral health sector. The risk differential arises because of contractual differences between equity and debt: Debthold- ers have a fixed claim against the cash flows and assets of the business, while stockholders have a residual claim to what is left after all other claimants have been paid. Not-for-profit providers have the same partitioning of risk, but the inherent risk of the business is split between the creditors who supply debt capital and the implied stockholders (the community at large). The second complicating factor is that the riskiness of an investment depends on the context in which it is held. For example, a stock held alone EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

Chapter 10: Financial Risk and Required Return 369 (in isolation) is riskier than the same stock held as part of an investor’s large portfolio (collection) of stocks. Similarly, a magnetic resonance imaging (MRI) system operated independently is riskier than the same system oper- ated as part of a large, geographically diversified business that owns and oper- ates numerous types of diagnostic equipment. 1. What complications arise when dealing with financial risk in a business setting? SELF-TEST QUESTION Returns on Investments The concept of return provides investors with a convenient way to express the financial performance of an investment. To illustrate, suppose you buy ten shares of a stock for $100. The stock pays no dividends, and at the end of one year, you sell the stock for $110. What is the return on your $100 investment? One way to express an investment’s return is in dollar terms: Dollar return = Amount to be received − Amount invested = $110 − $100 = $10. If, at the end of the year, you sell the stock for only $90, your dollar return will be −$10. Although expressing returns in dollars is easy, two problems arise. First, to make a meaningful judgment about the return, you need to know the scale (size) of the investment; a $100 return on a $100 investment is a great return (assuming the investment is held for one year), but a $100 return on a $10,000 investment is a poor return. Second, you also need to know the timing of the return; a $100 return on a $100 investment is a great return if it occurs after one year, but the same dollar return after 100 years is not very good. The solution to these scale and timing problems is to express investment results as rates of return or percentage returns. For example, when $1,100 is received after one year, the rate of return on the one-year stock investment is 10 percent: Rate of return = Dollar return ÷ Amount invested = $100 ÷ $1,000 = 0,10 = 10%. EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Gapenski’s Healthcare Finance 370 The rate of return calculation “standardizes” the dollar return by considering the annual return per unit of investment. Introduction to Financial Risk Generically, risk is defined as a hazard, a peril, or an exposure to loss or injury. Thus, risk refers to the chance that an unfavorable event will occur. If a person engages in skydiving, she risks injury or death. If a person gambles at roulette, he is not risking injury or death but is taking a financial risk . Even when a person invests in stocks or bonds, she risks losing money in the hope of earning a positive rate of return. Similarly, when a healthcare business invests in new assets such as diagnostic equipment or a new cancer hospital, it is taking a financial risk. To illustrate financial risk, consider two potential personal investments. The first investment consists of a one-year, $1,000 face value US Treasury bill (or T-bill) bought for $950. Treasury bills are short-term federal debt that are sold at a discount (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 = 0.0526, or 5.26%. This calculation can be done using a spreadsheet, as follows: A B C D 1 2 1 Nper Number of periods 3 $ (950.00) Pv Present value 4 $ 1,000 Fv Future value 5 6 7 8 5.26% =RATE(A2„A3,A4) (entered into Cell A8) 9 10 The $1,000 payment is fixed by contract (the T-bill promises to pay this amount), and the US government is certain to make the payment, unless a national disaster prevents it, which is unlikely. Thus, there is virtually a 100 percent probability that the investment will actually earn the 5.26 percent rate of return that is expected. In this situation, the investment is defined as being riskless or risk-free . 1 Now, assume that the $950 is invested in a biotechnology partnership that will be terminated in one year. If the partnership develops a new, com- mercially 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%: EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

Chapter 10: Financial Risk and Required Return 371 A B C D 1 2 1 Nper Number of periods 3 $ (950.00) Pv Present value 4 $ 2,000 Fv Future value 5 6 7 8 110.53% =RATE(A2„A3,A4) (entered into Cell A8) 9 10 On the other hand, if the partnership does not develop anything worthwhile, it will be worthless, no money will be received, and the rate of return will be ($0 − $950) ÷ $950 = −1.00 = −100%: A B C D 1 2 1 Nper Number of periods 3 $ (950.00) Pv Present value 4 $ 0.01 Fv Future value 5 6 7 8 –100.00% =RATE(A2„A3,A4) (entered into Cell A8) 9 10 Note that spreadsheets give no solution when the future value is zero, but if a small number—for example, 0.01—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 5.26 percent as the expected return on the T-bill investment: (0.50 × 110.53%) + [0.50 × (−100%)] = 5.26%. However, the biotechnology partnership is a far cry from being riskless. 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 chance of earning a return that is less than expected, the partnership investment is described as risky. Thus, financial 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 financial risk. We should note that defin- ing financial risk as the probability of earning a return far below that expected is somewhat simplistic. As we discussed previously, there are many different ways of viewing financial risk. However, the simple definition presented here financial risk In an investment context, the risk that the return on an investment will be less than expected. The greater the chance of earning a return far below that expected, the greater the risk. In a capital structure context, it is the risk added to a business (more precisely, to the business’s owners) when debt financing is used. The greater the proportion of debt financing, the greater the financial risk. EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

Gapenski’s Healthcare Finance 372 is a good starting point for discussing the types of risk that are most relevant to decisions made within health services organizations. 1. What is a generic definition of risk? 2. Give an example of financial risk. SELF-TEST QUESTIONS Risk Aversion Why are defining and measuring financial risk so important? It is because, for the most part, both individual and business investors dislike risk. Suppose you were given the choice between taking a sure $1 million and flipping a coin for either zero or $2 million. You—and just about everyone else—would likely take the sure $1 million. A person who 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 financial risks, so all of us at some time typically exhibit risk-seeking behavior. How- ever, most individuals would never put a sizable portion of their wealth at extreme risk, and most health services managers would never “bet the busi- ness.” Most people are risk averse when it really matters. What are the implications of risk aversion for financial decision- making? First, given two investments with similar returns but different risk, investors will favor the lower-risk alternative. Second, investors will require higher returns on higher-risk investments. These typical outcomes of risk aversion have a significant impact on many facets of financial decision-making and appear over and over in the remainder of this book. 1. What does the term risk aversion mean? 2. What are the implications of risk aversion for financial decision-making? SELF-TEST QUESTIONS 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 risk aversion The tendency of individuals and businesses to dislike risk. The implication of risk aversion is that riskier investments must offer higher expected rates of return to be acceptable. EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Chapter 10: Financial Risk and Required Return 373 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 roll of a die, the probability distribution looks like this: Outcome Probability 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 0.1667 = 16.67% 1.0000 = 100% All possible outcomes (i.e., the number of dots showing after the die roll) are listed in the left column, and the probability of each outcome is listed in the right column, in both decimal and percentage format. For a complete probability 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—of 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 than 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 probability of dividends and stock prices having subpar per- formance, the higher the probability that the return will be significantly less than expected and hence the greater the risk. To illustrate the concept using a business investment, consider a radi- ology group evaluating the purchase of a new CT scanner. The cost of the scanner is an investment, and the net cash inflows that stem from patient utilization provide the return. The net cash inflows, in turn, depend on the number of procedures performed, the 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 of the technology, local market conditions, labor and supplies costs, and so on. The radiology group faces a probability distribution of returns rather than a single return known with certainty. The greater the probability of returns far below the return anticipated, the greater the risk of the scanner investment. probability distribution All possible outcomes of a random event along with their probabilities of occurrence; for example, the probability distribution of rates of return on a proposed investment. capital gain (loss) The profit (loss) from the sale of certain investments at more (less) than their purchase price. EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

Gapenski’s Healthcare Finance 374 1. What is a probability distribution? 2. How are probability distributions used in financial decision-making? SELF-TEST QUESTIONS Expected and Realized Rates of Return To be most useful, the concept of financial risk must be defined more pre- cisely than “the chances of a return far below that anticipated.” Exhibit 10.1 contains the estimated return distributions developed by the financial staff of Suffolk Community Hospital for two proposed projects: an emergency room (ER) expansion and a walk-in clinic. Here, each economic state reflects a combination of factors that dictate each project’s profitability. For example, for the ER project, the very poor economic state signifies a competitive mar- ket and low general utilization, low reimbursement rates, high usage by unin- sured patients, high operating costs, and so on. Conversely, the very good economic state assumes high utilization by insured patients, high reimburse- ment rates, low operating costs, and so on. The economic states are defined in a similar fashion for the walk-in clinic. The expected rate of return , defined in the statistical sense, is the weighted average of the return distribution, where the weights are the prob- abilities of occurrence. For example, the expected rate of return on the ER expansion, E ( R ER ), is 10 percent: E ( R ER ) = Probability of Return 1 × Return 1 + Probability of Return 2 × Return 2 + Probability of Return 3 × Return 3 + . . . = [0.10 × (−10%)] + (0.20 × 0%) + (0.40 × 10%) + (0.20 × 20%) + (0.10 × 30%) = 10%. 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, and each faced the return distribution given in exhibit 10.1, the average return on the 1,000 investments would be 15 percent, assuming that the returns in expected rate of return The return expected, in a statistical sense, on an investment when the purchase is made. EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

Chapter 10: Financial Risk and Required Return 375 each area are independent of one another (i.e., 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 ER investment) is risky. Expected rate of return expresses an expectation for the future. When the managers at Suffolk Community Hospital analyzed the ER 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 occurs. In this case, the realized rate of return , which is the rate of return that the investment produced as measured at termination, would be –10 percent. It is the potential of realizing a –10 percent return on an investment that has an expected return of +10 percent that produces financial risk. Note that in many situations, especially those arising in classroom examples, 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 that the given distribution truly reflects all of the potential investment outcomes, there is zero probability of actually realizing the 10 percent expected rate of return. Still, the 10 percent expected rate of return is the best measure of the profitability of the investment at the time it is being analyzed. Key Equation: Expected Value The expected value of a probability distribution can be found by sum- ming the products of the value in each state multiplied by its probabil- ity (Prob.) of occurrence. To illustrate, assume that the returns on an investment have the following probability distribution: Probability of Occurrence Rate of Return 0.20 0% 0.60 10 0.20 25 The expected rate of return, E ( R ), is 11.0 percent: E ( R ) = (Prob. 1 × Return 1) + (Prob. 2 × Return 2) + (Prob. 3 × Return 3) = (0.20 × 0%) + (0.60 × 10%) + (0.20 × 25%) = 0.0% + 6.0% + 5.0% = 11.0%. realized rate of return The return achieved on an investment when it is terminated. EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Gapenski’s Healthcare Finance 376 1. How is the expected rate of return calculated? 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? SELF-TEST QUESTIONS Stand-Alone Risk We can look at the two return distributions in exhibit 10.1 and conclude that the clinic is riskier than the ER expansion because the clinic has a chance of a 20 percent loss, while the worst possible loss on the ER is only 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 Suffolk’s only asset (operated in isola- tion). In the next section, portfolio effects are introduced, but for now, we can continue our discussion of stand-alone risk. Stand-alone risk depends on the “tightness” of an investment’s return distribution—that is, the amount of dispersion (size) of the 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 tight distribution of returns, and hence low stand-alone risk, but its expected rate of return might be very low, say, 2 percent. In this situation, the investment may not be financially attrac- tive, in spite of its low risk. Similarly, a high-risk investment with a sufficiently high expected rate of return would be attractive. 2 stand-alone risk The riskiness of an investment that is held in isolation as opposed to held as part of a portfolio (collection of investments). Probability Rate of Return if State Occurs Economic State of Occurrence ER Clinic Very poor 0.10 –10% –20% Poor 0.20 0 0 Average 0.40 10 15 Good 0.20 20 30 Very good 0.10 30 50 1.00 EXHIBIT 10.1 Suffolk Community Hospital: Estimated Returns for Two Proposed Projects EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

Chapter 10: Financial Risk and Required Return 377 It is also important to recognize that the size of an investment does not affect its risk. For example, a $1,000 investment in Tenet Healthcare (www.tenethealth.com) bonds has the same risk as a $1,000,000 invest- ment. The amount of capital (money) at risk does not change the riskiness of the investment—the bonds have a certain amount of risk regardless of the amount bought. What may differ, however, is the investor’s ability to bear that risk. An individual with modest income may be able to bear the risk of the $1,000 investment but not the $1,000,000 investment. On the other hand, a large fixed-income mutual fund would easily be able to bear the risk of the $1,000,000 investment. Remember, though, that the size of an invest- ment does not change its inherent risk. To be truly useful, any definition of risk must be measurable, so we need some numerical way to specify the “degree of tightness” of an invest- ment’s return distribution. One such measure is the variance . Variance is a measure of the dispersion of a distribution around its mean (expected value), but it is less useful than standard deviation because its measurement unit is percent (or dollars) squared, which has no economic meaning. Standard deviation , which is often denoted by the symbol σ (low- ercase Greek sigma), is a common statistical measure of the dispersion of a distribution around its mean—the smaller the standard deviation, the tighter the distribution and the lower the stand-alone risk of the investment. To illustrate the calculation of standard deviation, consider the estimated returns of the ER investment, as listed in exhibit 10.1. Here are the steps: 1. The expected rate of return on the ER, E ( R ER ), is 10 percent. 2. The variance of the return distribution is determined as follows: Variance = (Probability of Return 1 × [Rate of Return 1 − E ( R ER )] 2 ) + (Probability of Return 2 × [Rate of Return 2 − E ( R ER )] 2 ) + . . . = (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. 3. The standard deviation is defined as the square root of the variance: ( ) σ = = = ≈ Standard deviation Variance 120.00 10.95% 11%. Most spreadsheet programs have built-in functions that calculate stan- dard deviation. However, these functions assume that the distribution values standard deviation ( σ ) A statistical measure of the variability (dispersion) of a probability distribution about the mean (expected value). EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

Gapenski’s Healthcare Finance 378 that are entered have equal probabilities of occurrence and are not usable with the types of distributions contained in exhibit 10.1. The spreadsheet functions are designed to handle historical data, such as annual returns over the past five years, as opposed to forecast distributions with unequal prob- abilities of occurrence. Using the same procedure, the clinic investment listed in exhibit 10.1 was found to have a standard deviation of returns of about 18 percent. Because the standard deviation of returns of the clinic investment is larger than that of the ER investment, the clinic investment has more stand-alone risk than the ER investment. 1. What is stand-alone risk? 2. Are risk and return separate attributes of an investment? Explain. 3. Does the amount of money at risk affect the riskiness of an investment? Explain. 4. Define and explain the most common measure of stand-alone risk. SELF-TEST QUESTIONS Portfolio Risk and Return The preceding section introduced a risk measure—standard deviation—that applies to investments held in isolation. There is a problem, however, with viewing risk in the stand-alone sense: Most investments are held not in iso- lation but as part of a collection, or portfolio , of investments. Individual investors typically hold portfolios of 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 real- ized rate of return on the entire portfolio. Similarly, the stand-alone risk of each individual investment in the portfolio is not important to the investor; what matters is the aggregate risk of the portfolio, or portfolio risk . Thus, the nature of risk and how it is defined and measured changes when invest- ments are held not in isolation but as part of a portfolio. Portfolio Returns Consider the realized returns for the seven investment alternatives listed in exhibit 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 Joyce Dunne. The remaining three alternatives in exhibit 10.2 are portfolios. portfolio A number of individual investments held collectively. portfolio risk The riskiness of an individual investment when it is held as part of a diversified portfolio (collection of investments) as opposed to held in isolation. EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Chapter 10: Financial Risk and Required Return 379 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. Port- folio AD is an equal-weighted portfolio of investments A and D. As shown in the bottom of the exhibit, 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 percent), while invest- ments C and D have greater stand-alone risk than A and B. The rate of return on a portfolio , R P , is the weighted average of the expected returns on the assets that make up the portfolio, with the weights reflecting the proportion of the total portfolio invested in each asset: ( R P ) = [ w 1 × E ( R 1 )] + [ w 2 × E ( R 2 )] + [ w 3 × E ( R 3 )] + . . . . In this case, w 1 is the proportion of investment 1 in the overall port- folio, E ( R 1 ) is the expected rate of return on investment 1, and so on. Thus, the expected rate of return on portfolio AB is 10 percent: R AB = ( w A × R A ) + ( w B × R B ) = (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 13.5 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 exhibit 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: R AC = [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 that calculated from the expected rates of return of the two portfolio components: R AC = (0.5 × 10%) + (0.5 × 15%) = 12.5%. EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

Gapenski’s Healthcare Finance 380 Portfolio Risk: Two Investments When an investor holds a portfolio of investments, the portfolio is in effect a stand-alone investment, so the riskiness of the portfolio is measured by the standard deviation of portfolio returns , the previously discussed measure of stand-alone risk. How does the riskiness of the individual invest- ments 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 individual 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 investments may even be riskless. A simple example can be used to illustrate this concept. Suppose an individual is given the following opportunity: Flip a coin once; if it comes up heads, he wins $10,000, but if it comes up tails, he 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 indi- vidual has a 50 percent chance of losing $8,000. Thus, risk aversion would cause most people to refuse the gamble, especially if the $8,000 potential loss would result in financial hardship. Alternatively, suppose an individual is given the opportunity to flip the coin 100 times, and she wins $100 for each heads but loses $80 for each tails. It is possible, although extremely unlikely, that she could flip all heads and win $10,000. It is also possible, and also extremely unlikely, that she could flip all tails and lose $8,000. But the chances are high that she will flip close to 50 heads and 50 tails and net about $1,000. Even if she flipped a few more tails than heads, she would still make money on the gamble. Rate of Return if State Occurs Year A B C D AB AC AD 1 –10% 30% –25% 15% 10% –17.5% 2.5% 2 0 20 –5 10 10 –2.5 5.0 3 10 10 15 0 10 12.5 5.0 4 20 0 35 25 10 27.5 22.5 5 30 –10 55 35 10 42.5 32.5 Rate of return 10.0% 10.0% 15.0% 17.0% 10.0% 12.5% 13.5% Standard deviation 15.8% 15.8% 31.6% 13.5% 0.0% 23.7% 13.3% EXHIBIT 10.2 Realized Returns for Four Individual Investments and Three Portfolios EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

Chapter 10: Financial Risk and Required Return 381 Although each flip is risky in the stand-alone sense, collectively, the flips are not risky at all. In effect, the multiple flipping creates a portfolio of investments; each flip of the coin can be thought of as one investment, so the individual now has a 100-investment portfolio. Furthermore, 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 flip of the coin regard- less of the results of the previous flips. By combining the flips into a single gamble (i.e., into an investment portfolio), the risk associated with each flip of the coin is reduced. In fact, if the gamble consisted of a large enough number of flips, 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 profit. The key to the risk reduction inherent in the portfolio is that the negative consequences of tossing tails can be offset by the positive consequences of tossing heads. To examine portfolio effects in more depth, consider portfolio AB in exhibit 10.2. Each investment (A or B) has a standard deviation of returns of 10 percent and is quite risky when held in isolation. However, a portfolio comprising 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 verified by the value of zero for portfolio AB’s standard deviation of returns. The reason investments A and B can be combined to form a riskless portfo- lio is that their returns move exactly oppo- site to one another: 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. For Your Consideration The Gambler’s Fallacy On August 18, 1913, at the casino in Monte Carlo, a rare event occurred at the roulette wheel. (A roulette wheel contains black and red spaces, and the ball has roughly a 50/50 chance of landing on either color each time the wheel is spun.) On that day, the ball landed on a black space a record 26 times in a row. As a result, beginning when black had come up a phenomenal 15 times, there was a wild rush to bet on red. As each spin continued to come up black, players doubled and tripled their stakes, and, after 20 spins, many expressed the belief that there was not a chance in a million of another black. By the time the run ended, on the 27th spin, the casino was millions of francs richer. The players at the roulette wheel that day committed what is known as the “gambler’s fal- lacy,” also called the “Monte Carlo fallacy.” Sim- ply put, the fallacy rests on the assumption that some result must be “due” simply because what previously happened departed from what was expected to happen. In this example, one spin of the roulette wheel does not affect the next spin. Thus, each time the ball is rolled, there is (ideally) a 50 percent chance of landing on a black and a 50 percent chance of landing on a red. Even after many blacks have occurred in succession, there is still a 50/50 chance of a red because the results of previous spins have no bearing on the outcome of the next spin—the roulette wheel has no memory. What do you think of the gambler’s fal- lacy? What conditions must hold for the fallacy to apply? Are there situations that might occur that would cause the fallacy to be correct (past results predict future results)? If people are smart enough to recognize the fallacy, why do Las Vegas casino owners make so much money? EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Gapenski’s Healthcare Finance 382 The movement relationship of two variables (i.e., their tendency to move either together or in opposition) is called correlation . The correla- tion coefficient , represented by the variable 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. For every state in which 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 invest- ments move up and down together as the economic state changes. When the returns on two investments are perfectly positively correlated, combining the investments into a portfolio will not lower risk because the standard devia- tion of the portfolio is simply the weighted average of the standard deviations of the two components. To illustrate the impact of perfect positive correlation, consider port- folio AC in exhibit 10.2. Its rate of return, R AC , is 12.5 percent, while its standard deviation is 16.4 percent. Because of perfect positive correlation between the returns on A and C, portfolio AC’s standard deviation is the weighted average standard deviation of its components: σ AC = (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 contains two investments that have positive, but not perfectly positive, correlation? Combining the two invest- ments can eliminate some, but not all, risk. To illustrate, consider portfolio AD in exhibit 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 the component standard deviations, (0.5 × 11%) + (0.5 × 12.1%) = 11.6%, it also is less than the standard deviation of each component. The correlation coefficient between the return distribu- tions for A and D is 0.53, which indicates that the two investments are posi- tively correlated, but they are not perfectly correlated because the coefficient is less than +1.0. Combining two investments that are positively correlated, but not perfectly so, lowers risk but does not eliminate it. 3 Because correlation is the factor that drives risk reduction, a logi- cal question arises: What is the correlation among returns on “real-world” correlation The movement relationship between two variables. correlation coefficient A standardized measure of correlation that ranges from − 1 (variables move perfectly opposite of one another) to +1 (variables move in perfect synchronization); denoted by r . EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

Chapter 10: Financial Risk and Required Return 383 investments? Generalizing about the correlations among real-world invest- ment alternatives is difficult. However, it is safe to say that the return distri- butions of two randomly selected investments—whether they are real assets in a multispecialty group practice’s portfolio of service lines or financial assets in an individual’s investment portfolio—are virtually never perfectly correlated, so correlation coefficients are never −1.0 or +1.0. In fact, it is almost impossible to find actual investment opportunities with returns that are negatively correlated with one another or even to find investments with returns that are uncorrelated ( r = 0). Because all investment returns are affected to a greater or lesser degree by general economic conditions, invest- ment returns tend to be positively correlated with one another. However, because investment returns are not affected identically by general economic conditions, returns on most real-world investments are not perfectly posi- tively correlated. The correlation coefficient between the returns of two randomly chosen investments will usually fall in the range of +0.3 to +0.8. Returns on investments that are similar in nature, such as two inpatient projects in a hospital or two stocks in the same sector, will typically have correlations at the upper end of this range. Conversely, returns on dissimilar projects or securi- ties will tend to have correlations at the lower end of the range. For real-world correlations, consider exhibit 10.3, which shows the correlation coefficients between several investment classes. The base for all correlations is the Standard & Poor’s (S&P) 500 index, which is a diversified portfolio of large-firm stocks. Adding asset classes with negative or low cor- relation to traditional investments such as the S&P 500 can increase portfolio diversification. Portfolio Risk: Many Investments Businesses are not restricted to two projects, and individual investors are not restricted to holding two-security portfolios. Most companies have tens, or even thousands, 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 a multitude of individual securities. Thus, what is most relevant to investment decision-making is not what happens when two investments are combined into portfolios, but what happens when many investments are combined. To illustrate the impact on risk of creating large portfolios, consider exhibit 10.4. The graph illustrates the risk inherent in holding randomly selected portfolios of one investment, two investments, three investments, four investments, and so on, considering the correlations that occur among real-world investments. The plot is based on historical annual returns on common stocks, but the conclusions apply to portfolios made up of any type EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

Gapenski’s Healthcare Finance 384 of risky investment, including portfolios of real-asset investments held by healthcare providers. The plot shows the average standard deviation of all one-investment (one-stock) portfolios, the average standard deviation of all possible two-investment portfolios, the average standard deviation of all pos- sible three-investment portfolios, and so on. The risk inherent in holding an average one-investment portfolio is relatively high, as measured by its standard deviation of annual returns. The average two-investment portfolio has a lower standard deviation, so hold- ing an average two-investment portfolio is less risky than holding a single investment of average risk. The average three-investment portfolio has an even lower standard deviation of returns, so an average three-investment portfolio is less risky than an average two-investment portfolio. As more investments are randomly added to create larger and larger portfolios, the riskiness of the portfolio decreases. However, as more and more investments are added, the incremental risk reduction of adding even more investments decreases, and regardless of how many are added, some risk always remains in the portfolio—even with a portfolio of thousands of investments, substan- tial risk remains. 4 All risk cannot be eliminated by creating a large portfolio because 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 financial, are affected to a lesser or greater degree by general economic conditions. If the economy is booming, all investments tend to do well, while in a recession, all investments tend to do poorly. It is the positive correlation among real-world investment returns that prevents investors from creating riskless portfolios. Investment 1 Investment 2 Correlation Coefficient Investment-grade bonds S&P 500 –0.06 Cash S&P 500 –0.08 Commodities S&P 500 0.52 Currencies S&P 500 –0.52 Global indexes S&P 500 0.97 Hedge funds S&P 500 0.74 Real estate investment trusts S&P 500 0.73 S&P 500 S&P 500 1.00 EXHIBIT 10.3 Correlations Among Selected Asset Classes, January 2009– December 2018 Source: Guggenheim Investments, “Asset Class Correlation Map,” January 2009–Decem- ber 2018, www.guggenheiminvestments.com/mutual-funds/resources/interactive-tools/ asset-class-correlation-map. EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Chapter 10: Financial Risk and Required Return 385 Diversifiable Risk Versus Market Risk Exhibit 10.4 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 inher- ent in an individual investment can be eliminated if it is held as part of a large portfolio. For example, if a stock investor wanted to eliminate as much stand-alone risk as possible, he would have to own more than 6,500 stocks. Fortunately, it is not necessary to purchase all the stocks individu- ally because mutual funds that mimic all of the major stock indexes are available. 5 A portfolio that consists of a large number of stocks is called the market portfolio because it consists of the entire stock market, or at least one entire segment of the stock market. Studies have found that the market portfolio has only about one-half the standard deviation of an average stock. However, it is not necessary for individual investors to own the market portfolio to take advantage of the risk-reducing benefits of diversification. As illustrated in exhibit 10.4, most of the benefit can be obtained by hold- ing about 50 randomly selected stocks. Such a collection of investments is called a well-diversified portfolio . That part of the stand-alone risk of an individual investment that can be eliminated by diversification (i.e., by holding it as part of a well-diversified market portfolio A portfolio that contains all publicly traded stocks; often proxied by some market index, such as the S&P 500. 1 10 20 30 40 50 Number of Investments in the Portfolio Market Risk Diversifiable Risk Stand- Alone Risk Risk, σ p (%) EXHIBIT 10.4 Portfolio Size and Risk EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

Gapenski’s Healthcare Finance 386 portfolio) is called diversifiable risk . That part of the stand-alone risk of an individual investment that cannot be eliminated by diversification is called market risk . Every investment, whether it is the stock of Tenet Healthcare held by an individual investor or a CT scanner operated by a radiology group, has some diversifiable risk that can be eliminated and some portfolio risk that cannot be diversified away. Not all investments benefit to the same degree from portfolio risk-reducing effects, and some portfolios are not truly well diversified. In general, however, any investment will have some of its stand- alone risk eliminated when it is held as part of a portfolio. Diversifiable risk, as seen by individuals who invest in stocks, is caused by events that are unique to a single business, such as the introduction of a new service, a labor strike, or a lawsuit. Because these events are essentially random, their effects can be eliminated by diversification. When one stock in a large, well-diversified portfolio does worse than expected because of a nega- tive event that is unique to that firm, another stock in the portfolio will do better than expected because of a firm-unique positive event. On average, bad events in some companies will be offset by good events in others, so lower than expected returns on one stock will be offset by higher-than-expected returns on another, leaving the investor with an overall portfolio return closer to that expected than would be the case if only a single stock were held. Key Equation: Diversifiable Versus Portfolio Risk The stand-alone risk of an investment can be broken down into two parts: Stand-alone risk = Diversifiable risk + Market risk. Diversifiable risk is that portion of the stand-alone risk of an investment that can be eliminated by placing it in a well-diversified portfolio, while market risk is the risk that remains (cannot be diversified away). The same logic can be applied to a business with a portfolio of proj- ects. Perhaps 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. Note that if a hospital offered both inpatient and outpatient surgery, it would be hedging against the trend toward more outpatient procedures because reduced demand for inpatient surgery would be offset by increased demand for outpatient surgery. The point here is that the negative impact of random events that are unique to a particular business, or to a particular service within a business, can be offset by positive events in other businesses or in other products or diversifiable risk The portion of the risk of an investment that can be eliminated by holding the investment as part of a diversified portfolio. market risk The risk of an individual investment when it is held as part of a diversified portfolio as opposed to held in isolation. EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

Chapter 10: Financial Risk and Required Return 387 services. Thus, the risk caused by random, unique events can be eliminated by portfolio diversification. Individual investors can diversify by holding many different securities, and health services businesses can diversify by offering many different services. All investments do not benefit to the same degree from portfolio risk reduction. Some investments have a large amount of diversifiable risk and hence have a great deal of risk reduction when they are added to a portfo- lio. Others do not benefit nearly as much from portfolio risk reduction—it depends on the characteristics of both the investment and the portfolio. For example, consider the addition of a hospital management company’s stock to two portfolios. The first portfolio consists of the stocks of 50 health- care providers. The second portfolio consists of the stocks of 50 randomly selected companies from many different industries. Much less risk reduction will occur when the hospital stock is added to the healthcare portfolio than when it is added to the randomly selected portfolio. The reason should be obvious: Hospital returns are more highly correlated with healthcare provid- ers than with firms in other industries, and the higher the correlation, the less the risk reduction. This logic tells us that there is more risk reduction potential inherent in adding a nursing home to a hospital business than there is in adding it to a long-term care business that already owns a large number of such invest- ments. However, recognize that it is probably more difficult for managers of a hospital to manage a nursing home than it is for managers that specialize in such businesses. Unfortunately, not all risk can be diversified away. Market risk—the risk that remains even in a well-diversified portfolio—stems from factors (e.g., wars, inflation, recessions, high interest rates) that systematically affect all stocks in portfolio or all products and services produced by a system. For example, the increasing market power of managed care organizations or gov- ernment payers could lower reimbursement levels for all services offered by a hospital. No amount of diversification by the hospital into different health- care service lines could eliminate this risk. Implications for Investors The ability to eliminate a portion of the stand-alone risk inherent in indi- vidual investments has two significant implications for investors, whether the investor is an individual who holds securities or a healthcare business that offers service lines: 1. Holding a single investment is not rational. Holding a large, well-diversified portfolio can eliminate much of the stand-alone risk inherent in individual investments. Investors who are risk averse should EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Gapenski’s Healthcare Finance 388 seek to eliminate all diversifiable risk. Individual investors can easily diversify their personal investment portfolios by buying many individual securities or mutual funds that hold diversified portfolios. Businesses cannot diversify their investments as easily as individuals can, but healthcare businesses that offer a diverse line of services are less risky than those that rely on a single service. 2. Because an investment held in a portfolio has less risk than one held in isolation, the traditional stand-alone risk measure of standard deviation is not appropriate for individual assets when they are held as parts of portfolios. Thus, it is necessary to rethink the definition and measurement of financial risk for such investments. (Note, though, that standard deviation remains the correct measure for the riskiness of an investor’s entire portfolio because the portfolio is, in effect, a single investment held in isolation.) 1. What is a portfolio of assets? 2. What is a well-diversified 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, diversifiable risk, and market risk. 5. Why should all investors hold well-diversified 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? SELF-TEST QUESTIONS The Relevant Risk of a Stock If investors are risk averse, they will demand a premium for bearing risk—that is, the higher the risk of a security, the higher its expected return must be to induce investors to buy it or to hold it. All risk except that related to broad market movements can easily be eliminated through diversification. This implies that investors are primarily concerned with the risk of their portfolio rather than the risk of the individual securities in the portfolio. How, then, should the risk of an individual stock be measured? The capital asset pric- ing model (CAPM) provides one answer to that question. A stock might be quite risky if it is held by itself, but—because diversification eliminates about capital asset pricing model (CAPM) An equilibrium model that specifies the relationship between a stock’s value and its market risk, as measured by beta ( β ). EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Chapter 10: Financial Risk and Required Return 389 half of its risk—the stock’s relevant risk is its contribution to a well-diversified portfolio’s risk, which is much smaller than the stock’s stand-alone risk. A well-diversified portfolio has only market risk. Therefore, the CAPM defines the relevant risk of an individual stock as the amount of risk that the stock contributes to the market portfolio, which is a portfolio containing all stocks. The relevant measure of risk is called the beta coefficient , or simply beta (denoted by β ). The beta of stock i is calculated as: β i = ( σ i / σ m ) × r im , where σ i = standard deviation of stock i ’s returns, σ m = standard deviation of the market’s returns, and r im = correlation coefficient between the returns on stock i and the market’s returns. This equation shows that a stock with a high standard deviation, σ i , will have a high beta, which means that, all else held constant, the stock con- tributes a lot of risk to a well-diversified portfolio. This makes sense, because a stock with high stand-alone risk will tend to destabilize a portfolio. Note, too, that a stock with a high correlation with the market, r im , will also tend to have a large beta and hence greater risk. This also makes sense, because a high correlation means that diversification is not helping much—the stock will perform well when the portfolio is performing well, and the stock will perform poorly when the portfolio is performing poorly. The beta of any portfolio of investments is simply the weighted aver- age of the individual investments’ betas: β Portfolio = ( w 1 × β 1 ) + ( w 2 × β 2 ) + ( w 3 × β 3 ) + ( w i × β i ), and so on. Here, β Portfolio is the beta of the portfolio, which measures the volatility of the entire portfolio; w i is the fraction of the portfolio invested in each particu- lar asset; and β i is the beta coefficient of that asset. For ease of illustration, assume that an investor has only the four stocks listed in the following table: Stock Beta ( β i ) Weight in Portfolio ( w i ) 1 0.6 25% 2 1.0 25% 3 1.4 25% 4 1.8 25% beta coefficient ( β ) A measure of the risk of one investment relative to the risk of a collection (portfolio) of investments. EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Gapenski’s Healthcare Finance 390 The weighted average of the stock betas is the portfolio beta: β Portfolio = (0.6 × 0.25) + (1.0 × 0.25) + (1.4 × 0.25) + (1.8 × 0.25) = 1.20. Four key points about beta are important to know. 1. Beta determines how much risk a stock contributes to a well-diversified portfolio. If all the stocks’ weights in a portfolio are equal, then a stock with a beta that is twice as big as another stock’s beta contributes twice as much risk. 2. The average of all stocks’ betas is equal to 1; the beta of the market also is equal to 1. Intuitively, this is because the market return is the average of all the stocks’ returns. 3. A stock with a beta greater than 1 contributes more risk to a portfolio than does the average stock, and a stock with a beta less than 1 contributes less risk to a portfolio than does the average stock. 4. Most stocks have betas that are between about 0.4 and 1.6. Estimating Beta The CAPM is an ex ante model, which means that all of the variables repre- sent before-the-fact, expected values. In particular, the beta coefficient used by investors should reflect the relationship between a stock’s expected return and the market’s expected return during some future period. However, people generally calculate betas using data from some past period and then assume that the stock’s risk will be the same in the future as it was in the past. Many analysts use four to five years of monthly data, although some use 52 weeks of weekly data. To illustrate, we use the four years (2015–2018) of monthly returns to calculate the betas of Community Health Systems, Tenet Healthcare, and HCA Healthcare using the equation introduced ear- lier for calculating beta: Market (S&P 500) Community Health Systems Tenet Healthcare HCA Healthcare Symbol GSPC CYH THC HCA Standard deviation 3.3% 20.7% 14.4% 7.0% Correlation with the market 0.37 0.36 0.36 β i = ( σ i / σ m ) × ri m 2.32 1.58 0.77 EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Chapter 10: Financial Risk and Required Return 391 Suppose you plot a stock’s returns on the y axis of a graph and the market portfolio’s returns on the x axis. The formula for the slope of a regres- sion line is exactly equal to the formula for calculating beta. Therefore, to estimate beta for a security, you can estimate a regression with the stock’s returns on the y axis and the market’s returns on the x axis using the follow- ing equation: r it = a i + β i r Mt + e it , , where r it and r Mt are the actual returns for the stock and the market for date t ; a i and β i are the estimated regression coefficients; and e it is the estimated error at date t . Exhibit 10.5 illustrates this approach for Community Health Systems and HCA Healthcare. The blue dots represent each of the 48 data points, with the stock’s returns plotted on the y axis and the market’s returns on the x axis. We use the Trendline feature in Excel to show the regression equation on the charts (these are colored blue). It is also possible to use Excel’s SLOPE function to estimate the slope from a regression: SLOPE (known_ y ’s, known_ x ’s). Exhibit 10.5 shows that Community Health Sys- tems has an estimated beta of 2.32 and HCA Healthcare has an estimated beta of 0.77, the same values that we calculated earlier using the equation for beta. The black line is the plot of market versus market (a 45-degree line with a slope of 1.00). Notice that the slope of the regression line of Community Health Systems (2.32) is steeper than the slope of the market line (1.00), suggesting that Community Health Systems is riskier than the market. Conversely, the slope of the regression line of HCA (0.77) is flatter than the market, suggest- ing that HCA is less risky than the market. It is important to remember that beta cannot be observed; it can only be estimated. Community Health Systems has an estimated beta of 2.32. What does that mean? By definition, the average beta for all stocks is equal to 1, so Community Health Systems contributes 132 percent more risk to a well-diversified portfolio than a typical stock (assuming the two have the same portfolio weight). HCA Healthcare has an estimated beta of 0.77, meaning that it contributes 23 percent less risk to a well-diversified portfolio than a typical stock. When the market is doing well, a high-beta stock such as Community Health Systems tends to do better than an aver- age stock, and when the market does poorly, a high-beta stock also does worse than an average stock. The opposite is true for a low-beta stock: When the market soars, the low-beta stock tends to go up by a smaller amount; when the market falls, the low-beta stock tends to fall less than the market. EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Gapenski’s Healthcare Finance 392 The Relationship Between Risk and Return and the Capital Asset Pricing Model This chapter contains much discussion of the definition and measurement of financial risk. However, the ability to define and measure is of no value in financial decision-making unless risk can be related to return. Planners must consider the question, how much return is required to compensate inves- tors for assuming a given level of risk? In this section, we focus on setting required rates of return on stock investments because the basic theory of risk and return was developed with these in mind. In later chapters, the focus is on setting required rates of return on individual projects within firms. 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 CAPM. To begin, some basic definitions are needed: • E ( R i ) = Expected rate of return on stock i (any stock). • R ( R i ) = Required rate of return on stock i . • RF = Risk-free rate of return. In a CAPM context, RF is generally measured by the return on long-term US Treasury bonds (T-bonds). • β i = Market beta coefficient of stock i . The market beta of an average- risk stock is β A = 1.0. • R ( R M ) = Required rate of return on a portfolio that consists of all stocks, which is the market portfolio. R ( R M ) is also the required rate of return on an average ( β A = 1.0) stock. • RP M = Market risk premium = R ( R M ) − RF . This is the additional return over the risk-free rate required to compensate investors for assuming average ( β A = 1.0) risk. • RP i = Risk premium on stock i = [ R ( R M ) − RF ] × b i = RP M × b i . Stock i ’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 β i = β A = 1.0, then RP i = RP M . –60% –40% –20% 0% 20% 40% 60% 60% 40% 20% 0% –20% –40% –60% 60% 40% 20% 0% –20% –40% –60% –60% –40% –20% 0% 20% 40% 60% Community Health Systems HCA Healthcare y = 2.322x – 0.0431 S&P 500 S&P 500 y = 0.7697x + 0.0103 EXHIBIT 10.5 Monthly Stock Returns of Community Health Systems, HCA Healthcare, and S&P 500, 2014–2018 EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Chapter 10: Financial Risk and Required Return 393 Using these definitions, the CAPM relationship between risk and required rate of return is expressed by the following equation, which specifies a line called the security market line (SML) : Required return on stock i = Risk-free rate + Risk premium for stock i Required return on stock i = Risk-free rate + (Beta of stock i ) × (Market risk premium) R ( R i ) = RF + [ R ( R M )] − RF ) × β i = RF + ( RP M × β i ). The SML equation shows that a stock’s required return is determined by the risk-free rate, the market risk premium, and beta. The Risk-Free Rate A stock’s required return begins with the risk-free rate. To induce investors to take on a risky investment, they will need a return that is at least as big as the risk-free rate. The yield on long-term Treasury bonds is often used to mea- sure the risk-free rate when estimating the required return with the CAPM. The Market Risk Premium The market risk premium is the difference between the required rate of return on the market, R ( R M ), and the risk-free rate, RF : Market risk premium = R ( R M ) − RF Market risk premium = RP M . The market risk premium, RP M , is the extra rate of return that inves- tors require to invest in the stock market rather than purchase risk-free secu- rities. The size of the market risk premium depends on the degree of risk aversion that investors have on average. When investors are very risk averse, the market risk premium is high; when investors are less concerned about risk, the market risk premium is low. In this example, T-bonds yield an RF of 6 percent, and an average share of stock has a required rate of return, R ( R M ), of 10 percent, so RP M is 4 percentage points. If the degree of risk aversion is increased, R ( R M ) might increase to 12 percent, which would cause RPM to increase to 6 percentage points. Thus, the greater the overall degree of risk aversion, the higher the required rate on the market and hence the higher the required rates of return on all stocks. The Risk Premium for an Individual Stock The risk premium for an individual stock, RPi , is equal to the product of the stock’s beta and the market risk premium: security market line (SML) The portion of the capital asset pricing model (CAPM) that specifies the relationship between market risk and required rate of return. EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Gapenski’s Healthcare Finance 394 Risk premium for stock i = (Beta of stock i ) × (Market risk premium) Risk premium for stock i = [ R ( R M )– RF ] × β i Risk premium for stock i = ( RP M × β i ). For an illustration of the use of the SML, assume that RF is 6 percent and R ( R M ) is 10 percent. For a low-risk stock with β Low = 0.5, the risk pre- mium for the stock RP Low is: = [ R ( R M )– RF ] × β i = ( RP M × β i ) = (10% − 6%) × 0.5 = (4%) × 0.5 = 2%. The Required Rate of Return for an Individual Stock According to the SML, the required rate of return on the low-risk stock is: R ( R Low ) = Risk-free rate + Risk premium for low-risk stock R (R Low ) = RF + RP Low R ( R Low ) = 6% + 2% R ( R Low ) = 8%. If a high-risk stock has β High = 2.0, then its required rate of return is 14 percent, as we see here: R ( R High ) = 6% + (10% − 6%) × 2.0 R ( R High ) = 6% + (4%) × 2.0 R ( R High ) = 6% + 8% R ( R High ) = 14%. An average stock with β Average = 1.0 has a required return of 10 percent, the same as the market return, which can be seen if we figure it in the fol- lowing way: R ( R Average ) = 6% + (10% − 6%) × 1.0 R ( R Average ) = 6% + (4%) × 1.0 R ( R Average ) = 6% + 4% R ( R Average ) = 10% = R ( R M ). EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Chapter 10: Financial Risk and Required Return 395 The SML is often depicted in graphical form, as in exhibit 10.6, which shows the SML when RF = 6 percent and R ( R M ) = 10 percent. Here are the relevant points concerning the graph: • 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 β i = 0; therefore, RF is the vertical axis intercept. • The slope of the SML reflects the degree of risk aversion in the economy. The greater the average investor’s aversion to risk, (1) the steeper the slope of the SML, (2) the greater the risk premium for any stock, and (3) the higher the required rate of return on stocks. • The y axis intercept reflects the level of expected inflation. The higher inflation expectations are, the greater both RF and R ( R M ) are, and thus the higher the SML plots on the graph. • The values previously calculated for the required rates of return on stocks with β i = 0.5, β i = 1.0, and β i = 2.0 agree with the values shown on the graph. Both the SML and a firm’s position on it change over time because of changes in interest rates, investors’ risk aversion, and individual firms’ betas. Thus, the SML, as well as a firm’s risk, must be evaluated on the basis of cur- rent information. The SML, its use, and how its input values are estimated are covered in greater detail in chapter 13. 18% 12% 6% 0% Required Rate of Return 0.00 0.50 1.00 1.50 2.00 2.50 Beta SML: R ( R i ) = RF + [ R ( R M ) – RF )] β i = 6% + (10% – 6%) β i EXHIBIT 10.6 Security Market Line EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Gapenski’s Healthcare Finance 396 Required Return and Changes in Expected Inflation or Risk Aversion Change in Expected Inflation Exhibit 10.7 shows the impact on the SML of a 2 percent increase in expected inflation. The change causes the risk-free interest rate to increase from 6 percent to 8 percent, but there is no change in the market risk pre- mium (10% − 6% = 12% − 8% = 4%). This happens because as the risk-free rate changes, so will the required return on the market, and this will, other things held constant, keep the market risk premium stable. Exhibit 10.7 also shows that the increase in expected inflation leads to an identical increase in the required rate of return on all assets, because the same risk-free rate is built into the required rate of return on all assets. For example, R ( R M ) (and the average stock) increases from 10 percent to 12 percent. Other risky securities’ returns also rise by 2 percentage points. Change in Risk Aversion The slope of the SML reflects the extent to which investors are averse to risk: The steeper the slope of the line, the greater the average investor’s aversion to risk. Suppose all investors were indifferent to risk—that is, suppose they were not risk averse. If RF = 6 percent, then risky assets would also provide an expected return of 6 percent. If there were no risk aversion, then there would be no risk premium, and the SML would be a horizontal line. As risk aversion increases, so does the risk premium, and this causes the slope of the SML to become steeper. Exhibit 10.8 illustrates the impact of an increase in risk aversion on the SML. In this case, R ( R M ) rises from 10 percent to 12.5 percent. The returns on other risky assets also rise, and the effect of this shift 18% 12% 6% 0% Required Rate of Return 0.00 0.50 1.00 1.50 2.00 2.50 Beta SML: R ( R i ) = 8% + (12% – 8%) β i SML: R ( R i ) = 6% + (10% – 6%) β i EXHIBIT 10.7 Shift in the Security Market Line Caused by an Increase in Expected Inflation EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Chapter 10: Financial Risk and Required Return 397 in risk aversion is greater for riskier securities. For example, the required return on a stock with β i = 0.5 increases by only 1.25 percentage points, from 8.5 percent to 9.75 percent; the required return on a stock with β i = 1.0 increases by 2.5 percentage points, from 10.0 percent to 12.5 percent; and the required return on a stock with β i = 1.5 increases by 3.75 percentage points, from 12.0 percent to 15.75 percent. Comparison of Required Return to Expected Return The CAPM conceptualizes the required return on stock i , R ( R i ), as the risk- free rate plus the extra return (i.e., the risk premium) needed to induce an investor to hold the stock. That is, for a given level of risk as measured by beta, what rate of expected return does an investor require to compensate them for bearing that risk? Here, R ( R i ) can be considered the minimum expected return that is required by an average investor. How do investors use R ( R i )? If the expected rate of return, E ( R i ), is less than R ( R i ), an investor would not purchase stock i or would sell it if it is owned. If E ( R i ) is greater than R ( R i ), an investor would purchase stock i , and an individual would be indifferent about the purchase if E ( R i ) = R ( R i ). Suppose an investor is considering purchase shares of Regis Health- care, which has a beta of 1.1 and an expected rate of return, E ( R Regis ), of 15 percent. If RF is 6 percent and R ( R M ) is 10 percent, then the required rate of return on Regis Healthcare, R ( R Regis ), is calculated as follows: 24% 18% 12% 6% 0% Required Rate of Return 0.00 0.50 1.00 1.50 2.00 2.50 Beta SML: R ( R i ) = 6% + (12.5% – 6%) β i SML: R ( R i ) = 6% + (10% – 6%) β i EXHIBIT 10.8 Shift in the Security Market Line Caused by an Increase in Investor Risk Aversion EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Gapenski’s Healthcare Finance 398 R ( R i ) = RF + [ R ( R M ) – RF ] × b i R ( R Regis ) = 6% + (10% – 6%) × 1.1 R ( R Regis ) = 10.4%. In this circumstance, the expected rate of return is greater than the required rate of return for Regis Healthcare, so an investor would purchase the stock. In summary, • If E ( R i ) > R ( R i ), an investor would purchase stock i. • If E ( R i ) = R ( R i ), an investor would be indifferent about purchase of stock i . • If E ( R i ) < R ( R i ), an investor would not purchase stock i . Some Final Thoughts About Beta and the Capital Asset Pricing Model The CAPM is more than just an abstract theory described in textbooks. It is widely used by analysts, investors, and corporate managers. However, despite its intuitive appeal, at least serious concerns are prompted by the CAPM: 1. It is built on a restrictive set of assumptions that do not conform well to real-world conditions. 2. It is impossible to prove. Studies that do demonstrate the linear relationship between market risk and required return prove nothing because the results stem from the mathematical properties of the model, not because it is theoretically correct. 3. Some studies find no relationship between stocks’ returns and market betas. 4. The market betas that are actually used in the CAPM measure the historical relative volatility of a stock, but conditions often change. Thus, its future volatility, which is of real concern to investors, might be quite different from its past volatility. Despite these concerns, the CAPM is extremely appealing because it is simple and logical. It focuses on the impact that a single investment has on a portfolio, which in most situations is the correct way to think about risk. Furthermore, it tells us that the required rate of return on an investment comprises the risk-free rate, which compensates investors for time value, plus a risk premium that is a function of investors’ attitudes toward risk in the aggregate and the specific portfolio risk of the investment being evaluated. EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Chapter 10: Financial Risk and Required Return 399 Because of these points, the CAPM is an important conceptual tool. How- ever, its actual use to set required rates of return must be viewed with some caution. We have more to say about CAPM in practice in chapter 13. 1. What is the capital asset pricing model (CAPM)? 2. What is the appropriate measure of risk in the CAPM? 3. Write out the equation for the SML, and graph it. 4. How do changes in risk aversion and inflation expectations affect the SML? 5. What are the pros and cons regarding the CAPM? SELF-TEST QUESTIONS Key Concepts This chapter covers the important topics of financial risk and required return. The key concepts of this chapter are as follows: • Risk definition and measurement are important in healthcare finance because decision makers, in general, are risk averse and require higher returns from investments that have higher risk. • Financial risk is associated with the prospect of returns less than anticipated. The higher the probability of a return being far less than anticipated, the greater the risk. • The risk of investments held in isolation, called stand-alone risk , can be measured by the dispersion of the rate-of-return distribution about its expected value . The most commonly used measure of stand-alone risk is the standard deviation of the return distribution. • Most investments are not held in isolation but as part of a portfolio . Individual investors hold portfolios of securities, and businesses hold portfolios of projects (i.e., products and services). • When investments with returns that are less than perfectly positively correlated are combined in a portfolio, risk is reduced. The risk reduction occurs because less than expected returns on some investments are offset by greater than expected returns on other investments. However, among real-world investments, it is (continued) EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Gapenski’s Healthcare Finance 400 This concludes the discussion of basic financial management concepts. The next chapter begins our coverage of long-term financing. impossible to eliminate all risk because the returns on all assets are influenced to a greater or lesser degree by overall economic conditions. • That portion of the stand-alone risk of an investment that can be eliminated by holding the investment in a portfolio is called diversifiable risk , while the risk that remains is called portfolio risk . • Market risk is the risk of business projects (or of the stocks of entire businesses) when they are considered as part of an individual investor’s well-diversified portfolio of securities. • Market risk is measured by a project’s or stock’s market beta , which reflects the volatility of a project’s (or stock’s) returns relative to the volatility of returns on a well-diversified stock portfolio. • Stand-alone risk is most relevant to investments held in isolation; corporate risk is most relevant to projects held by not-for-profit firms and by small, owner-managed, for-profit businesses; and market risk is most relevant to projects held by large investor- owned corporations. • The beta coefficient ( β ) of a portfolio of investments is the weighted average of the betas of the components of the portfolio, where the weights are the proportion of the overall investment in each component. Therefore, the weighted average of corporate betas of all projects in a business must equal 1.0, while the weighted average of all projects’ market betas must equal the market beta of the firm’s stock. • The capital asset pricing model (CAPM) is an equilibrium model that describes the relationship between market risk and required rates of return. • The security market line (SML) provides the actual risk–required rate of return relationship. The required rate of return on any stock is equal to the risk-free rate plus the market risk premium times the stock’s market beta coefficient: R ( R i ) = RF + [ R ( R M ) – RF ] × β = RF + ( RP M × β ). (continued from previous page) EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Chapter 10: Financial Risk and Required Return 401 Questions 10.1. When considering stand-alone risk, the return distribution of a less risky investment is more peaked (“tighter”) than that of a riskier investment. What shape would the return distribution have for an investment with (a) completely certain returns and (b) completely uncertain returns? 10.2. Stock A has an expected rate of return of 8 percent, a standard deviation of 20 percent, and a market beta of 0.5. Stock B has an expected rate of return of 12 percent, a standard deviation of 15 percent, and a market beta of 1.5. Which investment is riskier? Why? (Hint: Remember that the risk of an investment depends on its context.) 10.3. a. What is risk aversion? b. Why is risk aversion so important to financial decision-making? 10.4. Explain why holding investments in portfolios has such a profound impact on the concept of financial risk. 10.5. Assume that two investments are combined in a portfolio. a. In words, what is the expected rate of return on the portfolio? b. What condition must be present for the portfolio to have lower risk than the weighted average of the two investments? c. Is it possible for the portfolio to have lower risk than that of either investment? d. Is it possible for the portfolio to be riskless? If so, what condition is necessary to create such a portfolio? 10.6. Explain the difference between market risk and diversifiable risk. 10.7. What are the implications of portfolio theory for investors? 10.8. a. What are the two types of portfolio risk? b. How is each type defined? c. How is each type measured? 10.9. Under what circumstances is stand-alone and market risk most relevant? 10.10. a. What is the capital asset pricing model (CAPM)? The security market line (SML)? b. What are the weaknesses of the CAPM? c. What is the value of the CAPM? EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Gapenski’s Healthcare Finance 402 Problems 10.1. Consider the following probability distribution of returns estimated for a proposed project that involves a new ultrasound machine: State of the Economy Probability of Occurrence Rate of Return Very poor 0.10 –10.0% Poor 0.20 0.0 Average 0.40 10.0 Good 0.20 20.0 Very good 0.10 30.0 a. What is the expected rate of return on the project? b. What is the project’s standard deviation of returns? c. In what situation is this risk relevant? 10.2. Suppose that a person won the Florida lottery and was offered a choice of two prizes: (1) $500,000 or (2) a coin-toss gamble in which he would get $1 million for heads and zero for tails. a. What is the expected dollar return on the gamble? b. Would the person choose the sure $500,000 or the gamble? c. If he chooses the sure $500,000, is this person a risk averter or a risk seeker? 10.3. Refer to exhibit 10.2. a. Construct an equal-weighted (50/50) portfolio of investments B and C. What are the expected rate of return and standard deviation of the portfolio? Explain your results. b. Construct an equal-weighted (50/50) portfolio of investments B and D. What are the expected rate of return and standard deviation of the portfolio? Explain your results. 10.4. Suppose that the risk-free rate, RF , is 8 percent and the required rate of return on the market, R ( R M ), is 14 percent. a. Write out the security market line (SML) equation, and explain each term. b. Plot the SML on a sheet of paper. c. Suppose that inflation expectations increase such that the risk- free rate, RF , increases to 10 percent and the required rate of return on the market, R ( R M ), increases to 16 percent. Write out and plot the new SML. EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Chapter 10: Financial Risk and Required Return 403 d. Return to the original assumptions in this problem. Now, suppose that investors’ risk aversion increases and the required rate of return on the market, R ( R M ), increases to 16 percent. (There is no change in the risk-free rate because RF reflects the required rate of return on a riskless investment.) Write out and plot the new SML. 10.5. Several years ago, the Value Line Investment Survey reported the following market betas for the stocks of selected healthcare providers: Company Beta Quorum Health 0.90 Beverly Enterprises 1.20 Health South Corporation 1.45 United Healthcare 1.70 At the time these betas were developed, reasonable estimates for the risk-free rate, RF , and required rate of return on the market, R ( R M ), were 6.5 percent and 13.5 percent, respectively. a. What are the required rates of return on the four stocks? b. Why do their required rates of return differ? c. Suppose that a person is planning to invest in only one stock rather than a well-diversified stock portfolio. Are the required rates of return calculated above applicable to the investment? Explain your answer. 10.6. Suppose that Apple Health Services has four different projects. These projects are listed below, along with the amount of capital invested and the market betas: Project Amount Invested Market Beta Walk-in clinic $ 500,000 1.1 MRI facility 2,000,000 1.5 Clinical laboratory 1,500,000 0.8 X-ray laboratory 1,000,000 1.0 $5,000,000 a. What is the overall market beta of Apple Health Services? b. How does the riskiness of Apple’s stock compare with the riskiness of an average stock? EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Gapenski’s Healthcare Finance 404 c. Would stock investors require a rate of return on Apple that is greater than, less than, or the same as the return on an average- risk stock? 10.7. Assume that Community Health Systems is evaluating the feasibility of building a new hospital in an area not currently served by the company. The company’s analysts estimate a market beta of 1.1 for the hospital project, which is somewhat higher than the 0.8 market beta of the company’s average project. Financial forecasts for the new hospital indicate an expected rate of return on the equity portion of the investment of 20 percent. If the risk-free rate, RF , is 7 percent and the required rate of return on the market, R ( R M ), is 12 percent, is the new hospital in the best interest of Community Health Systems’ shareholders? Explain your answer. 10.8. Yahoo Finance reported the following market betas for the stocks of selected health insurers: Company Beta UnitedHealth Group 0.70 Humana 1.16 Aetna 0.63 Cigna 0.71 Assume that the risk-free rate, RF , and required rate of return on the market, R ( R M ), are 2.0 percent and 8.5 percent, respectively. a. What are the required rates of return on the four stocks? b. Which of the stocks is riskiest for investors? Explain your answer. 10.9. Assume that Everly Healthcare, a provider of skilled nursing facility services, is evaluating the feasibility of building a new facility to replace one of its aging facilities in a small, low-volume market. The company’s analysts estimate a market beta for the project of 0.8, which is somewhat lower than the 0.91 market beta of the company’s average project. The corporate beta for the project is estimated to be 0.5. Financial forecasts for the new facility indicate an expected rate of return on the equity portion of the investment of 7 percent. If the risk-free rate, RF , is 2 percent and the required rate of return on the market, R ( R M ), is 10 percent, is the new facility in the best interest of Everly’s shareholders? Explain your answer. EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Chapter 10: Financial Risk and Required Return 405 Selected Case One case in Gapenski’s Cases in Healthcare Finance , sixth edition, is appli- cable to this chapter: Case 12: Mid-Atlantic Specialty, Inc. Notes 1. If inflation is considered, the T-bill investment is not truly risk-free. The real return , which recognizes inflation effects, is uncertain because it depends on the amount of inflation realized over the year. 2. In markets that are efficient , low-risk investments will have low expected returns, while high-risk investments will have high expected returns. However, not all markets are efficient. See chapter 12 for a complete discussion of market efficiency. 3. A portfolio of two investments has lower risk than that of either component only when the correlation coefficient between the returns on the two investments is less than the ratio of the standard deviations, when this ratio is constructed with the lower standard deviation in the numerator. For example, for portfolio AD to have less risk than both A and D, the correlation coefficient between the returns on A and D must be less than σ A / σ D = 11.0% ÷ 12.1% = 0.91. The actual correlation coefficient is 0.53, so the condition is met in this example. 4. Although stocks can be combined with complex investments (derivatives) to form riskless portfolios, our emphasis here is on real- asset investments. 5. The Wilshire 5000 Index, also called the Total Stock Market Index, mimics the returns of all publicly traded US stocks. Resources Bannow, T. 2018. “ACOs Reluctant to Move to Advanced Risk-Taking Models.” Modern Healthcare . Published October 8. www.modernhealthcare.com/article /20181006/NEWS/181009954/acos -reluctant -to -move -to -advanced -risk-taking-models. Beck, W., C. Kelly-Aduli, and B. B. Sanderson. 2018. “Protecting Revenue at Risk.” Healthcare Financial Managemen t 72 (4): 62–67. Brigham, E. F., and P. R. Daves. 2017a. “Risk and Return: Part I.” In Intermediate Financial Management , 13th ed., by E. F. Brigham and P. R. Davies, 56–111. Mason, OH: South-Western Cengage Learning. EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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Gapenski’s Healthcare Finance 406 . 2017b. “Risk and Return: Part II.” In Intermediate Financial Management , 13th ed., by E. F. Brigham and P. R. Davies, 112–48. Mason, OH: South- Western Cengage Learning. Friesen, C. A. 2018. “Assessing Risk in Healthcare Finance Calls for Stewardship.” Healthcare Financial Management 72 (4): 1–2. Kotecki, L. 2018. “Reaping the Benefits of an Actuarial Mindset.” Healthcare Finan- cial Management 72 (4): 50–54. Luthi, S. 2018. “Azar Hints at ‘Bold’ Risk-Based Medicare Payment Models.” Mod- ern Healthcare . Published September 6. www.modernhealthcare.com/article /20180906/NEWS/180909962/azar -expect -bold -risk -based -medicare -payment-models. Puchley, T., and C. Toppi. 2018. “ERM: Evolving from Risk Assessment to Strategic Risk Management.” Healthcare Financial Management 72 (4): 44–49. EBSCOhost - printed on 2/7/2024 8:34 PM via UNIVERSITY OF MARYLAND GLOBAL CAMPUS. All use subject to https://www.ebsco.com/terms-of-use

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