Foundations of Finance Chapter 4

The Meaning and 6 CHAPTER Measurement of Risk and Return Learning Objectives LO1 Define and measure the expected rate of return of Expected Return Defined and an individual investment....

The Meaning and 6 CHAPTER Measurement of Risk and Return Learning Objectives LO1 Define and measure the expected rate of return of Expected Return Defined and an individual investment. Measured LO2 Define and measure the riskiness of an individual Risk Defined and Measured investment. LO3 Compare the historical relationship between Rates of Return: The Investor’s risk and rates of return in the capital markets. Experience LO4 Explain how diversifying investments affects the Risk and Diversification riskiness and expected rate of return of a portfolio or combination of assets. LO5 Explain the relationship between an investor’s The Investor’s Required Rate of required rate of return on an investment and the Return riskiness of the investment. O ne of the most important concepts in finance is risk and return, which is the total focus of our third principle of finance—Risk Requires a Reward. You only have to look at what happened in the stock markets during the recent Great Recession. For example, assume that in October 2007 you had invested $10,000 in a portfolio consist- ing of stocks making up the Standard & Poor’s 500 Index. A mere 18 months later (March 2009), your portfolio would have been worth only $4,918. In fact, you would have lost money in 12 of the 18 months you had owned the stocks. Believe us when we say there were a lot of worried and anxious investors, including college professors with retirement accounts. But if you had continued to hold the portfolio, you would have begun feeling better in time. By January 2018, we had experienced a nine-year general market rise, and your portfolio was worth $18,120. We can summarize what happened as follows: 1. From October 2007 through March 2009 (18 months), your portfolio went from being worth $10,000 to $4,918, a decline in value of 51 percent, or equivalently a 41 percent loss on an annualized basis. 186  2. From March 2009 through January 2018 (106 months), your portfolio increased in value from $4,918 to $18,120, an increase of 269 percent, or the equivalent of growing at an annu- alized rate of 16 percent. But even during this positive period when you experienced a big increase in your portfolio value, you lost money in 28 months of the total 106 months! It was just that the good months far more than offset the negative months. 3. If we look at the two periods taken together, from October 2007 through January 2018 (124 months), we see an increase in RE ME MBE R YO U R P RIN C I PLES value of 81 percent, or an average annual return of 6 percent. This chapter has one primary objective, that of helping you understand Principle 3: Risk Requires a Reward. Portfolio Value Number of Change in Percentage Time Period Months Beginning Ending Value Change Annual Rate First period: Oct. 07–Mar. 09 18 $10,000 $ 4,918 ($5,082) (51%) (41%) Second period: Mar. 09–Jan. ‘18 106 $ 4,918 $18,120 13,202 269% 16% First and second periods combined: Oct. 07–Jan ‘18 124 $10,000 $18,120 $8,120 81% 6% So, owning stocks is sometimes not for the faint of heart, where in a single day you can earn as much as 5 percent on your investment or lose even more. The market crash and extreme volatility in stock prices in 2008 and 2009 are what Nassim Nicholas Taleb would call a black swan—a highly improbable event that has a massive impact.1 As a consequence, many investors and business managements alike have become much more cognizant of the financial risks. In this chapter, we will help you understand the nature of risk and how risk should relate to expected returns on investments. We will look back beyond the past few years to see what we can learn about long-term historical risk-and-return relation- ships. These are topics that should be of key interest to us all in this day and age. 1 In his book The Black Swan: The Impact of the Highly Improbable (New York: Random House, 2007), Nassim Nicholas Taleb uses the analogy of a black swan, which represents a highly unlikely event. Until black swans were discovered in Australia, everyone assumed that all swans were white. Thus, for Taleb, the black swan symbolizes an event that no one thinks possible. 187 188 PART 2 The Valuation of Financial Assets The need to recognize risk in financial decisions has already been apparent in earlier chapters. In Chapter 2, we referred to the discount rate, or the interest rate, as the opportunity cost of funds, but we did not look at the reasons why that rate might be high or low. For example, we did not explain why in January 2018 you could buy bonds issued by Dow Chemical that promised to pay a 4.15 percent rate of return, or why you could buy First Data bonds that would give you an 8.4 percent rate of return, provided that both firms make the payments to the investors as promised. In this chapter, we learn that risk is an integral force underlying rates of return. To begin our study, we define expected return and risk and offer suggestions about how these important concepts of return and risk can be measured quantitatively. We com- pare the historical relationship between risk and rates of return. We then explain how diversifying investments can affect the expected return and riskiness of those invest- ments. We also consider how the riskiness of an investment should affect the required rate of return on an investment. Let’s begin our study by looking at what we mean by the expected rate of return and how it can be measured. LO1 Define and measure the expected rate of return of Expected Return Defined and Measured an individual investment. The expected benefits, or returns, an investment generates come in the form of cash flows. Cash flow, not accounting profit, is the relevant variable the financial manager uses to measure returns. This principle holds true regardless of the type of security, whether it is a debt instrument, preferred stock, common stock, or any mixture of these (such as convertible bonds). To begin our discussion about an asset’s expected rate of return, let’s first understand holding-period return (historical or how to compute a historical or realized rate of return on an investment. It may also be realized rate of return) the rate of called a holding-period return. For instance, consider the dollar return you would have return earned on an investment, which equals the dollar gain divided by the earned had you purchased a share of Google on April 17, 2017, for $837.17 and sold it one amount invested. week later on April 24 for $862.76. The dollar return on your investment would have been $25.59 ($25.59 = $862.76 - $837.17), assuming that the company paid no dividend. In addition to the dollar gain, we can calculate the rate of return as a percentage. It is useful to summarize the return on an investment in terms of a percentage because that way we can see the return per dollar invested, which is independent of how much we actually invest. The rate of return earned from the investment in Google can be calculated as the ratio of the dollar return of $25.59 divided by your $837.17 investment in the stock, or 3.06 percent (3.06% =.0306 = $25.59 , $837.17). We can formalize the return calculations using equations (6-1) and (6-2): Holding-period dollar gain would be: Holding@period cash distribution = priceend of period + - price beginning of period (6-1) dollar gain, DG (dividend) and for the holding-period rate of return Holding@period dollar gain = rate of return, r pricebeginning of period priceend of period + dividend - pricebeginning of period = (6-2) pricebeginning of period The method we have just used to compute the holding-period return on our invest- ment in Google tells us the return we actually earned during a historical time period. However, the risk–return trade-off that investors face on a day-to-day basis is based not on realized rates of return but on what the investor expects to earn on an investment in the future. We can think of the rate as one that will come from making a risky investment in expected rate of return the terms of a range of possible return outcomes, much like the distribution of grades for this arithmetic mean or average of all possible outcomes where those class at the end of the term. To describe this range of possible returns, it is customary to outcomes are weighted by the use the average of the different possible returns. We refer to the average of the possible probability that each will occur. rates of return as the investment’s expected rate of return.  CHAPTER 6 The Meaning and Measurement of Risk and Return 189 TABLE 6-1 Measuring the Expected Return of an Investment Probability of Cash Flows from Percentage Returns State of the Economy the Statesa the Investment (Cash Flow 4 Investment Cost) Economic recession 20% $1,000 10% ($1,000 ÷ $10,000) Moderate economic growth 30% 1,200 12% ($1,200 ÷ $10,000) Strong economic growth 50% 1,400 14% ($1,400 ÷ $10,000) a The probabilities assigned to the three possible economic conditions have to be determined subjectively, which requires manag- ers to have a thorough understanding of both the investment cash flows and the general economy. Accurately measuring expected future cash flows is not easy in a world of uncer- tainty. To illustrate, assume you are considering an investment costing $10,000, for which the future cash flows from owning the security depend on the state of the economy, as estimated in Table 6-1. In any given year, the investment could produce any one of three possible cash flows, depending on the particu- RE ME MBE R YO U R P RIN C I PLES lar state of the economy. With this information, how should we select the cash flow estimate that means the most for Principle Remember that future cash flows, not reported earnings, determine the investor’s rate of return. That is, measuring the investment’s expected rate of return? One Principle 1: Cash Flow Is What Matters. approach is to calculate an expected cash flow. The expected cash flow is simply the weighted average of the possible cash flow outcomes where the weights are the probabilities of the various states of the economy occurring. Stated as an equation: cash flow probability cash flow probability Expected = ° in state 1 * of state 1 ¢ + ° in state 2 * of state 2 ¢ cash flow, CF (CF1) (Pb1) (CF2) (Pb2) cash flow probability + c + ° in state n * of state n ¢ (6-3) (CFn) (Pbn) For the present illustration: Expected cash flow = (0.2)($1,000) + (0.3)($1,200) + (0.5)($1,400) = $1,260 In addition to computing an expected dollar return from an investment, we can also calculate an expected rate of return earned on the $10,000 investment. Similar to the expected cash flow, the expected rate of return is an average of all the possible returns, weighted by the probability that each return will occur. As the last column in Table 6-1 shows, the $1,400 cash inflow, assuming strong economic growth, repre- sents a 14 percent return ($1,400 , $10,000). Similarly, the $1,200 and $1,000 cash flows result in 12 percent and 10 percent returns, respectively. Using these percent- age returns in place of the dollar amounts, we can express the expected rate of return as follows: rate of return probability rate of return probability Expected = ° for state 1 * of state 1 ¢ + ° for state 2 * of state 2 ¢ rate of return, r (r1) (Pb1) (r2) (Pb2) rate of return probability + c + ° for state n * of state n ¢ (6-4) (rn) (Pbn) 190 PART 2 The Valuation of Financial Assets CAN YOU DO IT? Computing Expected Cash Flow and Expected Return You are contemplating making a $5,000 investment that would have the following possible outcomes in cash flow each year. Probability Cash Flow What is the expected value of the future cash flows and the 0.30 $350 expected rate of return? 0.50 625 (The solution can be found on page 191.) 0.20 900 In our example: r = (0.2)(10,) + (0.3)(12,) + (0.5)(14,) = 12.6, To this point, we have learned how to calculate a historical holding-period return, expressed in both dollars and percentages. Also, we have seen how to estimate dollar and percentage returns that we expect to earn in the future. These financial tools may be summarized as follows: F I N A N C I A L D E CISIO N TO OLS Name of Tool Formula What It Tells You Holding@period Measures the dollar gain on an investment for a = priceend of period dollar gain, DG period of time Holding-period dollar gain cash distribution + - pricebeginning of period (dividend) Holding@period dollar gain Calculates the percentage rate of return for a security = held for a period of time rate of return, r pricebeginning of period Holding-period rate of return priceend of period + dividend - pricebeginning of period = pricebeginning of period Expected cash flow probability Estimates the cash flows that can be expected from cash flow, CF = ° in state 1 * of state 1 ¢ an investment, recognizing that there are multiple (CF1) (Pb1) possible outcomes from the investment cash flow probability Expected cash flow + ° in state 2 * of state 2 ¢ (CF2) (Pb2) cash flow probability + c + ° in state n * of state n ¢ (CFn) (Pbn) Expected rate of return probability An estimate of the expected rate of return on an rate of return, r = ° for state 1 * of state 1 ¢ investment, recognizing that there are multiple pos- (r1) (Pb1) sible outcomes from the investment rate of return probability Expected rate of return + ° for state 2 * of state 2 ¢ (r2) (Pb2) rate of return probability + c + ° for state n * of state n ¢ (rn) (Pbn)  CHAPTER 6 The Meaning and Measurement of Risk and Return 191 With our concept and measurement of expected returns, let’s consider the other side of the investment coin: risk. Concept Check 1. When we speak of “benefits” from investing in an asset, what do we mean? 2. Why is it difficult to measure future cash flows? 3. Define expected rate of return. Risk Defined and Measured LO2 Define and measure the riskiness of an individual investment. Because we live in a world where events are uncertain, the way we see risk is vitally important in almost all dimensions of our lives. The Greek poet and statesman Solon, writing in the sixth century b.c., put it this way: There is risk in everything that one does, and no one knows where he will make his landfall when his enterprise is at its beginning. One man, trying to act effectively, fails to foresee something and falls into great and grim ruination, but to another man, one who is acting ineffectively, a god gives good fortune in everything and escape from his folly.2 Solon would have given more of the credit to Zeus than we would for the out- comes of our ventures. However, his insight reminds us that little is new in this world, including the need to acknowledge and compensate as best we can for the risks we encounter. In fact, the significance of risk and the need for understanding what it means in our lives is noted by Peter Bernstein in the following excerpt: What is it that distinguishes the thousands of years of history from what we think of as modern times? The answer goes way beyond the progress of science, technology, capital- ism, and democracy. The distant past was studded with brilliant scientists, mathematicians, inventors, tech- nologists, and political philosophers.... [T]he skies had been mapped, the great library of Alexandria built, and Euclid’s geometry taught.... Coal, oil, iron, and copper have been at the service of human beings for millennia.... The revolutionary idea that defines the boundary between modern times and the past is the mastery of risk.... Until human beings discovered a way across that boundary, the future was a mirror of the past or the murky domain of oracles and soothsayers.3 DID YOU GET IT? Computing Expected Cash Flow and Expected Return (On page 190 you were asked to compute your $5,000 investment’s expected cash flow and expected return. Below is the solution.) Possible Expected Probability Cash Flow Return Cash Flow Return 0.30 $350 7.0% $105.00 2.1% 0.50 625 12.5% 312.50 6.3% 0.20 900 18.0% 180.00 3.6% Expected cash flow and rate of return $597.50 12.0% The possible returns are equal to the possible cash flows divided by the $5,000 investment. The expected cash flow and return are equal to the possible cash flows and possible returns multiplied by the probabilities. 2 From Iambi et Elegi Graeci ante Alexandrum Cantati, vol. 2. Edited by M. L. West, translated by Arthur W.H. Adkins (Oxford, UK: Clarendon Press, 1972). 3 From “Introduction” in Against the Gods: The Remarkable Story of Risk by Peter Bernstein, published by John Wiley & Sons, Inc., New York: 1998. 192 PART 2 The Valuation of Financial Assets In our study of risk, we want to consider these questions: 1. What is risk? 2. How do we know the amount of risk associated with a given investment; that is, how do we measure risk? 3. If we choose to diversify our investments by owning more than one asset, as most of us do, will such diversification reduce the riskiness of our combined portfolio of investments? Without intending to be trite, risk means different things to different people, depending on the context and on how they feel about taking chances. For the stu- dent, risk is the possibility of failing an exam or the chance of not making his or her best grades. For the coal miner or the oil field worker, risk is the chance of an explo- sion in the mine or at the well site. For the retired person, risk means perhaps not being able to live comfortably on a fixed income. For the entrepreneur, risk is the chance that a new venture will fail. While certainly acknowledging these different kinds of risk, we limit our risk potential variability in future attention to the risk inherent in an investment. In this context, risk is the potential cash flows. variability in future cash f lows. The wider the range of possible events that can occur, the greater the risk. 4 If we think about it, this is a relatively intuitive concept. To help us grasp the fundamental meaning of risk within this context, consider two possible investments: 1. The first investment is a U.S. Treasury bond, a government security that matures in 10 years and promises to pay an annual return of 2 percent. If we purchase and hold this security for 10 years, we are virtually assured of receiving no more and no less than 2 percent on an annualized basis. For all practical purposes, the risk of loss is nonexistent. 2. The second investment involves the purchase of the stock of a local publishing company. Looking at the past returns of the firm’s stock, we have made the fol- lowing estimate of the annual returns from the investment: Rate of Return Chance of Occurrence on Investment 1 chance in 10 (10% chance) −10% 2 chances in 10 (20% chance) 5% 4 chances in 10 (40% chance) 15% 2 chances in 10 (20% chance) 25% 1 chance in 10 (10% chance) 30% Investing in the publishing company could conceivably provide a return as high as 30 percent if all goes well, or a loss of 10 percent if everything goes against the firm. However, in future years, both good and bad, we could expect a 14 percent return on average.5 4 When we speak of possible events, we must not forget that it is the highly unlikely event that we cannot anticipate that may have the greatest impact on the outcome of an investment. So, we evaluate invest- ment opportunities based on the best information available, but there may be a black swan that we cannot anticipate. 5 We assume that the particular outcome or return earned in 1 year does not affect the return earned in the subsequent year. Technically speaking, the distribution of returns in any year is assumed to be indepen- dent of the outcome in any prior year.  CHAPTER 6 The Meaning and Measurement of Risk and Return 193 FIGURE 6-1 The Probability Distribution of the Returns on Two Investments 0.4 Probability of occurrence 1 Probability of occurrence 0.35 0.9 0.8 0.3 0.7 0.25 0.6 0.2 0.5 0.4 0.15 0.3 0.1 0.2 0.1 0.05 0 0 1 2 3 4 –10 5 15 25 30 Possible returns (%) Possible returns (%) Treasury bond Publishing company Expected return = (0.10)(-10,) + (0.20)(5,) + (0.40)(15,) + (0.20)(25,) + (0.10)(30,) = 14, Comparing the Treasury bond investment with the publishing company invest- ment, we see that the Treasury bond offers an expected 2 percent annualized rate of return, whereas the publishing company has a much higher expected rate of return of 14 percent. However, our investment in the publishing firm is clearly more “risky”—that is, there is greater uncertainty about the final outcome. Stated some- what differently, there is a greater variation or dispersion of possible returns, which in turn implies greater risk.6 Figure 6-1 shows these differences graphically in the form of discrete probability distributions. Although the return from investing in the publishing firm is clearly less certain than for Treasury bonds, quantitative measures of risk are useful when the difference between two investments is not so evident. We can quantify the risk of an investment by computing the variance in the possible investment returns and its square root, the standard deviation (σ). For the case where there are n possible returns (that is, states standard deviation a statistical of the economy), we calculate the variance as follows: measure of the spread of a probability distribution calculated by squaring the difference between each outcome and its expected value, weighting each Variance in rate of return expected rate 2 probability value by its probability, summing over rates of return = £ ° for state 1 - of return ¢ * of state 1 § all possible outcomes, and taking the square root of this sum. (σ2) (r1) r (Pb1) rate of return expected rate 2 probability + £ ° for state 2 - of return ¢ * of state 2 § (r2) r (Pb2) rate of return expected rate 2 probability + g + £ ° for state n - of return ¢ * of state n § (rn) r (Pbn) (6-5) 6 How can we possibly view variations above the expected return as risk? Should we even be concerned with the positive deviations above the expected return? Some would say “no,” viewing risk as only the negative variability in returns from a predetermined minimum acceptable rate of return. However, as long as the distribution of returns is symmetrical, the same conclusions will be reached. 194 PART 2 The Valuation of Financial Assets TABLE 6-2 Measuring the Variance and Standard Deviation of the Publishing Company Investment State of the Chance or World Rate of Return Probability Step 1 Step 2 Step 3 A B C D = B * C E = (B - r )2 F = E * C 1 - 10% 0.10 - 1% 576% 57.6% 3 5% 0.20 1% 81% 16.2% 4 15% 0.40 6% 1% 0.40% 5 25% 0.20 5% 121% 24.2% 6 30% 0.10 3% 256% 25.6% Step 1: Expected Return (r ) = 14% Step 4: Variance = 124% Step 5: Standard Deviation = 11.14% For the publishing company’s common stock, we calculate the standard deviation using the following five-step procedure: STEP 1 Calculate the expected rate of return of the investment, which was calcu- lated previously to be 14 percent. STEP 2 Subtract the expected rate of return of 14 percent from each of the possible rates of return and square the difference. STEP 3 Multiply the squared differences calculated in step 2 by the probability that those outcomes will occur. STEP 4 Sum all the values calculated in step 3 together. The sum is the variance of the distribution of possible rates of return. Note that the variance is actually the average squared difference between the possible rates of return and the expected rate of return. STEP 5 Take the square root of the variance calculated in step 4 to calculate the standard deviation of the distribution of possible rates of return. Table 6-2 illustrates the application of this process, which results in an estimated standard deviation for the common stock investment of 11.14 percent. This compares to the Treasury bond investment, which is risk-free, and has a standard deviation of zero percent. The more risky the investment, the higher is its standard deviation. CAN YOU DO IT? Computing the Standard Deviation In the preceding “Can You Do It?” on page 190, we computed the expected cash flow of $597.50 and the expected return of 12 percent on a $5,000 investment. Now let’s calculate the standard deviation of the returns. The probabilities of possible returns are given as follows: Probability Returns 0.30 7.0% 0.50 12.5% 0.20 18.0% (The solution can be found on page 198.)  CHAPTER 6 The Meaning and Measurement of Risk and Return 195 FINANCE at WORK A Different Perspective of Risk The first Chinese symbol shown here represents danger; the second stands for opportunity. The Chinese define risk as the combination of danger and opportunity. Greater risk, according to the Chinese, means we have greater opportunity to do well but also greater danger of doing badly. Alternatively, we could use equation (6-5) to calculate the standard deviation in investment returns as follows: 1 (-10% - 14%)2(0.10) + (5% - 14%)2(0.20) 2 σ = £ + (15% - 14%)2(0.40) + (25% - 14%)2(0.20) § + (30% - 14%)2(0.10) = 2124, = 11.14, Although the standard deviation of returns provides us with a quantitative measure of an asset’s riskiness, how should we interpret the result? What does it mean? Is the 11.14 percent standard deviation for the publishing company investment good or bad? First, we should remember that statisticians tell us that two-thirds of the time, an event will fall within 1 standard deviation of the expected value (assuming the distribution is normally distributed; that is, it is shaped like a bell). Thus, given a 14 percent expected return and a standard deviation of 11.14 percent for the publishing company investment, we can reasonably anticipate that the actual returns will fall between 2.86 percent and 25.14 percent (14, { 11.14,) two-thirds of the time. In other words, there is not much certainty with this investment. A second way of answering the question about the meaning of the standard deviation comes by comparing the investment in the publishing firm against other investments. Obviously, the attractiveness of a security with respect to its return and risk cannot be determined in isolation. Only by examining other avail- able alternatives can we reach a conclusion about a particular investment’s risk. For example, if another investment, say, an investment in a firm that owns a local radio station, has the same expected return as the publishing company, 14 percent, but with a standard deviation of 7 percent, we would consider the risk associated with the publishing firm, 11.14 percent, to be excessive. In the technical jargon of modern portfolio theory, the radio station investment is said to “dominate” the publishing firm investment. In commonsense terms, this means that the radio station investment has the same expected return as the publishing company investment but is less risky. What if we compare the investment in the publishing company with one in a quick-oil-change franchise, an investment in which the expected rate of return is an attractive 24 percent but the standard deviation is estimated at 13 percent? Now what should we do? Clearly, the oil-change franchise has a higher expected rate of return, but it also has a larger standard deviation. In this example, we see that the real chal- lenge in selecting the better investment comes when one investment has a higher expected rate of return but also exhibits greater risk. Here the final choice is determined by our attitude toward risk, and there is no single right answer. You might select the pub- lishing company, whereas I might choose the oil-change investment, and neither of us would be wrong. We would simply be expressing our tastes and preferences about risk and return. To summarize, the riskiness of an investment is of primary concern to an investor, where the standard deviation is the conventional measure of an investment’s riski- ness. This decision tool is represented as follows: 196 PART 2 The Valuation of Financial Assets F I N A N C I A L D E CISIO N TO OLS Name of Tool Formula What It Tells You Standard deviation A measure of risk, as determined Variance in rate of return expected rate 2 probability by the square root of the variance in rates of return rates of return = £ ° for state 1 - of return ¢ * of state 1 § = 2variance of cash flows or rates of returns, (σ2) (r1) r (Pb1) which measures the volatility of rate of return expected rate 2 probability cash flows or returns + £ ° for state 2 - of return ¢ * of state 2 § (r2) r (Pb2) rate of return expected rate 2 probability + c + £ ° for state n - of return ¢ * of state n § (rn) r (Pbn) EXAMPLE 6.1 MyLab Finance Video Computing the Expected Return and Standard Deviation You are considering two investments, X and Y. The distributions of possible returns are shown below: Possible Returns Probability Investment X Investment Y 0.05 - 10% 0% 0.25 5% 5% 0.40 20% 16% 0.25 30% 24% 0.05 40% 32% Compute the expected return and standard deviation for each investment. Would you have a preference for one investment over the other if you were making the decision? STEP 1: Formulate a Solution Strategy To compute the expected return for each investment, equation (6-4) is used, where there are five possible outcomes or states: rate of return probability of Expected rate = ° for state 1 * state 1 ¢ of return, r (r1) (Pb1) rate of return probability of rate of return probability of + ° for state 2 * state 2 ¢ + c + ° for state 5 * state 5 ¢ (r2) (Pb2) (r5) (Pb5) To calculate the riskiness of each investment, as measured by the standard deviation of returns, we rely on equation (6-5) for five possible outcomes: Variance in rate of return expected rate 2 probability rates of return = £ ° for state 1 - of return ¢ * of state 1 § (σ2) (r1) r (Pb1) rate of return expected rate 2 probability + £ ° for state 2 - of return ¢ * of state 2 § (r2) r (Pb2)  CHAPTER 6 The Meaning and Measurement of Risk and Return 197 rate of return expected rate 2 probability + g + £ ° for state 5 - of return ¢ * of state 5 § (r5) r (Pb5) Standard deviation (σ) = 2variance STEP 2: Crunch the Numbers For investment X: Expected rate = (0.05)(-10,) + (0.25)(5,) + (0.40)(20,) of return, r + (0.25)(30,) + (0.05)(40,) = 18.25, Variance in rates of returns = (0.05)(-10, - 18.25,)2 + (0.25)(5, - 18.25,)2 (σ2) + (0.40)(20, - 18.25,)2 + (0.25)(30, - 18.25,)2 + (0.05)(40, - 18.25,)2 = 143.19, Standard deviation of rates of returns = 2variance = 2143.19, = 11.97, (σ) For investment Y: Expected rate = (0.05)(0,) + (0.25)(5,) + (0.40)(16,) + (0.25)(24,) of return, r + (0.05)(32,) = 15.25, Variance in rates of returns = (0.05)(0, - 15.25,)2 + (0.25)(5, - 15.25,)2 (σ2) + (0.40)(16, - 15.25,)2 + (0.25)(24, - 15.25,)2 + (0.05)(32, - 15.25,)2 = 71.29, Standard deviation of rates of returns = 2variance = 271.29, = 8.44, (σ) STEP 3: Analyze Your Results The results are as follows: Investment Expected Return Standard Deviation Investment X 18.25% 11.97% Investment Y 15.25% 8.44% In this case, you will have to take on more risk if you want additional expected return. (There is no money tree.) Thus, the choice depends on the investor’s prefer- ence for risk and return. There is no single right answer. 198 PART 2 The Valuation of Financial Assets DID YOU GET IT? Computing the Standard Deviation Deviation (Possible Return 2 Deviation Probability 12% Expected Return) Squared Probability Deviation Squared - 5.0% 25.00% 0.30 7.500% 0.5% 0.25% 0.50 0.125% 6.0% 36.00% 0.20 7.200% 2 Sum of squared deviations * probability (variance σ ) 14.825% Standard deviation (σ) 3.85% Concept Check 1. How is risk defined? 2. How does the standard deviation help us measure the riskiness of an investment? 3. Does greater risk imply a bad investment? LO3 Compare the historical relationship between risk Rates of Return: The Investor’s Experience and rates of return in the capital So far, we have mostly used hypothetical examples of expected rates of return and markets. risk; however, it is also interesting to look at returns that investors have actually received on different types of securities. For example, Ibbotson Associates publishes the long-term historical annual rates of return for the following types of investments beginning in 1926 and continuing to the present: 1. Common stocks of large companies 2. Common stocks of small firms 3. Long-term corporate bonds 4. Long-term U.S. government bonds 5. Intermediate-term U.S. government bonds 6. U.S. Treasury bills (short-term government securities) Before comparing these returns, we should think about what to expect. First, we would intuitively expect a Treasury bill (short-term government securities) to be the least risky of the six portfolios. Because a Treasury bill has a short-term maturity date, the price is less volatile (less risky) than the price of an intermediate- or long- term government security. In turn, because there is a chance of default on a corporate bond, which is essentially nonexistent for government securities, a long-term gov- ernment bond is less risky than a long-term corporate bond. Finally, the common stocks of large companies are more risky than corporate bonds, with small-company stocks being more risky than large-firm stocks. With this in mind, we could reasonably expect different rates of return to the holders of these varied securities. If the market rewards an investor for assuming risk, the average annual rates of return should increase as risk increases. A comparison of the annual rates of return for five portfolios and the inflation rates for the years 1926 through 2016 is provided in Figure 6-2. Four aspects of these returns are included: 1. The nominal average annual rate of return; 2. The standard deviation of the returns, which measures the volatility, or riskiness, of the returns; 3. The real average annual rate of return, which is the nominal return less the infla- tion rate; and 4. The risk premium, which represents the additional return received beyond the risk-free rate (Treasury bill rate) for assuming risk.  CHAPTER 6 The Meaning and Measurement of Risk and Return 199 FIGURE 6-2 Historical Rates of Return Nominal Real Standard Risk Average Average Securities Deviation Premiumb Annual Annual of Returns Returns Returnsa Small- company 16.6% 31.9% 13.7% 13.2% stocks Large- company 12.0% 19.9% 9.1% 8.6% stocks Intermediate-term government 5.3% 5.6% 2.4% 1.9% bonds Corporate bonds 6.3% 8.4% 3.4% 2.9% U.S. 3.4% 3.1% 0.5% 0.0% Treasury bills Inflation 2.9% 3.0% aThe real return equals the nominal returns less the inflation rate of 2.9 percent. bThe risk premium equals the nominal security return less the average risk-free rate (Treasury bills) of 3.4 percent. Source: Data from Summary Statistics of Annual Total Returns: 1926 to 2016 Yearbook, Ibbotson Associates Inc. Looking at the first two columns of nominal average annual returns and standard deviations, we get a good overview of the risk–return relationships that have existed over the 91 years ending in 2016. In every case, there has been a positive relationship between risk and return, with Treasury bills being least risky and small-company stocks being most risky. The return information in Figure 6-2 clearly demonstrates that only common stock has, in the long run, served as an inflation hedge and provided any substantial risk premium. However, it is equally apparent that the common stockholder is exposed to a sizable amount of risk, as is demonstrated by the 19.9 percent standard deviation for large-company stocks and the 31.9 percent standard deviation for small-company stocks. In fact, in the 1926 through 2016 time frame, common shareholders of large firms received negative returns in 22 of the 91 years, compared with only 1 (1938) in 91 years for Treasury bills. Concept Check 1. What is the additional compensation for assuming greater risk called? 2. In Figure 6-2, which rate of return is the risk-free rate? Why? Risk and Diversification LO4 Explain how diversifying investments affects the riskiness and expected rate of return More can be said about risk, especially about its nature, when we own more than one of a portfolio or combination of asset in our investment portfolio. For the moment, let’s look at Cooper Groves and assets. consider how risk is affected if we diversify our investment by holding a variety of securities. Cooper still remembers graduating from Baylor University in December 2016. Not only did he get the good job he had hoped for but he also finished the year with a little nest egg—not enough to take that summer fling to Europe like some of his college friends but a nice surplus. Besides, he suspected that they used credit cards to go any- way. So right after graduation, he used some of his nest egg to buy Harley-Davidson stock. He also purchased Starbucks stock. (As the owner of a Harley Softail motorcycle, he had a passion for riding since his high school days. He gave Starbucks credit for helping him get through many long study sessions.) But since making these invest- ments, he has focused on his career and seldom thought about his investments. His first 200 PART 2 The Valuation of Financial Assets extended break from work came in May 2017. After a lazy Saturday morning, he decided to get on the Internet to see how his investments had done over the previous several months. He bought Starbucks for $79, and the stock was now trading at almost $51, which scared him. But he then noticed that the firm had split stock 2 for 1, so he had twice as many shares. So actually the two shares he owned in place of one would be worth $102. “Not bad,” he thought. But then the bad news—Harley-Davidson was sell- ing for $56, compared to the $66 that he paid for the stock. Clearly, what we have described about Harley-Davidson and Starbucks were events unique to these two companies, and as we would expect, investors reacted accordingly; that is, the value of the stock changed in light of the new information. Although Cooper might have wished he had owned only Starbucks stock at the time, most of us would prefer to avoid such uncertainties; that is, we are risk averse. Instead, we would like to reduce the risk associated with our investment portfolio, without having to accept a lower expected return. Good news: It is possible by diversifying your portfolio! Diversifying Away the Risk If we diversify our investments across different securities rather than invest in only one stock, the variability in the returns of our portfolio should decline. The reduction in risk will occur if the stock returns within our portfolio do not move precisely together over time—that is, if they are not perfectly correlated. Figure 6-3 shows graphically what we could expect to happen to the variability of returns as we add additional stocks to the portfolio. The reduction occurs because some of the volatility in returns of a stock are unique to that security. The unique variability of a single stock tends to be countered by the uniqueness of another security. However, we should not expect to eliminate all risk from our portfolio. In practice, it would be rather difficult to cancel all the variations in returns of a portfolio because stock prices have some tendency to move together. Thus, we can divide the total risk (total unsystematic risk (company- variability) of our portfolio into two types of risk: (1) company-unique risk, or unique risk or diversifiable risk) unsystematic risk, and (2) market risk, or systematic risk. Company-unique risk the risk related to an investment return that can be eliminated through might also be called diversifiable risk in that it can be diversified away. Market risk is diversification. Unsystematic risk is the nondiversifiable risk; it cannot be eliminated through random diversification. These two result of factors that are unique to the types of risk are shown graphically in Figure 6-3. Total risk declines until we have particular firm. Also called company- unique risk or diversifiable risk. approximately 20 securities, and then the decline becomes very slight. The remaining risk, which would typically be about 40 percent of the total risk, is systematic risk (market risk or nondiversifiable risk) the risk related the portfolio’s systematic, or market, risk. At this point, our portfolio is highly cor- to an investment return that cannot related with all securities in the marketplace. Events that affect our portfolio now are be eliminated through diversification. not so much unique events as changes in the general economy, major political events, Systematic risk results from factors that affect all stocks. Also called market risk and societal changes. Examples include changes in interest rates, changes in tax leg- or nondiversifiable risk. islation that affect all companies, or increasing public concern about the effect of FIGURE 6-3 Variability of Returns Compared with Size of Portfolio Unsystematic, or diversifiable, risk (related to company-unique events) Variability in returns (standard deviation) Total Systematic or nondiversifiable risk risk (result of general market influences) 1 10 20 25 Number of stocks in portfolio  CHAPTER 6 The Meaning and Measurement of Risk and Return 201 business practices on the environment. Our measure of risk should, therefore, measure how responsive a stock or portfolio is to changes in a market portfolio, such as the New York Stock Exchange or the S&P 500 Index.7 Measuring Market Risk To help clarify the idea of systematic risk, let’s examine the relationship between the common stock returns of eBay and the returns of the S&P 500 Index. The monthly returns for eBay and the S&P 500 Index for the 12 months ending January 2018 are presented in Table 6-3 and Figure 6-4. These monthly returns, or holding-period returns, as they are often called, are calculated as follows8. priceend of month - pricebeginning of month Monthly holding return = pricebeginning of month priceend of month = - 1 (6-6) pricebeginning of month TABLE 6-3 Monthly Holding-Period Returns, eBay versus the S&P 500 Index, February 2017 through January 2018 eBay S&P 500 Index Month and Year Price Returns (%) Price Returns (%) 2017 January $31.83 $2,279 February 33.90 6.50% 2,364 3.72% March 33.57 - 0.97% 2,363 - 0.04% April 33.41 - 0.48% 2,384 0.91% May 34.30 2.66% 2,412 1.16% June 34.92 1.81% 2,423 0.48% July 35.73 2.32% 2,470 1.93% August 36.13 1.12% 2,472 0.05% September 38.46 6.45% 2,519 1.93% October 37.64 - 2.13% 2,575 2.22% November 34.67 - 7.89% 2,648 2.81% December 37.74 8.85% 2,674 0.98% 2018 January 40.58 7.53% 2,824 5.62% Average Monthly Return 2.15% 1.81% Standard Deviation 4.75% 1.64% 7 The New York Stock Exchange Index is an index that reflects the performance of all stocks listed on the New York Stock Exchange. The Standard & Poor’s (S&P) 500 Index is similarly an index that measures the combined stock-price performance of the companies that constitute the 500 largest companies in the United States, as designated by Standard & Poor’s. 8 For simplicity’s sake, we are ignoring the dividend that the investor receives from the stock as part of the total return. In other words, if Dt equals the dividend received by the investor in month t, the holding- period return would more accurately be measured as Pt + Dt r1 = - 1 (6-7) Pt - 1 202 PART 2 The Valuation of Financial Assets FIGURE 6-4 Monthly Holding-Period Returns, eBay versus the S&P 500 Index, February 2017 through January 2018 10.00% SP 500 HD 8.00% 6.00% 4.00% Percentage Change 2.00% 0.00% Feb-17 Apr-17 Jun-17 Aug-17 Dec-17 Jan-18 –2.00% Oct-17 –4.00% –6.00% –8.00% –10.00% Months Source: Data from Yahoo Finance For instance, the holding-period returns for eBay and the S&P 500 Index for February 2017 are computed as follows: stock price at end of Feburary 2017 eBay return = - 1 stock price at end of January 2017 $33.90 = - 1 = 0.0650 = 6.50% $31.83 index value at end of February 2017 S & P 500 Index return = - 1 index value at end of January 2017 $2,364 = - 1 = 0.0372 = 3.72% $2,279 At the bottom of Table 6-3, we have also computed the averages of the returns for the 12 months, for both eBay and the S&P 500 Index, and the standard deviations for these returns. Because we are using historical return data, we assume each observation has an equal probability of occurrence. Thus, the average holding-period return is found by summing the returns and dividing by the number of months; that is, Average holding@period return in month 1 + return in month 2 + g + return in last month = return number of monthly returns (6-8) and the standard deviation is computed as follows: Standard deviation (return in month 1 - average return)2 + (return in month 2 - average return)2 + g + (return in last month - average return)2 = D number of monthly returns - 1 (6-9)  CHAPTER 6 The Meaning and Measurement of Risk and Return 203 In looking at Table 6-3 and Figure 6-4, we notice the following things about eBay’s holding-period returns over the 12 months ending in January 2018. 1. eBay’s shareholders realized higher monthly holding-period returns on average than the general market, as represented by the Standard & Poor’s 500 Index (S&P 500). Over the 12 months, eBay’s stock had an average 2.15 percent monthly return compared to only 1.81 percent for the S&P 500 Index. 2. While eBay had higher average monthly holding-period returns than the general market (S&P 500 Index) for the 12-month period ending January 2018, the vola- tility (standard deviation) of the returns was significantly higher for eBay than for the market—4.75 percent for eBay versus 1.64 percent for the S&P 500 Index. 3. In 9 of the 12 months, eBay’s stock price increased when the value of the S&P 500 Index increased and vice versa. That is, there is a moderate positive relationship between eBay’s stock returns and the S&P 500 Index returns. With respect to our third observation—that there is a relationship between the stock returns of eBay and the S&P 500 Index—it is helpful to see this relationship by graphing eBay’s returns against the S&P 500 Index returns. We provide such a graph in Figure 6-5. In the figure, we have plotted eBay’s returns on the vertical axis and the returns for the S&P 500 Index on the horizontal axis. Each of the 12 dots in the figure represents the returns of eBay and the S&P 500 Index for a par- ticular month. For instance, the returns for September 2017 for eBay and the S&P 500 Index were 6.45 percent and 1.93 percent, respectively, which are noted in the figure. In addition to the dots in the graph, we have drawn a line of “best fit,” which we call the characteristic line. The slope of the characteristic line measures the aver- characteristic line the line of “best age relationship between a stock’s returns and those of the S&P 500 Index; or stated fit” through a series of returns for a firm’s stock relative to the market’s differently, the slope of the line indicates the average movement in a stock’s price in returns. The slope of the line, response to a movement in the S&P 500 Index price. We can estimate the slope of the frequently called beta, represents the line visually by fitting a line that appears to cut through the middle of the dots. average movement of the firm’s stock returns in response to a movement in We then compare the rise (increase of the line on the vertical axis) to the run the market’s returns. (increase on the horizontal axis). Alternatively, we can enter the return data into a financial calculator or in an Excel spreadsheet, which will calculate the slope based on statistical analysis. For eBay, the slope of the line is 0.748, which means that, on average, as the market returns (S&P 500 Index returns) increase or decrease 1 percent, the return for eBay on average increases or decreases 0.748 percentage points. We can also think of the 0.748 slope of the characteristic line as indicating that eBay’s returns are 0.748 times as volatile on average as those of the overall market (S&P 500 Index). This slope has come to be called beta 𝛃 in investor jargon, and it beta the relationship between an measures the average relationship between a stock’s returns and the market’s returns. It is a investment’s returns and the market’s returns. This is a measure of the term you will see almost any time you read an article written by a financial analyst investment’s nondiversifiable risk. about the riskiness of a stock. Looking once again at Figure 6-5, we see that the dots (returns) are scattered all about the characteristic line—most of the returns do not fit neatly on the char- acteristic line. That is, the average relationship may be 0.748, but the variation in eBay’s returns is only partly explained by the stock’s average relationship with the S&P 500 Index. Other driving forces unique to eBay also affect the firm’s stock returns. (Earlier, we called this company-unique risk.) If we were, however, to diversify our holdings and own, say, 20 stocks with betas of 0.748, we could essentially eliminate the variation about the characteristic line. That is, we would remove almost all the volatility in returns, except for that caused by the general 204 PART 2 The Valuation of Financial Assets FIGURE 6-5 Monthly Holding-Period Returns, eBay versus the S&P 500 Index, February 2017 through January 2018 Google 20.00% September Slope = Rise/Run = 7.48%/10% = 0.748 2017 15.00% returns 10.00% characteristic line Google Monthly Returns 5.00% Rise = 7.48% 0.00% -20.00% -15.00% -10.00% -5.00% 5.00% 10.00% 15.00% 20.00% -5.00% Run = 10% -10.00% -15.00% -20.00% S & P 500 Monthly Returns Calculating beta: Visual—the slope of a straight line can be estimated visually by drawing a straight line that best “fits” the scatter of eBay’s stock returns and those of the market index. The beta coefficient then is the “rise over the run.” For example, when the S&P 500 Index is 10 percent, shown on the horizontal axis (the run), eBay shown on the vertical axis (the rise) is 7.48%. Thus, beta is the rise divided by the run, or 7.48% ÷ 10%. Financial calculator—financial calculators have built in functions for computing the beta coefficient. However, since the procedure varies from one calculator to another, we do not present it here. Excel—Excel’s Slope function can be used to calculate the slope, =slope (return values for eBay, return values for S&P). Source: Data from Yahoo! Finance. market, which is represented by the slope of the line in Figure 6-5. If we plotted the returns of our 20-stock portfolio against the S&P 500 Index, the points in our new graph would fit nicely along a straight line with a slope of 0.748, which means that the beta (β) of the portfolio is also 0.748. The new graph would look something like the one shown in Figure 6-6. In other words, by diversifying our portfolio, we can essentially eliminate the variations about the characteristic line, leaving only the variation in returns for a company that comes from variations in the general market returns. So beta (β)—the slope of the characteristic line—is a measure of a firm’s market risk or systematic risk, which is the risk that remains for a company even after we have diversified our portfolio. It is this risk—and only this risk—that matters for investors who have broadly diversified portfolios. Although we have said that beta is a measure of a stock’s systematic risk, how should we interpret a specific beta? For instance, when is a beta considered low and when is it considered high? In general, a stock with a beta of zero has no sys- tematic risk; a stock with a beta of 1 has systematic or market risk equal to the “typical” stock in the marketplace; and a stock with a beta exceeding 1 has more market risk than the typical stock. Most stocks, however, have betas between 0.60 and 1.60. We should also realize that calculating beta is not an exact science. The final esti- mate of a firm’s beta is heavily dependent on one’s methodology. For instance, it  CHAPTER 6 The Meaning and Measurement of Risk and Return 205 FIGURE 6-6 Holding-Period Returns for a Hypothetical Portfolio and the S&P 500 Index Portfolio returns (%) 30 20 Characteristic line 10 Holding-period returns S&P 500 Index returns (%) –30 –20 –10 10 20 30 –10 –20 –30 matters whether you use 24 months in your measurement or 60 months, as most professional investment companies do. Take our computation of eBay’s beta. We said eBay’s beta is 0.748, but Nasdaq.com estimated eBay’s beta to be 0.77. Further, Nasdaq.com’s beta estimates for the following firms are as follows: Company Name Beta Amazon 1.49 Apple 1.06 Coca-Cola 0.25 PepsiCo 0.55 ExxonMobil 0.81 General Electric 0.61 IBM 0.97 Lowe’s 1.36 Merck 0.62 Nike 0.81 Wal-Mart 0.70 EXAMPLE 6.2 Calculating Monthly Returns, Average Returns, and Standard Deviation, MyLab Finance Video and Estimating Beta Given the following price data for GroveCo and the S&P 500 Index, compare the returns and the volatility of the returns, and estimate the relationship of the returns for GroveCo and the S&P 500 Index. What do you conclude? 206 PART 2 The Valuation of Financial Assets Month GroveCo S&P 500 Index Jan-17 $88.33 $1,924 Feb-17 89.34 1,960 Mar-17 88.10 1,931 Apr-17 92.49 2,003 May-17 93.09 1,972 Jun-17 96.17 2,018 Jul-17 100.10 2,068 Aug-17 94.56 2,059 Sep-17 93.78 1,995 Oct-17 98.98 2,105 Nov-17 95.62 2,068 Dec-17 95.12 2,086 Jan-18 97.20 2,128 STEP 1: Formulate a Decision Strategy The following equations are needed to solve the problem: Monthly holding-period returns: priceend of month - pricebeginning of month Monthly holding return = pricebeginning of month priceend of month = - 1 pricebeginning of month Average monthly returns: return in month 1 + return in month 2 + g + return in last month Average holding@period return = number of monthly returns Standard deviation of the returns: Standard deviation (return in month 1 - average return)2 + (return in month 2 - average return)2 + g + (return in last month - average return)2 = D number of monthly returns - 1 Determining the historical relationship between GroveCo’s stock returns and those of the S&P 500 Index requires that we estimate the line that best fits the average rela- tionship, or using a financial calculator, or using a spreadsheet. STEP 2: Crunch the Numbers Given the price data, we have calculated the monthly returns, the average returns, and the standard deviation of returns as follows, as well as the slope of the character- istic line (using an Excel spreadsheet):  CHAPTER 6 The Meaning and Measurement of Risk and Return 207 GroveCo S&P 500 Index Month Price Returns Price Returns Jan-17 $88.33 $1,924 Feb-17 89.34 1.14% 1,960 1.87% Mar-17 88.10 - 1.39% 1,931 - 1.48% Apr-17 92.49 4.98% 2,003 3.73% May-17 93.09 0.65% 1,972 - 1.55% Jun-17 96.17 3.31% 2,018 2.33% Jul-17 100.10 4.09% 2,068 2.48% Aug-17 94.56 - 5.53% 2,059 - 0.44% Sep-17 93.78 - 0.82% 1,995 - 3.11% Oct-17 98.98 5.54% 2,105 5.51% Nov-17 95.62 - 3.39% 2,068 - 1.76% Dec-17 95.12 - 0.52% 2,086 0.87% Jan-18 97.20 2.19% 2,128 2.01% Average Monthly Return 0.85% 0.87% Standard Deviation 3.38% 2.57% The relationship between the GroveCo returns and the S&P 500 Index is shown in the graph below: PepsiCo returns (%) 20 15 October Slope = Rise/Run = 10.5%/10% = 1.05 2017 return 10 Characteristic line 5 Rise = 10.5% S&P 500 index returns (%) –10 –5 5 10 15 20 Run = 10% –5 STEP 3: Analyze Your Results We can see that the average return for GroveCo was approximately the same as the returns for the S&P 500. (Remember, these are only monthly returns.) But the volatil- ity of the returns for GroveCo is higher than for the S&P 500 Index. Then when we regress the GroveCo returns on the market (S&P 500) returns, we see that the average relationship is 1.05, which is a measure of systematic risk. That is, for every 1 percent that the market returns move up or down, GroveCo’s returns will on average move 1.05 percent. To this point, we have talked about measuring an individual stock’s beta. We will now consider how to measure the beta for a portfolio of stocks. 208 PART 2 The Valuation of Financial Assets CAN YOU DO IT? Estimating Beta Below, we provide the end-of-month prices for USACo stock and the Standard & Poor’s 500 Index for July 2017 through January 2018. Given the information, compute the following for both USACo and the S&P 500: (1) the monthly holding-period returns, (2) the average monthly returns, and (3) the standard deviation of the returns. Next, graph the holding-period returns of USACo on the vertical axis against the holding-period returns of the S&P 500 on the horizontal axis. Draw a line on your graph similar to what we did in Figure 6-5 to estimate the average relationship between the stock returns of USACo and the returns of the overall market as represented by the S&P 500. What is the approximate slope of your line? What does this tell you? (In working this problem, it would be easier if you used an Excel spreadsheet including t