I need your help with this: Can you please quote from the attached chapters / resource if possible? Corporations often use different costs of capital for different operating divisions. Using an example, calculate the weighted cost of capital (WACC). What are some potential issues in using varying techniques for cost of capital for different divisions? If the overall company weighted average cost of capital (WACC) were used as the hurdle rate for all divisions, would more conservative or riskier divisions get a greater share of capital? Explain the reasoning. What are two techniques that you could use to develop a rough estimate for each division?s cost of capital?

Hickman, K. A., Byrd, J. W., & McPherson, M. (2013).Essentials of finance[Electronic version]. Retrieved from https://content.ashford.edu/

9 Don Nichols/Getty Images Risk and Return Learning Objectives Upon completion of Chapter 9, you will be able to: Explain the significance of required return and its building blocks. Describe the relationship between risk and return, and how to measure both. Understand the meaning of risk in a business setting and give examples of different types of risk. Learn how to estimate required return and utilize it in valuation. byr80656_09_c09_217-244.indd 217 3/28/13 3:34 PM Section 9.1 The Building Blocks of the Required Return CHAPTER 9 I nvestors come in many forms. They may be individuals who invest in corporate stocks, a speculator investing in a Texas oil well, retirement funds that invest in bonds, partnerships investing in apartment buildings, or corporations investing in productive projects. One thing all these investors have in common is their desire to increase their wealth, which is done by identifying projects whose value is expected to exceed their cost. If we invest $100 today in a project that produces cash flows worth $125 in today's terms, then we increase our wealth by $25. The basic formula for estimating the value of an investment is found by discounting the expected future cash flows (CFs) back to today's equivalent value (V0) at a rate of return that is appropriate given the investment's riskin other words, at the investor's required return, R(r), in the following equation: (9.1) N CFt V0 5 a t t51 1 1 1 R 1 r 2 2 One part of the formula that hasn't been covered is how to estimate the required return that is appropriate to use as the discount rate in the valuation calculation. Finding the required rate of return is the topic of this chapter (and is expanded upon in Chapter 10). 9.1 The Building Blocks of the Required Return I n Chapter 1, we introduced the idea that investors are assumed to be rational and risk averse. Because they are (mostly!) rational, investors will give up control of their money for a period of time by investing only if they expect to increase their wealth. Therefore, investors have an almost instinctual return requirement as they invest. For example, rational investors would always want to earn at least the risk-free rate of return when investing in some security or project. Otherwise, they would be settling for a return lower than what they could be assured of by simply depositing the funds in a savings account that is guaranteed by both the bank and the government though the FDIC (Federal Deposit Insurance Corporation). So we believe that the first building block for assessing a required return is the risk-free interest rate. A Closer Look: Different Types of Returns It's useful to do some thinking about different kinds of returns that investors might discuss when they are considering investment performance. One type of return is the historical return, also known as an actual or realized return. If you buy a share of stock for $20 and a year later you sell it for $22, you have earned a historical or realized return equal to 10% per year ($2 gain on a $20 investment). The actual return you earned is 10%. This may be the same as, or it may be quite different than, the expected return that you were hoping for when you bought the stock. Perhaps your friend who is a stockbroker told you that she had calculated a target selling price of $30 for the stock. If you believed her forecast, then you were expecting a 50% return when you decided to buy the stock. Clearly, if you were expecting a 50% return but the actual return was only 10%, then it's likely that you were disappointed in the result. But were you satisfied with the 10% that you earned? To answer that, we need to know what your required return for the stock was equal to. The estimation of the required return for an investment is the subject of this chapter, but it is generally acknowledged that risk contributes (continued) byr80656_09_c09_217-244.indd 218 3/28/13 3:34 PM Section 9.2 Risk and Return CHAPTER 9 A Closer Look: Different Types of Returns (continued) to one's required return. So if this was a very risky stock, you may have had a required return equal to 25%. In this case, you would have been pretty unhappy with the result. On the other hand, if the stock was considered a low-risk investment, then you might have had a return requirement of only 8%, and you were probably very satisfied with the 10% actual return, given the stock's low risk. For most investments, however, the risk-free rate is only the first component of the required return. Virtually all investments have some risk associated with them, so investors also require what is known as a risk premium to compensate them for this risk exposure. Recall that we assume that investors are risk averse, which implies that to bear risk, they require compensation in order to subject themselves to distasteful uncertainty. Now we have the two fundamental building blocks of the required return for an investment: the risk-free return and a risk premium. (9.2) R(r) 5 (Risk-Free Rate of Return) 1 (Risk Premium) Given these intuitive building blocks, we will now take a closer look at risk, returns, and their relationship to one another in order to fully develop the methods for more precisely estimating the required return for an investment. 9.2 Risk and Return T he trade off between risk and return is second nature to us: We instinctively understand that if we are going to invest in a bond issued by United Airlines, we would do so only if we expected to receive a rate of return greater than we would receive if we invested in a bond issued by the U.S. Treasury. Why? United Airlines is generally considered riskier than the U.S. government. One of the great intellectual challenges of finance over the past 50 years was to find a method for measuring risk, and then to find a formula that quantifies the relationship between risk and return. Using our example, we need to find a method for quantifying how much risk United Airlines has and then discover a method for estimating the return that investors should require given that level of risk. Risk We begin our discussion of risk by thinking of uncertainty. There are many, many sources of uncertainty in business. To organize our thinking about this uncertainty, it is useful to classify risk into three major categories: financial risk, business risk, and investors' risk. Financial risk was introduced in Chapter 8 and is associated with the use of debt (leverage) in a firm's capital structure. Business risk, on the other hand, has less to do with how the company is financed and more to do with what the business actually does and how it does it. That is, business risk depends on the uncertainties of the firm's activities as it produces and sells goods and services. byr80656_09_c09_217-244.indd 219 3/28/13 3:34 PM Section 9.2 Risk and Return CHAPTER 9 Finally, we recall that the objective of the firm is to increase the wealth of investors in the business. Naturally, the value of these investors' securities (i.e., their bonds, common stock, preferred stock) will depend in part on their riskthe uncertainty regarding their price changes. It is the uncertainty inherent in the value of these securities that defines investment risk. The most common way of defining business risk is the variability of operating income (or the variability of its earnings before interest and taxesits EBIT). Therefore, this risk depends on the fluctuations of sales and expenses. These, in turn, depend on factors that may be associated with the firm itself, with the industryincluding demand for the industry's products and the actions of competitorsand also economy-wide influences on the firm's operations. As these factors cause the firm's operating income to change, the effect of that change on the firm's profits may be magnified by the company's leverage as we saw in Chapter 8. As profits and cash flows change, the value of the securities that represent claims on those cash flows will also change. An example may help to illustrate this continuum of risk. Suppose that a grocery store sees its sales fall because of extensive road construction on the major boulevard leading to the business. This is an example of a firm-level business risk. This loss of sales is made worse because a drought in the Midwest has caused the costs of groceries to dramatically increase, causing a fall in demand and a narrowing of profit marginsan example of an industry-level business risk. Let's say that the level of sales has dropped by 20%. We will assume the store is highly leveraged, causing its profits to actually drop by nearly 80% once interest on the borrowed money is paid, a decline well above the 20% fall in profits. Here, financial risk is at play, dampening profitability. An investor in this grocery store would likely see the value of his or her holdings fall as a consequence of the company's bad fortunean example of the risk that investors face being a culmination of a variety of other risks. The fall in security prices implied by this story would lead to a negative investment return for this period. Measuring Risk Clearly, we are not certain what return we will receive in the future when we invest. With some investments, we feel a greater level of uncertainty than we do with others. For example, if an investor chooses to buy a 5-year certificate of deposit at a bank insured by the FDIC, most investors would feel there is very little uncertainty about how much their deposit will be worth after the 5-year period. However, if the funds were invested in Facebook common stock, there is a wide range of potential values that the stock could have 5 years after the investment is made. One might wonder, \"How much riskier is Facebook stock than a certificate of deposit? Is it twice as risky? Ten times as risky? Twenty times as risky?\" To answer that question, we need a metric for measuring risk. We begin by introducing the concept of an investment's total risk. We will define total risk as the variability of returns that is measured by their standard deviation. For simplicity's sake we will be using historical returns to measure risk because future returns are so difficult to predict, as was already discussed. Note that we are assuming in this case that past risk is a good predictor of future risk, which may be adequate, but as you become a more sophisticated analyst, this estimate may be adjusted up or down depending on what you know about the prospects of the firm or the investment that you're analyzing. byr80656_09_c09_217-244.indd 220 3/28/13 3:34 PM CHAPTER 9 Section 9.2 Risk and Return One way of measuring total risk is to use the standard deviation: (9.3) Total Risk 5 Standard Deviation 5 a1 N 1 Rt 2 E 1 R 2 2 2 / N where Rt 5 the return at time t, E(R) is the investment's expected return, and N is the number of return periods used in the calculation. As its name implies, the standard deviation is a way of measuring the typical distance (or deviation) of a return from the average (or expected) return. So a stock that has a standard deviation of 15% has more uncertainty regarding its returns than a stock with a standard deviation of only 10%. To see this, take a look at Figure 9.1, which illustrates the distribution of returns for two stocks: Peabody Coal and Pacific Gas and Electric (PG&E). These histograms show the frequency of weekly returns from March 2010 until March 2012. Notice that PG&E's returns are much more tightly clustered, whereas Peabody's have long tails, particularly a long tail to the left of its center. Both of these distributions have about the same average return (0.00), but there is much more uncertainty about the return of Peabody because the standard deviation of its returns is 0.068 per week while Pacific Gas and Electric's standard deviation is only 0.023. Therefore, judging by this historical data, risk averse investors would be much more concerned about owning Peabody stock because of the uncertainty surrounding its returns. It is important to keep in mind where the uncertainty illustrated in Figure 9.1 actually comes from: Risk is measured by the variability of returns, and returns are generated by price changes as we saw in Equation (9.1). Recall that price changes are caused by the arrival to the market place of new information, which investors and analysts anxiously a wait in order to adjust their view of the company's or investment's worth. Risk, therefore, has at its foundation information and the investment's sensitivity to that information. As an example, consider what might happen to a company's stock value, and therefore its returns, if the United States announces that it will impose a significant tax on carbon emissions. The prices of oil companies would likely fall dramatically as one would imagine gasoline costs increasing and demand decreasing, lowering oil company profits. However, a hydroelectric-based utility company might see little change in its value, as it does not produce carbon, so its cost and pricing structure would remain unchangedits value might actually increase as demand for clean energy would likely rise. The risk of adverse price movements can be decreased by diversification. For example, in the previous example, consider what would happen to an investor's portfolio (collection of investment assets) if the investor held both oil company stocks and hydroelectric utility stocks. The oil stock values would fall because of the carbon tax, but this risk would be mitigated by the positive response to the tax by hydroelectric firms. In this case, the fall in gas stock prices is offset by the positive response of the utility stocks. Risk is decreased in this case because of the different reactions by the two industries to the same information. When one has investments in a variety of companies, there is a good chance that what affects one company negatively may actually have little impact, or perhaps a positive impact, on the value of another stock in the portfolio. byr80656_09_c09_217-244.indd 221 3/28/13 3:34 PM CHAPTER 9 Section 9.2 Risk and Return Figure 9.1: The historical distribution of returns for Peabody Coal and Pacific Gas and Electric using weekly returns 3/2010-3/2012 Peabody Coal 30 25 20 15 10 5 0 -0.16 -0.12 -0.08 -0.04 0 0.04 0.08 0.12 0.16 Pacific Gas & Electric 60 50 40 30 20 10 0 -0.16 -0.12 -0.08 -0.04 0 0.04 0.08 0.12 0.16 Total risk can, therefore, be broken down into risk that may be diversified away (called diversifiable risk) and risk that cannot be avoided or mitigated (called systematic or nondiversifiable risk): (9.4) Total Risk 5 Diversifiable Risk 1 Nondiversifiable Risk Diversifiable risk is often characterized as \"firm-specific\" Risk and \"industry-specific\" risk and is also referred to as unique or unsystematic risk: (9.5) Diversifiable (Unsystematic) Risk 5 Firm-Specific Risk 1 Industry-Specific Risk. Nondiversifiable risk involves market risk and is also called systematic risk: (9.6) byr80656_09_c09_217-244.indd 222 Nondiversifiable (Systematic) Risk 5 Market Risk 3/28/13 3:34 PM CHAPTER 9 Section 9.2 Risk and Return It is fairly obvious that what investors are most concerned with is the nondiversifiable risk because they can eliminate a great deal of the diversifiable risk by simply holding a large number of different stocks. Table 9.1 summarizes the kinds of risk. Let's take a closer look at each of these. Table 9.1: Types of risk Risk that can be diversified away Risk that cannot be diversified away Industry and firm-specific risk Market risk Diversifiable risk Nondiversifiable risk Idiosyncratic risk Systematic risk Unsystematic risk Economy-wide risk Firm-specific or idiosyncratic risk is associated with events such as a company making a poor product decision, being sued, having a CEO get indicted or die, having a big fire at a factory, or having a competitor develop a new product. All of these would adversely affect the value of the company, but if an investor is well diversified, the impact will be minimal to the overall portfolio because such an event the impacts only a single firm. Also, with enough firms in a portfolio, there is a good chance that when a firm-specific bad event happens to Company A, you may have another company that experiences firm-specific good news. For example, suppose that on the same day that Firm A loses a lawsuit, Firm B discovers oil, so these events would tend to offset one another in your portfolio. Sometimes, a news event for one company ripples through the industry. If Apple announces a new, more powerful but less expensive iPad, that will almost certainly affect the prospects of other companies making tablet computers. If one airline company has several planes grounded for safety inspections, other airlines might benefit as passengers switch their flight plans. Industry-specific risks are also largely avoidable via diversification because events that harm a particular type of industry will not necessarily have a negative effect on other stocks in a portfolio that represent firms in other industries. For example, low interest rates may hurt the profits in the banking industry, yet they actually help the housingbuilding industry. Therefore, holding a portfolio of stocks (in other words, being diversified) enables the negative impact of low interest rates on one industry to be offset by the positive effect these rates have on other industries within the investor's portfolio. Nondiversifiable or market risks are difficult to avoid regardless of how many stocks you own and/or how diversified your investment portfolio becomes. Some events have negative effects that pervade the entire economy. For example, unemployment hurts almost all companies as consumer demand falls, lowering sales and profits, and as savings fall, making capital scarce. These kinds of far-reaching events are referred to as nondiversifiable or market risks. High inflation, war, economic recessions, and acts of terrorism all have a negative impact on almost all of the firms in your portfolio, regardless of how many stocks you own! byr80656_09_c09_217-244.indd 223 3/28/13 3:34 PM CHAPTER 9 Section 9.2 Risk and Return Because much of the risk of investing, including firm- and industry-specific risks, may be avoided simply by diversifying one's portfolio, it is argued that we need not concern ourselves with these diversifiable risks. It is, for example, just as easy to buy a mutual fund that holds shares of 500 different companies as it is to load up on a single firm's stock. Clearly, the mutual fund strategy avoids much of the risk that the investor in a single security faces. In fact, the standard deviation (the variability) of a diversified portfolio's returns can easily be reduced by about half compared to the average standard deviation of the individual stocks in the portfolio. The diversification effect of lowering risk is shown in Figure 9.2, where we can see that increasing the number of even randomly selected stocks in a portfolio can dramatically reduce the portfolio's standard deviation of returns. Figure 9.2: The diversification effect 35% Portfolio Standard Deviation 30% 25% 20% unique risk 15% 10% market risk 5% 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Number of Stocks Portfolio standard deviation of returns as the number of randomly selected stocks in the portfolio increases. Since investors can easily and inexpensively eliminate most diversifiable risk from their portfolios, we assume that everyone does so. Thus, the risk that is relevant to the investor is the nondiversifiable or market risk of an investment. The question becomes, how can we measure this risk? To measure market risk, we utilize a metric called beta. Beta measures a firm's typical responsiveness to information that impacts the entire market, such as information about economic growth, political news, inflationary expectations, the balance of trade, natural disasters, and so on. By definition, the average sensitivity to this kind of information would be measured by the responsiveness of the market portfolio. byr80656_09_c09_217-244.indd 224 3/28/13 3:34 PM CHAPTER 9 Section 9.2 Risk and Return The market portfolio theoretically would be totally diversified and would include virtually all the stocks that are traded and all the bonds, etc. In practice, there is no such thing as a true market portfolio, so a proxy isused as an approximation. Typically, the S&P 500 Index is used as that proxy, but other proxies can be used (see Table 9.2 for some other choices). The beta of the market portfolio is defined as being equal to positive one. Table 9.2: Indexes that can be used as proxies Index Description S&P 500 An index composed of 500 of the largest publicly traded stocks in the United States. Dow Jones Industrial Average An index created in 1896 composed of 30 significant publicly traded stocks; its composition changes as firms become less important and others increase in importance. Russell 2000 An index composed of 200 small-capitalization companies; used as a benchmark for small-cap stock and mutual fund performance. A firm may be more sensitive than the average to economic information, in which case this firm's beta would be greater than 1. A company that is twice as sensitive as the average firm to economic events will have a beta of 2.00, while a firm that is less sensitive than average will have a beta below 1. Here is an example: Take a firm that sells luxury goods, like a Porsche automobile dealership. We might assume that when the economy is booming, this business is doing really well, but when the economy is doing poorly, iStockphoto/Thinkstock luxury sports car sales suffer drastically. Let's suppose that Porsche A company that sells luxury goods may be more sensitive dealership's beta is 1.70, meaning than average to economic events. that it is 1.7 times as sensitive as the overall market to \"macroeconomic\" type events. So, if the government announces that economic growth is very strong, we might hear that the S&P 500 portfolio had a return of 5% that day in response to this good economic news. But because a Porsche dealership is more sensitive than average to such information, its stock would likely return around 8.5% on the same day (found by taking the product of 1.7 3 5% 5 8.5%). If, on the other hand, the market portfolio declines by 10% one month because of bad economic news, then the dealership's stock would probably fall by around 17%. byr80656_09_c09_217-244.indd 225 3/28/13 3:34 PM Section 9.2 Risk and Return CHAPTER 9 Of course, these are the expected returns for the Porsche dealership and may not be equal to the firm's actual returns on those days because there are always firm-specific factors that may affect a single stock's return. For example, on the day that the government announces strong economic growth (good news and we expect the 8.5% return), it may be the dealership also learns that the company is being sued, so the firm's stock could actually fall in value on the date because of this negative firm-specific announcement. Some businesses are less sensitive to market-level information than the average firm is. An example might be an electric utility company, say Pacific Gas and Electric. When the economy is doing well, Pacific Gas and Electric does well because there is more demand for electricity. And when times are bad, demand for power falls, but it doesn't fall too far because, unlike Porsche sports cars, electricity is close to being a necessity. So with this relatively low sensitivity, Pacific Gas and Electric has lower than average market risk and its beta is below 1. In June 2012, Yahoo! Finance reported PG&E's beta as 0.29. If the country goes into a recession and the market as proxied by the S&P 500 declines by 10%, PG&E, with its beta of 0.29, would see its stock price drop by only about 3% on average. Betas are typically estimated using historical returns and linear regression estimation. Linear regression is a statistical technique for estimating a best-fit line through points plotted in an x-y coordinate system like the graphs typically used in algebra. The idea is that the slope of this line will capture the average relationship between the x and the y variables. So if the slope is 1.5 for a regression line, that means that for each unit increase in the x value, the y value will (on average) increase by one and a half units. Regression is used to estimate a variety of relationships, like the effect that the time spent studying has on the grade point average of students. For our purposes, we use returns for the S&P 500 as our x values and the corresponding returns for the stock that we are interested in as our y values. The regression's slope, therefore, is an estimate of the stock's average responsiveness to market-wide returns, or its beta. Since betas are statistical estimates, they vary depending on the sample of data being used for the estimate. In June 2012 the PG&E beta reported by Yahoo! was 0.29 and is lower than the beta of 0.43 we estimate in Figure 9.3, using just 24 weeks of data from 2012. (The Yahoo! beta is based on 36 months of monthly return data). byr80656_09_c09_217-244.indd 226 3/28/13 3:34 PM CHAPTER 9 Section 9.2 Risk and Return Figure 9.3: S&P 500 returns versus PG&E returns 4% 3% y = 0.4327x + 0.0024 R2 = 0.2502 PG&E Returns 2% 1% -5% -4% -3% -2% -1% 0% 0% 1% 2% 3% 4% 5% -1% -2% -3% -4% S&P 500 Returns PG&E Return 5 0.433 3 S&P 500 Return 1 0.0024. Measuring Return We have been using the term \"return\" so far without precisely defining it. For simplicity's sake, we will use stocks to illustrate what we mean by return. A single period's historical return is given by the formula (9.7) Return over a Period 5 Rt 5 (Pricet 2 Pricet-1 1 Dividendt)/Pricet-1 For example, if you buy a share of stock for $40, hold it for 1 year during which you collect a dividend of $2 a share, and then sell the stock for $40.50, what was your return? The answer is (40.50 2 40 1 2)/40 5 2.50/40 5 0.0625 5 6.25% This stock formula can be generalized for any investment's return as (9.8) Return over a Period 5 Rt 5 (Valuet 2 Valuet-1 1 CashFlowt)/Valuet-1 In words, the return for the period is equal to the change in value of the asset during the period, plus any cash flows paid by the asset during the period, divided by the value of the asset at the beginning of the period. It would be really useful to predict returns (for one thing, you would get rich if you could consistently forecast returns!). Unfortunately, to predict returns, you would need to know what price changes will be in the future so these future prices could be plugged into the return formula in Equation (9.1). But, as you learned in Chapter 1, market efficiency byr80656_09_c09_217-244.indd 227 3/28/13 3:34 PM Section 9.2 Risk and Return CHAPTER 9 implies that competitive market prices reflect all available information. Therefore, we cannot say what future price changes will be and therefore what returns will be. This is because price changes will only reflect new information, and it's anybody's guess whether that information will be good news or bad news for the company or for the economy. Because it is nearly impossible to predict returns, we often use the historical average return as our best estimate of the expected future return. Bear in mind that, when using an average to predict the future, one should use a relatively long-run average since almost anything can happen in the short term. For example, between 1950 and 2010, the average annual return for the stock market (as proxied by the S&P 500 index) has been about 11% per year. This is considered a better estimate than, say, the 5 years average stock market return between 2007 and 2012, which averaged about zero! The Capital Asset Pricing Model Now we know how to measure nondiversifiable risk (by using beta) and how to measure returns. Next we need to learn how to utilize these metrics to estimate the required rate of return for an investment. Recall from the beginning of the chapter that the \"building blocks\" of a required return include the risk-free rate and a risk premium. These two elements are present in an equation called the Capital Asset Pricing Model (CAPM). This model is what we need to quantify the relationship between investor's required rate of return and the risk of an investment. Here is the principal equation for the CAPM as it was originally developed: (9.9) Required Return for an Investment 5 Rf 1 b(E(Rmkt) 2 Rf) where Rf 5 Risk-Free Rate of Return E(Rmkt) 5 Expected Return on the Market Portfolio b 5 Stock's Beta This theoretical relationship between risk and return was one of the groundbreaking achievements in economics in the 1960s, for which several academics were awarded the Nobel Prize. Like many theories, however, there are challenges when using the CAPM in practice. For example, no one knows what the expected return on the market, E(Rmkt), is equal to. In the model it is also assumed that there is a single, observable risk-free rate, when in reality there is no investment free of risk and there is more than one possible rate that can be used as a close proxy for risk-free. Because of these problems, most practitioners use a different form of the model, which is given below: (9.10) Required Return for an Investment 5 Rf 1 b(Market Risk Premium) Rf can be thought of as the rate that links the CAPM to current market conditions. This is important because interest rates are constantly changing because of changes in inflation, economic activity, or government policies. We use yields on outstanding debt issued by the U.S. government as a proxy for the risk-free rate, choosing the Treasury bill or bond that best matches the life of the asset we are evaluating. So, for stocks that have an almost perpetual life, long-term U.S. Treasury Bond yields are often used for the risk-free rate. The market risk premium is the amount of return yielded by the market portfolio over and above the treasury yield. It can be thought of as the return required for each additional byr80656_09_c09_217-244.indd 228 3/28/13 3:34 PM Section 9.2 Risk and Return CHAPTER 9 unit of risk as measured by beta. Often, the market risk premium (MRP) is assumed to equal its historical average, which is about 5% to 7%, depending on whose data you use. Here is an example of using the CAPM. Let's estimate the required return for Nordstrom's (Ticker: JWN) stock given that Nordstrom's beta is 1.58, as reported on Yahoo! Finance in June 2012. Nordstrom's has a fairly high beta because it is considered a high-end or almost luxury retailer, not dealing in necessity goods. Consequently, when times are tough, some people may discontinue patronizing Nordstrom's and may buy their shoes and clothing at a more moderately priced retailer. Let's also assume that U.S. Treasury bonds are yielding 4.5% per year, and the historical average market risk premium (the average return of the market portfolio over and above the risk-free return) is about 6%. Using this information, we may estimate the required return for Nordstrom's stock using the CAPM: RNordstroms 5 Rf 1 bNordstroms(Market Risk Premium) 5 0.045 1 1.58(0.06) 5 0.1398 5 13.98% A second example is required rate of return for PG&E's stock. Using the published beta of 0.29 and the risk-free rate and market risk premium above, we would compute PG&E's required rate of return as: RPG&E 5 Rf 1 bPG&E(Market Risk Premium) 5 0.045 1 0.29(0.06) 5 0.0624 5 6.24% Notice that the much lower beta of PG&E results in a much lower required rate of return compared to Nordstrom. Hence, the risk of investing in PG&E is lower when compared to investing in Nordstrom. A Closer Look: Multifactor Models The CAPM was developed independently by William Sharpe, John Linter, and Jan Mossin. It is known as a single factor model because it posits that the only relevant risk for investors is an asset's level of market risk. This risk is measured by beta. However, the CAPM is not without its critics. As we point out in the chapter, there are challenges in implementing the CAPM because many of its theoretical assumptions are not met in the \"real world\" (assumptions like the existence of a true market portfolio and risk-free asset). Other models have been developed. Probably the two most famous are the Arbitrage Pricing Theory (the APT) and the Fama French Model. The APT is similar to the CAPM in that it is theoretically derived (in this case by Stephen Ross). It asserts that there may be multiple risk factors that enter into the required return calculation. In fact, it can be shown that if it turns out that there is actually only one risk factor that matters, then the APT and the CAPM yield identical formulas and identical results. The difficulty with the APT is that it tells us neither the number of risk factors that influence prices nor what those factors are likely to be. The development of the APT kicked off a series of empirical studies that used actual data (rather than theory) to try to discover what the risk factors might be. The most famous of these studies were those done by Eugene Fama and Kenneth French in the 1990s. They added firm size and the book-to-market ratio to the traditional single index model (that used the market portfolio and beta) and found that their estimates were more accurate than those predicted by the CAPM. (Book-to-market is the ratio of the market value of a firm's equity to the book value of its equity). To learn more of the CAPM, the APT, and multifactor models like those of Fama and French, you can consult an MBA-level Investments or Portfolio Theory text, such as Modern Portfolio Theory and Investment Analysis by Elton, Gruber, Brown, and Goetzmann. byr80656_09_c09_217-244.indd 229 3/28/13 3:34 PM Section 9.2 Risk and Return CHAPTER 9 Try It: Calculator Key Strokes and Excel FunctionsFinding Beta Using Excel Finding an asset's beta using Excel is a two-step process: First, compute returns from prices; second, use the LINEST function to compute the beta (slope of a regression line). We downloaded stock prices for Dow Chemical (Ticker: DOW) from Yahoo! Finance using its historical price feature. We then downloaded the index data for the S&P 500 (Yahoo! Ticker: ^SPX). We use the adjusted close price to make sure dividends are included in the price. Our data are weekly and run from Friday, February 3, 2012 through Friday, June 22, 2012. We compute returns using the formula from the text: (Change in Price)/Beginning Price Because we are using adjusted prices, we don't need to explicitly include dividends in the returns equation. Here is the spreadsheet of prices showing the price series, the returns formulas, and the LINEST formula. Here is the same spreadsheet with the numerical results shown. Using this small sample of data, Dow's beta estimate is 1.63. (continued) byr80656_09_c09_217-244.indd 230 3/28/13 3:34 PM CHAPTER 9 Section 9.2 Risk and Return Try It: Calulator Key Strokes and Excel FunctionsBeta Calculation Using the CAPM (continued) Recall the CAPM formula: E(Ri) 5 Rf 1 b(E(Rmkt) 2 Rf) where E(Ri) is the expected return on a risky asset Rf is some risk-free rate, typically a short-term CD or treasury rate E(Rmkt) is the expected return on the market index, like the S&P 500 b is the sensitivity of the risky asset to the economy as a whole: b 5 Covariance (Ri, Rmkt)/Variance (Rmkt) That is, beta is the covariance of the risky asset with the market, divided by the variance of the market. In words, the covariance is the sum over every month of the products of the differences between the two factors and their means. The variance is the sum over every month of the squared differences between the market returns for each month and the average market return over the 5 months. This example illustrates the calculations: Suppose we have 5 months of returns data for two risky assets and the S&P. These are the returns: Month S&P 500 Risky Asset A Risky Asset B June 0.03 0.03 0.01 July 20.01 20.02 0.00 August 0.02 0.01 0.01 September 0.04 0.05 0.03 October 0.02 0.07 0.00 Five-month average returns 0.02 0.03 0.01 Obviously, Asset A \"moves around\" and is riskier than Asset B. That is the \"spirit\" captured by beta and the CAPM. The S&P 500 is a proxy for systematic risk. To calculate beta, these steps are followed: Covariance (Ra, Rmkt) 5 \u0007(0.03 2 0.02)(0.03 2 0.03) 1 (20.01 2 0.02)(20.02 2 0.03) 1 (0.02 2 0.02)(0.01 2 0.03) 1 (0.04 2 0.02)(0.05 2 0.03) 1(0.02 2 0.02)(0.07 2 0.03) 5 0 1 0.0015 1 0 1 0.0004 1 0 5 0.0019 Dividing that sum by n 2 1 5 0.0019/4 5 0.000475. Covariance (Rb, Rmkt) 5 \u0007(0.03 2 0.02)(0.01 2 0.01) 1 (20.01 2 0.02)(0.00 2 0.01) 1 (0.02 2 0.02)(0.01 2 0.01) 1 (0.04 2 0.02)(0.03 2 0.01) 1 (0.02 2 0.02)(0.00 2 0.01) 5 0 1 0.0003 1 0 1 0.0004 1 0 5 0.0007 Dividing that sum by n 2 1 5 0.0007/4 5 0.000175.\b (continued) byr80656_09_c09_217-244.indd 231 3/28/13 3:34 PM CHAPTER 9 Section 9.2 Risk and Return Try It: Calulator Key Strokes and Excel FunctionsBeta Calculation Using the CAPM (continued) Variance (Rmkt, Rmkt) 5 \u0007(0.03 2 0.02)(0.03 2 0.02) 1 (20.01 2 0.02)(20.01 2 0.02) 1 (0.02 2 0.02)(0.02 2 0.02) 1 (0.04 2 0.02)(0.04 2 0.02) 1 (0.02 2 0.02)(0.02 2 0.02) 5 0.0001 1 0.0009 1 0 1 0.0004 1 0 5 0.0014 Dividing that sum by n 2 1 5 0.0014/4 5 0.00035. Therefore ba 5 0.000475/0.00035 5 1.37 bb 5 0.000175/0.00035 5 0.5 bmkt 5 0.00035/0.00035 5 1 Portfolio Betas There are times when investors may want to estimate the required return for a portfolio or for stocks or other investment assets. For example, we may want to compare the actual, realized return on a portfolio to the required return on that portfolio in order to assess the performance of the manager who is in charge of the portfolio's investments. If the realized return exceeds the required return given the portfolio's risk, then the manager may be given a bonus or may use this information to attract new investment clients. If, on the other hand, the realized return is below the required return for the portfolio, then the manager may find his or her position in jeopardy. Portfolio betas are found by taking the weighted average of the betas of the assets held in the portfolio, where the weights are determined by the amount invested in each asset. For example, consider the portfolio in Table 9.3. Table 9.3: Sample portfolio Stock Stock's beta Amount invested Weight Acme Inc. 1.20 $100,000 0.10 XYZ Corp 1.50 $150,000 0.15 ABC Corp 0.70 $500,000 0.50 Delphi Inc. 1.00 $250,000 0.25 byr80656_09_c09_217-244.indd 232 3/28/13 3:34 PM Section 9.2 Risk and Return CHAPTER 9 The weights represent the proportion of total investment that is invested in each asset. For example, the total investment here is $1,000,000, so the investment of $150,000 in XYZ's stock represents 15% of the total, and the weight for XYZ is 0.15. In this case, the portfolio's beta is the sum: Portfolio Beta 5 (1.20)(0.10) 1 (1.50)(0.15) 1 (0.70)(0.50) 1 (1.00)(0.25) 5 0.945 To illustrate the usefulness of this number, we note that it could be used to estimate the risk of this portfolio versus another portfolio or a mutual fund. It could also be plugged into the CAPM to give an estimate of the required return for this portfolio. One warning is that beta is not a good measure of risk unless the investor is what is known as \"well diversified.\" Usually, if you own over 20 stocks, you are considered well diversified. However, if these stocks are concentrated in only a couple of industries, then you are probably not effectively diversified. Diversification is a subjective and relative term, so as a rule, it's better to be more diversified than to be less diversifiedin other words, it's probably prudent to invest in a lot of stocks and other assets (like bonds and real estate) that are not concentrated in only a few industries (or geographic locations). Correlation and Diversification In later finance courses you will learn a more advanced mathematical model of diversification's effect. You will learn that diversification depends on asset returns having imperfect correlation with one another. In other words, if all stocks in a portfolio went up and down together, then they would be perfectly correlated and there would be no point in diversifying. This is because if a disaster happened to one stock, it would also happen to all the other stocks because they are perfectly positively correlated to one another. This is why it is not a good idea to concentrate your portfolio in only one industry or even just a few industries because they would likely have high positive correlations with one another. On the other hand, stocks with lower correlations make great choices for forming a welldiversified portfolio. In fact, the most efficient diversification happens when we mix negatively correlated assets in our portfolios because then their risks will offset one another. To illustrate, consider this example: Suppose you owned stock A whose returns were perfectly negatively correlated with another stock, stock B. Whenever stock A's return went up, stock B's went down and vice versa. Imagine that they both varied around an average return of 10%. The situation is illustrated in Figure 9.4. If you were lucky enough to locate these two stocks, and you put your money in each one to form a special two-stock portfolio, your portfolio would earn exactly 10%, but you would experience zero variability. That is because every time stock A had a return below 10%, stock B would have a return above 10% because of its perfect negative correlation. In other words, you could have a risk-free portfolio with a 10% return. Of course, it is not easy (perhaps impossible) to find two stocks with perfect negative correlation. byr80656_09_c09_217-244.indd 233 3/28/13 3:34 PM CHAPTER 9 Section 9.2 Risk and Return Figure 9.4: Returns with perfect negative correlation Returns with Perfect Negative Correlation 0.06 0.04 0.02 Stock A 0 0 2 4 6 8 10 12 14 Stock B -0.02 -0.04 -0.06 The good news is that you can always reduce risk by mixing together assets whose returns are imperfectly correlated, and you do this without lowering their average return. Furthermore, since almost all investment assets are imperfectly correlated, you can get the risk-reducing benefits of diversification by simply mixing together even a randomly selected bunch of stocks (as was shown in Figure 9.2). Of course, with a little insight you can improve the benefits of diversification by being sure not to focus on one industry group and by mixing in, for example, a few international stocks. This is because the benefits of diversification are greater when you invest in firms whose returns have low correlations. In fact, if you can locate investments that are negatively correlated, then you can achieve dramatic reductions in risk exposure. International stocks added to a domestic portfolio tend to have lower correlations and therefore contribute a lot to risk reduction. byr80656_09_c09_217-244.indd 234 3/28/13 3:34 PM CHAPTER 9 Section 9.3 Required Returns and Valuation 9.3 Required Returns and Valuation E stimating required return and determining an asset's beta are important for evaluating whether to invest in a particular asset. We treat each of these in turn and show some examples of how to perform the calculations. Required Returns Required returns may now be estimated, and this is tremendously valuable because required returns are used as the discount rates in the valuation formulas when doing time value of money problems and security valuation. To illustrate, suppose that PG&E just paid an annual dividend of $1.82 per share, and we believe that dividends are expected to grow at a 2% annual rate in the future. In this case we can use the constant growth stock valuation formula from Chapter 5 to estimate the value of PG&E's stock: (5.4) Leo Cullum/The New Yorker Collection/http://www.cartoonbank.com P0 5 D 0 1 1 1 gN 2 r 2 gN where P0 is the value of PG&E stock, gN is long-run normal growth rate of dividends, and r is the required return. Note that we use the form of the model with D0 in the numerator because we were given the last dividend paid. Recalling that PG&E's required return was 6.24%, we get Value of PG&E Stock 5 ($1.82)(1.02)/(0.0624 2 0.02) 5 $1.8564/0.0424 5 $43.783 Suppose we did this estimation of value and looked up an actual quote for PG&E's stock on the Internet. If the price is currently $43.00, this would mean that the stock appears to be underpriced on the market, which would indicate that it is a bargain according to our estimates. However, before we run out to buy PG&E stock, which we think may be worth $0.78 more per share than its price, we need to consider market efficiency and market frictions. Remember that market efficiency says that the market price ($43.00) is the best available estimate of value. Now, we must decide whose estimate we put more faith in: our estimate ($43.78) or the market's ($43.00)? If we believe in market efficiency, we would probably not make the investment. There are times, however, when we need to value an asset or a closely held stock for which no market price exists: In this case we have little choice but to rely on our own estimates. In these cases, the ability to estimate the required return is essential. byr80656_09_c09_217-244.indd 235 3/28/13 3:34 PM CHAPTER 9 Section 9.3 Required Returns and Valuation What Determines an Asset's Beta? During the discussions of the Porsche dealership, Nordstrom's, and PG&E, we said that it is the nature of the products and services that are sold by a company that determine the company's risk. Luxury goods, like sports cars, tend to have more market risk than do necessities like electricity. Demand for durable goods, items that are long-lived and can be repaired like cars and appliances, will fluctuate more as the economy rises and falls than demand for food or medicine. If a company has invested in productive assets to make luxury or durable goods, then it is likely to have a high beta (higher than 1 or the market average beta). Similarly, if the factories and equipment make food or electricity or things that are necessary (or that have steady demand), then the company will have a lower beta (less than 1). Thus, it is the assets of a company and the products those assets make that determine the company's beta. Companies that produce similar goods sold in similar markets will have similar betas because those companies will be impacted similarly by the kind of macroeconomic news that creates market risk. In theory, the asset beta of a Porsche dealership should be very nearly the same as the asset beta of, say, a BMW dealership. This is because they are in very similar businesses. If both the Porsche and the BMW dealerships had no debt financing, then both of their asset betas would be identical to the betas of their stock. This is because the stock would represent the only claim against the assets, so the risk of the assets would translate directly to the risk of the stock. In this case, both dealerships' stock, being in the same business, would probably have almost identical betas, and both would also have almost identical required rates of return. However, most companies use debt financing in addition to equity financing. As we have already mentioned, this use of debt is also known as leverage. The use of leverage increases the risk of equity because debt, with its priority claim, forces equity holders to bear the risk that there will be lower cash flows available for them after debt payments are made. For this reason, the betas of stock differ even among firms in the same industry because of the varying amount of debt that the companies borrow. Asset betas, therefore, depend primarily on the nature of a company's businessthey reflect only the firm's undiversifiable business riskwhereas equity betas (the betas of investing in just a company's stock) depend on both a firm's asset beta and on its use of leverageboth the business and financial risk that stockholders must bear. Demonstration Problem 9.1: Standard Deviation of Two-Stock Portfolios a. A stock portfolio consists entirely of two stocks, Bach Corp. and Beethoven Inc. Using the following data, calculate the standard deviation of the portfolio. CORRBach-Beethoven 5 0.6 Bach byr80656_09_c09_217-244.indd 236 Beethoven 0.7 Portfolio weighting 0.3 0.11 Standard deviation 0.23 (continued) 3/28/13 3:34 PM CHAPTER 9 Section 9.3 Required Returns and Valuation Demonstration Problem 9.1: Standard Deviation of Two-Stock Portfolios (continued) b. If you wanted to increase the risk of your portfolio, what would you do? c. Show how your answer to part b would increase portfolio risk. Solution a. Calculate the portfolio standard deviation: SDp 5 # 3 1 0.7 2 2 1 0.11 2 2 1 1 0.3 2 2 1 0.23 2 2 1 2 1 0.7 2 1 0.3 2 1 0.6 2 1 0.11 2 1 0.23 2 4 5 0.130 b. Increase the weighting of Beethoven Inc. because it has the higher standard deviation. c. Any increase in the proportion of Beethoven Inc. would increase portfolio risk. In this example, we increase the proportion of Beethoven Inc. to 0.7. Recalculating portfolio standard deviation gives us SDp 5 # 3 1 0.3 2 2 1 0.11 2 2 1 1 0.7 2 2 1 0.23 2 2 1 2 1 0.7 2 1 0.3 2 1 0.6 2 1 0.11 2 1 0.23 2 4 5 0.183 Demonstration Problem 9.2: Risk and Return of Two-Stock Portfolios For stocks A and B, ra 5 0.13, rb 5 0.22, SDa 5 0.3, and SDb 5 0.5. a. What are the expected returns and standard deviations of portfolios consisting entirely of stock A and entirely of stock B? b. Calculate portfolio returns and standard deviations under the following assumptions: Wa 5 0.5, CORRAB 5 1 Wa 5 0.5, CORRAB 5 0 Solution a. No calculation is required. A portfolio consisting entirely of stock A would have ra 5 0.13 and SDa 5 0.3. A portfolio consisting entirely of stock B would have rb 5 0.22 and SDb 5 0.5. b. For Wa 5 0.5 and CORRAB 5 1, rP 5 0.175 and SDp 5 0.4. For Wa 5 0.5, and CORRAB 5 0, rP 5 0.175 and SDp 5 0.292. Demonstration Problem 9.3: Calculating Beta Calculate the beta for Emmett Corp. stock, using the following information: SDEmmett 5 0.08 SDS&P 500 5 0.65 CORREmmett-S&P 500 5 0.85 Solution Solve for beta: bs 5 CORREmmett2S&P 500SDEmmettSDS&P 500 1 SDS&P 500 2 2 bs 5 1 0.85 2 1 0.08 2 1 0.065 2 / 1 0.065 2 2 byr80656_09_c09_217-244.indd 237 3/28/13 3:34 PM Key Terms CHAPTER 9 Summary I n this chapter, the building blocks of required returns were introduced. Most of the chapter used stocks to illustrate and explore the relationships between risk and investors' return requirements. The calculation of returns, of total risk (standard deviation of returns), and of market risk (beta) were covered. The benefit of diversification by the elimination of certain types of risk was discussed, as was the effectiveness of diversification and how it is linked to the correlation between investments' returns. The Capital Asset Pricing Model, used to estimate the required return for an investment, was also covered, and its use was illustrated. The concept of an asset beta was introduced. Chapter 9 makes extensive use of the CAPM, and the asset beta will also be utilized as we attempt to discover the overall required return for the entire firm rather than just the return requirement for its equity as we focused on in this chapter. The theories and techniques explored in Chapter 9 will be vitally important for those of you who one day enter a career in the investments field, but the insights will also be important for everyone who becomes an investor, whether they are investing for retirement or for their child's college fund. Key Terms actual or realized return The actual return received from an asset, which may differ from its expected return. beta A measure of an asset's systematic risk, also known as its market risk; used to find the required return for an asset using the CAPM. business risk The possibility that a company will have lower-than-anticipated profits, or that it will experience a loss rather than a profit. Capital Asset Pricing Model (CAPM) A formula that quantifies the connection between an investment's market risk and its required rate of return. correlation A statistical measure of the comovement of asset returns; varies between negative and positive one, while a correlation of zero means that the two assets are \"uncorrelated\" and move independently. byr80656_09_c09_217-244.indd 238 diversifiable risk Risk that is unique to a certain asset or company. diversification The mixing of investments in a single portfolio, which can reduce risk exposure; the benefits of diversification are most dramatic when the correlations between assets in the portfolio are low or even negative. diversification effect The reduction in risk (standard deviation) that occurs through the blending of stocks into a portfolio. firm-specific or idiosyncratic risk Risk that is specific to an asset. Firm-specific risk has little or no correlation with market risk and therefore can be reduced by portfolio diversification historical average return The average past performance of a security or index. historical return The past performance of a security or index. 3/28/13 3:34 PM CHAPTER 9 Key Formulas industry-specific risks Risks that are inherent in each investment. Industryspecific risks can be reduced through diversification. risk averse A behavioral trait in which people focus more on losses than on equivalent gains. Risk aversion implies that investor must be paid to bear risk. investors' risk The uncertainty that is inherent in the value of securities. risk-free rate of return The theoretical rate of return for an investment that has zero risk. leverage A description of the proportion of debt used in a firm's capital structure. market portfolio A portfolio of all assets in an economy. In practice a broad market index, such as the S&P 500, is used to represent the market. market risks Macroeconomic, or economywide, sources of risk that affect the overall market. Also called systematic or nondiversifiable risk. market risk premium The difference between the rate of return on the market (e.g., S &P 500) and the risk-free return (e.g., Treasury bonds). portfolio betas A measure of the volatility, or systematic risk, of a portfolio in comparison to the market as a whole. risk premium The added return necessary to compensate investors for taking added risk. Standard & Poor's 500 Index (S&P 500) A price index of 500 stocks representing a broad cross section of industries often used to represent the entire stock market's activity. systematic or non-diversifiable risk Macro economic, or economy-wide, sources of risk that affect the overall market. Also called market risk. unique or unsystematic risk Risk that affects primarily one company or industry; unique risk may be mitigated by diversifying one's portfolio. Key Formulas a1 N 1 R 2 E 1 R 2 2 2/N (9.3) Total Risk 5 Standard Deviation 5 (9.4) Total Risk 5 Diversifiable Risk 1 Nondiversifiable Risk (9.5) \u0007Diversifiable (Unsystematic) Risk 5 Firm-Specific Risk 1 Industry-Specific Risk (9.6) Nondiversifiable (Systematic) Risk 5 Market Risk byr80656_09_c09_217-244.indd 239 3/28/13 3:34 PM Critical Thinking and Discussion Questions CHAPTER 9 (9.7) Return over a Period 5 Rt 5 (Pricet 2 Pricet21 1 Dividendt)/Pricet21 (9.8) \u0007Return over a Period 5 Rt 5 (Valuet 2 Valuet21 1 Cash Flowt)/Valuet21 (9.9) \u0007Required Return for an Investment 5 Risk-Free Rate of Return 1 Stock's Beta (Expected Return on the Market Portfolio 2 Risk-Free Rate of Return) (9.10) Required Return for an Investment 5 Rf 1 b(Market Risk Premium) Web Resources This chapter has introduced some measures of risk. On this website, an investment advisor discusses how he advises his clients to assess their tolerance to risk: http://www.youtube.com/watch?v59bMzdkYY-hk. Professor Aswath Damodaran of NYU maintains a list of betas for different industries. These may be viewed and the sectors compared at the following website. It might be interesting to compare the average betas in two sectors, like gambling versus natural gas utilities, to see if the betas conform to your intuition: http://pages.stern.nyu.edu/~adamodar/New_Home_Page/datafile/Betas.html. Critical Thinking and Discussion Questions 1. If the stock market is perfectly efficient, then what relationship should exist between the required return for a stock and the expected return when investing in the stock? 2. Name the two building blocks of a required return. 3. Total risk is measured by the standard deviation of returns. Jot down the formula for the standard deviation and then comment on what part of the formula has to do with deviations and what part is related to calculating the standard of these deviations. (Hint: Deviation may be thought of as departure from what is typical, and standard may be thought of as the average or what is expected). 4. The theoretical development of the CAPM calls for use of the risk-free rate. Swiss Government bonds have typically had a lower-risk rating than any other government bond. Why isn't the Swiss bond, therefore, the standard for use in the CAPM worldwide? 5. The average return on the S&P 500 has been in the neighborhood of 11%, and in 1980 U.S. Treasury Bonds were yielding about 15%. Imagine you were an investor in 1980. How do the circumstances at that time help explain why it is better to use the market risk premium (of around 6%) rather than [Rmkt 2 Rf] when estimating the required return with the CAPM? byr80656_09_c09_217-244.indd 240 3/28/13 3:34 PM Practice Problems CHAPTER 9 6. Beta is the measure of market risk. Look at the businesses listed below and see if you can identify one that could very likely have relatively high total risk but have a low beta. Explain your reasoning. a. The manufacturer of diamond-encrusted dog collars. b. A company that specializes in finding and salvaging old shipwrecks from the Age of Discovery (the 1500s and 1600s). c. A casino. 7. Pick three firms that you think may have low betas and three that you think may have high betas. Go to Yahoo! Finance, MSN Money, Value Line, or Google Finance and look up their betas. Are there any surprises? Can you think of some reasons that these betas are different from what you expected? 8. When betas are being estimated using historical data and linear regression, here are some of the choices that you must make: which index to use as a proxy for the market portfolio (the S&P 500, the Wilshire 5000, and the NYSE Composite Index are a few of the possibilities); the length of the return period (daily returns, weekly returns, and monthly returns are all used); the length of the historical record (2, 3, or 5 years are candidates). Each combination of these choices will yield a slight (or maybe even a major) difference in the estimated beta. How many different betas for a single firm could you and your classmates get on the same day by making different choices among these options? 9. Now consider finding the required return using the CAPM with the betas estimated in question 8. In the CAPM, you must choose security to use as a proxy for the risk-free rate. Candidates for this choice include Treasury bills, notes, and bonds, with maturities that could be 1 year, 5 years, or even 20 years in length. Next, the market risk premium must be estimated. Depending on what data and method you use, you may get values ranging from 4% all the way up to 8% (if you are curious about methods, you might look up \"A Note on Estimating the Historical Risk Premium,\" by Fuller and Hickman, in Financial Practice and Education, 1991). How does this task and question 8 support the statement that finance is \"some science and some art\"? 10. Security A has a standard deviation of returns equal to 20% and a beta of 1.50. Security B has a standard deviation of 16% and a beta of 1.80. Which security probably has the higher required return? Explain. Practice Problems 1. If you buy a share of stock for $50, hold it for 1year and sell it for $49, then what is your return if the dividend you receive during the year equaled each of the following values? a. $3.50 per share b.$0 c. $1.00 per share d.$0.50 2. If you buy a share of stock for $30 and collect an annual dividend of $3 per share each year for 6 years, and then sell it for $30 just after collecting the sixth dividend, then what was your annual return over the life of this investment? (Hint: You may calculate this as an IRR, or you may just reason it out without any calculations at all). byr80656_09_c09_217-244.indd 241 3/28/13 3:34 PM CHAPTER 9 Practice Problems 3. Suppose you buy a share of stock for $20, and then collect a dividend equal to $2 a share during the year. By the end of the year, the firm's fortunes have changed, and it has declared bankruptcy. The stock is now worthless. What rate of return did you earn on your investment? 4. Mini-Case: Returns, Risk, and Correlation A. A security earns returns of 5%, 18%, 6%, and 11% over 4 years from 2000 to 2003. Without using Excel (in other words, using a calculator and by hand), find the average annual return for this stock over the 4-year period as well as the standard deviation of returns over the period. B. Over the same 4 years as the security in part A, another investment had returns of 9%, 3%, 8%, and 8%. Using the same approach, find the average annual return and the standard deviation of returns for this investment. C. If you have $1,000 to invest and you put $500 in each stock at the beginning of 2000, calculate the value of your portfolio at the end of each year. D. Using the result from part C, calculate the return for each year, assuming that you keep all of the money in the portfolio (none of the returns are in the form of dividends, or if a firm pays a dividend, and then you reinvest it in the firm that paid the dividend). E. Now calculate the average return of the portfolio and the standard deviation of the portfolio. F. Do you think the two investments had positively correlated, negatively correlated, or uncorrelated returns? G.\tHow does your answer to part F affect what you found in part E compared to your answers to parts A and B? 5. Given the data below, calculate the returns and beta for ACME Corporation. Month Stock Price Dividends Market Index 1 $30.50 0 1259 2 $35.00 0 1277 3 $32.50 $1.50 1280 4 $33.05 0 1288 5 $29.50 0 1200 6 $33.50 $1.50 1250 7 $35.75 0 1350 6. If the market risk premium has average 7% and the risk-free rate is 3%, what is the required return for a stock with a beta equal to 0.80? 7. If the market risk premium has averaged 6.5% and a 1-year Treasury bill with a face value of $10,000 is selling for $9,560, what is the required return of a stock whose beta is 1.60? 8. Find the value of a share of stock whose beta is 1.20, that recently paid a dividend of $3 per share, and whose dividends were $1.50 ten years ago. Assume the market risk premium is 5.5%. Assume the risk-free rate of return equals 3%. byr80656_09_c09_217-244.indd 242 3/28/13 3:34 PM Practice Problems CHAPTER 9 9. What is the beta of a stock that is selling for $40 a share if the following assumptions hold: The next dividend it expects to be $2 per share; dividends are expected to grow at 7% annually; the market risk premium has averaged 7.5%; and the risk-free rate is 2% per year. 10. Some analysts argue that gold has a negative beta. Test this proposition by doing a bit of research. Find the price per ounce of gold over each of the last 30 years and find the level of the stock market for those 30 years. Note, you may find a year-end price for each asset, or an average value for each asset during the year in questionjust be consistent. Next, estimate a regression line to find beta. Does your result support or seem to be at odds with the negative beta hypothesis? Why might gold have a negative beta? byr80656_09_c09_217-244.indd 243 3/28/13 3:34 PM byr80656_09_c09_217-244.indd 244 3/28/13 3:34 PM 10 Nina Mourier/Getty Images Cost of Capital Learning Objectives Upon completion of Chapter 10, you will be able to: Understand the meaning of the weighted average cost of capital (WACC). Be able to estimate the weights in the WACC. Be able to estimate the cost of debt and how it is affected by taxes. Be able to estimate the cost of preferred stock. Know three approaches for estimating the cost of equity. Und