Estimating the Cost of Capital PDF

CH APTE R Estimating the Cost of Capital 12 EVALUATING INVESTMENT OPPORTUNITIES REQUIRES FINANCIAL NOTATION managers...

CH APTE R Estimating the Cost of Capital 12 EVALUATING INVESTMENT OPPORTUNITIES REQUIRES FINANCIAL NOTATION managers to estimate the cost of capital. For example, when executives at Intel ri  required return Corporation evaluate a capital investment project, they must estimate the ap- for security i propriate cost of capital for the project in order to determine its NPV. The cost of E [ Ri ]  expected return capital includes a risk premium that compensates Intel’s investors for taking on of security i the risk of the new project. How can Intel estimate this risk premium and, there- r f risk-free interest rate fore, the cost of capital? rwacc  weighted-average cost In the last two chapters, we have developed an answer to this question—the of capital Capital Asset Pricing Model. In this chapter, we will apply this knowledge to β i  beta of investment i compute the cost of capital for an investment opportunity. We begin the ­chapter with respect to the by focusing on an investment in the firm’s stock. We show how to estimate the market portfolio firm’s equity cost of capital, including the practical details of identifying the MV i  total market capitaliza- market portfolio and estimating equity betas. Next, we develop methods to es- tion of security i timate the firm’s debt cost of capital, based either on the debt's yield or its beta. E Value of Equity By combining the equity and debt costs of capital, we can estimate the firm’s D Value of Debt weighted ­average cost of capital. We then consider investing in a new project, and show how to estimate a project’s cost of capital based on the cost of capital α i alpha of security i of ­comparable firms. τc corporate tax rate βU unlevered or asset beta β E equity beta β D debt beta rE equity cost of capital rD debt cost of capital ru  unlevered cost of capital 445 M12_BERK6318_06_GE_C12.indd 445 26/04/23 6:33 PM  446 Chapter 12 Estimating the Cost of Capital 12.1 The Equity Cost of Capital Recall that the cost of capital is the best expected return available in the market on invest- ments with similar risk. The Capital Asset Pricing Model (CAPM) provides a practical way to identify an investment with similar risk. Under the CAPM, the market portfolio is a well-diversified, efficient portfolio representing the non-diversifiable risk in the economy. Therefore, investments have similar risk if they have the same sensitivity to market risk, as measured by their beta with the market portfolio. So, the cost of capital of any investment opportunity equals the expected return of available investments with the same beta. This estimate is provided by the Security Market Line equation of the CAPM, which states that the cost of capital of an investment oppor- tunity with beta, β i , is: The CAPM Equation for the Cost of Capital (Security Market Line) ri = r f + β i × ( E [ R Mkt ] − r f )  (12.1)  Risk premium for security i In other words, investors will require a risk premium comparable to what they would earn taking the same market risk through an investment in the market portfolio. As our first application of the CAPM, consider an investment in the firm’s stock. As we demonstrated in Chapter 9, to value a share of stock, we need to calculate the equity cost of capital. We can do so using Eq. 12.1 if we know the beta of the firm’s stock. EXAMPLE 12.1 Computing the Equity Cost of Capital Problem Suppose you estimate that Disney’s stock (DIS) has a volatility of 20% and a beta of 1.29. A simi- lar process for Domino’s Pizza (DPZ) yields a volatility of 30% and a beta of 0.62. Which stock carries more total risk? Which has more market risk? If the risk-free interest rate is 3% and you estimate the market’s expected return to be 8%, calculate the equity cost of capital for Disney and Domino’s. Which company has a higher cost of equity capital? Solution Total risk is measured by volatility; therefore, Domino’s stock has more total risk than Disney. Sys- tematic risk is measured by beta. Disney has a higher beta, so it has more market risk than Domino’s. Given Disney’s estimated beta of 1.29, we expect the price for Disney’s stock to move by 1.29% for every 1% move of the market. Therefore, Disney’s risk premium will be 1.29 times the risk premium of the market, and Disney’s equity cost of capital (from Eq. 12.1) is rDIS = 3% + 1.29 × ( 8% − 3% ) = 3% + 6.45% = 9.45% Domino’s has a lower beta of 0.62. The equity cost of capital for Domino’s is rDPZ = 3% + 0.62 × ( 8% − 3% ) = 3% + 3.10% = 6.10% Because market risk cannot be diversified, it is market risk that determines the cost of capital; thus Disney has a higher cost of equity capital than Domino’s, even though it is less volatile. While the calculations in Example 12.1 are straightforward, to implement them we need a number of key inputs. In particular, we must do the following: Construct the market portfolio, and determine its expected excess return over the risk- free interest rate Estimate the stock’s beta, or sensitivity to the market portfolio M12_BERK6318_06_GE_C12.indd 446 26/04/23 6:33 PM  12.2 The Market Portfolio 447 We explain how to estimate these inputs in more detail in the next two sections. CONCEPT CHECK 1. According to the CAPM, we can determine the cost of capital of an investment by comparing it to what portfolio? 2. What inputs do we need to estimate a firm’s equity cost of capital using the CAPM? 12.2 The Market Portfolio To apply the CAPM, we must identify the market portfolio. In this section we examine how the market portfolio is constructed, common proxies that are used to represent the market portfolio, and how we can estimate the market risk premium. Constructing the Market Portfolio Because the market portfolio is the total supply of securities, the proportions of each secu- rity should correspond to the proportion of the total market that each security represents. Thus, the market portfolio contains more of the largest stocks and less of the smallest stocks. Specifically, the investment in each security i is proportional to its market capitaliza- tion, which is the total market value of its outstanding shares: MV i = ( Number of Shares of i Outstanding ) × ( Price of i per Share ) (12.2) We then calculate the portfolio weights of each security as follows: Market Value of i MV i xi = = (12.3) Total Market Value of All Securities in the Portfolio ∑ MV j j A portfolio like the market portfolio, in which each security is held in proportion to its mar- ket capitalization, is called a value-weighted portfolio. A value-weighted portfolio is also an equal-ownership portfolio: We hold an equal fraction of the total number of shares outstanding of each security in the portfolio. This last observation implies that even when market prices change, to maintain a value-weighted portfolio, we do not need to trade un- less the number of shares outstanding of some security changes. Because very little trading is required to maintain it, a value-weighted portfolio is also a passive portfolio. Market Indexes If we focus our attention on U.S. stocks, then rather than construct the market portfolio ourselves, we can make use of several popular market indexes that try to represent the per- formance of the U.S. stock market. Examples of Market Indexes. A market index reports the value of a particular portfolio of securities. The S&P 500 is an index that represents a value-weighted portfolio of 500 of the largest U.S. stocks.1 The S&P 500 was the first widely publicized value-weighted index and is the standard portfolio used to represent “the market portfolio” when using the 1 Standard and Poor’s periodically replaces stocks in the index (on average about 20–25 per year). While size is one criterion, S&P tries to maintain appropriate representation of different sectors of the economy and chooses firms that are leaders in their industries. Also, starting in 2005, the value weights in the index are based on the number of shares actually available for public trading, referred to as its free float. M12_BERK6318_06_GE_C12.indd 447 26/04/23 6:33 PM  448 Chapter 12 Estimating the Cost of Capital Value-Weighted Portfolios and Rebalancing When stock prices change, so do the weights in a value-weighted portfolio. Yet there is no need to trade to rebalance the portfolio in response, making it very efficient from a transaction cost perspective. To see why, consider the following example. Suppose we invest $50,000 in a value-weighted portfolio of General Electric, Home Depot, and Cisco, as shown below: Market Data Our Portfolio Shares Outstanding Market Cap Percent Initial Shares Stock Stock Price (billions) ($ billion) of Total Investment Purchased General Electric $80.00 1.50 120 20% $10,000 125 Home Depot $250.00 1.20 300 50% $25,000 100 Cisco $50.00 3.60 180 30% $15,000 300 Total 600 100% $50,000 Note that our investment in each stock is proportional to each stock’s market capitalization. In addition, the number of shares purchased is proportional to each stock’s outstanding shares. Now suppose that the price of GE’s stock increases to $120 per share and Home Depot’s stock price drops to $220 per share. Let’s compute the new value weights as well as the effect on our portfolio: Shares Outstanding Market Cap Percent New Investment Stock Stock Price (billions) ($ billion) of Total Shares Held Value General Electric $120.00 1.50 180 28.8% 125 $15,000 Home Depot $220.00 1.20 264 42.3% 100 $22,000 Cisco $50.00 3.60 180 28.8% 300 $15,000 Total 624 100% $52,000 Note that although the value weights have changed, the value of our investment in each stock has also changed and remains proportional to each stock’s market cap. For example, our weight in GE’s stock is $15,000 $52,000 = 28.8%, matching its market weight. Thus, there is no need to trade in response to price changes to maintain a value-weighted portfolio. Rebalanc- ing is only required if firms issue or retire shares, or if the set of firms represented by the portfolio changes. CAPM in practice. Even though the S&P 500 includes only 500 of the roughly 4000 pub- licly traded stocks in the U.S., because the S&P 500 includes the largest stocks, it represents almost 80% of the U.S. stock market in terms of market capitalization. More recently created indexes, such as the Wilshire 5000, provide a value-weighted index of all U.S. stocks listed on the major stock exchanges.2 While more complete than the S&P 500, and therefore more representative of the overall market, its returns are very similar; between 1990 and 2022, the correlation between their weekly returns was nearly 99%. Given this similarity, many investors view the S&P 500 as an adequate measure of overall U.S. stock market performance. The most widely quoted U.S. stock index is the Dow Jones Industrial Average (DJIA), which consists of a portfolio of 30 large industrial stocks. While somewhat representa- tive, the DJIA clearly does not represent the entire market. Also, the DJIA is a price-weighted 2 The Wilshire 5000 began with approximately 5000 stocks when it was first published in 1974 but since then the number of stocks in the index has changed over time with U.S. equity markets ( as of 2022 the number had fallen to about 3600). Similar indices are the Dow Jones U.S. Total Market Index and the S&P Total Market Index. M12_BERK6318_06_GE_C12.indd 448 26/04/23 6:33 PM  12.2 The Market Portfolio 449 (rather than value-weighted) portfolio. A price-weighted portfolio holds an equal number of shares of each stock, independent of their size. Despite being nonrepresentative of the entire market, the DJIA remains widely cited because it is one of the oldest stock market indexes (first published in 1884). Investing in a Market Index. In addition to capturing the performance of the U.S. mar- ket, the S&P 500 and the Wilshire 5000 are both easy to invest in. Many mutual fund companies offer funds, called index funds, that invest in these portfolios. In addition, exchange-traded funds represent these portfolios. An exchange-traded fund (ETF) is a se- curity that trades directly on an exchange, like a stock, but represents ownership in a port- folio of stocks. For example, Standard and Poor’s Depository Receipts (SPDR, nicknamed “spiders”) trade on the American Stock Exchange (symbol SPY) and represent ownership in the S&P 500. Vanguard’s Total Stock Market ETF (symbol VTI, nicknamed “viper”) is based on the Wilshire 5000 index. By investing in an index or an exchange-traded fund, an individual investor with only a small amount to invest can easily achieve the benefits of broad diversification. Although practitioners commonly use the S&P 500 as the market portfolio in the CAPM, no one does so because of a belief that this index is actually the market portfolio. Instead they view the index as a market proxy—a portfolio whose return they believe closely tracks the true market portfolio. Of course, how well the model works will depend on how closely the market proxy actually tracks the true market portfolio. We will return to this issue in Chapter 13. The Market Risk Premium Recall that a key ingredient to the CAPM is the market risk premium, which is the expected excess return of the market portfolio: E [ R Mkt ] − r f. The market risk premium provides the benchmark by which we assess investors’ willingness to hold market risk. Before we can estimate it, we must first discuss the choice of the risk-free interest rate to use in the CAPM. Determining the Risk-Free Rate. The risk-free interest rate in the CAPM model corre- sponds to the risk-free rate at which investors can both borrow and save. We generally de- termine the risk-free saving rate using the yields on U.S. Treasury securities. Most investors, however, must pay a substantially higher rate to borrow funds. In mid-2018, for example, even the highest credit quality borrowers had to pay almost 0.35% over U.S. Treasury rates on short-term loans. Even if a loan is essentially risk-free, this premium compensates lend- ers for the difference in liquidity compared with an investment in Treasuries. As a result, practitioners sometimes use rates from the highest quality corporate bonds in place of Treasury rates in Eq. 12.1. The next consideration when determining the risk-free rate is the choice of maturity. As discussed in Chapter 5, when discounting risk-free cash flows we match the maturity of the interest rate to that of the cash flows. It is common to do the same when apply- ing the CAPM; that is, we use a short-term risk-free rate to evaluate a short-term invest- ment, and a long-term rate when evaluating a long-term investment.3 For example, when valuing a long-term investment with an indefinite horizon, such as a stock, most financial 3 As explained in the Chapter 11 appendix, according to the CAPM theory the precise rate to use depends on the horizons of investors and their relative propensity to borrow or save. As there is no easy way to ­accurately assess these investor characteristics, judgment is required. Throughout this chapter, we high- light gray areas like this and explain common practices. M12_BERK6318_06_GE_C12.indd 449 26/04/23 6:33 PM  450 Chapter 12 Estimating the Cost of Capital analysts report using the yields of long-term (10- to 30-year) bonds to determine the risk- free ­interest rate.4 The Historical Risk Premium. One approach to estimating the market risk premium, E [ R Mkt ] − r f , is to use the historical average excess return of the market over the risk- free interest rate.5 With this approach, it is important to measure historical stock returns over the same time horizon as that used for the risk-free interest rate. Because we are interested in the future market risk premium, we face a tradeoff in selecting the amount of data we use. As we noted in Chapter 10, it takes many years of data to produce even moderately accurate estimates of expected returns. Yet very old data may have little relevance for investors’ expectations of the market risk pre- mium today. Table 12.1 reports excess returns of the S&P 500 versus one-year and 10-year Treasury rates, based on data since 1926, as well as just the last 50 years. For each time period, note the lower risk premium when we compare the S&P 500 to longer-term Treasuries. This difference primarily arises because, historically, the yield curve has tended to be upward sloping, with long-term interest rates higher than short-term rates. Table 12.1 also suggests that the market risk premium has declined over time, with the S&P 500 showing a significantly lower excess return over the past 50 years than over the full sample. There are several potential explanations for this decline. First, more investors participate in the stock market today, so that the risk can be shared more broadly. Second, financial innovations such as mutual funds and exchange-traded funds have greatly reduced the costs of diversifying. Third, except for the recent increase in the wake of the 2008 financial crisis, the overall volatility of the market has declined over time. All of these reasons may have reduced the risk of holding stocks, and so diminished the premium in- vestors require. Most researchers and analysts believe that future expected returns for the market are likely to be closer to these more recent historical numbers, in a range of about 4%–6% over Treasury bills (and 3%–5% over longer term bonds).6 TABLE 12.1     Historical Excess Returns of the S&P 500 Compared to One-Year and Ten-Year U.S. Treasury Securities Period S&P 500 Excess Return Versus 1926–2015 1965–2015 One-year Treasury 7.7% 5.0% Ten-year Treasury* 5.9% 3.9% * Based on a comparison of compounded returns over a 10-year holding period. 4 See Robert Bruner, et al., “Best Practices in Estimating the Cost of Capital: Survey and Synthesis,” Financial Practice and Education 8 (1998): 13–28. 5 Because we are forecasting the expected return, it is appropriate to use the arithmetic average. See Chapter 10. 6 I. Welch, “Views of Financial Economists on the Equity Premium and on Professional Controversies,” Journal of Business 73 (2000): 501–537 (with 2009 update); J. Graham and C. Harvey, “The Equity Risk Premium in 2008: Evidence from the Global CFO Outlook Survey,” SSRN 2008; and Ivo Welch and Amit Goyal, “A Comprehensive Look at The Empirical Performance of Equity Premium Prediction,” Review of Financial Studies 21 (2008): 1455–1508. M12_BERK6318_06_GE_C12.indd 450 26/04/23 6:33 PM  12.3 Beta Estimation 451 A Fundamental Approach Using historical data to estimate the market risk premium suffers from two drawbacks. First, despite using 50 years (or more) of data, the standard errors of the estimates are large (e.g., even using data from 1926, the 95% confidence interval for the excess return is ±4.1% ). Second, because they are backward looking, we cannot be sure they are representative of current expectations. As an alternative, we can take a fundamental approach toward estimating the market risk premium. Given an assessment of firms’ future cash flows, we can estimate the expected return of the market by solving for the discount rate that is consistent with the current level of the index. For example, if we use the constant expected growth model presented in Chapter 9, the expected market return is equal to Div 1 r Mkt = + g = Dividend Yield + Expected Dividend Growth Rate (12.4) P0 While this model is highly inaccurate for an individual firm, the assumption of constant expected growth is more reasonable when considering the overall market. If, for instance, the S&P 500 has a current dividend yield of 2%, and we assume that both earnings and dividends per share are expected to grow 6% per year, this model would estimate the ex- pected return of the S&P 500 as 8%. Following such methods, researchers generally report estimates in the 3%–5% range for the future equity risk premium.7 CONCEPT CHECK 1. How do you determine the weight of a stock in the market portfolio? 2. What is a market proxy? 3. How can you estimate the market risk premium? 12.3 Beta Estimation Having identified a market proxy, the next step in implementing the CAPM is to determine the security’s beta, which measures the sensitivity of the security’s returns to those of the market. Because beta captures the market risk of a security, as opposed to its diversifiable risk, it is the appropriate measure of risk for a well-diversified investor. Using Historical Returns Ideally, we would like to know a stock’s beta in the future; that is, how sensitive will its future returns be to market risk. In practice, we estimate beta based on the stock’s historical sensi- tivity. This approach makes sense if a stock’s beta remains relatively stable over time, which appears to be the case for most firms. Many data sources provide estimates of beta based on historical data. Typically, these data sources estimate correlations and volatilities from two to five years of weekly or monthly returns and use the S&P 500 as the market portfolio. Table 10.6 on page 383 shows estimated betas for a number of large firms in different industries. 7 See e.g., E. Fama and K. French, “The Equity Premium,” Journal of Finance 57 (2002): 637–659; and J. Siegel, “The Long-Run Equity Risk Premium,” CFA Institute Conference Proceedings Points of Inflection: New Directions for Portfolio Management (2004). Similarly, L. Pástor, M. Sinha, and B. Swaminathan report a 2%–4% implied risk premium over 10-year Treasuries [“Estimating the Intertemporal Risk-Return Tradeoff Using the Implied Cost of Capital,” Journal of Finance 63 (2008): 2859–2897]. M12_BERK6318_06_GE_C12.indd 451 26/04/23 6:33 PM  452 Chapter 12 Estimating the Cost of Capital As we discussed in Chapter 10, the differences in betas reflect the sensitivity of each firm’s profits to the general health of the economy. For example, the demand for personal and household products has very little relation to the state of the economy. Firms pro- ducing these goods, such as Clorox and Procter & Gamble, tend to have very low betas (near 0.5). In contrast, many technology stocks, as well as makers of luxury and leisure goods, tend to have high betas (over 1.0) because demand for their products fluctuates with the business cycle: Companies and consumers tend to purchase their products when times are good, but cut back on these expenditures when the economy slows. Let’s look at Callaway Golf Company as an example. Callaway’s stock trades on the NYSE with the ticker symbol ELY (after the company’s founder, Ely Callaway Jr.). Figure 12.1 shows the weekly returns for Callaway and the monthly returns for the S&P 500 from the beginning of 2017 through 2021. Note the overall tendency for Callaway to have a high return when the market is up and a low return when the market is down. Indeed, Callaway tends to move in the same direction as the market, but with greater amplitude. The pattern suggests that Callaway’s beta is larger than 1. Rather than plot the returns over time, we can see Callaway’s sensitivity to the market even more clearly by plotting Callaway’s excess return as a function of the S&P 500 excess return, as shown in Figure 12.2. Each point in this figure represents the excess return of Callaway and the S&P 500 from one of the weeks in Figure 12.1. For example, for the week ending on December 21, 2018, Callaway was down 11% and the S&P 500 was down 7% (while risk-free Treasuries returned only 0.04%). Once we have plotted each week in this way, we can then plot the best-fitting line drawn through these points.8 FIGURE 12.1 Weekly Returns for Callaway Golf’s Stock and for the S&P 500, 2017–2021 25% S&P 500 Callaway (ELY) 20% 6 Nov 2020 RSP500 = 7.2% 15% RELY = 10.7% 10% 5% Return 0% –5% –10% –15% 21 Dec 2018 RSP500 = –7.0% –20% RELY = –11.0% –25% 2017 2018 2019 2020 2021 Week Callaway’s returns tend to move in the same direction, but with greater amplitude, than those of the S&P 500. 8 By “best fitting,” we mean the line that minimizes the sum of the squared deviations from the line. In Excel, it can be found by adding a linear trendline to the chart. M12_BERK6318_06_GE_C12.indd 452 26/04/23 6:33 PM  12.3 Beta Estimation 453 Identifying the Best-Fitting Line As the scatterplot makes clear, Callaway’s returns have a positive covariance with the mar- ket: Callaway tends to be up when the market is up, and vice versa. Moreover, from the best-fitting line, we can see visually that a 10% change in the market’s return corresponds to about a 17% change in Callaway’s return. That is, Callaway’s return moves about 1.7 times that of the overall market, implying a beta of about 1.7. More generally, Beta corresponds to the slope of the best-fitting line in the plot of the security’s excess returns versus the market excess return.9 To understand this result fully, recall that beta measures the market risk of a security— the percentage change in the return of a security for a 1% change in the return of the mar- ket portfolio. The best-fitting line in Figure 12.2 captures the components of a security’s return that we can explain based on market risk, so its slope is the security’s beta. Note though, that in any individual month, the security’s returns will be higher or lower than the best-fitting line. Such deviations from the best-fitting line result from risk that is not related to the market as a whole. These deviations are zero on average in the graph, as the points above the line balance out the points below the line. They represent firm-specific risk that is diversifiable and that averages out in a large portfolio. FIGURE 12.2 25% y = 1.7645x – 0.0002 Scatterplot of Weekly R 2 = 0.4092 Excess Returns for Callaway 20% ELY Excess Return Golf Versus the S&P 500, 2017–2021 15% Beta corresponds to the slope of the best-fitting line. Beta 10% measures the expected change 6 Nov 2020 in Callaway’s excess return per 1% change in the market’s 5% excess return. Deviations from the best-fitting line correspond 0% to diversifiable, non-market- –25% –20% –15% –10% –5% 0% 5% 10% 15% 20% 25% related risk. In this case, Market Excess Return Callaway’s estimated beta is 21 Dec 2018 –5% approximately 1.76. –10% Deviations from –15% best-fitting line = Diversifiable risk See the CAPM tool in the eText- –20% book and MyLab Finance to explore similar charts using –25% current data. 9 The slope can be calculated using Excel’s SLOPE( ) function, or by displaying the equation for the trend- line on the chart (R 2 is the square of the correlation between the returns). See appendix for further details. M12_BERK6318_06_GE_C12.indd 453 26/04/23 6:33 PM  454 Chapter 12 Estimating the Cost of Capital Using Linear Regression The statistical technique that identifies the best-fitting line through a set of points is called linear regression. In Figure 12.2, linear regression corresponds to writing the excess re- turn of a security as the sum of three components:10 ( Ri − r f ) = α i + β i ( R Mkt − r f ) + ε i (12.5) The first term, α i , is the constant or intercept term of the regression. The second term, β i ( R Mkt − r f ), represents the sensitivity of the stock to market risk. For example, if the market’s return is 1% higher, there is a β i percent increase in the security’s return. We refer to the last term, ε i , as the error (or residual) term: It represents the deviation from the best-fitting line and is zero on average (or else we could improve the fit). This error term corresponds to the diversifiable risk of the stock, which is the risk that is uncorrelated with the market. If we take expectations of both sides of Eq. 12.5 and rearrange the result, because the average error is zero (that is, E [ ε i ] = 0 ), we get  ( E [ Ri ] = r f + β i E [ R Mkt ] − r f ) + αi  (12.6) Expected return for i from the SML Distance above below the SML The constant α i , referred to as the stock’s alpha, measures the historical performance of the security relative to the expected return predicted by the security market line—it is the distance the stock’s average return is above or below the SML. Thus, we can interpret α i as a risk-adjusted measure of the stock’s historical performance.11 According to the CAPM, α i should not be significantly different from zero. Using Excel’s regression data analysis tool for the monthly returns from 2017–2021, Callaway’s estimated beta is 1.76, with a 95% confidence interval from 1.5 to 2.0. If we assume Callaway’s sensitivity to market risk is stable over time, we would expect Callaway’s beta to be in this range in the near future. With this estimate in hand, we are ready to esti- mate Callaway’s equity cost of capital. EXAMPLE 12.2 Using Regression Estimates to Estimate the Equity Cost of Capital Problem Suppose the risk-free interest rate is 3%, and the market risk premium is 5%. What range for Callaway’s equity cost of capital is consistent with the 95% confidence interval for its beta? Solution Using the data from 2017 to 2021, and applying the CAPM equation, the estimated beta of 1.76 implies an equity cost of capital of 3% + 1.76 × 5% = 11.8% for Callaway. But our estimate is uncertain, and the 95% confidence interval for Callaway’s beta of 1.5 to 2.0 gives a range for Callaway’s equity cost of capital from 3% + 1.5 × 5% = 10.5% to 3% + 2.0 × 5% = 13%. 10 In the language of regression, the stock’s excess return is the dependent (or y) variable, and the market’s excess return is the independent (or x) variable. 11 When used in this way, α i is often referred to as Jensen’s alpha. It can be calculated using the INTERCEPT( ) function in Excel (see appendix). Using this regression as a test of the CAPM was intro- duced by F. Black, M. Jensen, and M. Scholes in “The Capital Asset Pricing Model: Some Empirical Tests” in M. Jensen, ed., Studies in the Theory of Capital Markets (Praeger, 1972). M12_BERK6318_06_GE_C12.indd 454 26/04/23 6:33 PM  12.4 The Debt Cost of Capital 455 Why Not Estimate Expected Returns Directly? If the CAPM requires us to use historical data to estimate beta 49% 5 = 22%, leading to a 95% confidence interval of and determine a security’s expected return (or an investment’s 31% ± 44%! Even with 100 years of data, the confidence cost of capital), why not just use the security’s historical aver- bounds would be ± 9.8%. Of course, Callaway has not existed age return as an estimate for its expected return instead? This for 100 years, and even if it had, the firm today would bear method would certainly be simpler and more direct. little resemblance today to what it was like 100 years ago. As we saw in Chapter 10, however, it is extremely difficult Using the methods described in this section, however, to infer the average return of individual stocks from histori- we can infer beta from historical data reasonably accurately cal data. For example, consider Callaway’s stock, which had even with as little as two years of data. In theory at least, an average annualized return of 31%, and a volatility of 49%, the CAPM can provide much more accurate estimates of based on weekly data from 2017–2021. Given 5 years of data, expected returns for stocks than we could obtain from their the standard error of our estimate of the expected return is historical average return. The estimate of Callaway’s alpha from the regression is −0.02%. In other words, given its beta, Callaway’s average weekly return was 0.02% lower than required by the security market line, which is equivalent to −0.02% × 52 = −1.04% per year. The standard error of the alpha estimate is about 17% on an annual basis, however, so that statistically the estimate is not significantly different from zero. Alphas, like expected returns, are difficult to estimate with much accuracy without a very long data series. Moreover, the alphas for individual stocks have very little persistence (unlike betas).12 Thus, although Callaway’s return underperformed its required return over this time period, it may not necessarily continue to do so. In this section we have provided an overview of the main methodology for estimating a security’s market risk. In the appendix to this chapter, we discuss some additional practical considerations and common techniques for forecasting beta. CONCEPT CHECK 1. How can you estimate a stock’s beta from historical returns? 2. How do we define a stock’s alpha, and what is its interpretation? 12.4 The Debt Cost of Capital In the preceding sections, we have shown how to use the CAPM to estimate the cost of capi- tal of a firm’s equity. What about a firm’s debt—what expected return is required by a firm’s creditors? In this section, we’ll consider some of the main methods for estimating a firm’s debt cost of capital, the cost of capital that a firm must pay on its debt. In addition to being useful information for the firm and its investors, we will see in the next section that knowing the debt cost of capital will be helpful when estimating the cost of capital of a project. Debt Yields Versus Returns Recall from Chapter 6 that the yield to maturity of a bond is the IRR an investor will earn from holding the bond to maturity and receiving its promised payments. Therefore, if there is little risk the firm will default, we can use the bond’s yield to maturity as an estimate of investors’ expected return. If there is a significant risk that the firm will default on its obli- gation, however, the yield to maturity of the firm’s debt, which is its promised return, will overstate investors’ expected return. 12 Indeed, over the prior five-year period, 2012–2016, Callaway’s weekly returns had a positive alpha equiv- alent to 6.8% per year, significantly outperforming its required return, but as we have seen this positive alpha did not forecast superior future returns. M12_BERK6318_06_GE_C12.indd 455 26/04/23 6:33 PM  456 Chapter 12 Estimating the Cost of Capital COMMON MISTAKE Using the Debt Yield as Its Cost of Capital While firms often use the yield on their debt to estimate a market risk premium of 5%, an expected return of 20% their debt cost of capital, this approximation is reasonable would imply a debt beta greater than 3 for AMR, which is only if the debt is very safe. Otherwise, as we explained in unreasonably high, and higher even than the equity betas of Chapter 6, the debt’s yield—which is based on its promised many firms in the industry. payments—will overstate the true expected return from Again, the problem is the yield is computed using the holding the bond once default risk is taken into account. promised debt payments, which in this case were quite dif- Consider, for example, that in mid-2009 long-term ferent from the actual payments investors were expecting: bonds issued by AMR Corp. (parent company of American When AMR filed for bankruptcy in 2011, bondholders lost Airlines) had a yield to maturity exceeding 20%. Because close to 80% of what they were owed. The methods de- these bonds were very risky, with a CCC rating, their yield scribed in this section can provide a much better estimate greatly overstated their expected return given AMR’s sig- of a firm’s debt cost of capital in cases like AMR’s when the nificant default risk. Indeed, with risk-free rates of 3% and likelihood of default is significant. To understand the relationship between a debt’s yield and its expected return, consider a one-year bond with a yield to maturity of y. Thus, for each $1 invested in the bond today, the bond promises to pay $ ( 1 + y ) in one year. Suppose, however, the bond will default with probability p, in which case bond holders will receive only $ ( 1 + y − L ) , where L represents the expected loss per $1 of debt in the event of default. Then the expected return of the bond is13 rd = ( 1 − p ) y + p ( y − L ) = y − pL  = Yield to Maturity − Prob(default ) × Expected Loss Rate (12.7) The importance of these adjustments will naturally depend on the riskiness of the bond, with lower-rated (and higher-yielding) bonds having a greater risk of default. Table 12.2 shows average annual default rates by debt rating, as well as the peak default rates experienced during recessionary periods. To get a sense of the impact on the expected return to debt holders, note that the average loss rate for unsecured debt is about 60%. Thus, for a B-rated bond, during average times the expected return to debt holders would be approximately 0.055 × 0.60 = 3.3% below the bond’s quoted yield. On the other hand, outside of recessionary periods, given its negligible default rate the yield on an AA-rated bond provides a reasonable estimate of its expected return. TABLE 12.2    Annual Default Rates by Debt Rating* Rating: AAA AA A BBB BB B CCC CC-C Default Rate: Average 0.0% 0.1% 0.2% 0.5% 2.2% 5.5% 12.2% 14.1% In Recessions 0.0% 1.0% 3.0% 3.0% 8.0% 16.0% 48.0% 79.0% Source : “Corporate Defaults and Recovery Rates, 1920–2011,” Moody’s Global Credit Policy, February 2012. * Average rates are annualized based on a 10-year holding period; recession estimates are based on peak annual rates. 13 While we derived this equation for a one-year bond, the same formula holds for a multi-year bond a­ ssuming a constant yield to maturity, default rate, and loss rate. We can also express the loss in default according to the bond’s recovery rate R : ( 1 + y − L ) = ( 1 + y ) R , or L = ( 1 + y ) ( 1 − R ). M12_BERK6318_06_GE_C12.indd 456 26/04/23 6:33 PM  12.4 The Debt Cost of Capital 457 Debt Betas Alternatively, we can estimate the debt cost of capital using the CAPM. In principle it would be possible to estimate debt betas using their historical returns in the same way that we estimated equity betas. However, because bank loans and many corporate bonds are traded infrequently if at all, as a practical matter we can rarely obtain reliable data for the ­returns of individual debt securities. Thus, we need another means of estimating debt betas. We will develop a method for estimating debt betas for an individual firm using stock price data in Chapter 21. We can also approximate beta using estimates of betas of bond indices by rating category, as shown in Table 12.3. As the table indicates, debt betas tend to be low, though they can be significantly higher for risky debt with a low credit ­rating and a long maturity. TABLE 12.3    Average Debt Betas by Rating and Maturity* By Rating A and above BBB BB B CCC Avg. Beta < 0.05 0.10 0.17 0.26 0.31 By Maturity (BBB and above) 1–5 Year 5–10 Year 10–15 Year > 15 Year Avg. Beta 0.01 0.06 0.07 0.14 Source: S. Schaefer and I. Strebulaev, “Risk in Capital Structure Arbitrage,” Stanford GSB working paper, 2009. * Note that these are average debt betas across industries. We would expect debt betas to be lower (higher) for industries that are less (more) exposed to market risk. One way to approximate this difference is to scale the debt betas in Table 12.3 by the relative asset beta for the industry (see Figure 12.4 on page 463). EXAMPLE 12.3 Estimating the Debt Cost of Capital Problem In mid-2015, homebuilder KB Home had outstanding 5-year bonds with a yield to maturity of 6% and a B rating. If corresponding risk-free rates were 1%, and the market risk premium is 5%, estimate the expected return of KB Home’s debt. Solution Given the low rating of debt, we know the yield to maturity of KB Home’s debt is likely to sig- nificantly overstate its expected return. Using the average estimates in Table 12.2 and an expected loss rate of 60%, from Eq. 12.7 we have rd = 6% − 5.5% ( 0.60 ) = 2.7% Alternatively, we can estimate the bond’s expected return using the CAPM and an estimated beta of 0.26 from Table 12.3. In that case, rd = 1% + 0.26 ( 5% ) = 2.3% While both estimates are rough approximations, they both confirm that the expected return of KB Home’s debt is well below its promised yield. M12_BERK6318_06_GE_C12.indd 457 26/04/23 6:33 PM  458 Chapter 12 Estimating the Cost of Capital Note that both of the methods discussed in this section are approximations; more spe- cific information about the firm and its default risk could obviously improve them. Also, we have focused on the debt cost of capital from the perspective of an outside investor. The effective cost of debt to the firm can be lower once the tax deductibility of interest payments is considered. We will return to this issue in Section 12.6. CONCEPT CHECK 1. Why does the yield to maturity of a firm’s debt generally overestimate its debt cost of capital? 2. Describe two methods that can be used to estimate a firm’s debt cost of capital. 12.5 A Project’s Cost of Capital In Chapter 8, we explained how to decide whether or not to undertake a project. Although the project’s cost of capital is required to make this decision, we indicated then that we would explain later how to estimate it. We are now ready to fulfill this promise. As we did in Chapter 8, we will assume the project will be evaluated on its own, separate from any fi- nancing decisions. Thus, we assume for now that the project will be purely equity financed (with no new debt used to finance it) and consider project financing in Section 12.6. In the case of a firm’s equity or debt, we estimate the cost of capital based on the histori- cal risks of these securities. Because a new project is not itself a publicly traded security, this approach is not possible. Instead, the most common method for estimating a project’s beta is to identify comparable firms in the same line of business as the project we are considering undertaking. Indeed, the firm undertaking the project will often be one such comparable firm (and sometimes the only one). Then, if we can estimate the cost of capital of the assets of comparable firms, we can use that estimate as a proxy for the project’s cost of capital. All-Equity Comparables The simplest setting is one in which we can find an all-equity financed firm (i.e., a firm with no debt) in a single line of business that is comparable to the project. Because the firm is all equity, holding the firm’s stock is equivalent to owning the portfolio of its underlying assets. Thus, if the firm’s average investment has similar market risk to our project, then we can use the comparable firm’s equity beta and cost of capital as estimates for beta and the cost of capital of the project. EXAMPLE 12.4 Estimating the Beta of a Project from a Single-Product Firm Problem You have just graduated with an MBA, and decide to pursue your dream of starting a line of designer clothes and accessories. You are working on your business plan, and believe your firm will face similar market risk to Lululemon (LULU). To develop your financial plan, estimate the cost of capital of this opportunity assuming a risk-free rate of 3% and a market risk premium of 5%. Solution Checking Yahoo! Finance, you find that Lululemon has no debt. Using five years of weekly data, you estimate their beta to be 1.20. Using LULU’s beta as the estimate of the project beta, we can apply Eq. 12.1 to estimate the cost of capital of this investment opportunity as r project = r f + β LULU ( E [ R Mkt ] − r ) = 3% + 1.20 × 5% = 9% M12_BERK6318_06_GE_C12.indd 458 26/04/23 6:33 PM  12.5 A Project’s Cost of Capital 459 Thus, assuming your business has a similar sensitivity to market risk as Lululemon, you can estimate the appropriate cost of capital as 9%. In other words, rather than investing in the new business, you could invest in the fashion industry simply by buying LULU stock. Given this ­alternative, to be attractive, the new investment must have an expected return at least equal to that of LULU, which from the CAPM is 9%. Levered Firms as Comparables The situation is a bit more complicated if the comparable firm has debt. In that case, the cash flows generated by the firm’s assets are used to pay both debt and equity holders. As a result, the returns of the firm’s equity alone are not representative of the underlying a­ ssets; in fact, because of the firm’s leverage, the equity will often be much riskier. Thus, the beta of a levered firm’s equity will not be a good estimate of the beta of its assets and of our project. How can we estimate the beta of the comparable firm’s assets in this case? As shown in Figure 12.3 we can recreate a claim on the firm’s assets by holding both its debt and eq- uity simultaneously. Because the firm’s cash flows will either be used to pay debt or equity holders, by holding both securities we are entitled to all of the cash flows generated by the firm’s assets. The return of the firm’s assets is therefore the same as the return of a port- folio of the firm’s debt and equity combined. For the same reason, the beta of the firm’s assets will match the beta of this portfolio. The Unlevered Cost of Capital As we saw in Chapter 11, the expected return of a portfolio is equal to the weighted aver- age of the expected returns of the securities in the portfolio, where the weights correspond to the relative market values of the different securities held. Thus, a firm’s asset cost of capital or unlevered cost of capital, which is the expected return required by the firm’s investors to hold the firm’s underlying assets, is the weighted average of the firm’s equity and debt costs of capital: Asset or Unlevered  Fraction of Firm Value   Equity Cost  =     Cost of Capital  Financed by Equity   of Capital   Fraction of Firm Value   Debt Cost  +     Financed by Debt   of Capital  FIGURE 12.3 Using a Levered Firm as a Comparable for a Project’s Risk Comparable Firm If we identify a levered firm whose assets have comparable market risk Debt to our project, then we can estimate the project’s Assets 5 1 cost of capital based on a portfolio of the firm’s debt Project < Equity and equity. M12_BERK6318_06_GE_C12.indd 459 26/04/23 6:33 PM  460 Chapter 12 Estimating the Cost of Capital Writing this out, if we let E and D be the total market value of equity and debt of the com- parable firm, with equity and debt costs of capital rE and rD , then we can estimate a firm’s asset or unlevered cost of capital rU as follows:14 Asset or Unlevered Cost of Capital E D rU = rE + rD (12.8) E+D E+D Unlevered Beta. Because the beta of a portfolio is the weighted-average of the betas of the securities in the portfolio, we have a similar expression for the firm’s asset or unlevered beta, which we can use to estimate the beta of our project: Asset or Unlevered Beta E D βU = βE + βD (12.9) E+D E+D Let’s apply these formulas in an example. EXAMPLE 12.5 Unlevering the Cost of Capital Problem Your firm is considering expanding its household products division. You identify Procter & Gamble (PG) as a firm with comparable investments. Suppose PG’s equity has a market capital- ization of $300 billion and a beta of 0.56. PG also has $35 billion of AA-rated debt outstanding, with an average yield of 3.2%. Estimate the cost of capital of your firm’s investment given a risk- free rate of 3% and a market risk-premium of 5%. Solution Because investing in this division is like investing in PG’s assets by holding its debt and equity, we can estimate our cost of capital based on PG’s unlevered cost of capital. First, we estimate PG’s equity cost of capital using the CAPM as rE = 3% + 0.56 ( 5% ) = 5.8%. Because PG’s debt is highly rated, we approximate its debt cost of capital using the debt yield of 3.2%. Thus, PG’s unlevered cost of capital is 300 35 rU = 5.8% + 3.2% = 5.53% 300 + 35 300 + 35 Alternatively, we can estimate PG’s unlevered beta. Given its high rating, if we assume PG’s debt beta is zero we have 300 35 βU = 0.56 + 0 = 0.501 300 + 35 300 + 35 Taking this result as an estimate of the beta of our project, we can compute our project’s cost of capital from the CAPM as rU = 3% + 0.501( 5% ) = 5.51%. The slight difference in rU using the two methods arises because in the first case, we assumed the expected return of PG’s debt is equal to its promised yield of 3.2% (which overestimates the cost of debt, as we pointed out in Section 12.4), while in the second case, we assumed the debt has a beta of zero, which implies an expected return equal to the risk-free rate of 3% according to the CAPM (which underestimates the cost of debt, because PG’s debt is not risk free). The truth is somewhere between the two results. 14 For simplicity, we assume here that the firm in question maintains a constant debt-equity ratio, so that the weights E ( E + D ) and D ( E + D ) are fixed. As a result, Eq. 12.8 and Eq. 12.9 hold even in the pres- ence of taxes. See Chapter 18 for details and an analysis of settings with a changing leverage ratio. M12_BERK6318_06_GE_C12.indd 460 26/04/23 6:33 PM  12.5 A Project’s Cost of Capital 461 Cash and Net Debt. Sometimes firms maintain large cash balances in excess of their operating needs. This cash represents a risk-free asset on the firm’s balance sheet, and re- duces the average risk of the firm’s assets. Often, we are interested in the risk of the firm’s underlying business operations, separate from its cash holdings. That is, we are interested in the risk of the firm’s enterprise value, which we defined in Chapter 2 as the combined mar- ket value of the firm’s equity and debt, less any excess cash. In that case, we can measure the leverage of the firm in terms of its net debt: Net Debt = Debt − Excess Cash and Short-Term Investments (12.10) The intuition for using net debt is that if the firm holds $1 in cash and $1 in risk-free debt, then the interest earned on the cash will equal the interest paid on the debt. The cash flows from each source cancel each other, just as if the firm held no cash and no debt.15 Note that if the firm has more cash than debt, its net debt will be negative. In this case, its unlevered beta and cost of capital will exceed its equity beta and cost of capital, as the risk of the firm’s equity is mitigated by its cash holdings. EXAMPLE 12.6 Cash and Beta Problem In mid-2022, Garmin Ltd. had a market capitalization of $18.8 billion, $100 million in debt, and $1.6 billion in cash. If its estimated equity beta was 0.93, estimate the beta of Garmin’s underly- ing business enterprise. Solution Garmin has net debt of ( 0.1 − 1.6 ) = −$1.5 billion, and thus an enterprise value of ( 18.8 − 1.5 ) = $17.3 billion. Assuming Garmin’s debt and cash investments are both risk-free, we can estimate the beta of this enterprise value as E D 18.8 −1.5 βU = βE + βD = 0.93 + 0 = 1.01 E+D E+D 18.8 − 1.5 18.8 − 1.5 Note that in this case, Garmin’s equity is less risky than its underlying business activities due to its cash holdings. Industry Asset Betas Now that we can adjust for the leverage of different firms to determine their asset betas, it is possible to combine estimates of asset betas for multiple firms in the same industry or line of business. Doing so is extremely useful, as it will enable us to reduce our estimation error and improve the accuracy of the estimated beta for our project. 15 We can also think of the firm’s enterprise value V in terms of a portfolio of equity and debt less cash: V = E + D − C , where C is excess cash. In that case, the natural extension of Eq. 12.9 is E D C βU = βE + βD − βC E + D −C E + D −C E + D −C (and similarly for Eq. 12.8). The shortcut of using net debt is equivalent if the firm’s cash investments and debt have similar market risk, or if the debt beta reflects the combined risk of the firm’s debt and cash positions. Note that βC is not necessarily zero because sometimes firms invest their excess cash in risky securities. M12_BERK6318_06_GE_C12.indd 461 26/04/23 6:33 PM  462 Chapter 12 Estimating the Cost of Capital EXAMPLE 12.7 Estimating an Industry Asset Beta Problem Consider the following data for U.S. department stores in mid-2009, showing the equity beta, ratio of net debt to enterprise value (D/V), and debt rating for each firm. Estimate the average and median asset beta for the industry. Company Ticker Equity Beta D/V Debt Rating Dillard’s DDS 2.38 0.59 B JCPenney JCP 1.60 0.17 BB Kohl’s KSS 1.37 0.08 BBB Macy’s M 2.16 0.62 BB Nordstrom JWN 1.94 0.35 BBB Saks SKS 1.85 0.50 CCC Sears Holdings SHLD 1.36 0.23 BB Solution Note that D/V provides the fraction of debt financing, and ( 1 − D V ) the fraction of equity financing, for each firm. Using the data for debt betas from Table 12.3, we can apply Eq. 12.9 for each firm. For example, for Dillard’s: E D βU = βE + β D = ( 1 − 0.59 ) 2.38 + ( 0.59 ) 0.26 = 1.13 E+D E+D Doing this calculation for each firm, we obtain the following estimates: Ticker Equity Beta D/V Debt Rating Debt Beta Asset Beta DDS 2.38 0.59 B 0.26 1.13 JCP 1.60 0.17 BB 0.17 1.36 KSS 1.37 0.08 BBB 0.10 1.27 M 2.16 0.62 BB 0.17 0.93 JWN 1.94 0.35 BBB 0.10 1.30 SKS 1.85 0.50 CCC 0.31 1.08 SHLD 1.36 0.23 BB 0.17 1.09 Average 1.16 Median 1.13 The large differences in the firms’ equity betas are mainly due to differences in leverage. The firms’ asset betas are much more similar, suggesting that the underlying businesses in this industry have similar market risk. By combining estimates from several closely related firms in this way, we can get a more accurate estimate of the beta for investments in this industry. Figure 12.4 shows estimates of industry asset betas for U.S. firms. Note that businesses that are less sensitive to market and economic conditions, such as utilities and household product firms, tend to have lower asset betas than more cyclical industries, such as steel and high technology. CONCEPT CHECK 1. What data can we use to estimate the beta of a project? 2. Why does the equity beta of a levered firm differ from the beta of its assets? M12_BERK6318_06_GE_C12.indd 462 26/04/23 6:33 PM  12.6 Project Risk Characteristics and Financing 463 FIGURE 12.4 Industry Asset Betas (2022) Asset Beta 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 ck P G s Ca eal tne s & ert es Ca T gist g Ca Fu hn are d Eq g ns In ana Sto re ct er rs D du nm r F rink l sp ic s S nt t M en ftw il pl em ury ult s g g En nsu So e R es is c e in s an rs n l S e ea d d vi s ig d rv re lo es m s Eq ish gy ild Au efe ip. g P e od rts ni nu Air cts st tu s s Fo en g rn m s re I Ele t Re nt H H Bi Ha ura t pm s Al M es um e on fin l el Ho d G twe s & e m r As ar n l R es Em S Lux ons nic l te nt uc s in ia En P sh s te rod ing lth e ec dw s D at par Li ng til s o s Ph an Tob ods H ac tel o a ts to et rew ng re ru ics ie eis P ci ng ag ro old Pa pe eut lite r a ls oa d ts ot rv rs re th rs rv ail D i ac ati Re tee C ro i gy ta en s m a a vi m ft eta se tm So eta l & T ct ta e lth H Vin ce H age an Ser vc no R tain uct In c e um te ge re ui ing an o t In e nd ood & re li ie ea om ot r nt U ice ct n an s in oy ic G in Cr bu in M tille atio nta anc in to ns ru rin m a c um ed pe rm ica ra S to dc uc t a e Br Pro rke m Se e re A Se Ca r e e Se et iti ht v ic iv ic re rn olo D u c fa lin e c ep io e n ar d S ac Lo isi pl Ap uise ildi Au rn B sti Fa cki