Hurdle Rates
Risk Measurement And Hurdle Rates In Practice
Cost of Equity
The cost of equity is the rate of return that investors demand to invest in the equity of a company. All of the risk and return models talked about previously require a risk free rate and a risk premium (in the CAPM). We start by talking about those familiar inputs before directing our attention to the evaluation of risk parameters.
I. RiskFree Rate
The majority of risk and return models in finance begin with an investment that is identified as risk free and use the expected return on that investment as the riskfree rate. The expected returns on risky investments are then evaluated relative to the riskfree rate, with the risk generating an expected risk premium that is added on to the riskfree rate.
Prerequisites for an Asset to be RiskFree
We identified a riskfree investment as one for which the investor recognizes the expected returns with certainty. As a result, for an investment to be riskfree, that is, to have an actual return be equal to the expected return, two conditions have to be met:
 There has to be no default risk, which usually means that the investment must be circulated by a government. Note, though, that not all governments are defaultfree, and the existence of government or sovereign default risk can make it extremely hard to figure out riskfree rates in some currencies.
 There can be no doubt about reinvestment rates, which means that there are no intermediary cash flows. To demonstrate this example, suppose that you are struggling to figure out the expected return over a threeyear period and that you want a riskfree rate. A sixmonth Treasury bill rate, although defaultfree, cannot be riskfree, because there is the reinvestment risk of not knowing what the bill rate will be in six months. Even a threeyear Treasury bond is not riskfree, since the coupons on the bond will be reinvested at rates that cannot be calculated today. The riskfree rate for a threeyear time horizon has to be the expected return on a defaultfree (government) threeyear zero coupon bond.
This obviously has dire consequences for somebody doing corporate financial analysis, where expected returns generally need to be projected for extended periods over many years. A stickler's scrutiny of riskfree rates would then involve different riskfree rates for each period and different expected returns. As a sensible concession, however, it is worth recognizing that the present value effect of using riskfree rates that change from year to year tends to be small for most wellbehaved term structures.^{1} In these cases, we could use a duration matching strategy, where the duration of the defaultfree security used as the riskfree asset is matched up to the duration of the cash flows in the analysis.^{2} If, however, there are very large differences in either direction between shortterm and longterm rates, it does pay to use yearspecific riskfree rates in computing expected returns.
Cash Flows and RiskFree Rates: The Consistency Principle
The riskfree rate used to derive the expected returns should be calculated consistently with how the cash flows are calculated. If the cash flows are nominal, the riskfree rate should be in the same currency in which the cash flows are estimated. This also means that it is not where a project or firm is located that establishes the picking of a riskfree rate, but the currency in which the cash flows on the project or firm are anticipated. Thus, Granite can look at a future project in Mexico in dollars, using a dollar discount rate, or in pesos, using a peso discount rate. For the former, it would use the U.S. Treasury bond rate as the riskfree rate, but the latter would need a peso riskfree rate. Image 2.1 compares risk free rates in different currencies in early 2009:
Image 2.1: Risk Free Rates by Country 2021
Note that if these are truly default free rates, the number one factor determining the variation between currencies is expected inflation. The riskfree rate in Australian dollars is higher than the riskfree rate in Swiss Francs, because expected inflation is higher in Australia than in Switzerland.
Under conditions of elevated and volatile inflation, valuation is often estimated in real terms. Essentially, this suggests that cash flows are prepared using real growth rates and without allowing for the growth that comes from price inflation. To stay consistent, the discount rates used in these cases must be real discount rates. To get a real expected rate of return, we need to begin with a real riskfree rate. Even though government bills and bonds offer returns that are riskfree in nominal terms, they are not riskfree in real terms, because inflation can be unstable. The typical approach of subtracting an expected inflation rate from the nominal interest rate to arrive at a real riskfree rate presents at best only an estimate of the real riskfree rate. In the past, there were hardly any traded defaultfree securities offered to estimate real riskfree rates; but the innovation of inflationindexed Treasuries (called TIPs) has filled this void. An inflationindexed Treasury security does not offer a certain nominal return to buyers, but instead offers a guaranteed real return. In August 2021, for instance, the inflation indexed U.S. tenyear Treasury bond rate was 1.016 percent, radically lower than the nominal tenyear bond rate of 1.15 percent.
II. Risk Premium
The risk premium(s) is definitely an important input in all of the asset pricing models. In the next section, we will start by investigating the supporting determinants of risk premiums and then look at a number of useful method to estimating these premiums.
What Is the Risk Premium Supposed to Measure?
The risk premium in the CAPM measures the extra return that would be required by investors for moving their money from a riskless investment to the market portfolio or risky investments, on average. It should be a function of two variables:
1. Risk Aversion of Investors: As investors become more riskaverse, they should insist on a larger premium for changing from the riskless asset. Although a few of this risk aversion may be clear, some of it is also a function of economic prosperity (when the economy is doing great, investors are likely to be more enthusiastic about taking on more risk) and recent events in the market (risk premiums tend to surge after large market drops).
2. Riskiness of the Average Risk Investment: As the riskiness of the average risk investment rise, so should the premium. This will depend on what companies are actually traded in the market, their economic fundamentals, and how involved they are in dealing with risk.
Because each investor in a market is certain to have a different opinion of a normal equity risk premium, the premium will be a weighted average of these individual premiums, where the weights will be gauged on the capital the investor brings to the market. Put more directly, what Warren Buffett, with his sizeable wealth, believes is a suitable premium will be biased more into market prices than what you or I might think about the same measure.
3. Implied Equity Premiums
There is a substitute to projecting risk premiums that does not need past statistics or changes for country risk but does assume that the overall stock market is accurately priced. Think about, for example, a very straight forward valuation model for stocks
Value = Expected Dividends Next Period / (Required Return on Equity  Expected Growth Rate in Dividends)
This is basically the present value of dividends growing at a stable rate. Three of the four variables in this model can be obtained simplythe current level of the stock market (i.e., value), the estimated dividends next period, and the estimated growth rate in earnings and dividends in the long term. The one unknown is then the required return on equity; when we solve for it, we get an implied expected return on stocks. Subtracting out the riskfree rate will produce an implied equity risk premium.
To demonstrate, suppose that the current level of the S&P 500 Index is 4,400, the expected dividend yield on the index for the next period is 2.4 percent, and the expected growth rate in earnings and dividends in the long run is 3.5 percent. Solving for the required return on equity yields the following:
4,400 = 4,400 *(0.024) / r  .035
Solving for r,
r  0.035 = 0.024
r = 0.059 = 5.90%
If the current riskfree rate is 1.29 percent, this will yield a premium of 4.61 percent.
This calculation can be simplified to allow for high growth for a period and widened to deal with cash flowbased rather than dividendbased, models. To explain this, look at the S&P 500 Index on January 1, 2020. On December 31, 2019, the S&P 500 Index closed at 3140, and the dividend yield on the index was roughly 4.56%. In addition, the universal estimate of growth in earnings for companies in the index was about 9.7% for the next 5 years.^{3} Since the companies in the index have bought back enormous amounts of their own stock over the last few years, we included buybacks as part of the cash flows to equity investors. Worksheet 2.6 shows dividends and stock buybacks on the index, going back to 2014.
Worksheet 2.6: Dividends and Stock Buybacks on S&P 500 Index: 20142021
Year  Market value of index  Dividends  Buybacks  Cash to equity  Dividend yield  Buyback yield  Gross Cash Yield 
2014  2058.90  39.55  62.44  101.98  1.92%  3.03%  4.95% 
2015  2043.94  43.41  64.94  108.35  2.12%  3.18%  5.30% 
2016  2238.82  45.70  62.32  108.02  2.04%  2.78%  4.82% 
2017  2673.61  48.93  60.85  109.78  1.83%  2.28%  4.11% 
2018  2506.85  54.39  96.11  150.50  2.17%  3.83%  6.00% 
2019  3230.78  58.50  87.81  146.31  1.81%  2.72%  4.53% 
2020  3756.07  57.00  61.66  118.66  1.52%  1.64%  3.16% 
2021  4297.50  56.96  59.21  116.17  1.33%  1.38%  2.70% 
Average  2850.81  50.55  69.42  119.97  1.84%  2.61%  4.45% 
In 2020, for instance, companies collectively returned 3.16% of the index in the form of dividends (1.52%) and stock buybacks (1.64%). Buybacks are volatile, and dropped about 40% in the last quarter of 2020, relative to the last quarter of 2019, in the face of a market crisis and a slowing economy. Since this slowdown is likely to continue into 2021, we reduced the buybacks in 2021 by 16% to compute a normalized cash yield of 1.38% for the year (resulting in a total cash to equity of 116.17 for the year). In Worksheet 2.7, we estimate the cash flows to investors in the S&P 500 index from 20142021 by growing the normalized cash flow at 4% a year for the first five years and 1.25% (set equal to the riskfree rate) thereafter.
Worksheet 2.7: Cashflows on S&P 500 Index  



Year  Expected growth rate  Dividends+Buybacks on Index 
2020  118.66  
2021  4.00%  123.41 
2022  4.00%  128.34 
2023  4.00%  133.48 
2024  4.00%  138.82 
2025  4.00%  144.37 
2026  1.25%  146.17 
Using these cash flows to compute the expected return on stocks, we get the following:
$4,297.50 = 153.16/(1+ r) + 167.85/(1+ r)^{2} + 183.94/(1+ r)^{3} + 201.58/(1+ r)^{4} + 220.91/(1+ r)^{5} + 223.49/(r .0125)(1+ r)^{5}
Solving for the required return and the implied premium with the higher cash flows: Required Return on Equity = 5.678%
Implied Equity Risk Premium = Required Return on Equity  Riskfree Rate
= 5.678%  1.25% = 4.428%
We think that this estimate of risk premium (4.428%) is a more realistic value for July 1, 2021 than the historical risk premium of 5.50%. What makes this approach better is that it is marketdriven and forwardlooking and does not need historical data. Also, it will adjust in response to changes in market movements. Note that the S&P 500 a year prior was trading at 3100.36 and the implied equity risk premium on July 1, 2020 was 4.72%. The unusual movements are best viewed by graphing out implied premiums from the S& P 500 from 1960 in Image 2.2:
Image 2.2: Equity Risk Premiums US Equity Mkt 19602020
In terms of mechanics, we used analyst estimates of growth rates in earnings and dividends as our projected growth rates and a twostage dividend discount model (similar to the one that we used to compute the implied premium in the last paragraph). Looking at these numbers, we would draw the following conclusions.
Implied versus Historical Risk Premiums : For much of the last forty years, the implied equity premium has been lower than the historical risk premium, reflecting the long term upward movement in stock prices between 1981 and 2021. At the peak of dotcom boom at the end of 1999, the implied equity risk premium was 2% while the historical risk premium was about 6.5%. It is only in the last quarter of 2008 that implied premiums surged well above historical risk premiums.
Effects of inflation : The implied equity premium did increase during the 1970s as inflation skyrocketed. This poses some interesting questions about the movement of risk premiums. Instead of presuming that the risk premium is a constant and is unaffected by the level of inflation and interest rates, which is what we do with historical risk premiums, it may be more practicable to increase the risk premium as expected inflation and interest rates increase.
Mean Reversion : While implied equity risk premiums have changed notably over time, with a low of 2% in 1999 and a high of 6.69% at the end of 2018, there is support that they drift back to a historic average of between 4% and 4.5%. That turnaround, however, happens over long time periods.
III. Risk Parameters
The last set of information we need to put risk and return models into use are the risk factors for individual investments and projects. In the CAPM, the beta of the asset has to be estimated relative to the market portfolio.
Standard Procedures for Estimating CAPM Parameters, Betas and Alphas
To set up the general approach for calculating the beta in the CAPM, let's review the equation it provides for the expected return on an investment (R_{j}) as a measure of the beta of the investment (βj) riskfree rate (R_{f}) and the expected return on the market portfolio (R_{m}):
R_{j} = R_{f}+ β_{j} (R_{m}  R_{f})
This equation can be rewritten in one of two ways:
In terms of excess returns: R_{j}  R_{f} = β_{j} (R_{m}  R_{f})
In terms of raw returns: R_{j} = R_{f} (1 β_{j} )+ β_{j} R_{m}
These equations provide the templates for the two general approaches for estimating the beta of an investment, using past returns. In the first, we calculate the returns generated by an investment and a specific market index over previous time lines, in excess of the riskfree rates in each of the time lines, and regress the excess returns on the investment against the excess returns on the market:
Jensen's Alpha: This is the difference between the actual return on an investment and the return you would have expected it to make during a past period, given what the market did, and the investment's beta.
(R_{j} R_{f}) = ∝ + β_{j} (R_{m} R_{f})
In the second, we figure out the raw returns (not adjusted for the riskfree rate) earned by an investment and the market index over past time periods and regress the raw returns on the investment against the raw returns on the market:
R_{j} = ∝ + β_{j} R_{m}
In both regressions, the slope of the regression measures the beta of the stock and measures the riskiness of the stock. The intercept is a simple measure of stock price performance, relative to CAPM expectations, in each regression, but with slightly different interpretations. In the excess return regression, the intercept should be zero if the stock did exactly as predicted by the CAPM, and a positive (negative) intercept can be viewed as a measure that the stock did better (worse) than expected, at least during the period of the regression. In the raw return regression, the intercept has to be compared to the predicted intercept, R_{f} (1 β_{j} ), in the CAPM equation:
If ∝ > Rf (1  β) Stock did better than expected during regression period
∝ = Rf (1  β) Stock did as well as expected during regression period
∝ < Rf (1  β) Stock did worse than expected during regression period
This measure of stock price performance (∝ in excess return regression, and ∝  Rf (1  β) in the raw return regression) is called Jensen's alpha and provides a measure of whether the asset in question under or outperformed the market, after adjusting for risk, during the period of the regression.
R Squared (R^{2}): The R squared measures the proportion of the variability of a dependent variable that is explained by an independent variable or variables in a regression.
The third statistic that emerges from the regression is the R squared (R^{2}) of the regression.
Although the statistical reasoning of the R^{2} is that it suggests a measure of the goodness of fit of the regression, the financial basis for the R^{2} is that it provides an estimate of the proportion of the risk (variance) of a firm that can be attributed to market risk; the balance (1  R^{2}) can then be attributed to firm specific risk.
The last statistic worth mentioning is the standard error of the beta estimate. The slope of the regression, like any statistical calculation, is estimated with error, and the standard error exposes just how noisy the estimate is. The standard error can also be used to arrive at confidence intervals for the "true" beta value from the slope estimate.
The two methods should produce identical estimates for all of the inputs, but the excess return approach is generally more accurate, since it allows for the changes in riskfree rates from period to period. The basic return approach is easier to put into practice, simply because we need only the average risk free rate over the regression period.^{4}
Estimation Issues
There are three choices the analyst must decide on when setting up the regression described. The first involves the length of the estimation period. The tradeoff is straightforward: A longer time line provides more data, but the company itself might have changed in its risk characteristics over the time period. Granite Construction and Amazon have changed significantly in regards to both business mix and financial leverage over the past few years, and any regression that we run using past data will be exaggeragted by these changes.
The second estimation problem relates to the return period. Returns on stocks are available on annual, monthly, weekly, daily, and even intraday basis. Using daily or intraday returns will boost the number of observations in the regression, but it exposes the estimation process to a considerable bias in beta estimates linked to nontrading.^{5} For instance, the betas estimated for small firms, which are more prone to suffer from non trading, are biased downward when daily returns are used. Using weekly or monthly returns can lower the nontrading bias a lot.^{6}
The third estimation problem is what market index do we use in the regression. Since we are estimating the betas for the capital asset pricing model, the index that we are using, at least in theory, should be the market portfolio, which includes all traded assets in the market, held in proportion to their market values. While such a market portfolio may not exist today, the closer the preferred index comes to this model, the more important the beta estimate should be. Therefore, we should push away from limited indices (Dow 30, Sector indices or the NASDAQ) and towards larger and wider indices and away from equally weighted indices to value weighted indices. It should be no shock that the most commonly used market index by beta estimation services in the United States is the S&P 500. It may consist of only 500 stocks, but since they signify the leading market capitalization companies in the market, held in proportion to their market value, it does embody a notable percentage of the market portfolio, but only if we identify it narrowly as US equities. As asset classes multiply and global markets get bigger, we have to think about how best to broaden the index we use to reflect these excluded risky assets.
Example 2.3: Estimating CAPM Risk Parameters for Granite Construction
To evaluate how Granite Construction performed as an investment between June 2015 and July 2021 and how risky it is, we regressed monthly raw returns on Granite Construction against returns on the S&P 500 between June 2015 and July 2021. The returns on Granite Construction and the S&P 500 index are computed as follows:
1. The returns to a stockholder in Granite Construction are computed month by month from June 2015 and July 2021. These returns include both dividends and price appreciation and are defined as follows:
Return_{Granite Constructionj} = (Price_{Granite Constructionj}  Price_{Granite Constructionj1} + Dividends_{Granite Constructionj}) / Price_{Granite Constructionj1}
where Price_{Granite Constructionj} is the price of Granite Construction stock at the end of month j; and Dividends_{Granite Construction}_{j} are dividends on Granite Construction stock in month j . Note that Granite Construction pays dividends quarterly and that dividends are added to the returns of the month in which the stock went ex dividend.
2. The returns on the S&P 500 are computed for each month of the same time period, using the level of the index at the end of each month, and the monthly dividend yield on stocks in the index.
Market Return_{S&P} _{500,j} = (Index_{j}  Index_{j}_{1} + Dividends_{t})/Index_{j}_{1}
where Index_{j} is the level of the index at the end of month j and Dividend_{j} is the dividends paid on stocks in the index in month j. Although the S&P 500 is the most widely used index for U.S. stocks, they are at best imperfect proxies for the market portfolio in the CAPM, which is supposed to include all traded assets.
Image 2.3 graphs monthly returns on Granite Construction against returns on the S&P 500 index from July 2015 to July 2021.
Image 2.3 Granite Construction versus S&P 500: 20152021
The regression statistics for Granite Construction are as follows:^{7}
a. Slope of the Regression = 1.40. This is Granite Construction's beta, based on returns from 2015 to 2021. Using a different time period for the regression or different return intervals (weekly or daily) for the same period can result in a different beta.
b. Intercept of the Regression = 0.52 percent. This is a measure of Granite Construction's performance, but only when it is compared with R_{f }(1  β).^{8} Since we are looking at an investment made in the past, the monthly riskfree rate (because the returns used in the regression are monthly returns) between 2015 and 2021 averaged 1.02 percent, resulting in the following estimate for the performance:
R_{f } (1  β) = 1.02% (1  1.40) = 0.428%
Intercept  R_{f} (1  β) = 0.52%  (0.4281%) = 0.091%
This analysis suggests that Granite Construction's stock performed 0.091 percent worse than expected, when expectations are based on the CAPM, on a monthly basis between June 2015 and July 2021. This results in an annualized excess return of approximately .68 percent.
Annualized Excess Return = (1 + Monthly Excess Return)^{12}  1
= (1 +(.091))^{12}  1 = 0.068 or .68%
By this measure of performance, Granite did worse than expected during the period of the regression, given its beta and the market's performance over the period.
Note, however, that this does not say that Granite might be a good investment looking forward. It also does not give an analysis of how much of this excess return can be attributed to industrywide results and how much is specific to the firm. To make that breakdown, the excess returns would have to be calculated over the same period for other companies in the engineering and construction industry and compared with Granite's return. The difference would be then attributable to firmspecific eventss. In this case, for example, the average annualized excess return on other engineering and construction firms between 2015 and 2021 was 13.04 percent. This would imply that Granite Construction stock outperformed its peer group by 18.66 percent between 2015 and 2021, after adjusting for risk. (Firmspecific Jensen's alpha = 5.62%  (13.04%) = 18.66%)
c. R squared of the regression = 28 percent. This statistic implies that 28 percent of the risk (variance) in Granite Construction comes from market sources (interest rate risk, inflation risk etc.) and that the rest of 72 percent of the risk comes from firmspecific events. The latter risk should be diversifiable, and for that reason unrewarded. Granite Construction's R^{2} is somewhat higher than the median R^{2} of US companies against the S&P 500, which was roughly 24 percent in 2021.
d. Standard Error of Beta Estimate = 0.04 . This statistic suggest that the real beta for Granite Construction could span from 1.36 to 1.44 (subtracting or adding one standard error to the beta estimate of 1.41) with 67 percent confidence and from 1.32 to 1.48 (subtracting or adding two standard errors to the beta estimate of 1.40) with 95 percent confidence. These ranges may seem big, but they are not extraordinary for most U.S. companies. This suggests that we should judge regression estimates of betas from regressions with concern.
3. Determinants of Betas
The beta of a company is determined by three inputs: (1) the type of business or businesses the firm is in, (2) the degree of operating leverage in the firm, and (3) the firm's financial leverage.
Type of Business Since betas gauge the risk of a company relative to a market index, the more sensitive a business is to market conditions, the higher its beta. Thus, other things remaining equal, cyclical firms can be expected to have higher betas than noncyclical firms. Other things remaining equal, then, companies involved in housing and automobiles, two sectors of the economy that are very sensitive to economic conditions, will have higher betas than companies involved in food processing and tobacco, which are generally insensitive to business cycles.
Building on this point, we would also reason that the degree to which a product's purchase is, or is not necessary (discretionary or not) will change the beta of the company making the product. So, the betas of discount retailers, such as WalMart, should be lower than the betas of high end specialty retailers, such as Tiffany's or Gucci, since shoppers can hold off the purchase of the latter's products during dire economic times.
It is true that firms have only limited control over how discretionary a product or service is to their customers. There are firms, however, that have used this limited control to maximum effect to make their products less discretionary to buyers and by extension lowered their business risk. One approach is to make the product or service a much more integral and necessary part of everyday life, thus making its purchase more of a requirement. A second approach is to effectively use advertising and marketing to build brand loyalty. The objective in good advertising, as we see it, is to make discretionary products or services seem like necessities to the target audience. Thus corporate strategy, advertising, and marketing acumen can, at the margin, alter business risk and betas over time.
Cyclical Firm: A cyclical firm has revenues and operating income that tend to move strongly with the economyup when the economy is doing well and down during recessions.
Operating Leverage : A measure of the proportion of the operating expenses of a company that are fixed costs.
Degree of Operating Leverage The degree of operating leverage is a function of the cost structure of a firm and is usually defined in terms of the relationship between fixed costs and total costs. A firm that has high operating leverage (i.e., high fixed costs relative to total costs) will also have higher variability in operating income than would a firm producing a similar product with low operating leverage.^{9} Other things remaining equal, the higher variance in operating income will lead to a higher beta for the firm with high operating leverage.
Although operating leverage affects betas, it is difficult to measure the operating leverage of a firm, at least from the outside, because fixed and variable costs are often aggregated in income statements. It is possible to get an approximate measure of the operating leverage of a firm by looking at changes in operating income as a function of changes in sales.
Degree of Operating Leverage = % Change in Operating Profit/% Change in Sales For firms with high operating leverage, operating income should change more than proportionately when sales change, increasing when sales increase and decreasing when sales decline.
Can firms change their operating leverage? Although some of a firm's cost structure is determined by the business it is in (an energy utility has to build costly power plants, and airlines have to lease expensive planes), firms in the United States have become increasingly inventive in lowering the fixed cost component in their total costs. Labor contracts that emphasize flexibility and allow the firm to make its labor costs more sensitive to its financial success; joint venture agreements, where the fixed costs are borne by someone else; and subcontracting of manufacturing, which reduces the need for expensive plant and equipment, are only some of the manifestations of this phenomenon. The arguments for such actions may be couched in terms of competitive advantages and cost flexibility, but they do reduce the operating leverage of the firm and its exposure to market risk.
Example 2.5: Measuring Operating Leverage for Granite Construction
In Worksheet 2.11, we estimate the degree of operating leverage for Granite Construction from 2011 to July 2021 using earnings before interest and taxes (EBIT) as the measure of operating income.
Worksheet 2.11 Degree of Operating Leverage: Granite Construction
Year  Sales  % Change in Sales  Operating Income (EBIT)  % Change in EBIT 
2011  $ 2,009  $ 86  
2012  $ 2,082  3.63%  $ 49  43.02% 
2013  $ 2,267  8.89%  $ (14)  128.57% 
2014  $ 2,275  0.35%  $ 46  428.57% 
2015  $ 2,371  4.22%  $ 96  108.70% 
2016  $ 2,514  6.03%  $ 82  14.58% 
2017  $ 2,958  17.66%  $ 63  23.17% 
2018  $ 3,287  11.12%  $ 8  87.30% 
2019  $ 3,446  4.84%  $ (82)  1125.00% 
2020  $ 3,562  3.37%  $ (3)  96.34% 
2021  $ 3,596  0.95%  $ 32  1166.67% 
Average  6.11%  300.45%  





Operating Leverage 

(49.20) 


The degree of operating leverage can adjust radically from year to year, due to yeartoyear movements in operating income. Working with the average changes in sales and operating income over the period, allows us to calculate the operating leverage at Granite Construction:
Operating Leverage = % Change in EBIT/% Change in Sales
= 300.45%/6.11% = 49.20
There are two important comments that we can make about Granite Construction over the period. First, the operating leverage for Granite Construction is lower than the operating leverage for other engineering and construction firms, which we computed to be 1.15.^{10} This would mean that Granite Construction has lower fixed costs than its competitors. Second, the acquisition of Layne Water by Granite Construction in 2018 may be affecting the operating leverage. Looking at the numbers since 2015, we get a higher estimate of operating leverage:
Operating Leverage_{20152020} = 11.71%/9.91% = 1.18
We won't read too much into these numbers because Granite Construction has such a wide range of businesses. We would assume that Granite Construction's material business has higher fixed costs (and operating leverage) than its transportation.
Degree of Financial Leverage Other things remaining equal, an increase in financial leverage will increase the equity beta of a firm. Naturally, we would think that the fixed interest payments on debt to enhance earnings per share in good times and to drive it lower in bad times.^{11} Higher leverage amplifies the variance in earnings per share and makes equity investment in the firm riskier. If all of the firm's risk is borne by the stockholders (i.e., the beta of debt is zero),^{12} and debt creates a tax benefit to the firm, then
β_{L} = β_{u} (1 + (1  t)(D/E))
where
β_{L} = Levered beta for equity in the firm
β_{u} = Unlevered beta of the firm (i.e., the beta of the assets of the
firm)
t
= Marginal tax rate for the firm
D
/E = Debt/equity ratio
The marginal tax rate is the tax rate on the last dollar of income generated by the company and usually will not be equal to the effective or average rates; it is used because interest expenses save taxes on the marginal income. Intuitively, we see that as leverage increases (as calculated by the debt to equity ratio), equity investors bear higher levels of market risk in the company, leading to higher betas. The tax factor in the equation captures the benefit created by the tax deductibility of interest payments.
The unlevered beta of a company is identified by the kinds of businesses in which it operates and its operating leverage. This unlevered beta is sometimes defined as the asset beta because its value is recognized by the assets (or businesses) owned by the company. Consequently, the equity beta of a company is determined both by the riskiness of the business it operates in as well as the amount of financial leverage risk it has taken on. Because financial leverage multiplies the underlying business risk, it stands to reason that companies with elevated business risk should be hesitant to take on financial leverage. It also stands to reason that companies operating in somewhat established businesses should be better prepared to take on financial leverage. Utilities, for example, have traditionally maintained high debt ratios but not high betas, mostly because their fundamental businesses have been well established and rather predictable.
Breaking risk down into pieces  business and financial leverage, can give us some insight into why companies have high betas. Those companies with high betas generally operate in a risky business, or they take on too much financial leverage in a relatively stable business.
Should Small or HighGrowth Firms Have Higher Betas than Larger and More Mature Firms?
Though the answer might appear apparent at first glancethat smaller, highergrowth companies are riskier than larger firmsit is not a simple question to answer. If the question were posed in terms of total risk, smaller and highergrowth companies will generally be riskier simply because they have more unstable earnings (and their market prices shows it). When it is identified in terms of betas or market risk, smaller and higher growth firms should have higher betas only if the products and services they offer are more discretionary to their customers or if they have higher operating leverage. It is also likely that smaller companies might operate in smaller niche markets and sell products that customers can typically delay or defer buying and that the absence of economies of scales lead to elevated fixed costs for these companies. These companies should have higher betas than their larger counterparts. It is also sensible that neither condition holds for a particular small company. The answer will then depend on both the company in question and the industry in which it operates.
Example 2.6: Effects of Financial Leverage on Betas: Granite Construction
From the regression for the period 2015 to 2021, Granite Construction had a beta of 1.40. To estimate the effects of financial leverage on Granite Construction, we began by estimating the average debt/equity ratio between 2015 and 2021 using market values for debt and equity.
Average Market Debt/Equity Ratio between 2015 and 2021 = 35.76% The unlevered beta is estimated using a marginal corporate tax rate of 25%:^{13}
Unlevered Beta = Current Beta/(1 + [1  tax rate] [Average Debt/Equity])
= 1.40/(1 + [1  0.25] [0.357]) = 1.10
The levered beta at different levels of debt can then be estimated:
Levered Beta = Unlevered Beta * [1 + (1  tax rate) (Debt/Equity)]
For instance, if Granite Construction were to increase its debt equity ratio to 10 percent, its equity beta will be
Levered Beta (@10% D/E) = 1.10*(1+ (1  0.25) (0.10)) = 2.18
If the debt equity ratio were raised to 25 percent, the equity beta would be Levered Beta (@25% D/E) = 1.10 *[1 + (1  0.25) (0.25)] = 2.29.
Worksheet 2.12 summarizes the beta estimates for different levels of financial leverage ranging from 0 to 90 percent debt.
Worksheet 2.12 Financial Leverage and Betas
Debt to Capital  Debt/Equity Ratio  Beta  Effect of Leverage 
0.00%  0.00%  1.10  0 
10.00%  11.11%  2.18  1.08 
20.00%  25.00%  2.29  1.19 
30.00%  42.86%  2.42  1.32 
40.00%  66.67%  2.60  1.50 
50.00%  100.00%  2.85  1.75 
60.00%  150.00%  3.23  2.13 
70.00%  233.33%  3.85  2.75 
80.00%  400.00%  5.10  4.00 
90.00%  900.00%  8.85  7.75 
As Granite Construction's financial leverage increases, the beta increases concurrently.
BottomUp Betas
Breaking down betas into their business, operating leverage, and financial leverage components gives us another way of figuring out betas, whereby we don't need past prices on an individual company or asset to estimate its beta.
To build this improved method, we need to introduce an added feature that betas possess that proves invaluable. The beta of two assets put together is a weighted average of the individual asset betas, with the weights based on market value. So, the beta for a firm is a weighted average of the betas of all of the different businesses it is in. Thus, the bottomup beta for a company, its asset, or its project can be estimated as follows.
1. Identify the business or businesses segments within the company whose beta we are trying to figure out. Most companies give a summary of their revenues and operating income by types of business in their annual reports and financial filings.
2. Calculate the average unlevered betas of other publicly traded firms that are primarily or only in each of these businesses. In making this assumption, we have to think about the following estimation issues:
Comparable firms : In most businesses, there are at least a few similar companies and in some businesses, there can be hundreds. Start with a narrow description of comparable companies, and broaden it if the number of comparable companies is too small.
Beta Estimation : Once a listing of comparable companies has been prepared, we need to figure out the betas of each of these companies. Optimally, the beta for each company will be estimated against a common index. If that proves unrealist, we can use betas estimated against different indices.
Unlever First or Last : We can compute an unlevered beta for each firm in the comparable firm list, using the debt to equity ratio, and tax rate for that company, or we can compute the average beta, debt to equity ratio, and tax rate for the sector and unlever using the averages. Given the standard errors of the individual regression betas, we would suggest the latter approach.
Averaging Approach : The average beta across the comparable companies can be either a simple average or a weighted average, with the weights based on market capitalization. Statistically, the savings in standard error are larger if a simple averaging process is used.
Adjustment for Cash : Investments in cash and marketable securities have betas close to zero. So, the unlevered beta that we calculate for a business by looking at comparable companies might be changed by the cash assets of these companies. To figure out an unlevered beta cleansed of cash:
Unlevered Beta corrected for Cash = Unlevered Beta /(1  Cash/ Firm Value)
The resulting number is sometimes called a pure play beta, indicating that it measures the risk of only the business and not any other corporate holdings.
3. To calculate the unlevered beta for the company, we take a weighted average of the unlevered betas, using the proportion of company value derived from each business as the weights. These company values will have to be estimated because segments of a company generally do not have market values available.^{14} If these values cannot be estimated, we can use operating income or revenues as weights. This weighted average is called the bottomup unlevered beta. In general, it is a good exercise to compute two unlevered betas for a company, one for only the operating assets of the company, and one with cash and marketable securities treated as a separate business, with a beta of zero.
4. Calculate the current debt to equity ratio for the company, using market values if available. Otherwise, use the target debt to equity ratio specified by the management of the company or industry debt ratios.
5. Estimate the levered beta for the equity in the company (and each of its business segments) using the unlevered beta from Step 3 and the debt to equity ratio from Step 4.
Naturally, this approach rests on you being able to recognize the unlevered betas of individual businesses.
There are three benefits with using bottomup betas, and they are important:
We can estimate betas for companies that have no previous price history because all we need to do is identify the business or businesses segments they operate in. In other words, we can estimate bottomup betas for initial public offerings, private businesses, and divisions of companies.
 Since the beta for the business is found by averaging across a large number of regression betas, it will usually be more accurate than any individual company's regression beta estimate. The standard error of the average beta estimate will be a function of the number of comparable firms used in Step 2 and can be approximated as follows:
 So, the standard error of the average of the betas of 100 firms, each of which has a standard error of 0.25, will be only 0.025 (0.25/v100).
 The bottomup beta can point out current and even impending adjustments to a company's business mix and financial leverage, because we can alter the combination of businesses and the weight on each business in making the beta estimate.
σ_{Average Beta} = Average σ _{Beta} / √ Number of firms
Example 2.7: BottomUp Beta for Granite Construction
Granite Construction is an engineering and construction company with diverse holdings. The Transportation, Water and Specialty segments are predominantly in the public sector and include certain federal agencies, state departments of transportation, local transit authorities, county and city public works departments, school districts and developers, utilities and private owners of industrial, commercial and residential sites. Customers of our Materials segment include internal usage by our own construction projects, as well as thirdparty customers. The majority of business is in the United States. To estimate Granite Construction's beta, we broke their business into four major segments:
1. Transportation, has constructed some of the most iconic and complex transportation projects in the United States, developing urban and rural transit programs that connect millions of people, freight and products every day.
2. Specialty Construction, provides solutions to a variety of segments, including tunnel, power, mining, oil and gas, renewable energy and more.
3. Waterrelated Construction: Layne, A Granite Company (Layne) is a global water management, mineral exploration and drilling company. They provide responsible infrastructure solutions for natural resources in water, minerals and energy, while offering innovative, sustainable products and services with an enduring commitment to safety, operational excellence, and client satisfaction. In June 2018, Layne became a whollyowned subsidiary of Granite Construction, Inc.
4. Materials, Granite mines quality aggregates that fuel infrastructure, including asphalt concrete, aggregates, specialty sands and rock.
This segmentation replicates Granite Construction's reporting in its annual report. In reality, there are a number of smaller businesses that Granite Construction is in that are entrenched in these four businesses.
For the four businesses for which we have in depth information, we figured out the unlevered beta by searching for comparable companies in each business.^{15} Worksheet 2.13 sums up the comparables found and the unlevered beta for each of the businesses.
Worksheet 2.13 Estimating Unlevered Betas for Granite Construction's Business Area
Business  Comparable Firms  # of firms  Median Beta  Median MV Debt/Equity  Unlevered Beta  Cash/Firm Value  Beta (less cash) 
Transportation  Engineering & Construction  39  0.98  28.34%  0.77  5.50%  0.81 
Specialty  Construction Building Supplies  50  0.97  18.91%  0.95  6.16%  1.01 
Materials  Precious Metals, Coal  199  0.72  54.50%  0.87  2.70%  0.89 
Water  Water Utility  9  0.61  4.13%  0.66  1.74%  0.67 
To obtain the beta for Granite Construction, we have to estimate the weight that each business is of Granite Construction as a company. The value for each of the divisions was estimated by applying the typical revenue multiple at which comparable firm trade at to the revenue reported by Granite Construction for that segment in 2021.^{16} The unlevered beta for Granite Construction as a company in 2021 is a valueweighted average of the betas of each of the different business areas. Worksheet 4.14 summarizes this calculation.
Worksheet 2.14 Estimating Granite Construction's Unlevered Beta
Business  Revenues  EV/Sales  Estimated Value  Firm Value Proportion  Unlevered beta 
Specialty  $724  0.77  $557  6.78%  0.06 
Construction & Engineering Services  $2,017  1.17  $2,368  28.82%  0.22 
Materials  $381  2.90  $1,106  13.47%  0.12 
Water  $440  9.51  $4,186  50.94%  0.30 
Granite Operations  $3,562  $8,217  100.00%  0.70 
The equity beta can then be calculated using the current financial leverage for Granite Construction as a firm. Combining the market value of equity of $1,817 million with an estimated market value of debt of $399 million,^{17} we arrive at the levered (equity) beta for Granite Construction's operating assets:
Debt/Equity Ratio for Granite Construction = $399 / $1,817 = 21.95%
Equity Beta for Granite Construction's Operating Assets = 0.70 (1 + (1  0.25)(0.2195)) = 0.818 These are the estimates of unlevered beta and equity beta that we will be using for the rest of the book, when analyzing operating assets.
We can also compute an unlevered beta for all of Granite Construction's assets including its cash holdings and the resulting equity beta:
β_{Granite Construction} = β_{Operating Assets} [Value_{Operating Assets} / (Value_{Operating Assets} + Value_{Cash})] + β_{Cash} [Value_{Operating Assets} / (Value_{Operating Assets} + Value_{Cash})]
= 0.818 (8,217 / 8,217 + 528) + 0 (528 / 8,217 + 528) = 0.768
Equity Beta_{Granite Construction} _{as} _{company} = 0.768 (1 + (1  0.25)(0.2195)) = 0.894
We can compare this equity beta to the regression beta of 1.40. Clearly it is lower, it is also more exact (because of the averaging) and identifies Granite Construction's current mix of businesses. There will be far less call for us to use these cashadjusted beta values in analyses.^{18}
Example 2.8: BottomUp Beta for Capital Construction
We cannot estimate a regression beta for Capital Construction, a private firm, because it does not have a history of past prices. We can, however, estimate the beta for Capital Construction using the bottomup approach. We list the betas of these firms as well as debt, cash, and equity values in Worksheet 2.15.
Worksheet 2.15 Betas and Leverage of Publicly Traded Engineering and Construction Firms
Company Name  Industry Group  Beta  Market Debt to Equity ratio  Unlevered Beta  Cash/ Firm Value  Unlevered Beta Corrected for Cash 
Ameresco, Inc.  Engineering/Construction  0.59  31.32%  0.477  1.38%  0.48 
Amincor, Inc.  Engineering/Construction  0.85  0.00%  0.853  0.00%  0.85 
Argan, Inc.  Engineering/Construction  0.40  0.32%  0.401  50.44%  0.81 
AVEW Holdings Inc.  Engineering/Construction  0.00%    0.00%    
Befut Global, Inc.  Engineering/Construction  0.00%    0.00%    
BOTS, Inc.  Engineering/Construction  0.01  0.00%  0.013  0.07%  0.01 
Central Wireless, Inc.  Engineering/Construction  1.65  0.00%  1.651  0.00%  1.65 
CMARK International, Inc.  Engineering/Construction  0.34  0.00%  0.336  0.00%  0.34 
Conair Corporation  Engineering/Construction  0.00%    0.00%    
Concrete Pumping Holdings, Inc.  Engineering/Construction  0.79  180.96%  0.334  0.68%  0.34 
Dycom Industries, Inc.  Engineering/Construction  1.65  26.62%  1.375  0.39%  1.38 
Firemans Contractors, Inc.  Engineering/Construction  4.56  0.00%  4.563  0.00%  4.56 
Fluor Corporation  Engineering/Construction  2.01  89.17%  1.203  49.25%  2.37 
Galenfeha, Inc.  Engineering/Construction  1.03  0.00%  1.025  0.00%  1.03 
Go Solar USA, Inc.  Engineering/Construction  0.00%    0.00%    
Granite Construction Incorporated  Engineering/Construction  1.24  0.00%  1.238  0.00%  1.24 
HC2 Holdings, Inc.  Engineering/Construction  1.71  288.14%  0.542  16.88%  0.65 
Infrastructure and Energy Alternatives, Inc.  Engineering/Construction  1.11  115.72%  0.595  7.03%  0.64 
JNS Holdings Corporation  Engineering/Construction  0.91  0.00%  0.908  0.00%  0.91 
Kingfish Holding Corporation  Engineering/Construction  1.18  1.181  1.18  
Limbach Holdings, Inc.  Engineering/Construction  0.41  68.95%  0.272  23.99%  0.36 
Matrix Service Company  Engineering/Construction  1.17  13.54%  1.066  24.83%  1.42 
Moro Corporation  Engineering/Construction  0.32  0.00%  0.324  0.00%  0.32 
National Storm Management, Inc.  Engineering/Construction  0.00%    0.00%    
Northeast Development Corp., Inc.  Engineering/Construction  0.00%    0.00%    
Northwest Pipe Company  Engineering/Construction  0.76  8.81%  0.717  10.07%  0.80 
Orbital Energy Group, Inc.  Engineering/Construction  1.16  27.25%  0.963  4.79%  1.01 
Orion Group Holdings, Inc.  Engineering/Construction  0.93  60.72%  0.642  1.13%  0.65 
PlayBOX  Engineering/Construction  2.21  0.00%  2.211  0.00%  2.21 
Premier Pacific Construction, Inc.  Engineering/Construction  1.06  1.057  1.06  
Reliant Holdings, Inc.  Engineering/Construction  3.77%    7.57%    
ReneSola Ltd  Engineering/Construction  1.36  19.36%  1.188  2.00%  1.21 
Social Detention, Inc.  Engineering/Construction  0.26  1.00%  0.261  0.56%  0.26 
Texas Gulf Energy, Incorporated  Engineering/Construction  1.29  0.00%  1.294  0.00%  1.29 
The Peck Company Holdings, Inc.  Engineering/Construction  0.86  30.10%  0.705  0.29%  0.71 
Trans Global Group, Inc.  Engineering/Construction  0.00%    0.00%    
UGE International Ltd.  Engineering/Construction  0.77  8.90%  0.723  0.30%  0.73 
Victura Construction Group, Inc.  Engineering/Construction  0.00%    0.00%    
Williams Industrial Services Group Inc.  Engineering/Construction  0.43  74.07%  0.277  3.57%  0.29 
Median  0.98  28.34%  0.5947  0.07%  0.6519 
Even though the companies in this example are very different in terms of market capitalization, the betas are consistent. To estimate the unlevered beta for the sector, we first unlevered the beta for each firm and corrected each unlevered beta for the firm's cash holdings. The median value for the unlevered beta, corrected for cash holdings, is .6519.^{36}
Because the debt/equity ratios used in computing levered betas are market debt equity ratios, and the only debt equity ratio we can compute for Capital Construction is a book value debt equity ratio, we have assumed that Capital Construction is close to the book industry median debt to equity ratio of 28.34 percent. Using a marginal tax rate of 25 percent for Capital Construction, we get a levered beta of .79.
Levered beta for Capital Construction = .6519 [1 + (1  0.25) (0.2834)] = .79
I. Estimating the Cost of Equity
Having estimated the riskfree rate, the risk premium(s), and the beta(s), we can now estimate the expected return from investing in equity at any company. In the CAPM, this expected return can be shown as:
Expected Return = RiskFree Rate + Beta * Expected Risk Premium
where the riskfree rate would be the rate on a longterm government bond; the beta would be either the historical, fundamental, or accounting betas; and the risk premium would be either the historical premium or an implied premium.
The expected return on an equity investment in a company, given its risk, has key implications for both equity investors in the company and the managers of the company. For equity investors, it is the rate they need to make to be compensated for the risk that they have taken on investing in the equity of a firm. If after evaluating a stock, they find out that they cannot generate this return, they would not buy it; instread, if they determine they can make a higher return, they would make the investment. For company managers, the return that investors need to make to break even on their equity investments is the return that they need to deliver to keep these investors from becoming impatient and rebellious. In other words, it becomes the rate that they have to beat in terms of returns on their equity investments in individual projects. Thus, this is the cost of equity to the firm.
Example 2.13: Estimating the Cost of Equity
In Example 2.7, we estimated a bottomup unlevered beta for Granite Construction and each of its divisions. To calculate the levered beta for Granite Construction, we estimated a debt to equity ratio of 28.41%, based upon the total market value of equity ($1,817 million) and debt ($399 million). To estimate the levered beta for each of the divisions, we face a challenge in figuring out the debt to equity ratio at the divisional level, since we do not have market equity values for the individual divisions nor do we have full details on which divisions are responsible for the borrowing. We have two choices. One is to believe that Granite Construction debt to equity ratio applies to all of its individual divisions. The other is to try to make judgments about the debt to equity ratios for the individual divisions, based upon the information available. In Worksheet 2.20, we tried to do the latter:
Worksheet 2.20: Allocating Debt and Equity to divisions
Business  Estimated EV  Allocated Debt  Estimated Equity  D/E Ratio  D/E Ratio of comps  Estimated debt  Proportions 
Specialty  $ 557  $ 81  $ 476  17.02%  18.91%  $95  21.62% 
Construction & Engineering Services  $ 2,368  $ 226  $ 2,142  10.54%  28.34%  $250  56.90% 
Materials  $ 1,106  $ 43  $ 1,064  4.01%  54.50%  $44  10.11% 
Water  $ 4,186  $ 49  $ 4,136  1.19%  4.13%  $50  11.36% 
$439  100% 
We started with the estimates of enterprise value that we calculated in Worksheet 2.14, determined by multiplying the revenues in each division by the median EV/Sales ratio of comparable companies in the division. We then used the market Debt/Equity ratios of these same comparable companies to estimate the debt in each division in the second to last column and used the ratios computed from these estimated debt numbers to distibute the existing debt ($399 million) across the divisions.^{19} Lastly, we estimated the value of equity in each division by subtracting the debt from the estimated enterprise value.
Using the US dollar riskfree rate (from Example 2.1) and the equity risk premium estimated for mature markets (from Example 4.2), we estimate the cost of equity for Granite Construction's operating assets and for each of its divisions, listed in Worksheet 2.21.
Worksheet 2.21 Levered Beta and Cost of Equity: Granite Construction
Business  Unlevered Beta  D/E Ratio  Levered Beta  Cost of Equity 
Specialty  0.06  17.02%  0.792  4.57% 
Construction & Engineering Services  0.22  10.54%  0.757  4.42% 
Materials  0.12  4.01%  0.723  4.27% 
Water  0.30  1.19%  0.708  4.21% 
Granite Construction  0.70  14.96%  0.729  4.30% 
The costs of equity vary across the remaining divisions, with Specialty division having the highest beta (and cost of equity) and the Water division the lowest.
To estimate the cost of equity for Capital Construction, we will use the beta of 0.98 estimated from Example 4.8 together with the riskfree rate and risk premium for the United States: Cost of Equity = 1.25% + .98 (4.31%) = 5.47%
Inherent in the use of beta as a measure of risk is the assumption that the marginal investor in equity is a welldiversified investor. Although this is a defensible assumption when evaluating publicly traded companies, it turns out to be much more difficult to validate for private firms. The owner of a private firm usually has the majority of his or her wealth tied up in the business. Therefore, he or she carries virtually the total risk in the business rather than just the market risk. Thus, for a business like Capital Construction, the beta that we have estimated of 0.98 (leading to a cost of equity of 6.64 percent) will understate the risk carried by the owner. There are three possible answers to this problem:
1. Suppose that the business is managed with the shortterm goal of selling to a large publicly traded company. In this example, it is okay to use the market beta and cost of equity that comes from it.
2. Add a premium to the cost of equity to show the elevated risk created by the owner's failure to diversify. This could help explain the excessive returns that a number of venture capitalists demand on their equity investments in fledgling businesses.
3. Change the beta to show the total risk rather than market risk. This change is rather simple, because the R^{2} of the regression measures the proportion of the variance that is market risk. Dividing the market beta by the square root of the R^{2} (which yields the correlation coefficient) yields a total beta. In the Capital Construction example, the regressions for the comparable firms against the market index have an average correlation with the market of 24.03% (the average R ^{2} was 49.02%). The total beta for Capital Construction can then be computed as follows:
Total Beta = (Market Beta)/Correlation with the market = 0.98/0.2403= 4.078
Using this total beta would yield a much higher and more realistic estimate of the cost of equity.
Cost of Equity = 1.25% + 4.0782 (4.31%) = 18.83%
So, private businesses will generally have much higher costs of equity than their publicly traded counterparts, with diversified investors. Although a lot of them eventually give in by selling to publicly traded competitors or going public, a few companies elect to remain private and succeed. To do so, they must diversify on their own (as many familyrun businesses in Asia and Latin America did) or acknowledge the lower value as a cost paid for retaining total control.
From Cost of Equity to Cost of Capital
Equity is clearly a vital and essential ingredient of the financing mix for every business, but it is only one ingredient. Nearly all public businesses finance some or a good deal of their operations with debt or some mix of equity and debt. The costs of using these financing sources are generally quite different from the cost of equity. Therefore, the minimum acceptable hurdle rate for a project will reflect their costs as well, in proportion to their use in the financing mix. Intuitively, the cost of capital is the weighted average of the costs of the different mix of financingincluding debt, equity, and hybrid securitiesused by a companies to fund their day to day operations.
The Costs of Capital
Default Risk: The risk that a company might not be able to make its required debt payments, such as interest expenses or principal payments.
To estimate the cost of the funding that a company wants, we have to estimate the costs of all of the nonequity pieces. In this section, we look at the cost of debt first and then move our analysis to hybrids, including preferred stock and convertible bonds.
The Cost of Debt
The cost of debt measures the current cost to the company if they borrow money to fund projects. In general terms, it is determined by the following variables:
1. The current level of interest rates : As market interest rates rise, the cost of debt for all companies will also rise.
2. The default risk of the company : As the default risk of a company increases, lenders will charge higher interest rates (a default spread) to compensate for the added risk.
3. The tax advantage tied to debt : Since interest is taxdeductible, the after tax cost of debt is a function of the tax rate. The tax benefit that accrues from paying interest makes the aftertax cost of debt lower than the pretax cost. Furthermore, this benefit increases as the tax rate increases.
AfterTax Cost of Debt = (Riskfree rate + Default Spread) (1  Marginal Tax Rate)
The challenge in estimating the cost of debt is really one of estimating the correct default spread for a company.
Estimating the Default Risk and Default Spread of a Firm
The easiest way to estimate the cost of debt is if a company has long term bonds outstanding that are widely traded and have no special features, such as convertibility or first claim on assets, skewing interest rates. The market price of the bond, together with its coupon and maturity, can serve to calculate a yield we use as the cost of debt. For example, this approach works for companies that have a large number of outstanding bonds that are liquid and trade frequently.
Many firms have outstanding bonds that rarely trade on a regular basis. Given these firms are usually rated, we can determine their costs of debt by using their ratings and assigned default spreads. Thus, Granite Construction with a B rating can be expected to have a cost of debt approximately 3.28 percent higher than the Treasury bond rate, in July 2021, because this was the spread typically paid by B rated firms at the time.
A number of companies simply cannot get rated. Many smaller firms and most private businesses fall into this category. Ratings agencies have sprung up in many emerging markets, but there are still a number of markets in which companies are not rated on the basis of default risk. When there is no rating available to estimate the cost of debt, there are two options:
Recent Borrowing History: Countless companies that are not rated still borrow money from banks and other financial institutions. By looking at the most recent borrowings made by a company, we can get an ideal of what banks are charging this companies and the default spreads and use these information to come up with a cost of debt.
Estimate a Synthetic Rating and Default Spread : An alternative is to play the role of a ratings agency and assign a rating to a company based on its financial ratios; this rating is called a synthetic rating. To make this opinion, we start with rated firms and study the financial traits shared by companies in each ratings class. Let's look at a very simple example, using the operating income to interest expense ratio, which is, the interest coverage ratio, and we computed it for each rated firm. In Worksheet 2.24, we list the range of interest coverage ratios for manufacturing firms in each S&P ratings class, arranged by market capitalization into large (>$5 billion) and small (<$5 billion).^{20} We also show the standard default spreads for bonds in each ratings class in July 2021.^{21}
Worksheet 2.24 Interest Coverage Ratios and Ratings
Interest Coverage Ratio: Small market cap(<$5 billion)  Interest Coverage Ratio: Large market cap (>US $ 5 billion)  Rating  Typical Default 
> 12.5  >8.5  AAA  0.59% 
9.5012.50  6.58.5  AA  0.55% 
7.509.50  5.56.5  A+  0.58% 
6.007.50  4.25 5.5  A  0.61% 
4.506.00  3 4.25  A  0.84% 
4.004.50  2.53.0  BBB  1.06% 
3.504.00  2.252.5  BB+  1.53% 
3.003.50  2.02.25  BB  1.99% 
2.503.00  1.752.0  B+  2.65% 
2.002.50  1.51.75  B  3.31% 
1.502.00  1.251.5  B  4.45% 
1.251.50  0.81.25  CCC  5.58% 
0.801.25  0.650.8  CC  5.98% 
0.500.80  0.20.65  C  10.91% 
< 0.65  <0.2  D  13.58% 
Source: Federal Reserve https://fred.stlouisfed.org
Now, let's look at a private firm with $10 million in earnings before interest and taxes and $3 million in interest expenses; it has an interest coverage ratio of 3.33. Based on this interest coverage ratio, we would assign a synthetic rating of BB for the company and attach a default spread of 3.31 percent to the riskfree rate to come up with a pretax cost of debt. A large market cap firm with the same interest coverage ratio would be assigned a rating of A and a default spread of .84%.
By basing the synthetic rating on the interest coverage ratio alone, we run two risks. One is that using the previous year's operating income as the basis for the rating may generate too low or too high a rating for a company that had an unusually good or bad earnings years. We can answer that by using the average operating income over a period, say 5 years, to compute the coverage ratio. The other is that we risk overlooking the data that is available in the other financial ratios along with qualitative data used by ratings agencies. The counter to that is to widen the analysis to include other ratios. The initial step would be to compute a score based on multiple ratios. For example, the Altman z score, which is used as a proxy for default risk, is a collection of five financial ratios, which are weighted to generate a z score. The ratios and their relative weights are typically based on the historic track record of company defaults. The second step is to link the level of the score to a bond rating, much as we did in Worksheet 2.24, with interest coverage ratios. In making this choice, though, recognize that complexity comes at a cost. Credit or z scores may, in fact, generate better estimates of synthetic ratings than those based only on interest coverage ratios, but changes in ratings arising from these scores are much more difficult to explain than those based on interest coverage ratios. That is the reason we prefer the flawed but more visible ratings from interest coverage ratios.
Calculating the Weights of Debt and Equity Components
Once we have costs for each of the different financing pieces, all we need are weights on each piece to figure out a cost of capital. In this section, we consider the choices for weighting, the reason for using market value weights, and if the weights can adjust over time.
Choices for Weighting
In determining weights for debt, equity, and preferred stock, we have two options. We can take the accounting estimates of the value of each funding source from the balance sheet and calculate book value weights. Or, we can use or estimate market values for each piece and figure out weights based on relative market value. As a general rule, the weights used in the cost of capital calculation should be based on market values . This is because the cost of capital is a forwardlooking measure and captures the cost of raising new money to finance projects. When new debt and equity has to be raised in the market at prevailing prices, the market value weights is the right choice.
There are a few analysts who keep on using book value weights and defend them using four arguments, none of which are persuasive:
 Book value is more reliable than market value because it is not as volatile : Yes it is true that book value does not change as much as market value, this is more a indication of weakness than strength, because the true value of the firm changes over time as new information comes out about the firm and the overall economy. We would make a case that market value, with its volatility, is a much better indication of true value than is book value.^{22}
 Using book value rather than market value is a more conservative approach to estimating debt ratios . The book value of equity in most companies in developed markets is generally well below market value, while the book value of debt is typically close to the market value of debt. Since the cost of equity is much higher than the cost of debt, the cost of capital calculated using book value ratios will be lower than those calculated using market value ratios, making them less conservative estimates, not more so.^{23}
 Since accounting returns are established based on book value, consistency requires the use of book value in computing cost of capital: Although it may seem consistent to use book values for both accounting return and cost of capital calculations, it does not make economic sense. The moneys invested in these projects can be invested elsewhere, earning market rates, and the costs should therefore be computed at market rates and using market value weights.
Estimating Market Values
In a world where all funding was raised in financial markets and all securities were continuously traded, the market values of debt and equity should be easy to find. In practice, however, there are a number of stocks and bonds where limited market value information is available, even for large publicly traded firms, and none of the financing components are traded in private firms.
The Market Value of Equity
The market value of equity is generally the number of shares outstanding times the current stock price. Because it measures the cost of raising funds today, it is not good practice to use average stock prices over time or some other normalized version of the price.
With market value debt ratios: 15% (0.9) + 5% (0.1) = 14%
With book value debt ratios: 15% (0.7) + 5% (0.3) = 12%
Multiple Classes of Shares : If there is more than one class of shares outstanding, the market values of all of these securities should be combined and looked at as equity. Even if some of the classes of shares are not traded, market values have to be estimated for nontraded shares and added to the combined equity value.
Equity Options : If there other equity claims in the companywarrants and conversion options in other securitiesthese should also be valued and added on to the value of the equity in the company. In the past, the use of options as management compensation created a host of problems, because the value of these options had to be estimated.
How do we determine the value of equity for private businesses? We have two options. One is to estimate the market value of equity by looking at the multiples of revenues and net income of publicly traded companies. The other is to ignore the estimation process and use the market debt ratio of publicly traded firms as the debt ratio for private firms in the same business. This is the assumption we made for Capital Construction. We used the industry average debt to equity ratio for the engineering/construction business as the debt to equity ratio for Capital Construction.
The Market Value of Debt
The market value of debt is typically much more difficult to figure out directly because very few firms have all of their debt in the form of bonds. A number of companies use nontraded debt, like bank loans, which are in book value terms, not market value terms. To get around the problem, many analysts make the simplifying assumptions that the book value of debt is equal to its market value. Although this is not a bad assumption for mature companies in developed markets, it can be a mistake when interest rates and default spreads are unstable.
The simplist way to convert book value debt into market value debt is to treat the entire debt on the books as a coupon bond, with a coupon set equal to the interest expenses on all of the debt and the maturity set equal to the facevalue weighted average maturity of the debt, and to then value this coupon bond at the current cost of debt for the company. Thus, the market value of $1 billion in debt, with interest expenses of $60 million and a maturity of six years, when the current cost of debt is 7.5 percent can be estimated as follows:
Estimated Market Value of Debt = 60 [(1  (1 / 1.076)^{6}) / .075] + 1,000 / (1.076)^{6} = $930
This is an approximation; a more accurate computation would require valuing each debt issue separately using this process.
Can Financing Weights Change over Time?
Using the current market values to obtain weights will yield a cost of capital for the current year. But can the weights attached to debt and equity and the resulting cost of capital change from year to year? Absolutely, and especially in the following scenarios:
Young firms : Young firms often are all equityfunded largely because they do not have the cash flows (or earnings) to sustain debt. As they become larger, increasing earnings and cash flow usually allow for more borrowing. When analyzing firms early in their life cycle, we should allow for the fact that the debt ratio of the firm will probably increase over time toward the industry average.
Target debt ratios and changing financing mix : Mature firms sometimes decide to change their financing strategies, pushing toward target debt ratios that are much higher or lower than current levels. When analyzing these firms, we should consider the expected changes as the firm moves from the current to the target debt ratio.
As a general rule, we should view the cost of capital as a yearspecific number and change the inputs each year. Not only will the weights attached to debt and equity change over time, but so will the estimates of beta and the cost of debt. In fact, one of the advantages of using bottomup betas is that the beta each year can be estimated as a function of the expected debt to equity ratio that year.
Example 2.17: Market Value and Book Value Debt Ratios: Granite Construction and Capital Construction
Granite Construction has a number of debt issues on its books, with varying coupon rates and maturities. Worksheet 2.28 summarizes Granite Construction's outstanding debt, broken down by when the debt comes due; we treat the debt due in 2021 as due in 1 year, the debt due in 2022 as due in 2 years and so on. The debt due after 2033 is given a maturity of 10 years, based upon a perusal of the actual due dates on the long term debt.
Worksheet 2.28 Debt at Granite Construction: July 2021
Due in  Maturity  Amount due  % due 
2023  2  $200  60.42% 
2027  6  $131  39.58% 
Weighted Average  4  $331 
To convert the book value of debt to market value, we use the current pretax cost of debt for Granite Construction of 5.58 percent as the discount rate, the face value of debt ($399 million) in July 2021 as the book value of debt and the current year's interest expenses of $24 million as the coupon payment:
Estimated MV of Granite Construction Debt = 24 * [(1  (1 / 1.0558)^{5.38}) / .0558] + 399 / (1.0558)^{5.38} = $357
To this amount, we add the present value of Granite Construction's operating lease commitments of $42 million.
Adding the $42 million debt value of operating leases to the market value of debt of $357 million yields a total market value for debt of $399 million at Granite Construction.
For Granite Construction, the market value debt ratio of 22.74 percent is lower than the book value debt ratio of 42.41 percent.
Estimating and Using the Cost of Capital
With the estimates of the costs of the individual piecesdebt, equity and preferred stock (if any)and the market value weights of each of the pieces, the cost of capital can be calculated. So if E, D, and PS are the market values of equity, debt, and preferred stock in order, the cost of capital can be shown as follows:
Cost of Capital = k_{E} [E/(D + E + PS)] + k_{D} [D/(D + E + PS)] + k_{PS} [PS/(D + E + PS)]
The cost of capital is a gauge of the combined cost of raising money that a company faces. It will usually be less than the cost of equity, which is the cost of just equity funding.
It is a point of bewilderment to many analysts that both the cost of equity and the cost of capital are used as hurdle rates in investment analysis. The best way to disentangle this bewilderment is to understand when it is right to use one of the other.
 If we want to look at only the equity investors in a business or a project and figure out the returns earned just by these investors on their investment, the cost of equity is the correct hurdle rate to use. In measuring the returns to equity investors then, we have to look at only the income or cash flows left over after all other claimholders needs (interest payments on debt and preferred dividends, for instance) have been met.
 If the returns that we are measuring are composite returns to all claimholders, based on earnings before payments to debt and preferred stockholders, we should be looking at the cost of capital.
Although these principles are abstract, we will consider them later when we look at examples of projects.
Example 2.18: Estimating Cost of Capital
Completing the analysis in this section, we first determine the costs of capital for each of Granite Construction's divisions. In making these estimates, we use the costs of equity that we figured out at each division level in Example 2.13 and Granite Construction's cost of debt from Example 4.14. We also assume that all of the divisions are funded with the same mix of debt and equity as the parent company. Worksheet 2.31 provides estimates of the costs of capital for the divisions:
Worksheet 2.31 Cost of Capital for Granite Construction's Divisions
Business  Cost of Equity  Aftertax cost of debt  E/(D+E)  D/(D+E)  Cost of capital 
Specialty  4.57%  11.04%  75.00%  14.54%  5.51% 
Construction & Engineering Services  4.42%  11.04%  64.68%  9.54%  5.05% 
Materials  4.27%  11.04%  68.64%  3.85%  4.54% 
Water  4.21%  11.04%  80.84%  1.18%  4.29% 
Granite Construction  4.30%  11.04%  73.04%  4.85%  4.63% 
The cost of capital for Granite Construction's operating assets is 4.63 percent, but the costs of capital vary across divisions with a low of 4.29 percent for the waters division to a high or 5.51 percent for Specialty division.
^{1}By "wellbehaved term structures", I would include a normal upwardly sloping yield curve, where long term rates are at most 23 percent higher than shortterm rates.
^{2} In investment analysis, where we look at projects, these durations are usually between three and ten years. In valuation, the durations tend to be much longer, because firms are assumed to have infinite lives. The duration in these cases is often well in excess of ten years and increases with the expected growth potential of the firm.
^{3} We used the average of the analyst estimates for individual firms (bottomup). Alternatively, we could have used the topdown estimate for the S&P 500 earnings.
^{4} With weekly or daily return regressions, the riskfree rate (weekly or daily) is close to zero. Consequently, many services estimate betas using raw returns rather than excess returns.
^{5} The nontrading bias arises because the returns in nontrading periods is zero (even though the market may have moved up or down significantly in those periods). Using these nontrading period returns in the regression will reduce the correlation between stock returns and market returns and the beta of the stock.
^{6} The bias can also be reduced using statistical techniques.
^{7} The regression statistics are computed in the conventional way.
^{8} In practice, the intercept of the regression is often called the alpha and compared to zero. Thus a positive intercept is viewed as a sign that the stock did better than expected and a negative intercept as a sign that the stock did worse than expected. In truth, this can be done only if the regression is run in terms of excess returns, that is, returns over and above the riskfree rate in each month for both the stock and the market index.
^{9} To see why, compare two firms with revenues of $100 million and operating income of $10 million, but assume that the first firm's costs are all fixed, whereas only half of the second firm's costs are fixed. If revenues increase at both firms by $10 million, the first firm will report a doubling of operating income (from $10 to $20 million), whereas the second firm will report a rise of 55 percent in its operating income (because costs will rise by $4.5 million, 45 percent of the revenue increment).
^{10} To compute this statistic, we looked at the aggregate revenues and operating income of Engineering/Construction companies each year from 2014 to 2021.
^{11} Interest expenses always lower net income, but the fact that the firm uses debt instead of equity implies that the number of shares will also be lower. Thus, the benefit of debt shows up in earnings per share.
^{12} If we ignore the tax effects, we can compute the levered beta as b _{L} = b_{u} (1 + D/E). If debt has market risk (i.e., its beta is greater than zero), the original formula can be modified to take it into account. If the beta of debt is b_{D}, the beta of equity can be written as β_{L} = β_{u} (1 + (1  t)(D/E))  β_{D} (1  t) D/E.
^{13} The marginal federal corporate tax rate in the United States in 2021 was 20.9 percent. The marginal state and local tax rates, corrected for federal tax savings, is estimated by Granite Construction in its annual report to be 4% percent.
^{14} The exception is when you have tracking stocks with each division traded separately in financial markets.
^{15} We used a 25% marginal tax rate for the comparable firms.
^{16} We first estimated the enterprise value for each firm by adding the market value of equity to the book value of debt and subtracting out cash. We divided the enterprise value by the revenues of each firm to obtain the EV/Sales multiple and then used the median value of these estimates. We did not use the averages of these revenue multiples of the individual firms because a few outliers skewed the results.
^{17} The details of this calculation will be explored later in this chapter.
^{18} Alternate approaches for estimating the beta yielded similar values, with aggregate values for debt, equity and cash generating an unlevered beta of 1.00 for the sector and simple averages for the beta, debt to equity ratio and cash to firm value across the firms provided an estimate of 0.786 for the beta.
^{19} Some analysts use the industry average debt to equity ratios to estimate levered betas by division. The problem with doing this is that the sum total of the debt that they estimate for the divisions may not match up to the actual debt of the company. In the case of Granite Construction, for instance, the dollar debt that we would have obtained with this approach ($352 million) would have greater than the debt owed by the company ($342 million)
^{20} This table was first developed in early 2000, by listing all rated firms with market capitalization lower than $5 billion and their interest coverage ratios, and then sorting firms based on their bond ratings. The ranges were adjusted to eliminate outliers and to prevent overlapping ranges. It has been updated every two years since.
^{21} These default spreads are obtained from an online site, found at https://fred.stlouisfed.org. You can find default spreads for industrial and financial service firms; these spreads are for industrial firms.
^{22} There are some who argue that stock prices are much more volatile than the underlying true value. Even if this argument is justified (and it has not conclusively been shown to be so), the difference between market value and true value is likely to be much smaller than the difference between book value and true value.
^{23}To illustrate this point, assume that the market value debt ratio is
10 percent, and the book value debt ratio is 30 percent, for a firm
with a cost of equity of 15 percent and an aftertax cost of debt of 5
percent. The cost of capital can be calculated as follows:
With market value debt ratios: 15% * (0.9) + 5% X (0.1) = 14%
With book value debt ratios: 15% * (0.7) + 5% * (0.3) = 12%
Excerpts (reworded) from Aswath Damodarn, Corporate Finance 2009.
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