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. Risk-Free 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 risk-free rate. The expected returns on risky investments are then evaluated relative to the risk-free rate, with the risk generating an expected risk premium that is added on to the risk-free rate.

Prerequisites for an Asset to be Risk-Free

We identified a risk-free investment as one for which the investor recognizes the expected returns with certainty. As a result, for an investment to be risk-free, 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 default-free, and the existence of government or sovereign default risk can make it extremely hard to figure out risk-free 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 three-year period and that you want a risk-free rate. A six-month Treasury bill rate, although default-free, cannot be risk-free, because there is the reinvestment risk of not knowing what the bill rate will be in six months. Even a three-year Treasury bond is not risk-free, since the coupons on the bond will be reinvested at rates that cannot be calculated today. The risk-free rate for a three-year time horizon has to be the expected return on a default-free (government) three-year 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 risk-free rates would then involve different risk-free rates for each period and different expected returns. As a sensible concession, however, it is worth recognizing that the present value effect of using risk-free rates that change from year to year tends to be small for most well-behaved term structures.¹ In these cases, we could use a duration matching strategy, where the duration of the default-free security used as the risk-free 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 short-term and long-term rates, it does pay to use year-specific risk-free rates in computing expected returns.

Cash Flows and Risk-Free Rates: The Consistency Principle

The risk-free 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 risk-free 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 risk-free 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 risk-free rate, but the latter would need a peso risk-free 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 risk-free rate. Even though government bills and bonds offer returns that are risk-free in nominal terms, they are not risk-free 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 risk-free rate presents at best only an estimate of the real risk-free rate. In the past, there were hardly any traded default-free securities offered to estimate real risk-free rates; but the innovation of inflation-indexed Treasuries (called TIPs) has filled this void. An inflation-indexed 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. ten-year Treasury bond rate was -1.016 percent, radically lower than the nominal ten-year 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 risk-averse, 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 simply-the 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 risk-free 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 risk-free 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 flow-based rather than dividend-based, 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: 2014-2021

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 2014-2021 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 market-driven and forward-looking 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:

In terms of mechanics, we used analyst estimates of growth rates in earnings and dividends as our projected growth rates and a two-stage 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 dot-com 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:

(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.

The third statistic that emerges from the regression is the R squared (R²) of the regression.

Although the statistical reasoning of the R² is that it suggests a measure of the goodness of fit of the regression, the financial basis for the R² 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 - R²) 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 trade-off 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 non-trading.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 non-trading 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 Constructionj-1} + Dividends_{Granite Constructionj}) / Price_{Granite Constructionj-1}

where Price_{Granite Constructionj} is the price of Granite Construction stock at the end of month j; and Dividends_{Granite Constructionj} 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: 2015-2021

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 risk-free 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 industry-wide 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 firm-specific 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. (Firm-specific 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 firm-specific events. The latter risk should be diversifiable, and for that reason unrewarded. Granite Construction's R² is somewhat higher than the median R² 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 Wal-Mart, 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.

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 year-to-year 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_2015-2020 = 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 High-Growth Firms Have Higher Betas than Larger and More Mature Firms?

Though the answer might appear apparent at first glance-that smaller, higher-growth companies are riskier than larger firms-it is not a simple question to answer. If the question were posed in terms of total risk, smaller and higher-growth 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.

Bottom-Up 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 bottom-up 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 bottom-up 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 bottom-up 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 bottom-up 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:

σ_{Average Beta} = Average σ _Beta / √ Number of firms

• 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 bottom-up 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.

Example 2.7: Bottom-Up 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 third-party 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. Water-related 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 wholly-owned 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 value-weighted 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 cash-adjusted beta values in analyses.¹⁸

Example 2.8: Bottom-Up 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 bottom-up 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 risk-free 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 = Risk-Free Rate + Beta * Expected Risk Premium

where the risk-free rate would be the rate on a long-term 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 bottom-up 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 risk-free 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 well-diversified 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 short-term 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 R² of the regression measures the proportion of the variance that is market risk. Dividing the market beta by the square root of the R² (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 family-run 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 financing-including debt, equity, and hybrid securities-used by a companies to fund their day to day operations.

Default Risk: The risk that a company might not be able to make its required debt payments, such as interest expenses or principal payments.

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 equity-funded 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 year-specific 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 bottom-up 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 pieces-debt, 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

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.

¹By "well-behaved term structures", I would include a normal upwardly sloping yield curve, where long term rates are at most 2-3 percent higher than short-term 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 (bottom-up). Alternatively, we could have used the top-down 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 risk-free 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.

²³To 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 after-tax 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.