** May 2003**

**Estimating
Cost of Capital Using Bottom-up Betas**

*By Nancy L. Beneda, PhD, CPA, University of North Dakota*

Every business must assess where to invest its funds and regularly reevaluate the quality and risk of its existing investments. Investment theory specifies that firms should invest in assets only if they expect them to earn more than their risk-adjusted hurdle rates. Knowing a business’ cost of capital allows a comparison of different ways of financing its operations. For example, increasing the proportion of debt may allow a company to lower its cost of capital and accept more investments.

Knowing the cost of capital also permits a company to determine its value of operations and evaluate the effects of alternative strategies. In value-based management, a business’ current value of operations is calculated as the present value of the expected future free cash flows discounted at the cost of capital. This analysis is useful as a guide in decision making as well as for projecting future financing needs.

The cost of capital is also used in the computation of economic value added (EVA). EVA is useful to the managers of the company as well as to external financial analysts. It is a measure of the economic value created by a company in a single year. EVA is computed by subtracting a capital charge (operating invested capital multiplied by weighted average cost of capital) from after-tax operating income.

Financial theorists agree that using a correct risk-adjusted discount rate is needed to analyze a company’s potential investments and evaluate overall or divisional performance. Risk-adjusted discount rates should incorporate business and operating risk as well as financial risk. Business risk is measured by the nature of the products and services the business provides (discretionary vs. nondiscretionary), the length of the product’s life cycle (shorter life cycles create more risk), and the size of the company (economies of scale can reduce risk). Operating risk is determined by the cost structure of the firm (higher fixed assets relative to sales increases operating risk). Financial risk is determined by a company’s level of debt. For example, industries that exhibit high operating leverage and short life cycles, and have discretionary products such as technology, have very high beta measurements. Borrowing money will only exaggerate the impact of the risk.

**Why Use Bottom-Up Betas? **

Computing the cost of capital for a growth company, however, can be problematic. Changing product mixes, changing cost structures, rapidly changing capital structures, and increasing size are inherent qualities of growth firms. Furthermore, because growing companies typically do not pay dividends, using the constant dividend growth model to compute the cost of equity yields a cost of equity equal to the company’s growth rate. In light of this, there is a need to deal more explicitly with risk when establishing hurdle rates for growth companies. The abundance of information available on the Internet makes computing a risk-adjusted hurdle rate simple. Yet the literature to date has provided little to explain the computation of a risk-adjusted cost of capital using readily available information.

The use of a bottom-up beta in computing the cost of equity component of the cost of capital is an exceptional method of capturing all types of risk. An example using Community Health Systems, Inc., a hospital business, illustrates the procedures. The application presented would be especially useful to investors who hold growth stocks in their portfolios, equity research analysts, venture capitalists, and managers of growth firms.

A bottom-up beta is estimated from the betas of firms in a specified business, thereby addressing problems associated with computing the cost of capital. First, by eliminating the need for historical stock prices to estimate the firm’s beta, the standard error, created by regression betas, is reduced. Second, the problem of a changing product mix is eliminated because the business computes a different cost of capital for each product line. Third, the levered beta is computed from the company’s current financial leverage, rather than from the average leverage over the period of the regression.

Overall, bottom-up betas are designed to be a better measure of the market risk associated with the industry or sector of the business. Because betas measure the risk of a firm relative to a market index, the more sensitive a business is to market conditions, the higher its beta. Bottom-up betas also capture the operating and financial risk of a company. Intuitively, the more financial risk or operating risk a firm has, the higher its beta.

**Developing a Bottom-Up Beta **

Exhibit 1 illustrates the computation of a bottom-up beta for Community Health
Systems, Inc. (CHS). CHS has an average three-year historical sales growth
rate of 25.6%. The August 12, 2002, *Fortune* reported it to be one of
the top 40 companies traded, based on both value and growth indicators. It
went public on June 9, 2000, and does not have a reported beta.

The first step in estimating a bottom-up beta is to identify the business and a set of comparable established companies. Compustat, Value Line, and Hoovers.com, all report companies by industry and sector. Panel A of Exhibit 1 shows a set of eight comparable companies identified by Hoovers.

Second, the reported beta (reported betas are levered) and recent financial statements for each comparable company should be obtained. Value Line and Compustat provide reported betas. Financial statements can be obtained from Hoovers or Compustat. From the financial statements, the marginal tax rate and the debt-to-equity ratio is determined. Using the reported beta, the debt-to-equity ratio, and the tax rate, an unlevered beta is computed (Exhibit 1, Panel A). The computation of an unlevered beta removes the effects of financial leverage of the comparable firms. The unlevered beta for each comparable company is determined using the following:

*Equation 1:*

Bl = Bu x [1 + (1 – tax rate) x (D / E)], or

Bu = Bl / [1 + (1 – tax rate) x (D / E)]

The unlevered beta (Bu) in Equation 1 is the beta of a firm with no debt and is determined by the types of businesses in which it operates and its operating leverage (risk). The degree of operating leverage is a function of a company’s cost structure, and is usually defined in terms of the relationship between fixed costs and total costs. A company that has high operating leverage (high fixed costs relative to total costs) will also have higher variability in earnings before interest and taxes than a company producing a similar product with low operating leverage. Other things being equal, the higher variance in operating income will lead to a higher beta for companies with high operating leverage.

The debt to equity ratio (D/E) in Equation 1 represents the amount of financial leverage or the debt level of the company. Other things being equal, an increase in financial leverage will increase the beta. The obligated payments on debt increase the variance in net income, with higher leverage increasing income during good times and decreasing income during economic downturns. The tax advantage of debt financing is represented in the formula by (1 – tax rate). The higher the tax rate, the more favorable the debt financing.

The third step is to compute a weighted-average unlevered beta of the comparable companies, and from this, using Equation 1, compute the levered beta (BL) for the company being evaluated. The computation of the unlevered beta of the comparable company is weighted according to company size measured as the market value (MV) of equity plus debt. As shown in Panel B of Exhibit 1, the weighted average unlevered beta for the comparable companies is 0.72.

The levered beta (BL) for CHS is then computed using the weighted-average unlevered beta of the comparable companies. The debt-to-equity ratio and tax rate of the company under consideration is used to lever up the unlevered beta in Equation 1. This procedure adjusts the beta for the financial risk and tax benefits associated with the individual firm, project, or division in question. The levered beta (BL) of a firm is a function of its operating leverage, the type of businesses in which it operates, and its financial leverage.

The levered beta computed for CHS is 0.917. The debt-to-MV ratio of 0.42, obtained from CHS’s financial statements, is slightly higher than the debt-to-MV ratios for comparable companies and factored into the computation of CHS’s levered beta of 0.917.

Finally, operating leverage and growth should also be observed for comparable companies. Exhibit 1, Panel C, provides a detailed explanation of the computations and sources of data for the information in Panels A and B. The operating leverage of the entity being analyzed should be compared to the operating leverage of the comparable companies. While operating leverage affects betas, measuring the operating leverage of a company is difficult 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 company by looking at changes in operating income as a function of changes in sales. For companies with high operating leverage, operating income should change more than proportionately when sales change. Operating leverage can be computed using the following formula:

*Equation 2:*

Operating leverage = (% change in operating income) / (% change in sales)

If the company’s project or division being analyzed has a higher (lower) operating leverage than the comparable firms, the unlevered beta should be adjusted upward (downward). CHS’s operating leverage of 1.1 and three-year historical growth rate of 25.6 appear to be fairly consistent with the operating leverage and growth rates of the comparable companies. Thus the computed levered beta for CHS is a good estimate of the company’s market risk with regard to operating leverage and growth.

The unlevered betas appear to be associated with a combination of size and operating leverage. Because the largest companies have the highest operating leverage, it appears that these companies attempt to balance business risk with operating risk by increasing their operating leverage.

The unlevered betas, however, do not appear to correlate with growth rates. The annual growth rates for the most recent three years of the comparable companies range from 2.5% to 42.7%. The five-year annual projected growth rate for 2002 through 2007 of the hospital industry is reported by Yahoo Finance to be 16.3% (finance.yahoo.com; as of September 4, 2002).

Growth companies generally tend to have significant fixed costs associated with setting up infrastructure and developing new products. Once these costs have been incurred, however, the variable costs are relatively low. For growth companies in high-risk industries, such as technology, higher growth leads to higher fixed costs and higher betas. The low betas of companies in the hospital industry indicate a low market risk, even for growth companies. This can be explained by the industry’s nondiscretionary products and longer product life cycles.

**Capital Asset Pricing Model (CAPM) and the Cost of Equity**

Exhibit 2 shows the computation of the cost of equity for CHS using the capital asset pricing model (CAPM), Equation 3, and the bottom-up beta computed in Exhibit 1. The components that go into measuring the cost of equity using the CPM include the riskless rate, the market risk premium, and the beta of the firm, product, or division.

*Equation 3:*

Cost of equity = (riskless rate + beta) x market risk premium

A riskless asset is one in which the investor knows the expected return with certainty. Consequently, there is no default risk and no uncertainty about reinvestment rates. To eliminate uncertainty about reinvestment rates, the maturity of the security should be matched with the length of the evaluation. In practice, using a long-term government rate—which can be obtained from Bondsonline (www.bondsonline.com)—as a riskless rate in all types of analyses will yield a close approximation of the true value.

The market risk premium measures the extra return that would be demanded by investors for shifting their money from a riskless investment to an average-risk investment. It should be a function of how risk-averse the investors are and how risky they perceive stocks and other risky investments to be, in comparison to a riskless investment. The most common approach to estimating the market risk premium is to estimate the historical premium earned by risky investments (stocks) over riskless investments (government bonds). The average historical market risk premium over the period 1926 to 1999 for small companies is 12.1%, as reported by Ibbotson’s.

Exhibit 2 illustrates the computation of the cost of equity for CHS. Using a risk-free rate of 5.5%, a market risk premium of 12.1%, and the bottom-up beta of 0.917, the cost of equity for CHS is estimated to be 16.6%.

**Cost of Debt**

The cost of debt measures the current cost of borrowing funds to finance projects. The cost of debt is measured by the current level of interest rates, the default risk of the company, and the tax advantage associated with debt.

*Equation 4:*

Pre-tax cost of debt = Treasury bond rate + default spread of company’s debt

The default spread is the difference between the long-term Treasury bond rate and the company’s bond yield. Default spreads can be found at Bondsonline if the company has a bond rating. Bond ratings can be found at www.standardandpoors.com. For companies that are not rated (CHS is not rated by Standard & Poor’s), the rating may be obtained by computing the company’s interest coverage ratio and adjusting for industry standards or expected future interest coverage. The interest coverage ratio for CHS is 2, and the associated ratings for the comparable companies indicate a bond rating of B+ for CHS. The default spread for this rating from Bondsonline is 8.5%. Exhibit 3 illustrates the computation of the cost of debt for CHS, from Equation 4, as 14%.

**Cost of Capital**

The estimated cost of capital should be based on the market values of a company’s debt and equity, since a company has to earn more than its market value cost of capital to generate value. From a practical standpoint, using the book value cost of capital will tend to understate the cost for most companies, especially highly levered companies. These companies have more equity in their capital structures, and equity is more likely to have a higher market value than book value.

*Equation 5:*

Cost of Capital = ke [E / (D+E)] + kd [D / (D+E)]

The market value of equity (E) for CHS is $2,354 million, calculated as the number of shares outstanding times the stock price as of December 31, 2001. The stock price can easily be obtained from Yahoo Finance, and the number of shares outstanding is reported on the financial statements, which can be obtained from either Hoovers or Compustat. Generally the book value of debt is an adequate proxy for the market value unless interest rates have changed drastically. Exhibit 4 illustrates the computation of the cost of capital for CHS, which is 14.40%.

Robert H. Colson, PhD, CPA

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