Cost of Equity and the Capital Asset Pricing Model

The cost of equity (k_e) is the rate of return required to induce investors to purchase a firm’s equity. It is a return to shareholders after corporate taxes have been paid but before personal taxes. It may be estimated using the capital asset pricing model (CAPM), which measures the relationship between expected risk and return. Presuming investors require higher rates of return for accepting higher levels of risk, the CAPM states that the expected return on an asset is equal to a risk-free rate of return plus a risk premium.

A risk-free rate of return is one for which the expected return is free of default risk.⁸ Other types of risk remain such as the reinvestment rate (i.e., the rate of return that can be earned at the end of the investor’s holding period), the potential loss of principal if the security is sold before its maturity date (market risk) and the loss of purchasing power due to inflation (inflation risk). Despite widespread agreement on the use of US Treasury securities as assets that are free of default risk, analysts differ over whether a short- or long-term Treasury rate should be applied. Which rate should be used depends on how long the investor intends to hold the investment. The investor who anticipates holding an investment for 5 or 10 years should use either a 5- or 10-year Treasury bond rate.⁹ In this book, a 10-year Treasury bond rate is used as the risk-free rate, since it is most appropriate for a strategic or long-term acquirer.

Estimating Market Risk Premiums

The market risk, or equity premium, is the additional return in excess of the risk-free rate that investors require to purchase a firm’s equity. Intuitively, the risk premium is what the investor demands as compensation for buying a risky asset and is the only factor the basic CAPM model uses to approximate the incremental risk of adding a stock to a diversified portfolio. While the risk premium should be forward looking, obtaining precise estimates of future market returns is exceedingly difficult. Analysts often look to historical data, despite results that vary based on the time periods selected and whether returns are calculated as arithmetic or geometric averages. CAPM relates the cost of equity (k_e) to the risk-free rate of return and market risk premium as follows:

$\mathrm{CAPM}:k_e=R_f+\beta \left(R_m-R_f\right)$

where

• R_f = risk-free rate of return
• β = beta (see “Risk Assessment” section)¹⁰
• R_m = expected rate of return on equities
• R_m − R_f = 5.5% (i.e., risk premium equal to the difference between the return on a diversified portfolio of stocks and the risk-free rate)¹¹

Despite its intuitive appeal, studies show that actual returns on risky assets frequently differ significantly from those returns predicted by basic CAPM.¹² Since the CAPM measures a stock’s risk relative to the overall market and ignores returns on assets other than stocks, some analysts use multifactor models.¹³ Studies show that, of those variables improving the CAPM’s accuracy, firm size tends to be among the more important. The size premium serves as a proxy for factors such as smaller firms being subject to higher default risk and generally being less liquid than large-capitalization firms. Table 7.1 provides estimates of the adjustment to the cost of equity to correct for firm size based on actual data since 1963.¹⁴

Eq. (7.1) can be rewritten to reflect an adjustment for firm size as follows:

$\mathrm{CAPM}:k_e=R_f+\beta \left(R_m-R_f\right)+\mathrm{FSP}$

where FSP = firm size premium.

Applying CAPM in a Near Zero (or Negative) Interest Rate Environment

For more than a decade (as of this writing), interest rates in many developed countries have been at or near historical lows (and in some instances negative). Since central bank policies in many countries have been to inject massive amounts of liquidity into financial markets during this period, it is unclear whether interest rates have been more reflective of anemic global economic growth or central bank policies or both.

One school of thought argues that the low interest rate environment is more a result of weak financial market conditions due to expectations of lackluster economic growth as central banks exert only limited control of short-term interest rates and little influence on long-term interest rates.¹⁵ Others argue that the continuing low interest rate environment reflects the highly aggressive liquidity injections by central banks during this period.¹⁶

If we believe low interest rates reflect primarily financial market factors such as anemic or uncertain global economic growth, historically low or negative risk-free rates do not bias target firm valuations. Why? Historically low risk-free rates will be offset partially by increasing risk premiums on stocks keeping the magnitude of the cost of equity comparatively stable. If the expected return on stocks remains the same, equity premiums must widen by definition as risk free rates decline.¹⁷ To illustrate how changes in the risk-free rate are offset partly by changes in the risk premium, consider the following calculations. Assuming the risk-free rate is -.5%, the expected return on all stocks is 6%, and the target firm’s beta is 1.2, the cost of equity using CAPM is 7.3% [i.e., cost of equity = − 0.005 + 1.2(0.06 − (− 0.005)]. The cost of equity is little changed at 7.1% [i.e., cost of equity = 0.005 + 1.2(0.06 − 0.005)] if we use the same assumptions except for a positive 0.5% risk-free rate. This model is the same as the basic two-factor CAPM.

If we believe that interest rates are depressed by central bank policies and do not reflect financial market conditions, application of the CAPM can result in an underestimate of the cost of equity and target firm overvaluation. To minimize this bias, the risk-free rate should be the expected future risk-free rate, as the CAPM is predicated on future cash flows and their associated risk. However, any estimation of future interest rates is highly problematic. A more practical alternative is to use an historical average of an appropriate long-term risk-free rate over several decades in order to capture more than one interest rate cycle. Nonetheless, using an historical average as a proxy for future rates is subjective due to the difficulty determining the appropriate length of the historical time period and may overstate the cost of equity.¹⁸ Many European respondents to a recent survey indicated that they used a risk-free rate higher than their country’s 10 year government bond rate, with their estimates varying between 1.5% and 3%.¹⁹

In summary, current low interest rates reflect most likely both sluggish growth and central bank policies. It is doubtful the relative importance of each factor can be determined because they are interdependent. That is, many central banks continue to aggressively inject liquidity into the financial markets because of slow economic growth and growth is slow in part because of these bank policies.²⁰ Without compelling evidence that adjusting CAPM for the current low interest rates provides more reliable estimates, the author recommends using the basic two-factor CAPM, adjusted for firm size if such data are available, to estimate the cost of equity.

Pretax Cost of Debt

Interest is the cost of borrowing and is tax deductible by the firm; in bankruptcy, bondholders are paid before shareholders as the firm’s assets are liquidated. For these reasons, debt is generally cheaper than equity. Default risk is the likelihood the firm will fail to repay interest and principal when required. Interest paid by the firm on its current debt can be used as an estimate of the current cost of debt if nothing has changed since the firm last borrowed.

When conditions have changed, the analyst must estimate the cost of debt reflecting current market interest rates and default risk. To do so, analysts use the yield to maturity (YTM)²¹ of the company’s long-term, option-free bonds. This requires knowing the price of the security and its coupon value and face value.²² In general, the cost of debt is estimated by calculating the YTM on each of the firm’s outstanding bond issues. We then compute a weighted average YTM, with the estimated YTM for each issue weighted by its percentage of total debt outstanding. In Table 7.2, Microsoft’s weighted average YTM on the bulk of its long-term debt on January 24, 2011, was 2.4%. The source for the YTM for each debt issue was found in the Financial Industry Regulatory Authority’s (FINRA) Trace database: www.finra.org/marketdata.²³

YTM represents the most reliable estimate of a firm’s cost of debt as long as the firm’s debt is investment grade,²⁴ since the difference between the expected rate of return and the promised rate of return is small. The promised rate of return assumes that the interest and principal are paid on time. The yield to maturity is a good proxy for actual future returns on investment-grade debt, since the potential for default is low.

Non-investment-grade debt (rated less than BBB by Standard & Poor’s and Baa by Moody’s) represents debt whose default risk is significant due to the firm’s leverage, deteriorating cash flows, or both. Ideally, the expected yield to maturity would be calculated based on the current market price of the non-investment-grade bond, the probability of default, and the potential recovery rate following default.²⁵ Since such data are frequently unavailable, an alternative is to use the average YTM for a number of similarly rated bonds of other firms. Such bonds include a so-called default premium, which reflects the compensation that lenders require over the risk-free rate to buy non-investment-grade debt. For nonrated firms, the analyst could use the cost of debt for rated firms whose debt-to-equity ratios, interest coverage ratios, and operating margins are similar to those of the nonrated firm.²⁶

Cost of Preferred Stock

Preferred stock is similar to long-term debt, in that its dividend is generally constant and preferred stockholders are paid after debt holders but before common shareholders if the firm is liquidated. Because preferred stock is riskier than debt but less risky than common stock in bankruptcy, the cost to the company to issue preferred stock should be less than the cost of equity but greater than the cost of debt. The cost of preferred stock should also exceed the cost of debt because with debt investors are certain to receive the principal of the bond if they hold such bonds until maturity. In contrast, the likelihood the investor will recover their initial investment with preferred stock issued without any redemption date is uncertain.

Viewing preferred dividends as paid in perpetuity, the cost of preferred stock (k_pr) can be calculated as dividends per share of preferred stock (d_pr) divided by the market value of the preferred stock (PR) (see “The Zero-Growth Valuation Model” section). Consequently, if a firm pays a $2 dividend on its preferred stock, whose current market value is $50, the firm’s cost of preferred stock is 4% (i.e., $2 ÷ $50). The cost of preferred stock can be generalized as follows:

$k_{\mathrm{pr}}=\frac{d_{\mathrm{pr}}}{\mathrm{PR}}$

Cost of Capital

The weighted average cost of capital (WACC) is the broadest measure of the firm’s cost of funds and represents the return that a firm must earn to induce investors to buy its common stock, preferred stock, and bonds. The WACC²⁷ is calculated using a weighted average of the firm’s cost of equity (k_e), cost of preferred stock (k_pr), and pretax cost of debt (i):

$\mathrm{WACC}=k_e\frac{E}{D+E+\mathrm{PR}}+i\left(1-t\right)\frac{D}{D+E+PR}+k_{\mathrm{pr}}\frac{\mathrm{PR}}{D+E+\mathrm{PR}}$

where

• E = the market value of common equity
• D = the market value of debt
• PR = the market value of preferred stock
• t = the firm’s marginal tax rate

A portion of interest paid on borrowed funds is recoverable by the firm because of the tax deductibility of interest. For every dollar of taxable income, the tax owed is equal to $1 multiplied by t. Since each dollar of interest expense reduces taxable income by an equivalent amount, the actual cost of borrowing is reduced by (1 − t). Therefore, the after-tax cost of borrowed funds to the firm is estimated by multiplying the pretax interest rate, i, by (1 − t).

Note that the weights [E/(D + E + PR)], [D/(D + E + PR)], and [PR/(D + E + PR)] associated with the cost of equity, preferred stock, and debt, respectively, reflect the firm’s target capital structure or capitalization. These are targets since they represent the capital structure the firm hopes to achieve and sustain in the future. The actual market value of equity, preferred stock, and debt as a percentage of total capital (i.e., D + E + PR) may differ from the target. Market values rather than book values are used because the WACC measures the cost of issuing debt, preferred stock, and equity securities, which are issued at market and not book value. The use of the target capital structure avoids the circular reasoning associated with using the current market value of equity to construct the weighted average cost of capital, which is subsequently used to estimate the firm’s current market value. Non-interest-bearing liabilities, such as accounts payable, often are excluded from the estimation of the cost of capital for the firm to simplify the calculation of WACC.²⁸ Estimates of industry betas, cost of equity, and WACC are provided by firms such as Ibbotson Associates, Value Line, Standard & Poor’s, and Bloomberg.

Cost of Capital With Limited Interest Deductibility

Among other things, the Tax Cuts and Jobs Act of 2017 reduced the top US corporate tax rate from 35% to 21% beginning in 2018. The marginal corporate rate for US firms becomes 26%, consisting of the 21% maximum federal rate plus an average 5% state and local tax rate. Furthermore, net interest expense deductions are capped at 30% of earnings before interest, taxes, depreciation, and amortization (EBITDA) through 2022 and earnings before interest and taxes (EBIT) thereafter. To incorporate the capping of the tax deductibility of net interest expense into the calculation of WACC, separate total debt into that portion whose interest is tax deductible and that portion whose interest is not.²⁹ The cost of equity, preferred stock, and debt are denoted by k_e, k_pr, and i, respectively. Eq. (7.4) can be rewritten as follows:

$\begin{array}{c}\mathrm{WACC}=k_e\left\{E/\left(D_1+D_2+E+\mathrm{PR}\right)\right\}+i\left(1-t\right)\left\{D_1/\left(D_1+D_2+E+\mathrm{PR}\right)\right\}+i\left\{D_2/\left(D_1+D_2+E+\mathrm{PR}\right)\right\}\\ +k_{\mathrm{pr}}\left\{\mathrm{PR}/\left(D_1+D_2+E+\mathrm{PR}\right)\right\}\end{array}$

where

• E = the market value of common equity
• D₁ = the market value of debt whose interest is tax deductible
• D₂ = the market value of debt whose interest is not tax deductible
• PR = the market value of preferred stock
• t = the firm’s marginal tax rate

Note that if the firm’s net interest expense is expected to be less than or equal to 30% of EBIT, all interest expense is tax deductible and D₂ in Eq. (7.5) would be zero. EBIT rather than EBITDA is the metric to use to measure the portion of net interest expense that would be tax deductible, since WACC is a long-term concept and EBIT will be the measure required by law after 2022.

Illustrations in the remainder of this chapter use a marginal tax rate of 40% (reflecting the pre-2017 federal corporate tax rate of 35% plus a 5% state and local tax rate), as the magnitude of the cap on the deductibility of interest could fluctuate periodically with changes in the political parties in power. Changes in the cap do not change the methodology for calculating WACC shown in Eq. (7.5).

Effects of Financial and Operating Leverage on Beta

Leverage represents the additional increase (decrease) in profit once a firm’s revenue exceeds (falls short of) its fixed expenses and the variable expenses incurred to realize the incremental revenue.³¹ Fixed expenses related to a firm’s operating activities do not vary with output and include depreciation, rent, and obligations such as employee and vendor contracts that do not vary with production. Fixed expenses related to a firm’s financing activities include interest and lease expenses and principal repayments on debt. Operating leverage refers to the way in which a firm combines fixed and variable expenses often measured by the ratio of a firm’s fixed expenses to total cost of sales. Financial leverage describes the way in which a firm combines debt and equity to finance its operations often measured by the ratio of its debt to equity.

In the absence of debt, the β is called an unlevered β, denoted β_u. β_u is determined by the firm’s operating leverage and by the type of industry in which the firm operates (e.g., cyclical or noncyclical). If a firm borrows, the unlevered beta must be adjusted to reflect the risk that the firm may not be able to repay the debt when due. By borrowing, the firm is able to invest more in its operation without increasing equity, resulting in a proportionately larger return to equity holders. The resulting beta is called a leveraged or levered β, denoted β_l. Both operating and financial leverage increase the volatility of a firm’s profits and financial returns.

Table 7.3 illustrates the effects of operating leverage on financial returns. The three cases reflect the same level of fixed expenses but varying levels of revenue and the resulting impact on financial returns. The illustration assumes in Case 1 that the firm’s total cost of sales is 80% of revenue and that fixed expenses comprise 60% of the total cost of sales. Note the volatility of the firm’s return on equity resulting from fluctuations of 25% in the firm’s revenue in Cases 2 and 3. There is evidence that the degree of operating leverage can impact the timing of acquisition decisions.³²

Table 7.4 shows how financial leverage increases the volatility of a firm’s financial returns.³³ This is because equity’s share of total capital declines faster than the decline in net income as debt’s share of total capital increases. The three cases in the table reflect varying levels of debt but the same earnings before interest and taxes. Between Case 1 and Case 3, net income declines by one-fourth and equity declines by one-half, magnifying the impact on returns.

If a firm’s stockholders bear all the risk from operating and financial leverage and interest paid on debt is tax deductible, then leveraged and unleveraged betas can be calculated as follows for a firm whose debt-to-equity ratio is denoted by D/E:

${\beta }_{\mathrm{l}}={\beta }_u\left[1+\left(1-t\right)\left(D/E\right)\right]$

and

${\beta }_u={\beta }_{\mathrm{l}}/\left[1+\left(1-t\right)\left(D/E\right)\right]$

Shareholders view risk as the potential for a firm not to earn sufficient future cash flow to satisfy their minimum required returns. Eq. (7.6) implies that increases in a firm’s leverage, denoted by D/E, will increase risk, as measured by the firm’s levered beta because the firm’s interest payments represent fixed expenses that must be paid before payments can be made to shareholders. This increased risk is offset somewhat by the tax deductibility of interest, which increases after-tax cash flow available for shareholders. Thus, the levered beta will, unless offset by other factors, increase with an increase in leverage and decrease with an increase in tax rates.

In summary, β_u is determined by the characteristics of the industry in which the firm competes and the firm’s degree of operating leverage. The value of β_l is determined by the same factors and the degree of the firm’s financial leverage. Our objective is to estimate a beta reflecting the relationship between future risk and return. Estimating beta using historical data assumes the historical relationship will hold in the future, which often is not the case.

An alternative to using historical data is to estimate beta using a sample of similar firms and applying Eqs. (7.6) and (7.7). Referred to as the “bottoms-up” approach, this three-step process suggests that the target firm’s beta reflects the business risk (cyclicality and operating leverage only) of the average firm in the industry better than its own historical risk/return relationship. Step 1 requires selecting firms with similar cyclicality and operating leverage (i.e., firms usually in the same industry). Step 2 involves calculating the average unlevered beta for firms in the sample to eliminate the effects of their current financial leverage on their betas. Finally, in step 3, we relever the average unlevered beta using the debt-to-equity ratio and the marginal tax rate of the target firm to reflect its capital structure and tax rate.

Network equipment and storage company Brocade Communications Systems’ beta estimated using historical data is 0.88 and its current debt-to-equity ratio is 0.256. Assume analysts believe that the firm’s levered beta estimated in this manner is too low. Using a representative data networking and storage industry sample, the firm’s levered beta is estimated to be 1.50 using the “bottoms-up” methodology (Table 7.5).

Using Eqs. (7.6) and (7.7), the effects of different amounts of leverage on the cost of equity also can be estimated.³⁴ The process is as follows:

1. 1. Determine a firm’s current equity β* and (D/E)*;
2. 2. Estimate the unlevered beta to eliminate the effects of the firm’s current capital structure:

${\beta }_u={\beta }^{\ast }/\left[1+\left(1-t\right)\left(D/E\right)\ast \right]$

1. 3. Estimate the firm’s levered beta: β_l = β_u [1 + (1 − t)(D/E)**];
2. 4. Estimate the firm’s cost of equity for the new levered beta,

where β* and (D/E)* represent the firm’s current beta and the market value of the firm’s debt-to-equity ratio before additional borrowing takes place. (D/E)** is the firm’s debt-to-equity ratio after additional borrowing occurs, and t is the firm’s marginal tax rate.

In an acquisition, an acquirer may anticipate increasing the target firm’s debt level after the closing. To determine the impact on the target’s beta of the increased leverage, the target’s levered beta, which reflects its preacquisition leverage, must be converted to an unlevered beta, reflecting the target firm’s operating leverage and the cyclicality of the industry in which the firm competes. To measure the increasing risk associated with new borrowing, the resulting unlevered beta is then used to estimate the levered beta for the target firm (see Exhibit 7.1).

Free Cash Flow to the Firm (Enterprise Cash Flow)

Free cash flow to the firm represents cash available to satisfy all investors holding claims against the firm’s resources. Claims holders include common stockholders, lenders, and preferred stockholders. Consequently, enterprise cash flow is calculated before the sources of financing are determined and, as such, is not affected by the firm’s financial structure.³⁵

FCFF can be calculated by adjusting operating earnings before interest and taxes (EBIT) as follows:

$\begin{array}{c}\mathrm{FCFF}=\mathrm{EBIT}\left(1-\mathrm{Tax}\mathrm{Rate}\right)+\mathrm{Depreciation\; and\; Amortization}\\ -\mathrm{Gross\; Capital\; Expenditures}-\Delta \mathrm{Net}\mathrm{Working\; Capital}\end{array}$

Only cash flow from operating and investment activities, but not from financing activities, is included. Why? Because this represents the cash flow available to compensate all those providing funds to the firm. The tax rate refers to the firm’s marginal tax rate. Net working capital is defined as current operating assets (excluding cash balances in excess of the amount required to meet normal operating requirements) less current operating liabilities.³⁶ Depreciation and amortization expenses are not actual cash outlays and are added to operating income in calculating cash flow.

Selecting the Right Tax Rate

The appropriate tax rate is either the firm’s marginal rate (i.e., the rate paid on each additional dollar of earnings) or its effective tax rate (i.e., taxes due divided by taxable income). The effective rate is usually less than the marginal rate due to the use of tax credits to reduce actual taxes paid or accelerated depreciation to defer tax payments. Once tax credits have been used and the ability to further defer taxes has been exhausted, the effective rate can exceed the marginal rate in the future. Effective rates lower than the marginal rate may be used in the early years of cash flow projections, if the current favorable tax treatment is likely to continue into the foreseeable future, and eventually the effective rates may be increased to the firm’s marginal tax rate. It is critical to use the marginal rate in calculating after-tax operating income in perpetuity. Otherwise, the implicit assumption is that taxes can be deferred indefinitely.

Dealing With Operating Leases

Beginning in 2019 the Financial Accounting Standards Board will require any firms paying to lease real estate, office equipment, aircraft, or similar items to show such leases on the balance sheet. Leasing is likely to get a boost from the 2017 Tax Cuts and Jobs Act which caps interest expense. Consequently, equipment that might have otherwise been purchased will now be leased.

Future lease payments should be discounted to the present at the firm’s pretax cost of debt (i), since leasing equipment represents an alternative to borrowing, and the present value of the operating lease (PV_OL) should be included in the firm’s total debt outstanding. Once operating leases are converted to debt, operating lease expense (OLE_EXP) must be added to EBIT, because it is a financial expense and EBIT represents operating income before such expenses. Lease payments include both an interest expense component (to reflect the cost of borrowing) and a depreciation component (to reflect the anticipated decline in the value of the leased asset).

An estimate of depreciation expense associated with the leased asset (DEP_OL) then must be deducted from EBIT, as is depreciation expense associated with other fixed assets owned by the firm, to calculate an “adjusted” EBIT (EBIT_ADJ). DEP_OL may be estimated by dividing the firm’s gross plant and equipment by its annual depreciation expense. Studies show that the median asset life for leased equipment is 10.9 years.³⁷ The EBIT_ADJ then is used to calculate free cash flow to the firm. EBIT may be adjusted as follows:

${\mathrm{EBIT}}_{\mathrm{ADJ}}=\mathrm{EBIT}+{\mathrm{OLE}}_{\mathrm{EXP}}-{\mathrm{DEP}}_{\mathrm{OL}}$

If EBIT, OLE_EXP, PV_OL, and the useful life of the leased equipment are $15 million, $2 million, $30 million, and 10 years, respectively, EBIT_ADJ equals $14 million [i.e., $15 + $2 − ($30/10)].

Free Cash Flow to Equity Investors (Equity Cash Flow)

Free cash flow to equity investors is the cash flow remaining for returning cash through dividends or share repurchases to current common equity investors or for reinvesting in the firm after the firm satisfies all obligations. These obligations include debt payments, capital expenditures, changes in net working capital, and preferred dividend payments. FCFE can be defined as follows:

$\begin{array}{c}\mathrm{FCFE}=\mathrm{Net}\mathrm{Income}+\mathrm{Depreciation\; and\; Amortization}-\mathrm{Gross\; Capital\; Expenditures}\\ -\Delta \mathrm{Net}\mathrm{Working\; Capital}+\mathrm{New}\mathrm{Debt\; and\; Preferred\; Equity\; Issues}\\ -\mathrm{Principal\; Repayments}-\mathrm{Preferred\; Dividends}\end{array}$

Exhibit 7.2 summarizes the key elements of enterprise cash flow, Eq. (7.8), and equity cash flow, Eq. (7.10). Note that equity cash flow reflects operating, investment, and financing activities, whereas enterprise cash flow excludes cash flow from financing activities.

Enterprise Discounted Cash Flow Model (Enterprise or FCFF Method)

The enterprise valuation method, or FCFF, approach discounts the after-tax free cash flow available to the firm from operations at the weighted average cost of capital to obtain the estimated enterprise value. The firm’s enterprise value (often referred to as firm value) reflects the market value of the entire business. It can be viewed as a theoretical takeover price. That is, it represents the sum of investor claims on the firm’s cash flows from all those holding securities, including long-term debt, preferred stock, common shareholders, and non-controlling shareholders. Since it reflects all claims, it is a much more accurate estimate of a firm’s takeover value than simply the market value of a firm’s equity. For example, in addition to buying a target firm’s equity, an acquirer would generally have to assume responsibility for paying off the target firm’s debt and preferred stock.

Since the enterprise DCF model estimates the present value of cash flows, the enterprise value also can be estimated as the market value of the firm’s common equity plus long-term debt, preferred stock, and non-controlling interest less cash and cash equivalents. The firm’s common equity value can be determined by subtracting the market value of the firm’s debt and other investor claims on cash flow, such as preferred stock and non-controlling interest, from the enterprise value.⁴⁰ The enterprise method is used when information about the firm’s debt repayment schedules or interest expense is limited.

Equity Discounted Cash Flow Model (Equity or FCFE Method)

The equity valuation, or FCFE, approach, discounts the after-tax cash flows available to the firm’s shareholders at the cost of equity. This approach is more direct than the enterprise method when the objective is to value the firm’s equity. The enterprise, or FCFF, method and the equity, or FCFE, method are illustrated in the following sections of this chapter using three cash flow growth scenarios: zero-growth, constant-growth, and variable-growth rates.

The Zero-Growth Valuation Model

This model assumes that free cash flow is constant in perpetuity. The value of the firm at time zero (P₀) is the discounted or capitalized value of its annual cash flow.⁴¹ The subscript FCFF or FCFE refers to the definition of cash flow used in the valuation.

${\mathrm{P}}_{0,\mathrm{FCFF}}={\mathrm{FCFF}}_0/\mathrm{WACC}$

where FCFF₀ is free cash flow to the firm at time 0 and WACC is the cost of capital.

${\mathrm{P}}_{0,\mathrm{FCFE}}={\mathrm{FCFE}}_0/k_e$

where FCFE₀ is free cash flow to common equity at time 0 and k_e is the cost of equity.

While simplistic, the zero-growth method has the advantage of being easily understood by all parties to the deal. There is little evidence that more complex methods provide consistently better valuation estimates, due to their greater requirement for more inputs and assumptions. This method often is used to value commercial real estate transactions and small, privately owned businesses (Exhibit 7.3).

The Constant-Growth Valuation Model

The constant-growth model is applicable for firms in mature markets, characterized by a somewhat predictable rate of growth. Examples include beverages, cosmetics, personal care products, prepared foods, and cleaning products. To project growth rates, extrapolate the industry’s growth rate over the past 5–10 years. The constant-growth model assumes that cash flow grows at a constant rate, g, which is less than the required return, k_e. The assumption that k_e is greater than g is a necessary mathematical condition for deriving the model. In this model, next year’s cash flow to the firm (FCFF₁), or the first year of the forecast period, is expected to grow at the constant rate of growth, g.⁴² Therefore, FCFF₁ = FCFF₀ (1 + g):

$P_{0,\mathrm{FCFF}}={\mathrm{FCFF}}_1/\left(\mathrm{WACC}-g\right)$

$P_{0,\mathrm{FCFE}}={\mathrm{FCFE}}_1/\left(k_e-g\right)$

where FCFE₁ = FCFE₀ (1 + g)

This simple valuation model also provides a means of estimating the risk premium component of the cost of equity as an alternative to relying on historical information, as is done in the capital asset-pricing model. This model was developed originally to estimate the value of stocks in the current period (P₀) using the level of expected dividends (d₁) in the next period. This model estimates the present value of dividends growing at a constant rate forever. Assuming the stock market values stocks correctly and that we know P₀, d₁, and g, we can estimate k_e. Therefore,

$P_0=d_1/\left(k_e-g\right)\mathrm{and}{k}_e=\left(d_1/P_0\right)+g$

For example, if d₁ is $1, g is 10%, and P₀ = $10, then k_e is 20%. See Exhibit 7.4 for an illustration of how to apply the constant-growth model.

The Variable-Growth (Supernormal or Nonconstant) Valuation Model

Many firms experience periods of high growth followed by a period of slower, more stable growth. Examples include cellular phone, personal computer, and cable TV firms. Such firms experience double-digit growth rates for periods of 5–10 years because of low penetration early in the product’s life cycle. As the market becomes saturated, growth slows to a rate more in line with the overall growth of the economy or the general population. The PV of such firms is equal to the sum of the PV of the discounted cash flows during the high-growth period plus the discounted value of the cash flows generated during the stable-growth period. The discounted value of the cash flows generated during the stable-growth period is often called the terminal, sustainable, horizon, or continuing-growth value.

The terminal value may be estimated using the constant-growth model.⁴³ Free cash flow during the first year beyond the nth or final year of the forecast period, FCFF_n + 1, is divided by the difference between the assumed cost of capital and the expected cash flow growth rate beyond the nth-year forecast period. The terminal value is the PV in the nth year of all future cash flows beyond the nth year. To convert the terminal value to its value in the current year, use the discount rate employed to convert the nth-year value to a present value. Small changes in assumptions can result in dramatic swings in the terminal value and in the valuation of the firm. Table 7.6 illustrates the sensitivity of a terminal value of $1 million to different spreads between the cost of capital and the stable growth rate. Note that, using the constant-growth model formula, the terminal value declines dramatically as the spread between the cost of capital and expected stable growth for cash flow increases by 1 percentage point.⁴⁴

Using the definition of free cash flow to the firm, P_0,FCFF can be estimated using the variable-growth model as follows:

$P_{0,\mathrm{FCFF}}={\displaystyle \sum }_{t=1}^n\frac{{\mathrm{FCFF}}_0{\left(1+g_t\right)}^t}{{\left(1+\mathrm{WACC}\right)}^t}+\frac{P_n}{{\left(1+\mathrm{WACC}\right)}^n}$

where

$P_n=\frac{{\mathrm{FCFF}}_n\left(1+g_m\right)}{{\mathrm{WACC}}_m-g_m}$

• FCFF₀ = FCFF in year 0
• WACC = weighted average cost of capital through year n
• WACC_m = cost of capital assumed beyond year n (Note: WACC > WACC_m)
• P_n = value of the firm at the end of year n (terminal value)
• g_t = growth rate through year n
• g_m = stabilized or long-term growth rate beyond year n (Note: g_t > g_m)

Similarly, the value of the firm to equity investors can be estimated using Eq. (7.16). However, projected free cash flows to equity (FCFE) are discounted using the firm’s cost of equity.

The cost of capital is assumed to differ between the high-growth and the stable-growth periods when applying the variable-growth model. High-growth rates usually are associated with increased levels of uncertainty. A high-growth firm may have a beta above 1. However, when the growth rate stabilizes, it is reasonable to assume that the beta should approximate 1. A reasonable approximation of the discount rate to be used during the stable-growth period is to adopt the industry average cost of equity or weighted average cost of capital.

Eq. (7.16) can be modified to use the growing-annuity model to approximate the growth during the high-constant-growth period and the constant-growth model for the terminal period. This formulation requires fewer computations if the number of annual cash flow projections is large. As such, P_0,FCFF also can be estimated as follows:

$P_{0,\mathrm{FCFF}}=\frac{{\mathrm{FCFF}}_0\left(1+g\right)}{\mathrm{WACC}-g}\left[1-{\left(\frac{1+g}{1+\mathrm{WACC}}\right)}^n\right]+\frac{P_n}{{\left(1+\mathrm{WACC}\right)}^n}$

See Exhibit 7.5 for an illustration of how to apply the variable-growth model and the growing annuity model.

Determining the Duration of the High-Growth Period

Projected growth rates for sales, profit, and cash flow can be calculated based on the historical experience of the firm or industry or surveying security analyst projections.⁴⁵ Recent research suggests that Wall Street analyst forecasts tend to be more accurate in projecting financial performance than simply looking at past performance.⁴⁶ The length of the high-growth period should be longer when the current growth rate of a firm’s cash flow is much higher than the stable-growth rate and the firm’s market share is small. For example, if the industry is expected to grow at 5% annually and the target firm, which has only a negligible market share, is growing at three times that rate, it may be appropriate to assume a high-growth period of 5–10 years. If the terminal value constitutes more than 75% of the total PV, the annual forecast period should be extended beyond the customary 5 years to at least 10 years to reduce its impact on the firm’s total market value. Historical evidence shows that sales and profitability tend to revert to normal levels within 5–10 years, suggesting that the conventional use of a 5- to 10-year annual forecast before calculating a terminal value makes sense.⁴⁷

Determining the Stable or Sustainable Growth Rate

The stable growth rate generally is going to be less than or equal to the overall growth rate of the industry in which the firm competes or the general economy. Stable growth rates above these levels implicitly assume that the firm’s cash flow eventually will exceed that of its industry or the general economy. Similarly, for multinational firms, the stable growth rate should not exceed the projected growth rate for the world economy or a particular region of the world.

Determining the Appropriate Discount Rate

In evaluating projects most firms use the weighted average cost of capital methodology to discount future cash flows. However, they often increase it to reflect business specific risk, and firms adding the largest premiums to their WACC estimates also tend to hold the largest cash balances. By using higher discount rates, such firms fail to pursue projects that would have been attractive using their actual cost of capital in an effort to hoard cash in anticipation of future attractive opportunities. Even when firms are not financially constrained, firms may forgo attractive projects due to operational issues such as the time required to expand their workforce and the inability of current management to manage new projects effectively at that time. In these circumstances, firms inflate their discount rates above their cost of capital to account for these operational constraints.⁴⁸ Other firms may use discount rates below their actual cost of capital, because they are “impatient” and choose to justify an acquisition based on other criteria such as preventing a competitor from buying the target firm.⁴⁹

As with other investment opportunities, choosing the right discount rate is critical: one that is too low overvalues the target and risks overpayment while one that is too high undervalues the target and reduces the likelihood of being able to close the deal. Surveys show that many acquirers use a single firm-wide WACC to value target firms due to its simplicity. This is especially problematic when the acquirer has many lines of business and fails to use the WACC associated with each line of business.⁵⁰ The correct discount rate is the target’s cost of capital if the acquirer is merging with a higher-risk business. However, either the acquirer’s or the target’s cost of capital may be used if the two firms are equally risky and based in the same country.

Determining the Market Value of Long-Term Debt

The current value of a firm’s debt generally is independent of its enterprise value for financially healthy companies. This is not true for financially distressed firms and for hybrid securities.

Determining the Market Value of Operating Leases

Both capital and operating leases also should be counted as outstanding debt of the firm. When a lease is classified as a capital lease, the present value of the lease expense is treated as debt. Interest is imputed on this amount, which corresponds to debt of comparable risk and maturity and is shown on the income statement. Although operating-lease expenses are treated as operating expenses on the income statement, they are not counted as part of debt on the balance sheet for financial reporting purposes. For valuation purposes, operating leases should be included in debt. Future operating-lease expenses are shown in financial statement footnotes. The discount rate may be approximated using the firm’s current pretax cost of debt, reflecting the market rate of interest that lessors would charge the firm. The principal amount of the leases also can be estimated by discounting the current year’s operating-lease payment as a perpetuity using the firm’s cost of debt (see Exhibit 7.6).

Determining the Cash Impact of Deferred Taxes

Deferred tax assets and liabilities arise when the tax treatment of an item is temporarily different from its financial accounting treatment. Such taxes may result from uncollectible accounts receivable, warranties, options expensing, pensions, leases, net operating losses, depreciable assets, and inventories. Deferred taxes have a current and a future or noncurrent impact on cash flow. The current impact is reflected by adding the change in deferred tax liabilities and subtracting the change in deferred tax assets in the calculation of working capital. The noncurrent impact of deferred assets generally is shown in other long-term assets and deferred tax liabilities in other long-term liabilities on the firm’s balance sheet. A deferred tax asset is a future tax benefit, in that deductions not allowed in the current period may be realized in some future period. A deferred tax liability represents the increase in taxes payable in future years. The excess of accelerated depreciation taken for tax purposes over straight-line depreciation often used for financial reporting reduces the firm’s current tax liability but increases future tax liabilities when spending on plant and equipment slows. The amount of the deferred tax liability equals the excess of accelerated over straight-line depreciation times the firm’s marginal tax rate.

To estimate a firm’s equity value, the PV of net deferred tax liabilities (i.e., deferred tax assets less deferred tax liabilities) is deducted from the firm’s enterprise value.⁵⁴ The use of net deferred tax liabilities is appropriate, since deferred tax liabilities often are larger than deferred tax assets for firms in the absence of NOLs. The impact on free cash flow of a change in deferred taxes can be approximated by the difference between a firm’s marginal and effective tax rates multiplied by the firm’s operating income before interest and taxes. The analyst may assume the effective tax rate is applicable for a specific number of years before reverting to the firm’s marginal tax rate. For example, the effective tax rate for five years increases the deferred tax liability to the firm during that period as long as the effective rate is below the marginal rate. The deferred tax liability at the end of the fifth year is estimated by adding to the current cumulated deferred tax liability the additional liability for each of the next five years. This liability is the sum of projected EBIT times the difference between the marginal and effective tax rates. Assuming tax payments on the deferred tax liability at the end of the fifth year will be spread equally over the following 10 years, the PV of the tax payments during that 10-year period is then estimated and discounted back to the current period (see Exhibit 7.7).

Determining the Cash Impact of Unfunded Pension Liabilities

Deduct the PV of such liabilities from the enterprise value to estimate the firm’s equity value. Publicly traded firms are required to identify the PV of unfunded pension obligations; if not shown on the firm’s balance sheet, such data can be found in the footnotes to the balance sheet.⁵⁵

Determining the Cash Impact of Employee Options

Key employees often receive compensation in the form of options to buy a firm’s common stock at a stipulated price (i.e., exercise price). Once exercised, these options impact cash flow as firms attempt to repurchase shares to reduce earnings-per-share dilution resulting from the firm’s issuance of new shares to those exercising their options. The PV of these future cash outlays to repurchase stock should be deducted from the firm’s enterprise value.⁵⁶

Determining the Cash Impact of Other Provisions and Contingent Liabilities

Provisions (i.e., reserves) for future layoffs due to restructuring usually are recorded on the balance sheet in undiscounted form, since they usually represent cash outlays to be made in the near term. Such provisions should be deducted from the enterprise value because they are equivalent to debt. Contingent liabilities, whose future cash outlays depend on certain events, are shown not on the balance sheet but, rather, in footnotes. Examples include pending litigation and loan guarantees. Since such expenses are tax deductible, estimate the PV of future after-tax cash outlays discounted at the firm’s cost of debt, and deduct from the firm’s enterprise value.

Determining the Market Value of Non-controlling Interests

When a firm owns less than 100% of another business, it is shown on the firm’s consolidated balance sheet. That portion not owned by the firm is shown as a non-controlling interest. For valuation purposes the non-controlling interest has a claim on the assets of the majority-owned subsidiary and not on the parent firm’s assets. If the less than wholly-owned subsidiary is publicly traded, value the non-controlling interest by multiplying the non-controlling ownership share by the market value of the subsidiary. If the subsidiary is not publicly traded and you as an investor in the subsidiary have access to its financials, value the subsidiary by discounting the subsidiary’s cash flows at the cost of capital appropriate for the industry in which it competes. The resulting value of the non-controlling interest also should be deducted from the firm’s enterprise value.

Cash and Marketable Securities

Excess cash balances are cash and short-term marketable securities held in excess of the target firm’s minimum operating cash balance. What constitutes the minimum cash balance depends on the firm’s cash conversion cycle, which reflects the firm’s tendency to build inventory, sell products on credit, and later collect accounts receivable. The length of time cash is committed to working capital can be estimated as the sum of the firm’s inventory conversion period plus the receivables collection period less the payables deferral period.⁵⁷ To finance this investment in working capital, a firm must maintain a minimum cash balance equal to the average number of days its cash is tied up in working capital times the average dollar value of sales per day. The inventory conversion and receivables collection periods are calculated by dividing the dollar value of inventory and receivables by average sales per day. The payments deferral period is estimated by dividing the dollar value of payables by the firm’s average cost of sales per day. Exhibit 7.8 illustrates how to estimate minimum and excess cash balances.

Investments in Other Firms

Such investments, for financial reporting purposes, may be classified as non-controlling passive investments, non-controlling active investments, or majority investments. These investments need to be valued individually and added to the firm’s enterprise value to determine the total firm value. See the companion website to this book for an explanation of the valuation methodology in a document entitled “Investments in Other Firms.”

Unutilized and Undervalued Assets

Target firm real estate may have a market value in excess of its book value. A firm may have an overfunded pension fund. Intangible assets such as patents and licenses may have substantial value. In the absence of a predictable cash flow stream, their value may be estimated using the Black–Scholes model (see Chapter 8) or the cost of developing comparable technologies.

Patents, Service Marks and Trademarks

A patent without a current application may have value to an external party, which can be determined by a negotiated sale or license to that party. When a patent is linked to a specific product, it is normally valued based on the “cost avoidance” method. This method uses after-tax royalty rates paid on comparable patents multiplied by the projected future stream of revenue from the products whose production depends on the patent discounted to its present value at the cost of capital. Products and services, which depend on a number of patents, are grouped together as a single portfolio and valued as a group using a single royalty rate applied to a declining percentage of the future revenue. Trademarks are the right to use a name, and service marks are the right to use an image associated with a company, product, or concept. Their value is name recognition reflecting the firm’s longevity, cumulative advertising expenditures, the effectiveness of its marketing programs, and the consistency of perceived product quality.

Overfunded Pension Plans

Defined benefit pension plans require firms to hold financial assets to meet future obligations. Shareholders have the legal right to assets in excess of what is needed. If such assets are liquidated and paid out to shareholders, the firm has to pay taxes on their value. The after-tax value of such funds may be added to the enterprise value.

Discussion Questions

1. 1. Estimate the firm’s cost of equity and after tax cost of debt.
2. 2. Estimate the firm’s weighted average cost of capital. (Hint: Recall that the debt-to-total capital ratio is equal to the debt-to-equity ratio divided by one plus the debt-to-equity ratio.)
3. 3. What is the WAAC beyond 2022?
4. 4. Use the discounted cash flow method to determine the standalone value for Rockwell Collins. Show your work.
5. 5. Assuming the free cash flows from synergy will remain level in perpetuity, estimate the after-tax present value of anticipated synergy?
6. 6. What is the maximum purchase price United Technologies should pay for Rockwell Collins? Did United Technologies overpay?
7. 7. How might your answer to Question 5 change if the discount rate during the first five years and during the terminal period is the same as estimated in Question 2?
8. 8. What are the limitations of the discounted cash flow method employed in this case study?

Solutions to these case discussion questions are available in the Online Instructor’s Manual for instructors using this book (https://textbooks.elsevier.com/web/Manuals.aspx?isbn=9780128150757).