Calculating a firm’s cost of capital: three different methods of determining the weighted average cost of capital for Microsoft and general electric produce different results for each firm. Thus, careful judgment and sensitivity analysis are important components for developing reliable cost of capital estimates

Michael S. Pagano

The idea of the “cost of capital” is fundamental to what managerial finance and accounting professionals do, directly or indirectly, as part of their participation on cross-functional decision teams. They need to understand and apply techniques for estimating the cost of capital for long-term capital budgeting; merger and acquisition analysis; use of Economic Value Added (EVA[R]) as a firm-wide financial performance indicator; incentive systems for financial control, using residual income for evaluating financial performance; equity valuation analyses; and accounting for purchased goodwill. (1)

Here we offer readers an overview of theoretical and empirical issues involved in estimating a firm’s weighted average cost of capital (WACC), and we review and apply several methods for estimating WACC for two widely held U.S. firms: General Electric (GE) and Microsoft. The most difficult to estimate component of a firm’s WACC relates to the cost of equity capital ([K.sub.s]), a process complicated in practice by the need to make various assumptions and practical choices. Conventional methods for estimating WACC, therefore, can yield substantially different approximations depending on the assumptions used in estimating [K.sub.s], so good judgment and sensitivity analysis are required when attempting to estimate a firm’s cost of capital for applications in accounting and finance.

THEORETICAL FRAMEWORK

Conceptually, a firm’s cost of capital is an investor’s opportunity cost of investing his or her capital in that firm. An estimate of the firm’s WACC is an attempt to quantify the average return expected by all investors in the firm: creditors of short-term and long-term interest-bearing debt, preferred stockholders, and common stockholders. (2) The firm’s cost of capital is a weighted average where the weights are determined by the value of the various sources of capital. (3)

In Equation 1 we show the conventional formula for estimating a firm’s WACC.

EQUATION 1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where,

[w.sub.i] = the weight of the i-th source of capital (i = 1, …, N) based on that source’s aggregate market value in relation to the firm’s total value,

[[summation of].sub.i][w.sub.i] = 1, and

[K.sub.i] = the expected return on the i-th security.

The portion to the right of the equal sign in Equation 1 can be rewritten in a simplified equation when there are only two sources of capital: long-term interest-bearing debt and (common) equity, as shown in Equation 2.

EQUATION 2

WACC = [w.sub.d][K.sub.d](1 – T) = [w.sub.s][K.sub.s]

where,

[K.sub.d] is the expected cost of long-term debt,

T is the firm’s marginal income tax rate (combined federal and state),

[K.sub.s] is the expected cost of common stock, and

[w.sub.d] and [w.sub.s] are the weights of long-term debt and common stock in the firm’s capital structure. (4) Note that this could be either its target or self-professed optimal capital structure. (5)

When determining the weights of debt and equity, we use their market values rather than book values because market values are more reflective of the true worth of the company.

There are two models that can be used to estimate [K.sub.s]: (1) a single-factor model called the Capital Asset Pricing Model (CAPM) and (2) a multiple-factor model called the Arbitrage Pricing Model (APM). (6) Next, we briefly outline these models below and a third model, the “bond yield plus risk premium model,” that financial analysts frequently use.

ESTIMATING THE COST OF EQUITY WITH CAPM

To calculate Equation 2, we need a means of estimating the required returns ([K.sub.i]) for each component of the firm’s capital structure. An asset-pricing model such as the CAPM can provide a convenient and theoretically consistent set of return estimates. The standard CAPM method says the required return on a risky asset such as common stock is related linearly to a nondiversifiable risk, otherwise known as “systematic” risk. Systematic risk is the riskiness of the “market portfolio” of all risky marketable assets. This relationship can be summarized concisely, as shown in Equation 3.

EQUATION 3

[K.sub.i] = [K.sub.rf] + [[beta].sub.i] ([K.sub.m] – [K.sub.rf])

where,

[K.sub.rf] = the expected return on a riskless security;

[K.sub.m] = the expected return on the systematic risk factor, i.e., the market portfolio’s return, which is represented by the return on a large equity portfolio such as the S&P 500; and

[[beta].sub.i] = beta = a measure of the i-th security’s sensitivity to the systematic risk factor.

A firm’s beta can be estimated from a regression using historical data for the returns on the stock ([K.sub.s]) and a market portfolio proxy ([K.sub.m]). Typically, monthly returns data are used when estimating this regression. This CAPM beta will be biased when estimating a forward-looking cost of equity capital. A forward-looking estimate for [K.sub.s] is important for our analysis because the CAPM (as well as the APM) assumes that investors base their investment decisions on expected returns on all marketable securities. In deriving their published estimates of corporate betas, brokerage and analytical firms such as Merrill Lynch, Bloomberg, and Value Line have used Marshall Blume’s idea to reduce the bias in the estimated beta and, in theory, improve one’s ability to develop forward-looking return estimates. (7) Value Line’s adjustment technique is relatively simple, as shown in Equation 4.

EQUATION 4

[[??].sub.i] = 0.35 + 0.67 [[beta].sub.i]

where,

[[beta].sub.i] = beta estimated via Equation 3 using historical time-series data and

[[??].sub.i] = an adjusted beta to account for the mean-reversion bias in the estimated beta.

Estimating the other two key components of Equation 3, the risk-free rate ([K.sub.rf]) and the market risk premium ([K.sub.m] – [K.sub.rf]), also requires the analyst’s judgment. (8)

Since the advent of the CAPM in the 1960s to explain the pricing of assets, there have been numerous theoretical and empirical developments in asset pricing. In particular, CAPM’s single-factor relation described by Equation 3 can be generalized to accommodate multiple systematic risk factors using logic based on the concept of “financial arbitrage.” This newer approach, called the Arbitrage Pricing Model (APM), explicitly incorporates risk factors beyond the market portfolio factor, but the APM does not spell out what those extra factors should be, so researchers have been forced to rely on extensive empirical testing of numerous macroeconomic and financial variables to find additional factors that might improve the explanatory power of the CAPM. (9) S. David Young and Stephen F. O’Bryne suggest using growth rates of real GDP and inflation, as well as interest rates, as additional factors in estimating [K.sub.s]. (10)

Other researchers have popularized the use of a three-factor model consisting of: (1) the “excess return,” or risk premium, for the market portfolio ([K.sub.m] – [K.sub.rf] from Equation 3); (2) the return on a portfolio that represents the difference between the returns on a group of small capitalization stocks and the returns on a group of large capitalization stocks (referred to as a “Small Minus Big” portfolio, or an “SMB” factor, which is related to the size of the company); and (3) the return on a portfolio that represents the difference between the returns on a group of stocks with high market-to-book equity ratios and the returns on a group of stocks with low market-to-book equity ratios (referred to as a “High Minus Low” portfolio, or an “HML” factor, which is related to the relative valuation of the company). (11)

These two extensions of the CAPM are based on observed empirical relations and do not have a convincing theoretical justification for the additional factors. This has led many practitioners to stay with the more widely accepted and simpler CAPM approach. Yet there are advantages to using the APM, most notably the fact that it usually leads to greater explanatory power of real-world stock returns when compared to the CAPM.

THE “BOND YIELD PLUS RISK PREMIUM” METHOD

The other technique for estimating [K.sub.s] is the “Bond Yield Plus Risk Premium” (BY+P). The BY+P method is popular among some practitioners (most notably multibillionaire investor Warren Buffett) because of its simplicity and limited number of assumptions. The BY+P method is essentially an ad hoc empirical relation with no solid theoretical justification. Yet there appears to be some empirical validity in the notion that the return on a company’s stock can be estimated by taking the firm’s bond yield-to-maturity (YTM) and adding a fixed risk premium to this yield. So, for example, a firm with a current bond YTM of 7% would lead to an estimated [K.sub.s] of 10% once a fixed 3% risk premium is added to the YTM. (12) Although there is no theoretical reason for adding a 3% premium, it appears that this relation is just as good or better for many stocks than using a formal model such as the CAPM.

CALCULATING GE AND MICROSOFT’S WACC

Let’s see how Equations 1 through 4 can be used to estimate a firm’s WACC. Our analysis uses 60 months of recent stock return data from the Center for Research in Security Prices’ (CRSP) database for all common stocks. (13) We computed excess market returns (“market risk premiums”) by subtracting the return on one-month U.S. T-bills from the market portfolio returns. For the market portfolio factor, we used the monthly return on the CRSP Value-Weighted Stock Index. (14) In addition to the above data, we also collected financial variables to estimate Eugene F. Fama and Kenneth French’s three-factor model, the APM. (15)

In total, therefore, we use three models to derive our WACC estimates for GE and Microsoft: (1) the adjusted-beta CAPM, (2) the APM, and (3) the Bond Yield Plus Risk Premium approach.

COST OF EQUITY

In Table 1, we estimate the cost of equity capital, [K.sub.s], using the adjusted-beta CAPM approach and employing several different assumptions about the appropriate market risk premium ([K.sub.m] – [K.sub.rf]) and the relevant risk-free rate ([K.sub.rf]).

There is considerable debate about the appropriate forward-looking U.S. market risk premium to use for capital budgeting, investment planning, and other financial decisions. Financial economists Robert Shiller and Roger Ibbotson examined the same set of historical data to determine current expectations for the market risk premium, and they both came up with different answers. (16) One suggests that the market risk premium in the U.S. is no more than 2% per annum while the other expects an average of 4% for the market risk premium as we progress in the early 21st century. Indeed, both of these figures are lower than the historical average U.S. risk premium of 6% to 8% that academic researchers have calculated and reported.

There also is no consensus about which maturity of U.S. Treasury securities is appropriate for estimating a risk-free rate, [K.sub.rf], within the CAPM framework. (17) For example, finance textbooks typically use the yield on a short-term U.S. Treasury security such as a one-month, three-month, or one-year T-bill, while many corporate practitioners use yields on longer-term instruments, such as five- or 10-year Treasury notes, in order to match more closely the time horizon associated with long-term investment projects. (18)

So, which estimates should be used? As is typical in many situations where uncertainty exists, the analyst must rely on his or her best judgment and also perform sensitivity analyses to help quantify the impact of different assumptions on the output of whatever decision model is being used.

Table 1 presents some possible alternative combinations of assumptions related to an analyst’s choice of market risk premium and risk-free rate. We use three sets of estimates for the risk-free rate (i.e., the approximate levels of the one-month, five-year, and 10-year U.S. Treasury securities) and three different expected market risk premiums (based on the two TIAA-CREF analysts’ estimates and a higher estimate of 6% based on historical experience). As can be seen from the results of using just one asset-pricing model (the adjusted-beta CAPM), a firm’s estimated [K.sub.s] can vary substantially. For example, Microsoft’s estimated cost of equity can vary from a low of 4.0% to a maximum of 13.9% depending on the particular combination of risk-free rate and expected market risk premium. Likewise, GE’s estimated cost of equity also varies considerably, although less than Microsoft’s because of GE’s lower adjusted-beta estimate of 1.08 versus Microsoft’s 1.48.

Using the APM approach, one can simply repeat the steps for the CAPM except that the regression now contains multiple variables–the returns on the market, SMB, and HML portfolios–to estimate the regression model’s parameters. Then, as in the CAPM results reported in Table 1, the expected returns on these three portfolios are multiplied by their respective regression parameter estimates and summed to obtain an APM-based estimate of the firm’s [K.sub.s]. We followed this procedure using five years of monthly data and then used a five-year average of the three portfolios’ returns as our estimate of these variables’ expected returns. The [K.sub.s] estimates derived from our APM-based analysis are reported in Table 2.

Now, using the BY+P approach, we simply use an estimate of the expected return on the firm’s long-term bonds and add a fixed 3% risk premium to estimate a firm’s [K.sub.s]. Similar to the APM estimates, we report the BY+P’s estimates of the firms’ [K.sub.s] in Table 2.

COST OF DEBT AND CAPITAL STRUCTURES

After estimating the cost of equity capital, [K.sub.s], we can proceed to estimating the other components of GE’s and Microsoft’s WACC. For most firms, this entails estimating the after-tax cost of (interest-bearing) debt and debt’s relative weight within the capital structure. Some firms such as Microsoft, however, have no debt (or even preferred stock) in their capital structure. So estimating Microsoft’s WACC is simplified greatly because we only need its cost of equity capital. In fact, for an “all-equity” firm (a company with no debt or preferred stock), the firm’s WACC is equal to its cost of equity ([K.sub.s]). To see this, review Equation 2 and note that [w.sub.s] = 1.0 when the firm has only equity in its capital structure. In this case, [w.sub.d] = 0; therefore, Equation 2 simplifies to WACC = [K.sub.s]. Thus, for Microsoft, we can stop here because estimates of [K.sub.d] are not needed.

For most companies, however, the capital structure does include at least some form of short-term and/or long-term interest-bearing debt. For them, we must estimate an after-tax cost of debt as well as debt’s weight within the capital structure. As noted earlier, market values of debt and common equity are preferred over book values because market values are more reflective of the true worth of the company. Yet the market value of a firm’s debt can be difficult to obtain, especially if the firm has privately issued debt such as bank loans and private placements of long-term debt. (19) Thus, it is common for analysts to use book values for the short-term and long-term debt in the capital structure while also using market values for the firm’s common equity–assuming the firm is publicly traded. If the firm is privately held, then an analyst must also decide whether to use the book value of equity for calculating the weights in the firm’s capital structure or create a separate estimate of the market value of the firm’s equity using, for example, a DCF (Discounted Cash Flow) valuation model. If the latter approach is used, then a cost of equity capital must be estimated before the DCF equity valuation can be developed. In turn, this DCF estimate of the firm’s equity value can be used to develop the relevant weights of debt and equity in the firm’s capital structure. Once again, the way in which we estimate the value of equity of a privately held firm illustrates that calculating a firm’s WACC is both art and science.

Now we have an estimate of the firm’s capital structure and can turn to the final step in estimating GE’s and Microsoft’s WACC: estimating the after-tax cost of debt (referred to as [K.sub.d] (1-T) in Equation 2). It is typically accomplished by either (1) looking up the current yield-to-maturity (YTM) of a representative sample of publicly traded corporate bonds that are judged to be of equivalent credit risk to the firm or (2) using the current YTMs of the firm’s outstanding bonds or bank loans.

To obtain data for the latter approach, an analyst can turn to the firm’s latest 10-K report, checking the footnote about the firm’s indebtedness. Here the firm will typically report its weighted-average interest rate on its long-term debt (including publicly and privately held bonds, bank loans, mortgages, etc.). For the former approach, the analyst can look at the market-based YTM of the firm’s bonds (or bonds of other companies with similar bond ratings and debt maturities) if the bonds are publicly traded and if Moody’s or Standard & Poor’s has rated the firm’s debt.

Our two stocks (as of this writing) have strong bond ratings (AAA for GE and AA for Microsoft). Thus, we can use a current estimate of the Moody’s Industrial average YTM for long-term corporate bonds rated AAA for GE, which is 6.63%. We do not need to estimate [K.sub.d] for Microsoft’s after-tax cost of debt because the company does not have any. But, for completeness, we also include in Table 2 the YTM for AA Industrial bonds of 6.83% for Microsoft. (20) As for the tax rate, T, we used an effective marginal tax rate of 40% for both firms to simplify our task. In practice, an analyst can examine the firm’s tax situation in more detail by accounting for tax loss carry forwards and other tax-related items to compute a more precise marginal tax rate.

RESULTS

Table 2 reports a column for each firm in our analysis. The first three rows of the table present the capital structure (i.e., weights) and the after-tax cost of (longterm) debt estimates for each firm. The next three rows report the cost-of-equity estimates for each of our three models. For the adjusted-beta CAPM estimates, we used the overall average [K.sub.s] figure from Table 1 (9.3% for Microsoft and 7.7% for GE). We can view these estimates as ones based on an average market risk premium of 4% and an average risk-free rate of 3.33% (obtained by averaging the one-month, five-year, and 10-year Treasury yields from Table 1). As shown in Table 2, there are sizable differences in the [K.sub.s] estimates across the three models. For GE, the [K.sub.s] estimates range from 7.65% (for the adjusted-beta CAPM approach) to 12.06% (for the APM approach).

Each method of estimating a company’s weighted average cost of capital yields significant variation. It should also be noted that the APM estimates are based on regression models that appear to describe real-world stock returns more accurately than the CAPM’s regression results. For example, GE’s adjusted [R.sup.2] statistics for the CAPM and APM approaches are 0.51 and 0.66, respectively. (21) Microsoft shows a similar improvement in its [R.sup.2] statistic, rising from 0.34 to 0.49 when the APM approach is used. Thus, our results support the notion that the APM method often is more descriptive of actual stock returns than the CAPM.

The last three rows of Table 2 present the three sets of WACC estimates based on the adjusted-beta CAPM, APM, and BY+P methods. Given the variability in the cost-of-equity estimates in Table 1, it is not surprising that the three methods for calculating WACC also yield substantial differences. For example, GE’s WACC ranges from 6.06% (using the adjusted-beta CAPM) to 8.55% (based on the APM). Microsoft’s estimates are less dispersed than GE’s but also display a sizable range of 1.5 percentage points (from a low of 8.32% to a high of 9.83%). In general, deviations similar to the magnitude reported in Table 2 (if not greater) are commonly found when applying the theory of cost-of-capital estimation to real-world companies.

VARIABILITY IN WACC ESTIMATES

With such large variations in estimates of companies’ WACC, an analyst must exercise judgment when selecting a final estimate of the firm’s weighted average cost of capital. For example, the analyst might consider creating a simple average of the various WACC estimates. In theory, this averaging process could produce a more accurate final WACC estimate because errors in one estimation method might be canceled by errors attributable to another method. (22)

Alternatively, the analyst might consider performing sensitivity analyses by using each of the different WACC estimates within the particular decision model–such as the DCF capital budgeting model. The analyst might find that the net present value (NPV) calculation in the model is not affected greatly by the choice of either the high, low, or average WACC estimates. In this case, he or she can feel more confident about the capital budgeting decision and valuation estimate.

A serious dilemma arises, however, when different WACC estimates yield materially different “signals” from the valuation model. In this case, disclosure of the valuation model’s sensitivity to the choice of WACC might be appropriate. For example, when the valuation model’s estimates are highly sensitive to the choice of WACC, the analyst could report the minimum and maximum valuation estimates in order to provide some indication of the magnitude of this sensitivity.

GRAPPLING WITH IMPRECISE ESTIMATES

Finance and accounting professionals need to be able to estimate the weighted average cost of capital because it pervades their work. They need WACC estimates for implementing the asset-impairment requirements of Statement of Financial Accounting Standards (SFAS) No. 144, “Accounting for the Impairment of Long-Lived Assets,” for example. They also need WACC for capital budgeting and equity valuation analyses, and for computing financial performance metrics such as EVA[R] and residual income.

A way forward, therefore, is for corporate accounting and finance professionals to use several methods for estimating WACC and choose a simple average of the estimates. Also, because of the inherent subjectivity involved in the WACC estimation process, we recommend they use sensitivity analysis whenever such estimates are used in managerial decision models.

Table 1: Microsoft’s and GE’s Cost of Equity Capital

(Using the Adjusted-Beta Capital Asset Pricing Model)

Market Risk

Microsoft: Premium Assumption

Risk-Free Rate Adjusted Beta 2% 4% 6% Average

1-month Rate (1%) 1.48 4.0% 6.9% 9.9% 6.9%

5-year Rate (4%) 1.48 7.0% 9.9% 12.9% 9.9%

10-year Rate (5%) 1.48 8.0% 10.9% 13.9% 10.9%

Average 6.3% 9.3% 12.2% 9.3%

Market Risk

General Electric: Premium Assumption

Risk-Free Rate Adjusted Beta 2% 4% 6% Average

1-month Rate (1%) 1.08 3.2% 5.3% 7.5% 5.3%

5-year Rate (4%) 1.08 6.2% 8.3% 10.5% 8.3%

10-year Rate (5%) 1.08 7.2% 9.3% 11.5% 9.3%

Average 5.5% 7.7% 9.8% 7.7%

Table 2: Three Approaches to

Estimating Microsoft’s and GE’s

Cost of Capital

Microsoft GE

Cost of Capital Components

Weight of Debt ([w.sub.d]) 0.000 0.434

Weight of Equity ([w.sub.s]) 1.000 0.566

After-Tax Cost of Debt: [K.sub.d] (1 – T) 4.10% 3.98%

[K.sub.s] using Average Adjusted CAPM 9.26% 7.65%[K.sub.s] using APM 8.32% 12.06%[K.sub.s] using Bond Yield + Risk Premium 9.83% 9.63%Weighted Average Cost of Capital

WACC using Adjusted-Beta CAPM 9.26% 6.06%

WACC using APM 8.32% 8.55%

WACC using Bond Yield + Risk Premium 9.83% 7.18%

(1) Prior to the August 2001 issuance of Statement of Financial Accounting Standards (SFAS) No. 144, “Accounting for the Impairment of Long-Lived Assets,” goodwill (the excess of the purchase price over the fair market value of identifiable net assets acquired) was recorded on the books only during a “purchase” combination. In the past, purchased goodwill generally was amortized using the straight-line method over a period of 40 years. Now companies are required to “reevaluate” or “reassess” the “value” of goodwill each year and explicitly estimate impairment charges if the fair value of goodwill must be written down. Many companies are writing off a substantial part of what was previously carried on the balance sheet as an asset. “The new accounting treatment requires companies to test the value of goodwill they carry on their financial statements every year and write down its value if that figure is deemed excessive as a result of changing business circumstances,” according to Henny Sender, “Floods of Firms to Take Goodwill Write-Downs,” The Wall Street Journal, April 24, 2002, p. C5. The article continues: “The [new] accounting standard’s preferred approach for estimating fair value is a discounted cash-flow valuation model … such models will be highly dependent upon the assumptions management makes, in particular the cost of capital and cash-flow growth assumptions.” See, also, AAA Financial Accounting Standards Committee, “Equity Valuation Models and Goodwill Impairment,” Accounting Horizons, June 2001, pp. 161-170.

(2) Noninterest-bearing liabilities such as accounts payable and taxes payable are not included in this calculation because a firm’s financial management does not typically control them. These “spontaneous” liabilities are due to sales and spending activity, and they directly affect a firm’s operating cash flows rather than its WACC. From a DCF valuation perspective, these noninterest-bearing liabilities are treated as a factor influencing the firm’s cash flows rather than the discount rate or WACC that is used to discount these cash flows.

(3) By this definition, the WACC estimate should be forward-looking rather than based on past historical data. As we shall note later in this article, however, often the only practical way to form a forward-looking WACC estimate is by using past data and historical relationships.

(4) Note that target weights rather than those based on existing market or book values are justified by the assumption that the firm’s capital structure will, over time, gravitate toward these target values and, therefore, can be considered more appropriate than market or book values reported at one point in time. Yet in Robert F. Bruner, Kenneth M. Eades, Robert S. Harris, and Robert C. Higgins, “Best Practices in Estimating the Cost of Capital: Survey and Synthesis,” Financial Practice and Education, Spring/Summer 1998, pp. 13-28, we find that the vast majority of large firms used market values rather than book or target values when estimating the relative weights of debt, equity, and preferred stock.

(5) If the firm also has preferred stock in its capital structure, Equation 2 is amended to include the product of the weight of preferred stock in its capital structure, [w.sub.ps], and the expected return on the preferred stock, [K.sub.ps]. Typically, the market value of preferred stock is used to estimate [w.sub.ps], or a target percentage can be used, and the current dividend yield is used as an estimate of the expected return on the preferred stock, that is, [K.sub.ps] = [Dividend.sub.ps]/Market [Price.sub.ps].

(6) S. David Young and Stephen F. O’Byrne, EVA[R] and Value-Based Management: A Practical Guide to Implementation, McGraw-Hill, New York, N.Y., 2001, pp. 161-203; and Bruner, Eades, Harris, and Higgins, 1998, describe the assumptions underlying these two models and how to estimate [K.sub.s] using both approaches. The interested reader can consult these books for a detailed treatment of these two methods of estimating [K.sub.s].

(7) Marshall Blume, “Betas and Their Regression Tendencies,” Journal of Finance, June 1975, pp. 785-795. Blume’s key insight is that a firm’s beta measured over one time period tends to revert toward a mean close to 1.0 when measured over a subsequent time period. Thus, empirically estimated betas tend to revert over time to a mean of approximately 1.0. Blume’s 1975 article shows in detail how to adjust for this “mean-reversion” bias in estimated betas.

(8) See Young and O’Byrne, 2001, pp. 161-203, and Bruner, Eades, Harris, and Higgins, 1998, pp. 13-28. The issues related to estimating these other components are discussed later when we present our empirical results for GE and Microsoft.

(9) The search for extra factors has been motivated, in part, by the fact that empirical research has shown that the CAPM does not explain a good portion of the variation in common stock returns for individual firms. The CAPM, however, explains a great deal–typically 90% or more–of the variation of the returns on large portfolios such as mutual fund returns.

(10) Young and O’Byrne, 2001, pp. 161-203.

(11) Eugene F. Fama and Kenneth French, “Common Risk Factors in the Returns on Stocks and Bonds,” Journal of Financial Economics, February 1993, pp. 3-56.

(12) Some analysts will adjust this fixed risk premium downward if, in their judgment, a firm is less risky than the average firm; conversely, they will adjust the risk premium upward if they view the firm as being more risky than the average firm.

(13) The average return data and the CAPM beta estimates are based on monthly data over a five-year period. Many practitioners use a 60-month period to estimate a CAPM beta, but other sample period lengths and time frequencies (such as, daily and weekly returns for one to 10 years) have been used. The key trade-off between using daily returns versus monthly returns, for example, is the advantage of having more observations when daily data are used and the disadvantage that daily data are typically much “noisier” in a statistical sense. Thus, betas based on daily return data are typically less reliable in statistical and economic significance when compared to betas based on monthly return data, which are inherently less “noisy.” See Young and O’Byrne, 2001, for more details.

(14) Practitioners can and do use other market portfolio benchmarks, such as the Center for Research in Security Prices’ (CRSP) Equal-Weighted Stock Index, the S&P 500, the Russell 2000, the Morgan Stanley World Stock Index, etc. Typically, the analyst will use an index that is widely available and representative of alternative investments that the average investor would consider part of his or her portfolio of risky assets. The CRSP Value-Weighted Stock Index is advantageous because it is the most comprehensive index of the market value and return behavior for an investor interested in investing in U.S. stocks.

(15) Fama and French, 1993, pp. 3-56.

(16) Robert Shiller and Roger Ibbotson, “Measuring Equity Risk Premium,” TIAA-CREF Investment Forum, June 2002, pp. 10-12.

(17) Bruner, Eades, Harris, and Higgins, 1998, pp. 13-28.

(18) Ibid, pp. 13-28.

(19) Privately placed securities typically are sold directly to large institutional investors. These securities are cheaper to issue but have limited marketability because usually only large institutional investors are permitted to transact in this market.

(20) In reality, we need this estimate of Microsoft’s [K.sub.d] not for estimating [K.sub.d] (1 – T) but because later we want to estimate the firm’s cost of equity via the Bond Yield Plus Risk Premium approach.

(21) An [R.sup.2] statistic closer to 1.0 indicates the regression model is a better fit to the actual observed stock returns for a company. Therefore, [R.sup.2] represents a valid measure of a model’s “goodness-of-fit.”

(22) This averaging technique implicitly assumes that errors across models are independent of each other (which may not be true in practice).

Michael S. Pagano, Ph.D., CFA, is assistant professor of finance in the College of Commerce and Finance at Villanova University in Villanova, Pa. You can reach him at (610) 519-4389 or michael.pagano@villanova.edu.

David E. Stout, Ph.D., is the Andrews Chair in Accounting at the Williamson College of Business Administration at Youngstown State University in Youngstown, Ohio. You can reach him at (330) 941-3509 or destout@ysu.edu.

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