This chapter presents and illustrates key return measures relevant to valuation and is organized as follows. Section 2 provides an overview of return concepts. Section 3 presents the chief approaches to estimating the equity risk premium, a key input in determining the required rate of return on equity in several important models. With a means to estimate the equity risk premium in hand, Section 4 discusses and illustrates the major models for estimating the required return on equity. Section 5 presents the weighted average cost of capital, a discount rate used when finding the present value of cash flows to all providers of capital. Section 6 presents certain facts concerning discount rate selection. Section 7 summarizes the chapter, and practice problems conclude it.

Equation 2-1 assumes, for simplicity, that any dividend is received at the end of the holding period. More generally, the holding period return would be calculated based on reinvesting any dividend received between t= 0 and t = H in additional shares on the date the dividend was received at the price then available. Holding period returns are sometimes annualized—in other words, the return for a specific holding period may be converted to an annualized return, usually based on compounding at the holding period rate. For e xample, (1.008)³⁶⁵- 1 = 17.3271 or 1,732.71 percent, is one way to annualize a one-day 0.80 percent return. As the example shows, however, annualizing holding period returns, when the holding period is a fraction of a year, is unrealistic when the reinvestment rate is not an actual, available reinvestment rate.

Although professional investors often formulate expected returns based on explicit valuation models, a return expectation does not have to be based on a model or on specific valuation knowledge. Any investor can have a personal viewpoint on the future returns on an asset. In fact, because investors formulate expectations in varying ways and on the basis of different information, different investors generally have different expected returns for an asset. The comparison point for interpreting the investment implication of the expected return for an asset is its required return, the subject of the next section.

The valuation examples presented in this book will illustrate the use of required return estimates grounded in market data (such as observed asset returns) and explicit models for required return. We will refer to any such estimate of the required return used in an example as the required return on the asset for the sake of simplicity, although other estimates are usually defensible. For example, using the capital asset pricing model (CAPM—discussed in more detail later), the required return for an asset is equal to the risk-free rate of return plus a premium (or discount) related to the asset’s sensitivity to market returns. That sensitivity can be estimated based on returns for an observed market portfolio and the asset. That is one example of a required return estimate grounded in a formal model based on marketplace variables (rather than a single investor’s return requirements). Market variables should contain information about investors’ asset risk perceptions and their level of risk aversion, both of which are important in determining fair compensation for risk.

In this chapter, we use the notation r for the required rate of return on the asset being discussed. The required rate of return on common stock and debt are also known as the cost of equity and cost of debt, respectively, taking the perspective of the issuer. To raise new capital, the issuer would have to price the security to offer a level of expected return that is competitive with the expected returns being offered by similarly risky securities. The required return on a security is therefore the issuer’s marginal cost for raising additional capital of the same type.

The difference between the expected return and the required rate of return on an asset is the asset’s expected alpha (or ex ante alpha) or expected abnormal return:

Estimates of required returns are essential for using present value models of value. Present value models require the analyst to establish appropriate discount rates for determining the present values of expected future cash flows.

Expected return and required rate of return are sometimes used interchangeably in conversation and writing.²⁰ As discussed, that is not necessarily correct. When current price equals perceived value, expected return should be the same as the required rate of return. However, when price is below (above) the perceived value, expected return will exceed (be less than) the required return as long as the investor expects price to converge to value over his time horizon.

Given an investor’s expected holding period return, we defined expected alpha in relation to a required return estimate. In the next section, we show the conversion of a value estimate into an estimate of expected holding period return.

We can illustrate how expected return may be estimated when an investor’s value e stimate, V₀, is different from the market price. Suppose the investor expects price to fully converge to value over τ years. (V₀ - P₀)/P₀ is an estimate of the return from convergence over the period of that length, essentially the expected alpha for the asset stated on a per-period basis. With r τbeing the required return on a periodic (not annualized) basis and E(Rτ) the expected holding-period return on the same basis, then:

To illustrate, as of the end of the first quarter of 2007, one estimate of the required return for Procter & Gamble (NYSE: PG) shares was 7.6 percent. At a time when PG’s market price was $63.16, a research report estimated PG’s intrinsic value at $71.00 share. Thus, in the report author’s view, PG was undervalued by V₀-P₀ = $71 - $63.16 = $7.84, or 12.4 percent as a fraction of the market price ($7.84/$63.16). If price were expected to converge to value in exactly one year, an investor would earn 7.6% + 12.4% = 20%. The expected alpha of Procter & Gamble is 12.4 percent per annum. But if the investor expected the undervaluation to disappear by the end of nine months, then the investor might anticipate achieving a return of about 18 percent over the nine-month period. The required return on a nine-month basis (τ = 9/12 = 0.75) is (1.076)^0.75- 1 = 0.0565 or 5.65 percent, so the total expected return is

Active investors essentially second-guess the market price. The risks of that activity include the risks that (1) their value estimates are not more accurate than the market price, and (2) even if they are more accurate, the value discrepancy may not narrow over the investors’ time horizon. Clearly, the convergence component of expected return can be quite risky.

In principle, because of varying expected future inflation rates and the possibly varying risk of expected future cash flows, a distinct discount rate could be applicable to each distinct expected future cash flow. In practice, a single required return is generally used to discount all expected future cash flows.²³

Sometimes an internal rate of return is used as a required return estimate, as discussed in the next section.

In a model that views the intrinsic value of a common equity share as the present value of expected future cash flows, if price is equal to current intrinsic value—the condition of market informational efficiency—then, generally, a discount rate can be found, usually by iteration, which equates that present value to the market price. An IRR computed under the assumption of market efficiency has been used to estimate the required return on equity. An example is the historical practice of many U.S. state regulators of estimating the cost of equity for regulated utilities using the model illustrated in Equation 2-3b.²⁴ (The issue of cost of equity arises because regulators set prices sufficient for utilities to earn their cost of capital.)²⁵

To illustrate, the simplest version of a present value model results from defining cash flows as dividends and assuming a stable dividend growth rate for the indefinite future. The stable growth rate assumption reduces the sum of results in a very simple expression for intrinsic value:²⁶

If the asset is correctly valued now (market price = intrinsic value), given consensus estimates of the year-ahead dividend and future dividend growth rate (which are estimates of the dividend expectations built into the price), we can solve for a required return—an IRR implied by the market price:

Finally, obtaining an IRR from a present value model should not be confused with the somewhat similar-looking exercise that involves inferring what the market price implies about future growth rates of cash flows, given an independent estimate of required return: That exercise has the purpose of assessing the reasonableness of the market price.

Using the equity risk premium, the required return on the broad equity market or an average-systematic-risk equity security is

Required return on equity = Current expected risk-free return + Equity risk premium where, for consistency, the definition of risk-free asset (e.g., government bills or government bonds) used in estimating the equity risk premium should correspond to the one used in specifying the current expected risk-free return.

The importance of the equity risk premium in valuation is that, in perhaps a majority of cases in practice, analysts estimate the required return on a common equity issue as either

Typically, analysts estimate the equity risk premium for the national equity market of the issues being analyzed (but if a global CAPM is being used, a world equity premium is estimated that takes into account the totality of equity markets).

Even for the longest established developed markets, the magnitude of the equity risk premium is difficult to estimate and can be a reason for differing investment conclusions among analysts. Therefore, we next introduce the topic of estimation in some detail. Whatever estimates analysts decide to use, when an equity risk premium estimate enters into a valuation, analysts should be sensitive to how their value conclusions could be affected by estimation error.

Two broad approaches are available for estimating the equity risk premium. One is based on historical average differences between equity market returns and government debt returns, and the other is based on current expectational data. These are presented in the following sections.

In using a historical estimate to represent the equity risk premium going forward, the analyst is assuming that returns are stationary—that is, the parameters that describe the return-generating process are constant over the past and into the future.

The analyst’s major decisions in developing a historical equity risk premium estimate include the selection of four factors:

Analysts try to select an equity index that accurately represents the average returns earned by equity investors in the market being examined. Broad-based, market value-weighted indexes are typically selected.

Specifying the length of the sample period typically involves trade-offs. Dividing a data period of a given length into smaller subperiods does not increase precision in estimating the mean—only extending the length of the data set can increase precision.²⁸ Thus, a common choice is to use the longest available series of reliable returns. However, the assumption of stationarity is usually more difficult to maintain as the series starting point is extended to the distant past. The specifics of the type of nonstationarity are also important. For a number of equity markets, research has brought forth abundant evidence of nonconstant underlying return volatility. Nonstationarity—in which the equity risk premium has fluctuated in the short term, but around a central value—is a less serious impediment to using a long data series than the case in which the risk premium has shifted to a permanently different level (see Cornell 1999). Empirically, the expected equity risk premium is countercyclical in the United States—that is, the expected premium is high during bad times but low during good times (Fama and French 1989; Ferson and Harvey 1991). This property leads to some interesting challenges: For example, when a series of strong market returns has increased enthusiasm for equities and raised historical-mean equity risk premium estimates, the forward-looking equity risk premium may have actually declined.

Practitioners taking a historical approach to equity premium estimation often focus on the type of mean calculated and the proxy for the risk-free return. There are two choices for computing the mean and two broad choices for the proxy for the risk-free return.

The mean return of a historical set of annual return differences between equities and government debt securities can be calculated using a geometric mean or an arithmetic mean:

Dimson, Marsh, and Staunton (2008) presented authoritative evidence on realized excess returns of stocks over government debt (“historical equity risk premia”) using survivorship bias-free return data sets for 17 developed markets for the 108 years extending from 1900 through 2007.²⁹ Exhibit 2-1 excerpts their findings, showing results for the four combinations of mean computation and risk-free return representation (two mean return choices multiplied by two risk-free return choices). In the table, standard deviation is the standard deviation of the annual excess return series and minimum value and maximum value are, respectively, the smallest and largest observed values of that series.

The following excerpt from Exhibit 2-1 presents a comparison of historical equity risk premium estimates for the United States and Japan. This comparison highlights some of the issues that can arise in using historical estimates. As background to the discussion, note that as a mathematical fact, the geometric mean is always less than (or equal to) the arithmetic mean; furthermore, the yield curve is typically upward sloping (long-term bond yields are typically higher than short-term yields).

For the United States, estimates of the equity risk premium relative to long-term government bonds runs from 4.5 percent (geometric mean relative to bonds) to 7.4 percent (arithmetic mean relative to bills). The United States illustrates the typical case in which realized values relative to bills, for any definition of mean, are higher than those relative to bonds.

The premium estimates for Japan are notably higher than for the United States. Because the promised yield on long-term bonds is usually higher than that on short-term bills, the higher arithmetic mean premium relative to bonds compared to bills in the case of Japan is atypical. The analyst would need to investigate the reasons for it and believe they applied to the future before using the estimate as a forecast for the future. In virtually all markets, the geometric mean premium relative to long-term bonds gives the smallest risk premium estimate (the exception is Germany). Note the following:

The next two sections discuss choices related to the calculation of a historical equity risk premium estimate.

In actual professional practice, both means have been used in equity risk premium estimation.

The arithmetic mean return as the average one-period return best represents the mean return in a single period. There are two traditional arguments in favor of using the arithmetic mean in equity risk premium estimation, one relating to the type of model in which the estimates are used and the second relating to a statistical property. The major finance models for estimating required return—in particular the CAPM and multifactor models—are single-period models; so the arithmetic mean, with its focus on single period returns, appears to be a model-consistent choice. A statistical argument has also been made for the arithmetic mean: With serially uncorrelated returns and a known underlying arithmetic mean, the unbiased estimate of the expected terminal value of an investment is found by compounding forward at the arithmetic mean. For example, if the arithmetic mean is 8 percent, an unbiased estimate of the expected terminal value of a €1 million investment in 5 years is €1(1.08)⁵ = €1.47 million. In practice, however, the underlying mean is not known. It has been established that compounding forward using the sample arithmetic mean, whether or not returns are serially uncorrelated, overestimates the expected terminal value of wealth.³¹ In the example, if 8 percent is merely the sample arithmetic mean (used as an estimate of the unknown underlying mean), we would expect terminal wealth to be less than €1.47 million. Practically, only the first traditional argument still has force.

The geometric mean return of a sample represents the compound rate of growth that equates the beginning value to the ending value of one unit of money initially invested in an asset. Present value models involve the discounting over multiple time periods. Discounting is just the reverse side of compounding in terms of finding amounts of equivalent worth at different points in time; because the geometric mean is a compound growth rate, it appears to be a logical choice for estimating a required return in a multiperiod context, even when using a single-period required return model. In contrast to the sample arithmetic mean, using the sample geometric mean does not introduce bias in the calculated expected terminal value of an investment (Hughson et al. 2006). Equity risk premium estimates based on the geometric mean have tended to be closer to supply-side and demand-side estimates from economic theory than arithmetic mean estimates.³² For these reasons, the geometric mean is increasingly preferred for use in historical estimates of the equity risk premium.

A bond-based equity risk premium estimate in almost all cases is smaller than a bill-based estimate (see Exhibit 2-1). But a normal upward-sloping yield curve tends to offset the effect of the risk-free rate choice on a required return estimate, because the current expected risk-free rate based on a bond will be larger than the expectation based on a bill. However, with an inverted yield curve, the short-term yields exceed long-term yields, and the required return estimate based on using a risk-free rate based on a bill can be much higher.

Industry practice has tended to favor use of a long-term government bond rate in premium estimates despite the fact that such estimates are often used in one-period models such as the CAPM. A risk premium based on a bill rate may produce a better estimate of the required rate of return for discounting a one-year-ahead cash flow, but a premium relative to bonds should produce a more plausible required return/discount rate in a multiperiod context of valuation.³³

To illustrate a reason for the preference, take the case of bill-relative and bond-relative premia estimates of 5.5 percent and 4.5 percent, respectively, for a given market. Assume the yield curve is inverted: The current bill rate is 9 percent and the bond rate is 6 percent, respectively. The required return on average-risk equity based on bills is 14.5 percent (9% + 5.5%) compared with 10.5 percent based on bonds (6% + 4.5%). That 14.5 percent rate may be appropriate for discounting a one-year-ahead cash flow in a current high interest and inflation environment. The inverted yield curve, however, predicts a downward path for short-term rates and inflation. Most of the cash flows lie in the future and the premium for expected average inflation rates built into the long-term bond rate is more plausible. A practical principle is that for the purpose of valuation, the analyst should try to match the duration of the risk-free-rate measure to the duration of the asset being valued.³⁴ If the analyst has adopted a short-term risk-free rate definition, nevertheless, a practical approach to dealing with the situation just presented would be to use an expected average short-term bill rate rather than the current 9 percent rate. Advocates of using short-term rates point out that long-term government bonds are subject to risks, such as interest rate risk, that complicate their interpretation.

In practice, many analysts use the current YTM on a long-term government bond as an approximation for the expected return on it. The analyst needs to be clear that he is using a current yield observation, reflecting current inflation expectations. The yield on a recently issued (“on the run”) bond mitigates distortions related to liquidity and discounts/premiums relative to face value. The available maturities of liquid government bonds change over time and differ among national markets. If a 20-year maturity is available and trades in a liquid market, however, its yield is a reasonable choice as an estimate of the risk-free rate for equity valuation.³⁵ In many international markets, only bonds of shorter maturity are available or have a liquid market. A 10-year government bond yield is another common choice.

Valuation requires definite estimates of required returns. The data in Exhibit 2-1 provide one practical starting point for an estimate of equity risk premium for the markets given. As discussed, one mainstream choice among alternative estimates of the historical equity risk premium is the geometric mean historical equity risk premium relative to government bonds.

One issue is survivorship bias in equity market data series. The bias arises when poorly performing or defunct companies are removed from membership in an index, so that only relative winners remain. Survivorship bias tends to inflate historical estimates of the equity risk premium. For many developed markets, equity returns series are now available that are free or nearly free of survivorship bias (see Exhibit 2-1). When using a series that has such bias, however, the historical risk premium estimate should be adjusted downward. Guidance for such adjustment based on research is sometimes available.³⁶

A conceptually related issue with historical estimates can arise when a market has experienced a string of unexpectedly positive or negative events and the surprises do not balance out over the period of sampled data. For example, a string of positive inflation and productivity surprises may result in a series of high returns that increase the historical mean estimate of the equity risk premium. In such cases, a forward-looking model estimate may suggest a much lower value of the equity risk premium. To mitigate that concern, the analyst may adjust the historical estimate downward based on an independent forward-looking estimate (or upward, in the case of a string of negative surprises). Many experts believe that the historical record for various major world markets has benefited from a majority of favorable circumstances that cannot be expected to be duplicated in the future; their recommended adjustments to historical mean estimates is downward. Dimson, Marsh, and Staunton (2002) have argued that historical returns have been advantaged by repricings as increasing scope for diversification has led to a lower level of market risk. In the case of the United States, Ibbotson and Chen (2001) recommended a 1.25 percentage point downward adjustment to the Morningstar (Ibbotson) historical mean U.S. equity risk premium estimate, based on a lower estimate from a supply-side analysis of the equity risk premium.

Example 2-2 illustrates difficulties in historical data that could lead to a preference for an adjusted historical or forward-looking estimate.

In Example 2-2, one criticism that could be raised relative to any historical estimate is the shortness of the period in the data set—the post-1991 reform period—that is definitely relevant to the present. Sampling error in any mean estimate—even one based on clean data—would be a major concern for this data set. The analyst might address specific concerns through an adjusted historical estimate. The analyst may also decide to investigate one or more forward-looking estimates. Forward-looking estimates are the subject of the next section. A later section on international issues provides more information on equity risk premium estimation for emerging markets such as India.

Equation 2-6 is based on an assumption of earnings growth at a stable rate. An assumption of multiple earnings growth stages is more appropriate for very rapidly growing economies. Taking an equity index in such an economy, the analyst may forecast a fast growth stage for the aggregate of companies included in the index, followed by a transition stage in which growth rates decline, and then a mature growth stage characterized by growth at a moderate, sustainable rate. The discount rate r that equates the sum of the present values of the expected cash flows of the three stages to the current market price of the equity index defines an IRR. Letting PVFastGrowthStage(r) stand for the present value of the cash flows of the fast earnings growth stage with the present value shown as a function of the discount rate r, and using a self-explanatory notation for the present values of the other phases, the equation for IRR is as follows:

A consequence of the model underlying Equation 2-6, making assumptions of a constant dividend payout ratio and efficient markets, is that earnings, dividends, and prices are expected to grow at the dividend growth rate, so that the P/E ratio is constant. The analyst may believe, however, that the P/E ratio will expand or contract. Some analysts make an adjustment to the estimate in Equation 2-6 to reflect P/E multiple expansion or contraction. From a given starting market level associated with a given level of earnings and a given P/E ratio, the return from capital appreciation cannot be greater than the earnings growth rate unless the P/E multiple expands. P/E multiple expansion can result from an increase in the earnings growth rate and/or a decrease in risk.

To illustrate a supply-side analysis, the total return to equity can be analyzed into four components as explained by Ibbotson and Chen:⁴²

For example, a 2002 survey of global bond investors by Schroder Salomon Smith Barney found an average equity risk premium in the range of 2 to 2.5 percent, while a Goldman Sachs survey of global clients recorded a mean long-run equity risk premium of 3.9 percent (see Ilmanen et al. 2002; O’Neill et al. 2002).

The expression for the CAPM that is used in practice was given earlier as Equation 2-4:⁴⁵

In the typical case in which the equity risk premium is based on a national equity market index and estimated beta is based on sensitivity to that index, the assumption is being made implicitly that equity prices are largely determined by local investors. When equities markets are segmented in that sense (i.e., local market prices are largely determined by local investors rather than by investors worldwide), two issues with the same risk characteristics can have different required returns if they trade in different markets.

The opposite assumption is that all investors worldwide participate equally in setting prices (perfectly integrated markets). That assumption results in the international CAPM (or world CAPM) in which the risk premium is relative to a world market portfolio. Taking an equity view of the market portfolio, the world equity risk premium can be estimated historically based on the MSCI World index (returns available from 1970), for example, or indirectly as (U.S. equity risk premium estimate)/(beta of U.S. stocks relative to MSCI World) = 4.5%/0.9218 = 4.9%. Computing beta relative to MSCI World and using a national risk-free interest rate, the analyst can obtain international CAPM estimates of required return. In practice, the international CAPM is not commonly relied on for required return on equity estimation.⁴⁶

Examples 2-3 through 2-5 apply the CAPM to estimate the required return on equity.

When a share issue trades infrequently, the most recent transaction price may be stale and not reflect underlying changes in value. If beta is estimated based on, for example, a monthly data series in which missing values are filled with the most recent transaction price, the estimated beta will be too small and the required return on equity will be underestimated. There are several econometric techniques that can be used to estimate the beta of infrequently traded securities. ⁵⁰ A practical alternative is to base the beta estimate on the beta of a comparable security.

The procedure must take into account the effect on beta of differences in financial leverage between the nonpublic company and the benchmark. First, the benchmark beta is unlevered to estimate the beta of the benchmark’s assets—reflecting just the systematic risk arising from the economics of the industry. Then the asset beta is relevered to reflect the financial leverage of the nonpublic company.

Let β_E be the equity beta before removing the effects of leverage, if any. This is the benchmark beta. If the debt of the benchmark is high quality (so an assumption that the debt’s beta is zero should be approximately true), analysts can use the following expression for unleveraging the beta:⁵¹

The CAPM is a simple, widely accepted, theory-based method of estimating the cost of equity. Beta, its measure of risk, is readily obtainable for a wide range of securities from a variety of sources and can be estimated easily when not available from a vendor. In portfolios, the idiosyncratic risk of individual securities tends to offset against each other, leaving largely beta (market) risk. For individual securities, idiosyncratic risk can overwhelm market risk and, in that case, beta may be a poor predictor of future average return. Thus the analyst needs to have multiple tools available.

Whereas the CAPM adds a single risk premium to the risk-free rate, arbitrage pricing theory (APT) models add a set of risk premia. APT models are based on a multifactor representation of the drivers of return. Formally, APT models express the required return on an asset as follows: where (Risk premium)_i = (Factor sensitivity)_i x (Factor risk premium)_i.

Factor sensitivity or factor beta is the asset’s sensitivity to a particular factor (holding all other factors constant). In general, the factor risk premium for factor i is the expected return in excess of the risk-free rate accruing to an asset with unit sensitivity to factor i and zero sensitivity to all other factors.⁵³

One of the best known models based on multiple factors expands upon the CAPM with two additional factors. That model, the Fama-French model, is discussed next.

In 1993, researchers Eugene Fama and Kenneth French addressed these perceived weaknesses of the CAPM in a model with three factors, known as the Fama-French model (FFM). The FFM is among the most widely known nonproprietary multifactor models. The factors are:

We can illustrate the FFM using the case of the U.S. equity market. A current short-term interest rate is 4.1 percent. We take RMRF to be 5.5 percent based on Panel B of Exhibit 2-1. The historical size premium is 2.7 percent based on Fama-French data from 1926. However, over approximately the past quarter century (1980 to 2006) the realized SML premium has averaged about one-half of that. Therefore, the historical estimate is adjusted downward to 2.0 percent. The realized value premium has had wide swings, but absent the case for a secular decline as for the size premium, we take the historical value of 4.3 percent based on Fama-French data. Thus, one estimate of the FFM for the U.S. market as of 2007 is:

Consider the case of a small-cap issue with value characteristics and above-average market risk—assume the FFM market beta is 1.20. If the issue’s market capitalization is small we expect it to have a positive size beta; for example,β_i = 0.5. If the shares sell cheaply in relation to book equity (i.e., they have a high book-to-market ratio) the value beta is also expected to be positive; for example, β_i = 0.8. For both the size and value betas, zero is the neutral value, in contrast with the market beta, where the neutral value is 1. Thus, according to the FFM, the shares’ required return is slightly over 15 percent:

Returning to the specification of the FFM to discuss its interpretation, note that the FFM factors are of two types:

The regression fit statistics for both the CAPM and FFM in Example 2-7 are high. There is more to learn about the relative merits of the CAPM and FFM in practice, but the FFM appears to have the potential for being a practical addition to the analyst’s tool kit. One study contrasting the CAPM and FFM for U.S. markets found that whereas differences in the CAPM beta explained on average 3 percent of the cross-sectional differences in returns of the stocks over the next year, the FFM betas explained on average 5 percent of the differences (Bartholdy and Peare 2001). Neither performance appears to be impressive, but keep in mind that equity returns are subject to a very high degree of randomness over short horizons.

The concept of liquidity may be distinguished from marketability. With reference to equities, liquidity relates to the ease and potential price impact of the sale of an equity interest into the market. Liquidity is a function of several factors, including the size of the interest and the depth and breadth of the market and its ability to absorb a block (i.e., a large position) without an adverse price impact. In the strictest sense, marketability relates to the right to sell an asset. Barring securities law or other contractual restrictions, all equity interests are potentially marketable—that is, they can potentially be marketed for sale, in the sense of the existence of a market into which the security can be sold. However, in private business valuation, the two terms are often used interchangeably (Hitchner 2006, 390). The typical treatment in that context is to take a discount for lack of marketability (liquidity) from the value estimate, where justified, rather than incorporate the effect in the discount rate, as in the PSM (Hitchner 2006, 390 -391).

A specific example of macroeconomic factor models is the five-factor BIRR model, presented in Burmeister, Roll, and Ross (1994), with factor definitions as follows:

Standard approaches to estimating the required return on equity for publicly traded companies, such as the CAPM and the FFM, are adaptable for estimating the required rate of return for non-publicly traded companies. However, valuators often use an approach to valuation that relies on building up the required rate of return as a set of premia added to the risk-free rate. The premia include the equity risk premium and one or more additional premia, often based on factors such as size and perceived company-specific risk, depending on the facts of the exercise and the valuator’s analysis of them. An expression for the build-up approach was presented in Equation 2-5. A traditional specific implementation is as follows (see Hitchner 2006, 173): r_i = Risk-free rate + Equity risk premium + Size premium_i+ Specific-company premium_i

Exhibit 2-8 explains the logic for a typical case. The equity risk premium is often estimated with reference to equity indexes of publicly traded companies. The market’s largest market-capitalization companies typically constitute a large fraction of such indexes’ value. With a beta of 1.0 implicitly multiplying the equity risk premium, the sum of the risk-free rate and equity risk premium is effectively the required return on an average-systematic-risk large-cap public equity issue. In the great majority of cases, private business valuation concerns companies much smaller in size than public large-cap issues. Valuators often add a premium related to the excess returns of small stocks over large stocks reflecting an incremental return for small size. (The premium is typically after adjustment for the differences in the betas of small- and large-cap stocks to isolate the effect of size—a beta-adjusted size premium.) The level of the size premium is typically assumed to be inversely related to the size of the company being valued. When the size premium estimate is appropriately based on the lowest market-cap decile—frequently the case because many private business are small relative to publicly traded companies—the result corresponds to the return on an average-systematic-risk micro-cap public equity issue. An analysis of risk factors that are incremental to those captured by the previously included premia may lead the valuator to add a specific company premium. This risk premium sometimes includes a premium for unsystematic risk of the subject company under the premise that such risk related to a privately held company may be less easily diversified away.

Two additional issues related to required return estimation for private companies include (1) consideration of the relative values of controlling versus minority interests in share value and (2) the effect on share value of the lack of ready marketability for a small equity interest in a private company. Lack of marketability is the inability to immediately sell shares due to lack of access to public equity markets because the shares are not registered for public trading. (Marketability may also be restricted by contractual or other reasons.)

With respect to the potential adjustment for the relative control associated with an equity interest in a private company, any adjustments related to the type of interest (controlling or minority) are traditionally not made in the required return but, if appropriate, directly to the preliminary value estimate. The issues involved in such adjustments are complex with some diversity of viewpoints among practitioners. Given these considerations, a detailed discussion is outside the scope of this chapter.⁵⁸ Similarly, adjustments for lack of marketability are traditionally taken as an adjustment to the estimated value for an equity interest after any adjustment for the degree of control of the equity interest.

To illustrate, suppose an analyst is valuing a private integrated document management solutions company. The risk-free rate is 5 percent, the analyst’s estimate of the equity risk premium is 4.5 percent, and based on assets and revenues the company appears to correspond to the top half of the tenth decile of U.S. public companies, which is decile 10a in Exhibit 2-9 with market capitalizations of equity ranging from about $174 million to about $314 million.

Thus, ignoring any appropriate specific-company premium, an estimate of the required return on equity is 5% + 4.5% + 4.35% = 13.85%. A caution is that the size premium for the smallest decile (and especially the 10b component) may reflect not only the premium for healthy small-cap companies, but former large-cap companies that are in financial distress. If that is the case, the historical estimate may not be applicable without a downward adjustment for estimating the required return for a small but financially healthy private company.

A so-called modified CAPM formulation would seek to capture departures from average systematic risk. For example, if the analyst estimated that the company would have a beta of 1.2 if publicly traded, based on its publicly traded peer group, the required return estimate would be or 5% + 1.2 x 4.5% + 4.35% = 14.75%. This result could be reconciled to a simple build-up estimate by including a differential return of (1.2 - 1.0)(4.5%) = 0.9% in the specific-company premium.

In the first edition of the book from which this chapter was taken, IBM’s required return was estimated as 12.9 percent using the CAPM; the inputs used were an equity risk premium estimate of 5.7 percent, a beta of 1.24, and a risk-free rate of 5.8. Based on the YTM of 6.238 percent for the IBM 8.375s of 2019, a bond yield plus risk premium estimate was 9.2 percent.

Thus, updating Example 2-9 to 2007 shows that a lower equity risk premium estimate is offset by IBM’s higher current beta, leaving the required return on equity almost unchanged according to the CAPM. With IBM ’s credit rating unchanged, the lower level of interest rates in 2007 would have lowered the bond yield plus risk premium estimate, all else equal. Because a lower level of interest rates is consistent with lower opportunity costs for investors, that result would have been logical. Because IBM’s systematic risk had increased, a risk premium increase was justified and the cost of equity estimate was essentially unchanged.

The bond yield plus risk premium method can be viewed as a build-up method applying to companies with publicly traded debt. The estimate provided can be a useful check when the explanatory power of more rigorous models is low. Given that a company’s shares have positive systematic risk, the yield on its long-term debt is revealing as a check on the cost of equity estimate. For example, Abitibi-Consolidated Inc.’s 7.5 debentures (rated by Moody’s and Standard & Poor’s as B3 and B, respectively) mature in 2028 and were priced to yield approximately 11 percent as of mid-August 2007, so required return estimates for its stock (NYSE: ABY) not greater than 11 percent would be suspect.

The difficulty of required return and risk premium estimation in emerging markets has been previously mentioned. Of the numerous approaches that have been proposed to supplement or replace traditional historical and forward-looking methods, we can mention two.

The cost of capital is relevant to equity valuation when an analyst takes an indirect, total firm value approach using a present value model. Using the cost of capital to discount expected future cash flows available to debt and equity, the total value of these claims is estimated. The balance of this value after subtracting off the market value of debt is the estimate of the value of equity.

In many jurisdictions, corporations may deduct net interest expense from income in calculating taxes owed, but they cannot deduct payments to shareholders, such as dividends. The following discussion reflects that base case.

If the suppliers of capital are creditors and common stockholders, the expression for WACC is where MVD and MVCE are the current market values of debt and (common) equity, not their book or accounting values. Dividing MVD or MVCE by the total market value of the firm, which is MVD + MVCE, gives the proportions of the company’s total capital from debt or equity, respectively. These weights will sum to 1.0. The expression for WACC multiplies the weights of debt and equity in the company’s financing by, respectively, the after-tax required rates of return for the company’s debt and equity under current market conditions. “After-tax,” it is important to note, refers only to corporate taxes in this discussion. Multiplying the before-tax required return on debt (r_d) by 1 minus the marginal corporate tax rate (1 - Tax rate) adjusts the pretax rate r_d downward to reflect the tax deductibility of corporate interest payments that is being assumed. Because distributions to equity are assumed not to be deductible by the corporations, a corporation’s before- and after-tax costs of equity are the same; no adjustment to r involving the corporate tax rate is appropriate. Generally speaking, it is appropriate to use a company’s marginal tax rate rather than its current effective tax rate (reported taxes divided by pretax income) because the effective tax rate can reflect nonrecurring items. A cost of capital based on the marginal tax rate usually better reflects a company’s future costs in raising funds.

Because the company’s capital structure (the proportions of debt and equity financing) can change over time, WACC may also change over time. In addition, the company’s current capital structure may also differ substantially from what it will be in future years. For these reasons, analysts often use target weights instead of the current market-value weights when calculating WACC. These target weights incorporate both the analyst’s and investors’ expectations about the target capital structure that the company will tend to use over time. Target weights provide a good approximation of the WACC for cases in which the current weights misrepresent the company’s normal capital structure.⁶⁰

The before-tax required return on debt is typically estimated using the expected YTM of the company’s debt based on current market values. Analysts can choose from any of the methods presented in this chapter for estimating the required return on equity, r. No tax adjustment is appropriate for the cost of equity assuming payments to shareholders such as dividends are not tax deductible by companies.

A cash flow after more senior claims (e.g., promised payments on debt and taxes) have been fulfilled is a cash flow to equity. When a cash flow to equity is discounted, the required return on equity is an appropriate discount rate. When a cash flow is available to meet the claims of all of a company’s capital providers—usually called a cash flow to the firm—the firm’s cost of capital is the appropriate discount rate.

Cash flows may be stated in nominal or real terms. When cash flows are stated in real terms, amounts reflect offsets made for actual or anticipated changes in the purchasing power of money. Nominal discount rates must be used with nominal cash flows and real discount rates must be used with real cash flows. In valuing equity, we will use only nominal cash flows and therefore we will make use of nominal discount rates. Because the tax rates applying to corporate earnings are generally stated in nominal money terms—such and such tax rates applying at stated levels of nominal pretax earnings—using nominal quantities is an exact approach because it reflects taxes accurately.

Equation 2-14 presents an after-tax weighted average cost of capital using the after-tax cost of debt. In later chapters, we present cash flow to the firm definitions for which it is appropriate to use that definition of the cost of capital as the discount rate (i.e., rather than a pretax cost of capital reflecting a pretax cost of debt). The exploration of the topic is outside the scope of this chapter because the definitions of cash flows have not been introduced and explained.⁶¹

In short, in later chapters we will be able to illustrate present value models of stock value using only two discount rates: the nominal required return on equity when the cash flows are those available to common shareholders, and the nominal after-tax weighted average cost of capital when the cash flows are those available to all the company’s capital providers.

For two important approaches to estimating a company’s required return, the CAPM and the build-up model, the analyst needs an estimate of the equity risk premium. This chapter examined realized equity risk premia for a group of major world equity markets and also explained forward-looking estimation methods. For determining the required return on equity, the analyst may choose from the CAPM and various multifactor models such as the Fama-French model and its extensions, examining regression fit statistics to assess the reliability of these methods. For private companies, the analyst can adapt public equity valuation models for required return using public company comparables, or use a build-up model, which starts with the risk-free rate and the estimated equity risk premium and adds additional appropriate risk premia.

When the analyst approaches the valuation of equity indirectly, by first valuing the total firm as the present value of expected future cash flows to all sources of capital, the appropriate discount rate is a weighted average cost of capital based on all sources of capital. Discount rates must be on a nominal (real) basis if cash flows are on a nominal (real) basis.

Among the chapter’s major points are the following: