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Chapter 9 / Cost of Capital 129

9 COST OF CAPITAL WOLFGANG BALLWIESER AND JÖRG WIESE

GERMANY

(Revised version as of June 24, 2010)

Introduction The cost of capital comprises that of both debt and equity. Among other

things, it is needed for investment and financing decisions, business valuations, capital budgeting and determining “recoverable amounts” for impairment test-ing. While the cost of debt is normally determined by contracts (except for items such as defined benefit pension plans), the cost of equity, the expected risk-adjusted rate of return required by (often unknown) investors, is usually unobservable and has to be estimated by using financial models. The most im-portant of those are: the Capital Asset Pricing Model (CAPM), the Arbitrage Pricing Theory (APT), the Fama-French Three-Factor Model, numerous Build-Up models and Ex-ante or Implied-Cost-of-Capital (ICC) models.

Those models differ mainly with respect to their assumptions. CAPM is an equilibrium model with a large set of restrictive assumptions (e.g. perfect capital markets, risk-averse investors and a one-period planning horizon), whereas APT only requires the absence of arbitrage possibilities and is inde-pendent from equilibrium conditions. They also vary in their time perspective when implemented. Although CAPM is ex ante in theory, which means that it uses expectations of capital market participants to explain expected returns of financial assets in market equilibrium, practically it is used ex post through sta-tistical analyses to empirically estimate past betas and market risk premiums, which are extrapolated into the future. In this respect, past results determine or at least influence future expectations. This is totally different from ICC models which use as inputs market prices of shares at the valuation date and forecasts by financial analysts of future profits and dividends; the model then estimates the cost of equity.

The next section discusses theoretical factors which determine the cost of equity and that of debt. It is followed by short descriptions of the most im-portant models to empirically determine the cost of equity. Both components contribute to the Weighted Average Cost of Capital (WACC), which is em-ployed in discounting free cash flows and also to determine fair value (IAS 39 and IFRS 3) and value in use (IAS 36). Empirical data and examples are in-


cluded; additionally there is a discussion of the weaknesses of all models and a warning against drawing false conclusions.

Drivers Of Costs Of Capital This section deals first with the cost of debt, then with that of equity.

Cost of Debt

Lenders expect payments of their loans both on the due date and at the contracted amount. They face the risk that payments are made too late and in for a lesser amount, including the risk of a total loss of capital and interest. To protect themselves against such contingencies lenders restrict their loans to cer-tain amounts, require security and adjust their interest rates to compensate for the known risks; these adjustments are usually spreads over the risk-free rate of return reflecting the creditworthiness of the borrowers.

The risk-free rate is that an asset would yield without any default, timing or exchange rate risk; as such, it is a non-observable theoretical construct. It is usually measured by the rates of return on government securities, which have the lowest risk, in any particular currency. To derive the risk-free rate, two questions need to be answered: (a) which market data should be chosen? (b) since returns are time dependent, which maturity is appropriate?

Concerning the first, the obvious answer is an average of historical yields on government loans. That is, however, not appropriate, since invest-ment and financing decisions or business valuations are based on future cash flows. Therefore, they should be related to expected rates of return as there is no reason to assume historic rates can be reasonably extrapolated; the capital market environment at a valuation date will surely differ from past circum-stances.

Another possibility is the use of yields to maturity or internal rates of re-turn on government bonds at the valuation date. The problem is that their use implies (contrary to empirical data) a flat (time-independent) yield curve; con-sequently, the use of yields to maturity is not correct.

Therefore, spot rates, which explicitly reflect the term structure of inter-est rates, should be applied in discounting future cash flows; those are the yields to maturity of government zero-coupon bonds. In other words, they are defined by only two cash flows, one at the valuation date, the other at a later time; thus they are time dependent as is the yield curve. The latter shows the relationship between market interest rates and the time to maturity of zero-coupon bonds. In some markets, where so-called stripped government bonds


are traded, the relationship is directly observable. Stripping a bond means sepa-rating the coupons from the principal; a 10-year coupon bond generates ten interest strips with maturities of one to ten years and a single ten-year principal strip; each strip consists of two cash flows and thus represents a spot rate. However, the trading volume of government strips is normally much lower than that of equivalent coupon bonds; hence, the majority of the European cen-tral banks apply the Svensson method1 to estimate the yield curve from implied zero-coupon yields of a series of government coupon bonds with different ma-turities. In valuing businesses, German auditors are required to apply this method. In it, the relationship between the continuous spot rate ic at valuation date t and time to maturity T is estimated by the equation

1 1 2

1 2

T T TT T

c 0 1 2 31 1 2

1 e 1 e 1 ei t, t T,b e e

T T T

, (1)

where 0 1 2 3 1 2b , , , , , is the parameter vector to be estimated and

e is the Eulerian number (2,718…); this allows spot rates to be estimated at any time. The continuous spot rates can be mathematically easily converted into discrete spot rates. Exhibit 9.1 shows the yield curve as estimated by the Euro-pean Central Bank (ECB) as at 1 December 2008.

Exhibit 9.1: Yield curve as at 1 December 2008

1,85

2,35

2,85

3,35

3,85

4,35

0 5 10 15 20 25 30 35 40 45 50

maturity in years

spot

rat

es in

%

yield curve (Svensson-Method)

Source: Authors’ calculations, using parameter estimated by ECB (discrete spot rates).

To use bonds with sufficient market depth, the ECB covers maturities of from three months to 30 years; beyond that, an assumption on the shape of the curve has to be made; in exhibit 9.1, as is common, the extended curve is

1 Lars E. O. Svensson, “Estimating and Interpreting Forward Interest Rates: Sweden 1992-1994,”

NBER Working Paper No. 4871, Cambridge, MA, September 1994.


treated as flat. The resulting spot risk-free rates can be applied to develop indi-vidual discount rates for periodic future cash flows. Alternatively, the estimated yield curve can be aggregated to a single “flat” rate of return that is consistent with the Svensson method, but not time dependent. For this, a simplified dis-counted cash flow model is used; this assumes cash flows increase at a constant rate (g) over an infinite time horizon.2 This model is described by the following equation:

t

0 00 t

t 1 d,t

CF 1 g CF 1 gV i g

i g1 i

, (2)

where

V0 = business value,

CF0 = cash flow in t = 0,

id,t = discrete spot rate for period t,

i = flat risk-free rate of return,

g = constant growth rate of cash flows.

The first present value in equation (2) is calculated with period specific spot rates and represents the correct business value. Equating this amount with the simplified formula in the right hand side of (2) and solving the equation yields the flat rate (i). The value of CF0 can be chosen arbitrarily, because the cash flows in equation (2) cancel out.

On 1 December 2008, the method yields the following flat rates for cer-tain selected cash flow growth rates:

Exhibit 9.2: Risk-free rates of return as at 1 December 2008

Source: Author’s calculations. This method also answers the second question concerning the maturity

of bonds to be chosen: market data should be used when it is available for gov-ernment bonds with sufficient trading depth; in most countries, those with a maturity of less than 30 years meet this requirement.

The spread added to the risk-free rate of return depends on the risks of which the lender is aware. An indicator of the required spread is given by the grade given to a borrower by rating agencies. They differentiate between many

2 Martin Jonas, Heike Wieland-Blöse and Stefanie Schiffarth, Basiszinssatz in der Unternehmensbe-

wertung, Finanz Betrieb 7 (2005): 647–653

Growth Rate (g) Flat Rate (i)

0.00 % 4.17 %

1.00 % 4.21 %

2.00 % 4.25 %


categories of risk which are summarized as an investment or speculative grade; the most important rating agencies, Moody’s, Standard & Poor’s (S&P) and Fitch, use the following categories:

Exhibit 9.3: Rating Categories of Established Rating Agencies

Moody’s S&P Fitch Debt is

Aaa AAA AAA of highest credit quality. Assigned only in case of exceptionally strong capacity for payment of financial commitments.

Aa AA AA of very high credit quality. AA indicates very strong capacity for payment of financial com-mitments.

A A A of high credit quality. A denotes expectations of low credit risk. The capacity to meet its finan-cial commitments is still strong. However, this may be more vulnerable to changes in circum-stances or economic conditions than obligations in higher rated categories. in

vest

men

t gra

de

Baa BBB BBB of good credit quality. The lowest investment grade indicates expectations of low credit risk. The capacity for payment of financial commit-ments is considered adequate but adverse changes in circumstances and economic condi-tions are more likely to impair this capacity.

Ba BB BB of speculative credit quality. There is a possi-bility of credit risk developing, however, busi-ness or financial alternatives may be available to allow financial commitments to be met.

B B B of highly speculative credit quality. The bor-rower currently has the capacity to meet its fi-nancial commitments. Adverse conditions likely will impair its capacity or willingness to meet its commitments.

Caa CCC CCC vulnerable to non-payment within one year and depends on favorable conditions to meet the financial commitment.

Ca CC CC currently highly vulnerable to non-payment.

spec

ula

tive

gra

de

C C C extremely vulnerable to non-payment. Rating may be used to cover a situation where a bank-ruptcy petition has been filed or similar action has been taken but payments on this obligation are being continued.


D RD defaulted. The D or RD rating is not prospec-tive; rather, a default has actually occurred.

1,2,3 in the areas of Aa to

Caa

+/- in the

areas of AA

to CCC

+/- in the

areas of

AA to

CCC

Modifiers may be appended to a rating to denote relative status within the major rating categories.

The following exhibit shows credit spreads of 5-year government bonds denominated in Euro (€) and US dollars ($) compared to German government bonds and US Treasuries as of 27 February 2009.

Exhibit 9.4: Credit Spreads of Government Bonds from Several Countries Compared to German Government Bonds (€) and US Treasury Bonds ($)

Rating (S&P) Currency Credit De-fault Spreads (basis points)

Countries

AAA 0 Germany

AAA (3) – 273 France, Netherlands, Austria, Ireland

AA+, AA 51 – 133 Spain, Belgium, Slovenia

A+, A, A- 46 – 282 Portugal, Italy, Czech Republic, Greece, Poland

BBB, BBB- 238 – 713 Morocco, Lithuania

BB+

€

701 – 1000 Latvia, Romania

AAA 0 USA

AA+, AA 92 – 277 New Zealand, Abu Dhabi

A+,A, A- 160 – 350 China, Chile, Malaysia, Korea

BBB+, BBB, BBB-

210 – 673 Thailand, Mexico, South Africa, Russia, Bulgaria, Peru, Colombia, Brazil

BB+, BB- 399 – 569 Turkey, Egypt, Indonesia

B-

$

427 Lebanon Source: Bloomberg

In early 2003 S&P modified its treatment of pension obligations in the rating process. Previously it did not include them in its financial ratios; now, S&P regards any part of a firm‘s pension obligation not covered by pension assets as debt. This had a significant impact on many firms. “Most promi-nently, German industrial conglomerate ThyssenKrupp AG experienced a two-notch downgrade, resulting in the assignment of non-investment grade to its


bonds.”3 According to management, interest increased by EUR 20 million per year; two years later investment grade was restored.

Though ratings help in determining the interest spread, it has to be kept in mind that rating agencies concentrate their activities on big, very often global firms. Therefore, many borrowers in which (potential) lenders are inter-ested in are not covered. In recent times, because of the financial market crisis there has been great doubt on the reliability of the conclusions of rating agen-cies with respect to so-called structured products composed of mortgage-loans. Rational lenders consider opportunity costs; they only make a loan if the ex-pected rate of return is at least as high as it could be gained in the best alterna-tive.

Cost of Equity Equity investors receive income which remains after all claims from contrac-

tual and non-contractual sources (like taxation). This residual income is not stable with respect to time or amount and reflects numerous risks.

Equity Risks

The most significant risks incorporated in to the cost of equity are:

Operational (or investment) Financial (or capital structure) Systematic (in contrast to idiosyncratic) Currency Location (or country)

Operational risks result from the business model of the firm, independ-ently of its financing. They depend to some extent on the sector in which the firm operates, such as airlines or computer chips. However, this classification, on its own, is too coarse to be effective because various members of the same sector usually face different risks. One only has to compare Lufthansa and Ali-talia or Intel and Infineon.

Financial risks result from the capital structure, especially the rates and repayment terms of the debt portion of that in most cases has to be serviced independently of the uncertain state of product or service demand, competition and regulation.

3 Bernhard Pellens and Nils Crasselt, “Funding Strategies for Defined Benefit Plans and the Meas-

urement of Leverage Risk”, in: Ballwieser, Wolfgang (Ed.): Current Issues in Financial Reporting and Financial Statement Analysis, Special Issue 2/05 of Schmalenbach Business Review, (Düssel-dorf, Frankfurt am Main: Handelsblatt), pp. 3-33, p. 5.


Systematic risk is defined in CAPM. One of the assumptions of this model is that all investors construct efficient portfolios which have (a) mini-mum risk given a certain expected return or (b) the maximum expected return given a certain level of risk. Therefore, at the equilibrium a premium is only paid for those risks that are not diversifiable through portfolio composition. The non-diversifiable portion is systematic risk; the diversifiable portion, at-tributed to individual securities, is idiosyncratic risk.

Operational and financial risks are both idiosyncratic because the risk-bearing entity is not considered in a portfolio context. According to CAPM this is not relevant for determining the cost of equity capital. Instead the possibility of reducing it by means of portfolio composition has to be used in order to re-duce the overall risk; the remaining component is systematic risk.

Currency risks result from positions denominated in foreign currencies. Any investor in the United States with positions in Euro based entities has a foreign currency risk as the exchange rate between the US-$ and the Euro is not stable. The same is true for British firms with sales in the United States: during 2008 the amount for £ 1.00 declined 26% from $1.98 to $1.46. As classical CAPM cannot consider currency risks, a Multi-Currency-CAPM has been de-veloped, but is too complex to apply in practice.

Location risk is a consequence of country or region-specific circum-stances, especially with respect to employment rates, inflation, money transfer and political stability. Usually it is measured by means of scores which com-bine numerous very different factors.

It is almost impossible to combine all risk elements in a meaningful way. While idiosyncratic risk can be measured by standard deviations or vari-ances of probability distributions of cash flows other measures such as correla-tion coefficients are needed for systematic risk. Those can also be used for cur-rency risks but none of them can easily be combined with measures of country risk since the latter ones are scores using an ordinal instead of a cardinal scale.

Size-dependent risks4 are not part of those factors. This is important since practitioners sometimes use Small Stock Risk Premiums (SSRP), which depend on the size of a listed company.5 SSRP are not part of CAPM and go back to multi-factor models. This point is discussed in detail in the section on the Fama-French Three-Factor Model.

4 Rolf W. Banz, “The Relationship between Return and Market Value of Common Stocks”, Journal of

Financial Economics 9, (1981): 3–18. 5 Shannon P. Pratt and Roger J. Grabowski, Cost of Capital – Applications and Examples (Hobo-

ken, 3rd ed. NJ: John Wiley & Sons, 2008).


Cost of Equity Methods The five often discussed models to establish the cost of equity for an en-

tity are discussed in detail below:

Capital Asset Pricing Model Arbitrage Pricing Theory Farma-French Three Factor Model Build-Up Models Implied-Cost-of Capital Models

Capital Asset Pricing Model (CAPM)

CAPM is the most-often used but, as it depends on the assumption of perfect capital markets, not necessarily the most effective model for the cost of equity capital. Its dominance might be explained by it being easy-to-understand. The following equation determines the expected rate of return of a security jμ r in capital market equilibrium:

2j f j M f j jM Mμ r = r +β μ r r β σ σ , (3)

where

rf = riskless interest rate,

Mμ r = expected rate of return of the market portfolio M,

jβ = beta; risk factor of a share of company j,

jMσ = covariance between returns of security j and returns of the market portfolio,

2Mσ = variance of return on market portfolio.

The term M fμ r r is the market risk premium. Beta is intended to

measure the systematic risk of a share and shows how sensitive its expected return is to changes in that of the market portfolio Mμ r . CAPM is theoreti-

cally an ex ante model, but its application is usually based on ex post data, since its parameters are not observable.

For empirical application, equation (3) is written as a regression line with beta as its slope:

j,t f , t j f , t j M,t j,t j j M,t j,tr r β r β r e a b r e (4)

where t is a moment in time, a is the intercept of the regression line and e is a random error term with (by assumption) an expected value of zero and a constant variance

j

2eσ . As illustrated in exhibit 9.5, parameters a and b are esti-


mated by regression analyses from a sample of historical returns on the particu-lar asset and the market portfolio.

Exhibit 9.5: Characteristic line of security j.

Source: Authors’ construction.

To assess the statistical accuracy of the estimates, standard errors of the coefficients a and b as well as the R2 of the trend line have to be considered; furthermore, estimated coefficients should be tested for significance.

In determining market risk premium and beta, several critical decisions need to be made concerning the following:

1. Risk-free rate of return 2. Stock index proxy for the market 3. Intervals for measuring returns 4. Estimation period (including the base year and the length) 5. Calculating the average market risk premium 6. Historic or adjusted betas

The factors determining appropriate choices are:

Risk-Free Rate of Return Since CAPM is a single-period model, the use of long-term bond yields

is not appropriate for the market risk premium. Multi-period applications re-quire a single-period risk-free rate; furthermore the yield curve is usually not flat. To determine the risk-free rate, spot rates from the Svensson method

r M

r j

x

x

x

x

x

x

xx

xx

x

x

bj,t

1random error ej,t

x

x

aj


should be converted into implied forward rates, to be used consistently in cal-culating expected returns as well as the risk premium.

Another problem arises from unanticipated interest rate changes. While stocks usually benefit from rate cuts in terms of capital gains, bonds with a short remaining maturity do not necessarily benefit. Thus, the market risk pre-mium, the difference between stock and bond returns, may be overestimated if a short-term bond index is used for the risk-free interest rate.6

Stock Index Proxy for the Market The stock index chosen should be as broad as possible to serve as a good

proxy for the market portfolio which theoretically contains all risky assets worldwide. For an index to be representative of the market, it must be value-weighted. Possibilities are: in the US, the S&P 500 or the New York Stock Ex-change (NYSE) Composite; internationally, the MSCI World, which covers 23 countries. The problems of a U.S.-based index are that its capital market condi-tions may not be relevant for other countries or for other currencies (since the dollar is not stable with respect to e.g. the Euro or the Yen).

Interval for Measuring Returns Estimates of beta are affected by the intervals for measuring returns and

their frequency. Data is available and may be collected daily, weekly, monthly, quarterly or annually. While preferable for actively traded issues, daily data is not appropriate for illiquid shares, since the estimate may be biased down-wards7; in those cases, weekly or monthly intervals should be chosen. For short intervals there may be problems from a systematic relationship between the frequency and the resulting beta; the longer the interval, the smaller the prob-lem. Even if longer intervals are chosen, estimated betas for illiquid shares are often biased; figures of less than 0.5 are normally only observed for this group8. However, for a given estimation period, longer intervals imply an in-creased standard error. Thus, the estimation period and the measurement inter-val have to be determined simultaneously. For short intervals, a lesser estima-tion period is enough to obtain sufficient data for a valid beta. Since beta varies

6 Ekkehard Wenger, “Verzinsungsparameter in der Unternehmensbewertung - Betrachtungen aus

theoretischer und empirischer Sicht”, Die Aktiengesellschaft, Sonderheft (2005): 9–22. 7 Peter Zimmermann, Schätzung und Prognose von Betawerten (Bad Soden/Ts.: Uhlenbruch, 1997),

pp. 99-120. 8 Ibid.


over time, separate figures should be calculated for varying periods (e.g. 250 days, 52 weeks, 60 months) and numbers of observations. Longer periods are, in principle, preferable but bear the risk of structural breaks in the time series.

Estimation Period There is no theoretical base for selecting a particular historic estimation

period. To obtain a very large sample it is desirable for it to be as long as pos-sible, say from 1900 to 2008; however, there is a trade-off between the estima-tion period and possible biases from structural breaks in the time series such as the Great Depression and two world wars. Therefore starting shortly after 1950, when Europe was at peace and the Treasury-Fed Accord (1951) had terminated the wartime fixed interest rate policy in the US, seems appropriate. Stehle9 who did one of the most comprehensive studies on German market risk premiums, adopted 1955 to 2003. Tied to the estimation period is the choice of base year. Excluding a short period or even a single year can easily change the reported premium by 1.0% or 1.5%.

Dimson/Marsh/Staunton10, e.g., chose 1900 to 2005; exhibit 9.6 summa-rizes their results.

9 Richard Stehle, “Die Festlegung der Risikoprämie von Aktien im Rahmen der Schätzung des Wertes

von börsennotierten Kapitalgesellschaften“, Die Wirtschaftsprüfung 57: 906-927. 10 Elroy Dimson, Paul Marsh and Mike Staunton, “The Worldwide Equity Premium: A Smaller Puz-

zle”, working paper, London Business School, 7 April 2006.


Exhibit 9.6: Annualized Equity Premiums for 17 Countries, 1900-2005

Source: Dimson/Marsh/Staunton (2006), p. 18.

Another frequently used database of historical returns (1926-2008) is from Ibbotson Associates. However many practitioners chose shorter periods as they provide a more up-to-date figure, even though the premium estimate will suffer from greater noise. Damodaran11 thus concludes: “The cost of using shorter time periods seems … to overwhelm any advantages associated with getting a more updated premium.”

Calculating the Average Market Risk Premium

The appropriate average is subject to controversy.12 Academics and practitioners disagree as to which of (a) the (higher) arithmetic mean, (b) the (lower) geometric mean or (c) a mixture of both, is superior. The differences

11 Aswath Damodaran, Investment Valuation (Hoboken, NJ: John Wiley & Sons, 2002), 161. 12 Marshall E. Blume, “Unbiased Estimators of long-run expected Rates of Return”, Journal of Busi-

ness Finance and Accounting 69 (1974): 634–638; Ian Cooper, “Arithmetic versus geometric Mean Estimators: Setting Discount Rates for Capital Budgeting”, European Financial Manage-ment 2 (1996): 157–167.


are not negligible, as shown in exhibit 9.7 and the German data, calculated by Stehle set out below; this uses two different measures of the market DAX and Composite DAX (CDAX):

Exhibit 9.7: Estimated Market Risk Premium in Germany

1955-2003

Market Risk Premium Arithmetic Mean

Market Risk Premium Geometric Mean

CDAX 5.46 %

= 12.40 % - 6.94 %

2.66 %

= 9.50 % - 6.84 %

DAX 6.02 %

= 12.96 % - 6.94 %

2.76 %

= 9.60 % - 6.84 % Source: Stehle 2004, p. 921.

To answer the question of which is preferable for estimating the market risk premium one must investigate the stochastic process which stock returns follow. The arithmetic mean is correct if returns are (a) stochastically inde-pendent, (b) identically distributed over time and (c) their expected value must be known. Empirical evidence suggests that returns are negatively correlated or stochastically dependent13, therefore the correct amount is between the arithme-tic and geometric mean. Since assumptions concerning the stochastic process cannot be verified a pragmatic deduction from the arithmetic mean is recom-mended. However, stochastically dependent returns are incompatible with CAPM, while stochastically independent, identically distributed returns are consistent with the CAPM but imply in the long run extremely right skewed distributions of the cash flows from investment projects.14

Published Betas In the U.S., valuators can obtain public company Betas from Barra,

Morningstar, Merrill Lynch, Value Line, Standard & Poor’s, Bloomberg, Ib-botson and CompuServe. There is no uniformity regarding the market proxy, adjustments, and/or period and frequency of data used by these organizations. Accordingly, for the same entity they will vary, sometimes substantially.

13 Eugene F. Fama and Kenneth R. French, “Permanent and temporary Components of Stock Prices”,

Journal of Political Economy 81 (1988): 246–273. 14 Eugene F. Fama, “Discounting Under Uncertainty”, Journal of Business 69 (1996): 415–428.


Historic or Adjusted Betas

Raw betas, estimated by regression analysis, are often adjusted as em-pirical evidence suggests that they tend to revert over time to a mean.15 This might be the industry average, or 1.0 (that of the market); Bloomberg therefore reports an adjusted beta that assumes convergence towards the market by modi-fying, somewhat arbitrarily, the “raw” figure, as follows:

adjusted Beta 2 3 raw Beta 1 3 1 . (5)

Alternatively, the so-called Vasicek-adjustment used by Ibbotson ap-plies the Bayesian theory of estimation.16 The concept is that betas with high standard errors (often high-beta stocks) are more strongly adjusted toward a mean than those with a low standard error. Most beta services, e.g. Barra, Value Line, Morningstar or S&P, employ similar procedures.

Another adjustment is necessary for betas derived from a peer group. Although in theory, the only relevant figure is that for the entity, that of a peer group is often used for privately owned firms. A proper peer group has the same systematic risk as the business to be valued. However, their individual betas reflect the different capital structures of each member. These levered be-tas ( Lβ ) have to be adjusted to remove the effect of a firm’s leverage on its beta; therefore, so-called Hamada-betas are calculated.17

The first step is to convert Lβ into a beta of a virtual unlevered company ( Uβ ) by the following equation

U L Dβ β 1 1 T

E

, (6)

where

T = the peer’s (compound, effective) tax rate which reflects the benefits of a (partial) deductibility of interest,

D = the peer’s market value of debt,

E = the peer’s market value of equity. The second step is a adjust the unlevered beta to reflect the capital struc-

ture of the subject entity by the formula

L U Dβ β 1 1 T

E

, (7)

15 Roger Buckland and Patricia Fraser, “Political and Regulatory Risk: Beta Sensitivity in U.K. Elec-

tricity Distribution”, Journal of Regulatory Economics 19 (2001): 5–25. 16 Oldrich A. Vasicek, “A Note on Using Cross-Sectional Information in Bayesian Estimation of Secu-

rity Betas”, Journal of Finance 28 (1973): 1233-1239. 17 Robert S. Hamada, “The Effect of the Firm's Capital Structure on the Systematic Risk of Common

Stocks,” Journal of Finance 27 (1972): 435–452.


Example:

Assume the following published data for guideline company Ascendo SA:

Published levered Beta: 2.1; tax rate: 24%; capital structure: 60% debt, 40 % equity

βu = 2.1/[1 + (1-.24)(.60/.40)] = 2.1/[1 + 1.14] = 2.1/2.14 = 0.98.

Assuming the subject company has a capital structure of 35 % debt and 65 % equity both at market value and a 30% tax rate

βL = 0.98 [1 + (1-.30) (.35/.65)] = 0.98 [1+0.70 0.5385] = 0.98 1.3769 = 1.35.

The fundament of equations (6) and (7) are the results of Modi-gliani/Miller.18 According to Modigliani/Miller, the relationship between the cost of equity of the levered company L

Er and the unlevered company UEr is

formalized as

L U UE E E f

Dr r r r 1 T

E . (8)

Inserting CAPM (3) into (8) yields equations (6) and (7). Note that equa-tion (8) does only hold for the case of a perpetuity. In the case of variable cash flows, more complex adjustment formulas19 have to be applied. Debt-equity ratios in equations (6) through (8) are theoretically based on market values in terms of values derived by a discounted cash flow approach. In practice, book values of equity or the market capitalization are often used instead, which both are not equal to a market or business value necessary for using the above-mentioned equations.

CAPM is the most often used model to determine cost of equity. Its ad-vantages lie in its sound theoretical basis and the ability to use market data. There are, however, serious disadvantages; they include: restrictive assump-tions, unanswered empirical validity and the subjective assumptions needed to apply the model. These criticisms should not lead to a complete rejection as the alternatives, discussed below, also have serious shortcomings.

18 Franco Modigliani and Merton H. Miller, ”Corporate Income Taxes and the Cost of Capital: A

Correction,” American Economic Review 53 (1963): 433-443. 19 Isik Inselbag and Howard Kaufold, “Two DCF Approaches for Valuing Companies Under Alterna-

tive Financing Strategies (And how to Choose Between Them)”, Journal of Applied Corporate Fi-nance 10 (1997): 114-122.


Arbitrage Pricing Theory (APT) CAPM states that security rates of return are linearly related to a single

factor, the expected rate of return on the market. The Arbitrage Pricing Theory (APT) formulated by Ross20 is based on a similar intuition but assumes that several common risk factors generate rates of return:

j 0 j1 1 0 j2 2 0 jN N 0μ r = λ +β μ δ λ β μ δ λ ... β μ δ λ , (9)

where

jμ r = expected rate of return of security j,

0λ = constant,

kμ δ = expected return on a portfolio with unit sensitivity to factor k (k =

1,…,N) and zero sensitivity to all other factors,

k 0μ δ λ = risk premium associated with factor k,

jkβ = sensitivity of security j to factor k (factor loading).

APT is built on two key concepts: no-arbitrage principle and multifactor risk model21 with the constant 0λ equal to the riskless rate of return22. Although APT assumes several risk factors, it is not a generalization of CAPM as the two models are based on different sets of assumptions.23 While CAPM is applied by means of a single regression, APT is applied by multiple regression. Since the factors are undefined, they are mostly specified by macroeconomic or industry specific items, such as the term spread, growth in industrial production, unex-pected inflation or the credit spread (the yield difference between corporate and government bonds). The number of factors is arbitrary but must be much smaller than the assets under consideration.

Compared to CAPM, the assumptions underlying APT are less restric-tive. Furthermore, APT allows the expected rates of return to depend on more than one factor and the market portfolio does not necessarily play a role; it, unlike CAPM, can also be easily extended to a multi-period framework. The central shortcoming is the non-specification of the risk factors and there is no

20 Stephen A. Ross, “The Arbitrage Theory of Capital Asset Pricing”, Journal of Economic Theory

13 (1976): 341-360; Stephen A. Ross, “Risk, Return, and Arbitrage”, in: Friend, Irwin/Bicksler, James L. (Eds.): Risk and Return in Finance, Cambridge 1977, pp. 189-218.

21 Haim Levy and Thierry Post, Investments (Harlow et al.: Prentice Hall, 2005). 22 Stephen A. Ross, “The Arbitrage Theory of Capital Asset Pricing”, Journal of Economic Theory

13 (1976): 341–360. 23 Haim Levy and Thierry Post, Investments (Harlow et al.: Prentice Hall, 2005).


consensus about which have the best explanatory power. Since tests of APT, like those of CAPM, are always joint tests of the theory, methodology and quality of the data, unambiguous empirical evidence for or against APT is dif-ficult to find.24

Fama-French Three-Factor Model APT is a multifactor risk model using historical returns and factors.

Thus, factor loadings can be estimated by multiple regressions. Another multi-factor risk model that has gained great attention is the Three-Factor Model of Eugene Fama and Kenneth French25:

j f j M f j jμ r r b μ r r s μ SMB h μ HML (10)

This assumes that the excess return of the security j over the risk-free in-terest j fμ r r rate is a linear function of three factors:

(a) the excess return of a broad market index (as a proxy of the market portfolio) over

the risk-free rate M fμ r r ,

(b) the difference between the expected returns on a portfolio of small and large

stocks μ SMB (“small minus big”),

(c) the difference between the expected returns on a portfolio of high and low book-

to-market stocks μ HML (“high minus low”):

Parameters bj, sj and hj are factor sensitivities estimated by a time-series regression. While the first is CAPM’s risk premium, the other two are moti-vated by empirical findings. As a consequence, their economic interpretations remain unclear. The discussion of this model is combined with that of the small stock risk premium (SSRP) mentioned above. It must be borne in mind that: (a) a SSRP is not compatible with the CAPM and (b) SSRPs are not reliably meas-urable.

As investors are presumed to be perfectly diversified, in CAPM, on a conceptual level, a single risk factor (beta) captures all relevant, systematic, risks of the business. Consequently, there is no premium for any additional, unsystematic, risks which might be associated with smaller entities. This is

24 Ibid. 25 Eugene F. Fama and Kenneth R. French, “The Cross-Section of Expected Stock Returns”,Journal

of Finance 47 (1992): 472–465; Eugene F. Fama and Kenneth R. French, “Common Risk Factors in the Returns on Stocks and Bonds”, Journal of Financial Economics 33 (1993): 3–56; Eugene F. Fama and Kenneth R. French, “Multifactor explanations of asset pricing anomalies”, Journal of Finance 51 (1996): 55–84.


confirmed by Stehle26, who states CAPM has to be rejected when empirical size effects are found. Fama and French (1992, 1993) report evidence of a size effect using their Three-Factor Model, not CAPM; consequently SSRP is not associated with CAPM.

The size effect has not been proven empirically, as it “could simply be a coincidence”27. A recent paper by van Dijk28 states: “Although many of the early empirical studies identify a significant and consistent size premium in U.S. equity returns, the overall evidence of a size effect in the U.S. is not over-whelming and several papers report that the effect has disappeared after 1980. […] the instability of the size effect reinforces the concern that data snooping biases may have played a role in the discovery of the effect.” Further: “Con-cluding, we assess the empirical evidence for the size effect to be consistent at first sight, but frail at closer inspection. More empirical research is needed to establish the robustness of the size effect in international equity markets.” For these reasons the German Institute of Certified Public Accountants (IDW), de-clines to add a SSRP to the discount rate.29

Build-Up Models For small and medium-sized companies which are often not publicly

traded, valuators in practice often use Build-Up models.

Ibbotson’s Build-Up Method

The most common in the United States is Ibbotson’s Build-Up Method, which derives a cost of equity by adding together a market-based rate of return (systematic risk) and premiums for unsystematic risks associated with the sub-ject company; the basic formula is:

j f j jμ r = r +μ ERP +μ SSRP +μ SCR , (11)

where

ERP = expected equity risk premium for market,

26 Richard Stehle, “Der Size-Effekt am deutschen Aktienmarkt“, Zeitschrift für Bankrecht und

Bankwirtschaft 9 (1997): 237–260; also Rolf W. Banz, “The Relationship between Return and Market Value of Common Stocks”, Journal of Financial Economics 9 (1981): 3–18.

27 Richard A. Brealey, Stewart C. Myers and Franklin Allen, Principles of Corporate Finance (New York et al., 8. ed.: McGraw-Hill, 2006), p. 203.

28 Mathijs A. van Dijk, “Is Size Dead? A Review of the Size Effect in Equity Returns”, Working Paper, RSM Erasmus University February 2007, ssrn, pp. 29, 31.

29 IDW (Ed.), WP Handbuch 2008 – Wirtschaftsprüfung, Rechnungslegung, Beratung, Vol. II (Düs-seldorf, 13. ed.: IDW Verlag).


SSRP = small stock risk premium (size premium),

SCR = specific company risk.

ERP is the long-term premium over the risk-free rate an investor would expect to receive from investing in a portfolio of large equity securities, such as the S&P 500. The SSRP is adjusted for CAPM’s β , so that excess returns of small companies that can be explained by their higher β s are not included; specific entity risks are, as well as the size premiums unsystematic risk. The SCR premium measures the risk arising from specific factors, such as charac-teristics of the industry or the firm. These may include high volatility (meas-ured by the standard deviation) of returns, poor management, key person and supplier dependence, pending lawsuits, regulatory situations or lack of access to capital. As the CAPM assumes those unsystematic risks can be eliminated through diversification, the Build-Up Method is not consistent with CAPM.

The Risk Rate Component Model (RRCM)

The Risk Rate Component Model (RRCM) is a similar approach less widely used in the United States, except for valuing small companies or in the valuation process in countries where there is only unreliable information about the rates of return of publicly-traded companies. The RRCM begins with taking a risk-free rate of return and adds a weighted average risk premium for several risk factors to that rate. These risk factors include competition, financial strength, management ability and depth and profitability and stability of earn-ings. Risk premiums for these factors are reported in ranges. These ranges ob-viously require subjective conclusions.30

Implied-Cost-of-Capital Models (ICCM)

While applications of CAPM or APT are based on ex post data, Implied-cost-of-capital models determine cost of equity by means of a forward-looking approach. They use current share prices and forecasts of future profits or divi-dends by financial analysts and solve for the discount rate that equates the pre-sent value of the future cash flows to the share price. The method is not at all

30 For an application of RRCM see material from Chapter 5, “Capitalization/Discount Rates,” of the

course “Business Valuations: Universal and Fundamental Applications,” © 2007 IACVA.


new31 but is experiencing a revival through a variety of models based on divi-dends, residual income or earnings.

They use as inputs:

(1) (consensus-) forecasts of earnings per share (eps) provided by financial analysts (e.g. I/B/E/S data),

(2) forecasts of dividends per share (dps),

(3) expected growth rates of earnings, dividends or residual income for the time beyond the explicit forecast period,

(4) current share prices.

All of them share the dividend discount model as their theoretical basis; this determines the value of a share P as the present value of future dps:

jt

j0 tt 1

j

dpsP

1 r

. (12)

The cost of equity implied by the share price and the projected dividends is obtained by solving equation (12) for rj or by iteration. The equity risk pre-mium is rj less the risk-free rate of return. Since dps projections are only avail-able for three to five years, an assumption about their subsequent growth has to be made; assuming constant growth at rate g from the valuation date equation (12) reduces to (Gordon/Shapiro 1956)

j1j0 j

j

dpsP r g

r g

, (13)

so that cost of equity is:

j1j

j0

dpsr g

P . (14)

Alternatively, models based on residual income rps are used, e.g.

jt j jt 1 jtj0 j0 j0t t

t 1 t 1j j

eps r bps rpsP bps bps

(1 r ) (1 r )

, (15)

where

bps = book value per share,

eps = projected earnings per share.

Equation (15) is theoretically equivalent to the dividend discount model (12) if the so-called clean-surplus relation is fulfilled:

31 Myron J. Gordon and Eli Shapiro, “Capital Equipment Analysis: The Required Rate of Profit”,

Management Science 3 (1956): 102–110.


t t tt 1bps bps eps dps (16)

Unfortunately, this is not in conformity with IFRS. Equations (12) and (15) assume that residual income or dividends can be explicitly forecasted in-definitely, which is not true in practice. Hence, two or three stage models are used; in those (residual) earnings are forecasted explicitly for some periods and extrapolated at a fixed growth rate subsequently. Sometimes a reversion to the mean for earnings is assumed as an intermediate stage. Claus/Thomas32 for ex-ample employ the following two-stage residual income model

5jt j jt 1 j5 j j4

j0 j0 jt 5t 1 j j j

eps r bps (eps r bps )(1 g)P bps r g

(1 r ) (r g)(1 r )

, (17)

where g is the assumed long-term growth rate.

The following exhibit shows how the implied market risk premium var-ies over the years:

Exhibit 9.8: German implied market risk premiums according to the models of Claus/Thomas (CT), Gebhardt/Lee/Swaminathan (GLS), Ohlson/Jüttner-Nauroth (OJN) and Price-Earnings-Growth Model (PEG).

Source: Reese (2007), p. 102.33

32 James Claus and Jacob Thomas, “Equity Premia as Low as Three Percent?, Evidence from Ana-

lysts’ Earnings Forecasts for Domestic and International Stock Markets”, Journal of Finance 56 (2001): 1629–1666.

33 Raimo Reese, Schätzung von Eigenkapitalkosten für die Unternehmensbewertung (Frankfurt am Main: Peter Lang, 2007; William R. Gebhardt, Charles M.C. Lee and Bhaskaran Swaminathan, “Toward an Implied Cost of Capital”, Journal of Accounting Research 39: 135-176; James A. Ohlson and Beate Jüttner-Nauroth (2005), “Expected EPS and EPS Growth as Determinants of Value”, Review of Accounting Studies 10: 349-365.

-0,02

0,03

0,08

0,13

0,18

0,23

1988 1990 1992 1994 1996 1998 2000 2002 2004

rpCT rpGLS rpOJN rpPEG


The advantages of such models are its forward-looking perspective with both cash flows and discount rates determined by expectations. Their disadvan-tages are circularity, sensitivity to the particular model and dependency on con-sensus forecasts. The circularity results from the fact that valuations are not necessary, once implied capital costs have been derived. This cost of equity capital can only be calculated if new investments have the same risks as the current business (also applies to CAPM); that is not always the case. The sensi-tivity to the particular model is shown in Exhibit 9.8. The reason is that an as-sumed growth rate of “g” leads to different results when applied to dividends, eps or residual income. The use of market capitalization neglects the difference between this and the value of a business. In practice control premiums, often substantial, are paid which contradicts the basis of using market capitalization. Furthermore, there is evidence that financial analysts’ forecasts are systemati-cally biased.34

Weighted Average Cost of Capi-tal (WACC)

IAS 36.90 requires that a cash-generating unit to which goodwill has been allocated shall be tested for impairment at least annually, and also when-ever there is an indication that the unit may be impaired. This is done by com-paring its carrying amount, including goodwill, with the recoverable amount. For a cash-generating unit the recoverable amount is the higher of its “fair value less costs to sell” and its “value in use“.

Value in Use

IAS 36.30 states: “The following elements shall be reflected in the cal-culation of an asset’s value in use:

(a) an estimate of the future cash flows the entity expects to derive from the as-set;

(b) expectations about possible variations in the amount or timing of those fu-ture cash flows;

(c) the time value of money, represented by the current market risk-free rate of interest;

(d) the price for bearing the uncertainty inherent in the asset; and

34 James Claus and Jacob Thomas, “Equity Premia as Low as Three Percent?, Evidence from Ana-

lysts’ Earnings Forecasts for Domestic and International Stock Markets”, Journal of Finance 56 (2001): 1629-1666.


(e) other factors, such as illiquidity, that market participants would reflect in pricing the future cash flows the entity expects to derive from the asset.”

According to IAS 36.55 the discount rate should be pre-tax reflecting both current market assessments of the time value of money and the risks spe-cific to the asset for which the future cash flow estimates have not been ad-justed. IAS 36.A17 states as a starting point “the entity might take into account:

(a) the entity’s weighted average cost of capital determined using techniques such as the Capital Asset Pricing Model;

(b) the entity’s incremental borrowing rate; and

(c) other market borrowing rates.”

It goes on “Consideration should be given to risks such as country risk, currency risk and price risk”. (IAS 36.A18)

The rate to be chosen under the so-called Free Cash Flow Method is the WACC:

LE D

E DWACC r r (1 T)

D E D E

, (18)

where rD is the cost of debt. The hint to use WACC is astonishing, since it is influenced by (a) capital structure and (b) taxes, whereas according IAS 36 financing and taxes are to be ignored in determining the value in use.

Neglecting taxes (as required) and assuming the firm’s cost of debt is equal to the risk free rate (rD = rf), the WACC and the cost of equity of an unlevered firm are the same according to the Modigliani/Miller concept. In practice a firm’s cost of debt will exceed the risk-free rate; however this as-sumption follows from the Modigliani/Miller model, where no bankruptcy risk is accepted.

L U UE D E E D D

U UE E D D

UE

E D D E DWACC r r r r r r

D E D E E D E D E

E D D Dr r r r

D E D E D E D E

r .

(19)

In practice, pretax-WACC is usually calculated on an after tax basis (mostly, cost of equity is after tax) as follows:

LE

D

r E DWACC r

(1 T) D E D E

. (20)

Normally, the WACC of equation (18) is grossed up for taxes at the firm’s marginal rate, for example 40%, and either

Case A – a target capital structure of say 50% debt is used or


Case B – the actual capital structure is approximated by the book value of the debt ($56,117,000 at an average cost of 4.8%) and the market capitaliza-tion ($42,840,000 with a cost of equity of 12.31%).

In those circumstances the results of equation (19) and (20) are identical only by chance, whereas equation (19) follows consistently from the Modi-gliani/Miller model.

The different assumptions yield varying results for equation (20):

Cost of Debt = 4.8%

Net Cost of Equity = 12.31%

Case A Debt = 50%, Equity = 50%

Case B Debt = 56,117/98,957 =56.7% Equity = 43.3%

Case A WACC = 4.8% 50%+ 12.31%/(1-.40) 50% = 12.66%

Case B WACC = 4.8% 56.7% + 12.31%/(1-.40) 43.3% = 11.61%

At the valuation date the actual capital structure will differ from man-agement’s target; nor can it be assumed that the cost of equity is determined correctly, since equation (8) requires the perpetuity assumption that is normally not fulfilled in the special situation. Generally, the amount of the error cannot be determined, since it depends on a lot of factors.

IAS 36 recommends determining value in use by discounting free cash flows by WACC, but defines both measures differently from standard valuation theory. The free cash flows according to IAS 36 are not diminished by taxes, while WACC has no tax rate and is identical to the cost of equity for an unlevered firm. How the capital structure, essential for determining WACC, can reflect a calculation before financing is not discussed in the standard.

Conclusion Derivating the cost of capital is one of the most important issues in

valuation. There are nevertheless some open questions. On a conceptual level, there are difficulties in consistently combining the various components. For example, a SSRP is not consistent with the assumptions underlying the CAPM, which, on the other hand, may not capture all relevant risks. Since cost of capi-tal cannot be observed directly, it must be estimated. There is also a joint hy-pothesis problem, because tests and applications of CAPM, APT or ICCM are always tests of both the respective model and market efficiency. Even if a model, such as CAPM, is assumed to be true, the valuation formula cannot be required to calculate the cost of equity mechanically as several critical deci-sions concerning the selection of empirical data need to be made. Thus, the


derivation of cost of capital is not a “point estimate” but is always based on reasoned judgment of financial and other data as well as experience of the prac-titioner.

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