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Estimating the Cost of Capital


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1

The Cost of Capital

To value a company using enterprise DCF, we discount free cash flow by the weighted

average cost of capital (WACC). The WACC represents the opportunity cost that

investors face for investing their funds in one particular business instead of others with

similar risk.

In its simplest form, the weighted average cost of capital is the market-based weighted

average of the after-tax cost of debt and cost of equity:

To determine the weighted average cost of capital, we must calculate its three

components: (1) the cost of equity, (2) the after-tax cost of debt, and (3) the

companys target capital structure.

emd kV

E)T(1k

V

DWACC


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Successful Implementation Requires Consistency

The most important principle underlying successful implementation of the cost of

capital is consistency between the components of WACC and free cash flow. Toassure consistency,

It must include the opportunity costs from all sources of capital debt, equity,

and so onsince free cash flow is available to all investors.

It must weight each securitys required return by its market-based target weight,

not by its historical book value.

It must be computed after corporate taxes (since free cash flow is calculated in

after-tax terms). Any financing-related tax shields not included in free cash flow

must be incorporated into the cost of capital or valued separately.

It must be denominated in the same currency as free cash flow

It must be denominated in nominal terms when cash flows are stated in nominal

terms


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3

The Cost of Capital: An Example

Source of

capital

Debt

Equity

WACC

Proportionof total

capital

8.3%

91.7%

100.0%

Cost of

capital

4.7%

9.9%

Marginal

tax rate

38.2%

After-taxopportunity

cost

2.9%

9.9%

Contribution toweighted

average

0.2%

9.1%

9.3%

The Cost of Capital: Home Depot

The weighted average cost of capital at Home Depot equals 9.3%. The majority of

enterprise value is held by equity holders (91.7%), whose CAPM-based required

return equals 9.9%. The remaining capital is provided by debt holders at 2.9% of

an after-tax basis.

Lets examine the components of WACC one-

by-one, starting with the cost of equity


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The Cost of Equity

To estimate the cost of equity, we must determine the expected rate of return of the

companys stock. Since expected rates of return are unobservable, we rely on asset-

pricing models that translate risk into expected return.

The three most common asset-pricing models differ primarily in how they define risk.

The capital assets pric ing model (CAPM)states that a stocks expected return

is driven by how sensitive its returns are to the market portfolio. This sensitivity is

measured using a term known as beta.

The Fama-French three-factor modeldefines risk as a stocks sensitivity to

three portfolios: the stock market, a portfolio based on firm size, and a portfolio

based on book-to-market ratios.

The Arb itrage Pric ing Theory (APT)is a generalized multi-factor model, butunfortunately provides no guidance on the appropriate factors that drive returns.

The CAPM is the most common method for estimating expected returns, so we begin

our analysis with that model.


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The Capital Assets Pricing Model

Expected return

Percent

Beta (systematic risk)

The CAPM postulates that the

expected rate of return on acompanys stock equals the risk-

free rate plus the securitys beta

times the market risk premium:

E[Ri] = rf+ Bi (E[Rm] rf)

To estimate a stocks expected

return, you need to measure

three inputs:

1. The risk-free rate

2. The market risk premium

3. The stocks beta


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Component 1 of the CAPM: The Risk Free Rate

Percent

Source:Bloomberg Years to maturity

To estimate the risk-free rate, we look to government default-free bonds. For

simplicity, most valuation analysts choose a single yield to maturity from one

government bond that best matches the entire cash flow stream being valued.

For U.S.-based corporate valuation, the most common proxy is the 10-year

government bond rate. This rate can be found in any daily financial publication.

Yield to Maturity on Government Bonds

Ideally, each cash

flow should be

discounted using a

government bond witha similar maturity.


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Component 2 of the CAPM: The Market Risk Premium

Sizing the market risk premiumthe difference between the markets expected return

and the risk-free rateis arguably the most debated issue in finance.

Methods to estimate the market risk premium fall in three general categories:

1. Ex trapo late h isto r ica l excess re tu rns. If the risk premium is constant, we can

use a historical average to estimate the future risk premium.

2. Regress ion analysi s. Using regression, we can link current market variables,such as the aggregate dividend-to-price ratio, to expected market returns.

3. Use DCF to reverse engineer the r isk premium . Using DCF, along with

estimates of return on investment and growth, we can reverse engineer the

markets cost of capital and subsequently the market risk premium.

None of the methods precisely estimate the market risk premium. Still, based on

evidence from each of these models, we believe the market risk premium as of year-

end 2003 was approximately 5 percent.


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Method 1: Use Historical Excess Returns

Investors, being risk-averse,

demand a premium for holding

stocks rather than bonds.

If the level of risk aversion hasnt

changed over the last 100 years,

then historical excess returns are a

reasonable proxy for future

premiums. But many econometricissues quickly arise. For instance,

Which risk free rate should be

used to compute the excess

return?

Which method of averaging is

better, arithmetic or geometric?

Is a prediction based on U.S.

data too high?

Arithmetic Geometric Standard

Percent over bonds mean mean deviation

Japan 9.5 5.4 33.3

Germany 8.7 4.9 29.7

Australia 7.6 6.0 19.0

Italy 7.6 4.1 30.2

Sweden 7.2 4.8 22.5South Africa 6.8 5.2 19.4

United States 6.4 4.4 20.3

The Netherlands 5.9 3.8 21.9

Median 5.9 4.0 20.3

France 5.8 3.6 22.1

Canada 5.5 4.0 18.2

United Kingdom 5.1 3.8 17.0

Ireland 4.8 3.2 18.5

Spain 3.8 1.9 20.3Switzerland 2.9 1.4 17.5

Denmark 2.7 1.5 16.0

Source: Ibbotson Associates: Dimson-Marsh-Staunton (DMS), 2003

Historical Annual Market Risk Premium, 1900-2002


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Using Historical Excess Returns: Best Practices

To best measure the risk premium using historical data, you should:

Calculate the premium over long-term government bonds

Use long-term government bonds, because they match the duration of a

companys cash flows better than do short-term rates.

Use the longest period possible

If the market risk premium is stable, a longer history will reduce estimation

error. Since, no statistically significant trend is observable, we recommend the

longest period possible.

Use an arithmetic average of longer-dated intervals (such as five years)

Although the arithmetic average of annual returns is the best predictor of future

one year returns, compounded averages will be upward biased (too high).Therefore, use longer-dated intervals to build discount rates.

Adjust the result for econometric issues, such as survivorship bias.

Predictions based on U.S. data (a successful economy) are probably too high.


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Geometric Versus Arithmetic Average

Annual returns can be calculated using either an arithmetic average or a geometric

average. An arithmetic (simple) average sums each years observed premium and

divides by the number of observations:

1(t)r1

(t)r1

T

1AverageArithmetic

T

1t f

m

1(t)r1

(t)r1AverageGeometric

T1

T

1t m

m

A geometric (compounding) average compounds each years excess return and takesthe root of the resulting product:

Arithmetic averages always exceed geometric averages when returns are volatile.

So whic h averaging method best est imates the expected future rate of return?


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Problems with the Arithmetic Average

Scenario

1

2

3

4

Current

value

100

100

100

100

Return in

period one

1.2

1.2

0.9

0.9

Return in

period two

1.2

0.9

1.2

0.9

Future

value

144

108

108

81

Expected value

when returns

are

independent

25%

25%

25%

25%

100%

36.0

27.0

27.0

20.3

110.3

Expected value

when returns are

negatively

autocorrelated

15%

35%

35%

15%

100%

21.6

37.8

37.8

12.2

109.4

The arithmetic average of annual returns is the best predictor of future one year

returns, but compounded averages will be biased upwards (i.e. too high).

Consider a portfolio which can either grow by +20% or -10% in a given period. The

arithmetic average equals 5%. If you invested $100 in this portfolio, what is the

portfolios expected value after two years?

If returns are independent, the expected

value is $110.3, the same as if $100

had grown consistently at the arithmetic

average of 5% for two periods.

If returns are negatively autocorrelated,

i.e. high returns are more likely followed

by low returns, a compounded

arithmetic return is too high!


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When Possible, Use Long-Dated Holding Periods

Source: Ibbotson Associates; McKinsey analysis

Arithmetic mean of

Number of

observations

U.S.

stocks

U.S.

government

bonds

U.S.

excess

return

U.S.

excess

returns

Blume

estimator

1-year holding periods


4-year holding periods5-year holding periods


100

50

2520

10

11.3%

24.1

49.968.2

165.6

5.3%

10.9

23.129.5

72.1

6.2%

12.6

23.032.3

70.1

6.2%

6.1

5.35.8

5.5%

6.2%

6.1

6.05.9

5.6

Cumulative returns Annualized returns

Arithmetic Returns for Various Intervals, 1903-2002

To correct for the bias caused negative autocorrelation in returns, we have two choices.

First, we can calculate multi-period holding returns directly from the data, rather than

compound single-period averages. Alternatively, we can use an estimator proposed byMarshall Blume, one that blends the arithmetic & geometric averages.


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Method 2: Regression Analysis

Source: Lewellen (2004); Goyal and Welch (2003); McKinsey analysis

-5

-3

-1

1

3

5

7

9

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

Percent

Predicted Market Risk Premium

based on the dividend to price ratio Using advanced regression

techniques unavailable to earlierauthors, Jonathan Lewellen of

Dartmouth found that observable

variables, such as dividend

yields, do predict future market

returns.

Plotting the models predictions

reveals one major drawback: the

risk premium prediction can be

negative!

Other authors question the idea

of using financial ratios, arguingunconditional historical averages

predict better than more

sophisticated regression

techniques.


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Method 3: Reverse Engineer Discounted Cash Flow

Using the principles of discounted cash flow, along with estimates of growth, various

authors have attempted to reverse engineer the market risk premium.

We use the key value driver formula to reverse engineer the market risk premium.

After stripping out inflation, the expected market return (not excess return) is

remarkably constant, averaging 7.0%.

0

5

10

15

20

1962 1972 1982 1992 2002

P

ercent

Predicted Market Risk Premium

By reverse engineering market DCF

Subtracting the real

interest rate of 2.1%

from our estimate of

7.0% leads to a risk

premium just under 5%.


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Component 3 of the CAPM: Measuring Beta

S&P 500 monthly returns

HomeDepotmon

thly

stockreturns

Percent

According to the CAPM, a stocks

expected return is driven by beta,

which measures how much the

stock and market move together.

Since beta cannot be observed

directly, we must est imateits

value.

The most common regression

used to estimate a companys raw

beta is the market model:

The Beta for Home Depot

mi RR

Based on data from 1998-2003,

Home Depots beta is estimated

at 1.37

E ti ti B t B t P ti


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Estimating Beta: Best Practices

As can be seen on the previous slide, estimating beta is a noisy process. Based on

certain market characteristics and a variety of empirical tests, we reach several

conclusions about the regression process:

Raw regressions should use at least 60 data points (e.g., five years of monthly

returns). Rolling betas should be graphed to examine any systematic changes in a

stocks risk.

Raw regressions should be based on monthly returns. Using shorter return periods,such as daily and weekly returns, leads to systematic biases.

Company stock returns should be regressed against a value-weighted, well-

diversified portfolio, such as the S&P 500 or MSCI World Index.

Next, recalling that raw regressions provide only estimates of a companys true beta,

we improve estimates of a companys beta by deriving an unlevered industry beta

and then relevering the industry beta to the companys target capital structure.

Wh P ibl C t R lli B t


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When Possible, Compute a Rolling Beta

0.00

0.40

0.80

1.20

1.60

1985 1988 1991 1994 1997 2000 2003

Beta

IBM Market Beta, 1985-2004

Because estimates of beta are imprecise, plot the companys rolling 60-month beta to

visually inspect for structural changes or short-term deviations.

IBMs beta hovered near 0.7 in the 1980s but rose dramatically in the mid-1990s and now

measures near 1.3. This rise in beta occurred during a period of great change for IBM.

Le ering and Unle ering Betas


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Levering and Unlevering Betas

To improve the precision of beta estimation, use industry, rather than company-specific,

betas. Companies in same industry face similar operating risks, so they should have

similar operating betas.

Simply using the median of an industrys raw betas, however, overlooks an important

factor: leverage. A companys equity beta is a function of not only its operating risk,

but also the financial risk it takes.

The weighted average beta for operating assets (b

u- which is called the unlevered

beta) and financial assets (btxa) must equal the weighted average beta for debt (bd)

and equity (be). Our goal is to use this to solve for bu:

edtxa

txau

txau

txau

u bED

Eb

ED

Db

VV

Vb

VV

V

Because there are many unknowns and only one equation,

we must impose additional assumptions to solve for bu

operating assets tax assets debt equity



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Method 1: Assume btxa equals bu. If you believe the risk associated with tax shields(bu) equals the risk associated with operating assets (bu), the risk equation can be

simplified dramatically. Specifically,

Method 2: Assume btxa equals bd. If you believe the risk associated with tax shields(btxa) is comparable to the risk of debt (bd), the equation can once again be arranged to

solve for the unlevered cost of equity.

edu bED

Eb

ED

Eb

e

txa

d

txa

txau b

EV-D

Eb

EV-D

V-Db

e

m

u b

E

DT-11

1b

If the dollar level of debt is constant and debt is risk free,

ue bE

D1b

if bd = 0

Determining the Industry Beta


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Determining the Industry Beta

DebtOperating leases

Excess cash

Total net debt

Shares outstanding (Mil)

Share price ($)

Market value of equity

Debt/equity

Raw beta (step 1)Unlevered beta (step 2)

Industry average (step 3)

Relevered beta (step 4)

1,3656,554

(1,609)

6,310

2,257

35.49

80,101

0.079

1.371.27

1.14

1.23

Home Depot

3,7552,762

(948)

5,569

787

55.39

43,592

0.128

1.151.02

1.14

1.30

LowesCapital structure

Beta calculations Home Depot Lowes

To estimate an industry-adjusted

company beta:

1. First, regress each companys

stock returns against the S&P 500

to determine raw beta.

2. Next, to unlever each beta,

calculate each companys market-debt-to-equity ratio.

3. Determine the industry unlevered

beta by calculating the median (in

this case, the median and

average betas are the same).4. Relever the industry unlevered

beta is to each companys target

debt-to-equity ratio

Applying the CAPM


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Applying the CAPM

The CAPM postulates that the expected rate of return on a companys stock equals

the risk free rate plus the securitys beta times the market risk premium.

To estimate the risk-free rate in developed economies, use highly liquid, long-term

government securities, such as the 10-year zero-coupon strip.

Based on historical averages and forward-looking estimates, the appropriate

market risk premium is currently between 4.5 and 5.5 percent.

To estimate a companys beta, use industry derived betas levered to the companys

target capital structure.

For Home Depot:

E[Ri] = rf+ Bi (E[Rm] rf)

E[Ri] = 4.34% + 1.23 (4.5%) = 9.9%

An Alternative Model: Fama & French


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In 1992, Eugene Fama and Kenneth French published a paper in the Journal of

Finance that received a great deal of attention because they concluded,

In short, our tests do not support the most basic prediction of the

SLB [Sharpe-Lintner-Black] Capital Asset Pricing Model that average

stock returns are positively related to market betas.

Based on prior research and their own comprehensive regressions, Fama and French

concluded that:

Equity returns are inversely related to the size of a company (as measured by

market capitalization).

Equity returns are positively related to the ratio of the book value to market value

of the companys equity.

With this model, a stocks excess returns are regressed on excess market returns, the

excess returns of small stocks over big stocks (SMB), and the excess returns of high

book-to-market stocks over low book-to-market stocks (HML).



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Lets use the Fama-French three-factor model to continue our Home Depot example.

To determine the companys three betas, Home Depot stock returns are regressed

against the excess market portfolio, SMB, and HML (available from professionalservice providers).

Market risk premium

SMB premium

HML premium

Premium over risk free rate

Risk free rate

Cost of equity

Factor

0.25

0.36

Average

monthlypremium

Percent

4.5%

3.0%

4.4%

Average

annualpremium

Percent

1.35

(0.04)

(0.10)

Regression

beta

6.1%

(0.1)

(0.5)5.5

4.3

9.8%

Contribution toexpected return

Home Depot: Fama & French Expected Returns

For HD, the F&F

model leads to a

slightly smaller cost

of equity than the

CAPM.

An Alternative Model: The Arbitrage Pricing Theory


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An Alternative Model: The Arbitrage Pricing Theory

The Arbitrage Pricing Theory (APT) can be thought of as a generalized version of

the Fama-French 3-Factor model. In the APT, a securitys returns are fullyspecified

by k factors and random noise:

~F~

b...F~

bF~

bR~

kk2211i

By creating well-diversified factor portfolios, it can be shown that a securitys

expected return must equal the risk free rate plus its exposure to each factor times

the factors excess return (denoted by lambda):

kk2211fi b...bbr]E[R

Implementation of the APT however has been elusive, as there is little agreement

on either the number of factors, what the factors represent, or how to measure the

factors.

The Cost of Debt


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The Cost of Debt

The weighted average cost of capital represents the blended rate of return for a

companys investors, both debtholders and equity holders:

emd kV

E)T(1k

V

DWACC

To compute the WACC, we must estimate the cost of debt (kd). To do this we look to the

yield to maturity (YTM). Although YTM represents a promised yield, it is a good

approximation for expected return for investment grade companies.

To compute yield-to-maturity, you have two options:

1. Compute the yield-to-maturity on long-term bonds by reverse engineering thediscount rate needed to set DCF equal to the price.

2. Compute the yield-to-maturity indirectly by adding a default premium (based on thecompanys rating) to the risk free rate.

Lets examine the indirect method

Component 1 of YTM: The Risk Free Rate


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Component 1 of YTM: The Risk Free Rate

Percent

Source:Bloomberg Years to maturity

The yield-to-matrurity can be estimated by adding a default premium (based on the

companys rating) to the risk free rate. The first component of yield-to-maturity is the

risk free rate.

Regardless of the maturity structure for the companys debt, use a long-term risk free

rate when estimating a companys cost of capital. Using short-term debt yields to

approximate the cost of debt ignores the fact that future debt will have different yields.

Yield to Maturity on Government Bonds

In 2003, the 10-year yield

to maturity was 4.3% in

the U.S. and in Europe.

S&P and Moody Ratings Classes


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EXTREMELY STRONG capacity to meet its financial commitments. AAA is the highest Issuer CreditRating assigned by Standard & Poors.

VERY STRONG capacity to meet its financial commitments. It differs from the highest rated obligors

only in small degree.

STRONG capacity to meet its financial commitments but is somewhat more susceptible to the adverse

effects of changes in circumstances and economic conditions than obligors in higher-rated categories.

ADEQUATE capacity to meet its financial commitments. However, adverse economic conditions or

changing circumstances are more likely to lead to a weakened capacity of the obligor to meet its

commitments.

Speculative debt is rated BB, B, and CCC. In these case, YTM is a poor proxy for the cost of debt.

AAA / Aaa

AA / Aa

A/ A

BBB / Baa

y g

InvestmentGrade

S&P / Moodys

In order to be compensated for default risk, lenders charge a premium over the default-

free benchmark rate to risky customers. The higher the chance of default, the higher

the premium will be.

Professional firms, such as S&P and Moodys, rate the default risk of most bonds.

Lets examine the ratings defined by Standard & Poors:

Component 2 of YTM: The Corporate Yield Spread


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Source: Bloomberg

Rating

Aaa/AAA

Aa1/AA+

Aa2/AA

Aa3/AA

A2/A

Baa2/BBB

Ba2/BB

B2/B

Yield Spread in Basis Points, December 2003

1

34

37

39

40

57

79

228

387

3

35

33

34

36

49

96

260

384

5

21

34

35

37

57

108

257

349

7

22

40

42

43

65

111

250

332

10

28

29

34

37

48

102

236

303

30

50

62

64

65

82

134

263

319

Maturity in years

Once a bond rating has been

identified, convert the rating into ayield to maturity.

Lets examine U.S. corporate yield

spreads over U.S. government

bonds. All quotes are presented in

basis points, where 100 basispoints equals 1%.

Since Home Depot is rated Aa3

by Moodys and AA by S&P, we

estimate that the 10-year yield to

maturity is between 34 and 37basis points over the 10-year

Treasury.

Component 2 of YTM: The Corporate Yield Spread

The Cost of Debt at Distressed Companies


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p

Source: Lehman Brothers; Global Family of Indices, FixedIncome Research; Morgan Stanley Capital

International; U.S. Treasury; Paul Sweeting

Asset class

Treasury bonds

Investment-grade corporate debt

High-yield corporate debt

Beta

0.19

0.27

0.37

Yield to maturity is not an expected return. It is the return earned if the obligation is

paid on time and in full. Since distressed companys have a significant chance of

default, the yield-to-maturity is a poor proxy for expected return.

One alternative for computing expected return is the CAPM. Since most bonds dont

trade enough to generate a reliable beta, however, we compute index betas instead.

High yield debt has only a

slightly higher beta than

investment grade debt.

If the market risk premium

equals 5%, this difference

translates to onlya 50 basis

point differential in expected

return!

Beta by Bond Rating

Use Market-Based Target Weights


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g g

With our estimates of the cost of equity (ke) and cost of debt (kd), we can now blend

the two expected returns into a single number. To do this, we use the target weights of

debt (and equity) to enterprise value, on a market (not book) basis:

emd kV

E)T(1k

V

DWACC

To develop a target capital structure for a company,

1. Estimate the companys current market-value-based capital structure.

2. Review the capital structure of comparable companies.

3. Review managements implicit or explicit approach to financing the business and

its implications for the target capital structure.

D/V equals the companys

market-based, target

debt-to-value ratio

Typical Market Weights Across Industries


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Note: Market value of debt proxied by book value. Enterprise value proxiedby book value of debt plus market value of equity

22

26

30

33

47

19

15

13

12

4

0Information technology

Healthcare

Aerospace and defence

Industrial machinery

Consumer discretionary

Consumer staples

Oil and gas

Chemicals, paper, metals

Telecommunications

Airlines

Utilities

Median Debt-to-Value, 2003

In percent

To place the companys

current capital structure in

the proper context, compare

its capital structure with

those of similar companies.

Industries with heavy fixed

investment in tangible assets

tend to have higher debt

levels.

High-growth industries,

especially those with

intangible investments, tend

to use very little debt.

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