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Estimating the Cost of Capital
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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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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
2-year holding periods
4-year holding periods5-year holding periods
10-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
Levering and Unlevering Betas
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Levering and Unlevering Betas
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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An Alternative Model: Fama & French
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).
An Alternative Model: Fama & French
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An Alternative Model: Fama & French
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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