Cost of Capital
Estimation
Methods for Benchmarking the
Cost of Equity Capital
1.
2.
3.
Capital Asset Pricing Model using beta from a
regression analysis (top-down method)
Capital Asset Pricing Model weighting beta
estimates for individual divisions using industry
betas (bottom-up method)
Implied Cost of Equity using stock valuation
model, given stock price and expected growth
rates
Only Systematic Risk is Priced in
the Capital Asset Pricing Model

The key result of the CAPM is that the relevant risk of
any asset is the risk that it adds to the market
portfolio—the systematic risk



Systematic risk is exposure to market factors that affect all
securities to a greater or lesser degree (e.g. inflation, GDP
growth, interest rates, political events, etc.)
In well diversified portfolios exposure to firm-specific
(unsystematic) events is diversified away (e.g. management
changes, product announcements, litigation, etc.)
The systematic risk is measured by the security’s comovement with a broadly diversified portfolio
Risk measures
Standard deviation (stdev)

2
^


 ri  r 


i 1 
.
n 1
n
^
^



 rAi  r A  rBi  r B 



i 1 
Cov(AB) 
n2
n
Covariance (covar.s)
Correlation
Coefficient (correl)
 AB
Cov(AB)

AB
The standardization confines the ρ to values between –1 and +1
Portfolio standard deviation for a
two-security portfolio:
p  w 2A2A  w 2B2B  2w A w BCOVAB
Two-Security Portfolio
Variance-Covariance Matrix

The two-security portfolio  contains two
covariance (market risk) terms and two
variance (stand-alone risk) terms
2A
CovA.B
CovB.A
2B
Three-Security Portfolio
Variance-Covariance Matrix
 2A
CovA.B
CovA.C
CovB.A
 2B
CovB,C
CovC,A
CovC.B
 2C
Standard deviation and risk
The standard deviation of a single
security includes both systematic and
unsystematic risk
 For well diversified portfolios, the
standard deviation includes systematic
risk only

Efficient Portfolios


Combining assets with less than perfect
correlation improves portfolio efficiency by
reducing unsystematic risk
An efficient portfolio is one that offers:
 the
 the

most return for a given amount of risk, or
least risk for a give amount of return.
The collection of efficient portfolios is called
the efficient frontier
Expected
Portfolio
Return,
E(Rp)
Efficient Frontier
Risk, p
See two-security example
Capital Asset Pricing Model
The CAPM is an equilibrium model that
specifies a linear relationship between
risk and required rate of return for
assets held in well-diversified portfolios.
w mm
Adding a risk-free security


When a risk-free security with return rRF is
added, investors can create portfolios that
combine this security with a portfolio of risky
securities.
Since the risk-free asset has zero variance, its
covariance is also zero
 Thus
the standard deviation of a 2-security
portfolio of the risk-free asset and the market
portfolio, M, is:
wmσm
What impact does rRF have on
the efficient frontier?
Both the standard deviation and expected
return are linear functions of the weights
wrf and wm
 The straight line connecting rRF with M
(market), the tangency point between the
line and the old efficient frontier, becomes
the new efficient frontier.

Efficient Frontier with a Risk-Free
Asset
Expected
Return, rp
^
rM
.
M
The Capital Market
Line (CML):
New Efficient Frontier
rRF
M
Risk, p
^rp =
rRF +
^
rM - rRF
M
 p.
p is determined by selecting weights on:
the risk-free security (wrf)
the market portfolio (wm)
The Security Market Line (SML)


The equation for the Security Market Line, the
principal result of the Capital Asset Pricing Model,
gives the risk/return relationship for individual
securities.
Substitute the contribution of an individual security’s
risk to the market portfolio standard deviation, ρi,mσi ,
for σp:
^
^
i , m  i
r i  rrf  [r m  rrf ]
m
^
^
r i  rrf  [r m  rrf ]betai
Beta

 i i ,m i m Covi ,m
betai  i ,m


2
m
m
 m2
Beta intuition


Beta is simply a measure of relative systematic
risk, or relative exposure to the economic
variables that drive market returns.
For example, a security with a beta of 1.20
exhibits 20 percent more variability in response
to market returns as compared with a typical
security.
Result of the CAPM
SML:

^
r i  rrf  [r m  rrf ]betai
Expected return for stocks includes




^
a risk-free component
a risky component as determined by a risk premium
for the average stock, known as the ‘market risk
premium’, (rm - rrf)
and the beta of the individual stock, which measures
the degree of market risk exposure for the individual
security
The expected (required) return on the stock is
the issuing company’s cost of Equity
Security Markey Line
25%
Expected Return
20%
Expected return on
market
15%
= Rm - Rrf
10%
5%
Risk-free rate
0%
0
0.5
1
1.5
Beta
2
2.5
3
 Security
A
 Security B
Std. Deviation
20%
30%
Beta
1.25
0.95
Which security has more unsystematic
risk?
 Which security has more systematic risk?
 Which security should have the higher
required return?

Estimating the CAPM Inputs
The beta of the security
 The expected market risk premium
 The current risk-free rate

^
^
r i  rrf  [r m  rrf ]betai
Estimating Beta (top-down approach)

Standard approach is to regress stock returns
against those of a broad market index, where the
slope of the regression line is the beta coefficient:
 most
services use either 5 years (monthly returns) or
2-3 years (weekly) regressed on the S&P 500
 A 5-year interval insures against possible aberrant
shocks to the beta due to unusual short-term events
 A shorter interval may better reflect the company’s
current risk profile if its business or operating
environment have changed

Many services adjust beta toward 1.0
 Example:
Adj. Beta = 0.67*beta + 0.33*1.0
26
Bottom-up Betas

The beta of a portfolio is a market-value weighted
average of the betas of the assets comprising the
portfolio
 the
beta of a firm is a weighted average of the
individual divisions or projects in which the firm invests

A bottom-up beta estimates beta for each of the
divisions using industry betas, and uses a
weighted average of these division betas to
estimate the corporate beta
Advantages of Bottom-up Betas


Since the procedure involves averaging across
several regression betas, the estimation error is
lower, and the estimates are more stable
The bottom-up beta may provide a better
estimate of the true beta when the firm has
reorganized or restructured itself substantially
during the period of the regression
 Weight

the division betas based on the current mix
Division betas are required to make investment
decisions
Division Cost of Capital
The firm’s overall cost of capital cannot be applied to
individual divisions or projects when there are
differences in risk: 1) operating (business) risk; 2)
financial risk
Rate of Return
(%)
13.0
Project H
11.0
10.0
9.0
7.0
0
WACC
Division H’s WACC
Project L
Composite WACC
for Firm A
Division L’s WACC
RiskL
RiskAverage
RiskH
Risk
Operating Risk


Variability in Earnings Before Interest and Taxes
Two sources:
1) Industry Effects on sales


Cyclical companies have higher business risk than noncyclical firms
Firms which sell more high-cost and discretionary products
will have higher business risk
2) Operating leverage effects: Operating leverage
refers to the proportion of the total costs of the firm
that are fixed.
Other things equal, higher operating leverage results in
greater earnings variability
 Operating leverage measure =
% Change in EBIT / % Change in Revenues

Operating Leverage
Revenue
100
200
300
Variable costs (20%)
(20)
(40)
(60)
(100)
(100)
(100)
EBIT
(20)
60
140
Revenue
100
200
300
Variable costs (50%)
(50)
(100)
(150)
Fixed costs
(40)
(40)
(40)
10
60
110
Fixed costs
EBIT
Financial Risk
As firms borrow, they create fixed costs
(interest payments) that make their
earnings to equity investors more volatile
 This increased earnings volatility
increases the equity Beta
 As more variance is added, and some
fraction of this variance is correlated with
markets, beta increases

The Pure-Play approach to Beta
estimation

The typical approach is to find publicly traded
‘pure play’ companies operating primarily in
division’s business
 Can
expand to customers and suppliers if difficult to
find companies


They should have levels of operating risk (EBIT
variance) that are comparable to that of the
division since they are in the same industry
Their levels of financial risk, however, will vary
due to differences in financing choice
The Pure-Play approach to Beta
estimation

Process for dealing with financial leverage
differences:
 Unlever the betas of the pure-play firms
 removes the effects of their debt-equity mix on beta
 Take an average of these unlevered betas
 Relever
ratio
the betas at the division’s target debt-to-stock
The Cost of Equity at Different Levels
of Debt: Hamada’s Equation

bL = bU [1 + (1 - t)(D/S)]

bU is the beta of a firm when it has no debt (the unlevered beta)

Use this equation to unlever the observed pure play betas
(bL’s), then average the resulting bU’s


Use the debt-stock mix (D/S) and marginal tax rates (t) of these companies
to unlever

Divide bL by term in brackets
Relever average unlevered beta at the division’s target capital
structure (D/S) and marginal tax rate (t)

Multiply resulting average bU by term in brackets

Plug relevered beta into CAPM to yield rs

See AOL example
Levered Beta Calculation
Division's target capital structure (D/S) = 0.6
Tax Rate = 40%
Pure Play
Actual
Market
Value
Company
Beta (bL)
Debt
Equity
D/S
Beta (bu)
A
0.8
1000
1000
1.00
0.50
B
1.2
800
500
1.60
0.61
C
0.6
1500
2000
0.75
0.41
Average
0.51
Levered beta =
.51[1+(1-.4)*.6]
Result
0.69
Market Value
Unlevered
Second CAPM Input:
r r
The Market Risk Premium
^
i
rf
^
 [r m  rrf ]betai
The equity market risk premium is the
premium that investors demand for
investing in an average risk investment,
above the risk-free rate, (rm – rrf)
 The expected rate of return on the
average stock minus the risk-free rate at
any point in time.

Approaches to Estimating the
Market Risk Premium




Assume that the actual premium delivered over
long time periods is equal to the expected
premium - i.e., use historical data
Estimate the implied premium using today’s
security prices and expected growth in earnings
Forecasts future stock returns based on
fundamentals: payouts, multiples and growth
Survey data
The Historical Risk Premium
Approach
Defines a time period for the estimation
 Calculates the difference between average
returns on a stock index and average
returns on a riskless security during the
period
 Uses the difference as a premium looking
forward

The Historical Risk Premium
Approach

The limitations of this approach are:
 Assumes
that the risk aversion of investors has not
changed in a systematic way across time. (The risk
aversion may change from year to year, but it reverts
back to historical averages)
 Assumes that the riskiness of the “risky” portfolio
(stock index) has not changed in a systematic way
across time.
 Strange results, since after periods of high returns,
you conclude that investors have become risk averse,
when the opposite is probably true.
Risk Premium Based on
Forecasted Fundamentals
R =
≈ 7.5%
PAYOUT * E/P
≈ 50%
7%
+
(a P/E of 14)



G
≈ 4%
(2 real +
2 inflation)
Use P/E ratio to determine earnings yield, multiply by
payout which includes dividends and repurchases
Add an expected LT growth rate for earnings to arrive at
7.5% forecast yield for large stocks (S&P 500)
Subtracting long-term Treasury yield of 1.8% yields an
estimated risk premium of 5.7%
Implied Market Risk Premiums
2011 survey data on the
Market Risk Premium
43
Third CAPM Input:
The Risk-free Rate





^
r i  rrf  [r m  rrf ]betai
U.S. Treasuries are used as the risk-free rate
While open to debate, most favor using a long-term
Treasury rate for the following reasons:


^
The LT rate reflects an average of future expected short-term
rates over the life of the investment
The LT rate is much more stable than ST rates
The cash flows underlying stocks are long-lived
The 10-year Treasury is commonly used
The Treasury rates can be found on:

http://www.bloomberg.com/markets/rates/index.html
Implied Cost of Equity as another
benchmark
 As an alternative to the CAPM approach, bottom-up
or top-down, for corporate finance purposes the cost
of Equity can be estimated using the stock price and
expected growth in Free Cash Flow to Equity
Constant growth version:
P 0  FCFE1 /( r  gFCFE )
 Since stock price and consensus analyst growth
forecasts are known, the company can back into an
implied cost of Equity by solving for ‘r’ using a stock
valuation model.
Advantages of Implied Cost
Market-based measure
 Do not have to estimate beta
 Do not have to estimate market risk
premium
 These assumptions are implicit in the
market’s valuation

Weighted Average
Cost of Capital
The overall required rate of return % on
Invested Capital (Debt + Stock)
WACC = (D/V)rd(1-T) + (S/V)rs
rd = % weighted average yield-to-maturity on debt
rs = % required return on stock (cost of Equity)
D = $Debt market value
S = $Stock market value
T = % tax rate
V = $Enterprise value = D + S
Key Terms
Capital Asset Pricing Model
 Systematic risk
 Beta (unlevered and levered)
 Market risk premium
 Operating risk
 Financial risk

48