FNCE201 Corporate Finance
Lecture 2: Advanced Cost of Capital
Suggested Reading: Corporate Finance - Chapter 12
Ma Pengfei
Lee Kong Chian School of Business
Singapore Management University
Class Outline and Learning Objectives
▪ Weighted Average Cost of Capital (WACC)
▪ Cost of equity
▪ Approaches to estimate the cost of equity
▪ Capital Asset Pricing Model (CAPM)
▪Estimate equity risk premium
▪Techniques in estimating Beta
▪ Cost of debt and preferred stock
▪ Approaches to estimate the cost of debt
▪ Component weights for WACC
▪ Project’s cost of capital
▪ Unlevered Beta
Cost of Capital
▪ The rate of return that the suppliers of capital
(bondholders and owners) require as compensation
for their contribution of capital.
▪ It reflects the opportunity cost of the suppliers of
capital (investors), who will not voluntarily invest in a
firm unless its return meets or exceeds what they
could earn elsewhere in an investment of
comparable risk.
▪ Is a marginal cost i.e., the cost of raising additional
capital since is used mainly to evaluate investment
opportunities.
Weighted Average Cost of Capital
▪ Capital is raised by issuing equity, debt, and hybrid
instruments
▪ Each of these sources of capital has a cost referred
to as its component cost of capital.
▪ The cost of capital is, therefore, a weighted average
of the component costs of capital and is known as
the weighted average cost of capital (WACC).
▪ WACC is also called the marginal cost of capital,
since it is also the cost of raising additional capital,
with the weights representing the proportion of each
source of financing that is used.
Weighted Average Cost of Capital
▪ rwacc = wE × rE + wD × rD × (1 − 𝜏C ) + wP × rP
▪ The weights wE , wD and wP, refer to the proportion of equity,
debt and preferred stock, that the firm uses when it raises
new funds. 𝜏C is the marginal tax rate, rE is the cost of equity,
rD is the cost of debt, and rP is the cost of preferred stock.
▪ Interest on debt is tax-deductible, so rD must be
adjusted downwards:
▪ Debt holders receive a return of rD, while the firm
pays a net cost of rD(1 − 𝜏C )
▪ Payments to owners are not tax-deductible, so the
required rate of return on equity (whether rE or rP) is
the equity cost of capital.
Part 1
Cost of Equity
Cost of Equity (rE)
▪ The cost of common equity (rE) is the rate of return
required by the firm’s common shareholders.
▪ It is difficult to compute rE as there are no promised
or pre-specified returns based on a financial
contract.
▪ Three approaches to estimate rE:
1. Dividend discount model (DDM)
2. Bond yield plus risk premium
3. Capital asset pricing model (CAPM)
Dividend Discount Model (DDM)
▪ Use the constant-growth DDM and re-write it to
estimate rE: P0 = D1/(rE – g) => rE = D1/P0 + g
▪ g may be estimated by using third-party estimates
of the firm’s dividend growth or computing the
sustainable growth rate (SGR).
▪ SGR is the rate at which a company can maintain its
growth without requiring additional financing from
external sources.
▪ SGR the product of the historical ROE and the
retention rate: g =(1 − D/EPS)× ROE
▪ D/EPS is the assumed stable dividend payout ratio.
Example
▪ Brigham, Shome and Vinson (1985): cost of equity for
utility industry firms
▪ Value Line provides estimates of dividend for the
future 4 years, ROE, and retention rate (b)
▪ g = b × ROE
▪ DDM with Constant Long-Term Growth:
Div1
Div2
Div3
Div4
1
P0 =
+
+
+
+
1 + rE (1 + rE )2 (1 + rE )3 (1 + rE )4 (1 + rE )4
▪ We can then solve for rE
Div4 (1 + 𝑔)
rE − g
Bond Yield Plus Risk Premium (BYPRP)
▪ Requires adding a premium to the firm’s yield on its
debt: rE = rD + Risk Premium
▪ where rD is the before-tax cost of debt.
▪ This approach is based on the premise that the equity
of the firm is riskier than its debt, but that these sources
of capital move in tandem.
▪ The premium is often estimated using the historical
spread between bond yields and stock yields.
▪ In a developed country, a typical risk premium is
between 3 to 5 percent.
Equity Risk Premiums for Electric Utilities
▪ Survey question: assuming a double A, long-term
utility bond currently yields 12.5%, the common stock
for the same company would be fairly priced relative
to the bond if its expected return was [____]?
Capital Asset Pricing Model (CAPM)
▪ The CAPM states that the expected return on equity is
the sum of the risk-free interest rate and a premium for
bearing market risk, as follows: E(ri) = rF + βi[E(rM) − rF]
▪ where rF is the risk-free rate, βi is the beta of stock i,
E(rM) is the expected return on the market
▪ [E(rM) − rF] is the expected market risk premium or
equity risk premium.
▪ Risk-free rate. A risk-free asset is one with no default
risk. A common proxy for rF is the yield on a defaultfree government debt instrument.
▪ In general, the appropriate rF should be guided by
the duration of the projected cash flows
Example
▪ Suppose you estimate that Disney’s stock (DIS) has a
volatility of 20% and a beta of 1.29. A similar process
for Chipotle (CMG) yields a volatility of 30% and a
beta of 0.55. If the risk-free interest rate is 3% and you
estimate the market’s expected return to be 8%,
calculate the equity cost of capital for Disney and
Chipotle.
Solution
▪ We expect the price for Disney’s stock to move by
1.29% for every 1% move of the market. Therefore,
Disney’s risk premium will be 1.29 times the risk
premium of the market,
▪ rDIS=3%+1.29×(8%−3%)=3%+6.45%=9.45%
▪ Chipotle has a lower beta of 0.55,
▪ rGMG=3%+0.55×(8%−3%)=3%+2.75%=5.75%
▪ B/c market risk cannot be diversified, it is market risk
that determines the cost of capital; thus Disney has a
higher cost of equity, even though it is less volatile.
Expected Market Risk Premium
▪ Expected market risk premium. The premium that
investors demand for investing in a market portfolio
relative to rF:
▪ When using the CAPM to estimate rE, the beta is
typically estimated relative to an equity market index.
This means that the market risk premium, E(rM) − rF , is
actually an estimate of the equity risk premium (ERP).
▪ Approaches to estimate ERP:
1. Historical ERP approach
2. Implied ERP approach
3. Survey approach
Historical ERP
▪ Assumes that the realized ERP observed over a long
period of time is a good indicator of the expected
ERP.
▪ Requires historical data to compute the average rate
of return of a country’s market portfolio (rM) and the
average rate of return for rF.
▪ Typically obtained from external sources e.g.,
Ibbotson Associates, etc.
Historical ERP
▪ Determining the Risk-Free Rate
▪ The yield on U.S. Treasury securities
▪ Surveys suggest most practitioners use 10- to 30-year
treasuries
▪ Limitations of the historical ERP approach:
▪ Level of risk of the index may change over time.
▪ Risk aversion of investors may change over time.
▪ Estimates are sensitive to the method of estimation and the
historical period covered.
Implied Risk Premium Approach
▪ Also referred to as the DDM based approach. The
simplest approach uses the constant-growth DDM to
back out a value for rM from: P0 = D1/(rM − g)
▪ where P0 is the current value of the market index,
D1 are the expected dividends next period on the
index, and g is the expected growth rate of
dividends.
▪ The implied ERP is obtained by subtracting rF from
rM (used as a proxy for E(rM)).
▪ The advantage of the implied ERP is that it is marketdriven, up-to-date, and does not require historical
data. It is however very much model dependent.
Implied ERP for Singapore
▪ Source: http://www.market-risk-premia.com/sg.html
Model Parameters
Stock Beta (β)
▪ The stock beta is the sensitivity of stock returns to the
returns of the market portfolio. It measures the market
or systematic risk of the stock i.e., risk that cannot be
diversified away.
▪ β is estimated from a regression of excess stock return
against market risk premium: (ri - rF) = 𝛼ො + 𝛽መ (rM - rF)
▪ The R2 of the regression measures the goodness of fit
of the regression. It estimates the proportion of the risk
of the firm that can be explained by market risk.
Cisco versus S&P 500, 2000–2017
Length of the Estimation Period
▪ While a longer estimation period provides more data,
the firm’s risk characteristics may have changed over
time.
Stability of Beta
Length of Estimation Period
▪ Deciding on the estimation period involves a trade-off
between data richness captured by a long estimation
period versus firm-specific changes that are better
reflected with a shorter estimation period.
▪ A longer estimation periods is typically used for firms
with a long and stable operating history, while a
shorter estimation period is used for firms that have
recently undergone significant structural changes or
changes in their financial and operating leverage.
Outliers
▪ Beta estimates from a regression are significantly
affected by outlying observations.
Determinants of Beta
▪ The beta of a firm or project is affected by business
risk and financial risk. Both types of risk affect the
uncertainty of the cash flows of the firm or project.
▪ Total Cost = Fixed Cost + Variable Cost
▪ The greater the fixed cost, the greater the risk.
▪ Business Risk: uncertainty of revenues (sales risk) and
operating cost structure (operating risk).
▪ Financial Risk: uncertainty of net income and net
cash flows attributed to the use of financing that has
a fixed cost (i.e. debt and leases).
Operating Leverage
▪ The degree of operating leverage measures how
sensitive a firm (or project) is to its fixed costs.
▪ %Δin Profits ÷ %Δin Sales = 1 + Fixed Cost ÷ Profits =
Operating Leverage
▪ Operating leverage increases as fixed costs rise
▪ Q: how much does the profit change if sales change
by 10%, when
▪ Fixed Cost = 0
▪ Fixed Cost > 0
▪ Operating leverage magnifies the effect of cyclicality
on beta.
Operating Leverage and Beta
▪ Consider a project with expected annual revenues of
$120 and costs of $50 in perpetuity. The costs are
completely variable, so that the profit margin of the
project will remain constant.
▪ Suppose the project has a beta of 1.0, the risk-free
rate is 5%, and the expected return of the market is
10%. What is the value of this project?
▪ What would its value and beta be if the revenues
continued to vary with a beta of 1.0, but the costs
were instead completely fixed at $50 per year?
Solution
▪ The expected cash flow of the project is $120 − $50 =
$70 per year. Given a beta of 1.0, the appropriate
cost of capital is r = 5% + 1.0(10% − 5%) = 10%. Thus,
the value of the project if the costs are completely
variable is: $70/10% = $700
Solution
▪ If the costs are fixed, then we can compute the value
of the project by discounting the revenues and costs
separately.
▪ The revenues still have a beta of 1.0, and thus a
cost of capital of 10%, for a present value of:
$120/10% = $1,200
▪ The costs are fixed, we should discount them at the
risk-free rate of 5%, so their present value is $50/5%
= $1,000.
▪ Thus, with fixed costs the project has a value of only
$1200 − $1000 = $200.
What is the beta of the project?
▪ We can think of the project as a portfolio that is long
the revenues and short the costs. The project’s beta is
the weighted average of the revenue and cost betas:
R
C
1200
1000
βp =
βR −
βC =
1.0 −
0 = 6.0
R−C
R −C
1200 − 1000
1200 − 1000
▪ Given a beta of 6.0, the project’s cost of capital with
fixed costs is r = 5% + 6.0 (10% − 5%) = 35%.
▪ The present value of the expected profits is then:
$70/35% = $200
▪ As this example shows, increasing the proportion of
fixed versus variable costs can significantly increase a
project’s beta (and reduce its value).
Financial Leverage and Beta
▪ Operating leverage refers to the sensitivity to the firm’s
fixed costs of production.
▪ Financial leverage is the sensitivity to a firm’s fixed
costs of financing.
▪ If a firm raises it leverage, the return to equity will
become more volatile (higher equity β).
Part 2
Cost of Debt and Preferred Stock
Cost of Debt (rD)
▪ The cost of debt financing to a firm when it takes out
a bank loan or issues a bond. When the firm borrows
from a bank, rD, is simply the interest rate charged.
▪ Estimating the before-tax cost of debt when the firm
issues debt:
1. Yield-to-maturity (YTM). For long-term bonds that
are widely traded and have no special features.
2. Debt-rating. For long-term bonds that do not
trade on a regular basis but are rated.
3. Synthetic-rating. For long-term bonds that are not
rated. Mainly for many smaller companies and
most private businesses.
YTM v.s. Returns
▪ Yield to maturity is the IRR an investor will earn from
holding the bond to maturity and receiving its
promised payments.
▪ If there is little risk the firm will default, yield to maturity
is a reasonable estimate of investors’ expected rate
of return.
▪ If there is significant risk of default, yield to maturity will
overstate investors’ expected return.
Default Risk
▪ Consider a one-year bond with YTM of y. For each $1
invested in the bond today, the issuer promises to pay
$(1 + y) in one year.
▪ Suppose the bond will default with probability p, in
which case bond holders receive only $(1 + y − L),
where L is the expected loss per $1 of debt in the
event of default.
▪ So the expected return of the bond is:
▪ rD = (1-p)× y + p× (y-L) = y - p× L
▪ = YTM – Prob(default) × Expected Loss Rate
Annual Default Rates by Debt Rating
(1983–2011)
Rating:
AAA
AA
A
BBB
BB
B
CCC
CC−C
Default Rate:
Average
0.0%
0.1%
0.2%
0.5%
2.2%
5.5%
12.2%
14.1%
In Recessions
0.0%
1.0%
3.0%
3.0%
8.0%
16.0%
48.0%
79.0%
▪ The average loss rate for unsecured debt is 60%.
▪ During average times the annual default rate for B
rated bonds is 5.5%.
▪ So the expected return to B-rated bondholders during
average times is 0.055 × 0.60 = 3.3% below the bond’s
quoted yield.
Debt Betas
▪ Alternatively, we can estimate the debt cost of
capital using the CAPM .
▪ Debt betas are difficult to estimate because
corporate bonds are traded infrequently.
▪ One approximation is to use estimates of betas of
bond indices by rating category.
By Rating
A and above
BBB
BB
B
CCC
Avg. Beta
<0.05
0.10
0.17
0.26
0.31
By Maturity
(BBB and above)
1-5 year
5-10 year
10-15 year
>15 year
Avg. Beta
Blank
0.01
0.06
0.07
0.14
Example
▪ In mid-2015, homebuilder KB Home had outstanding
6-year bonds with a yield to maturity of 6% and a B
rating. If corresponding risk-free rates were 1%, and
the market risk premium is 5%, estimate the expected
return of KB Home’s debt.
Solution
▪ Given the low rating of debt, we know the yield to
maturity of KB Home’s debt is likely to signification
overstate its expected return. Using the average
estimates in previous Table and an expected loss rate
of 60%, we have: rD=6%−5.5%(0.60)=2.7%
▪ Alternatively, we can estimate the bond’s expected
return using the CAPM and estimated beta of 0.26. In
that case, rD =1%+0.26(5%)=2.3%
▪ Which both estimates approximation, they both
confirm that the expected return of KB Home’s debt is
well below its promised yield.
Synthetic-rating Approach
▪ Uses a synthetic rating to find the credit spread based
on the firm’s financial ratios e.g. interest coverage
ratios:
Cost of Preferred Stock
▪ A straight (i.e. nonconvertible) preferred stock pays
the holder a fixed dividend each period (quarterly)
forever. Because preferred stock are perpetual, its
value is: PP = DP/rP
▪ where PP is the current preferred stock price per
share, DP is the preferred stock dividend per share,
and rP is the return on preferred stock.
▪ Rearranging this equation, we obtain the cost of
preferred stock: rP = DP/PP
▪ The cost of preferred stock is not tax-adjusted
because dividends on preferred stock are not tax
deductible.
Part 3
Component Weights for WACC
Estimating the Component Weights
▪ Ideally, the weights to use when estimating the WACC
should reflect the relative proportions of each source
of capital that the firm intends to use in financing
investment projects. The weights are called the target
capital structure.
▪ The target capital structure cannot be observed but
may be estimated by:
▪ Using the firm’s current capital structure at market value
weights.
▪ Examining trends in the firm’s capital structure or statements
made by management regarding capital structure.
▪ Using the capital structure of comparable firms.
Estimating the Component Weights
▪ Weight of equity (wE). Based on market capitalization,
which is the market price per share times the number
of shares outstanding.
▪ Weight of debt (wD). Usually refers to interest-bearing
debt and based on market value if available.
However, market values are not easily available, so
book values are typically used.
Part 4
Project’s Cost of Capital
Estimating Beta
▪ The beta for publicly traded firms can be estimated
using easily accessible stock return data or obtained
from external vendors.
▪ Estimating the beta for a firm that is not publicly
traded or the beta for a project that is not the
average or typical project of the publicly traded firm,
is more difficult.
▪ To obtain a beta in these firms or projects, requires
proxying the beta using information on the firm or
project, combined with a beta of a publicly traded
company.
The Pure-Play Method
▪ This approach is used to estimate the beta for a
project from the betas of comparable publicly traded
firms, where a comparable firm is defined as a firm
with similar business risks.
▪ The beta of the comparable is first “unlevered” by
removing the effects of financial leverage, with the
resulting beta known as the firm’s unlevered beta or
asset beta, since it reflects the business risk of the
firm’s assets.
▪ After the unlevered beta is obtained, it is adjusted for
the capital structure of the project by “levering” this
asset beta, to arrive at an estimate of the equity beta
for the project.
Asset/Unlevered Beta
▪ Since the firm’s risk is shared between creditors and
owners, βA may be represented as the weighted
average of the creditors’ market risk (βD) and the
owners’ market risk (βE):
▪ 𝛽𝐴 (𝛽𝑈 ) =
Debt
Debt+Equity
× 𝛽𝐷 +
Equity
Debt+Equity
× 𝛽𝐸
▪ βA removes the effects of financial leverage
▪ Further, assuming βD = 0, we obtain the unlevering
formula
▪ 𝛽𝐴 =
Equity
Debt+Equity
× 𝛽𝐸 =
𝛽𝐸
Debt/Equity+1
Asset/Unlevered Beta
▪ Rewriting equation in terms of βE, results in the levering
formula:
▪ βE = βA[1 +
Debt
Equity
]
▪ The same unlevering and leveraging formulas may be
used to estimate the asset beta and equity beta for a
project.
▪ Note: to use this formula, we are assuming that the
firm/project maintains a constant D/E ratio, the
formula holds with the presence of tax.
Example
▪ Consider Grand Sport, Inc., which is currently allequity financed and has a beta of 0.90: βA = βE = 0.90
▪ The firm has decided to lever up to a capital structure
of 1 part debt to 1 part equity.
▪ Since the firm will remain in the same industry, its asset
beta should remain βA = 0.90.
▪ However, assuming a zero beta for its debt, its equity
beta would become twice as large:
▪ βA = 0.90 =
1
1+1
× βE
▪ βE = 0.90 × 2 = 1.80
Estimating βE Using Comparables
1. Estimate comparable firms’ βE and βD
2. Estimate their βA
Debt
Equity
▪
𝛽𝐴 =
▪
Firms in the same industry should have same βA
Debt+Equity
× 𝛽𝐷 +
Debt+Equity
× 𝛽𝐸
3. Average βA across comparable firms
4. Use the average βA and the target firm’s actual
leverage ratio to estimate βE
Example
▪ Suppose a medical diagnostic firm has Debt-Equity
ratio (D/E) of 0.2. The risk free rate is 5%. Company 2,
which makes similar class of therapeutic drugs, has an
equity beta of 2. Its D/E ratio is 0.4. The market risk
premium is 7%.
▪ What should be the cost of equity for the diagnostic
firm? Assume that the cost of debt for both firms is 6%.
▪ Hint: you need to solve debt beta using CAPM
Solution
E
D
a =
e +
d
E+D
E+D
rd = r f +  d [rm − r f ]
.06 = .05 +  d * .07
.01
d =
= .14
.07
1
.4
a =
*2+
* .14
1 + .4
1 + .4
= 1.47
E
D
a =
e +
d
E+D
E+D
1
.2
1.47 =
* e +
* .14
1 + .2
1 + .2
 e = 1.736
re = r f +  e [rm − r f ]
= .05 + 1.736 * .07 = .1715
re = 17.15%
Estimating the WACC for a Project
1. Estimate the project’s relative proportion of debt and
equity financing. Use the average D/E ratios of
comparable firms if the target D/E ratio is not
available, and rewrite them in terms of wE and WD.
2. Estimate the after-tax cost of debt. Use the credit
ratings and borrowing rates of the comparable firms
to compute rD. Apply the corporate tax rate to find
the after-tax cost of debt.
3. Unlever the project’s equity beta. Compute the
unlevered beta of the comparable firms.
Estimating the WACC for a Project
4. Compute the project’s equity beta. Find the average
unlevered beta and lever it to the project’s target
D/E ratio.
5. Estimate the project’s cost of equity. Use the CAPM
to find rE based on the project’s βE obtained in step 4.
6. Calculate the project’s WACC. Input all the WACC
variables to compute the project’s WACC:
▪ rWACC =
𝐸
𝐸+𝐷
× rE +
𝐷
𝐸+𝐷
× rD × (1 − 𝜏c)
Online Discussion
▪ Read chapter 10 “Best Practices” in Estimating the
Cost of Capital: An Update
▪ Robert F. Bruner, Kenneth M. Eades and Michael
J.Schill, Case Studies in Finance: Managing for
Corporate Value Creation, 8th International edition.
▪ Write some comments on what you have learned
from it.
Summary
▪ This class:
▪ Weighted Average cost of Capital (WACC)
▪ Cost of Equity
▪ Cost of Debt and Preferred Stock
▪ Component Weights for WACC
▪ Project’s Cost of Capital
▪ Next class:
▪ Capital structure