Principles of Finance
Topic 8: Cost of Capital
Lecturer: Yue (Lucy) Liu
Slides are based on Ch 12 in Berk and DeMarzo (3rd edition).
Outline
The Equity Cost of Capital
The Debt Cost of Capital
A Project’s Cost of Capital
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Outline
The Equity Cost of Capital
The Debt Cost of Capital
A Project’s Cost of Capital
3
The Equity Cost of Capital
Method 1: Estimation from historical data
Method 2: Fundamental approach
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The Equity Cost of Capital
Method 1: Estimation from historical data
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The Market Risk Premium
ri = r f + i ( E[ R Mkt ] − r f )
Market risk Premium
Risk Premium for Security i
▪ Estimate the risk premium (E[R Mkt]-rf) using the historical average excess return of
the market over the risk-free interest rate.
▪ Estimate Beta from Historical Returns.
Recall, beta is the expected percent change in the excess return of the security for a
1% change in the excess return of the market portfolio. Consider Cisco Systems
stock and how it changes with the market portfolio.
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Beta Estimation (Cont’d)
Scatter plot of Monthly Excess Returns for Cisco Versus the S&P 500, 1996–2009
✓ Cisco tends to be up when the market is up, and vice versa.
✓ A 10% change in the market’s return corresponds to about a 20% change in
Cisco’s return. So Cisco’s beta is about 2.
✓ Beta corresponds to the slope of the best-fitting line in the plot of the
security’s excess returns versus the market excess return.
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Beta Estimation (Cont’d)
◼ Use Linear Regression to identify the best-fitting line:
(Ri − r f ) = i + i (RMkt − r f ) + i
Input Input
– αi is the intercept term of the regression.
• It represents a risk-adjusted performance measure for the historical returns.
• Positive/negative αi can indicate the stock has performed better/worse than
predicted by the CAPM.
– βi (RMkt – rf) represents the sensitivity of the stock to market risk. When the market’s
return increases by 1%, the security’s return increases by βi%.
– εi is the error term and represents the deviation from the best-fitting line and is zero
on average.
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Beta Estimation (Cont’d)
Practical Considerations When Forecasting Beta
◼ Time Horizon (Practice: at least 2 years weekly return data or 5 years of monthly return data)
◼ The Market Proxy (e.g. international stocks)
◼ Beta Variation and Extrapolation, e.g. extrem value relative to norm
(historical or industry norm)
◼ Outliers
◼ Changes in the environment of the firm may cause the future to differ
from the past. (e.g. change industry)
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The Equity Cost of Capital
Method 2: Fundamental approach
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Fundamental Approach
◼ Using historical data has two drawbacks:
– Standard errors of the estimates are large
– Backward looking, so may not represent current expectations.
◼ A fundamental approach is to solve for the discount rate that is consistent with
the current level of the index.
Div1
rMkt = + g = Dividend Yield + Expected Dividend Growth Rate
P0
D1
P0 = D1 rE = + g
Example: rE − g P
The current dividend yield of S&P 500 is 2%, and both earnings and dividends are
expected to grow 6% per year, what is the expected return of the S&P 500?
Solution: rMkt = 2% + 6% = 8%
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Outline
The Equity Cost of Capital
The Debt Cost of Capital
A Project’s Cost of Capital
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The Debt Cost of Capital
Method 1: Use the debt yield as an estimate of the debt cost of capital
Method 2: CAPM (using the debt beta to estimate the debt cost of capital)
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The Debt Cost of Capital
Method 1: Use the debt yield as an estimate of the debt cost of capital
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The Debt Cost of Capital (cont’d)
◼ 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. So we need to make some adjustments.
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The Debt Cost of Capital (cont’d)
Adjustments need to be made when there is significant risk of default
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 - pL
= Yield to Maturity – Prob(default) X Expected Loss Rate
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Annual Default Rates by Debt Rating (1983–2008)
Example:
What is the expected return to BB-rated bondholders during average times, given that:
The bond’s quoted yield is 8.5%;
The average loss rate for unsecured debt is 60%.
Solution:
According to the above table, during average times the annual default rate for BB-rated
bonds is 2.1%. So the expected return to BB-rated bondholders during average times:
rd = Yield to Maturity – Prob (default) X Expected Loss Rate
= 8.5% - 2.1% X 0.6
= 7.24%
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The Debt Cost of Capital
Method 2: CAPM (using the debt beta to estimate the debt cost of capital)
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The Debt Cost of Capital (cont’d)
◼ We can estimate the debt cost of capital using the CAPM.
◼ Debt betas are difficult to estimate.
◼ One approximation is to use estimates of betas of bond indices by
rating category.
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The Debt Cost of Capital (cont’d)
Example
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The Debt Cost of Capital (cont’d)
Solution
Given the low rating of debt, as well as the recessionary economic conditions at
the time, we know the YTM of KB Home’s debt is likely to significantly overstate
its expected return. Assume the expected loss rate is 60%.
Method 1 (Debt Yields):
Using the recessionary estimates in the table, we have
rd = YTM – Prob (default) X Expected Loss Rate
= 8.5% - 8% (0.60)
= 3.7%
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The Debt Cost of Capital (cont’d)
Solution
Method 2 (CAMP):
We estimate the bond’s expected return using the CAPM and an estimated beta
of 0.17 from the table.
rd = rf + (Market risk premium)
= 3% + 0.17 (5%)
= 3.85%
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Outline
The Equity Cost of Capital
The Debt Cost of Capital
A Project’s Cost of Capital
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Purely equity financed project
◼ Method:
– Identify comparable firms in the same line of business
– Estimate the cost of capital of the assets of comparable firms
– Use that estimate as a proxy for the project’s cost of capital
◼ Comparable firms:
✓ Unlevered (all-equity financed) firm in a single line of business that is
Firm without debt.
comparable to the project.
✓ Levered firms as comparables
Firm with debt.
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Purely equity financed project
1. Using the unlevered firm as the comparable
Example
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Purely equity financed project (Cont’d)
Solution
Solution
● Rather than investing in the new coffee shop, you could invest in Peet’s
coffee shops simply by buying Peet’s stock.
● To be attractive, the new investment must have an expected return at least
equal to that of Peet’s stock, which from the CAPM is 7.15%.
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Purely equity financed project (Cont’d)
2. Using a Levered Firm as a Comparable
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Purely equity financed project (Cont’d)
◼ Asset (unlevered) cost of capital
– Expected return required by investors to hold the firm’s underlying assets.
– Weighted average of the firm’s equity and debt costs of capital
E D
rU = rE + rD
E+D E+D
◼ Asset (unlevered) beta
E D
βU = βE + βD
E+D E+D
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Purely equity financed project (Cont’d)
Example. Using the levered firm as the comparable
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Purely equity financed project (Cont’d)
Solution
Solution
E D
rU = rE + rD
E+D E+D
E D
βU = βE + βD
E+D E+D
● In the first method, we assumed the expected return of PG’s debt is its promised yield of 3.1%.
● In the second method, we assumed the beta of the debt is zero, which implies the expected return
of PG’s debt is the risk-free rate of 3% according to the CAPM.
● So we have slight difference in rU using the two methods.
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So far …….
Unlevered firm comparable rE
Purely equity financed project
Levered firm comparable rU
Not purely equity financed project ?
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Not purely equity financed project
(Mode of finance and WACC)
◼ How might the project’s cost of capital change if the firm uses leverage
to finance the project (i.e. not purely equity financed)?
◼ Perfect capital markets
In perfect capital markets, choice of financing does not affect cost of
capital or project NPV.
◼ Taxes – A Big Imperfection
When interest payments on debt are tax deductible, the net cost to the
firm is given by:
Effective after-tax interest rate = r(1-τC)
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Mode of finance and WACC (Cont’d)
◼ Weighted Average Cost of Capital (WACC)
E ED
rwacc = rE + rD ( 1-τ C )
E+D E+D
E D D
= rE + rD − rD C
E+D E+D E+D
rU
D
◼ Therefore, rwacc =rU - τ C rD
E+D
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Mode of finance and WACC (Cont’d)
Example
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Mode of finance and WACC (Cont’d)
Solution
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A Project’s Cost of Capital
Unlevered firm comparable rE
▪ Purely equity financed project
Levered firm comparable rU
▪ Not purely equity financed project Firm with the same financing and risk. rwacc
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Mode of finance and WACC (Cont’d)
Compare rwacc with rU
◼ Unlevered cost of capital rU (or pretax WACC):
• Expected return investors will earn by holding the firm’s assets
• In a world with taxes, it can be used to evaluate an all-equity project with the
same risk as the firm.
◼ Weighted average cost of capital rwacc (or WACC):
• Effective after-tax cost of capital to the firm.
• In a world with taxes, WACC is less than the expected return of the firm’s assets.
• In a world with taxes, it can be used to evaluate a project with the same risk and
the same financing as the firm.
● Same types (i.e. debt, equity or both)
● Same proportional allocations
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Project Risk Characteristics
◼ Firm asset betas reflect market risk of the average project in a firm.
◼ Individual projects may be more or less sensitive to market risk.
◼ Financial managers in multi-divisional firms should evaluate projects
based on asset betas of firms in a similar line of business
◼ Projects could have different market risk characteristics from the
firm’s other activities, even within a firm with a single line of business.
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