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Chapter 13
The Cost of Capital
Before You Go On Questions and Answers
Section 13.1
1. Why does the market value of the claims on the assets of a firm equal the market value of the
assets?
The investors who own the debt and equity claims on the assets of a firm have the right to
receive all of the after-tax cash flows that the assets of the firm produce. Since the market
value of the assets equals the present value of the cash flows the assets produce, the market
value of the assets must equal the value of the claims on those assets.
2. How is the WACC for a firm calculated?
The WACC is calculated as the weighted average of the different types of claims on the
firm’s assets. The weights in this calculation are the fractions of the total value of the
financing that is represented by each individual type of financing. Equation 13.2 is the
general form of the WACC calculation.
1 1 2 2 3 3
1
...n
firm i i n n
i
k x k x k x k x k x k
2
3. What does the WACC for a firm tell us?
The WACC tells us the average cost of the money that has been used to finance the firm.
Section 13.2
1. Why do analysts care about the current cost of long-term debt when estimating a firm’s cost
of capital?
Managers care about the current cost of long-term debt because the opportunity cost of
capital that is relevant when discounting future cash flows is the opportunity cost of capital
as of today. Managers focus on long-term debt because firms generally use it to finance their
long-term assets, and it is the long-term assets that they are concerned about when they think
about the value of a firm’s assets.
2. How do you estimate the cost of debt for a firm with more than one type of debt?
When a firm has more than one type of debt, its overall cost of debt is estimated as a
weighted average of the costs of each type of debt. The weights in this calculation are the
fractions of the total value of the debt represented by each individual type of debt.
3. How do taxes affect the cost of debt?
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In the United States, the ability of firms to deduct interest payments when they compute their
taxes actually reduces the cost of using debt. The after-tax cost of debt equals the pretax cost
of debt times one minus the firm’s marginal tax rate.
Section 13.3
1. What information is needed to use the CAPM to estimate kcs or kps?
In order to use the CAPM to estimate kcs or kps, you need to know the risk-free rate, the
market risk premium, and the beta for the stock.
2. Under what circumstances can you use the constant-growth dividend formula to estimate kcs?
The constant-growth dividend formula can be used to estimate kcs if you can observe the
current market price of the common stock and you can estimate the dividend that
stockholders will receive next period, D1, and the rate at which the market expects dividends
to grow over the long run, g. Of course, it only makes sense to use this model if dividends
are expected to grow at a constant rate for the foreseeable future and this growth rate is not
greater than the long-term growth rate of the economy.
3. What is the advantage of using a multistage-growth dividend model, rather than the constant-
growth dividend model, to estimate kcs?
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A multistage model allows dividends to grow at different rates over time, while the constant-
growth model allows for only a single growth rate in perpetuity.
Section 13.4
1. Do analysts use book values or market values to calculate the weights when they use
Equation 13.7? Why?
Analysts use market values because this is what the theory underlying the calculation says
they should use. Book values are relevant only if they just happen to equal the market values.
2. What kinds of errors can be made when the WACC for a firm is used as the discount rate for
evaluating all projects in the firm?
Using the WACC to discount cash flows for projects that are less risky than the firm can
result in managers rejecting positive-NPV projects. Using the WACC to discount cash flows
for projects that are more risky than the firm can result in managers accepting negative-NPV
projects.
3. Under what conditions is the WACC the appropriate discount rate for a project?
The WACC is the appropriate discount rate for a project when the project has the same level
of systematic risk as the firm and when the project will be financed with the same proportion
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of debt, preferred shares, and common shares that have been used to finance the assets of the
firm.
Self-Study Problems
13.1 The market value of a firm’s assets is $3 billion. If the market value of the firm’s liabilities
is $2 billion, what is the market value of the stockholders’ investment and why?
Solution:
Since the identity that Assets = Liabilities + Equity holds for market values as well as
book values, we know that the market value of the firm’s equity is $3 billion—$2 billion,
or $1 billion.
13.2 Berron Comics, Inc., has borrowed $100 million and is required to pay its lenders $8 million
in interest this year. If Berron is in the 35 percent marginal tax bracket, then what is the
after-tax cost of debt (in dollars as well as in annual interest) to Berron.
Solution:
Because Berron enjoys a tax deduction for its interest charges, the after-tax interest expense
for Berron is $8 million × (1 – 0.35) = $5.2 million, which translates into an annual interest
expense of $5.2/$100 = 0.052, or 5.2 percent.
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13.3 Explain why the after-tax cost of equity (common or preferred) does not have to be
adjusted by the marginal income tax rate for the firm.
Solution:
The U.S. tax code allows a deduction for interest expense incurred on borrowing.
Preferred and common shares are not considered debt and, thus, do not benefit from an
interest deduction. As a result, there is no distinction between the before-tax and after-tax
cost of equity capital.
13.4 Mike’s T-Shirts, Inc., has debt claims of $400 (market value) and equity claims of $600
(market value). If the after-tax cost of debt financing is 11 percent and the cost of equity is
17 percent, then what is Mike’s weighted average cost of capital?
Solution:
Mike’s T-Shirts’ total firm value = $400 + $600 = $1,000. Therefore,
Debt = 40 percent of financing
Equity = 60 percent of financing
WACC = xDebtkDebt(1-t) + xpskps + xcskcs
WACC = (0.4 × 0.11) + (0.6 × 0.17) = 0.146, or 14.6%
13.5 You are analyzing a firm that is financed with 60 percent debt and 40 percent equity. The
current cost of debt financing is 10 percent, but due to a recent downgrade by the rating
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agencies, the firm’s cost of debt is expected to increase to 12 percent immediately. How
will this change the firm’s weighted average cost of capital if you ignore taxes?
Solution:
The pretax debt contribution to the cost of capital is xDebt × kDebt, and since the firm’s
pretax cost of debt is expected to increase by 2 percent, we know that the effect on WACC
(pretax) will be 0.6 × 0.02 = 0.012, or 1.2 percent. Incidentally, if we assume that the firm
is subject to the 40 percent marginal tax rate, then the after-tax contribution to the cost of
capital for the firm would be 0.012 × (1 – 0.4) = 0.0072, or 0.72 percent.
Critical Thinking Questions
13.1 Explain why the required rate of return on a firm’s assets must be equal to the weighted
average cost of capital associated with its liabilities and equity.
Solution:
In order to conceptualize the answer to this question, it helps to think of the case in which
the firm has raised all of its capital needs from a single source who owns all of the
liability and all of the equity claims on the firm. Assume that this source has no other
investments. If we were to measure the rate of return on the combined portfolio of
investments for this source, we would find that it is exactly equal to the return on the total
assets of the firm since that is the ultimate source of the returns. The weighted return of
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that portfolio, which is the weighted average cost of capital for the firm (if for the time
being we abstract away tax effects), is the return on the assets of the firm.
13.2 Which is easier to calculate directly, the expected rate of return on the assets of a firm or
the expected rate of return on the firm’s debt and equity? Assume that you are an outsider
to the firm.
Solution:
As an outsider to the firm, you will not be privy to the complete information about the
projected cash flows of each of the firm’s assets, and so that is a somewhat difficult
proposition. However, the collective market has made an inference concerning the
expected cash flows of each of the financing claims of the firm, and by pricing those cash
flows has given us an expected return for each of those claims. Therefore, finding the
expected return on the debt and equity claims of the firm is much easier than finding the
expected return on the assets of the firm, although that return can then be calculated from
the expected return on the financing claims of the firm.
13.3 With respect to the level of risk and the required return for a firm’s portfolio of projects,
discuss how the market and firm’s management can have inconsistent information and
expectations.
Solution:
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Firm management will be fully informed concerning the firm’s project risks, but their
ability to accurately predict the required return for the firm’s projects depends on the
market’s assessment of those project risks. Alternatively, the collective market is not fully
informed (as outsiders) concerning the firm’s project risks and yet uses its incomplete
information set to dictate a required return for the firm’s projects. This suggests that if the
firm were able to better inform the market, and thereby reduce the market’s perceived risk
on the firm’s projects, then the firm might be able to reduce the required rate of return on
the firm’s projects.
13.4 Your friend has recently told you that the federal government effectively subsidizes the
cost of debt (vs. equity financing) for corporations. Do you agree with that statement?
Explain.
Solution:
Your friend is correct. Because interest expense on debt is tax deductible, whereas
dividend payments on equity are not, the firm effectively gets a rebate on interest paid
through a lowered tax bill. Two firms with identical EBIT amounts with different interest
expenses will have different cash flow available to its collective set of investors. The firm
with greater interest expense (assuming it is less than the EBIT amount) will have greater
cash flow available to all of its investors.
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13.5 Your firm will have a fixed interest expense for the next 10 years. You recently found out
that the marginal income tax rate for the firm will change from 30 percent to 40 percent
next year. Describe how the change will affect the cash flow available to investors.
Solution:
Let’s compare our firm to the firm with no interest expense. In order to make a concrete
example, assume that our firm has interest expense of $100 per year. Since the after-tax
cost of debt for the firm is equal to kdebt × (1 – t), then we can calculate the tax benefit to
using debt to be kdebt × t. In order to calculate that benefit in dollar terms, we would just
multiply (interest expense) × t. Therefore, the current dollar benefit to the interest expense
is $100 × 0.3 = $30. Next year, the dollar benefit is $100 × 0.4 = $40. The net benefit of
interest expense from the increased marginal corporate tax rate is $10, and that is a
positive benefit. Note that the analysis isolates the effect on debt and does not consider
the lower operating earnings figure caused by the increased tax rate. Overall, the increase
in tax rate will result in less cash flows available to investors, but for the leveraged (debt
holding) firm the reduction in cash flow is mitigated by the benefit from being able to
deduct interest expenses.
13.6 Describe why it is not usually appropriate to use the coupon rate on a firm’s bonds to
estimate the pretax cost of debt for the firm.
Solution:
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The pretax cost of debt for the firm is the current annual economic cost of borrowing for
the firm (before any tax effects). That cost is better measured by the current yield to
maturity on the firm’s debt than by the coupon rate that is currently paid on that debt.
Since most firms try to issue new bonds very close to par, the coupon rate on a bond is an
indication of the yield to maturity on the bond issue at the time of issue. Unless the
market-determined borrowing rate for the firm is the same as when the bond was issued,
then the current yield to maturity of a bond will not be equal to the current coupon rate on
the bond.
13.7 Maltese Falcone, Inc., has not checked its weighted average cost of capital for four years.
Firm management claims that since Maltese has not had to raise capital for new projects
since that time, they should not have to worry about their current weighted average cost
of capital since they have essentially locked in their cost of capital. Critique that
statement.
Solution:
That is a false statement. Maltese is assuming that since it does not have to raise capital
for new projects, then it has essentially locked in its cost of capital. However, in a liquid
capital market every firm competes for capital everyday since the firm’s investors have
the opportunity to sell their investments to other investors. If a firm does not provide
investors with an ample return, then the investors will sell their investments in the firm,
which, in aggregate, will have the effect of actually raising the cost of capital for the firm
(since the current price of the securities will move down). Therefore, a firm that ignores
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its current cost of capital by thinking that it has locked in a cost of capital might even be
raising its cost of capital by making that incorrect assumption.
13.8 Ten years ago, the Edson Water Company issued preferred stock with a price equal to the
par amount of $100. If the dividend yield on that issue was 12 percent, explain why the
firm’s current cost of preferred capital is likely not equal to 12 percent.
Solution:
Since the price of the preferred shares at issue was $100 and the dividend yield was 12
percent, then we know that the annual dividend on the shares is $12. We also then know
that the required rate of return at the time of issue was 12 percent. If during the last 10
years, the required rate of return on Edson preferred shares has changed at all, the current
required rate of return will not be 12 percent. This will, in turn, change the price of the
shares to some amount other than $100.
13.9 Discuss under what circumstances you might be able to use a model that assumes
constant growth in dividends to calculate the current cost of equity capital for a firm.
Solution:
In order to be completely correct, a firm must grow its dividends at a constant rate into
the indefinite future. If one expects the growth in dividends to change in the future, then
using a constant-growth dividend assumption is incorrect and only an estimation.
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13.10 Your boss just completed computing your firm’s weighted average cost of capital. He is
relieved because he says that he can now use that cost of capital to evaluate all projects
that the firm is considering for the next four years. Evaluate that statement.
Solution:
Your boss is incorrect. A firm is always subject to revisions to its cost of capital due to
current market and firm conditions. In addition, the firm could also be making an error by
using the same cost of capital for all of its future projects. For that particular error to not
be made, two conditions must be met. That is, future projects must be financed with the
same mix of capital (debt, preferred shares, and common shares) with which the entire
firm is currently financed. In addition, the future projects must contain the same level of
systematic risk as that of the average project that the firm is currently operating.
Questions and Problems
BASIC
13.1 Finance balance sheet: KneeMan Markup Company has total debt obligations with book
and market values equal to $30 million and $28 million, respectively. It also has total
equity with book and market values equal to $20 million and $70 million, respectively. If
you were going to buy all of the assets of KneeMan Markup today, how much should you
be willing to pay?
LO 1
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Solution:
The price you should be willing to pay for all of the assets of the firm is the market value
of those assets. Using the market price version of the balance sheet identity, we can add
the market price of the debt obligations and the equity to find the market price of the
assets. That is, $28 million + $70 million = $98 million.
13.2 WACC: What is the weighted average cost of capital?
LO 1
Solution:
The weighted average cost of capital (WACC) is the weighted average of the costs to the
different sources of capital used to fund a firm, The WACC is often used as an estimate
of the cost of financing a new project given the firm’s current mix of debt and equity.
13.3 Taxes and the cost of debt: How are taxes accounted for when we calculate the cost of
debt?
LO 2
Solution:
When we calculate the cost of debt for a U.S. firm, we must take into account the tax
subsidy given in the United States for interest payments on debt. For every dollar the firm
pays in interest, the firm’s tax bill will decline by ($1 * t), where t is the firm’s marginal
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tax rate. We adjust for this tax benefit by multiplying the pretax cost of debt by (1 - t).
This calculation gives us the after-tax cost of debt. We use the after-tax cost of debt for
cost of capital calculations such as when we calculate the WACC.
13.4 Cost of common equity: List and describe each of the three methods used to calculate
the cost of common equity.
LO 3
Solution:
1) The Capital Asset Pricing Model (CAPM) formula for the cost of common stock,
given in Equation 13.4, can be used to calculate the return investors will demand on
investment in the company’s common stock. See Chapter 7 for further discussion of
the CAPM.
2) The constant-growth dividend model can be used to calculate the cost of equity
implied by the firm’s current stock price. In an efficient market, the current price of
the company’s stock should reflect the cash flows (dividends) that investors will
receive in the future from holding equity in the firm, discounted by an appropriate
rate (the cost of equity). By knowing the current dividend paid by the firm and the
expected growth rate of dividends, we can use Equation 13.5 to compute the cost of
capital that is implied in the firm’s current stock price. The constant-growth dividend
model is only appropriate when there is a reasonable expectation that the firm’s
dividend will continue growing at approximately the same rate forever. For example,
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it might be used to calculate the cost of equity for a mature company whose growth
rate is similar to that of the economy.
3) The multistage-growth dividend model is very similar to the constant-growth
dividend model, but the multistage-growth dividend model can be applied in
situations when the growth rate is expected to change—for example, a small, fast
growing company whose growth will certainly slow as the company becomes larger.
See Section 13.3 for discussion of the calculation of cost of equity using the
multistage-growth dividend model.
13.5 Cost of common stock: Whitewall Tire Co. just paid a $1.60 dividend on its common
shares. If Whitewall is expected to increase its annual dividend by 2 percent per year into
the foreseeable future and the current price of Whitewall’s common shares is $11.66,
what is the cost of common stock for Whitewall?
LO 3
Solution:
The cost of common equity for Whitewall can be found using the constant-growth
assumption equation:
1 0
cs
D D ×(1+ ) $1.60×(1+ 0.02)P = = = = $11.66
- 0.02cs cs cs
g
k - g k - g k
Solving for kcs, we find it is equal to 0.16 or 16 percent.
13.6 Cost of common stock: Seerex Wok Co. is expected to pay a dividend of $1.10 one year
from today on its common shares. That dividend is expected to increase by 5 percent
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every year thereafter. If the price of Seerex is $13.75, then what is Seerex’s cost of
common stock?
LO 3
Solution:
We can use the formula to find the cost of common equity assuming constant growth.
1cs
cs
D $1.100.05 0.13, or 13%
P $13.75 k g
13.7 Cost of common stock: Two-Stage Rocket paid an annual dividend of $1.25 yesterday,
and it is commonly known that the firm’s management expects to increase its dividends
by 8 percent for the next two years and by 2 percent thereafter. If the current price of
Two-Stage’s common stock is $17.80, t what is the cost of common equity capital for the
firm?
LO 3
Solution:
2 2
1 1 1 1 1 1 2
2 2
2
D (1 ) D (1 ) D (1 ) (1 )P
1 1 1
cs
cs cs cs cs
g g g g
k k k g k
2 2
2 2
$1.25 (1 0.08) $1.25 (1 0.08) $1.25 (1 0.08) (1 0.02)$17.80
1 1 0.02 1
cs cs cs csk k k k
,
Using a spreadsheet to solve for the value of kcs, we find that the cost of common equity
capital is 10 percent.
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13.8 Cost of preferred stock: Fjord Luxury Liners has preferred shares outstanding that pay an
annual dividend equal to $15 per year. If the current price of Fjord preferred shares is
$107.14, then what is the after-tax cost of preferred stock for Fjord?
LO 3
Solution:
Using the equation for finding the cost of preferred equity, we have
ps
ps
D 150.14, or 14%
P 107.14 k
13.9 Cost of preferred stock: Kresler Autos has preferred shares outstanding that pay annual
dividends of $12, and the current price of the shares is $80. What is the after-tax cost of
new preferred shares for Kresler if the flotation (issuance) costs for preferred are 5
percent?
LO 3
Solution:
Kresler will only receive 95 percent of the proceeds, so we know that we can use the
equation to solve for the cost of preferred equity by adjusting the denominator for the
reduced proceeds from the sale of new equity. We then have:
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ps
D 12 120.158, 15.8%
P (1- F) 80(1 0.05) 76
psk or
13.10 WACC: Describe the alternatives to using a firm’s WACC as a discount rate when
evaluating a project.
LO 4
Solution:
There are two major reasons why WACC may not be used to discount new projects:
1. It is not appropriate to use a firm’s WACC to discount a project’s free cash flows if the
systematic risk of the project is very different from the systematic risk of the firm. To
account for this potential problem, some firms estimate discount rates that directly reflect
the risk involved in the project’s cash flows. For example, a risky project might be
assigned a discount rate that is significantly higher then the firm’s WACC.
2. It is not appropriate to use a firm’s WACC when a project that has the same systematic
risk as the firm is not being financed using the same mix of debt and equity as the firm—
for example, if a project will be financed entirely with equity. The project’s cash flows
should be discounted using the cost of equity rather than the firm’s WACC. These two
rates will be the same only if the firm has no debt.
13.11 WACC for a firm: Capital Co. has a capital structure, based on current market values,
that consists of 50 percent debt, 10 percent preferred shares, and 40 percent common
shares. If the returns required by investors are 8 percent, 10 percent, and 15 percent for
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debt, preferred equity, and common stock, respectively, what is Capital’s after-tax
WACC? Assume that the firm’s marginal tax rate is 40 percent.
LO 4
Solution:
cscspspsdebtdebt kxkxtkxWACC )1( =
WACC=0.5×0.08×(1-0.4)+0.1×0.10+0.4×0.15= 0.094, or 9.4%
13.12 WACC: What are direct out-of-pocket costs?
LO 4
Solution:
Direct out-of-pocket costs are the actual out-of-pocket costs that a firm incurs when it
raises capital. They include such things as fees paid to investment bankers and legal and
accounting expenses.
INTERMEDIATE
13.13 Finance balance sheet: Explain why the total value of all of the securities financing a
firm must be equal to the value of the firm.
LO 1
Solution:
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The value of the firm’s assets is equal to the present value of the future cash flows
expected to be generated by those assets. The cash flow claim on those assets is
prioritized by the financing of those assets. Therefore, the financing claims on the assets
of the firm fully account for the entire value of the assets, and the value of the financing
claims must equal the value of the assets that are carved up by those claims.
13.14 Finance balance sheet: Explain why the cost of capital for a firm is equal to the expected
rate of return to the investors in the firm.
LO 1
Solution:
If we view the firm as a conduit for the cash flows provided by the assets of the firm, then
it is easy to see that the cash flows provided by the assets of the firm must equal the cash
flows provided to the aggregate investor group of the firm. We also know that the capital
invested in the firm must equal the capital invested by the firm. Therefore, we then know
that the rate of return for the investors of the firm must equal the cost of capital provided
to the firm. The expected return to investors will also equal the expected cost of capital
for the firm.
13.15 Current cost of a bond: You know that the after-tax cost of debt capital for Bubbles
Champagne is 7 percent. If the firm has only one issue of five-year bonds outstanding,
what is the current price of the bonds if the coupon rate on those bonds is 10 percent?
22
Assume the bonds make semiannual coupon payments and the marginal tax rate is 30
percent.
LO 2
Solution:
We know the after-tax cost of debt, and from that we can find the pretax cost of debt by
multiplying by 1 minus the tax rate. This becomes 0.07 / (1 – .3) = 0.10.
Since the YTM on the bonds is equal to the coupon rate, then we know the bonds are
priced at par, or $1,000.
13.16 Current cost of a bond: Perpetual Ltd. has issued bonds that never require the principal
amount to be repaid to investors. Correspondingly, Perpetual must make interest
payments into the infinite future. If the bondholders receive annual payments of $75 and
the current price of the bonds is $882.35, what is the after-tax cost of debt for Perpetual if
the firm is in the 40 percent marginal tax rate?
LO 2
Solution:
Since the bonds represent a perpetuity, we know that the pretax cost of debt can be
solved using the following:
debt
Coupon Payment $75= = =0.085
Bond Price $882.35k
and the after-tax cost is 0.085 × (1 - .4) = 0.051, or 5.1%
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13.17 Current cost of a bond: You are analyzing the cost of debt for a firm. You know that the
firm’s 14-year maturity, 8.5 percent coupon bonds are selling at a price of $823.48. The
bonds pay interest semiannually. If these bonds are the only debt outstanding, what is the
after-tax cost of debt for this firm if it has a 30 percent marginal and average tax rate?
LO 2
Solution:
The current YTM for the bonds can be calculated as follows.
$823.48 = $42.50 × PVIFA(28, YTM/2) + $1,000 × PVIF(28, YTM/2)
Solving, we find that YTM = 0.11, and therefore the after-tax cost of debt is equal to:
0.11 × (1 – 0.3) = 0.077, or 7.7%
13.18 Taxes and the cost of debt: Holding all other things constant, does a decrease in the
marginal tax rate for a firm provide incentive for the firm to increase or decrease its use
of debt?
LO 2
Solution:
The after-tax cost of debt for the firm is equal kDebt x (1 – t). We can then calculate the
tax benefit to using debt to be kDebt x t. Therefore, the value of the tax benefit to debt
increases with the marginal tax rate. If the marginal tax rate decreases, then the tax
benefit to debt decreases as well. Therefore, the incentive to borrow actually decreases
with a decrease in the marginal tax rate.
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13.19 Cost of debt for a firm: You are analyzing the after-tax cost of debt for a firm. You
know that the firm’s 12-year maturity, 9.5 percent semi-annual coupon bonds are selling
at a price of $1,200. If these bonds are the only debt outstanding for the firm, what is the
after-tax cost of debt for this firm if it has a marginal tax rate of 34 percent? What if the
bonds are selling at par?
LO 2
Solution:
The current YTM for the bonds can be calculated as follows.
n
nB n
24
24
11
F(1 )P C
(1 )
11
$1,000(1 )$1,200.50 $47.50
(1 )
i
i i
i
i i
Solving, we find that YTM = 0.07008 or 7.008% and therefore the after-tax cost of debt
is equal to
0.07008 × (1 – 0.34) = 0.046253, or 4.63%
If the bonds are priced at par, then the YTM on the bonds is 9.5 percent and then the
after-tax cost of debt would be 6.27%
13.20 Cost of common stock: Underestimated Inc.’s common shares currently sell for $36
each. The firm’s management believes that its shares should really sell for $54 each. If
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the firm just paid an annual dividend of $2.00 per share and management expects those
dividends to increase by 8 percent per year forever (and this is common knowledge to the
market), what is the current cost of common equity for the firm and what does
management believe is a more appropriate cost of common equity for the firm?
LO 3
Solution:
The current cost of equity for the firm is
1
cs
D $2.00 1.080.08 0.14, or 14%
P $36.00
csk g
But the firm believes that its cost of capital is more appropriately
1
cs
D $2.00 1.080.08 0.12, or 12%
P $54.00
csk g
13.21 Cost of common stock: Write out the general equation for the price of the stock for a
firm that will grow dividends very rapidly for the four years after the next dividend and
thereafter at a constant, but lower rate. Discuss the problems in estimating the cost of
equity capital for such a stock.
LO 3
Solution:
2 3 4 4
1 1 1 1 1 1 1 1 1 1 1 2cs 2 3 4 55
cs cscs cs cs cs 2 cs
D D (1 ) D (1 ) D (1 ) D (1 ) D (1 ) (1 )P
1 k (1 )1 1 1 1
g g g g g g
kk k k k g k
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It is easy to see that in order to solve for a cost of capital, kcs, you must have a good idea
of what g1 and g2 are. If those growth rates are poor estimates, then the calculation for kcs,
will also be a poor estimate.
13.22 Cost of common stock: You have calculated the cost of common stock using all three
methods described in the chapter. Unfortunately, all three methods have yielded different
answers. Describe which answer (if any) is most appropriate.
LO 3
Solution:
Two of the methods involve an estimate of the growth rate in dividends for the firm. If
you are confident in your estimate of the growth rate, then those methods might be most
appropriate. Otherwise, utilizing the CAPM, which does not involve any dividend growth
rate estimates, would probably be the best. You may choose to average the results of all
three methods.
13.23 WACC for a firm: The managers of a firm financed entirely with common equity are
evaluating two distinct projects. The first project has a large amount of unsystematic risk
and a small amount of systematic risk. The second project has a small amount of
unsystematic risk and a large amount of systematic risk. Which project, if taken, is more
likely to increase the firm’s cost of capital?
LO 4
27
Solution:
Markets adjust the cost of capital according to the level of systematic risk in a project.
Therefore, the project with the greatest level of systematic risk will have the greatest
positive impact on the cost of capital for the firm, even if it has the lowest level of
nonsystematic risk.
13.24 WACC for a firm: The Imaginary Products Co. currently has debt with market value of
$300 million outstanding. The debt consists of 9 percent coupon bonds (semiannual
coupon payments) which have a maturity of 15 years and are currently priced at
$1,440.03 per bond. The firm also has an issue of 2 million preferred shares outstanding
with a market price of $12.00. The preferred shares pay an annual dividend of $1.20.
Imaginary also has 14 million shares of common stock outstanding with a price of $20.00
per share. The firm is expected to pay a $2.20 common dividend one year from today,
and that dividend is expected to increase by 5 percent per year forever. If Imaginary is
subject to a 40 percent marginal tax rate, then what is the firm’s weighted average cost of
capital?
LO 4
Solution:
Step 1: Total amount of debt, common equity, and preferred equity:
Debt = $300,000,000 (given)
Preferred equity = $12 × 2,000,000 = $24,000,000
Common equity = $20 × 14,000,000 = $280,000,000
28
Total capital = $604,000,000
xDebt = 300/604 = 0.4967
xps = 24/604 = 0.0397
xcs = 280/604 = 0.4636
Step 2: Cost of capital components:
Cost of debt:
$1,440.03 = $45 × PVIFA(30, YTM/2) + $1,000 × PVIF(30, YTM/2)
Solving, we find that YTM = 0.0484 (this is a pretax number).
Cost of preferred equity:
ps
ps
D $1.200.10
P $12.00 k
Cost of common equity:
1cs
cs
D $2.20g 0.05 0.16
P $20.00 k
Step 3: Combine using the WACC formula.
cscspspsdebtdebt kxkxtkxWACC )1( =
0.4967 0.0484 (1 0.4) 0.0397 0.10 0.4636 0.16 0.0926, or 9.26%WACC
13.25 Choosing a discount rate: For the Imaginary Products firm in Problem 13.24, calculate
the appropriate cost of capital for a new project that is financed with the same proportion
of debt, preferred shares, and common shares as the firm’s current capital structure. Also
assume that the project has the same degree of systematic risk as the average project that
29
the firm is currently undertaking (the project is also in the same general industry as the
firm’s current line of business).
LO 1
Solution:
Since Imaginary will be financing the project with the same mix of capital that the firm is
currently utilizing for its projects, we will have met the first restriction concerning
financing mix. In addition, the new project will have the same degree of systematic risk
(in addition to being in the same general line of business). Therefore, Imaginary can use
the 9.26 percent cost of capital to evaluate its project.
13.26 Choosing a discount rate: If a firm anticipates financing a project with a capital mix
different than the firm’s current capital structure, describe in realistic terms how the firm
is subjecting itself to a calculation error if its historical WACC is used to evaluate the
project.
LO 1
Solution:
Since the firm is financing the project with a different capital mix than it has historically
used, we know that the weights and rates for debt, preferred, and common shares in the
WACC formula will be different. We know that the cost of capital for each component is
a function of the individual weights and rates. Therefore, we know that the WACC will
30
be different for the overall firm versus that of the individual project. Therefore, using its
historical WACC can result in an error in the NPV estimate for the project.
ADVANCED
13.27 You are analyzing the cost of capital for MacroSwift Corporation, which develops
software operating systems for computers. The firm’s dividend growth rate has been a
very constant 3 percent per year for the past 15 years. Competition for the firm’s current
products is expected to develop in the next year, and MacroSwift is currently expanding
its revenue stream into the multimedia industry. Evaluate the appropriateness of
continuing to use a 3 percent growth rate in dividends for MacroSwift in your cost of
capital model.
LO 1
Solution:
While the growth in dividends has been extremely constant for Macroswift over the last
15 years, it is appropriate to assume a constant-growth rate only if that same rate will
continue in the future. Two factors will act to alter that growth in the future. MacroSwift
will have competition for its current product list in the near future, and that could alter the
firm’s growth rate. In addition, the firm is expanding its product line into an area that will
probably not yield the same level of growth. It is therefore, unlikely that MacroSwift’s
dividend growth rate will continue at a 3 percent annual rate. This suggests that we
should consider something other than constant growth in our modeling.
31
13.28 You are an external financial analyst evaluating the merits of a stock. Since you are using
a dividend discount model approach to calculate a cost of equity capital, you need to
estimate the dividend growth rate for the firm in the future. Describe how you might go
about that process.
LO 3
Solution:
One source for this data would be to measure the firm’s dividend growth rate in recent
history. If we could assume that such a growth in dividends will continue into the future,
then our measure would be reasonable. One additional source would be to read a financial
analyst’s report in which the author of the report may have a better estimate of the firm’s
future prospects.
13.29 You know that the return of Momentum Cyclicals’ common shares is 1.6 times as
sensitive to macroeconomic information as the return of the market. If the risk-free rate
of return is 4 percent and the market risk premium is 6 percent, then what is Momentum
Cyclicals’ cost of common equity capital?
LO 3
Solution:
We know that the beta for Momentum Cyclicals is 1.6, and we can use the remaining
information in the CAPM as follows:
32
E(R ) = R E(R ) - Rcs rf m rf
0.04 + 1.6(0.06) = 0.136 = 13.6%
+ β
=
13.30 In your analysis of the cost of capital for a common stock, you calculate a cost of capital
using a dividend discount model that is much lower than the calculation for the cost of
capital using the CAPM model. Explain a possible source for the discrepancy.
LO 3
Solution:
Comparing the two formulas for the two methods, we have:
E(R ) = R + E(R ) - Rcs rf m rfβ and gP
Dk
cs
cs 1
Given these two sources of information, we see that the only variable that we are not able
to get directly from the market is the growth rate in dividends (note that future dividends
are also a function of this growth rate), which is an estimate. Since our dividend discount
method provided a lower cost of capital than the CAPM, it seems likely that we estimated
the growth rate lower than what the aggregate market has assumed. Of course, this
assumes that the market is efficiently pricing the stock. If the market price is incorrect,
then this might lead to a difference.
13.31 RetRyder Hand Trucks has a preferred share issue outstanding that pays an annual
dividend of $1.30 per year. The current cost of preferred equity for RetRyder is 9 percent.
33
If RetRyder issues additional preferred shares that pay exactly the same dividend and the
investment banker retains 8 percent of the sale price proceeds, what is the cost of new
preferred shares for RetRyder?
LO 3
Solution:
The current cost of preferred shares for RetRyder is
ps
ps
D $1.30P $14.44
0.09
k
and then RetRyder would receive 92 percent of the proceeds. We could then adapt the
cost of preferred equity to the following:
ps
D $1.30 $1.30= = =0.0978, or 9.78%
P (1- F) $14.44(1-0.08) $13.29psk
13.32 Enigma Corporation’s management believes that the firm’s cost of capital (WACC) is too
high because the firm has been too secretive with the market concerning its operations.
Evaluate that statement.
LO 4
Solution:
The WACC is a function of the perceived risk involved in the cash flows of the projects
that the firm is currently operating. If the market perceives that risk to be higher than the
actual risk due to a lack of information concerning those projects, then the firm might be
34
able to lower that perceived risk by sharing more information with the market. That could
have the effect of lowering the firm’s WACC.
13.33 Discuss what valuable information would be lost if you decided to use book values in
order to calculate the cost of each capital component within a firm’s capital structure.
LO 4
Solution:
Market returns are impounded in market prices. If those prices are ignored, then the
efficiency of the market’s information process is essentially thrown away. Since the
market adjusts securities prices according to the expected return for investing in a
security, then ignoring that information is the same as ignoring what the market deems to
be an appropriate cost of capital for the firm.
13.34 Hurricane Corporation is financed with debt, preferred equity, and common equity with
market values of $20 million, $10 million, and $30 million, respectively. The betas for
the debt, preferred stock, and common stock are 0.2, 0.5, and 1.1, respectively. If the risk
free rate is 3.95 percent, the market risk premium is 6.01 percent, and Hurricane’s
average and marginal tax rates are both 30 percent, what is the company’s weighted
average cost of capital?
LO 4
Solution:
The fractions of the total financing represented by debt, common equity, and preferred
equity are:
Debt = $20 million
35
Preferred equity = $10 million
Common equity = $30 million
Total capital = $60 million
xdebt = $20/$60 = 0.3333
xps = $10/$60 = 0.1667
xcs = $30/$60 = 0.5000
The costs of debt, common equity, and preferred equity are:
E(RDebt) = Returnrf + βDebt (E(ReturnM) – Returnrf) = 0.0395 + (0.2 × 0.0601) = 0.0515, or 5.15%
E(Rps) = Returnrf + βps (E(ReturnM) – Returnrf) = 0.0395 + (0.5 × 0.0601) = 0.0696, or 6.96%
E(Rcs) = Returnrf + β (E(ReturnM) – Returnrf) = 0.0395 + (1.1 × 0.0601) = 0.106, or 10.6%
0.3333 0.0515 (1 0.3) 0.1667 0.0696 0.5 0.106 0.0766, or 7.66% WACC
13.35 You are working as an intern at Coral Gables Products, a privately owned manufacturing
company. Shortly after you read Chapter 13 in this book, you got into a discussion with
the Chief Financial Officer (CFO) at Coral Gables about weighted average cost of capital
calculations. She pointed out that, just as the beta of the assets of a firm equals a
weighted average of the betas for the individual assets (as shown in Equation 7.11):
Asset portfolio 1 1 2 2 3 3
1
. . .n
n i i n n
i
x x x x x
the beta of the assets of a firm also equals a weighted average of the betas for the debt,
preferred equity, and common equity of a firm:
Asset portfolio Debt Debt ps ps cs cs
1
n
n i i
i
x x x x
Why must this be true?
LO 3
Solution: Since, collectively, the debt and equity holders are entitled to receive all of the
cash flows that the assets of the firm are expected to produce, the systematic risk of the
36
cash flows that they are entitled to receive must be the same as the systematic risk of the
cash flows the assets are expected to produce.
13.36 The CFO described in problem 13.35 asks you to estimate the beta for Coral Gables’
common stock. Since the common stock is not publicly traded, you do not have the data
necessary to estimate the beta using regression analysis. However, you have found a
company with publicly traded stock that has operations which are exactly like those at
Coral Gables. Using stock returns for this pure-play comparable you estimate the beta for
the comparable company’s stock to be 1.06. The market value of that company’s
common equity is $45 million and it has one debt issue outstanding with a market value
of $15 million and an annual pretax cost of 4.85 percent. The comparable company has
no preferred stock.
a. If the risk free rate is 3.95 percent and the market risk premium is 6.01 percent, what
is the beta of the assets of the comparable company?
b. If the total market value of Coral Gables’ financing consists of 35 percent debt and 65
percent equity (this is what the CFO estimates the market values to be) and the pretax
cost of its debt is 5.45 percent, what is the beta for Coral Gables’ common stock?
LO 3
Solution: You can solve this problem using the equation:
Asset portfolio Debt Debt ps ps cs cs
1
n
n i i
i
x x x x
Since neither company has preferred stock, the equation in this case has only two
terms:
Asset portfolio Debt Debt cs cs n x x
37
a. To calculate the beta of the assets of the comparable company, you must first
estimate the beta of the comparable firm’s debt and the fractions of the total
market value of the comparable firm’s financing which is represented by the debt
and equity.
The beta of the comparable firm’s debt can be estimated using the CAPM formula
(Equation 7.10):
E(Ri) = Rrf + βi[E(Rm) – Rrf]
Since this formual can be used to calculate the expected return on debt, it can be
written as:
E(RDebt) = Rrf + βDebt[E(Rm) – Rrf]
If we assume that the market is pricing the debt correctly, the pretax cost of debt
equals the expected return on debt in this equation. Therefore, substituting the
information from the problem statement into this equation yields:
4.85% = 3.95% + βDebt[6.01%]
and the beta of the debt equals:
βDebt = (4.85% - 3.95%)/6.01% = 0.15
The fractions of the total financing represented by debt and common equity are:
Debt = $15 million
Common equity = $45 million
Total capital = $60 million
xDebt = $15/$60 = 0.25
xcs = $45/$60 = 0.75
Finally, the beta of the assets of the comparable company is:
Asset portfolio Debt Debt cs cs (0.25 0.15) (0.75 1.06) 0.8325n x x
38
b. Since the comparable company is in exactly like Coral Gables, this is a good
estimate of asset beta for Coral Gables. To calculate the beta for Coral Gables’
common equity, you use the following equation with the beta of the assets for the
comparable company and the beta of debt and financing fractions for Coral
Gables.
Asset portfolio Debt Debt cs cs n x x
The beta of the Coral Gables debt is:
E(RDebt) = Rrf + βDebt[E(Rm) – Rrf]
5.45% = 3.95% + βDebt[6.01%]
βDebt = (5.45% - 3.95%)/6.01% = 0.25
and the beta of the firm’s equity is:
Asset portfolio Debt Debt cs cs
cs
cs
0.8325 (0.35 0.25) (0.65 )
0.8325 (0.35 0.25) 0.7451.15
0.65 0.65
n x x
13.37 What is the weighted average cost of capital for Coral gables using your estimated beta
and the information in the problem statement in Problem 13.36? Assume that the average
and marginal tax rates for Coral Gables are both 25 percent.
LO 3, LO 4
Solution: The cost of equity is:
E(Rcs) = Rrf + βcs[E(Rm) – Rrf]
E(Rcs) = 3.95% + (1.15 × 6.01%) = 10.86%
39
The weighted average cost of capital is:
0.35 5.45% (1 0.25) 0.65 10.86% 8.49%WACC
CFA Problems
13.38 The cost of equity is equal to the
a. expected market return.
b. rate of return required by stockholders.
c. cost of retained earnings plus dividends.
d. risk the company incurs when financing.
LO 1
Solution:
b is correct. The cost of equity is defined as the rate of return required by stockholders.
13.39 Dot.Com has determined that it could issue $1,000 face value bonds with an 8 percent
coupon paid semiannually and a five-year maturity at $900 per bond. If Dot.Com’s
marginal tax rate is 38 percent, its after-tax cost of debt is closest to
a. 6.2 percent
b. 6.4 percent.
c. 6.6 percent.
d. 6.8 percent.
LO 2
Solution:
40
c is correct.
FV = $1,000; PMT = $40; N = 10; PV = $900
Solve for i. The six-month yield, i, is 5.3149%
YTM = 5.3149% 2 = 10.6298%
rd(1 – t)= 10.6298% × ( 1 – 0.38) = 6.5905%
13.40 Morgan Insurance Ltd. issued a fixed-rate perpetual preferred stock three years ago and
placed it privately with institutional investors. The stock was issued at $25 per share with
a $1.75 dividend. If the company were to issue preferred stock today, the yield would be
6.5 percent. The stock’s current value is
a. $25.00
b. $26.92
c. $37.31
d. $40.18
LO 3
Solution:
b is correct. The company can issue preferred stock at 6.5%.
Pp = $1.75/0.065 = $26.92
Note: Dividends are not tax deductible so there is no adjustment for taxes.
13.41 The Gearing Company has an after-tax cost of debt capital of 4 percent, a cost of
preferred stock of 8 percent, a cost of equity capital of 10 percent, and a weighted average
cost of capital of 7 percent. Gearing intends to maintain its current capital structure as it
41
raises additional capital. In making its capital-budgeting decisions for the average-risk
project, the relevant cost of capital is
a. 4 percent
b. 7 percent
c. 8 percent
d. 10 percent
LO 4
Solution:
b is correct. The weighted average cost of capital, using weights derived from the current
capital structure, is the best estimate of the cost of capital for the average-risk project of a
company.
13.42 Suppose the cost of capital of the Gadget Company is 10 percent. If Gadget has a capital
structure that is 50 percent debt and 50 percent equity, its before-tax cost of debt is 5
percent, and its marginal tax rate is 20 percent, then its cost of equity capital is closest to
a. 10 percent
b. 12 percent
c. 14 percent
d. 16 percent
LO 1
Solution:
c is correct.
re = ra + (ra – rd) (D/E)
Note:
If D/(D + E) = 0.50, then D/E = 1.0
42
re = 0.10 + (0.10 – 0.05)(1.0)(1 – 0.2)
re = 0.10 + [0.05(0.90)] = 0.10 + 0.04 = 0.14, or 14%
Sample Test Problems
13.1 The Balanced, Inc., has three different product lines of business. Its least risky product
line has a beta of 1.7, while its middle- risk product line has a beta of 1.8, and its most
risky product line has a beta of 2.1. The market value of the assets invested in each
product line is $1 billion for the least risky line, $3 billion for the middle risk line, and $7
billion for the riskiest product line. What is the beta of The Balanced, Inc.?
Solution:
Using the formula for the beta of an n asset portfolio, we know
1 1 2 2 3 3
1
...n
n asset portfolio i i n n
i
x x x x x
.
We have $1 billion + $3 billion + $7 billion = $11 billion, so
x1 = $1 billion / $11 billion = 0.09091
x2 = $3 billion / $11 billion = 0.27273
x1 = $7 billion / $11 billion = 0.63637
Then (0.09091 × 1.7) + (0.27273 × 1.8) + (0.63637 × 2.1) = 1.9818
43
13.2 Ellwood Corp. has a five-year bond issue outstanding with a coupon rate of 10 percent
and a price of $1,039.56. If the bonds pay coupons semiannually, what is the pretax cost
of the debt and what is the after-tax cost of the debt? Assume the marginal tax rate for the
firm is 40 percent.
Solution:
$1,039.56 = $50 PVIFA (i%/2, 10) + $1,000 PVIFA (i%/2, 10)
Using trial and error, we find that i%/2 = 4.5% ==> i% = 9% which is the pretax cost of
the debt. The after-tax cost of the debt is 9% × (1 - 0.4) = 5.4%.
Using the financial calculator, the results are the same.
13.3 Miron’s Copper Corp. expects its common stock dividends to grow 1.5 percent per year
for the indefinite future. The firm’s shares are currently selling for $18.45, and the firm
just paid a dividend of $3.00 yesterday. What is the cost of common stock for this firm?
Solution:
01 10
0 0
D 1+ $3.00 1 0.015D DP 0.015 0.18,or18%
P P $18.45
cs cs
cs
gk g g k
k g
13.4 Micah’s Time Portals has a preferred stock issue outstanding that pays an annual
dividend of $2.50 per year and is currently selling for $27.78 a share. What is the cost of
preferred stock for this firm?
44
Solution:
0 ps ps
ps ps 0
P = D /
= D / P
$27.78 $2.50 / $27.78 0.09,or 9%
k
k
13.5 The Old Time New Age Co. has a portfolio of projects that has a beta of 1.25. The firm is
currently evaluating a new project that involves a new product in a new competitive
market. Briefly discuss what adjustment Old Time New Age might make to its 1.25 beta
in order to evaluate this new project.
Solution:
As discussed in the chapter, the best method for evaluating the new project would be to
determine the level of systematic risk for the new project. If that is not ascertainable, the
firm might have a predetermined range of modifications for projects that do not have the
same level of systematic risk as the firm’s current portfolio of projects. That is, the firm
would adjust the beta downward, to the greatest extent, for an efficiency-type project,
with a lessened downward adjustment for product extension type project. The firm would
adjust its beta upward for a new market project and the greatest upward adjustment for
new products. The situation for Old Time New Age would dictate a large upward
adjustment. The exact amount of the adjustments would be determined by experienced
management of the firm.
