Week 2 Work Problems and Explanations
1. A company is 33% financed by risk-free debt. The risk-free interest rate is 1%, the
expected market risk premium is 7%, and the beta of the company’s common stock is
0.60. What is the company’s after-tax WACC, assuming that the company pays tax at
a 35% rate?
a.
b.
c.
d.
5.20%
8.00%
3.70%
5.57%
Explanation:
CAPM implies: rE = 0.01 + (0.60)(0.07) = 5.20%. After-tax WACC = (0.01)(1 −
0.35)(0.33) + (.0520)(1 – 0.33) = 3.70%.
2. EZCUBE Corp. is 51% financed with long-term bonds and 49% with common equity.
The bonds have a beta of 0.16 and the equity has a beta of 1.24. What is EZCUBE’s
unlevered (or asset) beta?
a.
b.
c.
d.
0.620
1.400
0.689
Cannot be determined
Explanation:
βU = 0.51 × 0.16 + 0.49 × 1.24 = 0.689.
3. Following information for Golden Fleece Financial:
Long-term debt outstanding:
Bond rating
Current yield to maturity (rD):
Number of shares of common stock:
Price per share:
Book value per share:
Expected rate of return on stock (rE):
$310,000
AAA
9%
10,500
$50.10
$26.00
16%
Calculate Golden Fleece's unlevered (all-equity) cost of capital. Assume continuous
leverage rebalancing.
a.
b.
c.
d.
13.40%
12.50%
14.10%
10.00%
Explanation:
The total market value of outstanding debt is $310,000. The cost of debt capital is 9%.
For the common stock, the outstanding market value is: $50.1 × 10,500 = $526,050.
The cost of equity capital is 16%. Thus, Golden Fleece’s company cost of capital is:
𝑟𝑟𝑈𝑈 =
𝐸𝐸
𝐷𝐷
526,050
310,000
(0.16) +
(0.09)
𝑟𝑟𝐸𝐸 +
𝑟𝑟𝐷𝐷 =
𝐸𝐸 + 𝐷𝐷
𝐸𝐸 + 𝐷𝐷
526,050 + 310,000
526,050 + 310,000
= 13.40%
4. Binomial Tree Farm’s financing includes $5.8 million of bank loans and equity capital.
The balance sheet in its latest annual report indicates an equity value $6.75 million. It
has 500,000 shares of common stock outstanding that trade on the Wichita Stock
Exchange at a value of $17.20 per share. Calculate the debt-to-value ratio appropriate
for calculating the after-tax weighted-average cost of capital.
a.
b.
c.
d.
46.2%
40.3%
50.0%
52.2%
Explanation:
We should use the market value of the stock, not the book value shown on the annual
report. This gives us an equity value of 500,000 shares times $17.20 per share = $8.6
million. Therefore the total value of the company is $8.6 million + $5.8 million = $14.4
million. So Binomial Tree Farm has a debt/value ratio of 5.8/14.4 = 0.403 or 40.3%.
5. Suppose a firm uses its company cost of capital to evaluate all projects. Will it
underestimate or overestimate the value of higher-risk projects?
a. Underestimate
b. Overestimate
c. Correctly estimate
Explanation:
Overestimate. Higher-risk projects will have a higher cost of capital than the
company’s existing securities. Therefore, the value of a high-risk project would be
overestimated if the company cost of capital is used.
6. Which one of the projects in the following pairs of projects is likely to have the higher
asset beta, other things equal?
i.
Project A’s sales force is paid a fixed annual salary. Project B’s sales force is paid
by commissions only.
ii. Project C is a first-class-only airline. Project D is a well-established line of
breakfast cereals.
a.
b.
c.
d.
Project A and Project C
Project B and Project C
Project A and Project D
Project B and Project D
Explanation:
Project A because a project with higher fixed costs generally has higher operating
leverage, which, in turn, leads to a higher beta. Project C because more cyclical
revenues lead to a higher beta.
7. The company cost of capital is the correct discount rate for all projects, because the
high risks of some projects are offset by the low risk of other projects.
a. True
b. False
Explanation:
False. The company cost of capital is the correct discount rate for new projects only if
the new projects have the same risk level as the existing business. If a new project is
riskier, a higher cost of capital should be used. If the new project is less risky, a lower
cost of capital should be used.
8. Graucho Services starts with an all-equity capital structure and a cost of equity of
12%. Suppose it subsequently refinances to the following market-value capital
structure:
Debt/Value
Equity/Value
49% with rD = 10%
51%
Calculate Graucho’s new cost of equity capital and new after-tax weighted-average
cost of capital. Assume Graucho continuously rebalances leverage and pays taxes at
a marginal rate of TC = 35%.
a.
b.
c.
d.
rE = 13.92%; WACC = 12.00%
rE = 13.92%; WACC = 10.29%
rE = 12.00%; WACC = 10.29%
rE = 22.00%; WACC = 18.50%
Explanation:
Initially, rE = rU = 12%. Leverage causes rE to increase to rE = rU + (rU – rD)(D/E) = 0.12
+ (0.12 – 0.10)(49/51) = 0.1392, or 13.92%.
After-tax WACC = 0.10(1 – 0.35)(0.49) + 0.1392(0.51) = 0.1028, or 10.29%.
9. Omega Corporation has 11.2 million shares outstanding, no debt, and an expected
rate of return to shareholders of 12.00%. Now, the firm issues $190 million of bonds
rated AAA yielding 9%. The firm invests the funds raised in new assets of identical
risk to the firm’s initial assets. After the debt issue, the firm’s shares trade at $53 per
share. Assume the firm pays taxes at a marginal rate of 35% and continuously
rebalances leverage.
Calculate Omega’s cost of equity capital (rE) and after-tax WACC after the debt issue?
a.
b.
c.
d.
rE = 21.01%; WACC = 24.25%
rE = 11.82%; WACC = 8.51%
rE = 12.00%; WACC = 12.00%
rE = 12.96%; WACC = 11.24%
Explanation:
E = $53 × 11.2 million = $593.6 million; V = D + E = $190 million + $593.6 million =
$783.6 million. D/V
= $190/$783.6
= 0.2425 or 24.25%.
rE = rU + (rU – rD)(D/E) = 0.12 + (0.12 – 0.09) × (190/593.6) = 0.1296 or 12.96%.
After-tax WACC = (0.09)(1 – 0.35) (0.2425) + (0.1296)(1 – 0.2425) = 0.1124 or
11.24%.
10. In perfect financial markets with no taxes or financial distress costs, which of the
following statements is correct?
i. The weighted-average cost of capital is constant for all leverage ratios.
ii. The cost of equity capital is constant for all leverage ratios.
iii. The cost of debt capital is constant for all leverage ratios.
a.
b.
c.
d.
i only
ii only
iii only
i, ii, iii
11. Whispering Pines, Inc., is all-equity-financed with an expected rate of return on the
company’s shares of 12.25%. Suppose the company issues debt, repurchases
shares, and moves to a 30.25% debt-to-value (D/V) ratio. What will the company’s
after-tax weighted-average cost of capital be at the new capital structure if the
borrowing rate on its debt is 7.75% and its marginal tax rate is 30%.
a.
b.
c.
d.
11.55%
12.25%
14.20%
7.75%
Explanation:
rE = 0.1225 + (0.1225 – 0.0775)(30.25/69.75) = 0.1420
WACC = 0.0775(1 – 0.30)(0.3025) + 0.1420(1 – 0.3025) = 0.1155, or 11.55%.
12. The table below shows a book balance sheet for the Wishing Well Motel chain. The
company’s long-term debt is secured by its real estate assets, but it also uses shortterm bank loans as a permanent source of financing. It pays 12% interest on the bank
debt and 10% interest on the secured debt. Wishing Well has 10 million shares of
stock outstanding, trading at $93 per share. The expected return on Wishing Well’s
common stock is 22%.
$ Millions
Accounts
receivable
Inventory
280
Current assets
450
Real estate
Other assets
Assets
170
2,350
130
2,930
Accounts
payable
Bank loan
Current
liabilities
Long-term debt
Equity
Liab & Equity
160
260
420
2,210
300
2,930
Calculate Wishing Well’s after-tax WACC. Assume that the book and market values of
Wishing Well’s debt are the same and the marginal tax rate is 35%.
a.
b.
c.
d.
20.20%
8.30%
7.85%
10.84%
Explanation:
Netting accounts payable against current assets and recognizing the bank debt is a
permanent source of financing, the market-value capital structure of the firm is:
Market Values
Bank loan
Long-term debt
Equity ($93/share
x 10 million
shares)
Liab & Equity
260
2,210
Market Value
%
7.65%
65.00%
Cost of Capital
%
12%
10%
930
27.35%
22%
3,400
100%
$ Million
WACC = 0.12(1 − 0.35)( 0.0765) + 0.10(1 − 0.35)(0.6500) + 0.22(0.2735) = 0.1084 or
10.84%.
13. The table below shows a simplified balance sheet for Rensselaer Felt. The debt has
just been refinanced at an interest rate of 7.50% (short term) and 9.50% (long term).
Short-term debt is a permanent part of the firm’s capital structure. The expected rate
of return on the company’s shares is 16.50%. There are 7,620,000 shares
outstanding, and the shares are trading at $40. The tax rate is 35%.
$ Thousands
Accounts
receivable
Inventory
124,700
Current assets
251,300
PP&E
Other assets
306,800
87,400
Assets
126,600
645,500
Accounts
payable
Short-term debt
Current
liabilities
Long-term debt
Deferred taxes
Sharehld’s
Equity
Liab & Equity
63,600
77,200
140,800
210,200
46,600
247,900
645,500
Calculate the company’s after-tax weighted-average cost of capital.
a.
b.
c.
d.
8.93%
10.77%
11.32%
9.32%
Explanation:
Three adjustments to the balance sheet are necessary: Deferred taxes are an
accounting entry and represent neither a liability nor a source of funds and should be
ignored. Accounts payable should be netted against current assets. The market value
of equity (7,620,000 shares x $40/share) should be used. Now, the market-value
capital structure of the firm is:
Market Values
Short-term debt
Long-term debt
Equity ($40/share
x 7,620,000
shares)
Liab & Equity
77,200
210,200
Market Value
%
13.0%
35.5%
Cost of Capital
%
7.5%
9.5%
304,800
51.5%
16.5%
592,200
100%
$ Thousands
WACC = 0.075(1 − 0.35) (0.130) + 0.0950(1 − 0.35)(0.355) + 0.1650(0.515) = 0.1132
or 11.32%
14. (Version 1) Abercrombie & Fitch, Inc. is attempting to estimate the cost of capital for
use in analyzing a proposed expansion into jewelry products. The firm hires you to
help them compute the cost of capital. You have collected the following data from
Bloomberg for jewelry companies:
Pure Players
βE
D/(E+D)
Tiffany & Co.
Signet Jewelers
Ltd.
Zales
2.00
1.90
20%
15%
Bond
Rating
BB
BB
1.80
10%
BB
Abercrombie targets a debt-to-value ratio of 25% and has a bond rating of BB and
a βE of 1.00 for current operations.
You estimate that the market risk premium (rM-rf) is 7 percent, the risk-free rate is
1%, and the marginal tax rate for all firms is 40%. Assume the beta on debt rated
BB is 0.17.
a. Calculate its after-tax weighted-average cost of capital for
Abercrombie’s current operations.
b. Calculate the unlevered beta for each pure player and estimate the
unlevered beta in the jewelry industry by averaging your estimates for
the pure players. Recall the formula:
𝑬𝑬
𝑫𝑫
𝜷𝜷𝑬𝑬 +
𝜷𝜷
𝑬𝑬 + 𝑫𝑫
𝑬𝑬 + 𝑫𝑫 𝑫𝑫
c. Calculate the after-tax weighted-average cost of capital appropriate for
Abercrombie’s proposed expansion into jewelry. Recall the formula:
𝜷𝜷𝑼𝑼 =
b:
𝒓𝒓𝑾𝑾𝑾𝑾𝑾𝑾𝑾𝑾 = 𝒓𝒓𝑼𝑼 −
1.900
c:
𝑫𝑫
𝒓𝒓 𝑻𝑻
𝑬𝑬 + 𝑫𝑫 𝑫𝑫 𝒄𝒄
i.
a:
6.35%
8.00%
ii.
a:
6.33% b:
1.637 c:
12.24%
iii.
a:
6.88%
b:
1.501
c:
12.46%
iv.
a:
7.00%
b:
1.612
c:
9.19%
Explanation:
a.
Abercrom
bie
TC
D/(E+
D)
βE
rE
βD
rD
WACC
40%
25.0%
1.00
8.00%
0.17
2.19%
6.33%
b.
Tiffany & Co.
Signet Jewelers
Ltd.
Zales
D/(E+D)
20%
βE
2.00
βD
0.17
βU
1.634
15%
1.90
0.17
1.641
10%
1.80
0.17
Average:
1.637
1.637
c.
D/(E+D)
Abercrombie 25.0%
βD
βD
βU
rU
rU – (D/(E+D))*rD*TC
0.17 2.19% 1.637 12.46%
WACC=12.24%
15. (Version 2) These results are estimates and may slightly differ from the estimates of
Version 1. Abercrombie & Fitch, Inc. is attempting to estimate the cost of capital for
use in analyzing a proposed expansion into jewelry products. The firm hires you to
help them compute the cost of capital. You have collected the following data from
Bloomberg for jewelry companies:
Pure Players
βE
D/(E+D)
Tiffany & Co.
Signet Jewelers
Ltd.
Zales
2.00
1.90
20%
15%
Bond
Rating
BB
BB
1.80
10%
BB
Abercrombie targets a debt-to-value ratio of 25% and has a bond rating of
BB and an equity beta of 1.00 for current (non-jewelry) operations.
You estimate that the market risk premium (rM-rf) is 7%, the risk-free rate is
1%, and the marginal tax rate for all firms is 40%. The bond yield is 3.5% for
BB-rated bonds, the probability of default for BB-rated bonds is 2.0%, and
the expected loss rate in default is 60% for all firms. Recall the following
formula:
𝒓𝒓𝑫𝑫 = 𝒀𝒀𝒀𝒀𝒀𝒀 − 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷(𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫) × (𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬𝑬 𝑳𝑳𝑳𝑳𝑳𝑳𝑳𝑳 𝑹𝑹𝑹𝑹𝑹𝑹𝑹𝑹)
a. Calculate Abercrombie’s after-tax weighted-average cost of capital for
current operations. (Do not round intermediate results and provide your
answer to 0.01%.).
b. Now, calculate the unlevered required return rU for each pure player and
estimate the unlevered required return in the jewelry industry by
averaging your estimates for the pure players. Recall the formula:
𝑬𝑬
𝑫𝑫
𝒓𝒓𝑬𝑬 +
𝒓𝒓
𝑬𝑬 + 𝑫𝑫
𝑬𝑬 + 𝑫𝑫 𝑫𝑫
c. Calculate the after-tax weighted-average cost of capital appropriate for
Abercrombie’s proposed expansion into jewelry. Recall the formula:
𝒓𝒓𝑼𝑼 =
𝒓𝒓𝑾𝑾𝑾𝑾𝑾𝑾𝑾𝑾 = 𝒓𝒓𝑼𝑼 −
𝑫𝑫
𝒓𝒓 𝑻𝑻
𝑬𝑬 + 𝑫𝑫 𝑫𝑫 𝒄𝒄
i.
a:
6.33%
b:
14.30% c:
8.00%
ii.
a:
6.88%
b:
8.25%
12.46%
iii.
a:
7.00%
b:
12.28% c:
9.20%
iv.
a:
6.35% b:
12.48%
c:
c:
12.25%
Explanation:
a.
TC
Abercrombie 40%
D/(E+D)
βE
rE
25.0%
1.00
8.00%
Bond
yield
3.5%
Prob
Default
2.0%
Loss
Rate
60%
rD
WACC
2.30%
6.35%
b.
D/(E+
D)
Tiffany & Co.
20%
Signet
Jewelers Ltd.
15%
Zales
10%
βE
rE
2.0
0
1.9
0
1.8
0
15.00
%
14.30
%
13.60
%
Bond
yield
Prob
Defaul
t
Loss
Rate
rD
rU
3.5%
2.0%
60%
2.30%
12.46%
3.5%
2.0%
60%
2.30%
12.50%
3.5%
2.0%
60%
2.30%
12.47%
Average:
12.48%
c.
D/(E+
D)
Abercrombie
25.0%
Bon
d
yield
Prob
Defau
lt
3.5%
2.0%
Loss
Rate
60%
rD
2.30
%
rU
12.48
%
rU –
(D/(E+D))*rD*TC
WACC=12.25%
16. Sharp Products, Inc. seeks a potential expansion into solar heating. Sharp has hired
you to help them compute estimates of the appropriate after-tax WACC for assessing
new solar heating projects. Based upon the technology of Sharp's proposed
expansion, you are able to find two companies engaged in similar solar heating
activities. However, each company also operates a division that manufactures home
generators:
Comparables
Advance Energy,
Inc.
NuEnergy, Inc.
βE
D/(E+D)
% Assets
Solar
Heating
% Assets
Home
Generators
1.10
20%
40%
60%
1.30
25%
60%
40%
Assume the debt betas for all companies are approximately zero, the corporate tax
rate is 35% for all companies, and Sharp desires a target debt-to-value ratio of
30%. In addition, you estimate that the market risk premium (rM-rf) is 5% and the
risk-free rate of interest is 1%.
HINT: You’ll need to solve a system of two equations in two unknowns. Here are
two links for a refresher on this topic. You might want to take a look before
proceeding with the problem:
http://www.algebra.com/algebra/homework/coordinate/Solution-of-the-lin-system-of-two-eqns-by-the-Substmethod.lesson
http://www.algebra.com/algebra/homework/coordinate/lessons/Solution-of-the-lin-syst-of-two-eqns-with-two-unknownsElimination-method.lesson
a. Calculate the unlevered beta βU for each comparable firm. Do not round
intermediate results, and provide numerical values to three decimal
places:
b. A firm is a portfolio of businesses and the beta of a portfolio is the
investment weighted average of the betas of each asset in the portfolio.
Here, each comparable firm is a portfolio of the assets of two businesses.
Hence, the unlevered beta for each comparable firm (calculated in b.) is
the asset weighted average of the unlevered betas for each of its two
businesses. Use a system of two equations in two unknowns (the
unlevered beta for each industry) to calculate the unlevered beta U fo r
the solar heating and home generator industries. (Do not round
intermediate results and provide numerical values to three decimal
places.)
c. Estimate the after-tax weighted-average cost of capital for Sharp that is
appropriate for projects in the solar heating industry. (Do not round
intermediate results and provide numerical values as a % to two decimal
places.)
i.
a: Advanced Energy: 1.050;
NuEnergy: 1.050
b: Solar Heating: 0.880; Home Generators: 0.975
c: 5.40%
ii.
a: Advanced Energy: 0.917;
NuEnergy: 1.040
b: Solar Heating: 0.917; Home Generators: 1.040
c: 6.83%
iii.
a: Advanced Energy: 0.880;
NuEnergy: 0.975
b: Solar Heating: 1.165; Home Generators: 0.690
c:
6.72%
iv.
a: Advanced Energy: 1.165;
NuEnergy: 0.690
b: Solar Heating: 1.050; Home Generators: 1.050
c: 6.25%
c.
Explanation:
rM-rf
5.0%
rf
1.0%
TC
35%
a.
Advance Energy
20%
βE
1.10
βD
0
βU
0.880
NuEnergy
25%
1.30
0
0.975
Advance Energy
0.880
=
0.40*βU_Solar + 0.60*βU_HomeGen
NuEnergy
0.975
=
0.60*βU_Solar + 0.40*βU_HomeGen
D/(E+D)
b.
Solving these two equations gives:
βU_Solar
1.165
βU_HomeGen
0.690
c.
Sharp
TC
D/(E+D)
35%
30.0%
βD
0
rD
1.00%
βU
1.165
rU
rU – (D/(E+D))*rD*TC
6.83%
WACC=6.72%
17. When using the overall firm’s weighted-average cost of capital (WACC) to discount
cash flows from a project, we assume which of the following?
I) The project's risks are the same as those of the firm's other assets and remain so
for the life of the project.
II) The project supports the same fraction of debt to value as the firm's overall capital
structure, and that fraction remains constant for the life of the project.
III) The cash flows from the project occur in perpetuity.
a.
b.
c.
d.
I only
II only
I and II only
I, II, and III
18. Which of the following is an important assumption required if using the WACC
formula?
a. Companies rebalance their capital structure to maintain a constant debt
ratio.
b. WACC must be used on public companies with actively traded securities.
c. Management bonuses must be added back to free cash flows.
d. The firm cannot issue any further debt.
Explanation:
A company can issue more debt and still use WACC, but an equity issue may be
required as well to maintain a constant leverage ratio.
19. The bonds of Casino, Inc. have a 12% coupon rate and yield to maturity of 14.0%.
However, investors expect Casino to default for a total loss with 3.2% probability
implying an expected return on debt of 10.8%. What cost of debt should be used in
Casino's WACC?
a.
b.
c.
d.
14.0%
10.8%
12.0%
9.1%
Explanation:
WACC uses the expected return on debt rather than the promised return, which is:
𝑟𝑟𝐷𝐷 = 𝑌𝑌𝑌𝑌𝑌𝑌 − 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃(𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷) × (𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅) = 14% − (3.2%)(100%) = 10.8%
20. Generally, which of the following is true?
a.
b.
c.
d.
βD > βA > βE
βE > βD > βA
βE > βA > βD
βA > βE > βD
21. An example of diversifiable risk that a financial manager should ignore when
analyzing a project's risk would include:
a.
b.
c.
d.
Commodity price risk
Labor cost risk
Firm’s stock price risk
Government rejection of the project for environmental risks
22. Projects with great amounts of diversifiable risk should generally have higher costs of
capital.
True
False
23. A sensible way for a manager to account for overoptimistic cash-flow forecasts is to
adjust the discount rate upward.
True
False
24. A manager who adjusts discount rates upward by using a "fudge factor" is more likely
to penalize short-term projects as opposed to long-term projects.
True
False