MØA 155
PROBLEM SET: Capital Structure
Exercise 1.
A firm currently own assets worth $4 mill that have a beta of 1. The risk free interest rate is 10% and the
market risk premium is 8%. Suppose the firm has the opportunity to invest in a project that will earn a
13% rate of return for certain into the indefinite future. The cost of the project is $1 mill. Should the firm
make the investment?
Exercise 2. Interest tax shields [4]
Calculate the present value of interest tax shields generated by the following three debt issues. The firm’s cost
of capital is 16%. Assume that only corporate tax rates are relevant, and the marginal tax rate (τ c )=50%.
1. A $1000 zero coupon loan due one year from now.
2. A $1000, 1 year loan at 10% interest.
3. A $1000 perpetuity at 10% interest.
Exercise 3. Arnold’s kindergartens [7]
As a financial consultant to Arnold’s kindergarten’s, you are given the following information about two
companies, one levered and one unlevered:
Before tax operating income
Interest on debt
Cost of capital (WACC)
Cost of Equity capital
Cost of Debt capital
Value of equity
Value of debt
Value of firm
Firm U
$10,000
0
?
?
?
0
?
Firm L
$10,000
2,500
16%
?
?
22,500
?
?
Find the missing figures in the above table. Assume that the corporate tax rate is 40% and all cash flows
are perpetual.
Hint: Start by computing the value of the levered firm from the information given in the table.
Exercise 4. Debt/Equity [7]
Firm Z and Y have identical cash flows. Firm Z is 40% debt financed and 60% equity financed, while firm Y
is 0% debt financed and 100% equity financed. The same required rate of return on their debt equals 10%.
(Assume debt is perpetual)
1. Next period’s cash flows for each firm are $100. Assume both firms pay out all excess cash in the form
of dividends. What cash flows go to the debt and equity holders of both firms? Assume no corporate
taxes. (Use Dz for the value of firm Z’s debt).
2. You own 10% of firm Z’s stock. What cash flow will you get in the future? What combination of other
assets will give you the same cash flow?
3. Suppose the value of firm Z is greater than firm Y. How can you become very rich? (You may assume
no transactions costs, or other market imperfections)
4. Now, suppose there is a corporate tax rate of 40%. What should the value of each firm be?
1
Exercise 5. Sweeny Pie [5]
Sweeny Pie Company’s common stock has a beta of 1.0 and a total market value of $12 million. The expected
risk premium on the market is 8% and the T-bill rate is 12%. If the company’s cost of capital is 18%, what
is the value of the company’s risk free debt?
Exercise 6. Charon Ferries
The market value of Charon Ferries’ common stock is $16 million, and the market value of its (risk-free)
debt is $4 million. The beta of the company’s common stock is 1.5 and the expected risk premium on the
market is 8%. If the Treasury Bill rate is 5%, what is the company’s cost of capital?
Exercise 7. Tax Rate Change [3]
In a M&M world with US–type taxation, the cost of equity is influenced by the tax rate. If the corporate tax
rate increases from 34% to 40%, would you expect the required rate of return on equity to increase, decrease
or stay the same?
Exercise 8. B&N (WC 15.1) [5]
The Bonner Company and the Kirkeby Company are identical except for their leverage ratios and the interest
rate on debt. Each has $10 million in assets, each earned $2 million before interest and taxes in 1992, and
each has a 40% corporate tax rate. Bonner, however, has a leverage ratio (debt/total assets) of 30% and
pays 10% interest on debt, while Kirkeby has a 50% leverage ratio and pays 12% on debt.
1. Calculate the rate of return on equity (net income/equity) for each firm.
2. Observing that Kirkeby has a higher return on equity, Bonner’s treasurer decides to raise the leverage
ratio from 30% to 60%. This will increase Bonner’s interest rate on debt to 15%. Calculate the new
rate of return on equity for Bonner.
2
Empirical
Solutions
MØA 155
PROBLEM SET: Capital Structure
Exercise 1.
The company-wide cost of capital currently is
r = 0.10 + 0.08 · 1.0 = 18%
If the firm compared the 13% return on the new project to its cost of capital, they would reject the project.
But this would be a mistake since the company-wide cost of capital ignores the risk of the project.
The NPV of the project is
$1, 000, 000 · 0.13
− $1, 000, 000
0.10
= $300, 000
NPV
=
Note that the appropriate discount rate for this project is rf = 10%.
If the firm issued stock worth $1 mill. to finance the project, the market value of the firms assets would
increase to $5.3 mill and the company-wide cost of capital would decline to 16.04%.
r=
1.3
4
· 0.18 +
· 0.10 = 16.04%.
5.3
5.3
Thus, the required rate of return on the firm’s equity would fall after the investment.
The reason that the required return on the firm’s equity falls to 16.04% is because the beta of the firm’s
assets decline to
4
1.3
β∗ =
· 1.0 +
· 0 = 0.755
5.3
5.3
Exercise 2. Interest tax shields [4]
1. 0.
2.
PV
($1, 000 · 10%) · 50%
1.16
(1000 · 0.1) · 0.5
=
1.16
= $43.10
=
3.
PV
=
∞
X
($1, 000 · 10%) · 50%
(1.16)t
t=1
=
=
(1000 · 0.1) · 0.5
100 · 0.5
=
0.16
0.16
$312.50
3
Exercise 3. Arnold’s kindergartens [7]
$10, 000(1 − 0.4)
= $37, 500
0.16
DL = VL − EL = $15, 000
$2, 500
DL = $15, 000 =
rD
⇒ rD = 16.67%
(10, 000 − 2, 500)(1 − 0.40)
EL = $22, 500 =
rE
⇒ rE = 20%
22, 500
15, 000
r = 16% = 0.20
+ 0.1667(1 − 0.4)
37, 500
37, 500
VU = VL − τ c D = $37, 500 − 0.4 · $15, 000 = $31, 500
10, 000(1 − 0.4)
VU = EU = 31, 500 =
rE
⇒ rE = r = 19.05%
VL =
Before tax operating income
Interest on debt
Cost of capital
Cost of Equity capital
Cost of Debt capital
Value of equity
Value of debt
Value of firm
Firm U
10,000
0
19.05%
19.05%
31,500
0
31,500
Firm L
10,000
2,500
16.0%
20.0%
16.67%
22,500
15,000
37,500
Exercise 4. Debt/Equity [7]
1. Cash flows.
Firm Z
Cash flow
100
Debt payment
0.10DZ
Equity payment 100 − .10DZ
Firm Y
100
−
100
2. Future cash flows:
0.10 · (100 − 0.10 · DZ )
Alternative assets to replicate these cash flows:
Buy 10% of Y
Borrow 10% of DZ
at 10% rate
Net
0.10 · 100
−0.1 · (0.10 · DZ )
0.1 · (100 − 0.1 · DZ )
3. We then recognise an arbitrage opportunity.
Buy Y
Short Z
BorrowDZ
Net
−VY
Cost
−VY
VZ − DZ
DZ
+ VZ > 0
4
Next period
100
−(100 − 0.1DZ )
−0.1DZ
0
4. New firm values after tax of 40%. Let VY , VZ be the no tax values of Y and Z.
Since Y has no debt, its value goes down by 40%, to 0.60 · VY .
For firm Z, its value is adjusted for the tax shelter of debt:
Firm Z goes to
0.6VZ + 0.4DZ
=
0.6VY + 0.4DZ
Exercise 5. Sweeny Pie [5]
Given
rf = 12%
E[rm ] − rf = 8%
E = $12mill
r∗ = 18%
Want to find D.
Remember
r∗ =
E
D
rD +
rE
D+E
D+E
Since the equity has a β of 1, it has the same return as the market E[rm ]:
E[rm ]
=
(E[rm ] − rf ) + rf
=
0.08 + 0.12
=
20%
Hence
rE = 20%
We are also given that the debt is risk-free. Hence
rD = 12%
Thus, we have everything but D in the equation:
r∗ =
0.18 =
E
D
rD +
rE
D+E
D+E
D
12
0.12 +
0.20
D + 12
D + 12
Solve for D, find
D = 4 mill
Exercise 6. Charon Ferries
First find the asset beta for the company:
β∗
=
=
=
D
E
βD +
βE
D+E
D+E
4
16
0+
1.5
4 + 16
4 + 16
16
1.5 = 1.2
20
5
Cost of capital:
r∗
= rf + (E[rm ] − rf )β ∗
=
0.05 + 0.08 · 1.2 = 14.6%.
Exercise 7. Tax Rate Change [3]
Decrease.
Exercise 8. B&N (WC 15.1) [5]
1. Each firm has assets V = 10 mill.
Bonner
Debt
3 mill
Equity
7 mill
Earnings
2 mill
Interest
300,000
Net earnings
1,700,000
Taxes
680,000
Net income
1,020,000
1,020
rE
7,000 = 14.57%
840
5,000
Kirkeby
5 mill
5 mill
2 mill
600,000
1,400,000
560,000
840,000
= 16.80%
2. b: Bonner changes leverage to 60%.
Debt
Equity
Earnings
Interest
Net earnings
Taxes
Net income
rE
Bonner
6 mill
4 mill
2 mill
900,000
1,100,000
440,000
660,000
660
=
16.5%
4,000
Kirkeby
5 mill
5 mill
2 mill
600,000
1,400,000
560,000
840,000
16.80%
6