Chapter Sixteen
Capital structure
Answers to Problems
Note: All problems in this chapter are available in MyLab Finance. An asterisk (*) indicates problems
with a higher level of difficulty.
Capital structure in perfect capital markets
For problems in this section assume no taxes or distress costs.
1. Consider a project with free cash flows in one year of $130 000 or $180 000, with each
outcome being equally likely. The initial investment required for the project is $100 000, and
the project’s cost of capital is 20%. The risk-free interest rate is 10%.
a. What is the net present value of this project?
b. Suppose that to raise the funds for the initial investment, the project is sold to
investors as an all-equity firm. The equity holders will receive the cash flows of the
project in one year. How much money can be raised in this way—that is, what is the
initial market value of the unlevered equity?
c.
Suppose the initial $100 000 is instead raised by borrowing at the risk-free interest
rate. What
are the cash flows of the levered equity, and what is its initial value
according to Modigliani and Miller?
Plan: In order to calculate the NPV of the project we must first compute the free cash flows for
that year by calculating the average of the two likely scenarios for cash flows that year. We can
then compute the NPV using the NPV formula. Knowing the free cash flows, the discount rate,
and the initial investment we can compute the NPV as well as the equity value. Finally, we can
compute the cash flows of the levered equity by computing the risk-free rate of the debt
payments and subtracting that from the two likely scenarios for the cash flows used in part (a).
Finally, the initial value of the project can be found by subtracting the debt payments of the
project from the equity value.
Execute:
a.
E CF1  = ( 0.5) (130, 000 ) + ( 0.5 ) (180, 000 ) = $155, 000
NPV =
155, 000
− 100, 000 = $29,167
1.20
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155, 000
= $129,167
1.20
b.
Equity value = PV ( CF1 ) =
c.
Value of debt = 100,000  Initial value = 129,167 – 100,000 = $29,167
Evaluate: The NPV rule states that to accept a project with positive NPV, such as this project,
therefore all else being equal, the company should undertake the project.
2. You are an entrepreneur starting a biotechnology firm. If your research is successful, the
technology can be sold for $30 million. If your research is unsuccessful, it will be worth nothing.
To fund your research, you need to raise $2 million. Investors are willing to provide you with $2
million in initial capital in exchange for 50% of the unlevered equity in the firm.
a. What is the total market value of the firm without leverage?
b. Suppose you borrow $1 million. According to Modigliani and Miller, what fraction of the
firm’s equity will you need to sell to raise the additional $1 million you need?
c.
What is the value of your share of the firm’s equity in cases (a) and (b)?
Plan: We can find the total market value of the firm without leverage by computing the total value
of equity knowing the $2 million in initial capital and the $2 million needed to fund your research.
We can compute the fraction of the firm’s equity you will need to sell to raise the additional $1
million you need using the total value of the firm, and the new value of equity after borrowing to
get the percentage of equity that must be sold. Finally, we can compute the firm’s value of equity
in both cases; knowing the fraction of the firm’s equity you will need to sell in both cases.
Execute:
a.
Total value of equity = 2 × 2,000,000 = $4,000,000.
b. MM says the total value of firm is still $4 million. $1 million of debt implies the total value of
equity is $3 million. Therefore, 33% of equity must be sold to raise $1 million.
c.
In (a), 50%  4,000,000 = $2,000,000. In (b), 2/3  3,000,000 = $2,000,000. Thus, in a perfect
market the choice of capital structure does not affect the value to the entrepreneur.
Evaluate: In this case, changing the capital structure does not affect the value to the owner of the
firm and therefore the owners have more flexibility with their capital structure.
3. Adelphi Industries owns assets that will have an 80% probability of having a market value of $50
million
in one year. There is a 20% chance that the assets will be worth only $20 million. The
current risk-free
rate is 5%, and Adelphi’s assets have a cost of capital of 10%.
a. If Adelphi is unlevered, what is the current market value of its equity?
b. Suppose instead that Adelphi has debt with a face value of $20 million due in one year.
According to Modigliani and Miller, what is the value of Adelphi’s equity in this case?
c.
What is the expected return of Adelphi’s equity without leverage? What is the expected
return of Adelphi’s equity with leverage?
d. What is the lowest possible realised return of Adelphi’s equity with and without leverage?
Plan: We can use Eq. 16.1 to compute the current market value of Adelphi’s equity. To determine
its expected return, we will compute the cash flows to equity. The cash flows to equity are the
cash flows of the firm net of the cash flows to debt (repayment of principal plus interest).
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Execute:
a.
E V1  = ( 0.8 ) (50 ) + ( 0.2 ) (20 ) = $44 million
Unlevered equity value =
20
b.
𝐷=
c.
Without leverage, 𝑟 =
1.05
44
= $40 million
1.10
= $19.048 million ∴ 𝐸 = 40 − 19.048 = $20.952 million
With leverage, 𝑟 =
40
44−20
20.952
d. Without leverage, 𝑟 =
With leverage, 𝑟 =
44
20
40
0
20.952
− 1 = 10%
− 1 = 14.55%
− 1 = −50%
− 1 = −100%
Evaluate: The current market value of Adelphi’s equity when unlevered is nearly double the
current market value of Adelphi’s equity when levered. The expected return is greater with debt
than without, yet the lowest possible realized return of Adelphi’s equity is less when unlevered as
opposed to levered.
Debt and taxes
10. Suppose the corporate tax rate is 30%. Consider a firm with $1000 EBIT each year with no risk.
The firm’s capital expenditures equal its depreciation expenses each year, and it will have no
changes to its net working capital. The risk-free interest rate is 5%.
a. Suppose the firm has no debt and pays out its net profit as a dividend each year. What
is the value of the firm’s equity?
b. Suppose instead the firm makes interest payments of $500 per year. What is the value
of equity? What is the value of debt?
c. What is the difference between the total value of the firm with leverage and without
leverage?
d. To what percentage of the value of the debt is the difference in part (c) equal?
Plan: We can use Eq. 16.3 to compute the value of the firm’s equity and debt. We can use our
answers in parts (b) and (c) to compute the percentage of the value of debt.
Execute:
a.
Net profit = 1000  (1 – 30%) = $700. Thus, equity holders receive dividends of $700 per year
with no risk.
E =
700
= $14, 000
0.05
350
= $7000
0.05
500
Debt holders receive interest of $500 per year  = D =
= $10, 000
0.05
b.
Net profit = (1000 − 500)  (1 − 0.3) = $350  E =
c.
With leverage, V = 7000 + 10,000 = $17,000. Without leverage, V = $14,000. Difference =
17,000 – 14,000 = $3,000
d. Percentage of the value of debt =
3000
= 30% = the corporate tax rate
10, 000
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Evaluate: MM Proposition I states that in a perfect capital market the total value of a firm is equal
to the market value of the free cash flows generated by its assets and is not affected by its choice
of capital structure. By adding leverage, the returns of the unlevered firm are effectively split
between low-risk debt and much higher risk levered equity. Returns of levered equity fall twice as
fast as those of unlevered equity if the cash flows decline. Leverage increases the risk of equity
even when there is no risk of default.
Copyright ©2018 Pearson Australia (a division of Pearson Australia Group Pty Ltd) –
9781488611001/Berk/Fundamentals of Corporate Finance/3e