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Answers and Solutions: 21 - 1
Chapter 21
Dynamic Capital Structures
ANSWERS TO END-OF-CHAPTER QUESTIONS
21-1 a. MM Proposition I states the relationship between leverage and firm value.
Proposition I without taxes is V = EBIT/rsU. Since both EBIT and rsU are constant,
firm value is also constant and capital structure is irrelevant. With corporate taxes,
Proposition I becomes V = Vu + TD. Thus, firm value increases with leverage and the
optimal capital structure is virtually all debt.
c. The Miller model introduces personal taxes. The effect of personal taxes is,
essentially, to reduce the advantage of corporate debt financing.
e. The value of the debt tax shield is the present value of the tax savings from the
interest payments. In the MM model with taxes, this is just interest x tax rate /
discount rate = iDT/r, and since i = r in the MM model, this is just TD. If a firm
grows and the discount rate isn’t r, then the value of this growing tax shield is
rdTDg/(1+rTS) where rd is the interest rate on the debt and rTS is the discount rate for
the tax shield. In the APV model, the tax savings are discounted at the unlevered cost
of equity.
Answers and Solutions: 21 - 2
21-2 Modigliani and Miller show that the value of a leveraged firm must be equal to the
value of an unleveraged firm. If this is not the case, investors in the leveraged firm
will sell their shares (assume they owned 10%). They will then borrow an amount
equal to 10% of the debt of the leveraged firm. Using these proceeds, they will
purchase 10% of the stock of the unlevered firm (which provides the same return as
the leveraged firm) with a surplus left to be invested elsewhere. This arbitrage
process will drive the price of the stock of the leveraged firm down and drive up the
price of the stock of the unlevered firm. This will continue until the value of both
stocks are equal.
The assumptions of the MM model are:
• Stocks and bonds are traded in perfect capital markets. Therefore, (a) there are no
brokerage costs and (b) individuals can borrow at the same rate as corporations.
Brokerage fees and varying interest rates will, in effect, lower the surplus
available for alternative investment.
21-3 MM without taxes would support AT&T, although if AT&T really believed MM,
they should not object to Gordon’s 50 percent debt ratio. MM with taxes would lead
21-4 The value of a growing tax shield is greater than the value of a constant tax shield.
This means that for a given initial level of debt a growing firm will have more value
21-5 If equity is viewed as an option on the total value of the firm with a strike price equal
to the face value of debt then the equity value should be affected by risk in the same
way that an option is affected by risk. An option is worth more if the underlying asset
Answers and Solutions: 21 - 4
SOLUTIONS TO END-OF-CHAPTER PROBLEMS
21-1 VL = VU = $500 million.
21-2 VL = VU + TD= $800 + 0.35($60) = $821 million.
21-4 VL = VU +
−)g
sU
r(
)
c
T)(D(
d
r
= $800 +
−)03.011.0(
)35.0)(60(05.0
= $800 + $13.125
= $813.125 million.
Answers and Solutions: 21 - 5
c. $2 Million Debt: VL = VU + TD = $10 + 0.25($2) = $10.5 million.
rsL = rsU + (rsU − rRF)(1 − T)(D/S)
= 15.625% + (15.625% − 10%)(0.75)($2/$8.5)
= 15.625 + 5.625% (0.75)($2/$8.5) = 16.62%.
d. $6 Million Debt: VL = $8.0 + 0.40($6) = $10.4 million.
rsL = 15.625% + 5.625%(0.60)($6/$4.4) = 20.23%.
21-6 a. VU =
sU
r
EBIT
=
10.0
million 2$
= $20 million.
b. rsU = 10.0%. (Given)
d. WACCU = rsU = 10%.
For Firm L, we know that WACC must equal rsU = 10% according to Proposition I.
e. VL = $22 million is not an equilibrium value according to MM. Here’s why.
Suppose you owned 10 percent of Firm L’s equity, worth 0.10($22 million - $10
million) = $1.2 million. Your cash flow is equal to 10% of the dividends paid by the
levered firm. Because it is a zero-growth firm, its dividends are equal to its net
income: Dividends = Net income = EBIT – rdD = $2,000,000 – 0.05($1,000,000) =
$1,500,000. Your 10% share is 0.10($1,500,000) = $150,000. Therefore, your annual
cash flow is $150,000.
Your cash stream would now be: (a) 10 percent of firm U’s dividends, which is
$200,000: 0.10(EBITU) = 0.10($2 million) = $200,000; plus (b) the return on the
extra $200,000 profit you invested in risk-free debt, which is $10,000: rd(profit) =
0.05($200,000) = $10,000; minus (c) the interest expense on the $1 million you
borrowed, which is $50,000: rd(loan) = 0.05($1 million)] = $50,000. Your net cash
flow from this strategy is $200,000 + $10,000 - $50,000 = $160,000.
Answers and Solutions: 21 - 7
21-7 a. VU =
sU
r
)T1(EBIT −
=
10.0
)4.01(2$ −
= $12 million.
VL = VU + TD = $12 + (0.4)$10 = $16 million.
c. SL =
sL
d
r
)T1)(DrEBIT( −−
=
15.0
6.0)]10($05.02[$ −
= $6 million.
21-8 a. VU =
)T1(r
)T1)(T1(EBIT
ssU
sC
−
−−
=
)01(10.0
)01)(4.01(2$
−
−−
= $12 million.
Answers and Solutions: 21 - 8
b. VU =
)T1(r
)T1)(T1(EBIT
ssU
sC
−
−−
=
)01(10.0
)01)(01(2$
−
−−
= $20 million.
Gain = VL − VU = $20 − $20 = $0.
c. VU =
)T1(r
)T1)(T1(EBIT
ssU
sC
−
−−
=
)01(10.0
)01)(4.01(2$
−
−−
= $12 million.
Answers and Solutions: 21 - 9
d. VU =
)T1(r
)T1)(T1(EBIT
ssU
sC
−
−−
=
)28.01(10.0
)28.01)(4.01(2$
−
−−
= $12 million.
21-9 a. VU = $500,000/(rsU – g) = $500,000/(0.13 - 0.09) = 12,500,000.
b.
million $16.0
0.09 - 0.13
million 5x 0.40x 0.07
million $12.5VL=
+=
. So since
D = 5, S = 16 – 5 = $11.0 million.
11
5
0.07)(0.130.13 rsL −+=
= 15.7%
21-10 a. VU = SU =
sU
r
EBIT
=
11.0
000,600,1$
= $14,545,455.
VL = VU = $14,545,455.
Answers and Solutions: 21 - 10
b. At D = $0:
rs = 11.0%; WACC = 11.0%
At D = $6 million:
At D = $10 million:
rsL = 11% + 5%
455,545,4$
000,000,10$
= 22.00%.
Answers and Solutions: 21 - 11
d. At D = $0:
WACC = (D/V)rd(1 - T) + (S/V)rs
=($6,000,000/$11,127,273)(6%)(0.6) + ($5,127,273/$11,127,273)(14.51%)
= 8.63%.
At D = $10 million:
Summary: (in millions)
D V D/V rs WACC
$ 0 $ 8.73 0% 11.0% 11.0%
e. The maximum amount of debt financing is 100 percent. At this level
D = V, and hence
15
14
13
Value (Millions of Dollars)
Answers and Solutions: 21 - 13
Since the bondholders are bearing the same risk as the equity holders of the unlevered
firm, rd is now 11 percent. Now, the total interest payment is $14,545,455(0.11) =
$1.6 million, and the entire $1.6 million of EBIT would be paid out as interest. Thus,
the investors (bondholders) would get $1.6 million per year, and it would be
capitalized at 11 percent:
f. (1) Rising interest rates would cause rd and hence rd(1 - T) to increase, pulling up
WACC. These changes would cause V to rise less steeply, or even to decline.
25
20
Cost of Capital (%)
kS
rs
Answers and Solutions: 21 - 14
21-11 a. The inputs to the Black and Scholes option pricing model are P = 5, X = 2, rRF = 6%,
= 50%, and t = 2 years. Given these inputs, the value of a call option is calculated
as:
t
t]2/r[)X/Pln(
d2
RF
1
++
=
=
8191.1
25.0
2]2/5.006.0[)2/5ln( 2=
++
.
b. The debt must therefore be worth 5-3.29 = $1.71 million. Its yield is
%1.881.0171.1/0.2 ==−
.
c. At a volatility of 30% d1 = 2.6547 and N(d1) = 0.9960. d2 = 2.2304 and N(d2) =
0.9871. This gives an option value of $3.23 million. The debt value is then 5.0 –
Answers and Solutions: 21 - 15
21-12 a. HVU,3 =
07.013.0
)07.1( 40$
−
= $713.33.
c. TS = (Interest expense)(T)
TS1 = $8(0.4) = $3.2
TS2 = $9(0.4) = $3.6
TS3 = $10(0.4) = $4.0
HVU,3 =
07.013.0
)07.1( 0.4$
−
= $71.33.
Answers and Solutions: 21 - 16
SOLUTION TO SPREADSHEET PROBLEM
21-13 The detailed solution for the problem is available in the file FM14 Ch21 P13 Build a
Model Solution.xls on the textbook’s Web site.
Mini Case: 21 - 17
MINI CASE
David Lyons, CEO of Lyons Solar Technologies, is concerned about his firm’s level of debt
financing. The company uses short-term debt to finance its temporary working capital
needs, but it does not use any permanent (long-term) debt. Other solar technology
companies average about 30 percent debt, and Mr. Lyons wonders why they use so much
more debt, and what its effects are on stock prices. To gain some insights into the matter,
he poses the following questions to you, his recently hired assistant:
a. Who were Modigliani and Miller (MM), and what assumptions are embedded in
the MM and Miller models?
Answer: Modigliani and Miller (MM) published their first paper on capital structure (which
assumed zero taxes) in 1958, and they added corporate taxes in their 1963 paper.
Modigliani won the Nobel Prize in economics in part because of this work, and most
subsequent work on capital structure theory stems from MM. Here are their
assumptions:
• Firms’ business risk can be measured by σEBIT, and firms with the same degree of
risk can be grouped into homogeneous business risk classes.
Mini Case: 21 - 18
b. Assume that firms U and L are in the same risk class, and that both have EBIT =
$500,000. Firm U uses no debt financing, and its cost of equity is rsU = 14%.
Firm L has $1 million of debt outstanding at a cost of rd = 8%. There are no
taxes. Assume that the MM assumptions hold, and then:
1. Find v, s, rs, and WACC for firms U and L.
Answer: First, we find Vu and VL:
VU =
sU
r
EBIT
=
14.0
000,500$
= $3,571,429.
Mini Case: 21 - 19
b. 2. Graph (a) the relationships between capital costs and leverage as measured by
D/V, and (b) the relationship between value and D.
Answer: Figure 1 plots capital costs against leverage as measured by the debt/value ratio.
Note that, under the MM no-tax assumption, rd is a constant 8 percent, but rs increases
with leverage. Further, the increase in rs is exactly sufficient to keep the WACC
Mini Case: 21 - 20
c. Using the data given in part B, but now assuming that firms L and U are both
subject to a 40 percent corporate tax rate, repeat the analysis called for in B(1)
and B(2) under the MM with-tax model.
Answer: With corporate taxes added, the MM propositions become:
Proposition I: VL = VU + TD.
Proposition II: rsL = rsU + (rsU – rd)(1 - T)(D/S).
4
Value of Firm, V
VUVL
($)
Figure 2
Thus, the use of $1,000,000 of debt financing increases firm value by T(D) =
$400,000 over its leverage-free value.
To find rsL, it is first necessary to find the market value of the equity:
D + SL = VL
$1,000,000 + SL = $2,542,857
SL = $1,542,857.
now,
The WACC is lower for the levered firm than for the unlevered firm when corporate
taxes are considered.
Figure 3 below plots capital costs at different D/V ratios under the MM model
Mini Case: 21 - 22
Figure 3
4
Value of Firm, V
($)
Figure 4
With Taxes
40%
45%
50%
rs
WACC
rd x (1-T)
Mini Case: 21 - 23
d. Now suppose investors are subject to the following tax rates:
TD = 30% and TS = 12%.
1. What is the gain from leverage according to the miller model?
Answer: To begin, note that Miller’s Proposition I is stated as follows:
VL = VU +
−
−−
−)T1(
)T1)(T1(
1D
SC
D.
Mini Case: 21 - 24
d. 2. How does this gain compare to the gain in the MM model with corporate taxes?
Answer: If only corporate taxes were considered, then
VL = VU + TCD = VU + 0.40D.
The net effect depends on the relative effective tax rates on income from stocks and
bonds, and on corporate tax rates. The tax rate on stock income is reduced vis-à-vis
d. 3. What does the Miller model imply about the effect of corporate debt on the value
of the firm, that is, how do personal taxes affect the situation?
Answer: The addition of personal taxes lowers the value of debt financing to the firm. The
underlying rationale can be explained as follows: the U.S. corporate tax laws favor
debt financing over equity financing, because interest expense is tax deductible while
Mini Case: 21 - 25
e. What capital structure policy recommendations do the three theories (MM
without taxes, MM with corporate taxes, and Miller) suggest to financial
managers? Empirically, do firms appear to follow any one of these guidelines?
Answer: In a zero tax world, MM theory says that capital structure is irrelevant--it has no
impact on firm value. Thus, one capital structure is as good as another. With
f. Suppose that Firms U and L are growing at a constant rate of 7% and that the
investment in net operating assets required to support this growth is 10% of
EBIT. Use the compressed adjusted present value (APV) model to estimate the
value of U and L. Also estimate the levered cost of equity and the weighted
average cost of capital.
Answer: If a firm is growing, the assumptions that MM made are violated. The extension to
the MM model shows how growth affects the value of the debt tax shield and the cost
of capital. The first difference in this situation is that the appropriate discount rate for
the debt tax shield is the unlevered cost of equity, not the cost of debt. The second
difference is that a growing debt tax shield is more valuable than a constant debt tax
shield.
Mini Case: 21 - 26
= 250,000/(0.14 – 0.07) = $3,571,429
Which is greater than in part C because the firm is growing.
In this case the increase in the firm’s value due to the debt tax shield as a percent of
its zero debt value is $457,143/$3,571,429 = 12.80%
This is less than the increase in the non-growing firm’s value as calculated using the
MM model: $400,000/$2,142,857 = 18.7%.
g. Suppose the expected free cash flow for Year 1 is $250,000 but it is expected to
grow unevenly over the next 3 years: FCF2 = $290,000 and FCF3 = $320,000,
after which it will grow at a constant rate of 7%. The expected interest expense
at Year 1 is $80,000, but it is expected to grow over the next couple of years
before the capital structure becomes constant: Interest expense at Year 2 will be
$95,000, at Year 3 it will be $120,000 and it will grow at 7% thereafter. What is
the estimated horizon unlevered value of operations (i.e., the value at Year 3
immediately after the FCF at Year 3)? What is the current unlevered value of
operations? What is the horizon value of the tax shield at Year 3? What is the
current value of the tax shield? What is the current total value? The tax rate and
unlevered cost of equity remain at 40% and 14%, respectively.
Answer: The unlevered horizon value of operations can be found by applying the constant
growth formula:
Mini Case: 21 - 27
HVU,3 = [FCF3(1+gL)]/(rsU – gL) = [$320(1.07)]/(0.14 – 0.07) = $4,891.43.
The horizon value of the tax shield can be found by applying the constant growth
formula:
HVTS,3 = [TS3(1+gL)]/(rsU – gL) = [$48(1.07)]/(0.14 – 0.07) = $733.71.
h. Suppose there is a large probability that L will default on its debt. For the
purpose of this example, assume that the value of L’s operations is $4 million
(the value of its debt plus equity). Assume also that its debt consists of 1-year,
zero coupon bonds with a face value of $2 million. Finally, assume that L’s
volatility, σ is 0.60 and that the risk-free rate rRF is 6%.
Answer: L’s equity can be considered as a call option on the total value of l with an exercise
price of $2 million, and an expiration date in one year. If the value of L’s operations
Mini Case: 21 - 28
than $2 million in one year, then management will repay the debt and the
stockholders will keep the company.
in this case, P = $4
X = $2
= 0.60
T = 1.0
R = 0.06
Mini Case: 21 - 29
i. What is the value of L’s stock for volatilities between 0.20 and 0.95? What
incentives might the manager of L have if she understands this relationship?
What might debtholders do in response?
Answer: The mini case model shows the calculations for the table below.
Value of Stock and Debt
for Different Volatilities
Volatility
Equity
Debt
0.20
2.12
1.88
0.25
2.12
1.88
0.30
2.12
1.88
0.35
2.12
1.88
0.40
2.13
1.87
The value of the equity increases as the volatility increases—and the value of the debt
decreases as well. A manager who knows this may choose to invest the proceeds
from borrowing in assets that are riskier than usual. This is called “bait and switch.”
This action decreases the value of the debt, because now its claim is riskier. It
increases the value of equity because the worse the stockholders can do is default on
the bonds, but the best they can do is potentially unlimited.
