1. The weighted average cost of capital is independent of the firm’s capital structure.
2. The WACC of a firm with debt is equal to the unlevered cost of equity.
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Value of Equity
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A B C D E F G H I J K
12/10/2012
Situation
Modigliani and Miller without Taxes
Input Data Firm U Firm L
No Debt Some Debt
EBIT $500,000 $500,000
Debt $0 $1,000,000
rsL = rsU + (rsU-rd)x (D/S)
WACC =
wd*rd + wce*rs = (D/V)*rd + (S/V)*rs
WACC = 2.24% + 11.8%
WACC = 14.00%
MM without Taxes
D V S D/V
rdrsWACC
0.0 $3.50 $3.50 0.00% 8.00% 14.00% 14.00%
Chapter 21. Mini Case
David Lyons, the 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 the difference occurs, 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. Business Week recently ran an article on companies’ debt policies, and the names Modigliani and Miller (MM) were mentioned
several times as leading researchers on the theory of capital structure. Briefly, who are MM, and what assumptions are embedded
in the MM and Miller models? Answer: See Chapter 21 Mini Case Show
(2.) Graph (a) the relationships between capital costs and leverage as measured by D/V, and (b) the relationship between value
and D.
1. 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:
a. Find V, S, rs, and WACC for Firms U and L.
Franco Modigliani and Merton Miller developed a model to examine the impact of debt on firm value. In this first version it is
assumed that taxes are zero.
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Value of Tax
Shield
Value of Stock = (EBIT – rdD)(1 – T)/rS
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A B C D E F G H I J K
Modigliani and Miller with CorporateTaxes
The MM results are different once corporate taxes are added in.
Firm U Firm L 40% Tax Rate 40% Tax Rate
Input Data No Debt Some Debt No Debt Some Debt
EBIT $500,000 $500,000 $500,000 $500,000
Debt $0 $1,000,000 $0 $1,000,000
WACC 14.00% 14.00% 14.00% 11.80%
Effects of Leverage: MM Models
MM with Corporate Taxes
Tc = 40.00%
DV S D/V
rdrd x (1-T) rsWACC
0.0 $2.14 $2.14 0.00% 8.00% 4.80% 14.00% 14.00%
V-No Taxes V-Taxes
$3.50 $3.50
$3.50 $3.70
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.
WACC = (D/V)rd(1-T) + (S/V)rs
$3,571,429
Value of Firm = Value of Unlevered Firm +T x Debt
$2,142,857
$2,542,857
Total Market
Value of Firm
$3,571,429
35%
40%
45%
50%
With Taxes rs
WACC
rd x (1-T)
$5.00
Relationship Between Value and Debt
50,000
7%
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growth 0% 7% 0% 7%
Unlev. Firm $2,142,857 $3,571,429 $2,142,857 $3,571,429 Value of Unlevered firm = FCF/(WACC – g)
Of Firm $2,142,857 $3,571,429 $2,371,429 $4,028,571 Value of Firm = Value of Unlevered Firm + Value of Tax Shield
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A B C D E F G H I J K
Relevant information from part c.
EBIT
500,000
Tax rate
40%
Unlevered cost of equity
14% = WACC if there is no debt
Cost of debt
8%
Firm U Firm U Firm L Firm L
Data for 40% Tax Rate 40% Tax Rate 40% Tax Rate 40% Tax Rate
Lyons zero Debt zero Debt some Debt some Debt
and no growth and 7% growth and no growth and 7% growth
exp. FCF 300,000$ 250,000$ 300,000$ 250,000$
Debt –$ –$ 1,000,000$ 1,000,000$
rd8.0% 8.0% 8.0% 8.0%
rsU 14.00% 14.00% 14.00% 14.00%
WACC = rsU if the firm is unlevered
Tax Rate 40% 40% 40% 40%
APV with growth: rTS = rsU.growth = 7.00%
T = 40.00%
DV S D/V Tax shield
rsL WACC
$4,028,571 $3,028,571 24.823% $457,143 15.981% 13.206%
This column will NOT be the same as the “40%
tax rate, some debt” column from part c
because we are discounting the tax shield at
rsU instead of rd.
This column is the same as the
“40% tax rate no debt” column
from part c.
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.
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?
(2.) How does this gain compare to the gain in the MM model with corporate taxes?
$0.00
$0.50
$1.00
Debt
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: See Chapter 21 Mini
(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: See Chapter 21 Mini Case Show
MM versus APV rsL and WACC
growth.
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11.20%
12.40%
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14.90% 12.88% 15.50% 13.36%
15.54% 12.32% 16.57% 13.04%
16.40% 11.76% 18.00% 12.72%
17.60% 11.20% 20.00% 12.40%
19.40% 10.64% 23.00% 12.08%
22.40% 10.08% 28.00% 11.76%
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Things to note:
2. The gain from debt is larger with growth than without growth.
rd8.0%
rsU 14.00%
Tax Rate 40%
D/V
50%
D/S 1
MM rsL MM WACC APV rsL APV WACC
D/V 17.60% 11.20% 20.00% 12.40%
0%
14.00% 14.00% 14.00% 14.00%
10%
14.40% 13.44% 14.67% 13.68%
1. The gain from the tax shield will be lower using the APV model than under MM because the APV model
discounts the interest tax shield at the unlevered cost of equity, which is larger than the cost of debt. The
MM model discounts the tax shield at the cost of debt.
3. The value of the firm, whether levered or not, will be larger with growth, provided ROIC is greater than
WACC. Although we don’t show it here, ROIC is greater than WACC, so the value of the firm increases with
g. Suppose the expected free cash flow for Year 1 is $250,000 but it is expected to grow faster than 7%
during 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.
15.00%
20.00%
25.00%
30.00%
Cost of Capital with growth
rsL
25.00%
30.00%
35.00%
40.00%
Costs of capital for MM and APV
MM rsL
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Unlevered Value = PV at rsU = $3,960.01 thousand
rsU – g
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A B C D E F G H I J K
Inputs: 1 2 3
Estimate the unlevered value of operations
Estimate the value of the tax shield
Tax Shield Value = PV at rsU = $584.94 thousand
Estimate the total value of operations
Vops = Tax shield value + Unlevered value = $4,544.95 thousand
If L’s debt is risky, then its equity is like a call option and can be valued with the Black-Scholes Option
Pricing Model (OPM). See Chapter 2 for details of the OPM.
Black-Scholes Option Pricing Model
Total Value of Firm 4.00 Analogous to the stock price from the BSOPM
Finally, assume that L’s volatility, σ is 0.60 and that the risk-free rate rRF is 6%.
Unlevered Horizon Value =
rsU – g
g. Suppose the expected free cash flow for Year 1 is $250,000 but it is expected to grow faster than 7%
during 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.
(Free Cash Flow)(1+g)
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2.12 1.88 6.48%
2.13 1.87 6.89%
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A B C D E F G H I J K
The value of L‘s equity must be $2.20 million. The value of its debt must be what is left over: $1.80 million.
This gives a yield of 10.88% for the debt.
Value of Stock and Debt for Different Volatilities
Equity Debt Debt yield
Volatility $ 2.20 $ 1.80 10.888%
0.20
2.12 1.88 6.18%
0.25
2.12 1.88 6.20%
0.30
2.12 1.88 6.27%
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: See below and the Chapter 21 Mini Case Show.
2.00
2.50
3.00
Values of Debt and Equity for Different Volatilities