Chapter 16
Capital Structure Decisions:
Part II
Topics in Chapter

MM models:
Without corporate taxes (1958)
 With corporate taxes (1963)
Miller model: (1977)
 With corporate and personal taxes




Extension to MM with growth and the
tax shield is risky
Equity as an option
16-2
Modigliani and Miller (MM)






Published theoretical papers that changed the
way people thought about financial leverage
Nobel prizes in economics
MM 1958
MM 1963
Miller 1977
The papers differed in their assumptions
about taxes.
16-3
Model Assumptions
1. No taxes
2. Business risk measured by σEBIT
Firms with same risk = “homogeneous
risk class”
3. Homogeneous expectations
All investors have same estimates of
firm’s future EBIT
16-4
Model Assumptions
4. Perfect capital markets
No transactions costs
All can borrow and lend at riskfree rate
5. Debt is riskless
Interest rate on all debt = rf
6. All cash flows are perpetuities
All firms are expect zero growth
All bonds are perpetuities
16-5
MM with Zero Taxes (1958)
No agency or financial distress costs.
VF = EBIT capitalized at WACC
L = Levered
rrL = levered return
U = Unlevered
rrU =unlevered return
Proposition I:
EBIT
EBIT
VL  VU 

WACC
rsU
(16-1)
16-6
MM (1958) Proposition I

Implications:



When there are no taxes, the value of the
firm is independent of its leverage
The WACC is completely independent of a
firm’s capital structure
Regardless of the amount of debt a firm
uses, its WACC = cost of equity that it
would have if it used no debt
16-7
MM with Zero Taxes (1958)
rrL = levered return
rrU =unlevered return
D = market value of firm’s debt
S = market value of firm’s equity
rd = constant cost of debt
Proposition II:
rsL = rsU + (rsU - rd)(D/S)
(16-2)
16-8
MM (1958) Proposition II

When there are no taxes:
(1) The cost of equity to an unlevered firm in the
same risk class, rsU, plus
(2) A risk premium depending on the difference
between an unlevered firm’s costs of debt and
equity and the amount of debt used

As debt increases, the cost of equity also
increases, and in a mathematically precise
manner.
16-9
MM (1958) Implications

Using more debt will noe increase the
value of the firm


The benefits of additional debt will be
exactly offset by the increase in the cost of
equity
In a world without taxes, both the value
of the firm and its WACC would be
unaffected by its capital structure.
16-10
MM (1958) Arbitrage Proof



Assume all firms = 0 growth
EBIT remains constant
All earnings paid out as dividends
Dividends Net Incom e (EBIT - rdD )
S


rsL
rsL
r sL
(16-3)
16-11
MM (1958) Arbitrage Proof







Firms U and L are in same risk class
EBIT (U,L) = $900,000
Firm U has no debt; rsU = 10%
Firm L has $4,000,000 debt at rd = 7.5%
All net income is paid out as dividends
No corporate or personal taxes
Both firms are “no growth” (g=0)
16-12
Before Any Arbitrage
EBIT  rd D
VU  SU 
 $9 ,000 ,000
rsU
EBIT  rd D $9 ,000 ,000  0.75 ($4 ,000 ,000 )
SL 

rsU
0.10
$600,000

 $6 ,000 ,000
0.10
VL  DL  SL  $4 ,000 ,000  $6 ,000 ,000
 $10 ,000 ,000
16-13
Before Any Arbitrage

VU = $9,000,000
VL = $10,000,000

Suppose you own 10% of L’s stock



Dis-equilibrium
Situation
Market value = $600,000
If VL >VU, then you can increase your
income without increasing your risk
16-14
Arbitrage Proof
1. Sell your 10% of L’s stock for $600,000
2. Borrow an amount = 10% of L’s debt
($400,000)
3. Buy 10% of U’s stock for $900,000
4. Invest the remaining $100,000 at 7.5%
16-15
Before & After Arbitrage
Old Portfolio
10% of L's $600,000
equity income
$60,000
TOTAL INCOME
$60,000
New Portfolio
10% of U's $900,000
equity income
90,000
Less 7.5% interest
on $400,000 loan
(30,000)
Plus 7.5% interest on
extra $100,000
7,500
TOTAL INCOME
$67,500
16-16
Arbitrage Proof
Propositions I and II



Substitute $400,000 of “homemade
leverage” for L’s leverage
Neither effective debt nor risk has
changed
Profit motive would force price of L’s
stock down and U’s up until market
values are equal.
16-17
Propositions I & II
Proposition I:
EBIT
EBIT
VL  VU 

WACC
rsU
Proposition II:
rsL = rsU + (rsU - rd)(D/S)
16-18
MM Relationships Between Capital Costs
and Leverage (D/V)
Without taxes
Cost of
Capital (%)
26
rs
20
WACC
14
rd
8
0
20
40
60
80
Debt/Value
100 Ratio (%)
16-19
MM Relationships Between Capital
Costs and Leverage (D/V)



The more debt the firm adds to its
capital structure, the riskier the equity
becomes and thus the higher its cost.
Although rd remains constant, rs
increases with leverage.
The increase in rs is exactly sufficient to
keep the WACC constant.
16-20
MM (1963) with Corporate Taxes
With corporate taxes added, the MM
propositions become:
Proposition I:
VL = VU + TD
Proposition II:
rsL = rsU + (rsU - rd)(1 - T)(D/S)
(16-4)
(16-6)
16-21
Tax Shield and Value of U
rD DT
TD  Tax Shield 
rD
EBIT ( 1  T )
VU  S 
rsU
(16-5)
16-22
Hamada’s Equation
b  bU [ 1  ( 1  T ) D S )]
(16-7)
Beta increases with leverage
16-23
Notes About the New
Propositions
1. When corporate taxes are added,
VL ≠ VU. VL increases as debt is added
to the capital structure, and the greater
the debt usage, the higher the value of
the firm.
2. rsL increases with leverage at a slower
rate when corporate taxes are
considered.
16-24
Frederickson Water Company





No debt
E(EBIT) = $2,400,000
No growth
All income paid out as dividends
If uses debt, rD=8%




Any debt would be used to repurchase stock
Beta = 0.80 (bU)
Risk-free rate = 8%
rsU = 12%
Market risk premium = 5%
16-25
Value of FWC (No Taxes)
With No Debt & No Taxes
EBIT
VU =
=
rsU
$2.4 m
= $20.0m
0.12
With $10.0m Debt & No Taxes
S=V-D = $20 m - $10 m = $10 m
rsL = rsU + (rsU - rd)(D/S)
= 12% + (12%-8%)($10/$10) = 16%
16-26
FWC’s WACC
WACC  ( D V )( rD )( 1  T )  ( S V )r sL
 ($10/$20)(8%)(1.0) ($10/$20)(16%)  12.0%
• Value
of the firm and the firm’s WACC are
independent of the amount of debt
16-27
FWCC with Corporate Taxes



Tax rate = 40%
Debt = $10 m
EBIT = $4,000,000*


Taxes will reduce net income and EBIT
EBIT increased to make comparison
easier
16-28
FWCC With Corporate Taxes
EBIT ( 1  T ) $4 m( 0.60 )
VU  S 

 $20 m
rsU
0.12
VL  VU  TD  $20 m  0.4 ($10 m )  $24 m
S  V  D  $24 m  $10 m  $14 m
16-29
FWCC with Corporate Taxes
r sL  r sU  ( r sU  rd )( 1  T )( D )
S
 12%  ( 12%  8%)( 0.6 )($10 m / $14 m )  13.71%
WACC  ( D / V )( rd )( 1  T )  ( S / V )r sL
 ($10/$24)(8%)(0.6)  ($14/$24)(13.71%)  10.0%
b  bU [ 1  ( 1  T ) D S ]
 0.80[1 (1- 0.40) $10m
$14m
 1.1429
16-30
MM: Capital Costs vs. Leverage
with Corporate Taxes
Cost of
Capital (%)
rs
26
20
14
8
0
20
40
60
80
WACC
rd(1 - T)
Debt/Value
100
Ratio (%)
16-31
MM: Value vs. Debt with
Corporate Taxes
Value of Firm, V (%)
4
VL
3
TD
VU
2
1
Debt
0
0.5
1.0
1.5
2.0
2.5 (Millions of $)
Under MM with corporate taxes, the firm’s value
increases continuously as more and more debt is used.
16-32
Miller Model with Personal Taxes
Miller’s Proposition I:
(1 - Tc)(1 - Ts)
VL = VU + 1 (1 - Td)
[
]D
(16-12)
Tc = corporate tax rate
Td = personal tax rate on debt income
Ts = personal tax rate on stock income
16-33
Tc = 40%, Td = 30%, Ts = 12%
[
]
(1 - 0.40)(1 - 0.12)
VL = VU + 1 D
(1 - 0.30)
= VU + (1 - 0.75)D
= VU + 0.25D
Value rises with debt; each $100 increase
in debt raises L’s value by $25.
16-34
Miller vs. MM Model with
Corporate Taxes



If only corporate taxes, then
VL = VU + TcD = VU + 0.40D
Here $100 of debt raises value by $40.
Personal taxes lowers the gain from
leverage, but the net effect depends on
tax rates.
16-35
Miller Model Implications
 ( 1  Tc )( 1  Ts ) 
VL  VU  1 
D
( 1  Td )


(16-12)
1. The right-hand term = gain from
leverage
2. If taxes ignored, then Miller=Original MM
3. If personal taxes ignored, then Miller =
MM with corporate taxes
16-36
Miller Model Implications
 ( 1  Tc )( 1  Ts ) 
VL  VU  1 
D
( 1  Td )


(16-12)
4. If Ts=Td, right-hand term = Tc
5. If (1-T)(1-T) =(1-T), right-hand term= 0

No gain to leverage
16-37
Criticisms of MM and Miller
No one believes the models holds precisely

Models assume personal and corporate
leverage are perfect substitutes

Homemade leverage puts stockholders
in grater risk of bankruptcy

Brokerage costs are assumed to be 0
16-38
Criticisms of MM and Miller
No one believes the models holds precisely
4.
Individuals cannot borrow at the riskfree rate
5.
For the Miller equilibrium to be reached,
the tax benefit from debt mustbe the
same for all firms
6.
MM and Miller assumed no cost to
financial distress
16-39
MM with Nonzero Growth &
A Risky Tax Shield

Under MM (with taxes; no growth)
VL = VU + TD
 This assumes the tax shield is
discounted at the cost of debt.


The debt tax shield will be larger if
the firms grow
16-40
MM with Nonzero Growth & a
Risky Tax Shield
Value of (growing) tax shield =
VTS
rd TD

rTS  g
(16-14)
Value of levered firm with growth:
 rd 
TD
VL  VU  
 rTS  g 
(16-15)
16-41
MM with Nonzero Growth & a
Risky Tax Shield

If rTS = rsU:
 rd 
TD
VL  VU  
 rsU  g 
rsL  rsU  ( rsU  r d ) D
S
D
b  bU  ( bU  bd )
S
(16-16)
(16-17)
(16-18)
16-42
Risky Debt

MM and Hamada assume riskless
debt


Βd = 0
If Bd ≠ 0:
rd  rRF  bd RPM
bd  ( rd  rRF ) / RPM
16-43
MM Extension with Growth
Peterson Power Illustration







E(FCF) = $1 m
G = 7%
rsU = 12%
T = 40%
VU = $20 m
$10 m debt
rd= 8%
16-44
Peterson Power
 rd 
0.08  0.40  $10 m 
TD  $20 m  
VL  VU  
  $26.4 m
0.12  0.07


 r sU  g 
S  VL  D  $26.4 m  $10 m  $16.4 m
r sL  rsU  ( r sU  r d ) D S  12%  ( 12%  8%)(
0.3788
)  14.44%
0.6212
WACC  0.3788( 1  0.40 )8%  ( 1.0  0.3788 )14.44%  10.78%
16-45
FWC vs. PPI
Debt
r(d)
E(EBIT)
E(FCF)
g
r(sU)
T
V(U)
V(L)
S
r(sL)
WACC
Frederickson
$10 m
8%
$4 m
0
12%
40%
$20 m
$24 m
$14 m
13.71%
10%
Peterson
$10 m
8%
$1 m
7%
12%
40%
$20
$26.4 m
$16.4 m
14.44%
10,78%
16-46
Cost of Capital for MM and
Extension
40%
35%
MM cost of equity
30%
MM WACC
25%
20%
Extension cost of
equity
Extension WACC
15%
10%
5%
0%
0% 10% 20% 30% 40% 50% 60% 70% 80%
D/V
16-47
Equity as an Option: Kunkel, Inc.



Firm value (Debt + Equity) = $20 m
Firm has $10 million face value of 5-year zero
coupon debt coming due soon
If the current value of the firm (D+S) = $9 m:


If firm value > $10 m:


Firm will default on debt; equity holders get 0
Firm pays off the debt; equity holders keep the
firm
Payoff to stockholders = Max(P-$10m,0)
16-48
Notation






V = Value of the option
P = Value of the firm (S+D)
X = strike = value of debt
rRF = risk-free rate
σ = volatility of the underlying asset
T = time to maturity in years
16-49
Kunkel Variables





P = $20 million (firm value)
X = $10 million (face value of debt)
T = 5 years (maturity of debt)
rRF = 6%
σ = 40%
16-50
The Black-Scholes Formulas
V  P [ N ( d 1 )]  X e  rRF T [ N ( d 2 ) ]
(16-19)
w he re:
d1 
ln(P / X )  ( rRF
d 2  d1   T
2
  / 2 )T
(16-20)
 T
(16-21)
16-51
Formula Functions



ln = natural log
N(x) = the probability that a normally
distributed variable with a mean of
zero and a standard deviation of 1 is
less than x
N(d1) and N(d2) denote the standard
normal probability for the values of d1
and d2.
16-52
BSOPM Kunkel Example
P = $20
X = $10
d1 
rRF = 6%
T=5
σ = 40%
ln(P / X )  ( rRF   2 / 2 )T
 T
ln(20 10 )  ( 0.06  0.40 2 2 )  5
d1 
 1.5576
0.40 5
d 2  d1   T
d 2  1.5576  0.40 5  0.6632
16-53
BSOPM
Call Price Example
d1 = 1.5576
N(1.5576) = 0.9403
d2 = 0.6632
N(0.6632) = 0.7464
V  P N ( d 1 )  X e  rT N ( d 2 )
V  20(0.9403) - 10e -.065 (0.7464)
Vs  $13.28  Value of Equity
Vd  $20m - $13.28m  $6.72 m
16-54
Zero-Coupon Debt Yield


Debt yield for 5-year zero coupon debt
= (Face value / Price)1/5 – 1
= ($10 million/ $6.72 million) – 1
= 8.27%
Yield on debt depends on:


Probability of default
Value of the option
16-55
Managerial Incentives


Managers can change a firm's  by
changing the assets the firm invests in.
Changing  can:


Change the value of the equity, even if it
doesn't change the expected cash flows
Transfer wealth from bondholders to
stockholders by making the option value of
the stock worth more, which makes what
is left, the debt value, worth less.
16-56
Effect on Option Values
Volatility = σ



Increased volatility increased upside
potential and downside risk
Increased volatility is NOT good for the
holder of a share of stock
Increased volatility is good for an option
holder


Option holder has no downside risk
Greater potential for higher upside payoff
16-57
Bait and Switch



Managers who know the effect of
volatility, might tell debtholders they are
going to invest in one kind of asset,
and, instead, invest in riskier assets.
“Bait and Switch”
Bondholders will require:


Higher coupon rates
Strict bond covenants as protection
16-58
Risky Coupon Debt


More complex analysis
With each coupon payment management
has an option on an option:


If it makes the interest payment then it
purchases the right to later make the
principal payment and keep the firm
This is called a compound option.
16-59
Capital Structure Theory
The Authors’ View
1.
2.
Debt financing has the benefit of
tax deductibility so firms should
have some debt in their capital
structure
Financial distress and agency costs
place limits on debt usage
16-60
Capital Structure Theory
The Authors’ View
“Pecking Order”
3.


Due to problems from asymmetric
information and flotation costs, lowgrowth firms should follow a pecking
order in raising funds (R/E, debt, new
equity)
High-growth firms whose growth involves
tangible assets should follow the same
pecking order (r/e, debt. Equity)
16-61
Capital Structure Theory
The Authors’ View
“Pecking Order”
3.

4.
High-growth firms whose growth is
primarily in intangible assets should
emphasize stock rather than debt
Firms should maintain reserve
borrowing power
16-62