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Capital Structure
Lakehead University
Winter 2005
Outline of the Lecture
• Modigliani and Miller’ Propositions
– With Taxes
– Without Taxes
• The Binomial Pricing Model
2
The Value of the Levered Firm
Assumptions:
• Capital markets are frictionless.
• Individuals can borrow and lend at the risk-free rate.
• No bankruptcy costs.
• Firms issue only two types of claims: risk-free debt and
(risky) equity.
• All firms are assumed to be in the same risk class.
3
The Value of the Levered Firm
Assumptions:
• Corporate taxes are the only form of government levy.
• No growth.
• No information asymmetry.
• Managers maximize shareholders’ wealth.
• Operating cash flows are unaffected by changes in capital
structure.
4
Modigliani and Miller’s Propositions
Irrelevance of the Capital Structure
Proposition I: In the absence of taxes, the market value of a
firm is constant regardless of the amount of leverage that it
uses to finance its assets.
Proposition II: In the absence of taxes, the expected return on a
firm’s equity is an increasing function of the firm’s leverage.
5
Analysis of M&M Proposition I
The value of a firm is given by the present value of all the cash
flows its assets are expected to generate in the future.
The value of a firm is equal to the value of its assets.
Unlevered Firm: VU = EU
Levered Firm: VL = D + EL.
6
Analysis of M&M Proposition I
M&M Proposition I states that
VU = VL.
Why?
Consider an all-equity firm with valueVU = EU .
Suppose there existed a way to finance this firm’s assets with
debt and equity such that
VL = D + EL > VU .
7
Analysis of M&M Proposition I
An arbitrageur could buyα shares of the above firm, place them
in a trust and sell debt and equity claims against these shares in
proportions such that
α(D+EL) > αEU ,
making then a riskless profit.
8
Analysis of M&M Proposition I
Similarly, someone could buy all of the firm’s shares forEU and
modify the firm’s capital stucture to have
VL = D + EL > EU
and then resell the firm’s assets for a riskless profit ofVL−VU .
9
Analysis of M&M Proposition I
In a frictionless market, this arbitrage opportunity would lead to
an increase in the value of the unlevered firm’s equity to the point
where
VU = EU = D + EL = VL
for any level ofD andEL.
10
Analysis of M&M Proposition I
In M&M world, the present value of an unlevered firm is given
by
VU =FCF
ρ=
EBITρ
,
whereρ is the cost of equity of the unlevered firm.
11
Analysis of M&M Proposition I
If two firms with the sameEBIT, one levered and the other
unlevered, are such thatVU = VL, then
VL =EBIT
ρ,
i.e. a firm’s WACC is not affected by its capital structure.
12
Analysis of M&M Proposition II
M&M Proposition II states that a firm’s expected return on
equity increases with leverage.
Take the firm’s WACC (there are no taxes here):
WACC = rD
(D
D+EL
)+ rLE
(EL
D+EL
).
13
Analysis of M&M Proposition II
Without taxes, a firm’s WACC,ρ, is independent of the capital
structure:
ρ = rD
(D
D+EL
)+ rLE
(EL
D+EL
).
Let’s factor outrLE.
14
Analysis of M&M Proposition II
ρ = rD
(D
D+EL
)+ rLE
(EL
D+EL
)
⇒ rLE
(EL
D+EL
)= ρ − rD
(D
D+EL
)rLE = ρ
(D+EL
EL
)− rD
DEL
rLE = ρ(
DEL
+1
)− rD
DEL
rLE = ρ +DEL
(ρ − rD) .
15
Analysis of M&M Proposition II
rLE = ρ +DEL
(ρ − rD)
Sinceρ is constant regardless ofD/EL, sinceρ− rD > 0 and
since debt is risk-free,rLE increases with leverage.
16
Analysis of M&M Proposition II
If debt is not risk-free, doesrLE increase withD?
WhenD/EL increases,rD also increases and thusρ− rD
decreases.
17
Analysis of M&M Proposition II
Incorporating more details in the analysis, it is possible to show
that
rLE ↑ asDEL↑ .
Thebinomial pricing model, for instance, can be used to show
this result.
18
The Binomial Pricing Model
Consider a firm that can take on two values at timeT, i.e.
VT =
VuT with probability p,
VdT with probability 1− p,
with VuT > Vd
T , whereu stands for “up” andd stands for “down”.
19
The Binomial Pricing Model
Let
V ≡ Current value of the firm’s assets
D ≡ Current value of the firm’s debt
E ≡ Current value of the firm’s equity
X ≡ Payment promised to debtholders at timeT
r f ≡ Risk-free rate of interest
20
The Binomial Pricing Model
What areD andE?
At time T, the value of the firm’s debt is
DT =
X if VT ≥ X,
VT if VT < X.
21
The Binomial Pricing Model: Risk-Free Debt
If VdT ≥ X, thenVu
T > X and thus debt is free of risk.
Risk-free assets are discounted at the risk-free rate, and thus
D =X
(1+ r f )T ,
which gives
E = V − D = V − X(1+ r f )T .
22
The Binomial Pricing Model: Risk-Free Debt
Using continuous discounting, this can be rewritten as
D = Xe−r f T
E = V−Xe−r f T .
23
The Binomial Pricing Model: Risk-Free Debt
Example
Consider a firm with present valueV = $400, future value
V3 =
Vu3 = $650 with probabilityp = 0.7,
Vd3 = $250 with probability 1− p = 0.3.
in three years, and a pure-discount debt issue that paysX = $200
in three years.
The risk-free interest rate isr f = 5%.
24
The Binomial Pricing Model: Risk-Free Debt
Example
Note that the above values give us a return on assets of
rA =
(pVu
3 +(1− p)Vd3
V
)1/3
−1
=(
.7×650+ .3×250400
)1/3
−1
= 9.83%.
Note thatrA≡ ρ.
25
The Binomial Pricing Model: Risk-Free Debt
Example
What is the current value of debt?
Debt is risk-free and thusX can be discounted at the risk-free
rate to findD:
D =X
(1.05)3 =200
(1.05)3 = 173.
The current value of equity is then
E = V − D = 400− 173 = 227.
26
The Binomial Pricing Model: Risk-Free Debt
Example
The return on equity in this case is
rLE =(
.7×450+ .3×50227
)1/3
−1 = 13.28%.
27
The Binomial Pricing Model: Risk-Free Debt
Example
What happens torLE if X increases to 250?
Debt is still risk-free and thus
D =250
(1.05)3 = 216.
28
The Binomial Pricing Model: Risk-Free Debt
Example
The value of the firm has a whole remainsV = 400 (M&M
Proposition I), we have
EL = 400− 216 = 184,
which gives
rLE =(
.7×400+ .3×0184
)1/3
−1 = 15.02%.
29
The Binomial Pricing Model: Risky Debt
Suppose now that debt is not risk-free. That is, suppose that
VdT < X < Vu
T .
The value of debt at timeT is then
DT =
X if VT = VuT ,
VdT if VT = Vd
T .
30
The Binomial Pricing Model: Risky Debt
The value of equity at timeT is
ET =
VuT −X if VT = Vu
T ,
0 if VT = VdT .
Let EuT = Vu
T −X and letEdT = 0.
31
The Binomial Pricing Model: Risky Debt
How can we findD andE in this case?
We have the risk-free discount rate and thus we can find the
present value of any risk-free asset.
Can we form a risk-free portfolio withV, D andE?
32
The Binomial Pricing Model: Risky Debt
A risk-free portfolio is a portfolio that provides the same payoff
in each state of the worldu andd.
How to make a portfolio that paysK, say, whetherVT = VuT or
VT = VdT ?
What canK be?
What payoff can we guarantee with certainty?
33
The Binomial Pricing Model: Risky Debt
Consider a portfolioP, which involves the purchase of all of the
firm’s assets and the short sale of a fractionδ of the firm’s equity.
The payoff of portfolioP at timeT is then
VuT − δEu
T in stateu,
VdT − δEd
T in stated.
34
The Binomial Pricing Model: Risky Debt
For portfolioP to be risk-free, we need
VuT − δEu
T = VdT − δEd
T ⇒ δ =Vu
T −VdT
EuT −Ed
T
.
The present value of portfolioP, V−δE, is then given by
V − δE =VT −δET
(1+ r f )T .
35
The Binomial Pricing Model: Risky Debt
With δ = VuT−Vd
TEu
T−EdT, we have
VT − δET = VuT − δEu
T = (1−δ)VuT + δX
= VdT − δEd
T = VdT
and thus
V − δE =Vd
T
(1+ r f )T .
36
The Binomial Pricing Model: Risky Debt
The current market value of equity is then
E =1δ
(V − Vd
T
(1+ r f )T
)and the current market value of debt is
D = V − E.
37
The Binomial Pricing Model: Risky Debt
Example
Consider a firm with present valueV = $400, future value
V3 =
Vu3 = $650 with probabilityp = 0.7,
Vd3 = $250 with probability 1− p = 0.3.
in three years, and a pure-discount debt issue that paysX = $400
in three years.
The risk-free interest rate isr f = 5%.
38
The Binomial Pricing Model: Risky Debt
Example
Same firm as before, except thatX = 400.
Debt is not default-free anymore.
What is the current value of debt and equity?
First thing to do: findδ in the portfolioV−δE such that the
latter be risk-free.
39
The Binomial Pricing Model: Risky Debt
Example
To haveVu3 −δEu
3 = Vd3 −δEd
3 , we need
δ =Vu
3 −Vd3
Eu3−Ed
3
,
where
Eu3 = max
{0 , Vu
3 −X}
= max{
0 , 650−400}
= 250
Ed3 = max
{0 , Vu
3 −X}
= max{
0 , 250−400}
= 0.
40
The Binomial Pricing Model: Risky Debt
Example
This gives
δ =Vu
3 −Vd3
Eu3−Ed
3
=650−250250−0
= 1.6
and thus
E =1δ
(V −
Vd3
(1+ r f )3
)=
11.6
(400− 250
(1.05)3
)= 184
andD = V−E = 400−184= 216.
41
The Binomial Pricing Model: Risky Debt
Example
Note that the return required by bondholders in this case is
rD =(
400216
)1/3
− 1 = 22.8%.
42
M&M Propositions with Taxes
Consider an unlevered firm, denotedU , that expects constant
earnings before interest and taxes, denotedEBIT, forever.
Each period, if the corporate tax rate isTc, shareholders receive
(1−Tc)EBIT
and the government receives
Tc×EBIT.
43
M&M Proposition I with Taxes
Let EU = VU denote the present value of the payment
(1−Tc)EBIT forever.
Let GU denote the present value of the paymentTc×EBIT
forever.
Let ρ denote the rate at which investors discountEBIT, i.e.
EU = VU =(1−Tc)EBIT
ρ.
44
M&M Proposition I with Taxes
Consider a levered firm, FirmL, with the sameEBIT asU , but
with a perpetual debt issueD with coupon ratei.
Interest payments are tax exempt.
Shareholders receive(1−Tc)(EBIT− iD) each period forever,
bondholders receiveiD each period forever, and
the government receivesTc(EBIT− iD) each period forever.
45
M&M Proposition I with Taxes
Let
QU = EU + GU and QL = EL + D + GL.
From M&M Proposition I without taxes, we know thatQU = QL
sinceEBIT is the same for both firms.EBIT should therefore be
discounted at the same rate for bothU andL, namelyρ.
46
M&M Proposition I with Taxes
Each period, the total cash flow to shareholders and bondholders
of Firm L is
(1−Tc)(EBIT− iD) + iD = (1−Tc)EBIT + TciD.
Assuming thatTciD is as safe as debt itself, we have
VL =(1−Tc)EBIT
ρ+
TciDrD
,
whererD is bondholders’ required return (risk-free rate here).
47
M&M Proposition I with Taxes
For debt to have been issued at par, we needi = rD and thus
VL =(1−Tc)EBIT
ρ+
TcrDDrD
= VU + TcD.
48
M&M Proposition II with Taxes
How doesrLE, the return required by the levered firm’s
shareholders, compare withρ?
Using
EL =(1−Tc)(EBIT− rDD)
rLE⇒ rLE =
(1−Tc)(EBIT− rDD)EL
and
(1−Tc)EBIT = ρVU = ρ(VL − TcD) = ρ(EL + (1−Tc)D),
49
M&M Proposition II with Taxes
we find
rLE =ρ(EL + (1−Tc)D) − (1−Tc)rDD
EL
= ρ +DEL
(1−Tc)(ρ− rD).
50
M&M Proposition II with Taxes
WACCL =DVL
(1−Tc)rD +EL
VLrLE
=DVL
(1−Tc)rD +EL
VL
(ρ +
DEL
(1−Tc)(ρ− rD))
=D(1−Tc)rD + ELρ + D(1−Tc)(ρ− rD)
VL
=(VL−D)ρ + D(1−Tc)ρ
VL
=VLρ − TcDρ
VL= ρ
(1− TcD
VU +TcD
)
51
M&M Proposition II with Taxes
WACCL = ρ(
1− TcDVU +TcD
)= ρ× VU
VU +TcD
52