Capital Structure - Lakehead...

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


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


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


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


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


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


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


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


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


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


If debt is not risk-free, doesrLE increase withD?

WhenD/EL increases,rD also increases and thusρ− rD

decreases.

17


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


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


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


Using continuous discounting, this can be rewritten as

D = Xe−r f T

E = V−Xe−r f T .

23


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


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


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


Example

The return on equity in this case is

rLE =(

.7×450+ .3×50227

)1/3

−1 = 13.28%.

27


Example

What happens torLE if X increases to 250?

Debt is still risk-free and thus

D =250

(1.05)3 = 216.

28


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 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


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


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


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


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


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 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


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


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


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


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


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


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


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


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


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


we find

rLE =ρ(EL + (1−Tc)D) − (1−Tc)rDD

EL

= ρ +DEL

(1−Tc)(ρ− rD).

50


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


WACCL = ρ(

1− TcDVU +TcD

)= ρ× VU

VU +TcD

52

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