FIRM VALUATIONFIRM VALUATION

Firm ValuationFirm Valuation

Assumptions:Assumptions: Corporate taxes - individual taxe rate is zeroCorporate taxes - individual taxe rate is zero Capital markets are frictionlessCapital markets are frictionless Individuals can borrow and lend at the Individuals can borrow and lend at the

risk-free raterisk-free rate There are no costs to bankruptcyThere are no costs to bankruptcy

Firms issue only two types of claims: Firms issue only two types of claims: risk-free debt & (risky) equity risk-free debt & (risky) equity

All firms are in the same risk class All firms are in the same risk class

No other taxes than corporate taxes No other taxes than corporate taxes All cash flow streams are perpetuitiesAll cash flow streams are perpetuities Everybody has the same informationEverybody has the same information No agency costsNo agency costs

CF CFi jCF CFi j

The value of an unlevered firm isThe value of an unlevered firm is

,where ,where

Expected future cash flowExpected future cash flow

Discount rate for an all - equity firm Discount rate for an all - equity firm of of equivalent riskequivalent risk

= Corporate tax rate= Corporate tax rate

tctc

E FCF( )E FCF( )

VE FCF tU c

( )( )1

V

E FCF tU c( )( )1

If the firm issues debt, then If the firm issues debt, then

,where,where

= The amount paid to the = The amount paid to the lenders, kd = interest rate,lenders, kd = interest rate,

D = amount of debtD = amount of debt

=interest on debt. If the debt is risk-free =interest on debt. If the debt is risk-free then then . .

VE NOI t k Dt

kL c d c

b

( )( )1

V

E NOI t k Dt

kL c d c

b

( )( )1

k Dtd ck Dtd c

kbkb

k rb fk rb f

If If

thenthen

In other words In other words

= Value of an unlevered firm + the = Value of an unlevered firm + the PV PV of the tax shield provided by of the tax shield provided by debt.debt.

Notice that if Notice that if thenthen

(The famous Modigliani-Miller hypothesis)(The famous Modigliani-Miller hypothesis)

Bk D

kd

b

Bk D

kd

b

V V t BL Uc V V t BL Uc

VLVL

tc 0tc 0 V VL UV VL U

This implies that This implies that

““The market value of any firm is The market value of any firm is independent of its capital structure and independent of its capital structure and is given by capitalizing its expected is given by capitalizing its expected return at the rate return at the rate appropriate to its risk appropriate to its risk class”class”

(Modigliani-Miller, American Economic Review, 1958 june)(Modigliani-Miller, American Economic Review, 1958 june)

When the firm makes an When the firm makes an investmentinvestmentI,I,its value will change its value will change according to (source)according to (source)

VI

t E NOI

It

BI

Lc

c

( ) ( )1

The above investment will affect the value of The above investment will affect the value of the levered firm:the levered firm:

Note that Equity = old + ds0+dsnNote that Equity = old + ds0+dsn Because the project has the same risk as Because the project has the same risk as

those already outstanding, the value of the those already outstanding, the value of the outstanding debt stays the same outstanding debt stays the same

. .

V S S B BL n n 0 0

( )B0 0( )B0 0

Because new project is financed with Because new project is financed with new debt, equity or bothnew debt, equity or both

Inserting Inserting I into the above formula (), I into the above formula (),

I S Bn n I S Bn n

VI

SI

S BI

L n n

0

VI

SI

S BI

L n n

0

VI

SI

L

0

1

VI

SI

L

0

1

This means that the project has to increase This means that the project has to increase the shareholders’ wealth, so thatthe shareholders’ wealth, so that

andand

SI

VI

L0

1 0

SI

VI

L0

1 0

VI

L

1VI

L

1SI

0

0SI

0

0

The Weighted Average Cost of CapitalThe Weighted Average Cost of Capital

Recall the formulaRecall the formula

as shown it should be greater than 1, soas shown it should be greater than 1, so

VI

t E NOI

It

BI

Lc

c

n

( ) ( )11

( ) ( )( )

11

t E NOI

It

BI

cc

WACC tBIc

1

This results in what is called “the Weighted This results in what is called “the Weighted Average Cost of Capital”, WACC, source.Average Cost of Capital”, WACC, source.

If there are no taxes the cost of capital is If there are no taxes the cost of capital is independent of capital structure.independent of capital structure.

What does What does mean ?mean ?

““If denotes the firm’s long run target If denotes the firm’s long run target debt ratio ...then the firm can assume, debt ratio ...then the firm can assume, that for any particular investment that for any particular investment

“ “ . .

BI

BI

BV

*

*

BV

*

*

dB

dI

B

V

*

*

dB

dI

B

V

*

*

An alternative definition of the An alternative definition of the weighted average cost of capitalweighted average cost of capital

Definition by Haley and Shall [1973]Definition by Haley and Shall [1973] Target leverage ratio Target leverage ratio

WACC tBVc

1

Reproduction Reproduction valuevalue

Reproduction value = PV of the stream of goods Reproduction value = PV of the stream of goods and services expected from the project.and services expected from the project.

How to calculate the cost of the two How to calculate the cost of the two components in WACC (debt & equity)components in WACC (debt & equity) Assumptions:Assumptions: The cost of debt = The cost of debt = The cost of equity capital is the return onThe cost of equity capital is the return on

S Sn0 S Sn0

NIS Sn0

NIS Sn0

This can be written as (C-W, p. 449):This can be written as (C-W, p. 449):

Since the total change in equity isSince the total change in equity is

, the cost of equity, the cost of equity

can be written ascan be written as

NIS S

t kB

S Sn c b n0 01

( )( )

NI

S St k

BS Sn c b n0 01

( )( )

S S Sn 0

kNISs

k t kBSs c b ( )( )1

If the firm has no debt in its capital If the firm has no debt in its capital structure, thenstructure, then

It can be shown that (C-W, 451) WACC It can be shown that (C-W, 451) WACC can be written as:can be written as:

k s k s

WACC t kB

B Sk

SB Sc b s

( )1WACC t k

BB S

kS

B Sc b s

( )1

tax shieldtax shield Percentage Percentage of equity in of equity in the capital the capital structurestructure

cost cost

of equityof equity

Percentage Percentage of debt in of debt in the capital the capital structurestructure

cost cost

of debtof debt

This formula is the same as the This formula is the same as the Modigliani-Miller definitionModigliani-Miller definition

The M-M and the traditional definition The M-M and the traditional definition are identical !are identical !

WACC tB

B Sc

1