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FIRM VALUATION WITH TAXES

Firm Valuation with Corporate Taxes
Assumptions:
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Only corporate taxes - individual tax rate is zero Capital markets are frictionless Individuals can borrow and lend at the risk-free rate There are no costs to bankruptcy

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n

n

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Firms issue only two types of claims: riskfree debt & (risky) equity All firms are in the same risk class

C Fi =λ C F j
n n n n

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

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The value of an unlevered firm is
V =
U

E(FCF )(1 − t c )

ρ

,where E(FCF ) = Expected future cash flow ρ = Discount rate for an all - equity firm of equivalent risk t c = Corporate tax rate

If the firm issues debt, then (C-W, p.442)

E(NOI)(1− t c ) k dDt c V = + ρ kb
L

,where

k dDt c = The amount paid to the lenders, k d = interest rate,
D = amount of debt (face value) k b =before tax cost of risk-free debt.

k dD If the market value of debt B = kb
then

V = V + t cB
L U

In other words L V = Value of an unlevered firm + the PV of the tax shield provided by debt. L U Notice that if t c = 0 then V = V(The famous Modigliani-Miller hypothesis)

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Note that in an economy with no risk for going bankrupt, the face value of the debt equals its market value, e.g. D = B.

This implies that “The market value of any firm is independent of its capital structure and is given by capitalizing its expected return at the rate ρ appropriate to its risk class”
(Modigliani-Miller, American EconomicReview, 1958 june)

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When the firm makes an investment I, its value will change according to (C-W, p. 445)

(1− t c )∗ ΔE(NOI) ΔV ΔB = + tc ρ∗ ΔI ΔI ΔI
L

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

ΔV = ΔS + ΔS + ΔB + ΔB
L 0 n 0 0

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Note that Equity = old + ΔS + ΔS n Because the project has the same risk as those already outstanding, the value of theoutstanding debt stays the same (ΔB 0 = 0) .
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Because the new project is financed with new debt, equity or both

ΔI = ΔS + ΔB
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(Previous slide (9))

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Inserting ΔI into the above formula,

ΔV ΔS ΔS + ΔB = + ΔI ΔI ΔI
L 0 n

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ΔV ΔS = +1 ΔI ΔI
L 0

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

ΔS ΔV = − 1> 0 ΔI ΔI
0 L

ΔV>1 ΔI
L

and

ΔS >0 ΔI
0

The Weighted Average Cost of Capital
(Copeland-Weston, 1988, p. 444) n

Recall the formula

(1− t c )∗ ΔE(NOI) ΔV ΔB = + tc >1 ρ ∗ ΔI ΔI ΔI
L n

as shown it should be greater than 1, so

(1− t c )ΔE(NOI) ΔB > ρ (1− t c ) ΔI ΔI

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This results in what is called “the Weighted Average Cost of Capital”, WACC,(C-W, p.446).

⎡1 − t ΔB ⎤ WACC = ρ c ⎢ΔI ⎥ ⎣ ⎦
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If there are no taxes the cost of capital is independent of capital structure.

What does
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B* V*

ΔB mean ? ΔI

“If denotes the firm’s long run target debt ratio ...then the firm can assume, that for any particular investment dB B * “ = dI V * (C-W, p. 446).

An alternative definition of the weighted average cost of capital
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Definition by Haley and Shall [1973]Target leverage ratio

⎡1− t ΔB ⎤ WACC = ρ c ⎢ ΔV ⎥ ⎣ ⎦
Reproduction value Reproduction value = PV of the stream of goods and services expected from the project.

How to calculate the cost of the two components in WACC (debt & equity)
n n n

ΔS + ΔS
0

Assumptions: The cost of debt = k b The cost of equity capital is the return on
n

ΔNI 0 n ΔS + ΔS

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

ΔNI ΔB = ρ + (1− t c )(ρ − k b ) 0 0 n ΔS + ΔS ΔS + ΔSn
Since the total change in equity is

ΔNI , the cost of equity k s = ΔS can be written as ΔB k s = ρ + (1− t c )(ρ − k b ) ΔS

ΔS = ΔS + ΔS
0

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If the firm has no debt in its capital structure, then k s = ρ It can be shown that (C-W, p. 451) WACC can be written as:
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B S WACC = (1− tc )kb + ks B+ S B+ S
tax...
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