FIRM VALUATION WITH TAXES
Firm Valuation with Corporate Taxes
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
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
The value of an unlevered firm is
E(FCF )(1 − t c )
,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
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
V = V + t cB
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)
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)
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
The above investment will affect the value of the levered firm:
ΔV = ΔS + ΔS + ΔB + ΔB
L 0 n 0 0
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) .
Because the new project is financed with new debt, equity or both
ΔI = ΔS + ΔB
(Previous slide (9))
Inserting ΔI into the above formula,
ΔV ΔS ΔS + ΔB = + ΔI ΔI ΔI
L 0 n
ΔV ΔS = +1 ΔI ΔI
This means that the project has to increase the shareholders’ wealth, so that
ΔS ΔV = − 1> 0 ΔI ΔI
ΔS >0 ΔI
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
as shown it should be greater than 1, so
(1− t c )ΔE(NOI) ΔB > ρ (1− t c ) ΔI ΔI
This results in what is called “the Weighted Average Cost of Capital”, WACC,(C-W, p.446).
⎡1 − t ΔB ⎤ WACC = ρ c ⎢ΔI ⎥ ⎣ ⎦
If there are no taxes the cost of capital is independent of capital structure.
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
Definition by Haley and Shall 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
Assumptions: The cost of debt = k b The cost of equity capital is the return on
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
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:
B S WACC = (1− tc )kb + ks B+ S B+ S
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