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More debt always increases firm value!
Prove MM1 with taxes (without is just a special case)
8 Proof of MM1 is Based on No Arbitrage
The strict definition of arbitrage is a strategy that
The
arbitrage
makes a positive amount of money with no risk
makes
There can’t be strict arbitrage opportunities in
There
financial market equilibrium. People would
replicate such strategies and make infinite profits.
replicate
In practice, arbitrage means going long and short
In
similar but not identical assets to make a positive
expected return. But there is some risk.
9 Proof of MM1 With Taxes: The Logic
We will prove MM1 by contradiction: Show that if it
We
does not hold, then there are arbitrage opportunities
does
Form two portfolios with same future cash flows & risk
– Portfolio 1 uses the levered firm, and thus inherits the
Portfolio
leverage of the firm
leverage
– Portfolio 2 uses the unlevered firm, and adds leverage
Portfolio
by borrowing on the investor’s account
by
Arbitrage opportunities will exist unless the two portfolios
Arbitrage
cost the same. This only happens if MM1 holds.
cost 10 Portfolio 1
Composition: a fraction α of equity in the levered
Composition:
levered
firm.
firm.
Initial cost: αSL
Initial
– Alpha is % of stock outstanding purchase
– C is operating cash flow Returns: perpetual cash flows of α(CrBBL)(1TC)
Returns:
11 Portfolio 2
Composition:
– Invest αVU dollars in the equity of the unlevered firm
Invest
unlevered
– Partially finance by borrowing αBL (1TC)
Partially
Initial net cost: αVU  αBL (1TC)
Returns: perpetual cash flows of
αC(1TC)
– From the equity:
From
– From the debt:
From  αrBBL(1TC) – Total perpetual return: α(C rBBL)(1TC)
Total
12 The Arbitrage Argument
The
The perpetual cash flows to portfolio 1 are equal to
the perpetual returns to portfolio 2.
the
What happens if VU + TCB > VL = SL + BL ?
Buy P1 (levered), sell P2 (unlevered)
Cash flow at time zero:
Cash
= (αVU  αBL(1TC))  αSL
))
= (αVU  αBL(1TC)  α(VLBL) = a (VU + TCB – VL) > 0
CFs in the future:
CFs
= α(C rBBL) (1TC) α(C rBBL) (1TC) = 0
)13 The Arbitrage Argument
What happens if VU + TCB < VL ?
Arbitrage in the other direction! Sell P1 and buy P2.
Arbitrage
Since arbitrage opportunities cannot exist:
VU + TCB = VL (MM1 with taxes)
Note that PV(tax shield)=TCrBB/rB=TCB becau...
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 Spring '13

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