Unformatted text preview: hen
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
makes a positive amount of money with no risk
There can’t be strict arbitrage opportunities in
financial market equilibrium. People would
replicate such strategies and make infinite profits.
In practice, arbitrage means going long and short
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
does not hold, then there are arbitrage opportunities
Form two portfolios with same future cash flows & risk
– Portfolio 1 uses the levered firm, and thus inherits the
leverage of the firm
– Portfolio 2 uses the unlevered firm, and adds leverage
by borrowing on the investor’s account
Arbitrage opportunities will exist unless the two portfolios
cost the same. This only happens if MM1 holds.
cost 10 Portfolio 1
Composition: a fraction α of equity in the levered
Initial cost: αSL
– Alpha is % of stock outstanding purchase
– C is operating cash flow Returns: perpetual cash flows of α(C-rBBL)(1-TC)
11 Portfolio 2
– Invest αVU dollars in the equity of the unlevered firm
– Partially finance by borrowing αBL (1-TC)
Initial net cost: αVU - αBL (1-TC)
Returns: perpetual cash flows of
– From the equity:
– From the debt:
From - αrBBL(1-TC) – Total perpetual return: α(C- rBBL)(1-TC)
12 The Arbitrage Argument
The perpetual cash flows to portfolio 1 are equal to
the perpetual returns to portfolio 2.
What happens if VU + TCB > VL = SL + BL ?
Buy P1 (levered), sell P2 (unlevered)
Cash flow at time zero:
= (αVU - αBL(1-TC)) - αSL
= (αVU - αBL(1-TC) - α(VL-BL) = a (VU + TCB – VL) > 0
CFs in the future:
= α(C- rBBL) (1-TC)- α(C- rBBL) (1-TC) = 0
)13 The Arbitrage Argument
What happens if VU + TCB < VL ?
Arbitrage in the other direction! Sell P1 and buy P2.
Since arbitrage opportunities cannot exist:
VU + TCB = VL (MM1 with taxes)
Note that PV(tax shield)=TCrBB/rB=TCB becau...
View Full Document