The ¯ Levering/Unlevering Formula
1. Start by deriving the MM WACC
A …rm will only invest if
PV of E(Return from project) + tax-shield bene…t > Investment Cost
(1 ¡ tc )E(NOI)
+ tc D
ru
>
I
N OI ´ Net Op. Income
r u ´ Unlevered Cost of Capital
tc D
(1 ¡ tc )E(NOI)
+ ru
> ru
I
I
multiply through by ru , divide through by I
µ
¿ cD
(1 ¡ tc )E(NOI)
> ru 1 ¡
I
I
¶
(0)
The right-hand side de…nes what MM call the WACC.
The expression says simply that the project return (left hand side) must be
greater than the WACC for a project to add to …rm value.
This expression modi…es the cost of capital to account for tax bene…t; however,
the expression could be modi…ed to adjust the project return (or project cash
‡ows) to account for tax shield.
In a MM world:
(a) rd = rf (return on debt is risk free) because debt is risk free. Remember
this for later.
(b) If the …rm is already at an optimal capital structure, the incremental
…nancing on this project will use this same optimal debt ratio. That is DI = D
.
V
The above expression can be modi…ed to
W ACC
MM
´r
u
µ
¿ cD
1¡
V
¶
(0.1)
(c) Finally, if there is a personal tax disadvantage to interest income, expression
(b) needs to be modi…ed. For example, in a Miller equilibrium (where there is no
tax advantage to debt)
(1 ¡ tc )E(NOI)
> ru
I
However, we assume there is no personal tax disadvantage to interest income.
2. Write down the “usual” formula for WACC
W ACC = (1 ¡ tc )rD
rE = W ACC
rE = r
u
D
E
+ rE
D+E
D+E
(D + E)
D
¡ (1 ¡ tc )rD
E
E
µ
D
1 ¡ ¿c
D+E
¶µ
after solving for rE
¶
D+E
D
¡ (1 ¡ tc )rD
E
E
after substituting in the MM de…nition of WACC.
rE = ru
µ
¶
D
D
D
+ 1 ¡ ru ¿ c ¡ (1 ¡ tc )rD
E
E
E
rE = ru + (ru ¡ rD )(1 ¡ tc )
rE = ru + (ru ¡ rD )(1 ¡ tc )
D
E
D
E
(2)
after recalling part (1a) and substituting in rD = rf :
3. Equate the CAPM expression for rE with the MM expression for rE in (2)
rE = rf + ¯ L (rm ¡ rf ) = ru + fru ¡ rf g(1 ¡ tc )
D
E
when ¯ L is levered equity ¯
rf + ¯ L (rm ¡ rf ) = rf + ¯ u (rm ¡ rf ) + frf + ¯ u (rm ¡ rf ) ¡ rf g(1 ¡ tc )
2
D
E
after substituting in CAPM expression for ru . ¯ u is unlevered equity ¯.
¯ L = ¯ u + ¯ u (1 ¡ tc )
¯L = ¯u
D
E
·
D
1 + (1 ¡ tc )
E
or
¯u =
¯L
1 + (1 ¡ tc )D=E
This is the levering/unlevering formula.
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