806  Appendix C
where
 
k
k
u
txa
=
=
unlevered cost of equity
cost of capital for the company’s interest tax shields
cost of debt
cost of equity
k
k
d
e
=
=
The four terms in this equation represent the proportional risk of operating 
assets, tax assets, debt, and equity, respectively.
Since the cost of operating assets (ku) is unobservable, it is necessary to 
solve for it using the equation’s other inputs. The required return on tax 
shields (ktxa) also is unobservable. With two unknowns and only one equation, 
it is therefore necessary to impose additional restrictions to solve for ku. If debt 
is a constant proportion of enterprise value (i.e., debt grows as the business 
grows), ktxa equals ku. Imposing this restriction leads to the following equation:
V
V
V
k
V
V
V
k
D
D
E k
E
D
E k
u
u
txa
u
txa
u
txa
u
d
e
+
(
) +
+
(
) =
+
(
) +
+
(
)
Combining terms on the left side generates an equation for the unlevered cost 
of equity when debt is a constant proportion of enterprise value:
k
D
D
E k
E
D
E k
u
d
e
=
+
(
) +
+
(
) 
(C.3)
Since most companies manage their debt-to-value ratio to stay within a 
particular range, we believe this formula and its resulting derivations are the 
most appropriate for standard valuation.
Unlevered Cost of Equity When ktxa Equals kd
Some financial analysts set the required return on interest tax shields equal to 
the cost of debt. In this case, Equation C.2 can be expressed as follows:
V
V
V
k
V
V
V
k
D
D
E k
E
D
E k
u
u
txa
u
txa
u
txa
d
d
e
+
(
) +
+
(
) =
+
(
) +
+
(
)
To solve for ku, multiply both sides by enterprise value:
V
k
V
k
D k
E k
u
u
txa
d
d
e
(
) +
(
) =
(
) + (
)
and move Vtxa(kd) to the right side of the equation:
V
k
D
V
k
E k
u
u
txa
d
e
(
) =
−
(
)
+ (
)

Appendix C  807
To eliminate Vu from the left side of the equation, rearrange Equation C.1 to 
Vu = D - Vtxa + E, and divide both sides by this value:
k
D
V
D
V
E k
E
D
V
E k
u
txa
txa
d
txa
e
=
−
−
+
(
) +
−
+
(
) 
(C.4)
Equation C.4 mirrors Equation C.2 closely. It differs from Equation C.2 
only in that the market value of debt is reduced by the present value of ex-
pected tax shields.
Unlevered Cost of Equity When Debt is Constant
Exhibit C.1 summarizes three methods to estimate the unlevered cost of eq-
uity. The two formulas in the top row assume that the risk associated with 
interest tax shields (ktxa) equals the risk of operations (ku). When this is true, 
whether debt is constant or expected to change, the formula remains the same.
The bottom-row formulas assume that the risk of interest tax shields equals 
the risk of debt. On the left, future debt can take on any value. On the right, 
an additional restriction is imposed that debt remains constant—in absolute 
terms, not as a percentage of enterprise value. In this case, the annual interest 
payment equals D(kd), and the annual tax shield equals D(kd)(Tm). Since tax 
shields are constant, they can be valued using a constant perpetuity:
PV Tax Shields
(
) =
(
)(
) =
(
)
D k
T
k
D T
d
m
d
m
Exhibit C.1  Unlevered Cost of Equity
Note: 
ke = cost of equity
kd = cost of debt
ku = unlevered cost of equity
ktxa = cost of capital for tax shields
Tm = marginal tax rate
D = debt
E = equity
Vtxa = present value of tax shields
Tax shields have
same risk as
operating assets
 
ktxa = ku
Dollar level of
debt fluctuates
Dollar level of
debt is constant
Tax shields have
same risk
as debt
 
ktxa = kd
ku =
kd +
ke
D
D + E
D + E
E
ku =
kd +
D – Vtxa + E
D – Vtxa
D – Vtxa + E
E
ke
ku =
kd +
D (1 – Tm)
D (1 – Tm) + E
E
D (1 – Tm) + E
ke
ku =
kd +
ke
D
D + E
D + E
E
 

808  Appendix C
Consequently, Vtxa in the formula in the bottom left corner is replaced with 
D(Tm). The equation is simplified by converting D - D(Tm) into D(1 - Tm). The 
resulting equation is presented in the bottom right corner.
Levered Cost of Equity
In certain situations, you will have already estimated the unlevered cost of 
equity and need to relever the cost of equity to a new target structure. In this 
case, use Equation C.2 to solve for the levered cost of equity, ke:
V
V
V
k
V
V
V
k
D
D
E k
E
D
E k
u
u
txa
u
txa
u
txa
txa
d
e
+
(
) +
+
(
) =
+
(
) +
+
(
)
Multiply both sides by enterprise value:
V
k
V
k
D k
E k
u
u
txa
txa
d
e
(
) +
(
) =
(
) + (
)
Next, subtract D(kd) from both sides of the equation:
V
k
D k
V
k
E k
u
u
d
txa
txa
e
(
) −
(
) +
(
) =
(
)
and divide the entire equation by the market value of equity, E:
k
V
E k
D
E k
V
E
k
e
u
u
d
txa
txa
=
(
) −
(
) +
(
)
To eliminate Vu from the right side of the equation, rearrange Equation C.1 to 
Vu = D - Vtxa + E, and use this identity to replace Vu:
k
D
V
E
E
k
D
E k
V
E
k
e
txa
u
d
txa
txa
=
−
+
(
) −
(
) +
(
)
Distribute the first fraction into its component parts:
k
D
E k
V
E
k
k
D
E k
V
E
k
e
u
txa
u
u
d
txa
txa
=
(
) −
(
) +
−
(
) +
(
) 
(C.5)
Consolidating terms and rearranging leads to the general equation for the cost 
of equity:
k
k
D
E k
k
V
E
k
k
e
u
u
d
txa
u
txa
=
+
−
(
) −
−
(
) 
(C.6)