Appendix C  809
If debt is a constant proportion of enterprise value (i.e., debt grows as the 
business grows), ku will equal ktxa. Consequently, the final term drops out:
k
k
D
E k
k
e
u
u
d
=
+
−
(
)
We believe this equation best represents the relationship between the levered 
cost of equity and the unlevered cost of equity.
The same analysis can be repeated under the assumption that the risk of 
interest tax shields equals the risk of debt. Rather than repeat the first few 
steps, we start with Equation C.5:
k
D
E k
V
E
k
k
D
E k
V
E
k
e
u
txa
u
u
d
txa
txa
=
(
) −
(
) +
−
(
) +
(
)
To solve for ke, replace ktxa with kd:
k
D
E k
V
E
k
k
D
E k
V
E
k
e
u
txa
u
u
d
txa
d
=
(
) −
(
) +
−
(
) +
(
)
Consolidate like terms and reorder:
k
k
D
V
E
k
D
V
E
k
e
u
txa
u
txa
d
=
+
−
(
) −
−
(
)
Finally, further simplify the equation by once again combining like terms:
k
k
D
V
E
k
k
e
u
txa
u
d
=
+
−
−
(
)
The resulting equation is the levered cost of equity for a company whose debt 
can take any value but whose interest tax shields have the same risk as the 
company’s debt.
Exhibit C.2 summarizes the formulas that can be used to estimate the le-
vered cost of equity. The top row in the exhibit contains formulas that assume 
ktxa equals ku. The bottom row contains formulas that assume ktxa equals kd. 
The formulas on the left side are flexible enough to handle any future capital 
structure but require valuing the tax shields separately. The formulas on the 
right side assume the dollar level of debt is fixed over time.

810  Appendix C
Levered Beta
Similar to the cost of capital, the weighted average beta of a company’s as-
sets, both operating and financial, must equal the weighted average beta of 
its financial claims:
V
V
V
V
V
V
D
D
E
E
D
E
u
u
txa
u
txa
u
txa
txa
d
e
+
(
) +
+
(
) =
+
(
) +
+
(
)
β
β
β
β
Since the form of this equation is identical to the cost of capital, it is pos-
sible to rearrange the formula using the same process as previously described. 
Rather than repeat the analysis, we provide a summary of levered beta in 
Exhibit C.3. As expected, the first two columns are identical in form to Exhibit C.2, 
except that the beta (β) replaces the cost of capital (k).
By using beta, it is possible to make one additional simplification. If debt is 
risk free, the beta of debt is 0, and βd drops out. This allows us to convert the 
following general equation (when βtxa equals βu):
β
β
β
β
e
u
u
d
D
E
=
+
−
(
)
into the following:
β
β
e
u
D
E
=
+




1
Exhibit C.2  Levered 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
ke = ku +
(ku – kd)
E
D
ke = ku +
(ku – kd)
E
D – Vtxa
ke = ku +
(ku – kd )
E
D
(ku – kd )
ke = ku + (1 – Tm) E
D

Appendix C  811
This last equation is an often-applied formula for levering (and unlevering) 
beta when the risk of interest tax shields (βtxa) equals the risk of operating as-
sets (βu) and the company’s debt is risk free. For investment-grade companies, 
debt is nearly risk free, so any errors using this formula will be small. If the 
company is highly leveraged, however, errors can be large. In this situation, 
estimate the beta of debt, and use the more general version of the formula.
Unlevered Beta and Pensions
Since stockholders are responsible for future pension payments and other re-
tirement obligations, the risks associated with these employee benefits can 
affect a company’s beta. If a company has significant pensions, especially un-
funded pensions, make sure to include them in the unlevering process.
If you believe the risk of pension assets matches the risk of future obliga-
tions, only the unfunded portion of benefit obligations affects the equity beta. 
In this case, use the unlevering equations in the preceding sections, but treat 
any unfunded benefit obligations identically to debt.
If you believe the risk of pension assets does not match the risk of future 
obligations, the unlevering formulas can be reworked such that the risk of 
Exhibit C.3  Levered Beta
Note: βe = beta of equity
 
βd = beta of debt
 
βu = unlevered beta of equity
 
βtxa = beta 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
 
βtxa = βu
Dollar level of
debt fluctuates
Dollar level of
debt is constant
and debt is risky
Tax shields have
same risk
as debt
 
βtxa = βd
βe = βu +
(βu – βd)
E
D
βe = (1 +
) βu
E
D
βe = [1 + (1 – Tm)
] βu
E
D
βe = βu +
(βu – βd)
E
D – Vtxa
Debt is
risk free
βe = βu +
(βu – βd)
E
D
(βu – βd)
βe = βu + (1 – Tm) E
D