812  Appendix C
pension assets and risk of benefit obligations are evaluated separately. To do 
this, start with the portfolio equation for beta:
V
V
V
V
V
V
D
V
E
V
u
u
pa
pa
pbo
pbo
d
e
β
β
β
β
β
+
=
+
+
where V
V
pa
pbo
=
=
value of pension assets
present value of pension benefit obligations
beta of pension assets
beta of pension benefi
β
β
pa
pbo
=
=
t obligations
sum of debt, equity, and benefit obligation
V =
s
Next, multiply both sides by V:
V
V
V
D
E
u
u
pa
pa
pbo
pbo
d
e
β
β
β
β
β
+
=
+
+
Subtract the term related to pension assets from both sides of the equation:
V
V
D
E
V
u
u
pbo
pbo
d
e
pa
pa
β
β
β
β
β
=
+
+
−
To isolate βu, divide both sides by Vu. This leads to the general equation for 
estimating unlevered beta, inclusive of pensions:
β
β
β
β
β
u
pbo
u
pbo
d
u
d
e
u
e
pa
u
pa
V
V
V
V
V
V
V
V
=
+
+
−
If debt and pension liabilities have the same beta, simplify the last equation 
by combining terms:
β
β
β
β
u
pbo
u
d
u
e
pa
u
pa
D
V
V
E
V
V
V
=
+
+
−
Chapter 23 discusses how to incorporate pensions into a valuation. In 
Exhibit 23.5, we use the equation above to unlever beta for a set of food 
manufacturers. Given the small size of each company’s pension relative to 
the respective company’s market value of equity, the difference in unlevered 
beta with and without the pension adjustment is minor. We therefore recom-
mend adjusting beta for pensions only when pension assets and liabilities 
are substantial. In most situations, the unlevering equations that classify the 
unfunded portion of pensions as debt will suffice.

813
Appendix D
Leverage and the 
Price-to-Earnings Multiple
This appendix demonstrates that the price-to-earnings (P/E) multiple of a le-
vered company depends on its unlevered (all-equity) P/E, its cost of debt, 
and its debt-to-value ratio. When the unlevered P/E is less than 1/kd (where 
kd equals the cost of debt), the P/E falls as leverage rises. Conversely, when 
the unlevered P/E is greater than 1/kd, the P/E rises with increased leverage.
In this proof, we assume the company faces no taxes and no distress costs. 
We do this to avoid modeling the complex relationship between capital struc-
ture and enterprise value. Instead, our goal is to show that there is a system-
atic relationship between the debt-to-value ratio and the P/E.
Step 1: Defining Unlevered P/E
To determine the relationship between P/E and leverage, start by defining 
the unlevered P/E (PEu). When a company is entirely financed with equity, its 
enterprise value equals its equity value, and its net operating profit after taxes 
(NOPAT) equals its net income:
PE
NOPAT
ENT
u
t
V
=
+1
where
 
V
t
ENT
enterprise value
NOPAT
net operating profit after taxes i
=
=
+1
n year t + 1

814  Appendix D
This equation can be rearranged to solve for the enterprise value, which we 
will use in the next step:
V
t
u
ENT
NOPAT
PE
=
(
)
+1

(D.1)
Step 2: Linking Net Income to NOPAT
For a company partially financed with debt, net income (NI) equals NOPAT 
less after-tax interest payments. Assuming the value of debt equals its face 
value, the company’s interest expense will equal the cost of debt times the 
value of debt, which can be defined by multiplying enterprise value by the 
debt-to-value ratio:
NI
NOPAT
ENT
t
t
d
V
D
V k
+
+
=
−




1
1
Substitute Equation D.1 for the enterprise value:
NI
NOPAT
NOPA
PE
t
t
t
u
d
T
D
V k
+
+
+
=
−
(
)



1
1
1
Factor NOPAT into a single term:
NI
NOPAT
PE
t
t
u
d
D
V k
+
+
=
−








1
1 1

(D.2)
Step 3: Deriving Levered P/E
At this point, we are ready to solve for the company’s price-to-earnings ratio. 
Since P/E is based on equity values, first convert enterprise value to equity 
value. To do this, once again start with Equation D.1:
V
t
u
ENT
NOPAT
PE
=
(
)
+1
To convert enterprise value into equity value, multiply both sides by 1 minus 
the debt-to-value ratio:
V
D
V
D
V
t
u
ENT
ENT
ENT
NOPAT
PE
1
1
1
−





=
(
)
−






+
Distribute VENT into the parentheses:
V
D
D
V
t
u
ENT
ENT
NOPAT
PE
−
=
(
)
−






+1
1

Appendix D  815
Replace enterprise value (VENT) minus debt (D) with equity value (E):
E
D
V
t
u
=
(
)
−






+
NOPAT
PE
ENT
1
1
Next, use Equation D.2 to eliminate NOPATt+1:
E
D
V
D
V k
t
u
u
d
=
(
)
−




−




+
NI
PE
PE
1
1
1
Divide both sides by net income to find the levered P/E:
E
D
V
D
V k
t
u
u
u
d
NI
PE
PE
PE
+
=
−




−




1
1
At this point, we have a relationship between equity value and net income, 
which depends on the unlevered P/E, the debt-to-value ratio, and the cost of 
debt. Debt-to-value, however, is in both the numerator and the denominator, 
so it is difficult to distinguish how leverage affects the levered P/E. To elimi-
nate the debt-to-value ratio in the numerator, use a few algebraic tricks. First, 
multiply both the numerator and denominator by kd:
E
k
D
V
k
k
D
V
k
t
u
d
u
d
d
u
d
NI
PE
PE
PE
+
=
(
) −



(
)
−



(
)




1
1
Next, subtract and add 1 (a net difference of 0) in the numerator:
E
k
D
V
k
k
D
V
t
u
d
u
d
d
u
NI
PE
PE
PE
+
=
(
) −

+
−



(
)




−




1
1
1
1
(
)




kd
After separating the numerator into two distinct terms, you can eliminate 
the components of the right-hand term by canceling them with the denomina-
tor. This allows you to remove debt-to-value from the numerator:
E
k
k
D
V
k
k
t
u
d
d
u
d
d
NI
PE
PE
+
=
(
) −
−



(
)




+
1
1
1
1