The Weighted Average Cost of Capital, Perfect Capital Markets, and Project Life:
A Clarification
James A. Miles; John R. Ezzell
The Journal of Financial and Quantitative Analysis, Vol. 15, No. 3. (Sep., 1980), pp. 719-730.
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JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS
Volume XV, No. 3 , September 1980
THE WEIGHTED AVERAGE COST OF CAPITAL, PERFECT CAPITAL MARKETS,
AND PROJECT LIFE:
A CLARIFICATION
J a m e s A . Miles and J o h n R . E z z e l l *
For f i n a n c i a l management t o make wealth maximizing c a p i t a l budgeting d e c i s i o n s , a model t h a t w i l l determine c o r r e c t l y t h e market v a l u e of a p r o j e c t ' s
l e v e r e d c a s h flows i s r e q u i r e d .
A c a p i t a l budgeting model should account n o t
only f o r t h e e f f e c t s of t h e investment d e c i s i o n , b u t a l s o f o r t h e e f f e c t s of
t h e f i n a n c i n g d e c i s i o n and t h e i n t e r a c t i o n s between t h e two d e c i s i o n s .
In per-
f e c t c a p i t a l markets a l l t h e e f f e c t s of t h e f i n a n c i n g d e c i s i o n p e r t a i n t o t h e
t a x s h i e l d c r e a t e d by d e b t f i n a n c i n g .
Thus, a s o r i g i n a l l y shown by Modigliani
and M i l l e r [81, t h e v a l u e of a p r o j e c t ' s l e v e r e d c a s h flow stream e q u a l s t h e
market v a l u e t h e stream would have i f it were unlevered p l u s t h e market v a l u e
of t h e stream of t a x s a v i n g s on i n t e r e s t payments a s s o c i a t e d w i t h t h e d e b t employed t o f i n a n c e t h e p r o j e c t .
While t h i s r e s u l t i s completely g e n e r a l w i t h
r e s p e c t t o t h e s p e c i f i c p r o c e s s e s u t i l i z e d by t h e market t o v a l u e t h e two components, MM s p e c i f i e d t h e v a l u e of t h e unlevered component a s t h e p r e s e n t v a l u e
of t h e unlevered c a s h flows d i s c o u n t e d a t t h e a p p r o p r i a t e r i s k a d j u s t e d unleve r e d c o s t of c a p i t a l and they s p e c i f i e d t h e v a l u e of t h e t a x s a v i n g s component
a s t h e p r e s e n t v a l u e of t h e t a x s h i e l d on i n t e r e s t d i s c o u n t e d a t t h e c o s t of
debt.'
Accordingly, t h e v a l u e of a p r o j e c t ' s l e v e r e d cash flows i s s p e c i f i e d
a s t h e sum of t h e s e two p r e s e n t v a l u e s , one r e p r e s e n t i n g t h e e f f e c t s of t h e i n vestment d e c i s i o n and t h e o t h e r c a p t u r i n g t h e e f f e c t s of t h e f i n a n c i n g d e c i s i o n .
The MM v a l u a t i o n model has been extended t o normative c a p i t a l budgeting analy-
s i s by Myers [91 i n terms of t h e a d j u s t e d p r e s e n t v a l u e (APV) model.
I n both t h e l i t e r a t u r e and p r a c t i c e of c a p i t a l budgeting i s observed, howe v e r , a p r e f e r e n c e f o r s p e c i f y i n g t h e v a l u e of t h e l e v e r e d c a s h flows i n terms
of a s i n g l e p r e s e n t v a l u e t h a t r e f l e c t s t h e combined e f f e c t s of both t h e i n v e s t ment and f i n a n c i n g d e c i s i o n s .
*
T h i s approach i s u s u a l l y o p e r a t i o n a l i z e d by
U n i v e r s i t y o f G e o r g i a and P e n n s y l v a n i a S t a t e U n i v e r s i t y , r e s p e c t i v e 1y
.
'see Modigliani and M i l l e r [81 f o r a d i s c u s s i o n of why i t might be approp r i a t e t o d i s c o u n t t h e t a x s h i e l d on i n t e r e s t a t t h e c o s t of d e b t . A condit i o n s u f f i c i e n t t o j u s t i f y t h i s v a l u a t i o n method i s t h a t b o t h t h e f i r m ' s d e b t
and t h e t a x s h i e l d s a r e r i s k l e s s .
d i s c o u n t i n g t h e unlevered c a s h flows a t a r a t e s p e c i f i e d a s a weighted average
T h i s o p e r a t i o n a l model of
of t h e f i r m ' s a f t e r - t a x c o s t s of d e b t and e q u i t y . 2
p r o j e c t v a l u a t i o n i s f r e q u e n t l y r e f e r r e d t o a s t h e "textbook" approach.
The p o p u l a r i t y of t h e textbook approach can probably be a t t r i b u t e d mainly
t o two f a c t o r s .
F i r s t , i f t h e p r o j e c t i n q u e s t i o n i s of t h e same r i s k c l a s s
a s t h e f i r m ' s p o r t f o l i o of e x i s t i n g p r o j e c t s , t h e c o s t s of d e b t and e q u i t y can
b e e s t i m a t e d on t h e b a s i s of t h e observed market r a t e s of r e t u r n on t h e f i r m ' s
d e b t and e q u i t y s e c u r i t i e s .
The f i r m ' s unlevered c o s t of c a p i t a l , on t h e o t h e r
hand, has no d i r e c t l y o b s e r v a b l e market c o u n t e r p a r t .
Second, t h e textbook ap-
proach f a c i l i t a t e s d e c e n t r a l i z e d c a p i t a l e x p e n d i t u r e a n a l y s e s and c h o i c e s where
t h e f i n a n c i n g and investment d e c i s i o n s a r e o r g a n i z a t i o n a l l y s e p a r a t e d .
Thus,
lower l e v e l managers a r e provided w i t h a s i n g l e d i s c o u n t r a t e which i s intended
t o r e f l e c t not only t h e p r o j e c t ' s operating r i s k , but a l s o the f i r m ' s financing
p o l i c i e s and which i s t o be used t o e v a l u a t e ,
a t a decentralized level, the
f i r m ' s investment o p p o r t u n i t i e s .
I n r e c e n t y e a r s , however, a p e r s i s t e n t c o n t r o v e r s y i n t h e c a p i t a l budgeti n g l i t e r a t u r e has been concerned w i t h t h e e f f e c t of p r o j e c t l i f e on t h e v a l i d i t y of t h e textbook approach t o p r o j e c t a n a l y s i s .
Although i t i s u s u a l l y ac-
c e p t e d t h a t t h e textbook weighted average c o s t of c a p i t a l (WACC) i s an approp r i a t e d i s c o u n t r a t e f o r e i t h e r of t h e p o l a r assumptions of (1) a one-year proj e c t l i f e o r ( 2 ) l e v e l p e r p e t u a l p r o j e c t c a s h f l o w s , a number of a u t h o r s have
argued t h a t t h e textbook approach does n o t g e n e r a l l y p r o v i d e c o r r e c t v a l u a t i o n s
of uneven f i n i t e c a s h f l o w s . 3
I n t h i s paper we examine t h e v a l i d i t y of t h e
textbook approach f o r uneven f i n i t e c a s h flows i n t h e c o n t e x t of t h e MM p e r f e c t
c a p i t a l markets r e s u l t
t h a t t h e market v a l u e of l e v e r e d cash flows i s e q u a l
t o t h e unlevered v a l u e p l u s t h e v a l u e of t a x s a v i n g s due t o d e b t f i n a n c i n g .
Our a n a l y s i s shows t h a t i f t h e unlevered c o s t of c a p i t a l , t h e c o s t of d e b t , t h e
t a x r a t e , and t h e market v a l u e l e v e r a g e r a t i o a r e c o n s t a n t f o r t h e d u r a t i o n of
a p r o j e c t , t h e n t h e v a l u e of a p r o j e c t ' s l e v e r e d c a s h flows can be o b t a i n e d by
d i s c o u n t i n g t h e unlevered cash flows a t a r a t e :
1. t h a t i s i n v a r i a n t w i t h r e s p e c t t o t h e time p a t t e r n and d u r a t i o n
of t h e l e v e r e d c a s h flows and
2.
t h a t i s e q u a l t o t h e textbook WACC.
Thus o u r a n a l y s i s i m p l i e s t h a t t h e textbook approach i s a s p e c i a l c a s e of t h e
L
See N a n t e l l and Carlson [ l o ] f o r a d i s c u s s i o n of c o s t of c a p i t a l s p e c i f i c a t i o n s . They r e f e r t o t h e textbook c o s t of c a p i t a l a s t h e "modern" s p e c i f i c a tion.
3
S e e , f o r example [ l l ,
[51, [91, and [111.
MM v a l u a t i o n r e s u l t and M y e r ' s APV c a p i t a l b u d g e t i n g r u l e .
4
The c r i t i c a l a s s u m p t i o n f o r e s t a b l i s h i n g t h e v a l i d i t y of t h e t e x t b o o k app r o a c h i s concerned w i t h t h e f i r m ' s f i n a n c i n g p o l i c y .
n o t an i s s u e .
The p r o j e c t ' s l i f e i s
When management a c t s t o m a i n t a i n a c o n s t a n t d e b t t o t o t a l v a l u e
r a t i o , i n t e r m s o f r e a l i z e d market v a l u e s , t h e i n v e s t m e n t d e c i s i o n i m p a c t s upon
t h e r i s k i n e s s a s w e l l a s t h e magnitude o f f u t u r e t a x s h i e l d s c r e a t e d by d e b t
financing.
Even though t h e f i r m might i s s u e r i s k l e s s d e b t , i f f i n a n c i n g p o l i c y
i s t a r g e t e d t o r e a l i z e d market v a l u e s , t h e amount o f d e b t o u t s t a n d i n g i n f u t u r e
p e r i o d s i s n o t known w i t h c e r t a i n t y ( u n l e s s t h e i n v e s t m e n t i s r i s k l e s s ) and,
c o n s e q u e n t l y , t h e magnitude o f t h e t a x s h i e l d s c a n n o t b e known w i t h c e r t a i n t y .
5
I f , f o r example, r e a l i z e d market v a l u e s a r e below o r i g i n a l e x p e c t a t i o n s , t h e n
t h e amount o f o u t s t a n d i n g d e b t w i l l be a d j u s t e d a c c o r d i n g l y c r e a t i n g a reduct i o n i n t h e magnitude o f r e a l i z e d t a x s a v i n g s from o r i g i n a l e x p e c t a t i o n s .
In
t h i s p a p e r we e s t a b l i s h t h e c o r r e c t l i n k a g e between t h e r i s k i n e s s of t h e i n v e s t ment, a s embodied i n t h e u n l e v e r e d d i s c o u n t r a t e , and t h e r i s k i n e s s o f t h e f u t u r e t a x s a v i n g s on i n t e r e s t payments when t h e market v a l u e l e v e r a g e r a t i o i s
held constant.
This c o r r e c t linkage provides t h e b a s i s f o r e s t a b l i s h i n g t h e
v a l i d i t y o f t h e t e x t b o o k WACC i n p e r f e c t c a p i t a l m a r k e t s .
S e c t i o n I o f t h e p a p e r i d e n t i f i e s t h e assumptions used i n S e c t i o n I1 t o
d e r i v e a model f o r o b t a i n i n g t h e v a l u e o f a l e v e r e d s t r e a m by d i s c o u n t i n g t h e
u n l e v e r e d s t r e a m by a s i n g l e d i s c o u n t r a t e .
c o u n t r a t e i s e q u a l t o t h e t e x t b o o k WACC.
S e c t i o n I11 p r o v e s t h a t t h i s d i s F i n a l l y , S e c t i o n I V summarizes t h e
paper.
4A number of w r i t e r s i n c l u d i n g Bar-Yosef
[21 , E z z e l l and P o r t e r [61 , and
Linke and K i m [71 have been a b l e t o show t h a t a c o n s t a n t c o s t o f e q u i t y , a cons t a n t c o s t of d e b t , a c o n s t a n t market v a l u e l e v e r a g e r a t i o , and a c o n s t a n t
weighted a v e r a g e c o s t o f c a p i t a l a r e i n t e r n a l l y c o n s i s t e n t c o n d i t i o n s f o r any
t i m e p a t t e r n o f c a s h f l o w s . But t h e s e a n a l y s e s d i d n o t a t t e m p t t o e s t a b l i s h
l i n k a g e between t h e MM p e r f e c t market v a l u a t i o n model and t h e t e x t b o o k weighted
a v e r a g e c o s t o f c a p i t a l . A l s o s e e Beranek [41 f o r t h e d e r i v a t i o n of a c a p i t a l
b u d g e t i n g model employing a book v a l u e weighted a v e r a g e c o s t of c a p i t a l .
5That f u t u r e t a x s h i e l d s a r e r i s k y when management a c t s t o m a i n t a i n a cons t a n t d e b t - t o - t o t a l v a l u e r a t i o was r e c o g n i z e d by Myers who n o t e d " . . . t h a t a l though t h e t a x s h i e l d a s s o c i a t e d w i t h any d e b t i n s t r u m e n t i s s a f e , t h e aggreg a t e value of t h e instrument o b t a i n a b l e i s uncertain."
( [ 9 , p . 221) For h i s
a n a l y s i s , however, Myers was a p p a r e n t l y u n w i l l i n g t o make t h e s t r o n g r e b a l a n c i n g assumption n e c e s s a r y t o m a i n t a i n a c o n s t a n t market v a l u e l e v e r a g e r a t i o .
I.
The Assumptions
We s h a l l assume t h e f o l l o w i n g :
1. C a p i t a l Markets a r e p e r f e c t .
Consequently, t h e market v a l u e of any
l e v e r e d c a s h flow stream e q u a l s t h e market v a l u e of t h e unlevered component p l u s t h e market v a l u e of t h e t a x s a v i n g s on i n t e r e s t payments.
To be a s g e n e r a l a s p o s s i b l e , we e x p r e s s t h e v a l u e of t h e l e v e r e d
L stream, Vk, a s
where
X
i = t h e expected unlevered cash flow a t time i ;
= t h e v a l u e of o u t s t a n d i n g d e b t a t time i-1;
Bi-l
r
= t h e c o s t of d e b t ;
T
= t h e f i r m ' s income t a x r a t e ;
dU
ij
= t h e market r a t e a p p r o p r i a t e f o r d i s c o u n t i n g t h e time i expected
unlevered c a s h flow i n p e r i o d j where f o r each i = k + 1,
k + 2,
i; and
T, j = k + l , k + 2 ,
...
d:.
...,
= t h e market r a t e a p p r o p r i a t e f o r d i s c o u n t i n g t h e time i expected
,T
t a x s a v i n g s i n p e r i o d j where f o r each i = k + 1, k + 2 ,
j = k + l , k + 2 ,
i.
...
...,
I n e q u a t i o n ( I ) , k r e p r e s e n t s t h e p o i n t i n time a t which market v a l u e
i s r e a l i z e d , t h e i t s r e p r e s e n t t h e p o i n t s i n time f o l l o w i n g k a t which
c a s h flows a r e r e a l i z e d , and t h e j ' s r e p r e s e n t t h e p e r i o d s between k
and each i .
2 . A t any time k , p o i s t h e a p p r o p r i a t e r a t e f o r d i s c o u n t i n g t h e time i
expected unlevered cash flow i n p e r i o d j where p
3.
5
i
is referred t o as
That i s , we assume, f o l l o w i n g MM,
" t h e unlevered c o s t of c a p i t a l . "
t h a t dyj = p o for 0 < k < j
0
5
T.
The f i r m m a i n t a i n s a c o n s t a n t l e v e r a g e r a t i o , L , s o t h a t B
L
i-1 = LVi-l.
I f , a t t h e end of any p e r i o d , t h e d e b t t o t o t a l v a l u e r a t i o does n o t
e q u a l L , we assume t h a t t h e f i r m u n d e r t a k e s f i n a n c i a l t r a n s a c t i o n s t o
6
r e s t o r e t h e r a t i o t o L.
'whether m a i n t a i n i n g a c o n s t a n t market v a l u e l e v e r a g e r a t i o , L , i s i n some
s e n s e t h e " b e s t " f i n a n c i n g p o l i c y i s n o t o u r concern. We make t h i s assumption
simply because t h e textbook weighted average c o s t of c a p i t a l i s s p e c i f i e d i n
4. At time k = i-1, the appropriate rate for discounting the time i tax
That is, d. = r for j = i = k + 1.
11 In view of assumption 3, it is only at time i-1 that B
is known for i-1 certain. Thus, unlike MM, we make no assumptions pertaining to the savings is r, the cost of debt.
value of dT for i > k + 1; rather we allow the appropriate values to
ij be deduced from the analysis. The objective of our analysis is to determine the composite set of rates pij
that will satisfy *
where p . is the appropriate rate for discounting the time i unlevered cash flow
11 Our analysis will show that in period j to obtain the levered market value, vL
0' if the firm maintains a constant leverage ratio in terms of realized market *
values, p . . is constant for each and every i and j and is equal to the so-called 13
textbook formulation of the weighted average cost of capital. 11.
Analysis of the Investment-Financing Interactions A clear understanding of the meaning of assumption 2 is critical to under- standing our analysis of the interactions between the firm's investment and Letting V [ I represent a generalized value operator
k that generates the time k market value of the cash flow stream within the financing decisions.
brackets, assumption 2 implies that -
X. for 0 (k
< i (T
since p
is the "correct" discount rate for all future periods.
0 Equation (3) in conjunction with the value additivity principle imples that time k.
vUk
"/
represents the value of the unlevered component at
Equations (3) and (4) together imply for 0 ( k < T where
Footnote 6 continued: terms of a constant L. Therefore, for an analysis to provide a meaningful basis on which to evaluate the conceptual validity of the textbook specifica- tion, L must be maintained constant. We should not be surprised to "discover" that the textbook approach does not give correct solutions when other financing patterns are followed. (5)
where 0
2
k < T-1
The v a l u a t i o n p r o c e s s embodied i n e q u a t i o n s ( 3 ) , ( 4 ) , a n d ( 5 ) i m p l i e s t h a t a t
t i m e T-1,
t h e v a l u e o f t h e u n l e v e r e d component i s
(6)
and a t t i m e T-2,
S u b s t i t u t i n g equation (6) i n t o ( 7 ) gives us
C o n t i n u i n g t h i s s i m p l e backward i t e r a t i o n p r o c e d u r e back t o t i m e 0 r e s u l t s i n
-
Since equation ( 9 ) is a d i r e c t implication of assumption 2 , equations ( 3 )
may seem t o b e l a b o r t h e o b v i o u s .
(8)
We i n c l u d e them t o s e t f o r t h c l e a r l y t h e
v a l u a t i o n p r o c e s s i m p l i e d by a s s u m p t i o n 2 and t h e r e b y t o s e t t h e s t a g e f o r d e a l ing w i t h t h e l e s s obvious problem of determining t h e v a l u e of t h e t a x s a v i n g s
on i n t e r e s t payments.
Employing o u r t h i r d a s s u m p t i o n t h a t t h e l e v e r a g e r a t i o , i n t e r m s o f r e a l i z e d m a r k e t v a l u e s , i s h e l d c o n s t a n t , e q u a t i o n (1) c a n now b e r e w r i t t e n a s
I t i s o b v i o u s from e q u a t i o n ( 1 0 ) t h a t i f t h e l e v e r a g e r a t i o i s h e l d c o n s t a n t i n
t e r m s of L =
Bi-l
,
vL1-1
.
t h e m a g n i t u d e o f t h e t a x s a v i n g s a t a n y t i m e i d e p e n d s upon
t h e v a l u e o f t h e l e v e r e d cash flow stream a t time i - 1 ,
vL1-1r b u t
L Vi-L
T upon t h e v a l u e o f t h e t a x s a v i n g s component a t t i m e ( i - l ) ,V:
Thus, $ c a n - 1-1' n o t g e n e r a l l y b e d e t e r m i n e d by s i m p l y a d d i n g a n i n d e p e n d e n t l y d e t e r m i n e d vT 0
0 when t h e f i r m a c t s t o m a i n t a i n a c o n s t a n t m a r k e t v a l u e l e v e r a g e r a t i o .
'dU
Rather
L
To account f o r t h e
it i s n e c e s s a r y t o determine V
and vT simultaneously.
O'L
0
we w i l l employ t h e backward i t e r a t i o n proceinterdependence between V Tand V
0
0
d u r e which has been shown t o be i m p l i c i t i n e q u a t i o n ( 9 ) .
We i n i t i a t e t h e backward i t e r a t i o n procedure by e x p r e s s i n g t h e v a l u e of
t h e l e v e r e d stream a t T-1 a s
X
(11)
+
L
T~LV
T- 1
where e q u a t i o n ( 3 ) i s u t i l i z e d t o v a l u e t h e unlevered component by d i s c o u n t i n g
t h e unlevered component a t t h e f i r m ' s unlevered c o s t of c a p i t a l , and assumption
4 i s u t i l i z e d t o d i s c o u n t t h e t a x s a v i n g s a t t h e f i r m ' s c o s t of d e b t .
e q u a t i o n (11) f o r
~ i - ~
Solving
g i v e s us
Equation ( 1 2 ) r e v e a l s t h a t
i s t h e a p p r o p r i a t e d i s c o u n t f a c t o r f o r v a l u i n g a l e v e r e d stream a t time T-1.
*
As
a s p e c i a l c a s e , i f ~ = le q,u a t i o n (13) shows t h a t p TT i s t h e c o r r e c t r i s k a d j u s t e d d i s c o u n t r a t e f o r t h e s i n g l e p e r i o d c a s h flow.
A t time T-2,
t h e v a l u e of t h e l e v e r e d cash flow i s
L
i s t h e a p p r o p r i a t e r a t e f o r d i s c o u n t i n g t h e v a l u e of t h e time
(T-1) (T-1)
L
The c o r r e c t s p e c i f i c a t i o n of d (T-1) (T-1)
T-1 l e v e r e d market v a l u e t o time T-2.
where d
i s a s t r a i g h t f o r w a r d i m p l i c a t i o n of o u r assumptions and a n a l y s i s t o t h i s p o i n t .
We can use e q u a t i o n ( 6 ) t o s u b s t i t u t e
vU
T- 1
(1 +
f o r ]i i n e q u a t i o n (12)
T
obtaining
Equation (15) shows t h a t
constant scale factor.
u
a r e q u a n t i t i e s t h a t d i f f e r o n l y by a
T- 1
L are
T h i s means t h a t a t any time k < T-1, v ~ 1- and vU
T- 1
vL
T-1
and V
p e r f e c t l y c o r r e l a t e d random v a r i a b l e s and must be valued a c c o r d i n g l y .
(5) shows t h a t vU
Equation
should be d i s c o u n t e d a t t h e f i r m ' s unlevered c o s t of c a p i t a l
T- 1
L
Consequently, VT-l should a l s o be d i s f o r v a l u a t i o n a t any time p r i o r t o T-1.
counted a t p
0
implying t h a t
S u b s t i t u t i n g e q u a t i o n s (12) and (16) i n t o e q u a t i o n (14) and s o l v i n g f o r V
L
T- 2
g i v e s us
Equation (17) r e v e a l s t h a t
Continuing w i t h s i m i l a r a n a l y s i s back t o time 0 shows t h a t
(19)
where
o r , solving f o r p
i s independent of t h e magnitude and timing
0
of t h e unlevered c a s h flow s t r e a m , t h e n t h e c o r r e c t d i s c o u n t r a t e f o r v a l u i n g
Equation ( 2 0 ) e s t a b l i s h e s t h a t i f p
a l e v e r e d stream i s a l s o independent of t h e magnitude and timing of t h e unleve r e d stream.
Thus, p
111.
i s dependent only upon p o l
7,
L , and r .
The Weighted Average Cost of C a p i t a l
The n e x t s t e p i s t o prove t h a t t h e textbook f o r m u l a t i o n of t h e weighted
a v e r a g e c o s t of c a p i t a l f o l l o w s from o u r p r e v i o u s a n a l y s i s .
such t h a t 0
5
k < T.
The expected end-of-period
L e t k b e any p e r i o d
c a s h flow t o e q u i t y can be
written a s
The f i r m r e c e i v e s t h e unlevered component p l u s t h e t a x s a v i n g s on i n t e r e s t payments.
Then, i n t e r e s t payments a r e deducted.
I f Bk+l < Bk,
t h e amobnt of d e b t
o u t s t a n d i n g i s reduced and c a s h i s used t o r e t i r e o u t s t a n d i n g d e b t .
o t h e r hand, Bk+l
I f , on t h e
> Bk, t h e r e i s a n e t i n c r e a s e i n c a s h f l o w .
Finally, equity
L
owns (1-L)vL
of t h e s t r e a m ' s f u t u r e v a l u e .
S u b s t i t u t i n g LvL f o r B and LV
k+l
k
k
k+l
f o r B ~ + we
~ ,can r e a r r a n g e terms s o t h a t t h e e x p e c t e d end-of-period c a s h flow
t o equity i s
Dividing t h i s e x p r e s s i o n by (1-L)vL t h e time k v a l u e of e q u i t y , y i e l d s
k'
i s t h e e x p e c t e d r a t e of r e t u r n on e q u i t y f o r t h e ( k + 1) p e r i o d .
(k+l)
*
The f i r s t term on t h e r i g h t - h a n d s i d e o f e q u a t i o n ( 2 1 ) i s simply (1 + p ) / ( l - L )
where p S
a s i m p l i e d by t h e f a c t t h a t p
stream.
i s t h e r e q u i r e d r a t e of r e t u r n f o r t h e l e v e r e d
Hence, e q u a t i o n ( 2 1 ) can be w r i t t e n a s
Equation ( 2 2 ) p r o v e s t h a t t h e c o s t of e q u i t y i s independent of t h e p a r t i c u l a r
time p e r i o d .
Solving equation (22) f o r p
*
yields
Equation ( 2 3 ) i s t h e t e x t b o o k f o r m u l a t i o n of t h e weighted a v e r a g e c o s t of
capital.
IV.
Summary
B a s i c a l l y , two approaches t o t h e v a l u a t i o n of a l e v e r e d s t r e a m have appeared i n t h e l i t e r a t u r e .
The t e x t b o o k approach assumes a c o n s t a n t c o s t of
e q u i t y , a c o n s t a n t c o s t of d e b t , and a c o n s t a n t l e v e r a g e r a t i o .
The Modigliani
and M i l l e r approach assumes a c o n s t a n t u n l e v e r e d c o s t of c a p i t a l and a c o n s t a n t
c o s t of d e b t .
The a d j u s t e d p r e s e n t v a l u e model developed by Myers i s a norma-
t i v e i m p l i c a t i o n o f t h e MM v a l u a t i o n approach.
Our a n a l y s i s shows t h a t t h e
t e x t b o o k approach i s a l s o an i m p l i c a t i o n of t h e MM approach and i s , t h e r e f o r e ,
a s p e c i a l c a s e of Myers' MM-based APV model.
I n p a r t i c u l a r , we have shown i n e q u a t i o n s ( 1 9 ) , ( 2 0 ) , and (23) t h a t i n
p e r f e c t c a p i t a l markets where t h e u n l e v e r e d d i s c o u n t r a t e ( p ) , t h e c o s t of
0
d e b t ( r ) , t h e t a x r a t e ( T ) , and t h e market v a l u e l e v e r a g e r a t i o a r e c o n s t a n t
f o r t h e d u r a t i o n o f an unlevered cash flow stream, t h e v a l u e of t h e l e v e r e d
c a s h f l o w s t r e a m c a n b e o b t a i n e d by d i s c o u n t i n g t h e u n l e v e r e d component a t a
c o n s t a n t r a t e t h a t i s e q u a l t o t h e t e x t b o o k WACC.
I n t h e d e r i v a t i o n o f equa-
t i o n s ( 1 9 ) , ( 2 0 ) , and ( 2 3 ) , no a s s u m p t i o n s were made r e g a r d i n g e i t h e r t h e t i m e
p a t t e r n o r t h e d u r a t i o n o f t h e u n l e v e r e d component.
Thus, t h i s p a p e r h a s demon-
s t r a t e d t h a t we c a n d i s p e n s e w i t h t h e two p o l a r s u f f i c i e n t a s s u m p t i o n s r e g a r d i n g p r o j e c t l i f e g e n e r a l l y i d e n t i f i e d w i t h t h e v a l i d i t y o f t h e t e x t b o o k WACC:
(1) t h e p r o j e c t h a s a o n e - y e a r l i f e ; o r ( 2 ) t h e p r o j e c t ' s c a s h f l o w s a r e a l e v e l
perpetuity.
The c r i t i c a l a s s u m p t i o n p e r t a i n s n o t t o p r o j e c t l i f e , b u t r a t h e r
t o t h e f i n a n c i n g p o l i c y f o l l o w e d by t h e f i r m .
Our a n a l y s i s assumes ( a s s u m p t i o n 3 ) t h a t t h e f i r m m a i n t a i n s a c o n s t a n t
l e v e r a g e r a t i o , L = Bi l / l ,
by a d j u s t i n g t h e amount o f o u t s t a n d i n g d e b t ,
L t o t h e r e a l i z e d l e v e r e d m a r k e t v a l u e , ViPl, i n e a c h p e r i o d . F o r a one1-1'
y e a r p r o j e c t l i f e t h e l e v e r a g e r a t i o i s L = B /vL a n d i s c o n s t a n t by d e f i n i t i o n
0 0
s i n c e n e i t h e r an o p p o r t u n i t y n o r a r e q u i r e m e n t a r i s e s t o r e b a l a n c e t h e c a p i t a l
B,
structure.
However, s i n c e f o r any c a s h f l o w d u r a t i o n e x c e e d i n g o n e y e a r , i n -
c l u d i n g a l e v e l p e r p e t u i t y , r e a l i z e d and e x p e c t e d l e v e r e d m a r k e t v a l u e s may
d i f f e r a s a c o n s e q u e n c e o f t h e r i s k i n h e r e n t i n t h e u n l e v e r e d component, a c t i v e
L
r e b a l a n c i n g by t h e f i r m i s r e q u i r e d t o m a i n t a i n a c o n s t a n t L = Bi-l/Vi-l.
This
r e b a l a n c i n g r e q u i r e m e n t i s o b s c u r e d when a l e v e l p e r p e t u i t y i s assumed.
In
t h i s c a s e , s i n c e b o t h t h e e x p e c t e d v a l u e o f d e b t , B, and t h e e x p e c t e d l e v e r e d
L
, a r e c o n s t a n t by i m p l i c a t i o n , t h e l e v e r a g e r a t i o , L = B/V L , i s a l s o
value, V
c o n s t a n t by i m p l i c a t i o n .
Nevertheless, i m p l i c i t i n t h i s constant leverage
r a t i o i s t h e rebalancing p o l i c y expressed i n terms of assumption 3 .
Therefore,
t h a t t h e t e x t b o o k WACC y i e l d s c o r r e c t v a l u a t i o n s f o r e i t h e r a s i n g l e - p e r i o d p r o j e c t o r a p r o j e c t with l e v e l p e r p e t u a l cash flows i s a consequence, n o t of proj e c t l i f e p e r s e , a s h a s been a r g u e d i n t h e l i t e r a t u r e , b u t r a t h e r o f m a i n t a i n ing i n d i r e c t l y a constant leverage r a t i o .
F u r t h e r m o r e , t h e f a i l u r e o f t h e lit-
e r a t u r e g e n e r a l l y t o p r o d u c e c o r r e c t v a l u a t i o n s f o r uneven f i n i t e c a s h f l o w s
with t h e textbook approach i s likewise n o t a matter of p r o j e c t l i f e , b u t i n s t e a d
t h e r e s u l t of assuming a d e b t t r a n s a c t i o n schedule t h a t allows t h e l e v e r a g e
r a t i o t o vary.
The c h o i c e between t h e t e x t b o o k and MM-APV a p p r o a c h e s t o c a p i t a l b u d g e t i n g
a n a l y s i s d e p e n d s on w h e t h e r i t i s (1) t h e d e b t - r e p a y m e n t s c h e d u l e o r ( 2 ) t h e
l e v e r a g e r a t i o t h a t i s exogenous w i t h r e s p e c t t o r e a l i z e d m a r k e t v a l u e s .
The
g e n e r a l p e r f e c t c a p i t a l m a r k e t s MM-APV model i s n e u t r a l w i t h r e s p e c t t o t h e
f i r m ' s f i n a n c i n g p o l i c y , b u t i t r e q u i r e s e x p l i c i t v a l u a t i o n of t h e t a x b e n e f i t s
of debt financing.
The t e x t b o o k a p p r o a c h i s a s p e c i a l c a s e of t h e g e n e r a l MM-
APV model when t h e l e v e r a g e r a t i o i s exogenous,
and i t e n a b l e s t h e f i r m t o v a l u e
t h e d e b t - r e l a t e d t a x b e n e f i t s i m p l i c i t l y simply by d i s c o u n t i n g t h e unlevered
c a s h flows a t t h e WACC.
However, t h i s approach assumes t h a t t h e f i r m i s a b l e
and w i l l i n g t o m a i n t a i n t h e l e v e r a g e r a t i o a t L = B/V by i s s u i n g more d e b t
when times a r e " b e t t e r " t h a n expected and r e t i r i n g d e b t when times a r e "worse"
t h a n expected.
With a n y o t h e r d e b t p o l i c y , t h e p e r i o d i c d e b t t r a n s a c t i o n s a r e
e i t h e r p a r t i a l l y o r completely exogenous w i t h r e s p e c t t o r e a l i z e d market v a l u e s
and, consequently, t h e l e v e r a g e r a t i o w i l l n o t be c o n s t a n t e x c e p t by chance.
Equations ( 2 1 ) and (22) show t h a t t h e c o s t of e q u i t y i n any g i v e n p e r i o d i s a
f u n c t i o n of t h a t p e r i o d ' s l e v e r a g e r a t i o .
Thus, we cannot determine t h e c o s t
of e q u i t y u n t i l we know t h e l e v e r a g e r a t i o , and we cannot determine t h e l e v e r age r a t i o u n t i l we know t h e l e v e r e d market v a l u e .
v a l u e i s t h e aim of o u r a n a l y s i s .
But, t h e l e v e r e d market
Thus, u n l e s s t h e l e v e r a g e r a t i o and, conse-
q u e n t l y , t h e c o s t of e q u i t y a r e exogenous t o r e a l i z e d market v a l u e s , t h e t e x t book WACC i s n o t an a p p r o p r i a t e d i s c o u n t r a t e f o r normative c a p i t a l budgeting
analysis.
Indeed, i t would seem t h a t t h e p r o s p e c t s of f i n d i n g any normatively
u s e f u l d i s c o u n t r a t e s p e c i f i e d i n terms of an average of e q u i t y and d e b t r a t e s
a r e remote i f t h e f i r m ' s d e b t t r a n s a c t i o n schedule i s exogenous w i t h r e s p e c t
t o r e a l i z e d market v a l u e s .
T h e r e f o r e , when i t i s assumed t h a t t h e d e b t t r a n s -
a c t i o n schedule r a t h e r t h a n t h e l e v e r a g e r a t i o i s exogenous, t h e a n a l y s i s of
c a p i t a l budgeting o p t i o n s should be conducted i n terms of t h e g e n e r a l MM-APV
v a l u a t i o n model.
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I m p l i c a t i o n s f o r C a p i t a l B u d g e t i n g . " J o u r n a l o f F i n a n c e , V o l . 29 (March
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[ill R e i l l y , R .
R.,
and W .
E . Wecker.
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p p . 123-126.