V O LU M E 2 0 | N U M B E R 2 | s p ring 2 0 0 8
Journal of
APPLIED CORPORATE FINANCE
A MO RG A N S TA N L E Y P U B L I C AT I O N
In This Issue: Valuation and Corporate Portfolio Management
Corporate Portfolio Management Roundtable
Presented by Ernst & Young
8
Panelists: Robert Bruner, University of Virginia; Robert Pozen,
MFS Investment Management; Anne Madden, Honeywell
International; Aileen Stockburger, Johnson & Johnson;
Forbes Alexander, Jabil Circuit; Steve Munger and Don Chew,
Morgan Stanley. Moderated by Jeff Greene, Ernst & Young
Liquidity, the Value of the Firm, and Corporate Finance
32
Yakov Amihud, New York University, and
Haim Mendelson, Stanford University
Real Asset Valuation: A Back-to-Basics Approach
46
David Laughton, University of Alberta; Raul Guerrero,
Asymmetric Strategy LLC; and Donald Lessard, MIT Sloan
School of Management
Expected Inflation and the Constant-Growth Valuation Model
66
Michael Bradley, Duke University, and
Gregg Jarrell, University of Rochester
Single vs. Multiple Discount Rates: How to Limit “Influence Costs”
in the Capital Allocation Process
79
The Era of Cross-Border M&A: How Current Market Dynamics are
Changing the M&A Landscape
84
Transfer Pricing for Corporate Treasury in the Multinational Enterprise
97
The Equity Market Risk Premium and Valuation of Overseas Investments
John Martin, Baylor University, and Sheridan Titman,
University of Texas at Austin
Marc Zenner, Matt Matthews, Jeff Marks, and
Nishant Mago, J.P. Morgan Chase & Co.
113
Stephen L. Curtis, Ernst & Young
Luc Soenen,Universidad Catolica del Peru, and
Robert Johnson, University of San Diego
Stock Option Expensing: The Role of Corporate Governance
122
Sanjay Deshmukh, Keith M. Howe, and
Carl Luft, DePaul University
Real Options Valuation: A Case Study of an E-commerce Company
129
Rocío Sáenz-Diez, Universidad Pontificia Comillas
de Madrid, Ricardo Gimeno, Banco de España, and
Carlos de Abajo, Morgan Stanley
Expected Inflation and the
Constant-Growth Valuation Model*
by Michael Bradley, Duke University, and Gregg A. Jarrell,
University of Rochester
T
he Constant-Growth model is a discounted cash
flow method of valuing companies and their
projects that is taught in all top-tier business
schools and used widely throughout the financial community. It is found in virtually all graduate-level
corporate finance textbooks and valuation manuals. But, as
we found during an extensive review of this literature, there
has no been careful, systematic analysis of the effects of inflation on this model.1 As we show in the pages that follow,
the failure to account properly for the effects of inflation
has led to what academics call a “misspecification,” and thus
an incorrect use, of the model in a particular set of cases—
those where a company is assumed either to make no net
new investments or to invest only in zero net present value
(NPV) projects. We show that the error produced by this
misapplication of the model is significant even at moderate
levels of expected inflation.
In addition to its reliance on operating cash flows rather
than earnings or P/E multiples, the main appeal of the
Constant-Growth valuation model is its simplicity. As shown
in Equation (1),
FCF1
V0 =
W − G (1)
the market value of the firm (V0) is a function of just three variables: the expected free cash flows in the next (or first future)
period (FCF1); the firm’s cost of capital (W); and the projected
growth rate of the firm’s future free cash flows (G).2 This model,
which can be found in virtually all finance textbooks, is always
written in nominal terms as in Equation (1).3
We have no quarrel with this equation. It is simply the
formula for a growing perpetuity. Rather, our quarrel is with
*
We thank Michael Barclay, John Coleman, Magnus Dahlquist, Doug Foster, Jennifer
Francis, John Graham, Campbell Harvey, David Hsieh, Albert “Pete” Kyle, Richmond
Mathews, Michael Moore, Stephen Penman, Michael Roberts, Frank Torchio, S. “Vish”
Viswanathan and Ross Watts for helpful comments. We have benefited from many discussions over the years regarding these and related issues with Robert Dammon, Tim
Eynon, and, especially, Al Rappaport.
1. The popular valuation texts, Rappaport (1998), Copeland, Koller and Murin (1994)
and Cornell (1993) all discuss various aspects of the effects of inflation on the valuation
process. However, none develops the effects of inflation on the Constant-Growth model
from first principles, as we do in this paper. This is also the case for the leading textbook
in the field, Brealey, Myers and Allen (2006). Our analysis most closely resembles that
of Rappaport. On page 47 he presents a formula for a “perpetuity with inflation” that is
equivalent to an important relation that we develop in this paper.
66
Journal of Applied Corporate Finance
•
Volume 20 Number 2
the incorrect transformations of this formula found throughout
the valuation literature for the value of a company that makes
no new investments or invests only in zero NPV projects.
Perhaps the most common application of the ConstantGrowth model is its use in estimating what is referred to in
the valuation literature as a company’s “continuation value” or,
alternatively, its “terminal value.” When valuing a company,
it is standard practice to estimate a company’s free cash flows
over a finite (say, five-year) forecast period, and then assume
that the firm will simply earn its cost of capital thereafter. The
justification for this practice is the standard assumption of
financial economists that the arrival of competitors, along with
technological innovation and obsolescence, cause above-normal
returns to become normal returns over time, and that, after the
forecast period, the firm will earn a normal rate of return on its
investments into perpetuity. The continuation value, as given
by Equation (1), is the present value of the expected free cash
flows beyond the forecast period into perpetuity.4
The assumption that the company will earn only normal
rates of return during the post-forecast period is equivalent to assuming that either the firm will make no new net
investments—that is, capital expenditures will equal depreciation—or that any investments that are made will have zero
NPVs. Using this “zero-rent” argument, financial economists
often assert that the terminal value of the firm can be estimated
by a simple perpetuity of next period’s free cash flows, with the
capitalization rate being the firm’s nominal cost of capital. This
formulation is equivalent to setting G, the nominal growth
rate in Equation (1), equal to zero:
FCF1
V0 W (2)
2. According to Brealey, Myers and Allen, p. 65, this formula was first developed in
1938 by J.B. Williams and rediscovered in 1956 by M.J. Gordon and E. Shapiro. It is
often called the Gordon Growth Model.
3. Throughout this paper we adopt the convention of stating nominal variables in upper-case letters and real variables in lower-case letters. Also, since the models developed
herein are forward-looking, all rates should be thought of as expectations.
4. Rappaport (1998) refers to the continuing value as the firm’s residual value, pp.
40-47. Also see Copeland et al. (1994), Chapter 9, “Estimating Continuing Value,” and
Cornell (1993), Chapter 6, “Estimating the Continuing Value at the Terminal Date.” It
should be noted that the continuing value as given by Equation (1) is the value of the
firm at the end of the forecast period. Thus, in order to find the present value as of today, the terminal value has to be discounted by (1+W)T, where T is the end of the last
forecast period.
A Morgan Stanley Publication • Spring 2008
We accordingly refer to this version of Equation (1) as the
Zero-Nominal-Growth (“ZNG”) model.
Use of the ZNG model, or the simple perpetuity formula,
is typically justified by the following reasoning: (1) with zero
net new investments, there will be no growth; and (2) growth
through the acceptance of zero NPV projects does not affect
the (present) value of the firm. Therefore, under either condition, it is appropriate to set G = 0 in Equation (1) and rely
on Equation (2). Although this logic might appear to be
sound, we show that this ZNG model is based on a mistaken
specification of the nominal growth in free cash flows—G in
Equation (1)—in the presence of inflation.
Specifically, the generally accepted expression for the value
of a “zero-investment” or a “zero-NPV” firm—as presented
throughout the finance literature and typically applied in
practice—ignores the effect of inflation on the company’s total
(accumulated) invested capital. In the traditional ConstantGrowth model without inflation, if there is no new investment,
there is no growth. However, in the presence of inflation, the
value of the initial invested capital will grow at the rate of inflation. And, assuming a constant real return on invested capital
and the replenishment of depreciated assets, the company’s
(nominal) free cash flows stemming from those investments
will grow at the same rate. In other words, the proper application of the Constant-Growth model is based on the market or
replacement value of assets, which of course is affected by inflation. By ignoring the effects of expected inflation, the generally
accepted expression—Equation (2)— understates the true
value of the firm. And the higher the rate of inflation relative
to the real cost of capital, the greater the understatement.
We develop the appropriate expression for the nominal
growth in cash flows in the presence of inflation and show that
the correct model assuming either zero investments or zero net
present value investments is not the ZNG model, but rather
the traditional Constant-Growth model, with the nominal
growth rate set equal to the rate of inflation. We also show that
this expression for the nominal growth term yields a valuation
model that is independent of expected inflation.5
A second contribution of this paper to the valuation
literature is an analysis of the effects of expected inflation
on a company’s weighted-average cost of capital (WACC),
the discount factor most often associated with the ConstantGrowth model. We find that the WACC methodology (at
least in its classic form, as developed by Miller and Modigliani) is incorrect if expected inflation is positive. Substituting
nominal values into the M&M WACC equation results in an
5. If by “inflation” we mean a proportionate increase in all nominal interest rates and
the prices of all goods and services, then expected inflation should have no effect on
present values, i.e., present values should be inflation neutral.
6. One might be tempted to speculate that these two errors are offsetting, since the
error in the ZNG model leads to an underestimate of the value of the firm and the error in the M&M WACC leads to an overestimate. However, as developed subsequently,
the adjustment to the ZNG model is to subtract Π, the rate of inflation, from W in the
denominator of Equation (2), whereas the adjustment to the M&M model is to add txΠL
Journal of Applied Corporate Finance
•
Volume 20 Number 2
overvaluation of the firm. We develop a correction factor that,
when added to the (nominal) M&M WACC formula, yields
a company’s true nominal WACC in the face of inflation.6
Finally, to complete our analysis, we show that the nominal
WACC model developed by two finance academics—James
Miles and John Ezzell—is compatible with any level of growth,
whether attributable to inflation or real investments.7
The Constant-Growth Model with Inflation
Our analysis of the effects of inflation on the ConstantGrowth model begins with a derivation of an expression for
the firm’s nominal free cash flows (FCFt). We then derive
an expression for the growth in nominal cash flows (G). We
defer our discussion of the appropriate discount rate (W)
until our later discussion of the firm’s WACC. For present
purposes, suffice it to say that in the subsequent analysis, we
assume that the firm’s nominal cost of capital (W) is consistent with the Fisher Equation:
W = w + 0 + w 0 (3)
where w is the firm’s real cost of capital and Π is the expected
rate of inflation.8
The Real Return on Investment, Net Cash Flows,
and Free Cash Flows
We begin our derivation of free cash flows with the following
definition of the real return on investment (r):
NCFt
r = (1 + 0 )K t −1 (4)
where NCFt is the firm’s net cash flow at the end of period t
and Kt-1 is its total capital stock at the beginning of the period.
Rearranging Equation (4) yields:
NCFt K t1r(1 0 ) (5)
Equation (5) states that the net cash flow to the firm in period
t is given by the amount of capital at the beginning of the
period (Kt-1), times the constant real return on investment
(r), times 1 plus the rate of inflation.9 Free cash flow in any
period equals the firm’s net cash flow for the period less any
net new investment:
FCFt NCFt NNIt (6)
to W in Equation 1, where tX is the firm’s tax rate and L is its debt to value ratio. Since
both tX and L are less than 1.0, the M&M adjustment is much smaller than the adjustment to the ZNG model.
7. Miles and Ezzell (1980)
8. Irving Fisher (1930). Also see Brealey, Myers and Allen, pages 116-118.
9. Recall that we have adopted the convention of expressing nominal variables in
upper-case letters and real variables in lower-case letters and that all rates and future
values should be interpreted as expectations.
A Morgan Stanley Publication • Spring 2008
67
Define k as the firm’s plowback rate, which is the fraction
of NCFt that is retained by the firm to finance net new
investments:10
NNIt
k NCFt (7)
Substituting Equation (7) into Equation (6) yields:
FCFt NCFt (1 k) (8)
The Growth in Free Cash Flows
Clearly, a critical parameter in the Constant-Growth model is
G, the projected constant growth in the firm’s free cash flows.
In Appendix A we derive the expression for the growth in
free cash flows based on two assumptions: (1) the (expected)
real return on investment, as defined in Equation (4) above,
remains constant; and (2) the Fisher Equation holds such that
the nominal return on investment (R) can be written as:
(9)
R r 0 0 r where r is the expected real return on investment and Π is
the expected rate of inflation.
Making these assumptions, we develop the following
expression for the growth in the firm’s free cash flows:
G =kR+(1-k) 0
(10)
the inflationary growth of the company’s invested capital will
be augmented by the nominal growth from new investments.
Finally, if k = 1, the firm is plowing back its entire net cash
flows. Consequently, the firm will generate no free cash flows
into perpetuity and therefore would be theoretically worthless,
since by assumption there would never be a distribution to
security holders.12 However, when k = 1, the value of the firm
will grow with the value of its capital stock, which, as given by
Equation (10), will equal its nominal return on investment R.
The value of the firm will continue to grow at this rate until
the firm is liquidated.
In addition to Equation (10), the analysis in Appendix A
develops three additional relations that will be helpful in our
subsequent analysis. Consistent with the Fisher Equation we
show that:
G g 0 0g (11)
G and
gG 0
that:
gg
0
k0gr 0g
kr
g k gr (12)
where k is the firm’s plowback rate, r is the real return on
investment and g is the real growth rate. Substituting EquationG(12)
krinto
0 Equation
0 k r (11) yields:
G0kr00k r 0 k r G kr
(13)
Transformations of the Constant-Growth Model
The first term in Equation (10) represents the growth in We now derive the appropriate expressions for the two most
the firm’s free cash flows from new investments. The second frequent transformations of the Constant-Growth model
term represents the increase in cash flows attributable to the found in the finance literature: the value of a company that
increase in the nominal value of the firm’s fixed and working either (1) makes no net new investments or (2) accepts only
capital that results solely from inflation. Thus, there are two zero NPV projects. As we demonstrate below, these two
forces that account for the growth in nominal cash flows. The conditions yield the same valuation expression.
first is the increase in cash flows due to the nominal return
on new investments. The second is the increase in nominal Zero Investments
cash flows due to the fact that the nominal value of the firm’s The value of the firm, according to the Constant-Growth
capital stock is higher by the rate of inflation. As we demon- model, assuming zero net new investment, is found by substistrate later, it is this second term that is ignored in the existing tuting Equation (8) into the numerator of Equation (1),
finance literature—in other words, according to the literature, substituting G from Equation (10) into the denominator of this
G = k R.
equation, and setting k, the reinvestment rate, equal to zero:
As indicated by Equation (10), the effect of inflation on
NCF1 (1 k)
NCF1 (1 k)
(14)
a company’s stock of capital will depend on the percentage of V0 V (14)
W0 (kR (1 k) 0)
cash flows that are reinvested in the firm (k). If the plowback
(14)
W (kR (1 k) 0)
rate is zero (k = 0), then G = Π and all growth is attributable to
NCF1
the nominal growth in the company’s initial fixed and working
NCF1
V0 (1(15)
5)
11
V00 (15)
W
capital stemming from inflation. Under these conditions,
W0
the firm’s total invested capital invested will remain constant
in real terms into perpetuity. If the plowback rate k > 0, then
10. It follows that (1-k) is the firm’s payout ratio.
11. See Appendix A for a demonstration of this proposition.
12. This is equivalent to the value of a zero coupon bond that never matures.
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Journal of Applied Corporate Finance
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Volume 20 Number 2
A Morgan Stanley Publication • Spring 2008
The intuition behind Equation (15) is that even though
the firm makes no net new investments (k = 0), it still makes
replacement investments sufficient to offset its economic
depreciation, so that the real net cash flows must be constant
over time.13 Thus, the future nominal cash flows (revenues,
costs, and profits) emanating from a stream of constant real
NCFs logically must be projected to grow at the rate of inflation, as is reflected in Equation (15).
We refer to this valuation equation as the Zero-RealGrowth (“ZRG”) model, since the formula is equivalent to
assuming that k = 0 and therefore g = 0 in Equation (12),
which implies that the real growth in cash flows is equal to
zero. This in turn implies that all observed growth is attributable to inflation, which justifies our designation of this model
as the ZRG model. (Of course, we could have just as easily
labeled this expression the Inflationary-Growth model.)
The ZRG model is inflation neutral in that it generates a
value for the firm that is independent of the level of expected
inflation. To see this, note that the formula can be written
exclusively in real terms. Given that W = w + Π + wΠ, it
follows that W – Π = w (1+Π). Substituting this expression
in the denominator and the expression for nominal cash flows
into Equation (15) yields the following:
which, again, is the ZRG or Inflationary-Growth model. Note
that real growth g is not necessarily zero under the zero NPV
assumption. Indeed, since g = kr, if r = w > 0, then real growth
g will be positive if k is positive. But any positive real growth
that may be projected will not increase the value of the firm
because all future investments have zero NPV (R = W and r
= w). Hence, g is absent from Equation (17).
The preceding analysis demonstrates that the expression
for the value of a zero-investment firm is the same as the
expression for a zero-NPV firm—that is, Equations (15) and
(17) are identical. This result—that the same expression is
relevant for firms that either invest only in zero-NPV projects
or invest nothing at all—will be useful later when we develop
the appropriate discount factor for the Constant-Growth
model under either of these conditions.
Review of the Literature
In this section we retrace the literature’s development of the
expressions for the value of the firm assuming (1) zero net new
investments and (2) investments only in zero NPV projects.
The purpose is to demonstrate that the formulas found in the
literature are incorrect when expected inflation is positive.
Zero Investments
ncf 1(1 0) ncf 1
V0 =
=
w(1 0)
w
(16) (16) The expression for the value of a firm that makes no net new
investments found in the literature is based on an erroneous
expression for the growth in cash flows. Although incorrect,
Zero Net Present Value Investments
it is nevertheless common to define the nominal growth in
The Constant-Growth model, under the assumption of zero net cash flows as:
net present value investments, is found by making the same
G kR (19)
substitutions as above and setting R = W:
NCF1 (1 k)
NCF1 (1 k)
V0 W (kR (1 k) 0) W(1 k) (1 k) 0
V0 NCF1
W0
where R is the firm’s nominal return on investment.14 The
correct expression for the growth in net cash flows, as we saw
earlier in Equation (10), is:
(17)
Thus, the expression for the value of the firm when r = w
(or equivalently R = W) is simply:
NCF1
ncf1 (1 0 ) ncf 1
V0 W0
w(1 0)
w (18)
13. See Appendix A.
14. Applying the Gordon Growth Model for the valuation of common stock, Brealey,
Myers and Allen write on page 65: P0= DIV1
r-g
where P0 is the price of the stock at time 0, DIV1 is the (expected) dividend in period 1,
r is the firm’s capitalization rate and g is the growth in dividends. It is important to note
that BMA do not indicate whether r or g is stated in real or nominal terms. However,
since DIV1 is presumably a nominal number, stated in period 1 dollars, we must presume
that both r and g are nominal values. Instead of deriving an expression for g, the nominal
growth in earnings, they simply assert on page 67:
“If Cascade earns 12 percent of book equity and reinvests 44 percent of income, then
book equity will increase by .44 x .12 = .053, or 5.3 percent. Earnings and dividends
Journal of Applied Corporate Finance
•
Volume 20 Number 2
G = k R + (1 - k) 0 (20)
Again, it is the second term in this equation that is ignored
in the literature. To derive what the literature describes as the
zero-investment formula, simply substitute G from Equation
(19) into Equation (1),
(18)
per share will also increase by 5.3%:
Dividend growth rate = g = plowback ratio x ROE.”
(Emphasis added.) Again, on page 69 they write: “Dividend growth rate = plowback
ratio x ROE.”
Other references that assert that G=kR include: Ross, Westerfield and Jaffee, p.
128, Grinblatt and Titman, p. 832, Rao, p. 403, Bodie and Merton, p. 123, Emery and
Finnerty, p. 149, Shapiro and Balbirer, p. 159, Benninga and Sarig, p. 9, Van Horn,
pp. 30-31, Martin, Petty, Keown and Scott, p.123. It is interesting to note that none of
these authorities feels compelled to prove this relation. They all rely on it as though it is
self-evident. Not only is this expression for the nominal growth rate not self-evident, it is
incorrect as we demonstrate by the derivation of Equation 10 in the text.
A Morgan Stanley Publication • Spring 2008
69
V0 NCF1 (1 k) W kR
(21)
(21)
and set k, the reinvestment rate, equal to zero.
V0 NCF1 W
(2(22)
2)
Equation (22) is what the finance literature defines as the
“no-growth” value of the firm. For example, the most recent
edition of a bestselling finance text describes Equation (22)
as the value of a company “that does not grow at all. It does
not plow back any earnings and simply produces a constant
stream of dividends.”15 Another popular text states that the
above expression “is the value of the firm if it rested on its
laurels, that is, if it simply distributed all earnings to the
stockholders.”16
While it may seem perfectly logical that with no net new
investments a firm would generate a constant stream of cash
flows into perpetuity, this reasoning ignores the effects of
inflation on the firm’s initial invested capital. If a company’s
investments simply keep up with obsolescence and economic
depreciation, its total invested capital stock will increase at
the rate of inflation; and, if we assume a constant real return
on investment, its cash flows will also increase at the rate of
inflation. Put simply, one cannot express the value of the firm
as a perpetuity in nominal terms—which is what Equation
(22) does—when inflation is positive. As we demonstrated
earlier in Equation (15), the presence of inflation requires
that the value of a zero-investment firm be expressed as a
growing perpetuity, with the growth rate set equal to the
rate of inflation.
To be clear, we are not implying that either the authors
of the texts cited above or the rest of the finance profession
are unaware of the effects of expected inflation on the value
of capital assets. Indeed, in the most influential text in the
profession, after the authors develop the Constant-Growth
model, they present the Fisher Equation and admonish
the reader to “Discount nominal cash flows at a nominal
discount rate. Discount real cash flows at a real rate. Never
mix real cash flows with nominal discount rates or nominal
flows with real rates.”17 But the authors never revisit the
Constant-Growth model and discuss how inflation impacts
the parameters of this model. Failure to do so has led to
confusion in the valuation literature, particularly regarding the value of G and the values of companies that make
no new investments or invest only in zero net present value
projects.
We earlier labeled Equation (22) the Zero-NominalGrowth model, since it is equal to Equation (1) with G,
the nominal growth rate, set equal to zero. We also use the
term to emphasize the internal inconsistency inherent in
this expression. A Zero-Nominal-Growth model in a world
of inflation is an economic oxymoron. Since inflation is the
only distinction between real and nominal values, what does
it mean to have inflation and zero nominal growth? Under
reasonable assumptions, you can’t have both at the same
time. Based on Equation (11) only under the highly unlikely
and therefore mostly irrelevant case of a negative real growth
rate—one that just offsets a positive rate of inflation—can
you have zero nominal growth and positive inflation.18
Since expected inflation is almost always positive, the
implication of negative real growth into perpetuity makes
the Zero-Nominal-Growth model useless for most realworld valuation applications. In addition, the theoretical
conditions under which real growth can be negative into
perpetuity are unlikely to occur in practice. As Equation
(13) indicates, a negative real growth rate implies that either
the real return on investment (r) or the plowback ratio (k) is
negative into perpetuity. Since the Constant-Growth model
is forward looking, companies cannot be expected to accept
a project with an ex ante negative return. Thus, a negative
real growth rate must be due to a negative k, which represents a steady liquidation of the firm over time. Obviously,
this condition is not appropriate for most corporate valuation applications.19
Zero Net Present Value Investments
The derivation of the expression for the value of a company
that invests only in zero NPV projects found throughout
the finance literature relies on the same mistaken expression for the growth in nominal cash flows given in Equation
(19). Zero NPV implies that R = W. Substituting Equations
(8) and (19) into Equation (1) and imposing this zero-NPV
condition (R = W) also yields the Zero-Nominal-Growth
formula:
15. Brealey, Myers and Allen (2006), p. 73.
16. Ross, Westerfield and Jaffee (1993), pp. 130-131. We have examined a number
of other texts and have been unable to find the expressions developed in this paper –
expressions that are necessary for an inflation-neutral valuation model.
17. Brealey, Myers and Allen, p.118.
18. Specifically, if g 0 , then G = 0 even though Π > 0.
10
19. Of course k can be negative or, with an influx of capital, even be greater than 1
over a finite period. However, only in very rare circumstances can these extreme values
of k persist into perpetuity.
70
Journal of Applied Corporate Finance
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Volume 20 Number 2
A Morgan Stanley Publication • Spring 2008
Table 1
Percentage Underestimate According to the Zero-Nominal-Growth Model
Expected Rate of Inflation
0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
11%
12%
13%
14%
15%
1%
0
50
2%
0
3%
0
4%
66
74
79
83
85
87
88
89
90
91
91
92
92
93
33
50
59
66
70
74
77
79
81
82
83
84
85
86
87
25
40
49
56
61
65
69
71
73
75
77
78
79
80
81
0
20
33
42
49
54
59
62
65
67
69
71
73
74
75
77
Real Cost of Capital
5%
0
17
28
37
43
49
53
57
60
62
65
66
68
70
71
72
6%
0
14
25
33
39
44
49
52
55
58
60
62
64
66
67
68
7%
0
12
22
29
35
40
45
48
51
54
56
59
60
62
64
65
8%
0
11
20
27
32
37
41
45
48
51
53
55
57
59
61
62
9%
0
10
18
24
30
35
39
42
45
48
50
52
54
56
58
59
10%
0
9
16
23
28
32
36
40
43
45
48
50
52
53
55
57
11%
0
8
15
21
26
30
34
37
40
43
45
47
49
51
53
54
12%
0
8
14
20
24
28
32
35
38
41
43
45
47
49
51
52
13%
0
7
13
18
23
27
30
33
36
39
41
43
45
47
49
50
14%
0
7
12
17
22
25
29
32
35
37
39
41
43
45
47
48
15%
0
6
12
16
20
24
27
30
33
36
38
40
42
43
45
47
NCF1 (1 k) NCF1 (1 k) NCF1 (1 k
V0 W kR
W kW
W(1 k)
inflation—what we have labeled the Zero-Real-Growth or
Inflationary-Growth model.
NCF1
The General Model
(23)
V0 W
Thus far, we have focused our analysis on two specific cases:
A number of financial economists have argued that (1) the value of a firm that makes no new investments; and (2)
Equation (23) is the expression for the value of a firm that the value of a firm that invests only in zero NPV projects. But
only invests in zero NPV projects.20 Although it may seem our criticisms of the treatment of inflation in the Constantlogical that since investments in zero NPV projects do not Growth model also apply to the estimation of G in general.
create value, G should not appear in Equation (23). Intuitively, As developed above, the expression for the growth in nominal
the discounted value of cash flows from additional investments cash flows that is found throughout the literature is:
just equals the present value of the new investments assuming
the return on investment equals the cost of capital. Thus, value
(24)
G kR
is not created by these additional investments. And of course
G kR
whereas the correct expression for the nominal growth
this logic is correct.
But, with positive inflation, the value of invested capital term is
will grow at the rate of inflation and, assuming a constant real G kR (1 k) 0
rate of return, the net cash flows from these investments (the
(25)
G kR (1 k) 0 numerator in Equation (23)) will also grow at the rate of inflation. The denominator in Equation (23) ignores this growth
To the extent that researchers and practitioners use the
former expression to calculate G, they are understating the true
and therefore understates the value of the firm.
In sum, the correct expression for the value of the value of the firm. Thus, our criticism of the Constant-Growth
firm that invests in only zero NPV projects also must be a model, as it is presented in the literature, goes beyond the two
growing perpetuity with the growth rate equal to the rate of special cases of zero NPV and zero investments. Our criticisms
20. Copeland, Koller and Murrin (1994), pp. 282-283. Rappaport (1998) p.42 also
asserts that Equation (23) is the value of the firm if all of its projects have zero NPV. Copeland et. al. pp. 513-515, derive the relation (stated in our notation) FCFt = NPVt ( 1 - k
). They then substitute G/R and write FCFt = NPVt ( 1 - G/R ). From this equation they
follow the derivation as outlined in the text. Thus, these authors erroneously assume, like
Brealey, Myers and Allen (2006), that k = G/R, when in fact the correct relation is k = g/r.
This substitution leads them to Equation (23). It is important to note that Copeland et. al.
Journal of Applied Corporate Finance
•
Volume 20 Number 2
do not specify whether their variables are nominal or real. The clear implication though,
is that they are all nominal values. Cornell (1993, p. 156) makes the same error. He
explicitly states that G is the nominal rate of growth in free cash flows and distinguishes
it from g, which he earlier defines on page 148 as the “long-run growth in real returns.”
Nevertheless, Cornell assumes that k=G/R. Weston et. al. (1998, p. 186) make the
same, erroneous substitution. Rappaport (1998, pp. 40-44) rationalizes Equation (23)
verbally, making the same arguments that are embodied in the algebra above.
A Morgan Stanley Publication • Spring 2008
71
also apply to the general model, provided the growth term is
calculated according to Equation (24) instead of Equation
(25).
The Error Rate of the Zero-Nominal-Growth Model
under Inflation
As noted above, the Zero-Real-Growth (ZRG) model is inflation-neutral, since the model can be written exclusively in real
terms, as we saw in Equation (18). But, somewhat ironically,
the Zero-Nominal-Growth (ZNG) model, as presented in the
literature, is not inflation-neutral. According to the model, the
value of the firm is inversely related to expected inflation—
that is, the higher expected inflation, the higher the discount
factor and hence the lower the value of the firm.
The percentage by which the Zero-Nominal-Growth (ZNG)
model understates the value of a firm with zero net new investments or zero NPV investments can be readily calculated by
sub­tracting the ratio of Equation (22) to Equation (15) from 1:
VL = V U + PVITS (27)
In general, the interest tax shield per period can be written
as:
ZNG
V
W– 0
Percentage Underestimate = 1 =1
V ZRG
w(1 0)
Percentage Underestimate = 1 w(1 0) 0
tion neutral when stated in nominal terms. We show that
the equation systematically understates the true WACC and
therefore overstates the value of the firm if expected inflation
is positive. We provide an expression for this underestimate
of the firm’s cost of capital that, when added to the M&M
WACC formula, generates the correct discount factor for the
Zero-Real-Growth (Inflationary-Growth) model. We show
that using this correction factor generates a value of the firm
that is independent of expected inflation.
There are two basic approaches to valuing the interest
tax shields of a levered firm into perpetuity: the Adjusted
Present Value (APV) method and the WACC method. The
APV method recognizes that the value of a levered firm (VL)
is equal to the value of the firm if it were un-levered (VU) plus
the present value of the interest tax shields (PVITS):
W
(26)
The entries in Table 1 illustrate the percentage undervaluation generated by the Zero-Nominal-Growth model for
various levels of the real cost of capital (w) and the expected
rate of inflation (Π). For example, if expected inflation is 2%
and the real rate of return is 3%, then the Zero-NominalGrowth model will underestimate the true value of the firm
by 40%! In other words if the true value of the firm was $100,
the Zero-Nominal-Growth formula will value the firm at $60.
Note also that the undervaluation is greater when the expected
rate of inflation is higher than the expected real interest rate.
In other words, the values in the north-east triangle of the
table are greater than those in the south-west triangle.
ITSt = t X WD D (28)
where t X is the corporate tax rate, W D is the nominal cost of
debt and D is the amount of debt outstanding at the beginning of the period.22
The main insight behind the alternative WACC methodology is that under certain circumstances, the value of a levered
firm can be found directly by discounting its future free cash
flows by its tax-adjusted, weighted-average cost of capital. In
other words, the value of a levered firm can be found by substituting the following definition of WACC for W in Equations
(15) and (17):
D
E
WACC (1 t X) WD W EL
VL
VL
(29)
Expected Inflation and the Zero-Real-Growth
Discount Factor
We now develop the appropriate discount factor for the ZeroReal-Growth model, W in Equations (15) and (17). Most of
the valuation literature advocates the use of the weightedaverage cost of capital (WACC) methodology developed by
Miller and Modigliani (M&M), which is designed to account
for the increase in the firm’s net cash flows stemming from
the tax deductibility of interest payments.21 As we show,
however, the M&M WACC model (equation) is not infla-
where E is the market value of equity, V L is the market value
of the levered firm (V L =D+E) and W LE is the firm’s nominal
cost of (or required rate of return on) its levered equity.23
Because of the tax subsidy to debt financing, the value of
the firm increases with greater leverage. Intuitively, as the firm
substitutes debt for equity, a greater weight is placed on the
first term of Equation (29). Since t X > 0 and W D < W LE, the
greater the debt-to-value ratio, the lower the WACC—and
the lower the WACC, the higher the value of the firm.24 As
already noted, under certain circumstances, the increase in the
21. “The appropriate rate for discounting the company’s cash flow stream is the
weighted average of the costs of debt and equity capital,” Rappaport, page 37. Also see
Brealey, Myers and Allen (2006), Chapter 19, Copeland et al., Chapter 8 and Cornell,
Chapter 7.
22. If the bond is issued at par, the cost of debt would be the bond’s coupon rate.
23. The analysis assumes that the firm can take full advantage of the tax deduction
of interest payments.
24. Of course, according to M&M, if markets are perfect and taxes are zero, then W LE
will adjust to offset exactly any change in weights, leaving WACC unchanged.
72
Journal of Applied Corporate Finance
•
Volume 20 Number 2
A Morgan Stanley Publication • Spring 2008
value of the firm caused by discounting its net cash flows by
this lower rate—namely its WACC—is exactly equivalent to
the present value of the debt tax shields created by the increase
in debt financing, provided the cost of debt is unaffected by
the increase in leverage.25
The Present Value of the M&M Tax Shields26
Under the assumptions of the M&M model, the firm’s cash
flows and the amount of debt outstanding are constant
into perpetuity. The interest tax shield in Equation (28) is
therefore a simple perpetuity. If we assume that the interest
payments are as risky as the debt itself, the present value of
this perpetuity is:
PVITS = ( t X WD D ) / WD = t X D. (30)
PVITS = ( t X WD D ) / WD = t X D.
Thus, substituting Equation (30) into (27), under
M&M,
VL = V U + t X D.
(31)
VL = V U + t X D.
The M&M WACC Model
The M&M WACC model posits a relation between a firm’s
cost of equity capital and its leverage. Specifically, the cost
of equity to a levered firm is equal to the cost of capital
if the firm was un-levered plus the difference between the
un-levered cost of capital and the (constant) cost of debt
times 1 minus the firm’s tax rate times the firm’s debt-toequity ratio. Although this formula can be found in most
corporate finance textbooks, like the Constant-Growth
model in general, the authors never state whether this relation holds for nominal or real variables.27 Unfortunately,
this silence and the context in which the M&M model is
typically presented in the literature clearly imply that the
relation holds in nominal terms. But, as we show below, this
is not correct.
The fact that the M&M model assumes fixed cash flows
and a fixed amount of debt outstanding immediately raises
doubts that the model holds in nominal terms. As we will
see, this intuition is correct. Since the M&M model is based
on the assumption of constant cash flows, it follows that the
M&M WACC provides the correct valuation when stated
in real terms. But, when stated in nominal terms, the M&M
model systematically understates the firm’s true nominal
WACC when expected inflation is positive. We provide an
adjustment factor that, when added to the nominal M&M
formula, makes it compatible with our Zero-Real-Growth
(Inflationary-Growth) model.
25. Note that the WACC methodology ignores bankruptcy costs and any other leverage-related costs. In fact the M&M model assumes that WD is equal to the risk-free rate
regardless of the degree of leverage. This is an obvious limitation to the M&M WACC
model since there is ample empirical evidence that leverage and the cost of debt are
positively related.
Journal of Applied Corporate Finance
•
Volume 20 Number 2
Real WACC Under the Zero-Real-Growth Assumption
Stating the M&M model in real terms, the cost of levered
equity is equal to:
D
w EL w U (w U w D)(1 t x )
(32)
(32)
E
where w U is the firm’s un-levered cost of capital.28 Substituting Equation (32) into Equation (29) and solving for the real
weighted average cost of capital yields:
wacc M&M w U (1 t X L) (33)
where L is the firm’s debt-to-value ratio, D . Now, according
VL
to the M&M model:
VL fcf1
VU t X D
wacc M&M
(34)
To see that Equation (34) holds, substitute Equation (33)
for waccM&M and show that the resulting expression is equal
to VL:
VL =
fcf1
wU ( 1 t X L )
Noting that fcf 1
VU
wU
VU
VL =
( 1 tX L )
(35)
(36)
(37)
Combining the above results, the value of a levered firm,
assuming zero real growth, can be written as either Equation
(34) or Equation (35), which implies the following:
fcf 1
(38)
VL = VU + t X D =
.
wacc M&M In other words, the M&M model holds in real terms.
26. This analysis follows the presentation of the M&M WACC model in Brealey, Meyers and Allen (2006) pp. 517-520.
27. See references in Footnote 3.
28. Note that we continue the convention of stating real variables in lower-case letters
and nominal values in upper-case letters.
A Morgan Stanley Publication • Spring 2008
73
Table 2
Percentage Overvaluation from Using M&M Nominal Parameters
Expected Rate of Inflation
0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
11%
12%
13%
14%
15%
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0%
10%
0.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
3.9
4.3
4.7
5.0
5.4
5.7
20%
0.0
0.9
1.7
2.6
3.5
4.3
5.2
6.0
6.9
7.7
8.6
9.4
10.3
11.1
12.0
12.8
30%
0.0
1.4
2.7
4.1
5.5
6.9
8.4
9.8
11.2
12.7
14.2
15.6
17.1
18.6
20.1
21.6
40%
0.0
1.9
3.9
5.9
7.9
10.0
12.1
14.2
16.4
18.7
20.9
23.3
25.6
28.1
30.5
33.1
50%
0.0
2.5
5.2
7.9
10.6
13.5
16.5
19.6
22.7
26.0
29.4
32.9
36.6
40.4
44.3
48.4
60%
0.0
3.2
6.6
10.1
13.8
17.7
21.8
26.0
30.5
35.3
40.3
45.5
51.1
57.1
63.3
70.0
70%
0.0
4.0
8.3
12.8
17.6
22.7
28.2
34.1
40.5
47.3
54.7
62.7
71.4
81.0
91.4
102.9
80%
0.0
4.9
10.2
15.9
22.1
28.9
36.3
44.5
53.5
63.5
74.8
87.4
101.7
118.0
136.9
158.9
Debt to Value
Nominal WACC under the Zero-Real-Growth
Assumption
The assumptions of the basic M&M model imply that it
is inflation neutral in that a change in expected inflation
will not have any effect on the value of the firm. However
the M&M equation, stated in nominal terms, is not inflation
neutral.
Inflation neutrality requires that rates of return adhere
to the Fisher Equation. Thus, for the nominal version of
the M&M WACC to be inflation neutral, it must be the
case that:
(39)
WACC
= wacc + 0 + 0 wacc
Substituting the definition of wacc from Equation (33)
into (39) yields:
True
WACC True w U (1 t x L) 0 0 w U (1 t XL) (40)
Stating the M&M WACC formula (Equation (33)) in nominal terms:
WACC M&M WU (1 t X L) (41)
Invoking the Fisher Equation and substituting for WU,
WACC M&M (wU 0 w U 0) (1 t X L)
(42)
Expanding Equation (42)
M&M
(43)
WACC
wU (1 t XL) 0 w U (1 t X L) 0 (1tX L)
Re-writing Equation (40)
(44)
So that,
(45)
(46)
WACC True = WACC M&M 0 t XL (39)
Equation (46) shows that the “true” nominal WACC is
equal to the M&M WACC plus the term ∏tX L. In other
words the M&M WACC under-states the appropriate value
by the factor ∏tX L. Intuitively, the nominal WACC model
assumes(40)
that an increase in inflation will increase the firm’s
tax shield from debt financing. However, this violates a basic
assumption of the M&M model that the amount of debt is
fixed and independent of its cost of capital as illustrated in
Equation (31).
Equation (46) also shows that it is easy to correct the
nominal WACC formula so that it is consistent with the Fisher
Equation and yields the correct value. One can simply add
the amount ∏tX L to the computed WACC using the standard
M&M nominal WACC formula. Adding the quantity
∏tX L to the standard M&M WACC yields the correct nominal
WACC, which is consistent with the Fisher Equation.29
In Table 2 above we illustrate that the error generated by
the nominal version of the M&M WACC equation is positively
related to the expected rate of inflation and that the error is
greater, the greater the firm’s leverage ratio. Intuitively, the
nominal M&M model assumes correctly that an increase in
29. Although not shown, it should be noted that the nominal version of the M&M
model is also vitiated if the real growth term is positive. Intuitively, the M&M model is
incompatible with growth of any sort, be it from inflation or new investment.
74
Journal of Applied Corporate Finance
•
Volume 20 Number 2
A Morgan Stanley Publication • Spring 2008
inflation will increase the firm’s WACC. However, the model
incorrectly assumes that the government will absorb the amount
∏tX L of the increase. In fact, as discussed above, an increase in
the expected rate of inflation—at the time of valuation—will
cause a proportionate increase in both the firm’s coupon rate
and the cost of debt, leaving the (present) value of the debt tax
shields, and hence the value of the firm, unaffected.
Table 2 illustrates the economic magnitude of the error
generated by the nominal M&M model. Each entry in the
table gives the percentage overvaluation caused by using the
nominal M&M WACC model instead of the correct model,
assuming zero real growth. Specifically we calculate the following expression:
V M&M
Percentage Overvaluation = True 1
(47)
V
assuming that the real cost of capital and the real return on
investment are both equal to 10%, the tax rate is 40%, and
the combination of the expected rate of inflation and the firm’s
leverage ratio are those given in the table. Casual inspection of
the table reveals that the error becomes material at combinations of 40% debt and 5% inflation. A 40% leverage ratio and
a 5% rate of inflation results in a 10% overvaluation of the
firm applying the M&M model in nominal terms.30
Note that with zero inflation (Π), zero leverage (L) or
a zero tax rate (tX), V M&M = V TRUE from Equation (46) and,
according to Equation (47), the percentage overestimate would
be zero. Note also that the percentage error is independent of
the level of free cash flows. Intuitively, since the free cash flows
in the first period are in the numerator of both valuation
formulas, they cancel out when calculating the ratio of the
two. Finally, recall that the “fix” for the nominal M&M model
is to add ∏tX L to the M&M WACC. Thus the percentage
error increases with increases in tX, L or Π.
Ezzell (M&E) to determine the appropriate discount factor for
a firm in which nominal growth is not zero. Note that since
the M&E analysis generates the appropriate WACC assuming a positive growth in nominal net cash flows, it does not
matter whether the growth is generated by real investment
or inflation.
Before moving on, it is important to clarify the distinction
between the analysis in the previous section and the analysis
in this section. In the previous section, the objective was to
determine the appropriate expression for the discount factor
under the assumption of zero real growth and positive expected
inflation—the W in Equations (15) and (17). In this section,
our objective is to determine the appropriate expression for the
discount factor when either real growth or expected inflation
is(47)
positive—the W in Equation (1).
The intuition behind the M&E analysis is that if debt is
a constant proportion of the value of the firm, then so is the
value of its interest tax shields. Moreover, since the amount of
debt outstanding is a constant fraction of the value of the firm,
the risk of these tax shields is equivalent to the risk of the firm’s
cash flows, which means that both should be discounted at the
same nominal rate. The M&E analysis relies on this logic.
In Appendix B, we demonstrate the validity of the M&E
WACC in nominal terms. We follow the same procedure as
we do in the previous section. We assume that the value of
a levered firm is equal to the value if it were un-levered plus
the present value of its interest tax shields.31 We define the
present value of the interest tax shields assuming a constant
debt-to-value ratio and add this to the present value of the
un-levered firm calculated according to Equation (1). We then
demonstrate that the M&E WACC, stated in nominal terms,
yields this same present value of the firm. As developed in
Appendix B, the M&E WACC formula is:
Expected Inflation and the Nominal-Growth
Discount Factor
In the previous section we developed the discount factor for
the ZRG model. There we argued that if the analysis were
done using real variables, the M&M WACC, stated in real
terms, was the appropriate discount rate. We also showed that
if the analysis were done using nominal variables, an adjustment to the nominal M&M WACC was necessary in order
to be consistent with Zero-Real-Growth.
Recall that the rationale behind the ZRG model is the
assumption that either the firm makes no new net investments or invests only in zero NPV projects. Thus, this model
is inappropriate if investment is not zero or if certain projects
have positive net present values. However, if one is willing to
assume that a firm’s ratio of debt to value is constant through
time, then one can rely on the analysis provided by Miles and
t W L ( 1 WU )
(48)
WACC M&E W U X D
1 WD
Conclusion
We have shown that the generally accepted expression for
the value of a firm that either makes no new investments or
invests only in zero net present value projects is misspecified if
inflation is positive. We identified the source of the misspecification and traced its origins to the misspecification of the
growth term in the Constant-Growth valuation formula. We
derived the correct expression for the growth term, and in
turn the correct expression for the value of the firm under
either of these conditions.
We have also shown that the most widely used method
of estimating a firm’s WACC is inappropriate when inflation
is positive, unless the analysis is performed in real terms. The
M&M equation for the WACC, stated in nominal terms,
underestimates the firm’s true WACC and thus overestimates
30. Although not shown, the overestimate is higher the lower the real cost of capital
(the real return on investment).
31. We reiterate that this literature, and hence the models in this paper, assume zero
bankruptcy and other leverage-related costs.
Journal of Applied Corporate Finance
•
Volume 20 Number 2
A Morgan Stanley Publication • Spring 2008
75
(48)
its value. The M&M model can be expressed in nominal terms
provided the real WACC is calculated first according to the
M&M formula, and then the nominal WACC is calculated
from the Fisher Equation. Alternatively, the nominal version
of the M&M model can be found by first calculating the
WACC in nominal terms, and then adding the correction
factor ∏tX L. Finally, we show that if the ratio of the firm’s debt
to value remains constant over time, then the nominal WACC
according to Miles & Ezzell is appropriate and requires no
adjustment.
Appendix A
Derivation of the Growth Rate in the
Constant-Growth Model
We begin our derivation of the growth in free cash flows
under the assumptions of the Constant-Growth model by
establishing the relation between the firm’s capital stock and
its net cash flows. The total capital stock at the beginning of
each period (Kt) is equal to the level of the capital stock at the
beginning of the previous period (Kt-1), less deterioration due
to wear, tear and obsolescence over the period (DETt-1), times
michael bradley is the F.M. Kirby Professor of Investment Bank- one plus the rate of inflation (Π), plus capital expenditures
ing, Fuqua School of Business and Professor of Law, Duke University K made
at time
tt(CAPX
):) 0
(K t(K
0
)CA
PX PX
(1) (1)
t
t K
1 DET
1) (1)(1
t
DET
CA
(bradley@duke.edu).
gregg jarrell is Professor of Finance and Economics, William E. K t
Simon Graduate School of Business Administration, University of Rochester (gregg@theJarrells.com).
t
t 1
t 1
t
(Ktt11)DET
(1DET
(1 PX
0) CAPX t
(K
tDET
0))t
1) CA
Ktt1K
(K
t 1
t 1 (1 0) tCAPX t
(1) (1)
We divide CAPX in any period into two components:
replacement expenditures (REPt) and net new investments
(NNI
):
t
(2) (2)
CAPX
t REP
t NNI
t
CAPX
t REP
t NNI
t
CAPX
REP t NNI t
CAPX tCAPX
REP
tNNI
(2) (2)
(2)(2)
tt REP tt NNI t
References
Brealey, R.A., S. C. Myers and F. Allen, Principles of Corporate Finance, 8th ed. (New York: McGraw-Hill, 2006).
Benninga, Simon and Oded Sarig, Corporate Finance: A
Substituting Equation (2) into Equation (1) yields:
Kt K
(K t(K
DE Tt 1T)(1 )(10) 0
REP
(3) (3)
1
t NNI
t
) REP
Valuation Approach (McGraw-Hill, 1997).
t
t 1 DE
t 1
t NNI
t
(3) (3)
Bodie, Z. and R. Merton, Finance (New Jersey: Prentice
(K
DE
)t0
REP
K t (KKt1K
tDE
T t11)(1
T0T))(1
REP
)REP
NNI
t
1 )(1
t NNIt (K
DE
0
t
t 1t t 1
tt NNIt
Hall, 2000).
Replacement expenditures are equal to the economic
Copeland, T., T. Koller and J. Murin, Valuation, 2nd ed.
depreciation in the firm’s fixed and working capital over any
(New York: John Wiley & Sons,1994).
Cornell, B., Corporate Valuation (New York: Irwin, arbitrary period, and are the costs of keeping the firm’s real
1993).
capital stock at a constant level throughout the perpetuity
Fisher, I., The Theory of Interest (New York: Augustus M. period. Note that we are replacing deteriorating assets at their
current, nominal costs.
Kelley, Publishers, 1965). Reprinted from the 1930 edition.
Martin, J., J.W. Petty, A. Keown, D. Scott, Basic Financial
DET
0) DETt REP
Management, 5th ed. (New Jersey: Prentice Hall, 1991).
(4) (4) (4)
t 1 (1
t . t . DET
t 1 (1 0 ) DET
t REP
Miles, J. and J. Ezzell, The Weighted Cost of Capital,
)0
) REP
DET
DET 1 (1
DET
0) (1
t 1DET
. t REP
(4)
t.
DET
(10
t DET
t 1 (4),
tt REP
t.
Perfect Capital Markets, and Project Life: A Clarification, Givent Equation
we
can rewrite
Equation
(3) as:
Journal of Financial and Quantitative Analysis 15 (1980), pp.
719-730.
(5) (5) (5)
K
KtK
0) NNI t
t K
1(1t t
1(1 0) NNI
t
Modigliani, F. and M. H. Miller, “Corporate Income Taxes
Kt K0K
)NNI
NNI t
K t that
K the
tNNI
1(1 )0
K
t 1(1
t investment
and the Cost of Capital: A Correction,” American Economic Given
realt)return
can be written as (5)
t 1(1 0on
t
Review, Vol. 53: 433-443 (June 1963).
NCFt
Myers, S., “Determinants of Corporate Borrowing,”
(6)
r
(1 0)K t 1 Journal of Financial Economics 5 (1977) 147-175.
NCFt
Rao, R., Financial Management, (Ohio: Southwestern
r
NCFt
(1 0(1
r )K
0
College Publishing, 1995).
ItNCF
follows
that
t 1)
r
K
t
t 1
(1 0)K t 1
Rappaport, A., Creating Shareholder Value, 2nd ed. (New
York: The Free Press, 1998).
(7)
NCF
t r K t 1 (1 0 )
NCFt r K t1 (1 0 )
Ross, S.A., R. Westerfield and J. Jaffe, Corporate Finance,
NCFt 1 NCFt (Kt K t 1 ) r (1 0)
(8)
and
3rd Edition, (Illinois: Irwin, 1993.
Shapiro, A. and S. Balbirer, Modern Corporate FinanceNCF NCF (K K ) r (1 0)
(8)
t 1
t
t
t 1
NCFt 1 NCFt (Kt K t 1 ) r (1 0)
(New Jersey: Prentice-Hall, 2000).
(8)
Stewart III, G.B., The Quest for Value (USA: Harper
Business, 1991).
76
Journal of Applied Corporate Finance
•
Volume 20 Number 2
(1)(1)
A Morgan Stanley Publication • Spring 2008
(3)(3)
(4)(4)
(5)(5)
(8)
Substituting Kt from Equation (5) into Equation (8) yields:
(9)
Note that the first and last variables in the bracketed term
on the RHS of Equation (9) are the same – Kt-1 . Expanding
Equation (9) and canceling out the Kt-1 terms yields:
NCFt 1 NCFt [K t10 NNIt ] r (1 0 )
(10)
Dividing Equation (10) by NCFt defines the growth in NCF
and, since FCFt = NCFt (1–k), the growth in FCF as well:
NCFt 1 NCFt FCFt 1 FCFt
G
FCFt
NCFt
0 K t 1 r (1 0)
NNIt r (1 0 )
G
K t 1 r (1 0)
K t 1 r (1 0)
(11)
(12)
WACC According to Miles and Ezzell
The WACC methodology developed by Miles and Ezzell
(M&E) does hold in nominal terms. This model assumes
that the ratio of debt to value remains constant through time.
Thus, if the nominal value of the firm grows with inflation,
so does the amount of debt outstanding and, as a result, the
per period
(10) interest tax shield. All else equal, the value of a
levered firm (the market value of its outstanding securities)
is positively related to the rate of inflation. The value of the
firm is not inflation neutral under the M&E model. Since
the value of the firm’s tax shields increases with inflation,
the M&E model is not correct if the parameters are stated
in real terms.
According to the M&E assumptions, the present value of
the firm’s interest tax shield is equal to:
t
t W L VL ( 1 G ) t-1
PVITS ¤ X D
( 1 WD ) ( 1 WU ) t-1
t 1
Recall that
NNIt
NNIt
k NCFt
K t1r(1 0 )
Appendix B
(19)
The numerator of Equation (19) is the interest tax shield
in period t – the tax rate times the interest rate times the debt
to value ratio times the value of the firm at the beginning
of the first period times the appropriate growth factor. Note
Substituting (13) into (12) yields:
that the interest tax shield is based on the beginning value of
(14)
the firm. The denominator of Equation (19) reflects the fact
G 0 + k r (1+ 0) = 0 + kr + k r 0
that under the M&E assumptions, the amount of debt in the
Define g as real growth in free cash flows. Since G = g when first period is known and therefore the interest tax shield is
discounted at the rate on debt, W D. Thereafter, the amount
Π = 0, it follows from Equation (14) that:
of debt outstanding depends on the value of the firm. Since
L is constant, the firm issues debt when the value of the firm
g = kr
(15)
rises and retires debt when the market value falls. Thus, the
Equation (15) states that the real growth in FCF is equal size of the interest tax shield rises and falls with the value of
to the plowback ratio times the real return on investment. the firm. Consequently, the interest tax shield is as risky as
To obtain the equation for nominal growth (G) in terms of the firm itself, which accounts for discounting the per period
nominal returns (R) and inflation (Π) add the quantity (+ tax shield at the un-levered cost of capital, W U, from period
kΠ – kΠ) to Equation (14)
t = 2 onward.
For purposes that will become clear shortly, we isolate the
G = 0 G+ =kr0 ++ krkr0++ krk 0
0 +- kk00 - k 0 (16)(16) first
(16)year’s tax shield
(13)
Gcollect
= kG(Gr=terms,
+=0k0(++r
+kr( 0
10-) +k+ )(k0
r kr
+00+) +r
10- k- k) 0
t W L VL
(17) (16)
(17)
PVITS X D
1 WD
t
(17)
G = k ( r + 0 +r 0 ) + ( 1 - k ) 0 (17) tX WD L VL ( 1 G ) t-1
¤
( 1 WD ) ( 1 WU ) t-1
t 2
and substitute R = r + Π + r Π from the Fisher Equation for
the nominal return to investment:
G = k RG+=(k1R- k+ )( 10- k ) 0
(20)
(18)(18) where,
(18) V L is the value of the levered firm in period 0. Adjusting the time subscripts in the second term:
This isGthe
= kgeneral
R + ( 1expression
- k ) 0 for the nominal growth in (18)
free cash flows.
Journal of Applied Corporate Finance
•
Volume 20 Number 2
A Morgan Stanley Publication • Spring 2008
77
t W L VL
(21)
PVITS x D
1 WD
t
t x WD L VL ( 1 G ) t
¤
( 1 WD ) ( 1 WU ) t
t 1
(27)
Substituting Q yields:
(28)
Define Q as:
(22)
Substituting WACCM&E into the denominator of the basic
valuation formula yields:
Substituting (22) into (21) and applying the formula for
the value of a growing perpetuity:
(23)
Recall that the value of a levered firm is equal to the value
of an un-levered firm plus the present value of interest tax
shields (PVITS):
Q (1 WU )
VL VU WU - G
Multiplying through by the denominator yields:
(30)
(31)
(32)
Dividing through by ( W U – G ) yields:
(25)
Substituting Equation 25 into the weighted into the
weighted average cost of capital formula:
(24)
We now demonstrate that the WACC model developed by
M&E yields this same expression. Under the M&E assumptions, the firm’s nominal cost of equity is:
(29)
(26)
(33)
(34)
Thus, Equation (29) equals Equation (24), which implies
that the M&E model is correct in nominal terms.
Note that since the M&E model assumes a constant ratio
of debt to equity, the firm’s debt, and hence its tax shields, will
increase at the rate of inflation, and the M&E model, stated in
real terms, cannot account for this increase in value.
yields:
78
Journal of Applied Corporate Finance
•
Volume 20 Number 2
A Morgan Stanley Publication • Spring 2008
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