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American Economic Association
Corporate Income Taxes and the Cost of Capital: A Correction
Author(s): Franco Modigliani and Merton H. Miller
Source: The American Economic Review, Vol. 53, No. 3 (Jun., 1963), pp. 433443
Published by: American Economic Association
Stable URL: http://www.jstor.org/stable/1809167
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COMMUNICATIONS
433
equanimity a writingdown of the value of their reserves, or unless one is
prepared to forego the possibility of exchangerate adjustment, any major
extension of the gold exchange standard is dependent upon the introduction
of guarantees. It is misleading to suggest that the multiple keycurrency system is an alternative to a guarantee, as implied by Roosa [6, pp. 57 and
912].
IV. Coniclusion
The most noteworthy conclusion to be drawn from this analysis is that the
successful operation of a multiple keycurrency system would require both
exchange guarantees and continuing cooperation between central bankers of
a type that would effectively limit their choice as to the form in which they
hold their reserves. Yet these are two of the conditions whose undesirability
has frequently been held to be an obstacle to implementation of the alterlnative proposal to create a world central bank. The multiple keycurrency proposal represents an attempt to avoid the impracticality supposedly associated
with a world central bank, but if both proposals in fact depend on the fulfillment of similar conditions, it is difficultto convince oneself that the sacrifice of
the additional liquidity that an almost closed system would permit is worth
while. Unless, of course, the object of the exercise is to reinforce discipline
rather than to expand liquidity.
JOHN WILLIAMSON*
REFERENCES
1. R. Z. ALIBER, “Foreign Exchange Guarantees and the Dollar: Comment,”
Am. Econ. Rev., Dec. 1962, 52, 111216.
2. S. T. 1E3ZAAND G. PATTERSON, “Foreign Exchange Guaranteesand the Dollar,” Am. Econ. Rev., June 1961, 51, 38185.
3.
AND
, “Foreign Exchange Guaranteesand the Dollar: Reply,”
Amn.Econ. Rev., Dec. 1962, 52, 111718.
4. F. A. LUTZ,The Problem of International Equilibrium. Amsterdam 1962.
5. R. NURKSE, International Currency Experience. Geneva 1944.
6. R. V. RooSA, “Assuring the Free World’s Liquidity,” Business Review
Supplement,Federal Reserve Bank of Philadelphia, Sept. 1962.
* The author is instructor in economics at Princeton University. He acknowledges the
helpful comments of Fritz Machlup. Views expressed are those of the author alone.
Corporate Income Taxes and the Cost of Capital:
A Correction
The purpose of this communication is to correct an error in our paper
“The Cost of Capital, Corporation Finance and the Theory of Investment”
(this Review, June 1958). In our discussion of the effects of the present
method of taxing corporations on the valuation of firms, we said (p. 272):
The deduction of interest in computing taxable corporate profits will
prevent the arbitrage process from making the value of all firms in a
given class proportional to the expected returns generated by their
434
THE AMERICAN ECONOMICREVIEW
physical assets. Instead, it can be shown (by the same type of proof
used for the original version of Proposition I) that the market values
of firms in each class must be proportional in equilibrium to their expected returns net of taxes (that is, to the sum of the interest paid and
expectednet stockholderincome). (Italics added.)
The statement in italics, unfortunately, is wrong. For even though one
firm may have an expectedreturn after taxes (our Xr) twice that of another
firm in the same riskequivalent class, it will not be the case that the actual
return after taxes (our X7) of the first firm will always be twice that of the
second, if the two firms have different degrees of leverage.’ And since the
distribution of returns after taxes of the two firms will not be proportional,
there can be no “arbitrage” process which forces their values to be proportional to their expected aftertax returns.2 In fact, it can be shownand
this time it really will be shownthat “arbitrage” will make values within
any class a function not only of expected aftertax returns, but of the tax
rate and the degree of leverage. This means, among other things, that the
tax advantages of debt financing are somewhat greater than we originally
suggested and, to this extent, the quantitative difference between the valuations implied by our position and by the traditional view is narrowed. It
still remains true, however, that under our analysis the tax advantages of
debt are the only permanent advantages so that the gulf between the two
views in matters of interpretation and policy is as wide as ever.
I. Taxes, Leverage,and the Probability Distribution of AfterTax Returns
To see how the distribution of aftertax earnings is affected by leverage,
let us again denote by the random variable X the (longrun average) earnings before interest and taxes generated by the currently owned assets of a
given firm in some stated risk class, k.3 From our definition of a risk class it
follows that X can be expressed in the form XZ, where X is the expected
value of X, and the random variable Z= X/X, having the same value for
all firms in class k, is a drawing from a distribution, sav fk(Z). Hence the
1 With some exceptions, which will be noted when they occur, we shall preserve here both
the notation and the terminology of the original paper. A working knowledge of both on the
part of the reader will be presumed.
2 Barring, of course, the trivial case of universal
linear utility functions. Note that in deference to Professor Durand (see his Comment on our paper and our reply, this Review,Sept. 1959,
49, 63969) we here and throughout use quotation marks when referring to arbitrage.
3 Thus our X corresponds essentially to the familiar EBIT concept of the finance literature.
The use of EBIT and related “income” concepts as the basis of valuation is strictly valid only
when the underlying real assets are assumed to have perpetual lives. In such a case, of course,
EBIT and “cash flow” are one and the same. This was, in effect, the interpretation of X we
used in the original paper and we shall retain it here both to preserve continuity and for the
considerable simplification it permits in the exposition. We should point out, however, that
the perpetuity interpretation is much less restrictive than might appear at first glance. Beforetax cash flow and EBIT can also safely be equated even where assets have finite lives as soon
as these assets attain a steady state age distribution in which annual replacements equal
annual depreciation. The subject of finite lives of assets will be further discussed in connection
with the problem of the cutoff rate for investment decisions.
COMMUNICATIONS
435
random variable XA, measuring the aftertax return, can be expressed as:
(1) Xr = (1 r)(XR)
where
r
+ R = (1 r)X
+ rR = (1 r)XZ
is the marginal corporate income tax rate (assumed equal to the
average), and R is the interest bill. Since E(XT) =XT=
X
(X
(1T)X+TR
we can
in (1) to obtain:
substitute XfTY’rRfor (1r)X
(2)
+ rR
X'( I
R)Z + rR
) Z+
Thus, if the tax rate is other than zero, the shape of the distribution of XK
will depend not only on the “scale” of the stream XTand on the distribution
of Z, but also on the tax rate and the degree of leverage (one measure of
which is R/XT). For example, if Var (Z)
Var (XY)
=
q2(X)2
= f2,
(1
–
we have:
r _
implying that for given Xr the variance of aftertax returns is smaller, the
higher r and the degree of leverage.4
II. The Valuation of AfterTax Returns
Note from equation (1) that, from the investor’s point of view, the longrun average stream of aftertax returns appears as a sum of two components: (1) an uncertain stream (1r)XZ; and (2) a sure stream rR.5
This suggests that the equilibrium market value of the combined stream
can be found by capitalizing each component separately. More precisely,
let p7 be the rate at which the market capitalizes the expected returns net
of tax of an unlevered company of size X in class k, i.e.,
pr
(1 – )X
F
Vu
~or
Vu=
(I)X
6
PT
4 It may seem paradoxical at first to say that leverage reduces the variability of outcomes,
but remember we are here discussing the variability of total returns, interest plus net profits.
The variability of stockholder net profits will, of course, be greater in the presence than in the
absence of leverage, though relatively less so than in an otherwise comparable world of no
taxes. The reasons for this will become clearer after the discussion in the next section.
5 The statement that rRthe tax saving per period on the interest paymentsis a sure
stream is subject to two qualifications. First, it must be the case that firms can always obtain
the tax benefit of their interest deductions either by offsetting them directly against other
taxable income in the year incurred; or, in the event no such income is available in any given
year, by carrying them backward or forward against past or future taxable earnings; or, in the
extreme case, by merger of the firm with (or its sale to) another firm that can utilize the deduction. Second, it must be assumed that the tax rate will remain the same. To the extent that
neither of these conditions holds exactly then some uncertainty attaches even to the tax
savings, though, of course, it is of a different kind and order from that attaching to the stream
generated by the assets. For simplicity, however, we shall here ignore these possible elements
of delay or of uncertainty in the tax saving; but it should be kept in mind that this neglect
means that the subsequent valuation formulas overstate, if anything, the value of the tax
saving for any given Dermanent level of debt.
6 Note that here, as in our original paper, we neglect dividend policy and “growth” in the
THE AMERICAN ECONOMICREVIEW
436
and let r be the rate at which the market capitalizes the sure streams generated by debts. For simplicity, assume this rate of interest is a constant
independent of the size of the debt so that
R
r=
or
D
D=*7
R
r
Then we would expect the value of a levered firm of size X, with a permanent level of debt DL in its capital structure, to be given by:
(I
VL = (=
(3)
–
TR
T)X
)DLX+
VU +
r
Pr
In our original paper we asserted instead that, within a risk class, market
value would be proportional to expected aftertax return XT(cf. our original
equation [1 1]), which would imply:
Xr
(4)
VL
(1
=
pr
r
TR
T)X
–
+
p
–
= VU +–TDL.
pr
pr
We will now show that if (3) does not hold, investors can secure a more
efficient portfolio by switching from relatively overvalued to relatively
undervalued firms. Suppose first that unlevered firms are overvalued or that
VL –
TDL < VU.
An investor holding m dollars of stock in the unlevered company has a right
to the fraction m/Vu of the eventual outcome, i.e., has the uncertain income
Yu
=

(1
(
)
Z.
Consider now an alternative portfolio obtained by investing m dollars as
follows: (1) the portion,
SL
VSL+ (1  'r)D
/
is invested in the stock of the levered firm, SL; and (2) the remaining portion,
(1
T)DL
SL + (1 
T)DL/
sense of opportunities to invest at a rate of return greater than the market rate of return. These
subjects are treated extensively in our paper, "Dividend Policy, Growth and the Valuation of
Shares," Jour. Bus., Univ. Chicago, Oct. 1961, 41133.
7Here and throughout, the corresponding formulas when the rate of interest rises with leverage can be obtained merely by substituting r(L) for r, where L is some suitable measure of
leverage.
8 The assumption that the debt is permanent is not necessary for the analysis. It is employed
here both to maintain continuity with the original model and because it gives an upper bound
on the value of the tax saving. See in this connection footnote 5 and footnote 9.
437
COMMUNICATIONS
is invested in its bonds. The stock component entitles the holder to a fraction,
m
SL + (1 
T)DL
of the net profits of the levered company or
(SL + (1 T)DL) [(1  r)(XZ  RL)].
The holding of bonds yields
m
(SL + (1 
T)DL)
~~[(1r)RL].
Hence the total outcome is
((SL + (1  T)DL))
[(
)XZ]
and this will dominate the uncertain income Yu if (and only if)
SL +
rDL
r)DL  SL + DL 
(1 
VL 
1DL < VU.
Thus, in equilibrium, Vu cannot exceed VL1DL, for if it did investors
would have an incentive to sell shares in the unlevered company and purchase the shares (and bonds) of the levered company.
Vu. An investment of m dollars in the stock
Suppose now that VLTDL>
of the levered firm entitles the holder to the outcome
YL
=
–
[(1
(M/SL)
= (m/SL) (1 –
T) (XZ – RL)]
)XZ
–
(1 –
(m/SL)
T)RL.
Consider the following alternative portfolio: (1) borrow an amount
for which the interest cost will be (Mn/SL)(1r)RL
(M/SL)(1 r)DL
(assuming, of course, that individuals and corporations can borrow at the
same rate, r); and (2) invest m plus the amount borrowed, i.e.,
m
m(1 – r)DL
+
SL
=
M
SL + (1 T)DL
=
(M/SL)1VL’TDL]
SL
in the stock of the unlevered firm. The outcome so secured will be
(MI/SL) (VL
UTDL)
)XZ.
(1
Subtracting the interest charges on the borrowed funds leaves an income of
Y u = (m/SL) (L
L) (1 –
T)XZ
–
(M/SL)(1
–
T)RL
which will dominate YL if (and only if) VL1DL> Vu. Thus, in equilibrium,
both VLT DL> Vu and VLT DL< Vu are ruled out and (3) must hold.
438
THE AMERICAN ECONOMIC REVIEW
III. Some Implications of Formula (3)
To see what is involved in replacing (4) with (3) as the rule of valuation,
note first that both expressions make the value of the firm a function of
leverage and the tax rate. The difference between them is a matter of the
size and source of the tax advantages of debt financing. Under our original
formulation, values within a class were strictly proportional to expected
earnings after taxes. Hence the tax advantage of debt was due solely to the
fact that the deductibility of interest payments implied a higher level of
aftertax income for any given level of beforetax earnings (i.e., higher by
the amount rRsince Yr= (1 r)Y+rR). Under the corrected rule (3), however, there is an additional gain due to the fact that the extra aftertax
earnings, rR, represent a sure income in contrast to the uncertain outcome
(1r)X. Hence rR is capitalized at the more favorable certainty rate,1/r,
rather than at the rate for uncertain streams, 1/p'.9
Since the difference between (3) and (4) is solely a matter of the rate at
which the tax savings on interest payments are capitalized, the required
changes in all formulas and expressions derived from (4) are reasonably
straightforward. Consider, first, the beforetax earnings yield, i.e., the ratio
of expected earnings before interest and taxes to the value of the firm.10
Dividing both sides of (3) by V and by (1r) and simplifying we obtain:
(31.c)
X
V
1
pr
1
DT
which replaces our original equation (31) (p. 294). The new relation differs
from the old in that the coefficient of D/V in the original (31) was smaller
by a factor of r/p7.
Consider next the aftertax earnings yield, i.e., the ratio of interest payments plus profits after taxes to total market value." This concept was discussed extensively in our paper because it helps to bring out more clearly
the differences between our position and the traditional view, and because
it facilitates the construction of empirical tests of the two hypotheses about
the valuation process. To see what the new equation (3) implies for this
yield we need merely substitute XTR for (1r)X in (3) obtaining:
9 Remember, however, that in one sense formula (3) gives only an upper bound on the value
of the firm since rR/r =rD is an exact measure of the value of the tax saving only where both
the tax rate and the level of debt are assumed to be fixed forever (and where the firm is certain to be able to use its interest deduction to reduce taxable income either directly or via
transfer of the loss to another firm). Alternative versions of (3) can readily be developed for
cases in which the debt is not assumed to be permanent, but rather to be outstanding only
for some specified finite length of time. For reasons of space, we shall not pursue this line of
inquiry here beyond observing that the shorter the debt period considered,the closer does the
valuation formula approach our original (4). Hence, the latter is perhaps still of some interest
if only as a lower bound.
10Following usage common in the field of finance we referred to this yield as the "average
cost of capital." We feel now, however, that the term "beforetax earnings yield" would be preferable both because it is more immediately descriptive and because it releases the term "cost
of capital" for use in discussions of optimal investment policy (in accord with standard usage
in the capital budgeting literature).
11We referred to this yield as the "aftertax cost of capital." Cf. the previous footnote.
439
COMMUNICATIONS
R
Xr 
(5)
V
Xr
+ rD =+
pT
r
pr

pr
D,
pT?
from which it follows that the aftertax earnings yield must be:
(11.c)
V



 r) /DV.
This replaces our original equation (11) (p. 272) in which we had simply
XIY/V=p'. Thus, in contrast to our earlier result, the corrected version
(11.c) implies that even the aftertax yield is affected by leverage. The
predicted rate of decrease of XTV with D/V, however, is still considerably
smaller than under the naive traditional view, which, as we showed, implied
See our equation (17) and the discussion
essentiallyTXr/V=pr(pr_r)D/V.
immediately preceding it (p. 277).12 And, of course, (11.c) implies that the
effect of leverage on XT/V is solely a matter of the deductibility of interest
payments whereas, under the traditional view, going into debt would lower
the cost of capital regardless of the method of taxing corporate earnings.
Finally, we have the matter of the aftertax yield on equity capital, i.e.,
the ratio of net profits after taxes to the value of the shares.13By subtracting D from both sides of (5) and breaking XT into its two componentsexpected net profits after taxes, *r7,and interest payments, RrDwe
obtaiin after simplifying:
S
VD
(1(6)
D.
p
p
From (6) it follows that the aftertax yield on equity capital must be:
7r
(I2.c)
X
=Pr + (1
T) [p'rr]DIS
(p. 272).
which replacesour originalequation (12), r7#S==pr+(prr)D/S
The new (12.c) implies an increase in the aftertax yield on equity capital
as leverage increases which is smaller than that of our original (12) by a
factor of (1 ...
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