Modern Economy, 2011, 2, 169-173
doi:10.4236/me.2011.22022 Published Online May 2011 (
Copyright © 2011 SciRes. ME
The Modigliani-Miller Theorem with Financial
John F. McDonald
Walter E. Heller College of Business Administra tio n, Roosevelt University, Chicago, USA
Received January 27, 2011; revised March 6, 2011; accepted April 3, 2011
This paper sh ows that, if firms borrow a t an interest rate that is g reater than the rate at wh ich they can lend,
the value of a firm declines with the amou nt bo rrowed. Th e model assumes the po ssibility that a f irm may go
bankrupt, which introduces the need for financial intermediation. A modified version of the homemade leve-
rage examples introduced by Modigliani and Miller [1] is used to introduce the concept. A state-preference
model is used for a more formal proof.
Keywords: Modigliani-Miller Theorem, Financial Intermediation, Valuation
1. Introduction
The purpose of this paper is to investigate the effect of
borrowing on the value of firms (and other financial in-
vestments). Even investment entities such as real estate
investment trusts that are not subject to income taxation
at the entity level borrow. For example, Chan, Erickson,
and Wang [1] found that 187 U. S. real estate investment
trusts in 2000 borrowed an average of 50% measured as
long-term debt to total capital and 46% measured as
long-term debt to total market capitalization. These are
entities for which there is no tax advantage in borrowing.
According to the classic Modigliani-Miller theorem [2],
borrowing by these untaxed entities has no effect on
market value. However, this conclusion depends upon
the assumption that the interest rate at which the real
estate investing entity can borro w equals the rate at which
the entity can lend. Alteration of this assumption pro-
duces a different conclusion; if the borrowing rate is
greater than the lending rate, then the value that the real
estate investing entity places upon a particular property
will decline with the amount borrowed.
The original article by Modigliani and Miller [2] very
recently was named by Arrow, et al. [3] one of the top
twenty articles that appeared in the first one hundred
years of the American Econom ic Review. The citation for
this award [2] reads in part.
The paper’s central result is that, in a setting with
complete capital markets and in the absence of tax-in-
duced distortions, a firm’s total market value is invariant
to its borrowing behavior. This powerful result can be
demonstrated constructively, by developing a straight-
forward set of borrowing or lending transactions that an
equity investor can undertake to offset the consequences
of changes in corporate borrowing. The analytical ap-
proach in this paper is one of the key foundations for the
modern field of financial economics.
The purpose of this paper is to add the service of fi-
nancial intermediation to the Modigliani-Miller model.
It is now widely recognized that the supply of finan-
cial intermediation services is an important element to
include in financial models. Woodford [4] has developed
a macroeconomic model that includes financial inter-
mediation in order to explain important aspects of the
most recent financial crisis and recession. The basic idea
is that there exists a loan production function that re-
quires the use of real resources, and that the volume of
lending by intermediaries involves diminishing returns to
increases in variable inputs (rising marginal costs) be-
cause of fixity of some inputs such as specialized exper-
tise. Models of this kind have been developed by Good-
friend and McCallum [5], for ex a mple.
Joseph Stiglitz [6] reexamined the Modigliani-Miller
propositions, and found that two assumptions of their
model are important for their proof; individuals and firms
borrow at the same interest rate, and there is no bank-
ruptcy. He states [6] that, “It should be clear that these
assumptions are not independent.” However, he did not
pursue the possibility that bankruptcy introduces the
need for financial intermediaries to provide the service of
Copyright © 2011 SciRes. ME
qualifying and monitoring borrowers and their invest-
ments. A model that incorporates these elements is pro-
vided in this paper.
Considerations of financial leverage make use of the
propositions of Modigliani and Miller [2]. MM Proposi-
tions I and II are as follows.
The market value of a firm is independent of its
capital structure. The basic proposition was dem-
onstrated assuming no taxation at the firm level, no
bankruptcy, and a constant borrowing and lending
rate (but also was demonstrated for the case in
which the borrowing and lending rate increases
with financial leverage in exactly the same rate for
all firms and individuals). Alternatively, the aver-
age cost of capital is indep endent of financial leve-
rage. Stiglitz [6] and Sargent [7] provided a more
general proof of Proposition I in the absence of
The expected rate of return to equity invested in
the firm
( )
is equal to the expected rate of
return in the absence of borrowing
( )
an amount that is a linear function of the ratio of
debt to equity. That function is
( )
( )( )()
= +−
, (1)
is the risk-free borrowing and lending rate, D is
debt, and S is equity.
( )
is the rate of return to the
asset in the absence of borrowing and
( )
ER r
the risk premium for the investment without leverage.
The expected rate of return to equity is increased by
borrowing if the expected rate of return to the investment
without borrowing exceeds the rate of interest on bor-
2. A Real Estate Investment Example
The central point in this paper is that the value of an in-
vestment is not independent of its capital structure be-
cause the borrowing rate is greater than the lending rate,
especially if the borrowing rate increases with the loan-
to-value ratio. The reason for this difference in the bor-
rowing and lending rates is fundamental. Much of the
lending in an economy is provided by financial institu-
tions that provide the service of financial intermediation;
transforming assets that are less desirable into asse ts that
are preferred by the public (i.e., their own liabilities).
Financial intermediaries are in the business of borrowing
on a short-term basis and making long-term loans (ma-
turity intermediation), risk reduction through diversifica-
tion, providing low-cost contracts, and facilitating pay-
ments. In doing so, they undertake risks—interest rate
risk and default risk. Lenders examine the quality of the
borrowers and the purposes for which they wish to bor-
row, and monitor the performance of the borrower. Other
lending is accomplished in the form of bonds issued by
firms. Firms use the services provided by financial insti-
tutions and bond rating agencies. The difference in the
borrowing and lending rates reflects the value of these
specialized services.
Consider a modification of the constructive demon-
stration of homemade leverage used by Modigliani and
Miller [2] f or MM Proposition I. This example introduc-
es a borrowing rate that is greater than the lending rate,
but does not introduce bankruptcy. Both of these ele-
ments are included in the model presented in Section 4
below. Suppose an investor owns a property (no bor-
rowing) with value 1
that produces expected annual
income 1
=, which is net operating income plus cap-
ital appreciation over the year. Then suppose that this
investor decides to sell this property and purchase a por-
tion of the equity in another property with expected an-
nual income X that is in the same “risk class,” and lends
the remaining amount of his/her funds to some other in-
vestor (e.g., purchases bonds). The investor’s return from
this alternative investment portfolio is
( )()
2 22DL
, (2)
is the investor’s equity investment, E2 is the
total equity in the p roperty, RD and RL are the borrowing
and lending rates, D is the amount that was borrowed on
the property, and L is the amount lent by the investor.
Under what conditions will the investor’s income from
the new portfolio equal X? We kn ow that V1 = e2 + L and
V2 = E2 + D.
Modigliani and Miller [2] propose homemade leverage
( )
2 221
e EVV=
( )
Substitution of these definitions into Equation (2)
() () ()
2 1212LD
. (3)
The arbitrage condition Y2 = X holds if RL = RD and
thus V1 = V2. This is MM Proposition I. However, if the
borrowing rate that was used exceeds the lending rate
that is available to the investor in question, then Y2 = X if
( )()
V VDXRR=− −<
. (4)
If the borrowing rate is greater than the lending rate
available to the investor, then the value of the property in
the new portfolio must be lower than the property with
no borrowing, and the reduction in value depends upon
the amount that was borrowed and the difference between
the two interest rates. Equation (4) can be rewri tten as
21 LD
. (5)
It is well known that the borrowing rate in real estate
Copyright © 2011 SciRes. ME
increases with th e loan-to-value ratio, so the lending rate
can equal the borrowing rate if somehow the investor
loans to some other real estate investors who have ap-
plied the same degree of leverage as was applied to the
property involved in this example. This is an unlikely
The expected rate of return to equity with a borrowing
rate that exceeds the lending rate is
( )
( )( )()()
ef BL
 . (6)
If the borrowing and lending rates are equal, then Eq-
uation (6) reduces to MM Proposition II. The expected
rate of return to equity must increase with leverage at a
greater rate than in MM Proposition II to compensate for
the higher cost of borrowing.
The other side of the Modigliani-Miller demonstra-
tion of Proposition I [2] involves investors who borrow
on personal account to replicate the leverage of a firm.
Consider an investor that owns a portion α of real estate
property number 2. The annual income of the investor
( )
= −, (7)
where α = e2/E2 (the share of equity owned by the inves-
tor), and D2 is the amount of debt that was used in the
acquisition the property.
Now suppose that the investor sells his/her share in the
investment s2 = αS2 and borrows amount αD2 in order to
purchase property number 1 outright. The income from
this new investment is
( )
12 21*2B
. (8)
Here RB* is the interest rate on the personal loan that is
used. As Modigliani and Miller [2] state,
He could do so by utilizing the amount αS2 realized
from his initial holding and borrowing an additional
amount αD2 on his own credit, pledging his new holdings
in company 1 as a collateral.
By definition V1 = S2 + D2 and V1 = S1.
However, “… pledging his new holdings in company
1 as collateral” is equivalent to taking out a mortgage
loan—as was done to purchase investment number 2.
This means that RB = RB*, and the investor has exactly
replicated investment number 2. The investor has sold
one investment in order to borrow and make another in-
vestment under exactly the same terms—the same bor-
rowing rate schedule and the same amount of leverage.
Under these conditions Y1 = Y2 and V1 = V2. This demon-
stration only shows that V2 cannot be greater than V1
because the two borrowing rates in question are equal.
The previous example shows that borrowing reduces
property value.
3. The Market for Investment Property
Thus far the paper has considered only examples with
two properties with identical expected annual incomes.
Now consider the market for such properties. The supply
of such assets is fixed at a large number, and the market
of perfectly competitive. There are many investing firms
that vary in their appetite for expected return to equity
and risk. Suppose that all of the investing firms are un-
taxed. Market equilibrium is established by the willing-
ness of the marginal investor to pay for the property.
Some untaxed investing entities such as pension funds
and some real estate investment trusts borrow little or
nothing and have high reservation prices for the property.
Other investors borrow a great deal of money because
they wish to increase the expected rate of return to equity
(or are equity constrained).
Figure 1 displays an example of equilibrium in the
market for property in a particular risk class. Supply is
fixed at S. The demand for this type of property (by the
untaxed investment entities) has a horizontal portion at
VD = 0 for those entities that do not borrow, and then the
demand price declines with quantity as entities that bor-
row are added—in the order of the amount that they
choose to borrow. Equilibrium is established at V*. The
market fixes the maximum amount D* that is borrowed
by the successful marginal bidder. This marginal bidder
earns a return to equity equal to
( )
( )( )( )( )
ef BL
. (9)
Market value declines with the amount borrowed by
the marginal buyer. The expected rate of return to the in-
vestment in the absence of borrowing is
( )
The fact that the marginal investor borrows enhances the
expected rate of return for the infra-marginal investors.
Val ue
D = 0
Figure 1. Market for investment property.
Copyright © 2011 SciRes. ME
4. A State-Preference Presentation
A more formal proof of the basic proposition can be
formulated using the state-preference approach adopted
by Stiglitz [6] and Sargent [7]. Assume that there is only
one date in the “future,” and that there are N possible
future states of the world. The index of future states of
the world is
1, 2,,N
. An individual has a concave
utility function U, and utility depends upon the future
state of the world and the amount of money M in his/her
possession at that time:
( )
. (10)
The individual has a set of subjective probabilities
over the states of the world
( )()()
π1,π2, ,
sum to 1.0. Individual are assumed to maximize expected
utility V:
( )()
πv UM
. (11)
The individual is assumed to have an endowment of
M0 at the present that is invested to provide for future
Consider a competitive economy in which there are N
markets for contingent (Arrow-Debreu) securities, where
each one promises to pay one dollar if the corresponding
state of the world θ occurs. The price of a security,
( )
, is the price of the claim on one dollar s hould s tate
θ occur. The units of
( )
are dollars in the current
period per dollar in state θ in the future. The price of a
certain dollar in the future is
( )
, which is the
reciprocal of one plus the risk-free interest rate. Perfect
markets for contingent secur ities in all states o f the world
mean that it is possible to insure against any risk.
Now consider firms that produce output that individu-
als purchase in the future. We assume an absence of tax-
es. A firm produces a return net of current labor and ma-
terials costs that depends upon the state of the world;
( )
. The firm issues bonds in the amount of B dollars,
and promises now to pay
( )
1B rc+−
to its bond hold-
ers at the future date, provided that the firm is not bank-
rupt at that time; i.e.,
( )()
1XB rc
≥ +−
. The rate at
which the firms borrow is r, and the rate that it is paid as
a lender is
. The firms goes bankrupt if
( )
( )
, so the realized returns to bonds depend upon
the state of the world as follows:
( )( )()
( )( )()
1 if 1
1 if 1
rc XBr
=+−≥ +
=− <+
The model includes possible bankruptcy so that there
is a need for financial intermediation. The amount cB is
the cost of providing the financial intermediation services
in which it was determined that the firm was in fact eli-
gible to borrow amount B. It is assumed that this cost
must be paid.
The value of the firm’s bonds is equal to the sum of
the values of the contingent securities on which the bond
consists implicitly. States of the world in which the firm
does not go bankrupt are indexed as
( )
, and states of
the world in which the firm goes bankrupt are indexed as
( )
. The value of the firm’s bonds to the lenders is:
( )( )
( )
( )
( )
{ }
( )
Br cBp
BXB cp
The value of the firm’s equity is:
( )()
( )
( )
E XrBp
= −+
. (14)
Therefore, the value of the firm V is:
( )()
, so
0 and 0
=+= −
∂∂< ∂∂<
The value of the firm decreases with both the amount
borrowed and the cost of financial intermediation. If the
borrowing and lend ing rates are equa l, then c = 0 and the
value of the firm does not depend upon borrowing. This
is, of course, MM Propo sition I. Introduction of a corpo-
rate income tax on the firm (with deductions for interest
payments) generates the conclusion that the cost of fi-
nancial intermediation tends to be offset by the tax ad-
vantages of borrowing.
5. Conclusion
This paper has shown that, if the borrowing rate exceeds
the lending rate (as in the case of financial intermediation
services), then the value of a firm declines with financial
leverage. The value of the firm is reduced by the cost of
the financial intermediation services. If the borrowing
rate and the lending rates are equal, then the value of the
firm is independent of financial leverage, as in Modig-
liani-Miller Proposition I. This proposition holds in the
presence of the possibility of firm bankruptcy, but it
would seem that the possibility of firm bankruptcy
creates the need for financial intermediation, which has
real resource costs creating a borrowing rate that exceeds
the lending rate.
6. References
[1] S. Chan, J. Erickson and K. Wang, Real Estate Invest-
ment Trusts: Structure, Performance, and Investment Op-
portunities,O xford University Press, New York, 2003.
[2] F. Modigliani and M. Miller, “The Cost of Capital, Cor-
Copyright © 2011 SciRes. ME
poration Finance, and the Theory of Investment,” The
American Economic Review, Vol. 48, No. 3, June 1958,
pp. 261-297.
[3] K. Arrow, B. Bernheim, M. Feldstein, D. McFadden, J.
Poterba and R. Solow, “100 Years of the American Eco-
nomic Review: The Top 20 Articles,” American Eco-
nomic Review, Vol. 101, No. 1, 2011, pp. 1-8.
[4] M. Woodford, “Financial Intermediation and Macroeco-
nomic Analysis,” Journal of Economic Perspectives, Vol.
24, No. 4, 2010, 21-44. doi:10.1257/jep.24.4.21
[5] M. Goodfriend and B. McCallum, “Banking and Interest
Rates in Monetary Policy Analysis: A Quantitative Ex-
ploration,” Journal of Monetary Economics, Vol. 54, No.
5, July 2007, pp. 1480-1507.
[6] J. Stiglitz, “A Re-examination of the Modigliani-Miller
Theorem,” The American Economic Review, Vol. 59, No.
5, December 1969, pp. 784-793.
[7] T. Sargent, Macroeconomic Theory,” 2nd E di t ion, Aca-
demic Press, Orlando, 1987.