Theoretical Economics Letters, 2012, 2, 361-364
http://dx.doi.org/10.4236/tel.[phone] Published Online October 2012 (http://www.SciRP.org/journal/tel)
The Modigliani-Miller Theorem for Equity Participation
John F. McDonald
Heller College of Business, Roosevelt University, Chicago, USA
Received June 19, 2012; revised July 16, 2012; accepted August 17, 2012
ABSTRACT
The paper shows that the use of an equity participation loan has no effect on the value of the firm, and that taxation of
the borrowing firm and lender reduces firm value. The paper includes the assumption that firms borrow at an interest
rate that is greater than the rate at which they can lend, so the value of the firm declines with the amount borrowed. Also,
it is assumed that the firm may go bankrupt, which introduces the need for financial intermediation, as discussed by
McDonald [1]. A state-preference model is employed.
Keywords: Equity Participation; Valuation; Modigliani-Miller Theorem
1. Introduction
The purpose of this paper is to investigate the effect of
equity participation and taxation on the value of firms
(and other financial investments). The paper also in-
cludes the assumption that the rate of interest at which a
firm can borrow exceeds the interest rate at which it can
lend. According to the classic Modigliani-Miller theorem
[2], borrowing by untaxed entities has no effect on mar-
ket value. However, this conclusion depends upon the
assumption that the firm’s borrowing rate equals its
lending rate. Alteration of this assumption produces a
different conclusion; if the borrowing rate is greater than
the lending rate, then the value of the firm will decline
with the amount borrowed. However, the value of the
untaxed firm is unchanged by equity participation even if
the borrowing rate and lending rate are unequal. Conver-
sion of some debt to equity increases the value of the
firm, and the value of the firm is reduced by taxation. A
state-preference model is employed to establish these pro-
positions.
It is now widely recognized that the supply of financial
intermediation services is an important element to in-
clude in financial models. Joseph Stiglitz [3] 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 bankruptcy. He states [3, p. 784] that, “It
should be clear that these assumptions are not independ-
ent.” However, he did not pursue the possibility that
bankruptcy introduces the need for financial intermedi-
aries to provide the service of qualifying and monitoring
borrowers and their investments. A model that reviews
literature and incorporates these elements was provided
by McDonald [1].
The topic of equity participation does not have an ex-
tensive literature. Theoretical studies consist of Schnabel
[4], Ebrahim [5], Alvayay, Harter, and Smith [6], and
Ebrahim, Shackketon, and Wojakowski [7]. Schnabel [4]
used an agency-theoretic model to demonstrate that an
equity-participation feature can increase the value of a
firm and control the under-investment problem. Ebrahim
[5] showed that equity participation can improve social
welfare through the sharing of ownership and risk.
Alvayay, Harter, and Smith [6] used option-pricing
theory to develop a model of the extent of lender equity
participation. Their model of lender behavior is based on
the assumption that a portion of standard debt is con-
verted to equity for the lender, and that the value of
lender’s asset remains constant. This procedure combines
two factors—the decision to undertake equity participa-
tion and the decision to reduce the amount of standard
debt. These actions are treated separately in this paper.
The Alvayay-Harter-Smith model is not used to explore
the effects of equity participation on the value of the
borrower’s asset and its expected rate of return. Nor is
the model used to investigate the effect of equity partici-
pation on the reservation value of the property involved.
Ebrahim, Shackleton, and Wojakowski [7] also employ
option-pricing theory to model the value of participation
mortgage loans. Their model of lender behavior also is
based on the assumptions that equity participation is
combined with a reduction in the size of the standard
loan and that the lender’s asset has a constant value with
or without equity participation. They find that a higher
level of participation is associated with a greater reduc-
tion in the interest rate on the loan, but that the reduction
C
opyright © 2012 SciRes. TEL

J. F. MCDONALD 362
in the interest rate is moderated by a higher loan-to-value
ratio. A major focus of their study is the effect of time to
maturity on the level of participation; the effect is
strongly negative. The question of the effect of time to
maturity is not addressed in this paper.
2. State-Preference Presentation of the Base
Case
A demonstration of the basic model can be formulated
using the state-preference approach as adopted by Stiglitz
[3], Sargent [8], and McDonald [1] to illustrate the Mod-
igliani-Miller Proposition I. This section is a more de-
tailed discussion of a model presented briefly in McDon-
ald [1]. 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 θ = 1, .
An individual has a concave utility function U, and util-
ity depends upon the future state of the world and the
amount of money M in his/her possession at that time:
2, N


UUM

 

(1)
The individual has a set of subjective probabilities
over the states of the world π(1), i(2), ,π(n) that sum
to 1.0. Individual are assumed to maximize expected util-
ity V:

 
πVUM


 




 (2)
The individual is assumed to have an endowment of M0
at the present that is invested to provide for future con-
sumption.
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,

p

, is the price of the claim on one dollar should state
θ occur. The units of

p

are dollars in the current
period per dollar in state θ in the future. The price of a
certain dollar in the future is

p


, which is the
reciprocal of one plus the risk-free interest rate. Perfect
markets for contingent securities in all states of the world
mean that it is possible to insure against any risk.
The individual faces the Arrow-Debreu budget con-
straint that states:
 
0
MpM


  (3)
Since it is assumed that a complete set of markets for
contingent securities exists, and if there is general agree-
ment about the probabilities for the future states of the
world, the prices for the contingent securities are actuari-
ally fair; i.e.,
   
ππ
ij i
pp
where


i
p

and


j
p

are the prices of the contin-
gent securities for states of the world i and j, and


πi

and


π
j

are the probabilities of states of the world i
and j. Maximization of utility subject to the budget con-
straint produces the condition for the marginal rate of
substitution between money in any two states of the world
is


    

 

π'π'
i
j
iij
ij
V
M
MRS
V
M
UM UM
pp


j

 














(5)
Since the market for contingent securities is actuarially
fair, this first-order condition reduces to





''
i
UM UM

j

  (6)
Assuming that the utility function is the same regard-
less of the future state of the world, Equation (6) means
that




i
MM
j

 . In short, the individual acts to
insure that the level of money in the future is the same
amount regardless of the future state of the world. The
individual invests M0 in Arrow-Debreu securities based
on equity in firms and bonds to generate this amount of
money in the future, and earns the risk-free rate of return.
It is important for understanding the model to recall that
individuals invest in a complete set of Arrow-Debreu
securities based on both stocks and bonds. Firms and
issuers of bonds are intermediaries that provide the in-
vestment vehicles upon which Arrow-Debreu securities
are based.
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;


X

. The firm issues bonds in the amount of B dollars,
and promises now to pay to its bond hold-
ers (the individual investors) at the future date, provided
that the firm is not bankrupt at that time; i.e.,

B1 rc





1
X
Brc
r

rc
. The rate at which firms and indi-
viduals borrow is r, and the rate that firms pay its lenders
is

. The firms goes bankrupt if


1

X
Br 

,
so the realized returns to bonds depend upon the state of
the world as follows.




  
11 if 1 or
1
rcrcX Br
X
BcifXB r


 

 
 
(7)
j

  (4)

J. F. MCDONALD 363
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 in full unless
 
1
X
Bc  ; in this
case the lender suffers a loss. The case in which the fu-
ture value of the firm is less than the outstanding balance
of the loan occurs if

X
B   (in real estate known as
being under water).
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 θ(a), and states of
the world in which the firm goes bankrupt are indexed as
θ(b). The value of the firm’s bonds to the lenders is:

  
() ()
1.
Lab
X
BrcBpB cp
B







 
 



 

(8)
The price vector p(θ) is set by the market so that the
lender earns the risk-free rate of return. The value of the
firm’s equity is:




–1 .
a
EXrBp


  

 (9)
Therefore, the value of the firm V is:




–
0 and 0.
L
VEBX cBp
VB Vc

  

 
so
(10)
The value of the firm decreases with both the amount
borrowed and the cost of financial intermediation. If the
borrowing and lending rates are equal, then c = 0 and the
value of the firm does not depend upon borrowing. This
is, of course, Modigliani-Miller Proposition I.
3. Equity Participation
Equity participation involves the lender accepting a re-
duction in the interest rate on the loan for a share of the
return to equity. Assume that the share of the return to
equity for the lender is s and the new (lower) interest rate
on the loan is r*. The value of that equity share EL is

 
,
La
EsXp


  (11)
and the value of the firm’s equity is now




**
11
a
EsXrB

.p





 (12)
The value of the firm’s bond to the lender is







**
1
.
La
b
BrcBp
BXBcp








 


 (13)
Therefore the value of the firm is now



** –.
LL
VE EBXcBp


  


(14)
The value of the firm is unchanged, and the terms of
the equity participation do not matter (i.e., the choices of
s and r*). Indeed, the interest rate on the loan r* could be
greater than r and still the value of the firm is unchanged.
The conversion of a portion of debt to equity is a trans-
action with two parts. As shown in Equation (10) above,
a reduction in debt increases the value of the firm if the
cost of financial intermediation is a function of the amount
of the debt. The equity participation portion of the trans-
action has no effect on firm value. So conversion of debt
to equity increases firm value in this model.
However, the lender will place a minimum condition
on the terms of equity participation such that the same
level of money is provided as with no equity participa-
tion. That condition is:
*
L
LL
BEB (15)
From Equations (8), (11), and (13), the lender’s condi-
tion reduces to









*
–0
aa
sXp rrBp

 



 (16)
In short, the value of the lender’s equity must equal the
reduction in the change in the value of the bond arising
from the change in the interest rate. This demonstrates
that the lender will charge a lower rate of interest in ex-
change for equity participation. Equation (16) implies an
explicit trade-off between and lender’s share of equity s
and the reduction in the interest rate (r – r*).
4. Taxation
This section adds the corporate income tax to the basic
model in Section 2. Both the lender and the borrowing
firm are subject to tax rate t. The value of the firm’s eq-
uity is:



 

0
111.
a
Ep
X
ttEr tB







(17)
Here E0 is the original equity investment. Equation (17)
states that the income and any capital gain for equity are
taxed at rate t, and that interest on the loan is a deductible
expense. The value of the bond for the lender is:







 


11
(1)
La
b
BrctBp
XcBtXBcp





  

  
 


(18)
Equation (18) shows that the lender pays taxes on in-
terest earned after expenses if the firm is not bankrupt,
pays taxes if




1
X
Bc  , and receives a tax de-
duction on the loss if


1
X
Bc  .

J. F. MCDONALD 364

B

 

The value of the firm is V = E + BL, or:

  




0
11
(1)
a
b
VpXttEct
XcBttBp




 

 


 (19)
 






0
–1
ab
XcBtp
tE ptBp






 




(20)
Compare Equation (20) to Equation (10), the value of
the firm in the absence of taxation. Taxation reduces the
value of the firm. If the firm is not bankrupt
  
1
X
Br  , so the return to the firm after expenses
exceeds the equity investment E0 and


tX cB  


 
1


exceeds tE0. If the firm is bankrupt

X
Br  and
it exists no longer. The value of a loan made to a bank-
rupt firm is a more complex question. The question in
this case is whether


tBt X  




cB


is less than or
greater than zero; if this expression is negative the value
of the firm clearly is smaller with taxation. However, if
is positive it is possible that the val-
ue of the firm is greater with taxation than in the ab-
sence of taxation. This condition can be rewritten as

tBt XcB  



 
10 oBcX   r0.
,
,
(21)
From Equation (7) there are four cases:
 
  
 

1()1,so 1
1,so 1
,so 1, and
,so 1.
BrXBc BcX
BcXB BcX
BXcBBc X
cB XBcX





 
 
 

(22)
In the first of these cases taxation reduces the value of
the firm, but in the last three cases taxation might actu-
ally increase the value of the firm. It seems unlikely that
these cases can outweigh the other cases so that taxation
produces an increase in the value of the firm. However, if
there is a high probability that the borrowing firm will go
bankrupt (so the equity has no value) and also be unable
to pay back the loan plus the cost of financial intermedia-
tion
 
B1 cX

 
 
, then the imposition of taxation
with deductions for losses will increase the value of the
loan to the lender. Do lenders that are subject to taxation
grant more “bad” loans than lenders that are not subject
to taxation? Empirical studies are needed. Equation (20)
reduces to Modigliani-Miller Proposition I if both c and t
are zero.
5. Conclusion
Equity participation is an arrangement in which a lender
agrees to reduce the interest rate on a loan to a borrowing
firm in exchange for a share of the return to equity. The
primary new result in the paper is that use of an equity
participation loan, instead of a conventional loan, has no
effect on the value of the firm. The paper derives the
condition that a lender would impose on an equity par-
ticipation deal; the value of the equity participation must
offset the decline in the interest on the loan. The paper
also shows that the value of the firm is reduced by taxa-
tion at the entity level, although the possibility of bank-
ruptcy makes the analysis somewhat complex. The paper
has included the result from McDonald [1] 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 finan-
cial leverage, as in Modigliani-Miller Proposition I. This
proposition holds in the presence of the possibility of
firm bankruptcy.
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