J. F. MCDONALD

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

bankruptcy.

• The expected rate of return to equity invested in

the firm

e

ER

is equal to the expected rate of

return in the absence of borrowing

ER

plus

an amount that is a linear function of the ratio of

debt to equity. That function is

( )( )()

ERERER rDS

= +−

, (1)

f

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

is

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-

rowing.

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)

where

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

where

and

.

Substitution of these definitions into Equation (2)

produces

() () ()

2 1212LD

YVVXVVDRR=+−

. (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

. (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

. (5)

It is well known that the borrowing rate in real estate