A Theory and Measurement of Cash Equivalents for Lease or
Buy Decisions
Working Paper
6/09/06
Keishiro Matsumoto, Ph.D.
University of the Virgin Islands
6501 Red Hook Plaza, Ste 201
St. Thomas, VI 00802
Phone 340-779-1261
E-Mail:[email protected]
1
A Theory and Measurement of Cash Equivalents for Lease or
Buy Decisions
ABSTRACT
The paper reviews the major techniques of lease or buy decisions over several
decades. The existing techniques are found to suffer from theoretical problems due to the
lack of any formal theory of risk in corporation finance.
To remedy the deficiency, the paper develops a new theory of risk based on the
anticipated return, the operating risk index, the financial risk index, and the expected
project life. Hence, an investment project is a four dimensional vector. The paper
stipulates that a financial manager’s preference relation on a set of investment projects is
a weak order. Imposing the behavioral risk postulates such as the presence of the most
preferred and least preferred project, the sure thing principle, and the generalized
Markowitz criteria, the paper asserts and proves that there is a continuous utility function
which relates the four return and risk attributes of an investment project to its cash
equivalent or coefficient: a cash equivalent or coefficient function.
The paper then illustrates how to measure a cash equivalent function or its
coefficient function from an experiment. It is based on presenting a hypothetical project
at 10 prices to a subject who must either accept or reject the purchase of the project at
each of the 10 prices. 160 dichotomous responses are elicited by repeating this
experiment at 10 hypothetical prices and 16 different experimental settings, By means of
logistic regression, it is possible to estimate the probability of acceptance and that of
rejection at the 16 experimental settings from which 16 cash equivalents or coefficients
are estimated. The cash equivalent function or its coefficient function are estimated by
fitting response surfaces over the 16 experimental settings to the 16 estimated cash
equivalents and coefficients. Finally, a lease or buy decision analysis is conducted where
the cash equivalent method of capital budgeting analysis is applied.
The paper should be of great interest not only to financial economists but also to
behavioral decision theorists and operations researchers, who work in the areas where
the measurement of a cash equivalent or its coefficient plays a key role.
2
A Theory and Measurement of Cash Equivalents for Lease or
Buy Decisions
1. Introduction
A lease analysis used to be one of major subjects in financial
management in 60s and 70s. However, its importance appears to have
greatly declined in the recent past. The topic appears to be no longer
regarded as theoretically stimulating to many researchers in corporation
finance.
This can be reflected in the following statements:
“a lease or buy decision is a straight forward application of
theories developed elsewhere…”.
………
“… the scant empirical literature on leasing
is reviewed.”1
In spite of the fact that leasing may have been viewed
no longer a frontline issue in corporate finance, truth is that there
are many issues left uninvestigated in lease or buy decisions.2 The most
critical issue which has been the main source of problem is the lack of
a theory of risk applicable to investment decision in corporation
finance. The purpose of this paper is to present a new theory of risk
and address unsolved issues in lease or buy decisions by means of the
new theory.
1
The two quotations are found in Chapter 17 of Copeland, Weston, and Shastri
2005.
2
It must be pointed that the lack of their interest does not necessarily imply
that lease or buy decisions are a topic actually exhaustively studied. Academic
researchers are at times akin in behavior to slash and burn farmers. The latter
are constantly on the lookout for new plots with full of nutrients. However,
they tend to desert their plots at the first sign that goings have become tough.
3
The organization of this work is as follows. Section 2 provides
the major reasons why this work is worthy of investigation. Section 3
describes a theory of cash equivalents, which establishes an axiomatic
foundation for cash equivalents. Section 4 discusses how to measure cash
equivalents or its cash equivalent coefficient by means of a lease or
buy case.
Section 5 shows how a lease or buy analysis should be
conducted by means of the cash equivalent method of capital budgeting.
Section 6 summarize what is accomplished in this work and presents
concluding remarks highlighting its key findings.
2. Motivation
Vancil 1963 appears to be one of the earliest researchers in
accounting that investigated lease or buy decisions. Other works published
in accounting journals include Bower, Herringer, Williamson 1966, Beechy 1969
and 1970. Mitchell 1970. Wyman 1973. A complicating feature of lease or buy
decisions arises due to the fact that incremental cash flows in lease or
buy decisions not only consist of changes in operating cash flows such
as after tax rents, lost deprecation tax savings, the lost salvage value
of an asset to be acquired but also those in financial cash flows such
as loan repayment and tax savings on interests. The purposes of Vancil’s
as well as of other accountants’ works are to identify appropriate
incremental cash flows and formulate lease or buy decisions as capital
budgeting problems. Researchers in finance also found these works to be
of their interest as well. They proposed a number of different
approaches to lease or buy decisions. Such lease or buy studies
pertinent to this works include
Johnson and Lewellen 1972, Roenfeld and
Osteryoung 1973, Bower 1973, and Myers, Dill,
and Bautista 1976.3
3 For additional works on financial leases, see Doenges 1971,
Gordon 1974.
Keller and Petersen 1974. Henderson 1976. Lewellen, Long, and McConnell 1976.
4
Earlier works on lease or buy decisions discounted lease rentals
at the before tax cost of debt and lost depreciation tax savings, tax
savings on lease rentals to be paid, and lost interest tax savings at
the after tax cost of debt in order to arrive at the net advantage of
leasing(henceforth denoted by NAL). Bower 1973 was credited with
establishing the practice of discounting the aforementioned incremental
cash flows at the after tax cost of debt.
In establishing their result,
Bower 1973 made use of an economic assumption later referred to as the
equivalent loan equation. Whereas, Myers, Bautista, and Dills 1976
employed the binding constraint on debt capacity and established the use
of the after tax cost of debt in lease or buy decisions similar to that
of Bower’s approach.
An interesting development in the analysis of lease or buy
decisions is the emergence of the modified risk adjusted discount rate
method (henceforth referred to as the MRADR method) under which that the
proponents of this approach have insisted decomposing the components of
incremental cash flows into its component cash flows and discounting
riskier operating cash flows, for instance, such as the lost salvage
value of an asset under the lease option at the weighted average cost of
capital and discounting less risky cash flows, for instance, such as tax
savings on interest at the after tax cost of debt.
Disputes arose as to which incremental cash flows should be
discounted by the after tax cost of debt and which ones by the weighted
average cost of capital.4,5 To bypass the disputes, some researchers such
Miller and Upton 1976. Frank and Hodges 1978. Levy and Sarnat 1979.
1984. Gutman and Yagel 1994.
Steele
4
For instance, see the survey by Bower 1973, Schall 1973, Johnson and Lewellin
1973, Clark, Jantorni, and Gann 1973, Lev and Orgler 1973.
5 It is necessary to realize that the tax rate gives rise to uncertainty
because it depends on the level of a firm’s profit. For this reason, the after
tax operating cash flows in lease or buy decisions such as depreciation charges,
5
as Roenfeld and Osteryoung 1973 resorted to the cash equivalent approach
in lease or buy decisions.
Whereas, some others such as Beechy’s 1969 and 1970. Mitchell
1970, as well as Doenges 1971 adopted the internal rate of return method
(henceforth abbreviated as the IRR method) and thus bypassed the use of
the MRADR method.6
However, Schall 1974 was vocal in rejecting Beechy’s IRR method
based on the ground that the risk of a project is not fully taken into
account in computing
the net advantage of leasing. The correct
procedure is the net present value method which maximizes shareholders’
wealth using the multiple discount rates under the MRARD method
according to his theory.
The point of interest in Schall’s 1974 paper
is that the additivity of a utility function was alluded as perhaps one
of the critical assumptions to rationalize the validity of the MRADR
method in lease or buy decisions.
The MRADR method appears to be originally attributed to Modigliani
and Miller (henceforth abbreviated as MM) in their papers 1958 and 1963.
Myers 1974 discussed a similar method of using multple discount rates
commensurate with the risks of cash flows involved in discounting.
He
referred to his method as the Adjusted Present Value method of capital
budgeting where after tax operating cash flows are discounted by the
weighted average cost of capital and tax savings on interest are
discounted by the before cost of debt on the belief that the latter are
more certain. The procedures of lease or buy analyses due to Bower 1973
lease payments, changes in operating expenses and so forth are also exposed to a
firm’s business risk at large.
6 Roenfeldt and Osteryoung 1973 used the after-tax cost of leasing which is
compared against the after tax cost of debt. The former is derived as an
internal rate of return. Hence, their method is similar to that of Beechy.
However, they are keenly conscious aware of risk differentials among cash flows
when they applies a certainty equivalent coefficient to the cash flow from the
salvage value of an asset bought. Mitchell 1970 and Doenges 1971 also derive the
after tax cost of leasing which is compared with the after tax cost of debt.
6
and Myers, Bautista, and Dills 1976 are highly popular in finance
literature but they rest on the assumptions such as the loan equivalent
equation and the binding constraint on the capital structure.
However, Johnson and Lewellen 1972 pointed out that that tax
savings on interest are not part of returns to investment (i.e.,
operating cash flows) to be discounted in arriving at the NAL. They
maintained that interest tax savings should reflected in the after tax
cost of debt. Hence, discounting interest tax savings at the after tax
cost of debt and additing the latter to the present value of operating
cash flows is a double counting. To quote from their work,
“the more basic error is including in the cash flow analysis any
form of interest charge or interest tax shield to begin with,”
In essence, they implied that MM’s theory is erroneous.
Johnson and Lewellen 1972 and Schall 1973 neither utilized the
debt equivalent equation nor the binding debt capacity constraint.
However, the problem with their approach to lease or buy decisions is
the use of the MRADR method. In what follows, the problem of the MRADR
method will be carefully reexamined to show that the addivitity
assumption is bound to fail in reality. Hence, so does the MRADR method.
Let a two tuple (x,y) respectively represent a firm’s perpetual
stream of tax savings on interest and that of the after tax operating
income y and u(x,y) the utility function which represents the value of
the cash flow streams to the firm. Let i be the interest rate on debt
and r the rate applicable to the after tax operating income y. Under the
MRADR method, the value of the two streams x and y is viewed as the sum
of the value v of the tax savings on interest x and the value w of the
after-tax operating income y as follows:
u(x,y)=v(x)+w(y)
where
7
v(x)=
x
i
and

w(y)=
y 7
.
r
The purpose of this discussion here is to demonstrate
by means of
a counter example that the additivity principle could readily fail.

Modigliani and Miller Paradox
Suppose that John Michaels, an MBA student in an accounting
course,
participated in an experiment where a subject is presented with
the following four pairs of perpetual streams of interest tax savings
and after-tax operating earnings in dollars: ($100,$50), ($10,$50),
($100,$500), and ($10,$500). In the first experimental session, John
prefers ($10,$50) to ($100,$50), since John finds the level of interest
implicit in the interest tax savings of $100
is too high in relation to
the after-tax operating earnings of $50 in the sense that the firm might
become insolvent, provided that the tax rate is, for instance, a
marginal corporate tax rate, for instance, 50%.
However, in the second
session, John prefers ($100,$500) to ($10,$500) because the level of the
after tax operating cash flow of $500 should be able to cover the
interest expenses implicit in the tax savings of $100: no insolvency
risk.
John’s preference order should imply the following inequalities in
utilities:
u(10,50)=v(10)+w(50) > u(100,50)=v(100)+w(50)
and
7 Recall that MM utilized this method to evaluate the value of their levered
firm in their 1958 and 1963. Hence, this is the well established practice in
corporation finance.
8
u(100,500)=v(100)+w(500) > v(10)+v(500)=u(10,500).
From the first inequality,
v(10) > v(100) 
100 10

i
i
which implies that i must be negative. Whereas, his second response
implies the inequality below.

v(100) > v(10) 
which means that i is positive.
10 100

i
i
Contradiction!.
It is pointed out that the additivity of a utility function cannot

hold because the value of tax savings on interest is dependent on the
level of the after tax operating cash flow stream. The latter is
equivalent to a violation of the independence axiom in the language of
utility theories.8
The modified version of the MM theory of capital costs insists
that the MM theory is valid when the debt ratio is moderate.
However,
the implication of this counter example is that the MM theory fails when
the level of operating cash flow is not sufficiently high to cover the
interest expenses, even when the debt ratio is moderate. What matters is
not a firm’s debt ratio but the level of operating earnings in relation
to interest expenses. Thus, the MRADR method is a theoretically invalid
method in general. This implies that a lease
or buy analysis based on the MRADR method is invalid.
Now, the cash equivalent method of capital budgeting utilized by
Roenfeld and Osteryoung 1973 is one method which has not been so far
unscathed in this survey of lease or buy analysis.
8 Indeed, should MM be given this choice, they would have preferred (100,500)
to (10,500). That is, they should maintain that the tax savings of 100 on
interest will be preferred to the tax savings of 10 on interest because they
stipulate that the firm under consideration belongs to a risk equivalent class:
no change in financial risk.
9
However, it is necessary to reformulate the cash equivalent method
so that it can be free of the additivity principle and can be applied to
lease or buy decisions without invoking the MRARD method. For this
reason, it becomes necessary to construct a new theory of risk from
which the cash equivalent method applicable to lease or buy decisions
can be rigorously derived as a formal risk theory.
3. A New Theory of Risk
The approach to find an alternative solution to lease or buy
decisions is inspired by the cash equivalent method of capital
budgeting.9 The key focus of this paper is to present the cash
equivalent method, whose axiomatic basis has not been carefully
investigated in corporation finance.10 Hence, what will be discussed in
this section is a foundation on which a new cash equivalent method of
capital budgeting will be constructed.11 The reminder of this section
present basic terms and definitions at the beginning and then state the
main assertions of the theory with the proofs.
§3.1: Projects
An investment project can be represented as a point on
the four dimensional Euclidean space. A decision maker (henceforth
denoted as DM) is to examine the return and risk attributes of a
project. Let a four tuple P=(,,,) represent a project.12 The first
9 See Chapter 11 of Financial Management by Keown, Martin, Petty, and Scott, Jr
2005 for the certainty equivalent approach to capital budgeting. In this work,
the cash equivalent method and the certainty equivalent method are the same.
10 No axiomatic theories of risk are available in corporation finance except
the expected utility theory and its variants.
11 See Luce and Suppes 1957, Rabin 1998, and Starmer 2001 onto be discussed in
this section are based.
12 Under the expected utility theory, the distribution of the return to
investment has been an object of choice. However, the new theory is deliberate
in choosing the four risk return attribute as an object of choice in this work.
It is in light of the fact that business decisions are typically made without
the assessment of any probability distribution involved. Hence  is the
10
attribute  is the anticipated level of return to a project. The second
risk attribute  is the business or operating risk index which results
from accepting the project at t=0. The third attribute is the
financial risk index at t=0. Let the fourth parameter  represent the
life of a project. All projects (,,,)s of interest in this work are
by assumption contained in a closed and bounded 4 dimensional Euclidean
space S specified as follows:
Definition 1(The Domain Space):
The domain space S is a set of projects contained in a closed and
bounded four parameter Euclidean Space S such that
S={(,,,),*≤≤, *≤≤, *≤≤, *≤≤}
where a superscript star * is the maximum value to be taken by each
variable and a subscript
*
is the minimum value of each variable.
A new theory of risk is concerned with assessing the cash
equivalent c of a risky project PS. To introduce this term precisely,
it is necessary to utilize the sure project PS. The following
definition is in order:
Definition 2(The Sure Project C):
C=(c,0,0,0)
project P and
is referred to as the sure project PS if a DM finds a
C=(c,0,0,0) as good.13
The definition of the cash equivalent c of a project P can be
formulated as follows.
Definition 3(Cash Equivalent c):
The cash equivalent of PS is the first element c of its sure project
C=(c,0,0,0) representing a certain amount of cash.
anticipated return from a project rather than the expected return which cannot
be obtained without the knowledge of the probability distribution.
13 A cash equivalent seems to have originated in connection with the St.
Petersburg game whose cash equivalent is log 2. See Levy 1998.
11
It is necessary to point out that the cash equivalent c of a
project PS does not always exist unless a DM’s risk preference
conforms a certain type of restrictions.
Hence, the next task is to
describe a set of such axioms or postulates which guarantee the
existence of the cash equivalent c of a project P.
A DM in this work is often a team of individuals
in charge of
corporation finance. It is stipulated that the team acts according to
the risk return preference of the corporation for which they work.
Hence, the cash equivalent c for any PS will be this firm’s or this
team’s cash equivalent c.
§3.2.1: Generic Axioms
Let Pa and Pb are two projects in S.
That Pa
is preferred to Pb
is written as {Pa>*Pb }. That Pb is preferred to Pa does not hold is
written as not{Pb>*Pa}. That Pa is as good as Pb is represented by
{Pa*Pb}, which is referred to as an indifference relation.
Let an arrow  indicate direction of implication. Let A and B
signify two statements. That A implies B will be thus denoted by A  B.
The converse of the latter is written as A  B. When both hold, A is
equivalent to B and written as A  B.
(Pa>*Pb or PaPb} is written as {Pa*Pb}. This relation is the
preference-indifference relation. The latter can be defined as the
negation of a preference: {Pa*Pb}=not{Pb >*Pa}. The preference relation
can be defined as the negation of the preference-indifference relation
as follows: {Pa>*Pb}=not{Pb*Pa}.
The indifference relation can be defined from the preferenceindifference relations as follows: {Pa*Pb and Pb*Pa}{Pa*Pb}. It is
seen that one relation can be defined by other relations.
12
Let us briefly discuss these properties of these relations.
1. Preference Relation>*:
A preference relation >* is said to be asymmetric:
{Pa>*Pb}not{Pb>*Pa} for any Pa and PbS, irreflexive: not{Pa>*Pa} for
any PaS and transitive: {Pa>*Pb} and {Pb>*Pc}  {Pa>*Pb} for any Pa, Pb
and PcS.
2. Indifference Relation ~*:
The indifference ~* is said to be symmetric: {Pa~*Pb}  {Pb~* Pa}
for any Pa and PbS, reflexive: {Pa~*Pa} for any Pa
in the set S and
transitive: {Pa~*Pb} and {Pb~*Pc}{Pa~*Pc} for any Pa, Pb
3. Connectedness
and PcS.
Pa Pb Pc
Given every pair of prospects P a and P b S, preference relations are
said to be connected:
1){Pa>*Pb }, 2) {Pa<* Pb} or 3){Pa*Pb}
4. Mixture Transitivity:


It is necessary to define the notations involving both preference and
indifference relations as follows:
{Pa<*Pb or Pa~*Pc}  {Pa<*Pc}
or
{Pa~Pb or Pa<*Pc}  {Pa<*Pc}.
5. Preference-Indifference *:
The preference-indifference relation is said to be symmetric:
{Pa*Pb}{Pb*Pa} for Pa and Pb S, reflexive: {Pa*Pa} for any PaS. It
is transitive: {Pa *Pb and Pb*Pc}{Pa*Pc}.
So, a stage is set for presenting five behavioral postulates that
are regarded as sufficient conditions.
§3.2.2: Behavioral Postulates
Definition 4(The Sure Thing Postulate):
13
For a sufficiently small cash , there exists the least preferred sure
project such that
C*=(,0,0,0)*P
for any project PS. For some large *, there exits the most preferred
sure project such that
P*(*,0,0,0).
Let the sure project C*=(*,0,0,0)S be the most preferred
project and included in set S. However, the least preferred project
(*,0,0,0) is not included in set S. Hence, the domain space S of
prospects is also a bounded and closed set in preference orders in the
following sense:
{C**P*C*|PS}.
The next four postulates are referred to as the generalized
Markowitz axioms, since they are in essence a 4 dimensional version of
the Markowitz criteria.
Definition 5(The Profit Maximization Postulate):
{(,,,)>*{(',',',')}{(<’)}
Definition 6(The Operating Risk Minimization Postulate):
{(,,,)>*{(',',',')}{(>’)}
Definition 7(The Financial Risk Minimization Postulate):
{(,,,)>*{(',',',')}{(>’)}
Definition 8(The Project Life Postulate):
{(,,,)>*{(',',',')}{(>’)}
Several comments are in order. In corporation finance, two types
of risk have been recognized: operating risk and financial risk.
Further, business investments are by nature intertemporal decisions and
14
hence it is necessary to take into account the time value of money. The
latter implies that the risk associated with a project rises as the life
of a project becomes longer due to the fact that individuals are
impatient to utilize money for immediate consumption as well as due to
the fact that the degree of uncertainty rises as the time to receive the
anticipated income is delayed as the project’s life becomes longer.
§3.3 The Theory of Cash Equivalents
The first assertion to be proved is the lemma which states an
important property of preference relations under the generalized
Markowitz criteria. The property will be needed in establishing the
continuity of the cash equivalent function.
Lemma 3(The Strictly Increasing Preference Structure):
For any ' and  such that 0'<1, a more preferred project
P=(p,p,p,p), and a less preferred project Q=(q,q,q,q) in set S,
'P+(1-')Q*<*P+(1-)Q*
where p>q,p<q.p<q,p<q.
Proof of lemma 1:
The proof can be established by comparing each component of P+(1-)Q*
with that of ’P+(1-’)Q*. In light of p>q,
p+(1-)q=q+(p-q)>'p+(1-')q=q+’(p-q)
whereas,
p+(1-)q=q+(p-q)>'p+(1-')q=q+’(p-q)
in light of p<q. A similar inequality should hold for the remaining
two risk parameters,  and . Thus, The Inequality of the assertion must
hold.
On any line emanating from the less preferred project toward a
more preferred project, the degree of preference will be strictly
15
increasing as a point on this line moves away from the less preferred
toward the more preferred.
The main theorem will be stated and its proof will be provided by
means of the two lemmas in this subsection
Theorem 1(The Representation Theorem):
There exists a cash equivalent function u which maps projects P and
P’S to its cash equivalents c and c’ R[,*] such that
u(P)=cu(P’)=c’P*P’
provided that the generic axioms 1, 2, 3, 4, and 5 of §3.2.1 on the
preference orders and the behavioral
postulates 1, 2, 3, 4, 5 of §3.2.2
hold.
The theorem directly follows from the two lemmas to be proved
next. The first task is to define the Euclidean norm which will be
needed in the process of establishing the lemmas.
Definition 9(The Euclidean Norm):
Given projects P=(,,,) and P’=(’,’,’,’), the Euclidean norm |PP’| is the distance between the two projects defined by the following:
|P-Q|=
(aa')2  (  ')  ( ')  (  ')2 .
2
2
In what follows: the proof of the lemma 2 will be presented in its
entirety. Whereas, that of the lemma 3 will not be depicted in detail

because the latter is similar to that of the lemma 2.
Lemma 2(The One to One Property):
For any PoS, there is a sure project (co,0,0,0),for co R=[, *], such
that
(co,0,0,0)*Po
provided that the same condition of the main theorem holds.
Proof of lemma 2 by the Bisection Search Algorithm
Search Step 0:
16
Let C*=(,0,0,0) and
C*=(*,0,0,0) are respectively the most
preferred project located at the North and the least preferred project
at the South of the sure thing postulate. The search area is this
straight line C={C*+(1-)C* for any , 01}. Set the starting search
vector to the following:
H(0)=C*.
{H(0)*Po}
must hold for all Pos , since H(0) is the least preferred
sure project located down at the South. This is the end of the search
step 0. Continue to the next search step.
Search Step 1:
It is necessary to formulate a new search vector H(1). To this end, let
D(0) denote the difference C*-C* as follow:
D(0)=(*-,0,0,0).
The Euclidean norm of the vector D(0) is the following:
D(0) =*-
The next search vector H(1) will be the mid point of the line C as
follows:

H(1)=H(0)+(0)
D(0)
2
where (0)=+1 because H(0) must be pushed up to the North or to the
direction of C* on the line C. Notice that H(1) is a sure project

(c1,0,0,0) where
c1     (0)

*
 

.
2
Suppose H(1)*Po. Discard all sure projects C=(,0,0,0)s out of C such
that C<*H(1). Let C redenote the remaining projects. Suppose

17
{Po*H(1)}. Again discard all Ps such that P>*H(1). Let C redenote the
remaining sure projects.
To form the nest search vector H(2), let D(1) be the difference
from H(1) to C* or from H(1) to C* as follows:
D(1)=
D(0)
.
2
Define the new search vector H(2) on the line C whose length |D(1)| is
one half of the original length |D(0)| as follows:

H(2)=H(1)+(1)
D(1)
2
where (1) is either +1 or -1 respectively depending on whether {H(1)*
Po} or {PoH(1)} holds or whether or not H(1) must be pushed up to the

direction of the North or pushed down to the direction of the South. The
search area will be reduced to one half again every time since one half
of the sure projects Cs that are found no longer needed will be
discarded. The new search vector H(2) is a sure prospect (c2,0,0,0)
where
c 2  c1   (1)
*
 

2
2
.
Continue the search to the next search step 3.
The process similar to
what is seen at the previous search step 2 must be repeated.

……
Suppose that the search vectors H(0), H(1), H(2),…, H(t-1) have
been generated. Continue the search to the search step t.
Search Step t:
Let the new search vector H(t)=(ct,0,0,0) be defined as follows:
H(t)=H(t-1)+(t-1)
D(t - 1)
2

18
where (t-1) is either +1 or -1 respectively depending upon whether or
not H(t-1) has to be pushed up to the direction of the North or pushed
down to the direction of the South that are left,
D(t-1)=
D(0)
2
t1
and
 t1   (t  1)
t 
*
 

2
t1
.
The search process should continue ad infinitum.
It is necessary to show that the bisection algorithm is

convergent.
To prove this, consider the Euclidean norm H(t  s)  H(t)
below:
H(t  s)  H(t)  c ts  c t .

Then, the following inequality holds:
H(t  s)  H(t)  H(t  s)  H(t  s  1)  H(t  s  1)  H(t  s  2)  ...

+ H(t + 2) - H(t + 1)  H(t  1)  H(t) .
Then,

H(t  s)  H(t)
  (t  s  1)
D(0)
2
t +s
  (t  s  2)
D(0)
2
  (t + 1)
D(0)
2
t 2
t +s
 ....
  (t)
D(0)
2
t 1
.
Since d(t)  1 for any t,

H(t  s)  H(t) 

*
 

2
t
1 1
1
1 
 s  s1  ...  2  1.
2 2
2
2 
Summing up the content of the brace,

19
H(t  s)  H(t)

*
   

2
t
*
1    
1  s t  .
 2  2
For any finite s and an arbitrarily small , >0, there is an
arbitrarily large natural n such that

*
 

2
t
< 
for any t> n. It is seen that H(t) is a Cauchy sequence that is
convergent. In words, H(t) will become as good as the prospect Po and

u(H(t))=ct converges to u(Po)=co as t passes to infinity for any Po  S.
Lemma 3(The Onto Property):
For any coR=[, *], there is a project PoS such that
(co,0,0,0)*Po
provided that the conditions of the representation theorem holds.
Proof of lemma 3
The bisection algorithm similar to the previous one will be
utilized to locate a project Po such that Po is as good as (co,0,0,0).
The line on which this search must be carried out will be the line
{P()=P*+(1-)P*,0l1}. The length of the line is D(0)=|P*-P*| Using a
similar bisection algorithm on this line, it is readily possible to
locate the project Po such that Po is as good as (co,0,0,0).
§3.4 The Continuity of a Cash Equivalent
The continuity of a cash equivalent as a function of the four
parameters is important. With this property, the latter can be
approximated, for instance, as a lower order polynomial function of the
four parameters. This property will enable us to significantly simply
the task of assessing a cash equivalent function by conducting a
20
behavioral experiment where cash equivalents to stimulus projects are
elicited from subjects.
Definition 10(The Continuity of a Real Valued Function):
A real valued function mapping a project Po in the interior of the
set S to its cash equivalent co into the open interval (c*,c*)
to be a continuous function at the project Po,
is said
provided that, for any
arbitrarily small real >0, there exists another small real  such
that
|P-Po|<
and for any such P in the above 4 dimensional sphere of radius , the
following inequality
must hold:
u(P)  u(P o) <  .
Now that the continuity of a function on the set S is formally defined,
a stage is set for proving the continuity of the cash equivalent
function.

Theorem 2(The Continuity Property):
The cash equivalent function u mapping a project P in the interior
of the set S into the open interval (,*) is a continuous function of
the return and risk attributes, , , , and .
Proof:
Consider any project Po in the interior of the set S with u(Po)=co.
Let  be a small real >0. Let the projects P(+) and P(-) denote the
most preferred and least preferred corner project of the 4 dimensional
cube with the half length of  and centered at the project Po as
follows:
P(+)=(o+,-,-,-)
and
21
P(-)=(o-,+,+,+).
The line P={P(+)+(1-)P(-),01} represents the line of
strictly increasing preference of the lemma 1. Set  to 1/2, P1/2 is Po.
P1 is P(+) and P0 is P(-). Hence, for a small , >0,
u{P(-)}<co-<u(Po)<co+<u{P(+)}.
Let ’ and ” be another small real constants such that 0<’<1 and
0<”<1. Let P(’) denote the project located on the straight line
connecting P(+) and Po such that
u{P(’)}=u{(o+’,-’,-’,-’)}=co+.
Let P(”) be another project located on the straight line connecting
P(-) and Po such that
u{P(’)}=u{(o-’,+’,+‘,+‘)}=co-.
The two projects P(’) and P(”)can be readily located by the
bisection algorithm of the lemma 3.
Let  be the smaller of the ’ and ”. Let P(+) and P(-) be
defined as follows:
P(+)=(o+,-,-,-)
and
P(-)=(o-,+,+,+).
The two projects above are the most preferred and least preferred corner
points of the 4 dimensional closed cube which in turn contained in the 4
dimensional closed box defined by P(’) and P(”). The former cube
whose half side length is  always contains the sphere of radius . Let
P be any project contained in this 4 dimensional sphere with radius .
Then,
co-<u(P)<co+
22
which is the following:
|u(P)-u(Po)|<.
The continuity of the cash equivalent function u is established.
Further, due to the lemma 1, the discountinuity of the second kind
is ruled out. It is useful to expand on this point. The cash equivalent
surface covering the set S is strictly increasing as a project moves
away to any direction from the least preferred corner project of the set
S. For any intermediate point c(,*), there is a project P by the
lemma 3 such that u(P)=c. Hence, there is no gap on the cash equivalent
surface suspended over the set S. That is, the surface is continuous
over the set S.
Several remarks are in order. The theory of cash equivalents is
flexible because the return and risk attributes can be increased to more
than four parameters, provided that the Markowitz postulate holds for a
new paramter. In corporation finance, business risk is often mentioned
on conjunction with operating risk. It can be readily incorporated into
this theory by increasing the number of parameters to five and adjust
the generalized Markowtiz postulates as follows: “the greater the
business risk index, the less preferred the project, holding the other
parameters constant.”
The second remark is with regard to the cash equivalent
coefficient associated with a cash equivalent. It is utilized with the
risk free rate of interest if. Let the cash equivalent coefficient, ,
be formally defined in the context of this work as follows:
Definition 11(The Cash Equivalent Coefficient):
The cash equivalent coefficient n is a constant applied to , the
anticipated return in year  which is related to the cash equivalent c
as follows:
23
c=
 
,
1 i 

f
or it can be defined as follows:

=
c
1 if 


.
Once the cash equivalent of a project is obtained, it is readily
possible to derive the cash equivalent coefficient. Since a cash

equivalent is a continuous function of , , , and , the cash
equivalent coefficient associated with it is also a continuous function
of the latter.
4. Measurement of Cash Equivalents
The section will illustrate how cash equivalents and
its
coefficients should be estimated in an experiment. The experiment is
carried out by means of the following lease or buy decision case.
Paradise Enterprise Case:
Paradise Enterprise, Inc. is a chain of
beach side bars and
restaurants in the Virgin Islands. Joe Dolittle, President, wants to
purchase a gaming terminal for $20,000 from Eden Technology. The bank is
willing to loan him $16,000 at 8%. The term of the loan is 4 years, the
expected life of the gaming machine. The terminal will be depreciated
under MARCS with the weights of 33%,
45%, 15% and 7% respectively in
the next four year. Paradise must get a maintenance contract for $80 a
year. The salvage value will be $800. Suppose that Paradise leases it
from Eden Tech. The rental will be $6,500 per year. At the end of the
expected life, the terminal must be returned to Eden. The firm’s
marginal tax rate is 40%.
To study whether or not the project is a worthy venture, a team of
four recent MBA graduates from top universities with major in finance
24
has been at work at Paradise. They
have participated in all capital
budgeting analyses conducted by the firm for a year. They are not full
informed of the firm’s risk preference over projects routinely
considered by the firm. They are now ready to advise Joe as to how the
cash equivalent method of capital budgeting should be conducted to solve
the lease or buy decision problem at Paradise.
They have already conducted the traditional analysis of lease or
buy decisions. It indicates that a gaming terminal is a worthy
investment. The next step is to analyze whether or not it should be
leased or purchased by the loan.
To conduct a lease or buy analysis, Joe wants to estimate the cash
equivalent coefficient function from cash equivalents elicited from the
four MBA graduates as experimental subjects. It is necessary to conduct
an experiment to obtained cash equivalents c from the subjects and then
estimate the coefficient function 4 for the firm.
Recall that the
project must be represented by the four tuple, (, , , ).. Let the
anticipated return  be the net terminal value of a project where it is
the future value of incremental cash flows less the future value of an
outlay. The use of the net terminal value is inspired by the modified
internal rate of return. It enables us to dispense the additivity
principle and bypass the problem discussed in connection with the MRADR
method.
It is necessary to specify the operating risk and financial risk
indices  and  , which will be proxied by the following two indices.
= max
F
(1  v)S
and

25
I = max
I
(1  v)S
where v is the variable cost per $1 in sales, F is the fixed cost, I the
interest, and S, the average sales, provided that the gaming terminal is
adopted.

The former of the two proxies is the ratio of the operating
break even sales coverage divided by the sales level. The latter is that
of the interest sales coverage divided by the sales level. The closer
the ratios to 1, the higher the operating risk or the financial risk.
See Matsumoto and Hoban, 2001 for an exposition of the breakeven sales
coverages. To determine these indices, the analyst team must prepare the
pro forma income statement and balance sheet under the assumption that a
project at issue is adopted.
The four MBAs are asked by Joe to become experimental subjects and
provide cash equivalents with hypothetical projects presented to them as
stimuli.
Joe wants to use
16 stimulus projects to elicit cash
equivalents because there are four factors and at least two levels per
factor must be utilized. Hence, a 24 factorial design ia utilized.
Insert table 1 approximately here
Observe table 1 which lists the four parameter values for four
projects i=1, 2, 15, 16 and responses yij, j=1,2,…,10. The factor i is
set to $1,000 and $2,000 in the experiment.
The operating risk index i
is set equal to 0.0 and 0.3. The financial risk index  is set equal to
0.1 and 0.2. The expected life i
are set to 3 and 5. The total number
of experiment settlings is thus 24=16.
The four subjects was randomly divided to a team of two. The
randomly chosen 8 projects are assigned to the first team. The remaining
8 projects are assigned to the second team.
Table 1 presents the first
two and last two experimental settings for illustration. The first
26
column is for the stimulus project number starting from 1 and going up
to 16. The second column shows the replication numbers starting from 1
to 10 each experimental setting.
Each team is required to answer whether or not they
want to
purchase the project (i,i,i,i) at the price cij listed on the third
column. Suppose that a team
than the price cij.
believes that each project is worth more
Its response is coded as yij=1, indicating that the
purchase offer is accepted by the team at the j-th replication. If not,
his response yij is coded 0. If the offer is rejected by the team, the
price cij is increased to a higher amount cij+1 at the next replication.
There are 10 prices cijs for each project i. The 10 prices at each
experimental setting must be properly spaced ranging from
a very low
price to a very high price so that the team is certain to switch from
yes to no at some price.
The 10 prices for the first experimental setting are listed on
the third 10 rows of table 1. The project 1 presented to a team is
($1,000, 0.1, 0, 3). If the team decides that the price c11 of $929 is
more expensive than its true worth. Thus, its response y11 is 0 in the
sense that the price is too high. The price c12 is then reduced to $864.
This team still considers the price too high. Its response y12 is again
0.
When the price is finally reduced to $805, the two find the project
($1,000, 0.1, 0, 3) to be
more valuable than
the price of $805. So,
the response y13 is 1 at the 3rd replication. This price of $805 is the
maximum price below which the subject accepts the investment
opportunity. Whereas,
$864 is its minimum price beyond which the
subject will no longer accept it. Table 1 presents the results of the 40
experimental sessions at which the two teams provide their 40
27
dichotomous responses. The remaining 120 responses are not presented in
table 1 for brevity.
Insert Table 2 approximately
In order to estimate the cash equivalents for the 16 experimental
sessions, a logistic regression is fitted to the 160 responses. The
panel A of Table 2 presents the estimated logistic regression equation
whose dependent variable is z as follows:
Z=2.253-0.001-5.316-8.284-0.403
All p-values of the estimated coefficients are significant at 5%. The
classification table is presented in the panel B of table 2. The fit of
the logistic regression is satisfactory in light to the fact that 70% of
160 cases are corrected classified.
The panel C of table 2 presents the estimated probabilities that
yij equals 1 and that yij equals 0. For illustration, consider the third
replication of the subject 1. The probability P that the response y13
equal 1, is obtained from the following:
P(y
13=1)=
1
1  exp 2.253  0.001 (1000 )  5.316 (0.1)  8.284 * 0  0.403 (3)
.
It will be readily verified that this P is .81914 at the maximum price
of $864. Whereas, P(y13=0)=1-P(y13=1) which is .18086 associated at the

minimum price of $805 at which the investment opportunity is rejected.
Behavioral theorists believe that a subject accepts a project with
0.5 and rejects it with 0.5 if the price is the cash equivalent. Refer
to the experimental setting 1 in the panel C of table 2. Let c be the
cash equivalent located between 895 and 805. At $805, Pr(y13=1)= 0.8194
whereas at $864, Pr{y13=0)=0.18086 as listed in the first row of the
table.
Thus, the cash equivalent c1 for the first experimental setting
can be interpolated from the following:
28
864  805
.18086  .81914

c1  805
0.5  .81914
.
The resulting cash equivalent c1 is $834. The remaining 15 estimated
cash equivalents are also interpolated by using a similar procedure as

done above. Panel C of Table 2 exhibits the complete list of all maximum
prices with their probabilities Pr{yij=1) and the minimum prices with
their probabilities Pr{yij=0) calculated from the estimated logistic
regression of the Panel A of table 2.
Insert Table 3 approximately here
Once a cash equivalent ci is estimated, it is also possible to
derive the corresponding cash equivalent coefficient n(i). To this end,
substitute ci and i at the experimental setting i. Then, the estimated
cash equivalent coefficient i at the i-th experimental setting can be
derived from definition 11. The following definition is in order:
Definition 12(The Cash Equivalent Coefficient):
n(i)=
ci
i
(1 if ) n .
Suppose that the risk free rate is 3%. Then, substiting a=1000,
c=834, if=0.03 into definition 12,

(1)=
834
(1.03) 3
1000
which is 0.9113.
The 16 estimated cash equivalents ci and the cash equivalent

coefficients n(i) are provided on the last two columns of the panel A
of Table 3.
Regressing the cash equivalent ci on these covariates i, i, i,
and i, it is possible to estimate the i-th cash equivalent function
over the four covariates. The panel B of table 3 presents the estimated
29
regression coefficients bis, its standard error, the t-statistics, and
the p-values.
Three coefficients are highly significant and the
coefficient of the operating risk index is significant. The R2 is 93.7%
and highly significant.
Regressing the estimated cash equivalent coefficients n(i) on the
four covariates i, i, i, and i listed on the columns 2, 3, 4, and 5
and the cash equivalent coefficients reported in the 7 th column of the
Panel A of Table 3,
it is also possible to obtained the estimated cash
equivalent coefficient function as shown on the Panel C of Table 3. The
estimated coefficients bis of the four covariates are all highly
significant with the R2 of 90%.
Regressing the estimated cash equivalent coefficients n(i) on the
same four covariates over the 16 experimental settings in the Panel A of
Table 3, the estimated cash equivalent coefficient function is obtained.
The panel C of Table 3 presents this result.
5. A Lease or Buy Analysis
This section illustrates how a lease or buy analysis should be
conducted under the cash equivalent method of capital budgeting. New
notations will be presented to formulate the lease or buy analysis.
A = asset value at t=0
Lt = lease rental at year t under the lease option
Dt = depreciation charge for year t under the purchase option
Ot = an increase in operating costs such as maintenance,
insurance expenses and the likes when an asset is purchased rather
than leased. If no change occurs, the latter should be set to 0.
R = proceeds from disposing the asset at the end of the expected
life
G = gains or losses if the asset is disposed
30
Bt=loan balance at t
d
A t =loan amortization at year end t
St=equity balance at t
e
A t =equity amortization at year end t

Pd=loan payment in any year t
e
P t =return to shareholders in year t

n=expected life=lease term
NAP = net advantage of purchasing

NTV = terminal value
wd = portion of debt
we = portion of equity
i = cost of debt
if= risk free rate
ke = internal cost of equity
r = the reinvestment rate=(1-tm)iwd+kewe
tm = marginal tax rate
These notations will be of great value in formulating a lease or
buy analysis with clarity. Two tables will be needed to discuss the
lease or buy case.
Table 4 presents the loan and equity amortization schedules in
panels A and B respectively.
Insert Table 4 approximately here
The loan payment Pd of $4,830.73 is constant whereas the loan
d
amortization A t and loan balance Bt vary as clearly seen from Panel B.
The t-th return to shareholders is the equity amortization A et plus
equity cost keSt. The outstanding equity St will also decline as the loan

balance Bt becomes smaller as shown in Panel B. However, the debt ratio
will remain at 80%, as seen in Panel C.

31
The loan payment less tax savings on interest to lenders plus the
t-th return to shareholders will be the total returns to investors on
the last column of Panel D.
Panel E shows the present value of the
total returns, which equal precisely the cost of the asset which is
$20,000.
Insert Table 5 approximately here
Panel A of Table 5 presents the depreciation schedules under
MARCS. Whereas, Panel B is the worksheet which systematically computes
the net terminal value of the project according to the formula below:
Definition 13(The Net Terminal Value):
N TV   t m D t  (1  t m ) L t  O t (1r)
n
n -t
 (R  t m G) - (1 + r) A .
n
t =1
In connection with the NTV computation, it is pointed out that the
weighted average cost of capital of r=6.84% is indeed the discount rate

which equates the present value of return to shareholders discounted at
this rate r. The following definition is in order:
Definition 14(The weighted average cost of capital):
P
A= 
n
t1
d

 t m i Bt  A t  k e S t
(1r)
e
t
where

r=(1-tm)iwd+kewe.
Recall in this regard, Panel E of the table 4 shows that the present
value of the returns to investors indeed sums up to the outlay of
$20,000.
The process of the NTV computation is depicted in the panel B of
the table 5. The first column presents the labels to identify
incremental cash flows involved. The second column shows before tax
figures whereas the 3rd column is for after tax figures. The fourth
32
column exhibits the timing of each cash flow. For instance, the first
cash flow of lease rentals are due at the beginning of each year for
three years. The entries on
the next column presents the future value
interest factors. The first interest factor on the top is the future
value of an annuity due. The future values of 7 cash flows are computed
by multiplying the the after tax figures on the 3rd column by the future
value interest factors on the fifth column in the last column of the
worksheet. Subtracting the future value of the project outlay , the net
terminal value is found to be $1.820.80.
The four MBA students are highly proficient in accounting and
hence prepared the pro forma income statement and balance sheet under
the assumption that the lease or buy option is accepted. They are able
to derive the % operating leverage index  and financial leverage index
, which turns out to be 0.2 and 0.15 respectively. Hence, the project
is ($1,820.8,0.2,0.15,4).
Substitute ($1,820,0.2,0.15,4) into the covariates of the cash
equivalent function 4 of the panel C of the table 3 as follows:
4=1.130+0.0001073(1,820.8)-0.525(0.2)-0.0929(0.15)-0.09(4)
which turns out to be 0.7210.
The final step of the lease or buy analysis is to evaluate the NAP
as follows:
Definition 15(the Net Advantage of Purchasing):
NAP=
 n NTV
(1if ) n
.
Substitute 4=0.7210, NTV=1,820.8, n=4, and if=0.03 into the NAP above
as follows:

NAP=
0.7210 (1820 .8)
1.03

4
33
which turns out to be 1,166.40. The NAP is positive and hence, the
purchase option should be accepted. Notice in this regard that the
rejection of the lease option can occur when the NTV is negative as well
as when  is negative. There is a direct relationship between the NTV,
the cash equivalent, and the required rate of return RR. The following
definition is in order:
Definition 15(The Required Rate of Return):
c

(1RR )
n
.
Substitute NAP=1166.40, =1,820.80, n=4 into the above. Solve for RR,
which turns out to be 11.8%. The implied required rate of return RR is

considerably higher than the weighted average cost of capital of 6.84%.
The NAP can be also obtained as the cash equivalent of
(1,820.8,0.2,0.150,4) directly from the estimated cash equivalent
function by substituting the parameters into the regression equation of
Panel B of table 3 as follows:
NAP=760.062+0.808(1802.80)-791.875(0.2)-1043.13(0.15)-175.062(4)
which turns out to be 1216.18. Using the same method utilized earlier,
the implied required rate of return RR is 10.6%.
In either instance, it is highly interesting to observe that the
MBA students have shown that their assessment of risks involved is much
greater than that implicit in the weighted average cost of capital
r=6.84% which supposedly reflects investors’ perception of risk
involved.
6. Summary and Concluding Remarks
The paper reviewed past studies of lease or buy decisions and
found that there is no satisfactory theory of lease or buy analysis.
Many researchers mistakenly view lease or buy decisions to be no longer
34
of any interest because all theoretical problems have been resolved. The
paper asserts that the only one approach left unexplored is the cash
equivalent method of capital budgeting to lease or buy decisions.
However, the cash equivalent method of capital budgeting did not become
a popular tool in lease or buy decisions in the past four decades
perhaps because cash equivalent coefficients are psychological entities
and finance researchers in 70s seemed not to be ready for an
experimental assessment of cash equivalents. Advances in behavioral
finance seems to have changed the climate significantly.
It is hoped
that researchers are now ready to embrace it, since the other approaches
failed.
With this background in mind, the paper presented an axiomatic
theory of cash equivalents. However, the axiomatic theory of risk is
still new to researchers in corporation finance because the expected
utility theory dominated the field of finance and no serious attempt has
been made to entertain any alternative approach. The new theory is
eclectic and inspired by the theory of consumer choice and advances in
behavioral decision theories.
The axiomatic risk theory without a technique of measurement is
incomplete. For this reason, the paper explored an experimental
measurement of cash equivalents for lease or buy decisions.
It has shown how Paradise Enterprise Case can be utilized to
elicit cash equivalents from the student subjects an how cash
equivalents for stimulus projects can be estimated from dichotomous
responses by means of the logistic regression and the linear
interpolation technique. It has shown that the response surface
regression can estimate the cash equivalent function and hence the cash
equivalent coefficient function. With the technique of measurement, the
cash equivalent method of capital budgeting is no longer a theoretical
35
construct but it can be utilized as an operational method in corporation
finance.
The contribution of this work is to open a new path toward
making the cash equivalent method as an operational tool in capital
budgeting.
The paper should be of great interest not only to financial
economists but also to behavioral decision theorists and operation
researchers who work in the areas where the theory and measurement of
cash equivalents or its coefficient play a key role.
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McConnell, John J. and James S. Schallheim, 1983. “Valuation of Asset
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38
Table 1: Experimental Settings
Project
no
i
replication
Prices
NTV
Operating
Risk Index
Financial
Risk Index
years
a team's
decisions
j
cij
ij
ij
ij
ij
yij
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
15
15
15
15
15
15
15
15
15
15
16
16
16
16
16
16
16
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
$929
$1,000
0.1
$864
$1,000
0.1
$805
$1,000
0.1
$751
$1,000
0.1
$702
$1,000
0.1
$658
$1,000
0.1
$616
$1,000
0.1
$579
$1,000
0.1
$544
$1,000
0.1
$512
$1,000
0.1
$884
$1,000
0.1
$784
$1,000
0.1
$697
$1,000
0.1
$621
$1,000
0.1
$555
$1,000
0.1
$497
$1,000
0.1
$446
$1,000
0.1
$402
$1,000
0.1
$363
$1,000
0.1
$328
$1,000
0.1
……………………………………………………………………….
……………………………………………………………………….
……………………………………………………………………….
$1,857
$2,000
0.3
$1,728
$2,000
0.3
$1,610
$2,000
0.3
$1,503
$2,000
0.3
$1,405
$2,000
0.3
$1,315
$2,000
0.3
$1,233
$2,000
0.3
$1,157
$2,000
0.3
$1,088
$2,000
0.3
$655
$2,000
0.3
$1,768
$2,000
0.3
$1,567
$2,000
0.3
$1,393
$2,000
0.3
$1,242
$2,000
0.3
$1,110
$2,000
0.3
$994
$2,000
0.3
$893
$2,000
0.3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
3
3
3
3
3
3
3
3
3
3
5
5
5
5
5
5
5
5
5
0
0
1
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
1
1
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
3
3
3
3
3
3
3
3
3
3
5
5
5
5
5
5
5
0
0
0
0
1
1
1
1
1
1
0
0
0
0
0
0
1
39
16
16
16
8
9
10
$725
$655
$655
$2,000
$2,000
$2,000
Table 2:
Measurement of Cash Equivalents
Panel A:
Estimated Logistic Regression
i
0
1
2
3
4
Variables
constant
NTV
Operating Risk Index
Financial Risk Index
Year
Estimated bi
2.253
0.001
-5.316
-8.284
-0.403
0.3
0.3
0.3
S.E.
0.000
1.887
1.929
187.000
1.029
Panel B: Quality of Classification by Estimated Logistic Regression
Predicted yi
0
1
obs. yi
0
33
31
1
17
79
Overall corrected prediction
total no of responses = 160
Panel C:List of Estimated Cash Equivalents
max
min
cash equicash equiSubvalent
estimated
valent
ject no
accepted
prob. P
reject
1
$805
0.81914
$864
2
$555
0.66918
$521
3
$658
0.46349
$702
4
$446
0.27842
$497
5
$702
0.61000
$751
6
$621
0.41127
$697
7
$544
0.22979
$579
8
$328
0.11759
$402
9
$1,728
0.84612
$1,875
10
$1,393
0.84612
$1,567
11
$1,728
0.92487
$1,805
12
$1,110
0.51192
$1,247
13
$1,704
0.44782
$1,867
14
$1,110
0.65505
$1,242
15
$1,315
0.44782
$1,495
16
$804
0.26589
$893
estimated
prob P
0.18086
0.33082
0.53651
0.72158
0.39000
0.58873
0.77021
0.88241
0.15388
0.15388
0.07513
0.48808
0.55218
0.34495
0.55218
0.73411
0.2
0.2
0.2
5
5
5
Wald TS
4.781
9.816
7.934
18.436
4.640
1
1
1
p=value
0.029
0.002
0.005
0.000
0.031
\
per cent
correct
51
82
70
Linear
Interpolated
cash equivalent
$834
$588
$680
$472
$678
$659
$562
$365
$1,802
$1,480
$1,767
$1,179
$1,786
$1,176
$1,360
$849
40
Tabl3: Estimating Cash Equivalent or Coefficient Functions
Panel A: Data Set
i


i

1
1000
0.10
2
1000
0.10
3
1000
0.10
4
1000
0.10
5
1000
0.30
6
1000
0.30
7
1000
0.30
8
1000
0.30
9
2000
0.10
10
2000
0.10
11
2000
0.10
12
2000
0.10
13
2000
0.30
14
2000
0.30
15
2000
0.30
16
2000
0.30
Panel B: Estimated Cash Equivalent Function
n
yi
0.00
0.00
0.20
0.20
0.00
0.00
0.20
0.20
0.00
0.00
0.20
0.20
0.00
0.00
0.20
0.20
3.00
5.00
3.00
5.00
3.00
5.00
3.00
5.00
3.00
5.00
3.00
5.00
3.00
5.00
3.00
5.00
834.00
588.00
680.00
472.00
678.00
659.00
562.00
365.00
1802.00
1480.00
1767.00
1179.00
1786.00
1176.00
1360.00
849.00
estimated bi
760.062
0.808
S.E.
200.240
0.072
t-ratio
3.796
11.228
p-value
0.003
0.000
791,875
359.643
-2.202
0.050
-1043.125
-175.062
359.641
35.964
-2.000
-4.868
0.014
0.000
0.9113
0.6817
0.7431
0.5472
0.7409
0.7640
0.6141
0.4231
0.9845
0.8579
0.9654
0.6834
0.9758
0.6817
0.7431
0.4921
Response variable=cash equivalent ci
Covariates:
constant
NTV
operating risk
index
financial risk
iindex
year
R2
0.937
P-value
0.000
Panel C: Estimated Cash Equivalent Coefficient Function
Response n
Covariates:
constant
NTV
operating risk
index
financial risk
index
year
R2
estimated bi
1.130
0.0001073
S.E.
0.040
0.000
t-ratio
13.519
3.572
p-value
0.000
0.004
-0.525
0.150
-3.494
0.005
-0.929
-0.090
0.150
0.015
-6.185
-6.022
0.000
0.000
0.900
41
P-value
0.000
Table 4: Loan, Equity, Capital Costs Tables
Panel A: Loan Amortization Schedule
years
balance
pmt
interest
1
$16,000.00
$4,830.73
$1,280.00
2
$12,449.27
$4,830.73
$995.94
3
$8,614.48
$4,830.73
$689.16
4
$4,472.90
$4,830.73
$357.83
Panel B: Equity Amortization Schedule
internal
equity
equity
equity
year
employed
pmt
cost
1
$4,000.00
$1,487.68
$600.00
2
$3,112.32
$1,425.55
$466.85
3
$2,153.62
$1,358.44
$323.04
4
$1,118.23
$1,285.96
$167.73
total
$0.00
Panel C: Annual Capital Structure Table
year
debt
equity
total
1
$16,000.00
$4,000.00 $20,000.00
2
$12,449.27
$3,112.32 $15,561.58
3
$8,614.48
$2,153.62 $10,768.09
4
$4,472.90
$1,118.23
$5,591.13
Panel D: Return to Investors
debt
tax savings
equity
year
pmt
interest
pmt
1
$4,830.73
-$512.00
$1,487.68
2
$4,830.73
-$398.38
$1,425.55
3
$4,830.73
-$275.66
$1,358.44
4
$4,830.73
-$143.13
$1,285.96
Panel E:P
Present Value of Return to Investors
PV of Total Cost Costs at WAC
year
total pmt
pvif @6.84% PV
1
$5,806.41 0.935979034
$5,434.68
2
$5,857.90 0.876056752
$5,131.85
3
$5,913.51 0.819970753
$4,848.91
4
$5,973.56 0.767475433
$4,584.56
$20,000.00
amortization
$3,550.73
$3,834.79
$4,141.57
$4,472.90
equity
amortization
$887.68
$958.70
$1,035.39
$1,118.23
$4,000.00
debt ratio
0.80
0.80
0.80
0.80
total
pmt
$5,806.41
$5,857.90
$5,913.51
$5,973.56
42
Table 5: Lease or Buy Analysis
Panel A:
MARCS Weights and Depreciation Charges
Gaming Machine Depreciation Schedule
year
Weights
Depreciation
1.00
0.33
$6,600.00
2.00
0.45
$9,000.00
3.00
0.15
$3,000.00
4.00
0.07
$1,400.00
Total
$20,000.00
Panel B:
Gaming Terminal Lease or Buy Worksheet
Paradise Enterprise, Inc.
labels
BT
lease rentals
$6,500.00
dep tax shields
1
$6,600.00
2
$9,000.00
3
$3,000.00
4
$1,400.00
maintenance
-$80.00
salvage value
$800.00
total future value
Outlay
NTV
AT
$3,900.00
$2,640.00
$3,600.00
$1,200.00
$560.00
-$48.00
$480.00
Time
0--3
1.00
2.00
3.00
4.00
0--3
4.00
IF
4.73240756
Future Value
$18,456.39
1.21955569
1.14147856
1.06840000
1.00000000
4.73240756
1.00000000
$3,219.63
$4,109.32
$1,282.08
$560.00
-$227.16
$480.00
$27,880.26
-$26,059.47
1,820.80
Note: rentals and maintenance costs due on the first day of each year
BT and AT before and after tax, time=cash flow timing IF=interest factor
43