A Theory and Measurement of Cash
Equivalents for Lease or Buy Decisions
FUR XII 2006
6/22--6/26
Rome, Italy
Keishiro Matsumoto,Ph.D.
University of the Virgin Islands
[email protected]
Motivation and Purposes of the Work:
--Points out that the lack of a formal theory of risk in corporation finance is
one of the major reasons why a lease or buy decision analysis suffers from
a variety of problems.
--Proposes a new theory for risk designed to deal with problems specific to
corporation finance such as lease or buy decisions where risk and return
attributes of projects are utilized in business decision making.
A New Theory of Risk for Corporation Finance
-- Key features of the new theory
1.Under the new theory, an investment project is represented by its
anticipated return(), operating risk index), financial risk
index(), and project life() as a 4 dimensional vector.
2.A decision maker’s preference order on investment projects
follows by stipulation an weak order. Namely, the preference order
is asymmetric, irreflexive, transitive, and connected.
3.Impose further the behavioral
postulates:


1.the existence of the least preferred and most preferred
projects,
2.the sure thing principle, and
3.the generalized Markowitz axioms
Measurement of Cash Equivalents (or Cash
Equivalent Coefficients)
-- Measurement of cash equivalents or cash equivalent
coefficients.
– Presents an experiment designed to elicit cash
equivalents (cash equivalent coefficients) from a
group of subjects for investment projects
– Estimate the cash equivalent or cash equivalent
coefficient as a function of the four parameters
and conduct a lease or buy analysis of investment
projects using the estimated cash equivalent or
cash equivalent coefficient function.
Target Groups:
The paper should be of great interest to the following
groups of people:
--corporation finance theorists
--operation researchers
--behavioral decision theorists
who are interested in the application of decision
theories to solving problems such as capital budgeting
problems, of which lease or buy decisions are part
Investment Opportunity Set=S
-- S={(,,,),*≤≤*,
0≤  ≤ *,0≤  ≤ *,
0 ≤ ≤ *}
-- This is a bounded closed 4 dimensional box
where every point in S represents an
investment project.
Weak Order
Generic Axioms:
--Preference relation * on set S is
asymmetric, irreflexive, transitive

and connected
--Indifference relation ~* on set S is
symmetric, reflexive, and transitive.
--Preference Indifference ≤* is the union of *
and ~* . It is symmetric, reflexive, and
transitive.
--The following transitive relations on ≤* and ~*
must hold on S.
If Pa≤*Pb, Pb~*Pc, Pa ≤* Pc
If Pa~*Pb, Pb≤*Pc, Pa ≤* Pc
Behavioral Postulate 1
--The existence of the most preferred project in
set S=(*,0,0,0)
--The existence of the least preferred project in
set S=(*,*,*,*)
Note that this follows partly from the boundedness of
set S which is closed. It is presented as a postulate
for clarity.
Behavioral Postulate 2
(the sure thing principle)
--(*,0,0,0) is preferred to any project in set S
--There is a cash equivalent project (,0,0,0)
such that
(,0,0,0)≤*(*,*,*,*)
where  < *.
Postulate 3
(Generalized Markowitz Criteria)
--1.(,,,) <*(’,,,) if ’ > 
--2. (,,,) <*(,',,) if ’ < .
--3. (,,,) <*(,,',) if ’< .
--4. (,,,) <*(,,,') if ’ > .
In words, holding the other risk parameters
constant, the greater the return, the more
preferred the project.
Holding the other return and risk parameters
constant, the smaller each risk index of the
project, the less preferred the project.
Lemma 1
(strictly increasing preference)
--For  and ’,0<< ’ <1,
P+(1- )P*<* ’P+(1- )P*
where P*=(*,*,*,*) is the least preferred
project and P is any other project in set S.
Proof:
(*,*,*,*)+(1-)(*,,,)'(*,*,*,*)
+(1-')(*,,,)
In light of the following:
+(-*}< +'(-*)
whereas,
+(-)>+'(-),
+(-)> +'(-),
and
+(-)> +'(-)
--On any straight line emanating from the
least preferred project P* , preference is
strictly increasing as a project moves
away from P* to any project P in S
--This property is needed to bypass the
discontinuity of the second kind.
Lemma 2
(one to one property)
--The cash equivalent function c=U(P) defined
on set S which maps a project P to its cash
equivalent c in R1 is a one to one function.
Proof:
Given any project Po,
(,0,0,0)<* Po <*(*,0,0,0)
by the sure thing principle.
--set the sure project P1 to be the mid point of the most
preferred sure project and the lease preferred sure
project.
P1=1/2(,0,0,0)+1/2(*,0,0,0)
--If P1≥*Po, discard all projects Ps such that
P>*P1. Let S2 be the remaining projects in S. It has
the most preferred sure project P1 and the least
preferred sure project (a,0,0,0). Let the new mid
point of the two be denoted as the sure project P2.
--If P1≤*Po, discard all projects Ps such that P<*P1. Let
S2 denote the remaining projects in S. S2 has the
least preferred sure project P1 and the most preferred
sure project (*,0,0,0)
--Set P2 be the point of these two least and most
preferred project.
--To start the next step of the search, compare Po with
P2 and determine which side Po is more or less
preferred to P2.
--Create a new set S3 by discarding the half of projects in set S2
that are no longer needed. Set the mid point of the most and
least preferred sure prospect of S3 as the next search vector P3.
………..
--Repeat the procedure similar to the above
until the the mid point vector Pt becomes as good as project Po.
--The sure project Po can be readily found as Pt becomes as good
as Po and St becomes a set of indifference projects as good as
Po=(co,0,0,0)
--Note that {Pt ,t=1,2,…} is a Cauchy sequence and convergent.
Lemma 3
(onto property)
The cash equivalent function u is an onto function
such that for any given co in its range, there is a
project Po in S such that
u(Po)=co
or
Po ~*(co,0,0,0).
Proof:
--Create a line between P* and P*
--Develop the search procedure similar to the one used
in the proof of lemma 1 on this line.
--Repeat the seach procedure until the the mid point
vector Pt becomes as good as project Po.
--The project Po can be readily found as Pt becomes as
good as Po and St becomes a set of indifference
projects as good as (co,0,0,0)
--Note that {Pt ,t=1,2,…} is a Cauchy sequence and
convergent again and Pt is no longer a sure project.
Main Theorem
(continuity of the cash equivalent func tion)
The cash equivalent function u(P)=c is a
continuous function of , , , and  That
isgiven any co =u(Po) for Po in the interior of
S and an arbitrarily small positive , there is a
small positive  such that the following
inequality holds for the Euclidian norm below:
|P-Po|<
and for any such project P, the following
holds:
|u(P)-co|<
Proof:
--Let Po be a project in the interior of S. let co be its
cash equivalent: u(Po)=co. Let  be a small positive
real such that for any sure project P in the open set of
sure projects
co-<u(P)<co+.
-- For ’ and ”, 0<’, ”<1, there exist P(’) and P(”)
by the onto property of lemma 2
where
and
P(’ o-’ o+’ o+’ o+’
P(” o+” o-"’ o-” o-”
such that for any P in the box defined by the above two projects,
co-= u{P(’<u( P)< co+= u{P(’
--Let  be the smaller of v’ and v”.
Let:
P(-)=(o-,o+,o+,o+)
and
P(+)=(o+,o-,o-,o-)
such that for any P in the sphere
co-=u{P(-)}<*u(P)<*u{P(+)}=co+.
These two projects define the 4 dimensional cube
centered at Po. There is always the 4 dimensional
sphere in the cube defined by the norm below:
|P-Po|<.
The sphere is in turn contained in the 4 dimensional
box defined by P(’) and P(”).
Hence , for any P in the sphere,
|u(P)-u(Po)|<
Thus, the u is a continuous function.
Due to the strictly increasing preference pointed out
in the lemma 1, the discontinuity of the second kind is
ruled out.
Measurement of cash equivalents
--A team of 4 consultants in a lease or buy decision
case is presented with 16 projects with a set of the
four parameters. They are requested to provide the
team’s cash equivalents for the 16 projects. The
following four parameters are set to two levels each.
--=the net terminal value NTV of a project
=future value of cash flows - future value of the
outlay where the reinvestment rate is the cost of
capital
--=a project life at issue
The operating risk index is a new index inspired by the
breakeven analysis. Assume that a project at issue
is accepted.
--=the operating breakeven point as a percentage of
the sales level.
OBEP=fixed cost/contribution margin per $1 in sales
=OBEP/Sales level
--The financial risk index is another new index
inspired by the breakeven analysis. Again
Assume that the project is accepted.
--=the financial breakeven point as a
percentage of the sales level
FBEP=interest cost/contribution margin per
$1 in sales
=FBEP/the sales level
16 experimental settings in the case
low
--anticipated return
--project life
--operating risk index
--financial risk index
high

$$2000
yrsyrs

0.1
0.3

experimental settings in total
For instance, (1000,0.1,0.0,3),(2000,0.3,0.2,5) etc…
Cash equivalent coefficients derived from cash
equivalents
c=cash equivalent quoted in an experiment
i = risk free rate=6%
cc=cash equivalent coefficient
cc=c(1+i)V
Recall that NTV is the net terminal value..
•
•
•
•
•
•
•
•
•
•
•
16 Cash Equivalents or Cash Equivalent Coefficients
no c
cc
no
c
cc
1 834 .9113
9 1802 .9845
2 588 .6817
10 1480 .8579
3 680 .7431
11 1767 .9654
4 472 .5472
12 1179 .6834
5 678 ,7409
13 1786 .9758
6 659 .7640
14 1176 .6817
7 562 .6141
15 1360 .7431
8 365 .4233
16
849 .4523
no=id c=cash equivalent cc=cash equivalent coefficient
--Note:
The team is presented with each project and asked if
they buy the project at a quoted price(cash
equivalent). They must indicate yes or no. If no, the
price is raised, the process is repeated until they say
yes. The cash equivalent for the project between the
two quoted prices is interpolated by using the
probability of acceptance and that of rejection for
each of the two quoted prices obtained from the
logistic regression.
--The next diagram shows that the cash
equivalent is interpolated to be c=675. The
two quoted prices are 650 and 700 with the
probabilities 0.6 and 0.4 respectively.
--the cash equivalent c is the price at which the
probability hits 1/2.
Interpolation
P(accept)=0.6
0.4=1-P(accept)
1/2
650
c
700
Estimating the cash equivalent coefficient function for
the team of the consultant in the case
yi=i-th dependent variable=i-th cash equivalent or its coefficient
i=i-th net terminal value
I=i-th operating breakeven point as a percentage of sales level
i=i-th financial breakeven point as a percentage of sales level
i=i-th project life
I =i-th response error term
The response surface to be fitted to 16 cash equivalent or its
coefficients
yi =bo+b1i+b2 I +b3I+b4 I+I
Estimated cash equivalent function
bi
estimates
t-values
i=0
i=1
i=2
i=3
i=4
760.062
0.808
791.875
1043.12
-175.06
3.796
11.288
-2.202
-2.000
-4.868
R2=0.94 significant at 5%
Estimated cash equivalent coefficient function
bi
i=0
i=1
i=2
i=3
i=4
estimates
1.130
0.0001075
-0.528
-0.929
-.090
R2=0.90
t-values
13.57
3.572
-3.494
-6.185
-6.022
significant at 5%
Lease or Buy Decisions
--Now that the cash equivalent coefficient for any
project can be computed by substituting the four
parameters of the project into , , , and  by usiing
the fitted equations.
It is possible to conduct a lease or buy decision by
means of the cash equivalent coefficient method
--The use of the net terminal value is of value since it is
not necessary to estimate the cash equivalent
coefficient for each annual cash flow. More
importantly, it enable us to bypass many tenuous
assumptions such as additivity, stationarity, etc..
associated with multiperiod discounting
--Lease or Buy Decisions can be conducted by using the
cash equivalents rather than the cash equivalent
coefficients. The required rate of return RR implicit in
a cash equivalent can be derived from the following
formula:
RR=(/c)1/-1
--The users of RR are biased typically against using the
cash equivalent or its coefficient. However, they
should not be adverse to the use of the cash
equivalent or its coefficient because they are
mathematically related as shown above. Using any
one method is equivalent to using any other method.
--The paper is the first one to clarify the foundation of
cash equivalents in corporation finance. People in
finance still live in the world of the expected utility
theory. This work demonstrates the the new risk
theory and measurement technique presented here is
a useful tool in corporation finance. It is hoped that
this work will stimulate interest in the application of
decision theories into corporate finance.