A Theory and Measurement of Cash Equivalents for Lease or Buy Decisions

FUR XII 20066/22--6/26Rome, Italy

Keishiro Matsumoto,Ph.D.

University of the Virgin Islands

Matsumoto507@aol.com

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 theory1.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, transitiveand 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 isgiven 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

P(’o-’o+’o+’o+’and

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 levelFBEP=interest cost/contribution margin per $1 in sales=FBEP/the sales level

16 experimental settings in the case

low high

--anticipated return $$2000--project life yrsyrs--operating risk index 0.1 0.3--financial risk index

experimental settings in totalFor 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

650 700

0.4=1-P(accept)

P(accept)=0.6

c

1/2

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+b1i+b2 I +b3I+b4 I+I

Estimated cash equivalent function

bi estimates t-values

i=0 760.062 3.796

i=1 0.808 11.288

i=2 791.875 -2.202

i=3 1043.12 -2.000

i=4 -175.06 -4.868

R2=0.94 significant at 5%

Estimated cash equivalent coefficient function

bi estimates t-values

i=0 1.130 13.57

i=1 0.0001075 3.572

i=2 -0.528 -3.494

i=3 -0.929 -6.185

i=4 -.090 -6.022

R2=0.90 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.