Sanches, Alexandre Leme; Pamplona, Edson de O. e Montevechi, José Arnaldo B. Capital Budgeting Using
Triangular Fuzzy Numbers. V Encuentro Internacional de Finanzas. Santiago, Chile, 19 a 21 de janeiro de 2005
CAPITAL BUDGETING USING
TRIANGULAR FUZZY NUMBERS
Alexandre Leme Sanches
Prof. M.Sc. IEPG – Universidade Federal de Itajubá (UNIFEI)
E-mail: alexandresanches@unifei.edu.br
Edson de Oliveira Pamplona
Prof. Dr. IEPG – Universidade Federal de Itajubá (UNIFEI)
E-mail: pamplona@unifei.edu.br
José Arnaldo Barra Montevechi
Prof. Dr. IEPG – Universidade Federal de Itajubá (UNIFEI)
E-mail: montevechi@unifei.edu.br
Summary:
The economic engineering the analysis involves uncertainty about future cash flows. To deal
quantitatively with imprecision or uncertainty, fuzzy set theory is primarily concerned with
vagueness in human thoughts and perceptions.
As an alternative to conventional cash flow models, where cash flows are defined as either
crisp numbers or vagueness, is proposed an engineering economic decision model in which
the uncertain cash flow and discount rates, specified as triangular fuzzy numbers.
The present value formulation of this fuzzy cash flow model is derived. The result of the
present value is also a fuzzy number with nonlinear membership function. The present value
can be approximated by a triangular fuzzy number.
Risk analysis involves the development of the probability distribution for the measure of
effectiveness. The uncertainty associated with an investment alternative is generally either
given as the “possibility” of an unfavorable value of the measure of effectiveness or measured
by the variance of the measure of effectiveness.
Keywords:
Investment evaluation; Fuzzy; Capital budgeting.
Code JEL:
D81 - Criteria for Decision-Making under Risk and Uncertainty
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Capital Budgeting Using Triangular Fuzzy Numbers
INTRODUCTION
Nowadays, the uncertainties associated with “investment analyses”, in all areas, create a
demand for alternatives methods to make possible the translation, to a mathematical language,
of the intangible values and human experience, improving the available resources in the
decision making process. Therefore, it becomes necessary the application of an investment
evaluation method which represents, in a realistic fashion, the economic viability of a given
investment subject to uncertainties.
One of the main problems found in the investment analyses is the measurement of the
analyses key variables. Most of the time, the evaluation of the numerical values of these
variables is directly associated with the analyst’s own abstraction.
Another problem common problem, and also of great relevance, is how people operate with
uncertain values and abstract ones, for, if the intuition of a uncertain value is complex,
working with them, intuitively, becomes an impractical task.
Fuzzy logic has the capability to represent, numerically, linguistic va lues, uncertain and
abstract, aiding significantly to the decision making process in investment analyses. Also, it
has the capability to work with those values, being, therefore, the resource herein explored.
This paper is about, specifically, the use of Fuzzy logic in the financial field, in what concerns
the uncertainties associated with predictions of future situations. Such predictions are input
data, in the calculation of fuzzy NPV (Net Present Value).
When it comes to the financial field, the user’s skill in the utilization of fuzzy logic might
enable such tool to surpass the deterministic methods or even the stochastic ones, for its
independence from the historical data to be successfully used.
OBJECTIVES
?
?
This paper has as its main objective the demonstration of the proper use of fuzzy logic
in the evaluation of investment projects, and its capacity to provide relevant
information towards the decision making process under uncertain conditions.
The secondary objective is the representation of a software prototype to calculate the
fuzzy NPV and analyses relative to the enterprise.
METHODOLOGICAL ASPECTS
The research method to be used is known as “quasi-experiment”. According to Bryman
(1989), the quasi-experiment is a research experience where the researcher doesn’t have total
control over the input variables of the system and there’s a non-random treatment of the
experiment. According to Trochin (2001), the quasi-experiment is similar to the experiment,
nonetheless there is no random designation. In what concerns internal validation, the quasiexperiment is inferior regarding the experimental method, considered to be the highest
internally validated research method.
Sill according to Trochin (apud Gonçalves (2003)), the quasi-experiment is a method which
incorporates a great part of the quantitative research works where the human behavior is
present, presenting various scopes.
In the current paper it is adopted the Proxy Pretest Design, which is characterized by the
realization of a pre-test and one post-test after the program has been implemented. This is
possible through the estimation of the studied variables for the group, before the beginning of
the program. Here it will be made an application of the deterministic method and latter the
application of the possibilistic method, finally both results will be compared.
2
Capital Budgeting Using Triangular Fuzzy Numbers
Investment Data (selected group)
Deterministic NPV Calculation – viability (pre -test)
Sensibility Analyses (uncontrolled)
M
e
t
h
o
d
Definition of the variables to be Fuzzyfied
Fuzzyfication of the selected variables (specialist)
Fuzzy NPV Calculation
F
u
z
z
y
L
o
g
i
c
Viability and possibilities analyses associated with the Fuzzyfied
NPVs (post-test).
Defuzzyfication of the NPV (if necessary)
Comparison with the Deterministic NPV - The Proxy Pretest Design
Figure 1 – Job/Work Sequence
Notwithstanding, it is possible to generalize certain research results, is terms of the
investment evaluation method used, for there is no great differences in how the models is
applied in other cases.
This paper has as its focus the “evaluation method” and not the “subject to be evaluated”.
Thus, the limitations mentioned above do not reduce, in an significant way, the capacity of
exploration and further attest the subject.
LITERATURE REVISION
The Fuzzy Logic
Fuzzy logic is a bridge which connects the human thinking to the machine’s logic.
In a fuzzy set, the transitions between a member or a non-member occur in a continuous ray,
being a membership degree associated between “0” (totally non- member) and “1” (totally
member).
The degree of membership “is not probability”, but a measure of compatibility between object
and the concept represented by the fuzzy set.
The concept of membership function is further better explained.
The uncertainty and subjectivity manipulation capacity, without the rigid barriers from the
classic logic, characterize the fuzzy numbers, in Martinez Jr. (2002) viewpoint.
3
Capital Budgeting Using Triangular Fuzzy Numbers
The Fuzzy Numbers
Fuzzy numbers are a subset from the real numbers set, representing the uncertain values. All
fuzzy numbers are related to degrees of membership which state how true it is to say if
something belongs or not to a determined set.
Membership Function
Source: Montevechi, J. A; Revisão Sobre Lógica Fuzzy
A diffuse set, cloudy or fuzzy “A” of an “E” universe of characterized by a membership
function ? A:E ? [0,1], which associates to each “x” from “E” a number ? A(x) in the [0,1]
interval, representing the degree of membership of “x” in “A”. To ? A(x) it is given the name
of membership function or simply membership, as it is called in most publications. To ? A(x)
near “1”, it is said that there is a high possibility that “x” ? “A”. On the other hand when
? A(x) is close to “0”, there is a low possibility that “x” ? “A”.
As a fuzzy set has a certain number of affinities relative to the “x” elements which constitute
it, it is feasible here to show how they must be represented, once the way it is shown in the
literature is completely particular.
A = [(x, ? A(x))] x ? E
(1)
According to Von Altrock (1995), to a “A” set, the function mA(x) “membership” is defined
as:
mA(x) = 1 if and only if x ? A
(2)
0 if and only if x ? A
mA(x) ? [0,1] ? x’ ? [x1 , xn ]
Analyzing the following diagram, where the left graph shows the Boolean logic, and the one
in the right shows, yet in a discrete fashion, the fuzzy logic, it can be stated that in the
Boolean logic the degree of pertinence/belonging of an element regarding a set is 0 or 1,
which means, the elements “d” and “b” belong to a set “A” (degree of pertinence / belonging
= 1), while the elements “c” and “d” do not belong to set “A” (degree of
pertinence/belonging = 0).
Now, in the fuzzy logic there may be an element “c” which partially belong to set “A” (0 ?
degree of pertinence/belonging ? 1), that is, mA(x)= 0,5.
4
Capital Budgeting Using Triangular Fuzzy Numbers
A
A
a
a
c
b
b
c
d
d
?A
?A
1
1
0.5
a
b
c
d
x
a
Boolean Logic (binary)
b
c
d
x
Fuzzy Logic
Figure 2 –“Membership” function
Source: http://if.kaist.ac.kr/lecture/cs670/lecture- note/Chapter1.ppt
The Many Forms of Fuzzy Numbers
Source: http://if.kaist.ac.kr/lecture/cs670/lecture- note/Chapter5.ppt
There are various types of fuzzy numbers and its nomenclature is, in general, associated with
its format, such as: sine numbers, bell shape, polygonal, trapezoids, triangular, and son on. In
this paper it is emphasized the triangular fuzzy numbers, once the are the most interesting for
the financial field.
General Shape of the Fuzzy Number
In Chiu and Park (1994), a fuzzy number is a fuzzy subset characterized by a Membership
function, which satisfies the following conditions:
? Normality: ? A(x) = 1, for, at least, each/one point/dot of x ? R
? Convexity: ? A(x’) = ? A(x 1 ) ? ? A(x2 ), where: ? A(x) ? [0 , 1] and ? x’? [x1 , x 2]
Adding to Kuchta (1996), where a fuzzy number is an element of hexadimentional
quantification in the following manner:
nf = (a1 ,a2 ,a3 ,a4 , f a (?), f a (?))
1
(3)
2
5
Capital Budgeting Using Triangular Fuzzy Numbers
Where: a1, a2, a3, a4, are real numbers and a1 ? a2? a3 ? a4
f 1a (?) is a continuous real function non decreasing defined in the interval [0, 1] such that:
f 1a ( 0 ) = a1 and f 1a ( 1 ) = a2
f 2a (?) is a continuous real function non increasing defined in the interval [0, 1] such that:
f 2a ( 1 ) = a3 , and f 2a ( 0 ) = a4
? A(x)
nf = (a1,a2,a3,a4, f a , f a )
1 2
1
?
?
f1a
f 2a
0
a1
x
a2
a4
Figure 3 – General Forma t of the Fuzzy Number
Triangular Fuzzy Numbers (TFN)
a
a
If f 1 (x) e f2 (x) are linear functions and also “a2 = a 3 ”, One can omit “a2 “ or “a3 ”, then
the membership function takes the following shape:
?0,
? x - a
1 ,
?
?? a - a 1
µ( A ) ( x ) = ? 2
a3 - x
?
,
? a3 - a2
?
?0,
x < a1
a1 ? x ? a2
(4)
a2 ? x ? a3
x > a3
6
Capital Budgeting Using Triangular Fuzzy Numbers
? (x)
1
0
a1
a2
x
a3
Figure 4 –Triangular Fuzzy Number Structure
The Fuzzyfication
Fuzzyfication is the mapping of the real numbers domain (generally discrete) to the fuzzy
domain.
Also it represents linguistic values assignments, vague descriptions or qualitative ones,
defined by a membership function to the various input variables.
Figure 5 displays the fuzzyfication of the ROR “universe” for a determined company.
ROR
Very Low
Low
Medium
High
Very High
1
0
5
10
15
20
25
30
35
40
%
Figure 5 - Fuzzyfication
Source: Montevechi (1998)
The Defuzzyfication
7
Capital Budgeting Using Triangular Fuzzy Numbers
Defuzzyfication is the operation/proceeding in which the value of the output linguistic value,
inferred by the fuzzy rules/regulations, will be translated to a discrete value (Shaw, 1999).
Figure 6 show the “universe” of the NPV’s classification for the investments of a given
company, where the real discrete value is obtained from the fuzzy result.
NPV
Bad
0
1000
(R$)
Regular
Good
3000
Great
5000
Excellent
7000
9000
Figure 6 – Defuzzyfication
Source: Montevechi (1998)
Traditional Investment Evaluation Methods
The investment eva luation methods, most commonly used and that have wider publicity,
involve the basic model of Discounted Cash Flow (DCF), with its main variants: NPV (Net
Present Value) and IRR (Internal Rate of Return) (Santos 2001). The traditional methods,
initially, may be seen as limited, when it comes to uncertainty dimension, nevertheless they
are the basis for the development of sophisticated techniques which, at the present time, have
been used with great success.
The Net Present Value (NPV)
According to Santos (2001), using financial mathematics, it is possible to carry a viability
analyses of the project. Thus it is necessary that a forecast of all future cash flows is made for
all the “n” future periods. This forecast must be made with as much accuracy as possible,
taking into which will it be the outflows and inflows in the next “n” periods, then, with a
predetermined rate, the ROR (rate of return), in order to obtain the value of that project at
period zero, where these values are added to the initial investme nt.
The NPV method has its basic format synthesized by the following equation:
(5)
8
Capital Budgeting Using Triangular Fuzzy Numbers
n
NPV
=?
i? 0
CFi
(1 ? r ) i
Where:
?
CF i = Cash Flow, expected for the given i period;
?
r = discount rate
?
i = 0, 1, 2, 3, ... , n (periods).
The existence of a positive NPV is defined as the basic criteria in the acceptance or rejection
of the determined project (Ross, Westerfield, 1995).
Evaluation of the Investments under Uncertainty Conditions Using Fuzzy Triangular
Numbers
In this topic it is presented the “possibilistic” method, based in Buckley (1987) and Souza
(1996), which use the Fuzzy set theory.
The membership function for NPV (Fuzzy), presented by Buckley (1987), is given:
n
-j
f n,i ((y)P)= ? f j,i ((y)F j )(1+ f k(j) ((y)r f ))
j=0
(6)
For:
i = 1, 2, where k = i for negative F and
k = 3 - i for positive F
(1) left side
(2) right side
REAL CASE APPLICATION
In the current case, a classic problem of investment viability under uncertainty conditions
evaluation occurs, in a single investment (no alternatives) scenario. The decision, here, is to
accept or reject the project.
Many uncertainties are inherent to this endeavor, some easily qua ntified, other not,
nonetheless all must be quantified somehow. The specialists forecasts must be added to the
historical data as well as a great dosage of “gut feeling”, taking us to adopt a possibility
array/ray for the input data of the endeavor, in order to latter manipulate them, finding the
possibilities array/ray for the NPV values.
The matter itself is to accept or reject a project, in an “undeterministic” fashion, which means,
quantify the chances of failure.
9
Capital Budgeting Using Triangular Fuzzy Numbers
To perform the involved calculation a software is presented (prototype) named “Fuzzyinvest
1.0”, which, at first, meets all the need of the studied case.
The main activity of the “Mining” company is the extraction of raw feldspar, the companies
which buy feldspar, here called clients of the mining company, purify and refine the feldspar,
which is destined to the construction market.
Observing the great expansion of its clients business, and having abundant available raw
material, the Mining company has shown interest in the feldspar processing, and in entering in
the market as a competitor of its clients.
Project Data
Fixed Investiment
R$12.874.035,00
Working Capital
R$2.376.000,00
Monthly Fized Cost
R$2.304.125,00
Variable Cost / unit
R$ 16/Ton
Forecasted Sales
100.000 Ton/ano
Price
R$ 98,00/Ton
Planning Horizon
10 years
Residual Value
"R$8.582.690,00"
ROR
15% year
Income Tax
35% year
Depreciation
10% year
Table 1 – “Project Data”
Deterministic NPV Calculation and Sensitivity Analyses
To calculate the deterministic NPV the Excel™ software is used. In the same sheet, exploring
the available resources, it is also carried a sensitivity analyses for the input variables involves
in the calculation.
10
Capital Budgeting Using Triangular Fuzzy Numbers
Figure 7 – “Deterministic NPV Calculation and Sensitivity Analyses Sheet”
On the aforementioned sheet, it is noticeable the presence of scroll bar in the input variables,
it can be noticed yet, the presence of macro which calculate the minimum price, the lowest
demand and the highest possible investment, where, based on this data, the investment
becomes viable.
Considering the didactical character/scope of this paper, it becomes relevant the fuzzyfication
of all the input data.
The NPV found is R$ 8,211,191.38, which justifies the acceptance of the investment, in the
case of a simple deterministic analysis.
It is advisable to say, as low as they can be, that all uncertainties affect the final result,
therefore a group of small uncertainties in all input variables may result in a great uncertainty
in the output variable, when it comes to fuzzy NPV.
Involved Uncertainties
The uncertainties forecasted by the company’s administration are:
?
Fixed Investment (Initial Outlay) : +/- 10%;
?
Working Capital: +/- 10%;
?
Annual Fixed Cost: +/- 10%;
?
Variable Cost / unit: +/- 13%;
?
Sales Forecast: -30% to +20%;
?
Price: -20% to +15%;
?
Life Time: -20% to +50%;
?
ROR: +/- 10%;
All the variables and its uncertainties, mentioned above, are fuzzyfied and used in the
calculation of the Fuzzy NPV.
11
Capital Budgeting Using Triangular Fuzzy Numbers
Presentation of the “Fuzzyinvest 1.0” Software (prototype) and Calculation of the
Possibilistic NPV
To carry out the calculation it was created a software “Fuzzyinvest 1.0”, in VBA (Excel, MS
Office XP) language, based on the aforementioned concepts, where, after the insertion of the
fuzzyfied data it calculates the fuzzy NPV as well as its graphic representation.
The software sho w, also, as a result, the “failure possibility” of the investment, that is, the
percentage of the triangle area (fuzzy NPV) which is in the negative region of the graph, on
the left of the vertical axis.
As an analyses option, it is possible to alter the endeavor/project’s data, through the graphics
themselves, and watch the consequences on the Fuzzy NPV graph, Still, as an option, the
graphical response can be altered, where the software questions which variable it is desired to
alter in order to obtain that response.
The main screen of “Fuzzyinvest 1.0” is the following:
Figure 8 - “Fuzzyinvest 1.0” Main Screen
This screen shows the fields where the input data can be inserted, and the fields that show the
output data, those output data are: Fuzzy NPV (graph) and the failure possibility of the
endeavor/project.
Besides the visualization of the fuzzyfied input data, some analyses may also be obtained
through the “Gráfico” (Graph) sheet such as the graphical sensibility analysis.
Tools such as Achieve Goal where the input data can be calculated by forcing the output, can
also be done with the “Gráfico” sheet. The main screen of “Fuzzyinvest 1.0” has a field
assigned for the “Gráfico” sheet. It is as follows:
12
Capital Budgeting Using Triangular Fuzzy Numbers
Figure 9 – “Gráfico” Sheet”
To visualize the involved data and further details about the Fuzzy NPV, it is also shown the
Sheet “Cálculos”, which can be taken as the most important part of this paper, for in it one
can find all the theory that’s been describe throughout the previous topics.
Figure 10 – “Cálculos” Sheet
13
Capital Budgeting Using Triangular Fuzzy Numbers
In the “Cálculos” sheet are all the “Scenarios” tools and Achieve Goal, which can also be
activated from the “Gráfico” sheet as previously mentioned.
Defuzzyfication of the Result and Comparison to a Fuzzy Pattern.
The fuzzy NPV value found is (-29.722.301, 8.211.191, 54.929.934) and the failure
possibility of the project is 27.51% . With these information at hand, the board of directors of
the company may take a more solid decision than it would with but a simple deterministic
NPV.
According to Shaw’s (1999) definition, a calculation of the failure possibility of 27.51% may
be considered as a defuzzyfication operation, for a fuzzy output variable is being translated
into a real number.
From this point on, there is a discrete numerical value which can be compared to a fuzzy
pattern. Such a pattern, defined by the high administration of the company, involves the
Universe of the Failure Possibilities classified as follows:
Very Low
Low
Medium
High
Very High
1
0
5
10
15
20
(27,51)
35
40
%
Figure 11 – Universe of the Failure Possibilities of a Project
The company also adopts its own acceptance or rejection criteria towards investments
according to the fuzzy pattern, such criteria is abstract and depend on the profile of the
investor, the amount of investment in question and even the tangible values defined by the
company administration itself.
The acceptance criteria of the company are:
Fuzzy classification array of the failure
Decision of the company
possibility of the investment
Very Low
Unconditionally Accept
Low
Accept with caution
Average
Accept under restrictions
High (27.51%)
Reject and review project
14
Capital Budgeting Using Triangular Fuzzy Numbers
Very High
Unconditionally Reject
Table 2 – “Investment Projects Acceptance Criteria”
Obs.: The endeavors/projects with failure possibility, with intermediary classifications, must
be proportionally considered, which means, for instance: a failure possibility of 17.5% is
classified as “0.5 Low / 0.5 Medium” in this pattern. Consequently the decision to accept or
reject must be wisely taken, which is “To accept with caution and restriction”.
RESULT ANALYSIS
In the present case, the failure possibility of the endeavor/project is classified as “High”,
therefore, the investment must be rejected and the whole project reviewed.
CONCLUS IONS
The most relevant conclusion, concerns the comparison of the deterministic NPV with the
Fuzzy NPV, being the “uncertainty” dimension made a go investment, in the deterministic
method, turn into a rejected one.
In spite of the low capacity of generalizatio n of the “quasi-experiment”, due to its nonrandomness inherent to the method, it is concluded that a comparison between the
deterministic NPV and the Fuzzy one takes us to relatively general conclusions.
The conclusions which may be considered general, are due to the exploration of the
investment evaluation method and not the object of analyses. The way to evaluate an
investment doesn’t change much, when applied to another object of analyses.
The real purpose of the “quasi-experiment” is the adaptation of the “experiment” method to
the administrative reality of the organizations, where certain characteristics of the
“experiment” do not apply.
An important conclusion about fuzzy logic, confirmed by Yager (1980), is regarding the noninversion of the operations, what many times can lead to mistakes. For instance: A+B=C does
not imply that C-B=A, which happens to real numbers.
Another conclusion is concerning the fact that this paper is restricted to triangular fuzzy
numbers, which are a restrict field of the fuzzy logic and that simplify a lot the operations,
Fuzzy logic, in general, goes way beyond the applications used in this paper.
One of the most relevant information, obtained from the fuzzy NPV, is the failure possibility
of the project, it is obtained from a proportion of the area seen under the membership curve,
which takes us to an analogy with the PDF (Probability Density Function) using statistical
methods.
There must be extreme caution when using this analogy, for, the origins of the membership
function and the PDF are totally distinct. It is also important to emphasize that the total area
under the PDF curve, in any given distribution, is always equal to 1, which doesn’t always
happen to the area below the membership curve.
The uncertainty associated with the fuzzy NPV, is characterized by the amplitude of the fuzzy
number that represents the fuzzy NPV, that is, “a3 – a1 ”, therefore, the “uncertainty associated
to the investment” and the “investment viability” are totally independent.
The concept of uncertainty must also be disassociated from a “bad” condition, once the same
way that uncertain conditions can lead to unfavorable conditions, they may also converge to
favorable ones.
15
Capital Budgeting Using Triangular Fuzzy Numbers
It is also important to point out the great visual analyses power of the fuzzy number, the
visualization of the membership graph takes us to another analyses dimension, improving
even more the decision making resources.
The computerized resources allow us to deal with possible difficulties found in the
calculation, with speed and accuracy, what happens with “Fuzzyinvest 1.0”.
The software, however, is not totally consistent, thus if absurd input data is inserted it won’t
tell the user, and it initiates the calculation. The solution to this problem may also be
presented as a suggestion to future works.
The user friendly feature of the VBA (Visual Basic Applications) makes the whole
programming task easier, which helps the calculation and allows them to be manipulated with
Excel sheets.
The software values the visual aspect and the relevant information, emphasizing the
membership graph and the failure possibility.
Many other observations may be noted when manipulating the software, according to the
application and the involved problem.
The fuzzy logic capacity to provide resources in the decision make process is unchallenged.
Nevertheless the need for special caution in the fuzzyfication of the input data and in the
general application of the method is also imperative.
Aware of the limitations and restriction of the method, it is possible to state that, from now on
its applications tend to increase significantly.
The application of fuzzy logic in other fields, besides the economic, has been increasing year
after year, what indicates a strong tendency of growth in the economic field as well.
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Capital Budgeting Using Triangular Fuzzy Numbers
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http://www.dc.ufscar.br/~fernandes/trabalhos/NebXPara.ppt
http://www.din.uem.br/ia/controle/fuz_cara.htm
http://members.tripod.com/geloneze/Fuzzy.htm
FUZZYTECH 5.5; User Manual, Information Software Corporation.
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