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UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
Prof. Alexandre Leme Sanches, MSc.
Prof. Edson de Oliveira Pamplona, Dr.
Prof. José Arnaldo Barra Montevechi, Dr.
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
Itajubá
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
Itajubá
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
Universidade Federal de Itajubá
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
1. Introduction
2. Objectives
3. Methodological aspects
4. Literature revision
5. Operations with Triangular Fuzzy Numbers (TFN)
6. Fuzzyfication and Defuzzyfication
7. The Net Present Value
8. Application of Fuzzy Numbers in Investiments Analysis
9. Analyzing the Fuzzy NPV
10. Real Case Aplication
11. Conclusions
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
Uncertainties associated with “Investment Analyses”
Alternatives methods
Decision making process
Optimization of financial resources
1.Introduction:
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
Main Objective:
Demonstrate the use of fuzzy logic in the evaluation of investment projects under uncertaint conditions;
2. Objectives:
Secondary Objective:
Presentation of a software prototype to calculate the fuzzy NPV and relative analyses.
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
3. Methodological aspects:
The research method to be used is known as “quasi-experiment”:
• Pre and Post Test, TROCHIN (2001).
• Doesn’t have total control over the input variables of the system, BRYMAN (1989).
• There’s a non-random treatment of the experiment, TROCHIN (2001).
• Where the human behavior is present, TROCHIN (apud GONÇALVES (2003)).
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
Investment Data (selected group)
Deterministic NPV Calculation – viability (pre-test)
Sensibility Analyses (uncontrolled)
Definition of the variables to be Fuzzyfied
Fuzzyfication of the selected variables (specialist)
Fuzzy NPV Calculation
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
L
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UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
4. 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 continuously;
• The degree of “membership is not probability”, but a
measure of compatibility between object and the
concept represented by the fuzzy set.
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
4.1. Membership Fuction - Example:
A
c
d
a b
A
a b c d x
1
A
d
ab
c
1
A
a b c d x
0.5
Boolean Logic (binary)
Fuzzy Logic (continuous)
A(x): Membership
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
4.2. Fuzzy Number – General Definition, KUCHTA (1996)
Where:
4,3,2,1 aaaa are real numbers and 4321 aaaa
)(1
af : is a continuous real function non decreasing defined in the interval [0,1], such that:
: is a continuous real function non increasing defined in the interval [0,1], such that:
))(),(( 2,4,3,2,11
affaaaafn a
42 )0( af a 32 )1( af a
11 )0( af a 21 )1( af a and
and
)(2
af
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4.3. Fuzzy Number:
a1 a4
xa2 a3
1
0
A(x) ),( 21,4,3,2,1aa ffaaaafn
af1 af2
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
4.4. Triangular Fuzzy Number (TFN):
If and are linear functions and a2 = a 3:
,0
,
,
,0
)(
3
32
23
3
21
12
1
1
)(
ax
axa
aa
xa
axa
aa
ax
ax
xA
a1 a3
A (x)
x
1
a2
A = (a1, a2, a3)
af1af2
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4.5. Fuzzy Number – Example I:
A “Fuzzy Set” representing the NPV: “Rates: Low/Medium/High”
Low
Medium
High
0
1
ROR26%18%10%
0.6
0.4
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4.6. Fuzzy Number – Example II:
A Fuzzy Set representing: (The value of one Dolar on 16/10/03): “Subjectivity”
0
1
Reais 2,6 2,7 3,0
0.5
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
5. Operations with Triangular Fuzzy Numbers (TFN):
Addition::If A = (a1, a2, a3) and B = (b1, b2, b3), so:
A (+) B = (a1, a2, a3) + (b1, b2, b3) = (a1 + b1, a2 + b2, a3 + b3), is a
TFN.
Example:
0
0,5
1
1 2 3 4 5 7 11
A B A + B
x
µ(x)
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Subtraction:
If A = (a1, a2, a3) and B = (b1, b2, b3), so:
A (-) B = (a1, a2, a3) - (b1, b2, b3) = (a1 - b3, a2 - b2, a3 - b1), is a TFN.
Example:B
0,5
1
-6 -3 0 1 2 4 5 7
A - B A B
x
µ(x)
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
Aproach by Chiu e Park (1994)
Multiplication:
Using the line equations:
A * B = [[Al(y)* Bl(y), Ar(y)*Br(y)] is not a TFN.
Example:
0
0,5
1
1 2 3 4 5 7 10 28
A B A x B
x
µ(x)
. . . . . . . . . . . . . . . . .
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
Division: (two diferents cases)
1) If A and B are both positives:
A / B = [Al(y)/ Br(y), Ar(y)/Bl(y)]
2) If A is positive and B is negative:
A / B = [Al(y)/ Bl(y), Ar(y)/Br(y)]
The result in the first case is a positive fuzzy number and in
the second case is a negative fuzzy number.
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
n, ,1 2 3n n nA a a a
An: (where n is a real number)
AB: (where B is a TFN (b1, b2, b3))
1 2 3, ,1 2 3B b b bA a a a
undefined
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
n, ,1 2 3n n nA a a a
An: (where n is a real number)
AB: (where B is a TFN (b1, b2, b3))
1 2 3, ,1 2 3B b b bA a a a xundefined
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
6. Fuzzyfication and Defuzzyfication:
Fuzzyfication: Is the maping of real numbers domain (generally discrete) to the fuzzy domain.
Defuzzyfication: Is the proceeding in which the value of the output linguistic, inferred by the fuzzy rules, will be transletad to a discrete value.
SHAW I. S. (1999)
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Fuzzyfication’s example:
1Very Low Low Medium High Very High
0 5 10 15 20 25 30 35 40 ROR (%)
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
Defuzzyfication’s example:
0 1000 3000 5000 7000 9000 NPV
Bad Medium Good Very good Very bad
8000
1
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7. The Net Present Value:
n
ii
i
r
CFCFNPV
10
)1(
Where:
NPV: net present value
CF0: first cash flow
CFi: cash flow on period i (i=1...n)
n: number of periods
r: discount rate
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
8. Application of Fuzzy Numbers in Investiments Analysis:
The Fuzzy Net Present Value:
According to BUCKLEY (1987) the Membership Function to NPV is givem:
n
j
jfjkjijin ryfFyfPyf
0)(,, )))((1)()(())((
To i = 1, 2, ...where k = i if F is negative and k = 3 - i if F is positive.
n
ii
i
r
CFCFNPV
10
)1(Comparing:
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9. Analyzing the Fuzzy NPVValor Presente Líquido Fuzzy
0,00
0,10
0,20
0,30
0,40
0,50
0,60
0,70
0,80
0,90
1,00
(35) (24) (14) (3) 7 18(Mi lhões)
VPL
“Investiment Sure and Viable”
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
Valor Presente Líquido Fuzzy
0,00
0,10
0,20
0,30
0,40
0,50
0,60
0,70
0,80
0,90
1,00
(35) (24) (14) (3) 7(Mi lhões)
VPL
“Investiment Sure and Unviable”
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
Valor Presente Líquido Fuzzy
0,00
0,10
0,20
0,30
0,40
0,50
0,60
0,70
0,80
0,90
1,00
(35) (24) (14) (3) 7 18 29 39 50 60 71 82(Mi lhões)
n=8
n=10
n=15
VPL
“Investiment Unsure and Viable”
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
Valor Presente Líquido Fuzzy
0,00
0,10
0,20
0,30
0,40
0,50
0,60
0,70
0,80
0,90
1,00
(35) (24) (14) (3) 7(Mi lhões)
n=8
n=10
n=15
VPL
“Investiment Unsure and Unviable”
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
Valor Presente Líquido Fuzzy
0,00
0,10
0,20
0,30
0,40
0,50
0,60
0,70
0,80
0,90
1,00
(35) (24) (14) (3) 7 18 29 39 50 60(Milhões)
VPL
Negative area Positive area
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
10.Real Case Aplication: 10.1. The Problem:
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.
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
Fixed Investiment R$12.874.035,00
Working Capital R$2.376.000,00
Yearly Fixed 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$8.582.690,00"
ROR 15% year
Income Tax 35% year
Depreciation 10% year
10.2. Project Data
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The value of NPV found is R$ 8.211.191,38. Therefore, in a simple Deterministic evaluation, the investiment could be acepted.
10.3. NPV Calculations Using Software Excel
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10.4. Analisyng the uncertainties involveds
Fixed Investiment: +/ - 10%; Working Capital: +/ - 10%; Yearly Fixed Cost: +/ - 10%; Variable Cost/ Unit: +/ - 13%; Forecast Sales: -30% a +20%; Price: -20% a +15%; Planning Horizon: –20% a +50%; ROR: +/ - 10%.
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“Fuzzyinvest 1.0” Main Screen
10.5. “Fuzzynvest 1.0” presentation:
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“Gráfico” Sheet
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“Cálculos” Sheet
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“Fuzzyinvest 1.0” Main Screen
10.6. Analysing the results.
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
1Very Low Low Medium High Very High
0 5 10 15 20 (27,51) 35 40 %
The failure possibility of the project.
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
Fuzzy classification array of the failure possibility of
the investment
Decision of the company
Very Low Unconditionally Accept
Low Accept with caution
Average Accept under restrictions
High (27.51%) Reject and review project
Very High Unconditionally Reject
“Investment Projects Acceptance Criteria”
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11.11.Conclusions:Conclusions:
1) 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.
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
Conclusions:Conclusions:
2) The way to evaluate an investment doesn’t change much, when applied to another object of analyses.
3) 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.
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
Conclusions:Conclusions:4) 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.
5) 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.
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASILV Encontro Internacional de Finanças - CHILE
Conclusions:Conclusions:
6) 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 values the visual aspect and the relevant information, emphasizing the membership graph and the failure possibility.
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Questions?