UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASIL
V 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 – BRASIL
V Encontro Internacional de Finanças - CHILE
Itajubá
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASIL
V Encontro Internacional de Finanças - CHILE
Itajubá
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASIL
V Encontro Internacional de Finanças - CHILE
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASIL
V Encontro Internacional de Finanças - CHILE
Universidade Federal de Itajubá
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASIL
V 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 – BRASIL
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1.Introduction:
Uncertainties associated with “Investment
Analyses”
Alternatives methods
Decision making process
Optimization of financial resources
UNIVERSIDADE FEDERAL DE ITAJUBÁ – BRASIL
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2. Objectives:
Main Objective:
Demonstrate the use of fuzzy logic in the evaluation of
investment projects under uncertaint conditions;
Secondary Objective:
Presentation of a software prototype to calculate
the fuzzy NPV and relative analyses.
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3. Methodological aspects:
The research method to be used is known as “quasiexperiment”:
•
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)).
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Investment Data (selected group)
Deterministic NPV Calculation – viability (pre-test)
Sensibility Analyses (uncontrolled)
M
E
T
H
O
Definition of the variables to be Fuzzyfied
Fuzzyfication of the selected variables (specialist)
Fuzzy NPV Calculation
D
F
L
U
O
Z
G
Z
I
Y
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
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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.
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4.1. Membership Fuction - Example:
A
a
c
A
a
b
c
b
d
d
A
A
1
1
0.5
a
b
c
Boolean Logic (binary)
d
x
a
b
c
d
Fuzzy Logic (continuous)
A(x): Membership
x
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4.2. Fuzzy Number – General Definition, KUCHTA (1996)
fn  (a1, a2, a3, a4, f1 a ( ), f 2a ( ))
Where:
a1, a2, a3, a4
are real numbers and
a1  a2  a3  a4
is a continuous real function non decreasing
f1 ( ) :defined
in the interval [0,1], such that:
a
f1a (0)  a1
and
f1a (1)  a 2
is a continuous real function non increasing
f 2 ( ) :defined
in the interval [0,1], such that:
a
f 2a (1)  a3
and
f 2a (0)  a 4
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4.3. Fuzzy Number:
fn  (a1, a2, a3, a4, f , f )
A(x)
a
1
1
f
f
a
1
a1
0
a2
a3
a
2
a4
x
a
2
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4.4. Triangular Fuzzy Number (TFN):
If
a and
1
f
f
a
2
are linear functions and a2 = a 3:
0,


 x  a1 ,
 a2  a1
 ( A) ( x)  
 a x
 3
,
 a3  a2

0,
A (x)
1
a1
a2
a3
x
A = (a1, a2, a3)
x  a1
a1  x  a2
a2  x  a3
x  a3
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4.5. Fuzzy Number – Example I:
A “Fuzzy Set” representing the NPV: “Rates: Low/Medium/High”
1
0.6
High
Low
Medium
0.4
0
10%
18%
26%
ROR
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4.6. Fuzzy Number – Example II:
A Fuzzy Set representing: (The value of one Dolar on
16/10/03): “Subjectivity”
1
0.5
0
2,6 2,7
3,0
Reais
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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:
µ(x)
A
1
B
A+ B
0,5
0
1
2
3
4
5
7
11
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:
A- B
µ(x)
B
A
1
B
0,5
-6
-3
0
1
2
4
5
7
x
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Multiplication:
Using the line equations:
A * B = [[Al(y)* Bl(y), Ar(y)*Br(y)] is not a TFN.
Example:
µ(x)
A
1
B
AxB
0,5
.................
0
1
2
3
4
5
7
10
28
x
Aproach by Chiu e Park (1994)
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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.
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An: (where n is a real number)

n
n
n
n
A  a1 ,a2 ,a3

AB: (where B is a TFN (b1, b2, b3))
undefined

B
b
1
b
2
b
3
A  a ,a
,a
1
2
3

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An: (where n is a real number)

n
n
n
n
A  a1 ,a2 ,a3

AB: (where B is a TFN (b1, b2, b3))
undefined

B
b
1
b
2
b
3
A  a ,a
,a
1
2
3

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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:
Very Low
Low
Medium
High
Very High
1
0
5
10
15
20
25
30
35
40 ROR (%)
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Defuzzyfication’s example:
1
Very bad
0
1000
Bad
3000
Medium
5000
Good
7000
8000
Very good
9000
NPV
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7. The Net Present Value:
n
NPV  CF 0  
CFi
i 1 (1  r )
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
i
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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
f n,i (( y) P)   f j ,i (( y) Fj )(1  f k ( j ) (( y)rf ))
j
j 0
To i = 1, 2, ...
where k = i if F is negative and k = 3 - i if F is positive.
n
Comparing:
NPV  CF 0  
CFi
i 1 (1  r )
i
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9.
Analyzing the Fuzzy NPV
Valor Presente Líquido Fuzzy
1,00
VPL
0,90
0,80
0,70
0,60
0,50
0,40
0,30
0,20
0,10
0,00
(35)
(24)
(14)
(3)
“Investiment Sure and Viable”
7
( M i l hõe s )
18
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Valor Presente Líquido Fuzzy
1,00
VPL
0,90
0,80
0,70
0,60
0,50
0,40
0,30
0,20
0,10
0,00
(35)
(24)
(14)
(3)
“Investiment Sure and Unviable”
( M i l hõe s )
7
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Valor Presente Líquido Fuzzy
n=8
1,00
n=10
0,90
n=15
0,80
VPL
0,70
0,60
0,50
0,40
0,30
0,20
0,10
0,00
(35)
(24)
(14)
(3)
7
18
29
39
50
“Investiment Unsure and Viable”
60
(71
M i l hõe s )
82
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Valor Presente Líquido Fuzzy
n=8
1,00
n=10
0,90
n=15
0,80
VPL
0,70
0,60
0,50
0,40
0,30
0,20
0,10
0,00
(35)
(24)
(14)
(3)
“Investiment Unsure and Unviable”
( M i l hõe s )
7
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Valor Presente Líquido Fuzzy
VPL
1,00
0,90
0,80
0,70
0,60
0,50
0,40
0,30
0,20
0,10
0,00
(35)
(24)
(14)
(3)
Negative area
7
18
29
39
Positive area
50 ( M ilhõ e s )
60
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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.
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10.2. Project Data
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/T on
Forecasted Sales
100.000 T on/ano
P rice
R$ 98,00/T on
P lanning Horizon
10 years
Residual Value
"R$8.582.690,00"
ROR
15% year
Income T ax
35% year
Depreciation
10% year
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10.3. NPV Calculations Using Software Excel
The value of NPV found is R$ 8.211.191,38. Therefore, in a simple
Deterministic evaluation, the investiment could be acepted.
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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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10.5. “Fuzzynvest 1.0” presentation:
“Fuzzyinvest 1.0” Main Screen
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“Gráfico” Sheet
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“Cálculos” Sheet
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10.6. Analysing the results.
“Fuzzyinvest 1.0” Main Screen
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The failure possibility of the project.
1
0
Very Low
5
Low
10
15
Medium
High
20
(27,51)
Very High
35
40
%
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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.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.
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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.
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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.
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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?