prof. alexandre leme sanches, msc

No Slide TitleUNIVERSIDADE 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
Itajubá
Universidade Federal de Itajubá
1. Introduction
2. Objectives
6. Fuzzyfication and Defuzzyfication
8. Application of Fuzzy Numbers in Investiments Analysis
9. Analyzing the Fuzzy NPV
10. Real Case Aplication
Uncertainties associated with “Investment Analyses”
Alternatives methods
Main Objective:
Demonstrate the use of fuzzy logic in the evaluation of investment projects under uncertaint conditions;
Objectives:
Secondary Objective:
Presentation of a software prototype to calculate the fuzzy NPV and relative analyses.
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)).
Investment Data (selected group)
Sensibility Analyses (uncontrolled)
Fuzzyfication of the selected variables (specialist)
Fuzzy NPV Calculation
Defuzzyfication of the NPV (if necessary)
Comparison with the Deterministic NPV - The Proxy Pretest Design
L
O
G
I
C
F
U
Z
Z
Y
M
E
T
H
O
D
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.
4.1. Membership Fuction - Example:
4.2. Fuzzy Number – General Definition, KUCHTA (1996)
Where:
are real numbers and
: 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:
and
and
4.3. Fuzzy Number:
4.4. Triangular Fuzzy Number (TFN):
If and are linear functions and a2 = a 3:
a1
a3
4.5. Fuzzy Number – Example I:
A “Fuzzy Set” representing the NPV: “Rates: Low/Medium/High”
Low
Medium
High
0
1
ROR
26%
18%
10%
0.6
0.4
4.6. Fuzzy Number – Example II:
A Fuzzy Set representing: (The value of one Dolar on 16/10/03): “Subjectivity”
0
1
Reais
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
11
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
A - B A B
Aproach by Chiu e Park (1994)
Multiplication:
A * B = [[Al(y)* Bl(y), Ar(y)*Br(y)] is not a TFN.
Example:
0
0,5
1
A B A x B
x
µ(x)
Division: (two diferents cases)
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.
An: (where n is a real number)
AB: (where B is a TFN (b1, b2, b3))
undefined
An: (where n is a real number)
AB: (where B is a TFN (b1, b2, b3))
x
undefined
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)
Fuzzyfication’s example:
0 5 10 15 20 25 30 35 40 ROR (%)
Defuzzyfication’s example:
Bad
Medium
Good
The Net Present Value:
n: number of periods
Application of Fuzzy Numbers in Investiments Analysis:
The Fuzzy Net Present Value:
According to BUCKLEY (1987) the Membership Function to NPV is givem:
To i = 1, 2, ...
where k = i if F is negative and k = 3 - i if F is positive.
Comparing:
Analyzing the Fuzzy NPV
“Investiment Sure and Viable”
CustoVar
0
1
0
INVEST.FUZZY
VALORES
RESULTADOS
(Investimento +/- 10%, o preço de venda +/- 20%, o volume de vendas -20% a +10% e vida -20% a +50%.)
VARIÁVEIS
a1
a2
a3
a1
a2
a3
ANO
FAIR
DEPREC.
f
a1
a2
a3
a1
a2
a3
a1
a2
a3
a1
a2
a3
a1
a2
a3
0
-13,130,000
-13,130,000
-13,130,000
0
0
0
0
0
0
0
0
0
-13,130,000
-13,130,000
-13,130,000
1
5,696,042
5,400,000
5,106,235
1,100,000
1,100,000
1,100,000
4,596,042
4,300,000
4,006,235
1,608,615
1,505,000
1,402,182
4,293,860
3,895,000
3,497,621
2
5,696,042
5,400,000
5,106,235
1,100,000
1,100,000
1,100,000
4,596,042
4,300,000
4,006,235
1,608,615
1,505,000
1,402,182
4,293,860
3,895,000
3,497,621
3
5,696,042
5,400,000
5,106,235
1,100,000
1,100,000
1,100,000
4,596,042
4,300,000
4,006,235
1,608,615
1,505,000
1,402,182
4,293,860
3,895,000
3,497,621
4
5,696,042
5,400,000
5,106,235
1,100,000
1,100,000
1,100,000
4,596,042
4,300,000
4,006,235
1,608,615
1,505,000
1,402,182
4,293,860
3,895,000
3,497,621
5
5,696,042
5,400,000
5,106,235
1,100,000
1,100,000
1,100,000
4,596,042
4,300,000
4,006,235
1,608,615
1,505,000
1,402,182
4,293,860
3,895,000
3,497,621
6
5,696,042
5,400,000
5,106,235
1,100,000
1,100,000
1,100,000
4,596,042
4,300,000
4,006,235
1,608,615
1,505,000
1,402,182
4,293,860
3,895,000
3,497,621
7
5,696,042
5,400,000
5,106,235
1,100,000
1,100,000
1,100,000
4,596,042
4,300,000
4,006,235
1,608,615
1,505,000
1,402,182
4,293,860
3,895,000
3,497,621
8
5,696,042
5,400,000
5,106,235
1,100,000
1,100,000
1,100,000
4,596,042
4,300,000
4,006,235
1,608,615
1,505,000
1,402,182
4,293,860
3,895,000
3,497,621
9
5,696,042
5,400,000
5,106,235
1,100,000
1,100,000
1,100,000
4,596,042
4,300,000
4,006,235
1,608,615
1,505,000
1,402,182
4,293,860
3,895,000
3,497,621
10
15,159,376
14,863,333
14,569,569
1,100,000
1,100,000
1,100,000
11,929,376
11,633,333
9,139,569
4,175,281
4,071,667
3,198,849
11,960,527
10,791,667
10,394,287
ANO
FAIR
DEPREC.
“Investiment Sure and Unviable”
CustoVar
0
1
0
INVEST.FUZZY
VALORES
RESULTADOS
VARIÁVEIS
a1
a2
a3
a1
a2
a3
ANO
FAIR
DEPREC.
f
a1
a2
a3
a1
a2
a3
a1
a2
a3
a1
a2
a3
a1
a2
a3
0
-32,812,300
-15,250,035
-13,725,031
0
0
0
0
0
0
0
0
0
-32,812,300
-15,250,035
-13,725,031
1
-364,574
-4,045,635
-3,893,997
1,158,663
1,287,404
3,019,870
-3,384,444
-5,333,039
-5,052,660
-1,184,555
-1,866,564
-1,768,431
1,403,857
-2,179,072
-2,709,441
2
-364,574
-4,045,635
-3,893,997
1,158,663
1,287,404
3,019,870
-3,384,444
-5,333,039
-5,052,660
-1,184,555
-1,866,564
-1,768,431
1,403,857
-2,179,072
-2,709,441
3
-364,574
-4,045,635
-3,893,997
1,158,663
1,287,404
3,019,870
-3,384,444
-5,333,039
-5,052,660
-1,184,555
-1,866,564
-1,768,431
1,403,857
-2,179,072
-2,709,441
4
-364,574
-4,045,635
-3,893,997
1,158,663
1,287,404
3,019,870
-3,384,444
-5,333,039
-5,052,660
-1,184,555
-1,866,564
-1,768,431
1,403,857
-2,179,072
-2,709,441
5
-364,574
-4,045,635
-3,893,997
1,158,663
1,287,404
3,019,870
-3,384,444
-5,333,039
-5,052,660
-1,184,555
-1,866,564
-1,768,431
1,403,857
-2,179,072
-2,709,441
6
-364,574
-4,045,635
-3,893,997
1,158,663
1,287,404
3,019,870
-3,384,444
-5,333,039
-5,052,660
-1,184,555
-1,866,564
-1,768,431
1,403,857
-2,179,072
-2,709,441
7
-364,574
-4,045,635
-3,893,997
1,158,663
1,287,404
3,019,870
-3,384,444
-5,333,039
-5,052,660
-1,184,555
-1,866,564
-1,768,431
1,403,857
-2,179,072
-2,709,441
8
-364,574
-4,045,635
-3,893,997
1,158,663
1,287,404
3,019,870
-3,384,444
-5,333,039
-5,052,660
-1,184,555
-1,866,564
-1,768,431
1,403,857
-2,179,072
-2,709,441
9
-364,574
-4,045,635
-3,893,997
1,158,663
1,287,404
3,019,870
-3,384,444
-5,333,039
-5,052,660
-1,184,555
-1,866,564
-1,768,431
1,403,857
-2,179,072
-2,709,441
10
10,356,516
6,913,055
7,302,293
1,158,663
1,287,404
3,019,870
4,723,046
3,249,651
1,687,904
1,653,066
1,137,378
590,766
9,765,750
5,775,677
5,649,227
ANO
FAIR
DEPREC.
“Investiment Unsure and Viable”
CustoVar
0
1
0
INVEST.FUZZY
VALORES
RESULTADOS
VARIÁVEIS
a1
a2
a3
a1
a2
a3
ANO
FAIR
DEPREC.
f
a1
a2
a3
a1
a2
a3
a1
a2
a3
a1
a2
a3
a1
a2
a3
0
-16,775,038
-15,250,035
-13,725,031
0
0
0
0
0
0
0
0
0
-16,775,038
-15,250,035
-13,725,031
1
8,516,582
13,739,910
15,033,504
1,158,663
1,287,404
1,416,144
7,100,439
12,452,507
13,874,841
2,485,153
4,358,377
4,856,194
3,660,388
9,381,533
12,548,351
2
8,516,582
13,739,910
15,033,504
1,158,663
1,287,404
1,416,144
7,100,439
12,452,507
13,874,841
2,485,153
4,358,377
4,856,194
3,660,388
9,381,533
12,548,351
3
8,516,582
13,739,910
15,033,504
1,158,663
1,287,404
1,416,144
7,100,439
12,452,507
13,874,841
2,485,153
4,358,377
4,856,194
3,660,388
9,381,533
12,548,351
4
8,516,582
13,739,910
15,033,504
1,158,663
1,287,404
1,416,144
7,100,439
12,452,507
13,874,841
2,485,153
4,358,377
4,856,194
3,660,388
9,381,533
12,548,351
5
8,516,582
13,739,910
15,033,504
1,158,663
1,287,404
1,416,144
7,100,439
12,452,507
13,874,841
2,485,153
4,358,377
4,856,194
3,660,388
9,381,533
12,548,351
6
8,516,582
13,739,910
15,033,504
1,158,663
1,287,404
1,416,144
7,100,439
12,452,507
13,874,841
2,485,153
4,358,377
4,856,194
3,660,388
9,381,533
12,548,351
7
8,516,582
13,739,910
15,033,504
1,158,663
1,287,404
1,416,144
7,100,439
12,452,507
13,874,841
2,485,153
4,358,377
4,856,194
3,660,388
9,381,533
12,548,351
8
8,516,582
13,739,910
15,033,504
1,158,663
1,287,404
1,416,144
7,100,439
12,452,507
13,874,841
2,485,153
4,358,377
4,856,194
3,660,388
9,381,533
12,548,351
9
8,516,582
13,739,910
15,033,504
1,158,663
1,287,404
1,416,144
7,100,439
12,452,507
13,874,841
2,485,153
4,358,377
4,856,194
3,660,388
9,381,533
12,548,351
10
19,237,672
24,698,600
26,229,794
1,158,663
1,287,404
1,416,144
15,207,929
21,035,197
20,615,405
5,322,775
7,362,319
7,215,392
12,022,281
17,336,281
20,907,019
ANO
FAIR
DEPREC.
“Investiment Unsure and Unviable”
CustoVar
0
1
0
INVEST.FUZZY
VALORES
RESULTADOS
VARIÁVEIS
a1
a2
a3
a1
a2
a3
ANO
FAIR
DEPREC.
f
a1
a2
a3
a1
a2
a3
a1
a2
a3
a1
a2
a3
a1
a2
a3
0
-16,775,038
-15,250,035
-13,725,031
0
0
0
0
0
0
0
0
0
-16,775,038
-15,250,035
-13,725,031
1
-4,694,537
-3,904,125
-482,439
1,158,663
1,287,404
1,416,144
-6,110,681
-5,191,529
-1,641,102
-2,138,738
-1,817,035
-574,386
-4,120,151
-2,087,090
1,656,299
2
-4,694,537
-3,904,125
-482,439
1,158,663
1,287,404
1,416,144
-6,110,681
-5,191,529
-1,641,102
-2,138,738
-1,817,035
-574,386
-4,120,151
-2,087,090
1,656,299
3
-4,694,537
-3,904,125
-482,439
1,158,663
1,287,404
1,416,144
-6,110,681
-5,191,529
-1,641,102
-2,138,738
-1,817,035
-574,386
-4,120,151
-2,087,090
1,656,299
4
-4,694,537
-3,904,125
-482,439
1,158,663
1,287,404
1,416,144
-6,110,681
-5,191,529
-1,641,102
-2,138,738
-1,817,035
-574,386
-4,120,151
-2,087,090
1,656,299
5
-4,694,537
-3,904,125
-482,439
1,158,663
1,287,404
1,416,144
-6,110,681
-5,191,529
-1,641,102
-2,138,738
-1,817,035
-574,386
-4,120,151
-2,087,090
1,656,299
6
-4,694,537
-3,904,125
-482,439
1,158,663
1,287,404
1,416,144
-6,110,681
-5,191,529
-1,641,102
-2,138,738
-1,817,035
-574,386
-4,120,151
-2,087,090
1,656,299
7
-4,694,537
-3,904,125
-482,439
1,158,663
1,287,404
1,416,144
-6,110,681
-5,191,529
-1,641,102
-2,138,738
-1,817,035
-574,386
-4,120,151
-2,087,090
1,656,299
8
-4,694,537
-3,904,125
-482,439
1,158,663
1,287,404
1,416,144
-6,110,681
-5,191,529
-1,641,102
-2,138,738
-1,817,035
-574,386
-4,120,151
-2,087,090
1,656,299
9
-4,694,537
-3,904,125
-482,439
1,158,663
1,287,404
1,416,144
-6,110,681
-5,191,529
-1,641,102
-2,138,738
-1,817,035
-574,386
-4,120,151
-2,087,090
1,656,299
10
6,026,553
7,054,565
10,713,851
1,158,663
1,287,404
1,416,144
1,996,809
3,391,162
5,099,462
698,883
1,186,907
1,784,812
4,241,741
5,867,658
10,014,968
ANO
FAIR
DEPREC.
Negative area Positive area
ANO
FAIR
DEPREC.
f
a1
a2
a3
a1
a2
a3
a1
a2
a3
a1
a2
a3
a1
a2
a3
0
-16,775,038
-15,250,035
-13,725,031
0
0
0
0
0
0
0
0
0
-16,775,038
-15,250,035
-13,725,031
1
765,463
5,895,875
10,506,288
1,158,663
1,287,404
1,416,144
-650,681
4,608,472
9,347,625
-227,738
1,612,965
3,271,669
-2,506,206
4,282,910
10,734,026
2
765,463
5,895,875
10,506,288
1,158,663
1,287,404
1,416,144
-650,681
4,608,472
9,347,625
-227,738
1,612,965
3,271,669
-2,506,206
4,282,910
10,734,026
3
765,463
5,895,875
10,506,288
1,158,663
1,287,404
1,416,144
-650,681
4,608,472
9,347,625
-227,738
1,612,965
3,271,669
-2,506,206
4,282,910
10,734,026
4
765,463
5,895,875
10,506,288
1,158,663
1,287,404
1,416,144
-650,681
4,608,472
9,347,625
-227,738
1,612,965
3,271,669
-2,506,206
4,282,910
10,734,026
5
765,463
5,895,875
10,506,288
1,158,663
1,287,404
1,416,144
-650,681
4,608,472
9,347,625
-227,738
1,612,965
3,271,669
-2,506,206
4,282,910
10,734,026
6
765,463
5,895,875
10,506,288
1,158,663
1,287,404
1,416,144
-650,681
4,608,472
9,347,625
-227,738
1,612,965
3,271,669
-2,506,206
4,282,910
10,734,026
7
765,463
5,895,875
10,506,288
1,158,663
1,287,404
1,416,144
-650,681
4,608,472
9,347,625
-227,738
1,612,965
3,271,669
-2,506,206
4,282,910
10,734,026
8
765,463
5,895,875
10,506,288
1,158,663
1,287,404
1,416,144
-650,681
4,608,472
9,347,625
-227,738
1,612,965
3,271,669
-2,506,206
4,282,910
10,734,026
9
765,463
5,895,875
10,506,288
1,158,663
1,287,404
1,416,144
-650,681
4,608,472
9,347,625
-227,738
1,612,965
3,271,669
-2,506,206
4,282,910
10,734,026
10
11,486,553
16,854,565
21,702,578
1,158,663
1,287,404
1,416,144
7,456,809
13,191,162
16,088,189
2,609,883
4,616,907
5,630,866
5,855,687
12,237,658
19,092,695
ANO
FAIR
DEPREC.
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.
Plan1
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
10.4. Analisyng the uncertainties involveds
· Fixed Investiment: +/- 10%;
· Working Capital: +/- 10%;
· Price: -20% a +15%;
· ROR: +/- 10%.
“Fuzzyinvest 1.0” Main Screen
10.5. “Fuzzynvest 1.0” presentation:
“Gráfico” Sheet
“Cálculos” Sheet
“Fuzzyinvest 1.0” Main Screen
10.6. Analysing the results.
1
0 5 10 15 20 (27,51) 35 40 %
The failure possibility of the project.
“Investment Projects Acceptance Criteria”
Fuzzy classification array of the failure possibility of the investment
Decision of the company
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.
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.
Conclusions:
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.
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.
Conclusions:
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.
Questions?
å
·

prof. alexandre leme sanches, msc

Documents