universidade federal de itajubÁ – brasil v encontro internacional de finanças - chile prof....

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.


Itajubá


Universidade Federal de Itajubá


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


Uncertainties associated with “Investment Analyses”

Alternatives methods

Decision making process

Optimization of financial resources

1.Introduction:


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.


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)).


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

O

G

I

C

F

U

Z

Z

Y

M

E

T

H

O

D


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.


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


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


4.3. Fuzzy Number:

a1 a4

xa2 a3

1

0

A(x) ),( 21,4,3,2,1aa ffaaaafn

af1 af2


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


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


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


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)


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)


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)

. . . . . . . . . . . . . . . . .


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.


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


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


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)


Fuzzyfication’s example:

1Very Low Low Medium High Very High

0 5 10 15 20 25 30 35 40 ROR (%)


Defuzzyfication’s example:

0 1000 3000 5000 7000 9000 NPV

Bad Medium Good Very good Very bad

8000

1


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


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:


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”


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”



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”



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”



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


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.


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


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%; Yearly Fixed Cost: +/ - 10%; Variable Cost/ Unit: +/ - 13%; Forecast Sales: -30% a +20%; Price: -20% a +15%; Planning Horizon: –20% a +50%; 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.


1Very Low Low Medium High Very High

0 5 10 15 20 (27,51) 35 40 %

The failure possibility of the project.


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”


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.


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.


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.


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.


Questions?

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