CONSISTENT INTERPOLATIVE FUZZY LOGICCONSISTENT INTERPOLATIVE FUZZY LOGICand

INVESTMENT DECISION MAKING

DD k K č ićk K č ić P t S k lićP t S k lić d Al k dd Al k d R kiR kić ićć ić

INVESTMENT DECISION-MAKING

DDarko Kovačevićarko Kovačević, , Petar SekulićPetar Sekulić and Aleksandar and Aleksandar RakiRakićeviććević

National bank of SerbiaNational bank of SerbiaBelgrade, Jun 23th, 2011Belgrade, Jun 23th, 2011

The structure of the presentation

• Classical Fuzzy logic

• Problems with classical approach

• Consistent Interpolative approach• Consistent Interpolative approach

• Practical implementation

Multi-valued logic approach short history

• Jan Lukasiewicz,“Logic changes in the fundamentals if we assume that besides trueLogic changes in the fundamentals, if we assume that besides trueand false there is a third logical value.”, (1920)

Lotfy Zadeh• Lotfy ZadehTheory of fuzzy sets, (1965)Objective: mathematical tool for calculating with wordsFuzzy logic is a superset of conventional (Boolean) logic that hasbeen extended to handle the concept of fuzziness, where the truthvalue may range between completely true and completely false“Fuzzy logic is not fuzzy but a precise logic of gradation, whichdoes not apply the principle of excluded middle”, (2006)

• Is this justified from the point of Boolean algebra?

Inconsistency

Fuzzy set A Complement of Fuzzy set A

Union

F A Complement IntersectionFuzzy set A Complement of Fuzzy set A

Intersection

Generalized Boolean polynomial

Folloving an approach proposed by Radojević (2008)

or

∑ ⊗⊗ = ))(()()( xSSx ασϕ ϕ

∪)(

)()(Ω∈

=PS

SS ασϕ ϕ

∑Ω∈ )(

))(()()(PS

ϕ ϕ

XxxaSxPS

iSPK SKa

K

i

∈⊗−= ∑ ∑Ω∈ Ω∈ ∪∈

⊗

)( )\(

|| ))()1()()( ϕσϕ

where denotes generalized product⊗

Interpolative approach

• Atomic structure

1 1 1 11

1 1 0 1 1 0 1 1 0 1 1 11 1 1 0

1 0 0 11 0 1 01 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1

C b

a b ba ⇔ ba ∨ Cb Ca

a b∪ a Cb∪ Ca b∪ Ca Cb∪

G li d l i l

1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

0 0 0 00

a b∩ a Cb∩ Ca b∩ Ca Cb∩

• Generalized polynomial

1

a b∪ a Cb∪ Ca b∪ Ca Cb∪ ba ⊗−1b

a⊗

+−1b +−1ba −+

1

a b∩ a Cb∩ Ca b∩ Ca Cb∩

a b ba ⇔ ba ∨ Cb Ca a b

ba ⊗ baa⊗−

bab⊗−

baba

⊗+−−1

a−1b−1baba

⊗+−−

21

baba⊗−+

2

ba⊗ba⊗ba⊗

0 0

Aggregation

)(),(),()( )(1

Ω∈Ω∈=∑=

BAPSSS i

m

ii i

ϕσωμ ϕσ

mii

m

ii ...1,0,1

1=≥=∑

=

ωω

( ) ∑⊗ ( ) ||||||||...|||| 1 iia

n aaaAggri

add ∑Ω∈

⊗ = μμ

Generalized product

• Product boundaries

• T-norm definition

))(),(min()()()1)()(,0max( xbxaxbxaxbxa ≤⊗≤−+

⎪⎪⎫

⎪⎪⎧ −

==

rmproduct nonormGodel t

pabpba

01),min(

⎪⎪⎭

⎪⎬

⎪⎪⎩

⎪⎨ −=−+

==⊗=

z t-normLukasiewicrmproduct no

otherwisebafpbapab

babaT

p

p

),(1)1,0max(

0),(

where p represents dependency parameter

⎭⎩

Generalized product – contd.

• Definition of Frank T-norm

⎪⎪⎪⎪⎫

⎪⎪⎪⎪⎧

=

=

pab

pba

1

0),min(

⎪⎪⎪

⎪⎪⎬

⎪⎪⎪

⎪⎪⎨

+∞=−+

==

pba

pabbaT

ba

p

))((

)1,0max(

1),(

⎪⎪⎪

⎭⎪⎪⎪

⎩ −−−

+ otherwisep

pp ba

p 1)1)(1(1(log

Static dependency parameter

E.g. properties height and weight

2

3

-1

0

1

-4 -3 -2 -1 0 1 2 3 4

Based on historical values -rank correlation - Spearman rho

-3

-2

Based on historical values -rank correlation - Spearman rho

))(),((),( 21 YFXFYXXY ρρ =

3),(12),(1

2121

1

−== ∫∫ duduuuCYXXYρρ 3),(12),(0

21210∫∫ duduuuCYXXYspearman ρρ

Static dependency parameter-cont

• Transformation of rank correlation

⎫⎧

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

∞→→==→−→

−

−

−

normTrank

normTrank

normTrank

pp

p

11001

ρρρ

• Obtained by Monte Carlo simulation on symmetrically quintile distribution

⎪⎭⎪⎩

spearman

spearman

spearmanp ρ

ρ

ρ +−= 12

)1(

Generalized product - validation

1,2

0 6

0,8

1

0,2

0,4

0,6

-0,2

00 10 20 30 40 50 60 70 80 90 100

( ) ( )=⊗⊗⊗ DCBA 321( ) ( )BCDA ''' 321

⊗⊗⊗

The formulation of IF-THEN rules

• in order to the proposed method works, because of rank correlation, membership functions must satisfy the principle of strict monotonicity.

• For the membership functions will use the generalized sigma functionc

lli i ti cbaxF ⎟⎞

⎜⎛==

1),,,(μ

Preposition 1:

bxacllinguisti ecbaxF ⎟

⎠⎜⎝ + −− )(1

),,,(μ

• Any verbal specification or derivative of two different basic linguistic properties does not change the rank correlation between the properties.

( ) ( ) ( )212121tdeterminan

2tdeterminan

1 cc⎟⎞

⎜⎛

⎟⎞

⎜⎛ ⎟

⎟⎞

⎜⎜⎛

⎟⎟⎞

⎜⎜⎛

Proof can be found in working paper

( ) ( ) ( )212121propertyproperty

2property

1propertypropertyderivedpropertyderived

,,, FFFFFF cc ρρρ =⎟⎟

⎠⎜⎜

⎝=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ⎟⎟⎠

⎜⎜⎝⎟⎟

⎠⎜⎜⎝

The formulation of IF-THEN rules

1

1,2

0,6

0,8

a=10; b=1;c=0.5 higha=10;b=2;c=0.5 very high

10 b 0 5 0 5 h hi h

0,2

0,4

a=10;b=0.5;c=0.5 somehow high

00 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

But what about the non-monotonic functions

• As we said membership functions must satisfy strict monotonicity

Preposition 2:

Any bounded, non-monotonic and differentiable derived property can beAny bounded, non monotonic and differentiable derived property can be expressed over generalized products of bounded, monotonic and differentiable derivatives from the same property.

Proof can be found in working paper

But what about the non-monotonic functions

• A is around A is very low and A is somewhat high• Generalized product enabled us to work with non-monotonic

features• By changing the parameters we can obtain any membership

f tifunction

1,2

0,6

0,8

1

a=30; b=1;c=0.2 very low

0,2

0,4

,a=-30;b=0.2;c=0.4 somehowhighvery low and somehow high

-0,2

00 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Defuzzification

• Starting from

( ) |||||||||||| aaaAggr ∑⊗ = μ

mii

m

ii ...1,0,1

1=≥=∑

=

ωω

( ) ||||||||...|||| 1 iia

n aaaAggri

add ∑Ω∈

= μμ

• We can evaluate every component of THEN part of the rule. This allows us insight into the structure of the obtained results

high (low high (highvery high

high (low risk)

high (high risk) low very low Rating

33,35% 21,87% 8,45% 66,55% 75,65% 28,17%

28,07% 11,13% 4,66% 64,52% 76,30% 22,42%

27,58% 17,31% 0,00% 72,68% 86,55% 24,74%

15,00% 1,96% 0,00% 32,84% 55,70% 11,04%

7,21% 0,00% 0,00% 30,07% 72,33% 10,96%

25,06% 0,00% 8,81% 59,32% 80,31% 19,55%5,06% 0,00% 8,8 % 59,3 % 80,3 % 9,55%

35,88% 18,44% 19,07% 58,57% 68,85% 29,46%

Examples of rules using a new approach

if (Liq is high and FinLev is low and Risk is low) or (Prof is somewhatif (Liq is high and FinLev is low and Risk is low) or (Prof is somewhat high and Dividend is somewhat high) then Rating is very high

Approach support any logical implication or verbal definition and satisfies all the Boolean axioms

The postulation of the problem

• We want to deal with investors decision problem to form a portfolio ofcompanies securities according to own risk preference.companies securities according to own risk preference.

• Companies which the investor takes into account are listed on S & P500 index.

• On example of around 60 companies which are listed on the S & P 500• On example of around 60 companies which are listed on the S & P 500index we will present a dynamic model of decision-making and multi-criteria ranking.

For the purpose of this conference we have introduced simple rating model that is open to multiple upgrades

in terms of both input and output logic!p p g

Financial ratios

• We will use a set of 32 publicly available financial indicators that will be divided into 5 groupsthat will be divided into 5 groups

• Liquidity Ratiosq y• Asset Turnover Ratios• Financial Leverage Ratios• Profitability Ratios• Profitability Ratios• Dividend Policy Ratios

For the mutual comparability of used indicators we will use the• For the mutual comparability of used indicators we will use thenormal standardization

)( secII averagetorl − )1,0(~)(

)( sectan, N

IStDevI

l

averagetorlls =

Results – no risk awareness

rule outputHigh

(low risk) very highHigh

(high risk) low very low

weight 0 0 333 0 0 333 0 333weight 0 0,333 0 0,333 0,333

65%

55%

60%

Microsoft CorporationIBM Corp.Google Inc.Exxon Mobilein

g

50%

Chevron Corp.Amazon.com InceBay inc.The Coca Cola CompanyColgate-Palmolive Company3M CompanyAbbott Laboratories

Ran

ki

40%

45%

i k f0 0,2 0,4 0,6 0,8 1 Risk preference

Results – some risk awareness with equal preferences

rule outputHigh

(low risk) very highHigh

(high risk) lowvery low

weight 0 2 0 2 0 2 0 2 0 2weight 0.2 0.2 0.2 0,2 0,2

50%

40%

45%

Microsoft CorporationIBM Corp.Google Inc.in

g

35%

gExxon MobileChevron Corp.Amazon.com InceBay inc.The Coca Cola CompanyColgate-Palmolive Company3M Company

Ran

ki

25%

30%

0 0 2 0 4 0 6 0 8 1

p yAbbott Laboratories

i k f0 0,2 0,4 0,6 0,8 1 Risk preference

Results – high risk awareness

rule outputHigh

(low risk) very highHigh

(high risk) lowvery low

weight 0 35 0 1 0 35 0 1 0 1weight 0.35 0.1 0.35 0,1 0,1

40%

30%

35%

Microsoft CorporationIBM Corp.Google Inc.Exxon Mobilein

g

20%

25%

Chevron Corp.Amazon.com InceBay inc.The Coca Cola CompanyColgate-Palmolive Company3M CompanyAbbott Laboratories

Ran

ki

10%

15%

0 0 2 0 4 0 6 0 8 1 i k f0 0,2 0,4 0,6 0,8 1 Risk preference

Model consideration

• Main points for the consideration of the model:– Relies on subjective perception of the user– Relies on subjective perception of the user.– Leans more toward ranking rather than rating system – thus

considering options within the same rating grade or sector.Ability to track similar company behavior within the different– Ability to track similar company behavior within the different sectors.

– Easily adjustable model with respect to linguistic rules.– Soft information manipulation – easily adjustable to change in

perception.– Ability to overcome structural breakdown in time series.

Thank you for your attentiony y

Questions?