and INVESTMENT DECISION-MAKING

Transcription

1 CONSISTENT INTERPOLATIVE FUZZY LOGIC and INVESTMENT DECISION-MAKING Darko Kovačević, č Petar Sekulić and dal Aleksandar Rakićević National bank of Serbia Belgrade, Jun 23th, 20

2 The structure of the resentation Classical Fuzzy logic Problems with classical aroach Consistent Interolative aroach Practical imlementation

3 Multi-valued logic aroach short history Jan Lukasiewicz, Logic changes in the fundamentals, if we assume that besides true and false there is a third logical value., 920 Lotfy Zadeh Theory of fuzzy sets, 965 Objective: mathematical tool for calculating with words Fuzzy logic is a suerset of conventional Boolean logic that has been extended to handle the concet of fuzziness, where the truth value may range between comletely true and comletely false Fuzzy logic is not fuzzy but a recise logic of gradation, which does not aly the rincile of excluded middle, 2006 Is this justified from the oint of Boolean algebra?

4 Inconsistency Fuzzy set A Comlement of Fuzzy set A Union Fuzzy set A Comlement of Fuzzy set A Intersection

5 Generalized Boolean olynomial Folloving an aroach roosed by Radojević 2008 or = x S S x α σ ϕ ϕ Ω = P S S S α σ ϕ ϕ Ω P S ϕ ϕ X x x a S x P S i S P K S K a K i = Ω Ω \ ϕ σ ϕ where denotes generalized roduct

6 Interolative aroach Atomic structure a b a Cb Ca b Ca Cb a b a b a b Cb Ca a b a Cb Ca b Ca Cb Generalized olynomial l a b a Cb Ca b Ca Cb a + b a b b + a b a + a b a b a b a b a b Cb Ca a b a b + 2a b a + b 2a b b a a b a Cb Ca b Ca Cb a b a a b b a b a b + a b 0 0

7 Aggregation m = μ S ωσ i σ ϕ S, S P Ω, ϕi BA Ω i m i= i= ωi =, ω 0, i =... m Aggr i a... a n = μ a μ add i ai Ω i

8 Generalized roduct Product boundaries max 0, a x + b x a x b x min a x, b x T-norm definition T a, b = a b min a, b ab = max0, a + b f a, b = = 0 = otherwise Godel t norm roduct norm Lukasiewicz t-norm where reresents deendency arameter

9 Generalized roduct contd. Definition of Frank T-norm = = ab b a 0, min = + + = = b a ab b a T b a max0,, + otherwise b a log

10 Static deendency arameter E.g. roerties height and weight Based on historical values -rank correlation - Searman rho ρ XY X, Y = ρ F X, F2 Y ρ searman = ρ XY X, Y = 2 C u, u 2 du du

11 Static deendency arameter-cont Transformation of rank correlation ρ rank ρ ρ rank rank = 0 T norm T norm T norm 0 = Obtained by Monte Carlo simulation on symmetrically quintile distribution = ρ searman ρsearman 2 + ρ searman

12 Generalized roduct - validation,2 0,8 06 0,6 0,4 0,2 0-0, B C D= A D ' ' C ' B 2 3 A 2 3

13 The formulation of IF-THEN rules in order to the roosed method works, because of rank correlation, membershi functions must satisfy the rincile of strict monotonicity. For the membershi functions will use the generalized sigma function F llinguistici = μ x, a, b, c = + e a x b c Preosition : Any verbal secification or derivative of two different basic linguistic roerties does not change the rank correlation between the roerties. ρ F = determinant determinant c c , F ρ F c =, F c 2 ρ F, F derived roerty derived roerty roerty roerty roerty roerty Proof can be found in working aer

14 The formulation of IF-THEN rules,2 0,8 0,6 a=0; b=;c=0.5 high a=0;b=2;c=0.5 very high a=0;b=0.5;c=0.5 somehow high 0,4 0, , 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

15 But what about the non-monotonic functions As we said membershi functions must satisfy strict monotonicity Preosition 2: Any bounded, non-monotonic and differentiable derived roerty can be exressed over generalized roducts of bounded, monotonic and differentiable derivatives from the same roerty. Proof can be found in working aer

16 But what about the non-monotonic functions A is around A is very low and A is somewhat high Generalized roduct enabled us to work with non-monotonic features By changing the arameters we can obtain any membershi function,2 0,8 0,6 0,4 a=30; b=;c=0.2 very low a=-30;b=0.2;c=0.4 somehow high very low and somehow high 0,2-0, , 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

17 Defuzzification Starting from Aggr m i= μ add a... a n = μ a ωi =, ωi 0, i =... m i ai Ω i We can evaluate every comonent of THEN art of the rule. This allows us insight into the structure of the obtained results very high high low high high risk risk low very low Rating 33,35% 2,87% 8,45% 66,55% 75,65% 28,7% 28,07%,3% 4,66% 64,52% 76,30% 22,42% 27,58% 7,3% 0,00% 72,68% 86,55% 24,74% 5,00%,96% 0,00% 32,84% 55,70%,04% 7,2% 0,00% 0,00% 30,07% 72,33% 0,96% 25,06% 0,00% 8,8% 59,32% 80,3% 9,55% 35,88% 8,44% 9,07% 58,57% 68,85% 29,46%

18 Examles of rules using a new aroach if 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 Aroach suort any logical imlication or verbal definition and satisfies all the Boolean axioms

19 The ostulation of the roblem We want to deal with investors decision roblem to form a ortfolio of comanies securities according to own risk reference. Comanies which the investor takes into account are listed on S & P 500 index. On examle of around 60 comanies which are listed on the S & P 500 index we will resent a dynamic model of decision-making and multicriteria ranking. For the urose of this conference we have introduced simle rating model that is oen to multile ugrades in terms of both inut and outut logic!

20 Financial ratios We will use a set of 32 ublicly available financial indicators that will be divided into 5 grous Liquidity Ratios Asset Turnover Ratios Financial Leverage Ratios Profitability Ratios Dividend Policy Ratios For the mutual comarability of used indicators we will use the normal standardization I s tan, l = I l I sec tor StDev I average l ~ N0,

21 Results no risk awareness rule outut High low risk very high High high risk low very low weight 0 0, ,333 0,333 65% 60% ing Ranki 55% 50% Microsoft Cororation IBM Cor. Google Inc. Exxon Mobile Chevron Cor. Amazon.com Inc ebay inc. The Coca Cola Comany Colgate-Palmolive Comany 3M Comany Abbott Laboratories 45% 40% 0 0,2 0,4 0,6 0,8 Risk ikreference

22 Results some risk awareness with equal references rule outut High low risk very high High high risk low very low weight ,2 0,2 50% 45% ing Ranki 40% 35% 30% Microsoft Cororation IBM Cor. Google Inc. Exxon Mobile Chevron Cor. Amazon.com Inc ebay inc. The Coca Cola Comany Colgate-Palmolive Comany 3M Comany Abbott Laboratories 25% ,2 04 0,4 06 0,6 08 0,8 Risk ik reference

23 Results high risk awareness rule outut High low risk very high High high risk low very low weight , 0, 40% 35% ing Ranki 30% 25% 20% Microsoft Cororation IBM Cor. Google Inc. Exxon Mobile Chevron Cor. Amazon.com Inc ebay inc. The Coca Cola Comany Colgate-Palmolive Comany 3M Comany Abbott Laboratories 5% 0% ,2 04 0,4 06 0,6 08 0,8 Risk ikreference

24 Model consideration Main oints for the consideration of the model: Relies on subjective ercetion of the user. Leans more toward ranking rather than rating system thus considering otions within the same rating grade or sector. Ability to track similar comany behavior within the different sectors. Easily adjustable model with resect to linguistic rules. Soft information maniulation easily adjustable to change in ercetion. Ability to overcome structural breakdown in time series.

25 Thank you for your attention Questions?