Fuzzy logic

The fuzzy logic methodology was developed in the mid-1970s by Lotfy A. Zadeh at Berkeley University (California), known at first as the incompatibility principle and subsequently recognized as fuzzy logic, and which description is: "As the complexity of a system increases, our capability to be precise and to create instructions regarding its behavior decreases until the threshold beyond which precision and meaning are excluding characteristics". ^{(Zadeh, 1994, p. 80)}

Figure 1 describes the interpretation of information for traditional logic and fuzzy logic, allowing to observe the strong change in the transition curve between the proposed ranges. The search for order within chaos leads to bifurcation, however, fuzzy logic produces a symmetry rupture point that has a traditional geometry in fractal terms that describes a geometric object, with wide scale ranges ^{(Gil, 2000}, ²⁰⁰⁵⁾. This means that it emphasizes the blurriness of the variables, particularly on the everyday and corporate bases due to the dissonance with reality ^{(Restrepo & Vanegas, 2015)}.

Source: Own elaboration.

Figure 1 Interpretation of traditional logics and fuzzy logic. 

The processes in which the qualities or competences of the set are compared must be carried out after defining the fuzzy set or subsets ^{(Ávila & Galeana, 2013)}. According to ^{Cardona (2015)}, when the conjugations of variables are of the "if..., then..." type, the fuzzy logic model establishes categories (linguistic values) and membership functions for each input and output variable (denominated linguistic variable). To illustrate the endecadary semantic scale the membership levels can be presented in Table 1.

Table 1 Membership levels. 

Source: ^{Ávila and Galeana (2013)}.

Types of fuzzy logic models

There are three classifications for this type of logic: (1) models in fuzzy continuous-time (MFC), used to estimate real financial options through the use of trapezoidal numbers; (2) fuzzy pay-off method (FPOM), works with triangular distributions, the value of which emerges from the representative fraction of the positive value area divided for the total area of possible values of the triangle and the possible average value of the fuzzy landscape; (3) models in fuzzy discrete-time (MFD), which adapt the binomial model to the fuzzy logic allowing to estimate the upward and downward movements ^{(Milanesi, 2014)}.

Fuzzy logic and financial risk analysis

^{Rico and Tinto (2008)} present the application of the fuzzy sets in five areas of business organizations related to accounting, where we find problems concerning: portfolio selection, financial mathematics, capital budget, technical analysis, credit analysis, and financial analysis.

Management control is a tool on which a financial institution relies in order to measure its performance. Risk indicators are indispensable tools in measuring said performance through formulas and mathematical calculations applied to the financial statements, the results of which help us measure the health of individual financial institutions ^{(Valencia & Restrepo, 2016)}.

There are many systems that allow measuring the performance of lending institutions, and from their application, the credit ratings are created. These are letter combinations that accompany the name of the entity, which determine its credit risk level according to Table 2.

Table 2 Rating types. 

Source: ^{Mascareñas (2008)}.

^{Medina (2006)} proposes fuzzy logic as a tool to solve financial problems, since it is useful in the optimal selection of investment portfolios as well as in dealing with the uncertainties of financial assets in the stock market. The traditional methodology requires extensive information due to the historical sequences requirement that determines the usual analysis models, whereas the uncertainty methodology handles a more flexible form due to its possibilistic approximation ^{(Kaufmann & Gilaluja, 1988)}.

The fuzzy inference models were proposed by ^{Medina and Paniagua (2008)} to measure the credit study by determining its viability to minimize the counterparty risk in the credit granting processes, since it could be measured through indicators applied to the client such as payment capacity, debt capacity, credit rating that feeds the fuzzy system as an evaluation tool, providing consistent answers regarding the amount and the concession period ^{(González, Flores, & Gil, 2010)} .

CAMEL model

The CAMEL model allows identifying financial difficulties in institutions, particularly banking institutions. Its acronym stands for: Capital (C), Asset quality (A), Management (M), Earnings (E), and Liquidity (L), "it is defined as a uniform rating system for financial institutions" ^{(Crespo, 2011)} . The process to measure credit risk is done based on models that allow measuring the performance ^{(Velez, 2003)} through the application of financial ratios.

The CAMEL method can develop a type of financial analysis that is sustained on the construction of financial reasons, which originate in the balances derived from the financial institutions.

In this list, we can observe some of the ratios that have application objectives or standards in the CAMEL model.

Table 4 presents the reference values that an institution must reach for each indicator. A credit rating described in Table 1 is determined with the financial reasons applied to the consolidated statements. Traditional logic would understand it as follows:

CAC "X" AAA+ Met the required objectives

CAC "Y" BBB+ Below the required average

CAC "Z" CCC Far below the required average

Table 4 Indicators of the CAMEL model. 

FD, available funds; SP, patrimonial sufficiency; PP, average capital; AP, average asset; CCCMV, maturing commercial credit portfolio; CCCNV, maturing consumer loan portfolio; GO, operating expense; GPO, estimated personnel costs; MCCCM, default commercial credit portfolio; MCCCN, default consumer loan portfolio.

Source: Own elaboration.

Fuzzy logic based systems

A fuzzy logic based system is comprised by Figure 2.

Source: ^{Benito and Duran (2009)}.

Figure 2 Structure of a fuzzy system. 

For its understanding, the diffuser block is placed according to the membership degree to each of the fuzzy sets through the characteristic function. Subsequently, the data of the variable to be analyzed is entered with its concrete values, obtaining as outputs the membership degrees to the studied sets.

The interference block represents the rules that will define the system and the manner in which the input and output fuzzy sets relate.

The method that allows obtaining the concrete result with degrees of security from the fuzzy sets, after applying mathematical defuzzification methods, is called defuzzification ^{(Rico & Tinto, 2008)}.

Its implementation is done through a fuzzy logic based system, which functions as a program that is executable through a conventional microprocessor (or microcontroller), which unlike other systems, the calculation resources used are relatively low ^{(Lorenzana, Barberá, & Terceno, 2001)}. Considering that there are various fuzzy logic based systems, we present some LD application software such as: FuzzyTECH, MATLAB, TILShell and FIDE, among others. Where MATLAB is currently the most complete environment since it allows working from a single environment with both classic and innovating techniques.

However, it was analyzed that the Xfuzzy software is an accessible application that allows identifying the development planning and execution processes according to the stated objectives ^{(Morillas Raya, 2006)}.

The aim of the study proposed is to interpret the financial risk indicators from the perspective of fuzzy logic, aiming to determine the credit rating membership levels. This process helps us measure their performance level from a perspective that values the qualities more than the quantities. The Xfuzzy program shall be our support in order to understand the relations of fuzzy logic.

Cooperativism in Ecuador

Cooperativism in Ecuador begins with the formation of human society, whose practices have survived the test of time, particularly, indigenous organizations created with the purpose of building roads and housing, among others. Other organizations that standout are unions and artisans, whose capacity has demonstrated forms of cooperativism.

The beginnings of organized cooperativism in Ecuador emerge toward the end of the 19th century and beginnings of the 20^th century, with the creation of the first Savings Bank of the Artisans Society, lovers of progress in 1886. By 1937 to 1963, laws and standards were created to regulate cooperativism, classifying them into four cooperative classes: (1) Production, (2) Credit, (3) Consumer, and (4) Mixed. Subsequently, from 1964 to 1988 the boom of the cooperative sector took place, creating the National Cooperative Council (NCC) and a great interaction from the Federation of Savings and Credit Cooperatives (FECOAC for its acronym in Spanish). Consecutively, from 1989 to 2006, free-market policies emerged, which modify the General Law for Institutions of the Financial System, the result of which was the financial crisis of 1999 and the dollarization and emergence of the National Association of Savings and Credit Cooperatives (ASOCOAC for its acronym in Spanish) due to the closing of various sector entities, thus leading Ecuador toward a new horizon of cooperative management ^{(Miño, 2013)}.

From 2007 onward, the new constitution mandates Ecuador to create standards for the regulation and control of the cooperative sector, such as the Organic Law of the Popular and Supportive Economy and the regulatory body of the Popular and Supportive Economy Superintendence (SEPS for its acronym in Spanish), which began their functions in June 2012. These institutions are classified by segments that go from one to five. Their status is identified by their contribution in the sector, transaction volume, number of associates, number and geographical location of operational offices throughout the country, amount of assets, and capital ^{(Superintendencia de Economía Popular y Solidaria, 2013)}.

Case study

As has been described in the literature, the cooperative sector of Ecuador is comprised by five segments. For the application of the fuzzy methodology, a cooperative called Cooperativa Coprogreso from segment one was analyzed, as it had available information, with the following financial indicators: earnings, adequacy, available funds, and capital for the year 2015. With the identified data, the input operational variables were applied.

The cooperative selected for our study, which we shall call Coop "X", had the risk rating and analysis applied through both traditional and fuzzy logic in order to identify the advantages that this tool could have in the financial sector.

Application of the traditional logic

Table 10 presents the indicators that the SEPS (control body in the country) defines to qualify the ratings of each subset along with its respective evaluation, using three ranges, both for traditional logic and for fuzzy logic (Table 11).

Table 10 Interpretation of the results using traditional logic. 

Source: ^{SEPS (2015)}.

Table 11 Credit ratings traditional logic. 

Source: Own elaboration.

As can be observed in Table 2, this provides the ranges defined for each indicator, detecting that Coop "X" has a valuation of 51 points, allowing to verify in Table 3 that the category granted by its credit quality rating is AA or a Good management quality, with strong protective factors, modest risk, very high credit rating, and protection from acceptable investors.

Table 3 CAMEL study parameters. 

Source: ^{Crespo (2011)}.

Application of fuzzy logic

The application of the non-traditional logic in Table 4 shows the ranges of the fuzzy indicators, obtaining the result that Coop "X" has indicators such as: IF688, IF1111, IF202, IF295, IF293, with values that belong to two fuzzy subsets, providing a score of Good and Very Good due to the 46 points it shows in Table 5. The fuzzy methodology places the cooperative in two credit ratings, with 0.8 (over one) being a good rating, where the protective factors are strong, the risk is modest, the credit quality is very good, and the protective factors from the investors are very strong (Tables 12 and 13).

Table 5 Reference values of the objectives of the CAMEL model. 

Source: Own elaboration.

Table 12 Interpretation of the result using fuzzy logic. 

Source: Own elaboration.

Table 13 Fuzzy logic credit ratings. 

Source: Own elaboration.

However, said score also grants a membership level of 0.4 to the very good plus rating, since it places it at a superior level with excellent risk and credit ratings.