**3. Research Assumptions and Forecasting Method**

The objective of this research was to develop a model for forecasting changes in the volume of nonperforming consumer loans in the specific country studied. The analyzed country was Poland. The forecasting horizon spanned twenty years: from 2000 to 2020. The main limitation of this research was the prevalence of changes in personal bankruptcy

laws. In Poland, consumer bankruptcy legislation was only introduced in 2014. Thus, the definition of bankruptcy in this research was broadened to include so-called nonperforming loans. In banking, commercial loans are considered nonperforming if a borrower's payment is 90 days past due. Nonperforming loans are also a good measure of household insolvencies. Such a broad definition is often used in studies of countries where personal bankruptcy laws have recently been introduced.

To address the complexity of the households' insolvency phenomenon, in forecasting the financial risk of consumers, the methodology of fuzzy logic was chosen for the model's development.

Most of the methodologies in use nowadays, in all areas, must have advanced decisionrendering capabilities (e.g., engineering, finance, law, etc.). They must have the ability to provide an answer to a complex question. Some of them are based on classical (conventional) logic, which will always correspond to either affirmative or non-affirmative answers such as "white" or "black," "no" or "yes," "high" or "low," etc. These sets of answers are considered sets of truth values {0, 1} [40]. The idea behind the fuzzy logic theory is to replace the set of truth values {0, 1} with the entire interval (0, 1) as a practical approach to addressing a complex problem.

The fuzzy set of universe X is represented by a membership function that maps each element according to its degree of membership within the interval (0, 1). The membership function is a generalized form of the characteristic function, and it is associated with fuzzy logic. Considering the "high"/"low" example, a sentence in this universe according to the classical logic theory can have two possible values, but, using the fuzzy logic theory, the provided answer may have any of a large number of values, which are evaluated in the following manner: "how high/low regarding the highest/lowest value." The fuzzy sets, therefore, solve the problem of quantifying vague linguistic terms.

Membership functions can be present in any form and may be arbitrarily determined by the analyst. In the literature, the most common functions take one of three forms: triangular, trapezoidal, or Gaussian.

The general issues to be taken into consideration before designing a forecasting model based on fuzzy logic are as follows [40]:


The developed model is based on five different entry variables and it forecasts the number of non-performing loans with the use of macro-economic factors. For each entry variable in the model, the author identified three fuzzy sets (which are subsets of a set of values of the entry variable) and their corresponding membership functions. The fuzzy sets and the thresholds for all the membership functions are presented in Table 1. It is important to note that the author developed the model using the entry variables in the dynamic form (the rate of change), not in the static form (the value of a variable during a specific period). There are two advantages to employing such a research approach. First, it increases the usefulness of the proposed model when discussing various countries. Most countries are characterized by varied combinations of economic conditions and of economic variables. Implementing the rate of change instead of using the static form of economic data increases the universality of the model. Second, the volume of personal bankruptcies is determined within a dynamic system. It is difficult to define a reference state that has been influenced by the static values of other economic variables. For example, the value of the interest rate itself at a specific moment may not influence changes in the volume of nonperforming loans, but an increase/decrease in interest rates can increase/decrease the macro-economic risk of consumer insolvencies. Thus, it is more reliable and representative of the actual situation to explore and forecast this phenomenon from a dynamic perspective. Figure 3 presents an example of fuzzy sets defined with membership functions for the growth rate of the GDP (variable "X4" in the model).


**Table 1.** The entry variables in the forecasting model. Source: based on the author's own studies.

**Figure 3.** Fuzzy sets for the variable "X4"—the growth rate of the GDP with membership functions. Source: based on the author's own studies.

With such defined subsets, the boundary between the values believed to have a positive or negative effect on the volume of nonperforming loans is fuzzified—specific variable values are "partially positive" and "partially negative." There is no such possibility in the case of classical logic, which is bivalent.

The output of the model is a variable representing the forecast of trends in the volume of nonperforming loans (consumer insolvencies) in the country studied. This variable has a value from −30% to +30%, and it is represented by three membership functions (Figure 4):


In other words, the function "increase" represents a worsening situation for the credit market, showing an increase in nonperforming consumer loans within a country. The function "steady" indicates the stabilization of the credit market, with a more-or-less stable risk of customer insolvencies. The function "decrease" represents an improvement in a country's situation from the perspective of the number of nonperforming loans and households at risk of bankruptcy.

**Figure 4.** Defined membership functions for the variable "output" representing the forecast of the trends in the volume of nonperforming loans. Source: based on the author's own studies.

#### **4. Results and Discussion**

To conduct this study, the author programmed the fuzzy logic model with the structure presented in Figure 5. The model consisted of five inputs (the variables presented in Table 1) and one rule block where the set of decision rules was stored. The model's output was a variable representing a forecast of the fluctuations in the volume of nonperforming household loans within the country studied. The model was based on a set of rules written by the author in the form of "IF—THEN," in which expert knowledge was encapsulated. As there were five entry variables (X1, X2, X3, X4, and X5) with three possible states ("decrease," "steady," and "increase"), and there was a set of 243 possible decision rules. Due to space constraints, only the 30 most important decision rules are presented in Table 2.

**Figure 5.** The structure of the fuzzy logic model. Source: based on the author's own studies.


**Table 2.** The exemplary set of decision rules of the fuzzy logic model. Source: based on the author's own studies.

Based on the set of decision rules, the model was used to evaluate the country's macro-economic situation, which had a direct influence on households' credit-related decisions. There are five variables analyzed in the rule block, and the rules are constructed in consideration of the specific influence that each variable has on the risk of consumer insolvencies. An increase in the interest rate (variable X1) has a negative influence on consumers' degree of solvency. The bigger the increase in the interest rate is, the greater its negative influence on the volume of nonperforming consumer loans is (causing an increase in the number of such loans). In the same negative way, an increase in the inflation rate (variable X2) affects the output of a rule block. We must also remember that these two variables are very strictly dependent on each other. The third factor, changes in the unemployment rate (variable X3), negatively influences the financial situation of households as it increases. This variable often is negatively correlated with the fourth variable, which is the growth rate of the GDP. Variable X4 is believed to have a positive influence on the creditworthiness of households (the higher the increase in the GDP, the better). The last factor affecting the volume of nonperforming consumer loans in the country examined was fluctuations in the exchange rate. In this model, the fluctuation of PLN against EUR was represented. It was assumed that the appreciation of EUR against PLN would cause a higher risk of consumer insolvencies, as it could be expected to lead to an increase in the cost of living and is also often positively correlated with CHF. In Poland, there is a large group of consumers holding credits denominated in CHF, which has a direct, negative influence on their creditworthiness.

The author tested the developed fuzzy logic model using the data representing the fluctuations in the number of nonperforming household loans in Poland from 2000–2020. To evaluate the effectiveness of the model, two measures were considered—the mean absolute error (MAE) and mean absolute percentage error (MAPE). In Figure 6, the real and forecasted yearly fluctuations in the volume of nonperforming loans are presented. Based on the obtained data (Figure 6), the first measure (MAE) was 8.29%, and the second (MAPE) was 33.01%. The idea of using such a fuzzy logic model for predicting the macro-economic risk of consumer insolvencies in a country is a new development in the literature. Thus, the author could not find any data to compare with the obtained results; however, looking at Figure 6, it can clearly be seen that, during the entire analyzed period, the real and forecasted lines representing the percentage change in the number of nonperforming loans always conformed to the same positive/negative trend. This finding indicates that the model correctly predicted the trends in the volume of such loans in the country for all years. The observed MAE and MAPE values also made it possible to draw the conclusion that the errors generated were small and acceptable.

**Figure 6.** Yearly fluctuations in the volume of nonperforming loans in Poland (real versus forecasted). Source: based on the author's own calculations.

It is also worth underlining a few unique features of the proposed fuzzy logic model:


Concluding the results and discussion, in the presented study there are three types of scientific deliberations that constitute direct contribution to the literature:

• theoretical discussion—the author presented an assessment of the main macro- and microeconomic factors affecting the risk of consumer insolvency and explained the phenomenon of overlapping social factors. Based on these theoretical considerations, four common profiles of consumer behavior related to financial risk have also been proposed.

