The Basel Accords opened the way for and encouraged credit entities to implement their own models for measuring financial risks. In this paper, we focus on the use of internal ratings-based (IRB) models for the assessment of credit risk and, specifically, on one of the approaches to model the probability of default (PD).

The traditional methods used for modeling credit risk, such as discriminant analysis and logit and probit models, start with several statistical restrictions. The rough set methodology avoids these limitations and is presented as an alternative to the classic statistical methods.

The objective of our study is to apply the rough set methodology to a database composed of 106 companies that are debtors of the same financial entity to obtain the ratios that best discriminate between healthy and bankrupt companies. A second objective is to find a series of decision rules that will help detect potentially failing credit operations as a first step in modeling the probability of default. Finally, we compare the results obtained using the rough set methodology to those obtained using classic discriminant analysis and logit models. We conclude that the rough set methodology presents good risk classification results.

This paper is structured as follows. Section 2 reviews the most significant empirical studies. Section 3 introduces the theory of rough sets. In section 4, we continue with a description of the sample of companies used for the empirical study. The empirical application is described in section 5 wherein we first use the rough set methodology to determine the variables that may explain the default, and then compare the results obtained using this methodology to those obtained using classic discriminant analysis and logit models. Finally, in section 6, we draw a series of conclusions.

The models for predicting business failure and estimating the probability of default (PD) required by the Basel Accords have been the subject of several studies, conducted not only by academics but also by the financial sector itself. All of the theoretical effort has been focused on the modeling of the stochastic process associated with insolvency and on determining the variables that must be included in these models. Among these traditional models, we can distinguish between univariate and multivariate models. Univariate models examine the behavior of each variable separately to explain any insolvency. One of the classic studies using this method was conducted by Beaver (1966), who found a number of financial ratios that could discriminate between healthy and bankrupt companies during the 5-year period prior to the occurrence of the actual default. Other notable studies were those conducted by Courtis (1978) and Altman (1993).

Unlike the univariate models, the multivariate models combine the information provided by a set of variables. The study that pioneered this method was performed by Altman in 1968, in which he proposed a discriminant analysis that combined the information provided by 25 financial ratios. A wide variety of studies have been based on discriminant analysis, including those by Dambolena (1980) and Laitinen (1991). In Spain, Cabedo et al. (2004) presented an adaptation of the discriminant model to calculate the probability of default in companies. The model was applied to a portfolio of hypothetical borrowers from the same sector to calculate the regulatory capital according to the foundational IRB method of the Basel Accord. The importance of this technique is demonstrated in the bibliographic review by Dimitras (1996). After analyzing 158 articles on the prediction of business insolvency during the period from 1932 to 1994, Dimitras concluded that discriminant analysis is the model that is most frequently used to resolve this type of problem.

Other authors have opted for logit and probit analysis. Ohlson (1980) was the first to apply this type of technique to predict company insolvency. Wilson (1997) developed the Credit Portfolio View model for McKinsey, establishing a discrete process with multiple periods. With this methodology, the probability of default is obtained through logit functions of indices of macroeconomic variables that, in some ways, represent the functioning of the economy (Zmijewski, 1984). Dimitras (1996) found that the logit model is the second most frequently used model for resolving the problem of company bankruptcy.

Fernández (2005), in an attempt to combine univariate and multivariate analysis, conducted an empirical study in which he used a prior univariate analysis to select those ratios with greater discriminant power within each of the categories of ratios established from among the 23 ratios initially considered.1 Subsequently, he performed a logit and probit multivariate analysis to obtain scores for each company; these scores enabled a rating system to be established and default probabilities to be assigned. Trucharte et al. (2002) developed a system for rating borrowers by estimating a logistic regression model that utilizes economic and financial information. The scores obtained are used to establish homogeneous categories in which the various borrowers are classified or rated and the probability of default that can be assigned to each category.

More recently, Altman and Sabato (2007) developed a distress predictor model specifically for the SME sector and analyzed its effectiveness against that of a generic corporate model. They

These categories were liquidity, leverage, activity, debt cover and productivity.

used a logit regression model technique on panel data from more than 2,000 US firms over the period from 1994 to 2002.

In parallel with these studies, other methods have been explored to overcome the restrictive hypotheses that models of statistical inference impose on the variables. These hypotheses usually do not conform to reality and distort the results obtained. For these reasons, Eisenbeis (1977), Ohlson (1980) and Zavgren (1983) questioned the validity of the traditional models. In particular, techniques originating from the field of artificial intelligence have begun to be used; programs have been produced that are capable of generating knowledge from empirical data and then utilizing that knowledge to make inferences based on new data. Within this approach, we can distinguish techniques that seek knowledge by identifying patterns in the data. Among these are various classes of neural networks and other techniques that infer decision rules from the base data. The rough set methodology belongs to this last group of techniques. Authors such as Dimitras et al. (1998) and Daubie et al. (2002) have applied this technique to the classification of commercial loans. Other authors, such as Ahn et al. (2000), have combined the rough set methodology with neural networks to predict company failure. In Spain, various studies can similarly be found that apply the rough set methodology for the prognosis of company insolvency. Segovia et al. (2003) applied this technique to the prediction of insolvency in insurance companies, and Rodríguez et al. (2005) utilized it for the same purpose in a sample of small- and medium-sized enterprises (SMEs).

Rough set theory, first proposed by Z. Pawlak in 1982, is considered as an appropriate tool for handling cases in which there is considerable vagueness and imprecision. More specifically, the method is effective at working with problems of multidimensional classification (Pawlak et al., 1994). The basic idea rests on the indiscernibility relation that describes elements that are indistinguishable from each other. Its principal objective is to find basic decision rules that enable the acquisition of new knowledge. Its key concepts are discernibility, approximation, reducts and decision rules.

The point of departure for the method is the existence of an information/decision table in which each element is characterized by a set of variables (attributes) and a decision variable that classifies the element into one of two or more categories. Indiscernibility exists when two elements are characterized by the same properties for all variables, but the categories in which they are classified do not coincide. This is the basis of the rough set theory. In such a case, for each class of decision or category X and for each subset B that contains variables, two sets are constructed; these sets are called the set of the lower approximation of the decision class and the set of the upper approximation of the decision class, respectively. The set of the lower approximation of decision class X with respect to the variables B, which is called BX, is given by the group of all elements that, being characterized by B, belong to class X with complete certainty. The set of the upper approximation of decision class X, which is called BX , is given by the group of elements that, based on the information B that we possess, may belong to class X, but about which we cannot be sure. The elements that are different between the two sets form the "doubtful" elements, which are those elements that, using only the information contained in B, are not known with complete certainty to belong to class X. When these different elements exist (i.e., when the difference is not zero), it is said that class X is a rough set with respect to the subset of variables B. This set can be characterized numerically by the quotient between the cardinal of the set of lower approximation and the cardinal of the set of upper approximation. This quotient is known as the "accuracy of approximation". If various decision classes exist, the sum of the cardinals of the lower approximations divided by the total of all elements is known as the "quality of approximation of the classification, by means of set B", and this is the percentage of the elements that have been correctly classified.

Another important aspect of this technique is the reduction of the initial table of data, which eliminates the redundant information. This process is performed using the reducts. A reduct is a minimum set of variables that conserve the same capacity for classifying the elements as the full table of information. A reduct is thus an essential part of the information and constitutes the most concise way of differentiating between the decision classes.2

The final stage of the rough set analysis is the creation of decision rules, rules that allow us to determine whether a given element belongs to particular decision classes. These rules represent knowledge and are generated by combining the reducts with the values of the data analyzed. A decision rule is a logical statement of the following type: "IF particular conditions are met, THEN the element belongs to a particular decision class". These rules allow us to classify new elements easily..3

The reducts were obtained based on the equivalence classes that defined the indiscernibility relation on the set of observations. For more detail on the formal mathematical aspects of the methodology, Komorowski et al. (1999) may be consulted.

A. Selection of the Sample.

We adopted the following approach for both selecting the sample and choosing the independent variables. Following Altman (1968), we have paired a number of healthy and bankrupt companies of similar size and sector, thus taking a sample in which the bankrupt companies represent 50% of the total. When selecting the sample, the data under consideration should be obtained for the same period of time for healthy and bankrupt companies alike. However, the companies in bankruptcy or suspension of payments tend to delay the presentation of their accounting data in the time period prior to their declarations of insolvency. To overcome this inconvenience, we have collected the accounting data from the last full year prior to the bankruptcy from the most recent data available.

In the development of our model, we employed a database, provided by a Spanish savings bank that contains information on companies that requested and obtained a loan from the bank. These companies were divided into two groups: healthy and bankrupt. In particular, the sample of bankrupt companies used for the analysis only included those companies with loans whose unpaid debt, whether of interest or principal, amounted to a percentage of more than 10% of the full risk accepted. The computation date was December 31, 2003.

The group of healthy companies, i.e., those that did not default in the time horizon considered, was selected using the individual pairing technique, controlled by those characteristics that could affect the relationships between financial ratios and failure. Each company in the failed group was matched with a healthy company of the same industry and same approximate size. In relation to the sector, the pairing was achieved at a level of four digits of the C.N.A.E. (National Classification of Economic Activities) of 1993. The criterion adopted for pairing by size was the total assets.

As a homogenizing factor for all of the companies, we controlled for the conditions that the total amount of the customer's operations with the savings bank, i.e., their live risk, should exceed 60,120 euros and that the companies should all be public limited companies (PLCs), which facilitates access to their accounting statements.

A. Application of the Rough Set Mrthodology

For the empirical application, the values of the 25 economic/financial ratios shown in Table 1 were calculated for each of the 53 bankrupt companies for the financial year before entering into default, and a similar procedure was adopted for each matched healthy company. This process produced a table of information containing 106x25 data items. An additional column that indicates whether the company in question is in a situation of bankruptcy or health is included in the table of information. Thus, we have assigned the value 0 to bankrupt companies and the value 1 to the matched healthy companies. We thus obtain an information-decision table with 106x26 data items.

From these data, we determine which ratio or ratios of the 25 variables serve to explain a company’s state of default as the first step in calculating the probabilities of default.

First, Napierian logarithms of the values of the ratios were computed to avoid problems with the normality of the variables when applying the discriminant analysis. Then, given the nature of the variables considered, we proceeded to discretize the values. This is not an essential requirement for the application of the technique, but it facilitates the interpretation of the results; it is a more consistent way to identify bankrupt or healthy companies when the values of the variables considered fall within the same range but do not coincide exactly. For this, we utilized the codification given in Table 2.5

The next step is to determine the accuracy provided by the explanatory variables using the ROSE software. The quality of the approximation is 1.6

We discretized the variables by grouping them into four ranges based on the number of observations belonging to each range. For this, we utilized the ROSE software, provided by the Institute of Computing Science of the Poznan University of Technology, and we thank the Institute for making this software available to us. The quality of the approximation is expressed by the ratio between the number of companies classified correctly and the total number of companies that comprise the sample.

Napierian logarithms of the values of the ratios are first computed. Then, the values are discretized according to the ranges identified in this table.

Variables 0 1 2 3

Next, we construct the reducts. Because there are correlations between the explanatory variables, the program produces many reducts. Specifically, the program produced 18,241 reducts with and between 6 to 12 variables. Importantly, there are no core elements; there is no variable that is essential for the classification or is more relevant than any other. The frequency of appearance of each variable is shown in Table 3.

reduct should contain the smallest possible number of variables; second, the variables should present a high frequency of appearance in the different reducts; and finally, they should be formed by the smallest number of ratios for each category considered. The ratios belonging to each of the reducts are shown in Table 4.

quotients that have high percentages of appearance and contain a ratio in each of the categories studied. The third reduct was chosen because it presents ratios from all of the categories. We limited our search to those reducts formed from a maximum of 7 ratios (one more than the number of categories considered).

With each of the 3 reducts, decision rules were generated using the lem2 procedure;7 these are shown in Table 5. The following points can be noted regarding the items in the table. We can observe that the number of rules varies from one reduct to another; we find that 24, 28 and 22 rules are required to classify 100% of the observations correctly. The first reduct is the one that

Chan et al., (1994).

requires the fewest rules to identify the bankrupt companies. We can see also that the first reduct is the one that has the rules with the greatest power of classification. Thus, the rules 11 [(R22 = 0) => (D1 = 0)] and 12 [(R10 = 1) & (R22 = 2) & (R25 = 3) => (D1 = 1)] of the first reduct classify, respectively, 37.74% of the bankrupt companies and 49.06% of the healthy ones. No other individual rule of the second or third reducts reaches such high percentages.

The above 2 rules tell us, first, that if the value of the Napierian logarithm of the profitability ratio R22 (Pre-tax profits / Net Worth) is less than 1.16457e-005, then the company must be classified as bankrupt. Second, if the value of the Napierian logarithm of the ratio of indebtedness R10 (Financial Costs /Gross Profits) is between 0.00032564 and 0.00170486, that of the profitability ratio R22 (Pre-tax profits / Net Worth) is between 0.000203684 and 0.00417805 and that of the ratio R25 (Gross Profits / Total Assets) is greater than 0.000161758, the company must be classified as healthy.

percentages of correct classification. The third reduct, in addition to meeting the previously imposed requirements, is the one that presents fewest type I and II errors (13.42% and 11.18%, respectively, for the validation sample).

Given the above results, we are thus able to confirm that the rough set methodology leads to good results in the classification of healthy and bankrupt companies and indicates which variables are the most relevant of those considered. Thus, by applying these rules to new credit operations, a bank would be able to detect possible defaults.

still remains the most frequently utilized technique for measuring company insolvency. Therefore, we have compared the results of the rough set methodology with those produced by the discriminant analysis.

We used the same set of data to perform this comparison8. First, we utilized Box's M- statistic to test the equality of the variance-covariance matrices between the two groups. From the results, we can accept this hypothesis at a 5% level of significance. The discriminant function (obtained using the ascending stepwise method, with Snedecor's F criterion for entry between 0.06 and 0.09) and the statistics associated with the model are shown in Table 7.

liquidity and indebtedness ratios are the most relevant items for determining the possible

The Napierian logarithms of the data were used, and their values were typified. insolvency of a company. The liquidity ratio positively affects a company’s probability of being classified as healthy. With respect to the ratios of indebtedness, their influence will depend on their definition. The ratio R6 (Net Worth / Total liabilities) has a positive influence, while the ratio R9 (Financial Costs / (Gross Profits + provision for amortization)) has a negative influence on the probability of the company in question being considered healthy.

Table 8 gives the results of the classification with the discriminant function applied to the original sample and validated using the cross procedure.

useful tool than discriminant analysis for the classification of the defaulting companies in the database considered for our empirical application. With the correct choice of reducts, only 6 of the 25 variables considered in the analysis are required to correctly classify 100% of the original observations. Furthermore, more validation samples are correctly classified, giving type I and II errors of 13.42% and 11.18%, respectively, compared to the errors of 28.3% and 17%, respectively, that are given by applying discriminant analysis and utilizing the forward method for the twenty-five variables.

In this study we present an alternative methodology to the classic discriminant analysis and logit model for determining the variable(s) that serve to explain the failure of a company to meet its debt repayments as the first step in determining the probability of default (PD).

Basing our arguments on a sample of sound and bankrupt companies and on a set of 25 financial ratios that are potential explanatory factors for the defaults occurring in the sample, we have shown that the rough set methodology can be a valid alternative to discriminant analysis and to the logit model when there is a need to classify objects into two different classes. In addition to obtaining acceptable percentages of correct classification using rough sets, there is no need to assume any type of prior statistical behavior of the variables involved in the classification, unlike discriminant analysis, which requires normality in the distributions and equality in the variance-covariance matrices. Furthermore, the variables are included as they are presented, with no need for any transformation. Among the more significant advantages of this methodology is that it eliminates redundant information, and it expresses the dependencies between the variables considered and the results of the classification through decision rules for which the language is closer to that normally utilized by the experts.

Authors Information: Reyes Samaniego Medina is the corresponding author and is with Pablo de Olavide University, Department of Financial Economics and Accounting, Carretera de Utrera, Km. 1 (41013) Seville, Spain. Tel.: +34 954 34 98 45. E-mail address: rsammed@upo.es. M. José Vázquez Cueto, Departamento de Economía Aplicada III, Universidad de Sevilla (41018) Sevilla, Spain (España).