Credit risk assessment using a multicriteria hierarchical discrimination approach: A comparative analysis

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1 European Journal of Operational Research 138 (2002) Credit ris assessment using a multicriteria hierarchical discrimination approach: A comparative analysis M. Doumpos a, K. Kosmidou a, G. Baourais b, C. Zopounidis a, * a Technical University of Crete, Department of Production Engineering and Management, Financial Engineering Laboratory, University Campus, Chania, Greece b Mediterranean Agronomic Institute of Chania, Department of Economic and Management Sciences, Chania, Greece Abstract Corporate credit ris assessment decisions involve two major issues: the determination of the probability of default and the estimation of potential future benefits and losses for credit granting. The former issue is addressed by classifying the firms seeing credit into homogeneous groups representing different levels of credit ris. Classification/discrimination procedures commonly employed for such purposes include statistical and econometric techniques. This paper explores the performance of the M.H.DIS method (Multi-group Hierarchical DIScrimination), an alternative approach that originates from multicriteria decision aid (MCDA). The method is used to develop a credit ris assessment model using a large sample of firms derived from the loan portfolio of a leading Gree commercial ban. A total of 1411 firms are considered in both training and holdout samples using financial information through the period A comparison with discriminant analysis (DA), logit analysis (LA) and probit analysis (PA) is also conducted to investigate the relative performance of the M.H.DIS method as opposed to traditional tools used for credit ris assessment. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Credit ris assessment; Multicriteria decision aid; Classification; Case study 1. Introduction Credit ris assessment is a significant area of financial management which is of major interest to practitioners, financial and credit analysts. On a daily basis credit/financial analysts have to investigate an enormous volume of financial and non-financial data of firms, estimate the corresponding credit ris, and finally mae crucial decisions regarding the financing of firms. Considerable attention has been devoted in this field from the theoretical and academic points of view during the last three decades. Financial and operational researchers have tried to relate the characteristics * Corresponding author. Tel.: , 69551; fax: , addresses: ostas@ergasya.tuc.gr, ostas@cha.forthnet.gr (C. Zopounidis) /02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S (01)

2 M. Doumpos et al. / European Journal of Operational Research 138 (2002) of a firm (financial ratios and strategic variables) to its credit ris. According to this relationship the components of credit ris are identified, and decision models are developed to assess credit ris and the corresponding creditworthiness of firms as accurately as possible. Decisions regarding credit ris assessment concern the evaluation of the firms financial and non-financial characteristics in order to mae optimal decisions which incorporate a tradeoff between the potential ris of loss and the probability of profits from granting credit (Srinivasan and Kim, 1987; Srinivasan and Ruparel, 1990). Actually, credit-granting decisions are usually realized by credit and financial analysts as sorting (classifying) the firms seeing financing from bans or credit institutions into categories according to their creditworthiness (i.e., creditworthy and insolvent firms). During the credit evaluation process there are two major problems which are usually encountered (Bergeron et al., 1996). The first one concerns a plethora of factors which should be examined. Factors which affect the assessment of credit ris include the financial characteristics of firms, strategic variables of qualitative nature which affect the general operation of the firm and its relation with the maret, and even macroeconomic factors (i.e., inflation, interest rates, etc.). The credit analysts have to identify the most relevant factors for credit ris evaluation, and focus their further analysis on the examination of these factors. The second major problem concerns the aggregation of the factors which have been selected in the previous phase, in order to mae a final decision. Usually, factors affecting credit ris assessment lead to conflicting results and decisions. The credit/financial analysts, when performing credit ris analysis, implicitly consider the tradeoffs between the conflicting criteria, according to their global preference system. In this way, they conclude on an appropriate aggregation of the partial evaluations of firms on each one of the evaluation criteria, and derive the optimal decision. This complexity of the credit ris assessment process has necessitated the construction of credit ris assessment models, based on the sorting approach, which can be used by financial and credit analysts both as evaluation systems of new firms seeing financing as well as screening tools of the firms which are included in the loan portfolio of a ban or a credit institution (Lane, 1972; Altman et al., 1981; Grablowsy and Talley, 1981; Srinivasan and Kim, 1987; Srinivasan and Ruparel, 1990). A comprehensive review of credit ris assessment over the last two decades is presented by Altman and Saunders (1998). The main purpose of this paper is to investigate the potentials and the applicability of a new discrimination method in credit ris assessment, based on the methodological framewor of multicriteria decision aid (MCDA). The M.H.DIS method that is proposed (Zopounidis and Doumpos, 2000) employs a hierarchical discrimination procedure to determine the class to which the firms under consideration belong. The method leads to the development of a set of additive utility functions, which are used to classify each firm into a specific group. The method is compared to discriminant analysis (DA), logit analysis (LA) and probit analysis (PA) using a sample of firms derived from the loan portfolio of a leading Gree commercial ban. The article is organized as follows. Section 2 outlines the basic characteristics, features, mathematical formulation and operation of the M.H.DIS method. Section 3 discusses the data used in the application along with some preliminary findings. Section 4 presents the results obtained from the application of the M.H.DIS method, while in Section 5 these results are compared to DA, LA and PA. Finally, Section 6 concludes the article, summarizes the main findings of this research and proposes some future research directions. 2. The Multi-group Hierarchical DIScrimination method 2.1. General scheme The development of credit ris assessment models in this case study is performed through the M.H.DIS method. The general scheme of the procedure used to develop the credit ris assessment model through the

3 394 M. Doumpos et al. / European Journal of Operational Research 138 (2002) Fig. 1. General scheme of model development in the M.H.DIS method. M.H.DIS method is illustrated in Fig. 1. Initially, a reference set A consisting of n firms a 1 ; a 2 ;...; a n, classified into q ordered classes C 1 C 2 C q (C 1 is preferred to C 2, C 2 is preferred to C 3, etc.) is used for model development (i.e., training sample). The firms are described (evaluated) along a set of m evaluation criteria x ¼fx 1 ; x 2 ;...; x m g. The evaluation of a firm a j on criterion x i is denoted as x ij. The set of criteria may include both criteria of increasing and decreasing preference. Without loss of generality, the subsequent discussion involves only the case of increasing preference criteria. The development of the classification model is performed so as to respect the pre-specified classification, as much as possible. In this regard, the model developed should be able to reproduce the classification of the firms considered in the training sample. Once, this is achieved, the classification model can be used for extrapolation purposes involving the classification of any new firm not included in the training sample. This is a common model development procedure that is widely used in statistics and econometrics (e.g., in DA, LA and PA), as well as in other MCDA preference disaggregation approaches too. Such regression-based techniques are used for model development in the UTA method (Jacquet-Lagreze and Sisos, 1982), for raning problems in the UTADIS method (a variant of the UTA method for sorting problems; Jacquet- Lagreze, 1995; Zopounidis and Doumpos, 1999), as well as in the context of the ELECTRE-TRI method (Mousseau and Slowinsi, 1998), a well-nown outraning relations approach for addressing classification problems (Yu, 1992). The major characteristic of the M.H.DIS method during the development of credit ris assessment models as opposed to other discrimination methods is that it employs a hierarchical procedure in classifying the firms into the predefined classes. In particular the discrimination procedure employed in M.H.DIS

4 proceeds progressively in the classification of the firms, starting from class C 1 (lowest ris group). In the first stage, the firms found to belong to class C 1 (correctly or incorrectly) are excluded from further consideration. The objective of the second stage is to identify the firms that belong to class C 2. Once again, all the firms found to belong to this class (correctly or incorrectly) are excluded from further consideration, and the same procedure continues until all firms are classified into the predefined classes. The number of stages in this hierarchical discrimination procedure is q 1 (where q is the number of classes). The decision regarding the classification of the firms is based on the development of two additive utility functions at each stage of the aforementioned hierarchical discrimination process. The form of these utility functions is the following: 1 U ðxþ ¼ Xm M. Doumpos et al. / European Journal of Operational Research 138 (2002) h i u i ðx i Þ and U ðxþ ¼ Xm h i u i ðx i Þ: The former utility function U ðxþ characterizes all firms belonging to class C whereas the latter utility function U ðxþ characterizes all firms belonging to lower (worse) classes than C at stage of the hierarchical discrimination process. The corresponding marginal utility functions for each criterion x i are denoted as u i ðx i Þ and u i ðx i Þ which are normalized between 0 and 1, while the criterion weights h i and h i sum-up to 1, i.e., P m h i ¼ 1and P m h i ¼ 1. The marginal utility functions u i ðx i Þ are increasing functions on the criterion scale for all criteria x i that are negatively related to credit ris (e.g., profitability ratios, liquidity ratios, etc.) and decreasing functions for all criteria x i that are positively related to credit ris (e.g., solvency ratios, expenses ratios). Similarly, the marginal utility functions u i ðx i Þ are decreasing functions for all criteria x i that are negatively related to credit ris and increasing functions for all criteria x i that are positively related to credit ris. Both utility functions assign a global utility between 0 and 1 to each firm. If the global utility of a firm according to the utility function U ðxþ is higher than the global utility estimated according to the utility function U ðxþ, then the firm is assigned to class C. Otherwise, if the global utility of a firm according to the utility function U ðxþ is higher than the global utility estimated according to the utility function U ðxþ, then the classification decision is not to assign the firm in class C. Such a case indicates that the firm should be classified into one of the classes C þ1 ; C þ2 ;...; C q (the specific classification will be determined during the subsequent stages of the hierarchical discrimination process). Fig. 2 illustrates the hierarchical discrimination process employed in M.H.DIS Estimation of the utility functions The estimation of the additive utility functions in M.H.DIS is accomplished through mathematical programming techniques. Two linear programs and a mixed-integer one are used in M.H.DIS to estimate optimally the utility functions for the classification of the firms included in the training sample. The solution to these problems at each stage of the discrimination procedure has a twofold objective. First, to minimize the overall misclassification cost through the development of a pair of utility functions that facilitates the discrimination between group C and the lower groups C þ1 ; C þ2 ;...; C q (henceforth denoted as C ). Secondly, to calibrate the developed utility functions in order to maximize the clarity of the classification. This objective is similar to the among-groups variance maximization in DA. These two 1 These expressions are equivalent to U ðxþ ¼ P m u iðx i Þ and U ðxþ ¼ P m u iðx i Þ if u i ðx i Þ and u i ðx i Þ are not normalized in the interval ½0; 1Š.

5 396 M. Doumpos et al. / European Journal of Operational Research 138 (2002) Fig. 2. The hierarchical discrimination process in the M.H.DIS method. objectives are addressed through a lexicographic approach. First, the minimization of the overall misclassification cost is pursued, and then the maximization of the clarity of the classification is sought. Pursuing the first objective on the development of the two utility functions (i.e., minimization of the overall misclassification cost) requires the minimization of the following function: EC ¼ w 1 N X 8a j2c I j! 1 þ w N X 8a j2c I j! ; ð1þ where I j and I j are 0 1 variables representing the classification status of each firm belonging to groups C and C, respectively (0 indicates correct classification, whereas 1 indicates misclassification). N represents the number of firms belonging to credit ris group C, whereas N represents the number of firms belonging to the set of groups C. The weighting parameters w and w should be defined on the basis of the misclassification costs and the a priori default probabilities: w ¼ p MC, w ¼ p MC such that w þ w ¼ 1 and w P 0, w P 0, where p and p are the a priori probabilities that a firm belongs to the credit ris groups C and C, respectively, whereas MC and MC are the misclassification costs

6 M. Doumpos et al. / European Journal of Operational Research 138 (2002) associated with the classification errors of the forms C! C and C! C. The definition of w and w depends on the decision maer. In the credit ris assessment problem, usually two classes of firms are considered, i.e., the financially sound firms (class C 1 ) and the firms that face financial problems (financially distressed firms; class C 2 ). In this case, the cost of misclassifying a distressed firm ðmc 1 Þ is higher than the cost of misclassifying a healthy one (i.e., MC 1 > MC 1 ). However, the number of distressed firms is considerably lower than the number of healthy firms, implying that the a priori probability that a firm is distressed is smaller than the a priori probability that a firm is healthy (i.e., p 1 < p 1 ; Theodossiou et al., 1996). Therefore, setting w 1 ¼ w 1 ¼ 0:5 is a reasonable choice. This specification is the one used in the application regarding credit ris assessment presented later on, in this paper. The development of a pair of utility functions that minimize the overall misclassification cost (1) (let EC min denote the minimum overall misclassification cost) requires the use of mixed-integer programming techniques. However, solving mixed-integer programming formulations in cases where there are many integer variables is a computationally intensive procedure. Even in cases of samples consisting of 50 firms (i.e., 50 integer variables) the development of the optimal classification rule could be a highly time-consuming process if there is a significant degree of group overlap. To address this issue, M.H.DIS initially employs an alternative error function EC 0 that approximates the overall misclassification cost: EC 0 ¼ w 1 N X 8a j2c e j! 1 þ w N X 8a j2c e j! : ð2þ The error variables e j and e j are surrogates of the 0 1 error variables I j and I j in (1). Both these classification errors are positive real numbers representing the magnitude of the violation of the classification rules employed during model development (x j denotes the vector consisting of the performances of the firm a j on all the evaluation criteria): n o e j ¼ max 0; U ðx j Þ U ðx j Þ ; n o ð3þ e j ¼ max 0; U ðx j Þ U ðx j Þ : The minimization of the function EC 0 is performed through the solution of the following mathematical programming problem: LP1: Minimization of the overall classification error Min EC 0 subject to: u i ðx ij Þ Xm u i ðx ij Þ Xm u i ðx ij Þþe j P s 8a j 2 C ; ð4þ u i ðx ij Þþe j P s 8a j 2 C ; ð5þ u i ðx i Þ¼0; u i ðx i Þ¼1; u i ðx i Þ¼1; u i ðx i Þ¼0; ð6þ

7 398 M. Doumpos et al. / European Journal of Operational Research 138 (2002) u i ðx i Þ increasing function; u i ðx i Þ decreasing function; u i ðx i Þ P 0; u i ðx i Þ P 0; e j P 0; e j P 0: LP1 is a simple linear programming problem that can be easily solved even for large data sets. In constraints (4) and (5) s is a small positive constant used to ensure the strict inequalities presented in definition (3) of the error variables e i and e i. Constraint set (6) is used to normalize the results of global utilities functions in the interval ½0; 1Š. In these constraints x i and x i represent the least preferred value and the most preferred one for the criterion x i. Solving LP1 yields an initial pair of utility functions that minimize the total classification error function EC 0 (let EC 0 min denote the minimum total classification error obtained after solving LP1). If these utility functions classify correctly all firms, then the error variables e i and e i will all be zero. Therefore, EC 0 min ¼ EC min ¼ 0. However, this is not always the case. Usually, EC 0 min 6¼ 0 and consequently EC min 6¼ 0. In such cases, bearing in mind the fact that EC 0 is an approximation of EC, it becomes apparent that the utility functions corresponding to EC 0 min will not necessarily yield the minimum overall misclassification cost EC min. For instance, consider that in a sample consisting of four firms classified into two groups C 1 and C 2 (low-ris and high-ris, respectively) the utility functions obtained after solving LP1 lead to two misclassified firms i (low-ris firm) and j (high-ris firm) with the following classification errors: e 1i ¼ 0:2 and e 1j ¼ 0:1. In this case EC 0 min ¼ 0:075 and EC ¼ 0:5 (assuming w 1 ¼ w 1 ¼ 0:5). However, an alternative solution that classifies j correctly (i.e., e 1j ¼ 0) but assigns a misclassification error to firm i equal to 0.5 is clearly preferred. In this case EC 0 ¼ 0:125 > EC 0 min, but EC min ¼ 0:25 < EC. Thus, through this simple example it becomes apparent that it could be possible to find an alternative pair of utility functions than the one developed through LP1. The latter one yields a classification error EC 0 P EC 0 min, but provides a lower overall misclassification cost. In M.H.DIS this possibility is explored through the solution of MIP. MIP: Minimization of the overall misclassification cost X N mis Min EC ¼ A 1 X N mis þ A subject to: I N mis j j¼1 N mis j¼1 I j ð7þ ð8þ u i ðx ij Þ Xm u i ðx ij Þ Xm u i ðx ij Þ Xm u i ðx ij Þ Xm u i ðx ij Þ P s; u i ðx ij Þ P s; u i ðx ij ÞþI j P s; u i ðx ij ÞþI j P s; a j ¼ 1; 2;...; N cor ; a j ¼ 1; 2;...; N cor ; a j ¼ 1; 2;...; N mis ; a j ¼ 1; 2;...; N mis ; ð9þ ð10þ u i ðx i Þ¼0; u i ðx i Þ¼1; u i ðx i Þ¼1; u i ðx i Þ¼0; ð11þ

8 M. Doumpos et al. / European Journal of Operational Research 138 (2002) u i ðx i Þ increasing function; u i ðx i Þ decreasing function; u i ðx i Þ P 0; I j ; I j integers: ð12þ ð13þ Starting with the initial utility functions developed through LP1, MIP explores the possibility to modify these utility functions so that the overall misclassification cost is minimized. This minimization is performed without changing the correct classifications obtained by LP1 (i.e., all firms correctly classified by the initial pair of utility functions are retained as correct classifications; cf. constraints (9)). Note that the 0 1 error variables I j and I j are not associated to all firms, but only to the ones misclassified by LP1 (constraints (10)). The number of firms actually belonging to group C which are misclassified by LP1 is denoted as N mis whereas N mis denotes the number of firms actually belonging to the set of groups C, which are classified by LP1 into group C. Similarly, N cor and N cor denote the number of corresponding correct classifications obtained by LP1. All these correct classifications are retained (constraints (9)). Since, in most cases, the number of firms misclassified by LP1 ðn mis þ N mis Þ is a small part of the whole sample, the number of integer variables in MIP is small, thus facilitating its easy solution. The pair of utility functions developed after solving initially LP1 and then MIP is optimal in terms of the overall misclassification cost. However, the ultimate purpose of the utility functions developed through M.H.DIS is to be used for credit ris assessment. Of course, it is difficult to ensure high predictability during model development. However, utility functions that clearly distinguish firms belonging to different credit ris groups are expected to have higher predictability than utility functions that yield the same overall misclassification cost but achieve a marginal discrimination during model development. Traditional DA addresses this issue through the maximization of the among-groups variance. In M.H.DIS, the measure employed to assess the distance between the two groups of firms according to the developed discrimination model (utility functions) is the minimum difference d between the global utilities of the correctly classified firms identified after solving MIP ðd > 0Þ. where (N cor0 d ¼ minfd 1 ; d 2 g; d 1 ¼ min j¼1;2;...;n cor0 fu ðx j Þ U ðx j Þg and d 2 ¼ min, j¼1;2;...;n cor0 fu ðx j Þ U ðx j Þg and N cor0 denote the number of firms belonging to groups C and C, respectively, classified correctly by MIP). The maximization of d is achieved through the solution of the following linear programming formulation (LP2). LP2: Maximization of the minimum distance Max d subject to: u i ðx ij Þ Xm u i ðx ij Þ Xm u i ðx ij Þ d P s; u i ðx ij Þ d P s; a j ¼ 1; 2;...; N cor0 ; a j ¼ 1; 2;...; N cor0 ; ð14þ

9 400 M. Doumpos et al. / European Journal of Operational Research 138 (2002) u i ðx ij Þ Xm u i ðx ij Þ Xm u i ðx ij Þ 6 0; u i ðx ij Þ 6 0; a j ¼ 1; 2;...; N mis0 ; a j ¼ 1; 2;...; N mis0 ; ð15þ u i ðx i Þ¼0; u i ðx i Þ¼1; u i ðx i Þ¼1; u i ðx i Þ¼0; ð16þ u i ðx i Þ increasing function; u i ðx i Þ decreasing function; ð17þ ð18þ u i ðx i Þ P 0; d P 0: LP2 begins with the utility functions obtained after solving MIP. N mis0 and N mis0 denote the number of firms actually belonging to groups C and C, respectively, misclassified by MIP. LP2 sees to modify the utility functions developed through MIP in order to maximize the distance measure d. All firms misclassified by the utility functions developed through MIP are retained as misclassified. Thus, the utility functions developed through LP2 do not affect the overall misclassification cost, since all correct classifications and misclassifications resulted after solving MIP are retained (constraints (14) and (15), respectively). The pair of utility functions obtained after solving LP2 is the one used for credit ris assessment purposes An illustrative example To illustrate the functionality of the M.H.DIS method, consider a simple example consisting of four firms F 1, F 2, F 3 and F 4, evaluated along two financial ratios (earnings before interest and taxes/total assets: x 1, current assets/current liabilities: x 2 ). The firms are classified into two groups as healthy (group C 1 ) and distressed (group C 2 ). Table 1 illustrates the performances of the firms according to each ratio and their predefined classification. Since this is a two-group classification problem, only two utility functions need to be developed; the functions U 1 ðxþ and U 1 ðxþ. On the basis of these functions the corresponding global utilities of the firms are expressed as follows: Firm F 1 : U 1 ðx 1 Þ¼u 11 ð10%þþu 12 ð2:97þ and U 1 ðx 1 Þ¼u 11 ð10%þþu 12 ð2:97þ; Firm F 2 : U 1 ðx 2 Þ¼u 11 ð7:5%þþu 12 ð1:05þ and U 1 ðx 2 Þ¼u 11 ð7:5%þþu 12 ð1:05þ; Table 1 Data of the illustrative example x 1 x 2 Group F 1 10% 2.97 C 1 F 2 7.5% 1.05 C 1 F 3 8% 0.80 C 2 F 4 3% 1.10 C 2

10 M. Doumpos et al. / European Journal of Operational Research 138 (2002) Firm F 3 : U 1 ðx 3 Þ¼u 11 ð8%þþu 12 ð0:80þ and U 1 ðx 3 Þ¼u 11 ð8%þþu 12 ð0:80þ; Firm F 4 : U 1 ðx 4 Þ¼u 11 ð3%þþu 12 ð1:10þ and U 1 ðx 4 Þ¼u 11 ð3%þþu 12 ð1:10þ: On the basis of these formulations regarding the estimation of the global utilities of the firms, LP1 is expressed as follows ðs ¼ 0:001Þ: subject to: Min EC 0 ¼ 0:5 1ðe 2 11 þ e 12 Þ þ 0:5 1ðe 2 13 þ e 14 Þ () Minðe 11 þ e 12 þ e 13 þ e 14 Þ Firm F 1 : ½u 11 ð10%þþu 12 ð2:97þš ½u 11 ð10%þþu 12 ð2:97þš þ e 11 P 0:001; Firm F 2 : ½u 11 ð7:5%þþu 12 ð1:05þš ½u 11 ð7:5%þþu 12 ð1:05þš þ e 12 P 0:001; Firm F 3 : ½u 12 ð8%þþu 12 ð0:80þš ½u 11 ð8%þþu 12 ð0:80þš þ e 13 P 0:001; Firm F 4 : ½u 11 ð3%þþu 12 ð1:10þš ½u 11 ð3%þþu 12 ð1:10þš þ e 14 P 0:001; u 11 ð10%þþu 12 ð2:97þ ¼1; u 11 ð3%þþu 12 ð0:80þ ¼0; u 11 ð3%þþu 12 ð0:80þ ¼1; u 11 ð10%þþu 12 ð2:97þ ¼0; u 11 ð10%þ P u 11 ð8%þ P u 11 ð7:5%þ P u 11 ð3%þ P 0; u 11 ð3%þ P u 11 ð7:5%þ P u 11 ð8%þ P u 11 ð10%þ P 0; u 12 ð2:97þ P u 12 ð1:10þ P u 12 ð1:05þ P u 12 ð0:80þ P 0; u 12 ð0:80þ P u 12 ð1:05þ P u 12 ð1:10þ P u 12 ð2:97þ P 0; e 11 ; e 12 ; e 13 ; e 14 P 0: The solution to this linear program and the estimated global utilities of the firms on the basis of the obtained solution are presented in Table 2. According to the estimated global utilities of firms F 1 and F 2 are assigned to category C 1 while firms F 3 and F 4 are assigned to category C 2. Since there is no misclassification, the procedure proceeds with the solution of LP2, which for this illustrative example is formulated as follows: Table 2 Solution of the problem LP1 for the data of the illustrative example Criterion x 1 Criterion x 2 Global utilities u 11 u 11 u 12 u 12 U 1 U 1 3% F % F % F % F

11 402 M. Doumpos et al. / European Journal of Operational Research 138 (2002) Max d subject to: Firm F 1 : ½u 11 ð10%þþu 12 ð2:97þš ½u 11 ð10%þþu 12 ð2:97þš d P 0:001; Firm F 2 : ½u 11 ð7:5%þþu 12 ð1:05þš ½u 11 ð7:5%þþu 12 ð1:05þš d P 0:001; Firm F 3 : ½u 12 ð8%þþu 12 ð0:80þš ½u 11 ð8%þþu 12 ð0:80þš d P 0:001; Firm F 4 : ½u 11 ð3%þþu 12 ð1:10þš ½u 11 ð3%þþu 12 ð1:10þš d P 0:001; u 11 ð10%þþu 12 ð2:97þ ¼1; u 11 ð3%þþu 12 ð0:80þ ¼0; u 11 ð3%þþu 12 ð0:80þ ¼1; u 11 ð10%þþu 12 ð2:97þ ¼0; u 11 ð10%þ P u 11 ð8%þ P u 11 ð7:5%þ P u 11 ð3%þ P 0; u 11 ð3%þ P u 11 ð7:5%þ P u 11 ð8%þ P u 11 ð10%þ P 0; u 12 ð2:97þ P u 12 ð1:10þ P u 12 ð1:05þ P u 12 ð0:80þ P 0; u 12 ð0:80þ P u 12 ð1:05þ P u 12 ð1:10þ P u 12 ð2:97þ P 0; d P 0: The solution of this linear problem, presented in Table 3, provides the final model to discriminate among the firms of the two categories. The discriminant model consists of the following two additive utility functions: U 1 ðxþ ¼0:333u 11 ðx 1 Þþ0:667u 12 ðx 2 Þ; U 1 ðxþ ¼u 11 ðx 1 Þ: The associated discrimination rule is to assign a firm to category C 1 if U 1 ðxþ > U 1 ðxþ and to category C 2 otherwise. Table 3 Solution of the problem LP2 for the data of the illustrative example Criterion x 1 Criterion x 2 Global utilities u 11 u 11 u 12 u 12 U 1 U 1 3% F % F % F % F

12 3. Data and preliminary findings 3.1. Data M. Doumpos et al. / European Journal of Operational Research 138 (2002) The data used in this article are derived from the loan portfolio of the Commercial Ban of Greece, one of the leading Gree commercial bans. Overall 1411 firms are considered from different business sectors. These firms are included in two data sets. The first one was provided by the ban for model development purposes (training sample). It consists of the financial data of 200 firms over the period On the basis of the latest information available for these firms (year 1997), the credit officers of the ban assigned half of them as firms of high credit ris. The remaining 100 firms of the training sample were evaluated as firms of low credit ris. Thus, the credit ris assessment model to be developed will be used to discriminate between these two groups of firms. The second data sample (holdout sample) consists of 1211 firms classified in the same two groups as the training sample. This holdout sample is used to validate the credit ris assessment model in order to evaluate its generalizing ability and classification performance on corporate data of firms that differ from the ones used for model development. The holdout sample consists of 1093 firms of low credit ris and 118 firms of high credit ris. On the basis of the available financial data of the firms 11 financial ratios are used as adequate measures of corporate credit ris (Table 4). The selection of these ratios has been performed with the collaboration of expert credit ris analysts from the Commercial Ban of Greece in order to consider the credit ris policy of the ban and the financial analysis approach employed in the daily practice of credit ris analysts. It should also be noticed that according to the international financial literature (Courtis, 1978) the selected ratios cover all aspects of the corporate financial performance, including profitability, solvency and managerial performance Preliminary findings Among the financial ratios considered, earnings before interest and taxes/total assets (EBIT/TA), net income/net worth (NI/NW), sales/total assets (SALES/TA), net income/woring capital (NI/WC) and gross profit/total assets (GP/TA) are related to the profitability of the firms. High values of these ratios correspond to profitable firms. Thus, all these ratios are negatively related to credit ris. The financial ratios quic assets/current liabilities (QA/CL) and cash/current liabilities (CASH/CL) involve the liquidity of the Table 4 List of financial ratios Codification EBIT/TA NI/NW SALES/TA GP/TA NI/WC TD/TA LTD/(LTD+NW) QA/CL CASH/CL CL/NW TD/WC Financial ratio Earnings before interest and taxes/total assets Net income/net worth Sales/total assets Gross profit/total assets Net income/woring capital Total debt/total assets Long-term debt/(long-term debt+net worth) (Current assets)inventories)/current liabilities Accounts receivable/current liabilities Current liabilities/net worth Total debt/woring capital

13 404 M. Doumpos et al. / European Journal of Operational Research 138 (2002) firms. Firms having enough liquid assets (current assets except for inventories) are in better liquidity position and they are more capable of meeting their short-term obligations to their creditors. Thus, these ratios are also negatively related to credit ris. The ratios total debt/total assets (TD/TA), long-term debt/ (long-term debt + net worth) [LTD/(LTD+NW)], and total debt/woring capital (TD/WC) are related to the solvency (financial leverage) of the firms. High values indicate severe indebtedness, that is the firms have Table 5 t-test for the differences in the means of financial ratios for each group of firms in the training sample Financial ratios EBIT/TA Healthy Distressed ) ) ) ) t-value (4.47) (5.32) (5.49) (1.60) NI/NW Healthy ) Distressed ) ) ) ) t-value (1.54) (3.82) (4.16) (1.16) SALES/TA Healthy Distressed t-value (3.98) (4.81) (4.84) (8.96) GP/TA Healthy Distressed ) t-value (3.16) (2.74) (3.58) (6.69) NI/WC Healthy ) ) Distressed ) ) ) ) t-value (0.88) (1.37) (2.99) (1.81) TD/TA Healthy Distressed t-value ()4.32) ()5.20) ()5.71) ()6.76) LTD/(LTD+NW) Healthy Distressed t-value ()1.99) ()1.65) ()2.57) ()1.97) QA/CL a Healthy Distressed t-value (1.17) ()0.01) (2.26) (2.20) CASH/CL Healthy Distressed t-value (0.99) (0.44) (1.84) (1.65) CL/NW Healthy Distressed t-value ()1.15) ()1.59) ()1.34) ()1.18) TD/WC Healthy Distressed t-value (0.74) ()1.27) ()0.71) ()1.31) Note: Parentheses include the t-values for testing the null hypothesis that the means of the financial ratios in the two considered groups of firms are equal. * Statistically significant at 5% level. ** Statistically significant at 10% level. a Quic assets ¼ Current assets)inventories.

14 M. Doumpos et al. / European Journal of Operational Research 138 (2002) to generate more income to meet their obligations and repay their debt. Consequently these ratios are positively related to credit ris. Table 5 presents the results of a t-test regarding the differences in the means of the financial ratios for the healthy and distressed firms in the training sample. The results indicate that the differences in the means of most ratios between the two groups of the firms are statistically significant at the 5% level. The profitability ratios sales/total assets (SALES/TA) and gross profit/total assets (GP/TA) as well as the solvency ratios total debt/total assets (TD/TA) and long-term debt/(long-term debt+net worth) [LTD/(LTD+NW)] are Table 6 t-test for the differences in the means of financial ratios for each group of firms in the holdout sample Financial ratios EBIT/TA Healthy Distressed ) ) ) ) t-value (6.84) (3.15) (2.82) (4.67) NI/NW Healthy Distressed ) ) ) t-value (3.30) ()0.59) (4.62) (5.34) SALES/TA Healthy Distressed t-value (6.39) (8.30) (0.19) (8.22) GP/TA Healthy Distressed ) t-value (3.41) (4.99) (5.95) (4.64) NI/WC Healthy Distressed ) ) ) ) T-value (1.48) (1.29) (1.12) (2.38) TD/TA Healthy Distressed t-value ()2.85) ()3.34) ()3.91) ()5.07) LTD/(LTD+NW) Healthy Distressed t-value ()1.03) ()1.50) ()2.57) ()3.10) QA/CL Healthy Distressed t-value ()0.53) ()1.00) ()0.42) (2.67) CASH/CL Healthy Distressed t-value ()0.772) ()1.18) (0.98) (2.13) CL/NW Healthy Distressed t-value ()2.07) ()1.16) ()1.74) ()2.97) TD/WC Healthy Distressed t-value ()0.49) ()0.54) (0.93) ()0.79) Note: Parentheses include the t-values for testing the null hypothesis that the means of the financial ratios in the two considered groups of firms are equal. * Statistically significant at 5% level. ** Statistically significant at 10% level.

15 406 M. Doumpos et al. / European Journal of Operational Research 138 (2002) significant throughout all years. EBIT/TA is significant at the 5% level during the years , while the ratios net income/net worth (NI/NW), net income/woring capital (NI/WC) and quic assets/current liabilities (QA/CL) are significant in two out of the four years. Table 6 presents the results of a t-test regarding the differences in the means of the financial ratios for each group of firms in the holdout sample. Similarly to the results in the training sample the profitability ratio GP/TA and the solvency ratio TD/TA are found significant throughout all four years. Furthermore, EBIT/TA is significant in all years, while in the training sample this ratio was significant in three out of the four years. Finally, the ratio SALES/TA alie with the training sample is also significant in the holdout sample with the exception of year Most of the other ratios are found significant for at least one of the years in the considered period. 4. Results obtained through the M.H.DIS method In order to develop the credit ris assessment models, the data of the training sample regarding the year 1997 were used. Two additive utility functions are developed, since there are only two groups of firms (healthy and distressed). The procedure leading to the development of these utility functions proceeds in the following way. Initially LP1 is solved to determine an initial pair of utility functions to explore whether it is possible to classify correctly all firms in year 1997 of the training sample for model development. According to the developed utility functions only one firm is misclassified as distressed while actually being healthy. This solution is optimal in terms of the error functions EC 0 and EC. Therefore, MIP is not solved. Thus, the utility functions developed by LP1 and the classification of the firms remain unchanged. Finally, LP2 is employed to find a pair of utility functions that do not change the obtained classification, but maximize the minimum difference d between healthy and distressed firms. This leads to a new pair of utility functions, which differ from the ones initially developed through LP1. Table 7 presents the final set of weights of the financial ratios in the two additive utility functions developed for credit ris assessment purposes. The utility function U 1 ðxþ characterizes the firms of low credit ris, whereas the utility function U 1 ðxþ characterizes the high-ris firms. Fig. 3 illustrates the marginal utility functions of the considered financial ratios in these two utility functions (the dotted lines correspond to the function developed for the low-ris firms and the solid line corresponds to the function developed for the high-ris firms). The obtained results indicate that the healthy firms are characterized by high values on the profitability ratios EBIT/TA, SALES/TA and GP/TA, and low values on the solvency ratio CL/NW. Table 7 Financial ratios weights in the utility functions developed through M.H.DIS Financial ratios h 1i (%) h 1i (%) EBIT/TA NI/NW SALES/TA GP/TA NI/WC TD/TA LTD/(LTD+NW) QA/CL CASH/CL CL/NW TD/WC

16 M. Doumpos et al. / European Journal of Operational Research 138 (2002) Fig. 3. Marginal utility functions of the financial ratios in the credit ris assessment model developed through M.H.DIS. On the other hand, distressed firms are characterized by low values on the profitability ratios EBIT/TA, SALES/TA and NI/WC, as well as by low values on the liquidity ratio QA/CL. Therefore, on the basis of the weights of the financial ratios in the two utility functions it is possible to identify three major differences. The ratio GP/TA is significant in the utility function corresponding to the

17 408 M. Doumpos et al. / European Journal of Operational Research 138 (2002) Fig. 3. (continued). financially healthy firms, but its weight in the function developed for the distressed firms is very low. This is also the case for the ratio CL/NW. On the contrary, QA/CL is significant in the case of the financially distressed firms but its importance to identify low-ris firms is limited. These results can be interpreted as follows: high values on the ratio GP/TA and low values on the ratio CL/NW are both significant characteristics of low-ris firms. However, the opposite does not hold, i.e., low values of GP/TA or high values of CL/NW are not significant indications that a firm is of high credit ris. On the contrary, although low values of the liquidity ratio QA/CL can be considered a significant indication of high-ris firms, high values do not indicate low-ris (often high liquidity indicates that a firm does not use appropriately its available funds to improve its profitability). The credit ris assessment model developed for the year 1997 is applied in the previous three years of the training sample as well as to all the four years regarding the holdout sample. This extrapolation test enables the evaluation of the efficiency of the model in performing correct credit ris assessment estimations as early as possible. The obtained results are reported in Table 8. The type I error corresponds to the classification of firms of high ris into the low-ris group, whereas the type II error corresponds to the classification of low-ris firms into the high-ris group. The total error is measured as the average of type I and type II error rates. Although the cost associated with the type I error is higher than the cost associated with the type II Table 8 Classification results obtained through the M.H.DIS method (error rates) Training sample Holdout sample 1997 (%) 1996 (%) 1995 (%) 1994 (%) 1997 (%) 1996 (%) 1995 (%) 1994 (%) Type I error Type II error Total error

18 M. Doumpos et al. / European Journal of Operational Research 138 (2002) error, the a priori probability that a firm belongs to the low-ris group is considerably lower than the probability that a firm belongs to the high-ris group (defaulters outnumber non-defaulters). In this regard, the assumption that both types of errors contribute equally to the total error is not an unreasonable choice (for more details on the manipulation of the probabilities and the costs associated with the type I and II errors see Theodossiou et al., 1996; Bardos, 1998). According to the obtained results the credit ris assessment model developed through the M.H.DIS method provides low error rates in both the training and holdout samples. Even for the year 1994 (three years prior to the data used for model development, i.e., 1997), the total error is 23% for the training sample and 24% for the holdout sample. The fact that type I error is significantly higher than the type II error throughout the years in both the training and holdout samples is not surprising. Generally, firms that are in a financially healthy position are expected to have good financial characteristics over time. On the contrary, firms that face problems in some specific point of time have a gradual deterioration of their financial characteristics over the preceding years. Therefore, it is possible that some of these firms may have similar financial characteristics to financially healthy firms few years prior to the occurrence of financial problems. Thus, it is generally easier to identify the firms of low credit ris from the ones of high credit ris. 5. Comparison with discriminant analysis, logit analysis and probit analysis DA can be considered as the first approach to tae into account multiple factors in discriminating among different groups of objects (Altman, 1968). DA is a multivariate statistical technique that leads to the development of a linear discriminant function maximizing the ratio of among-group to within-group variability, assuming that the variables follow a multivariate normal distribution and that the dispersion matrices of the groups are equal (in the linear case). Despite these assumptions and the criticism that they caused, DA has been widely used in the past in addressing a variety of financial decision maing problems, including credit ris assessment (see Altman et al., 1981 for a comprehensive review). On the other hand, LA and PA are alternative approaches to DA. The major advantage of both LA and PA over DA is that they overcome the statistical assumptions of DA. Both LA and PA provide the probability F ða þ bx i Þ that a firm belongs to the low-ris group on the basis of its performance on the financial ratios X i. The developed LA model has the form of the cumulative logistic probability function F ða þ bx i Þ¼1=ð1 þ e ðaþbxiþ Þ. Based on this probability a firm is classified as healthy or financial distressed, using a cut-off probability. Maximum lielihood estimation procedures are employed to determine the parameters a and b. The developed PA model is computed from the standardized normal cumulative distribution function F ða þ bx i Þ¼ Z aþbxi 1 1 ð2pþ 1=2 e z2 =2 dz: The consideration of LA and PA in this comparative study complements the obtained results, since their advantages mae them more appealing in credit ris assessment than DA. Furthermore, both methods have been widely used in the past in several applications related to credit ris assessment and financial distress prediction (Ohlson, 1980; Zavgren, 1985; Casey et al., 1986; Keasey et al., 1990; Sogsvi, 1990). DA, LA and PA are applied following the same methodology used for the development of credit ris assessment model through the M.H.DIS method. More specifically, the year 1997 is used for model development purposes. The application of the credit ris assessment models developed through DA, LA and PA in the holdout sample as well as in the remaining years of the training sample is based on the selections the appropriate cut-off point/probability so as to minimize the total misclassification cost (i.e., the total classification error). On the basis of the results for the training sample, which consists of the firms used during model development, the best cut-off point/probability was specified at 0.23 for the DA model, 0.21

19 410 M. Doumpos et al. / European Journal of Operational Research 138 (2002) for the LA model and for the PA credit ris assessment model. Table 9 presents the credit ris assessment model developed through DA, LA and PA, whereas Table 10 presents the estimates for error rates in the training and holdout samples of the developed models. On the basis of the above results the efficiency of the M.H.DIS method as opposed to traditional statistical and econometric techniques for developing credit ris assessment model becomes apparent. In particular, both DA and PA provide consistently higher error rates throughout all years both in the training and holdout samples. On the other hand, LA s results in the training sample are similar to the ones of M.H.DIS. M.H.DIS performed better in 1997 and 1995, while LA performed better in 1996 (in 1994 both methods provided the same classification results in terms of the total error). However, the results of the credit ris assessment model developed through LA, in the holdout sample, are inferior to the results of the M.H.DIS method. In terms of the individual error types all the credit ris assessment models developed through DA, LA and PA are biased towards higher type I error rates (high-ris firms classified into the low-ris group). The same result was also found in the case of the M.H.DIS method. However, the type I error rates of the DA, LA and PA models are consistently higher than the ones of the credit ris assessment model developed through M.H.DIS. Table 9 Credit ris assessment models developed through DA, LA and PA DA LA PA EBIT/TA NI/NW ) ) ) SALES/TA GP/TA ) ) NI/WC TD/TA ) ) ) LTD/(LTD+NW) ) QA/CL CASH/CL CL/NW ) ) ) TD/WC Constant Table 10 Error rates for the DA, LA and PA models Method Error type Training sample 1997 (%) 1996 (%) 1995 (%) 1994 (%) Holdout sample DA Type I Type II Total LA Type I Type II Total PA Type I Type II Total (%) 1996 (%) 1995 (%) 1994 (%)

20 M. Doumpos et al. / European Journal of Operational Research 138 (2002) Concluding remars and future perspectives Credit ris assessment is a complex financial problem that consists of two major aspects: the identification of firms that are liely to default on their credit and the estimation of the future benefits and losses from credit granting. This paper focused on the former issue. The objective was the development of credit ris assessment models discriminating the financially healthy firms from the financially distressed ones. Such a discrimination supports credit analysts in identifying potential defaulters, thus facilitating creditgranting decisions. The approach employed in this paper for developing credit ris assessment models is based on the MCDA approach. The M.H.DIS method was employed for this purpose. The method employs mathematical programming techniques to develop, optimally, discriminant models that have the form of a set of additive utility functions. Each utility function characterizes a set of objects (firms) belonging to the same class, thus facilitating the identification of the characteristics (variables) distinguishing each class of objects. Furthermore, this utility-based form of the discriminant models developed through the M.H.DIS method enables the consideration of non-quantifiable variables. This is of major importance to credit ris assessment, since non-financial data such as the management quality of the firms, their organization, their research and development level, the maret trend, etc., are often crucial factors in credit-granting decisions (Zopounidis, 1987). The application presented involved a large sample consisting of firms belonging to the loan portfolio of a leading Gree commercial ban. The results obtained through the application of the M.H.DIS method illustrated its ability to support the credit-granting process through the development of credit ris assessment models that discriminate financially healthy firms from financially distressed ones. The model developed provided high classification accuracy throughout the four years of the analysis ( ) in both the sample used for model development (training sample), and the sample used for model validation (holdout sample). Furthermore, the comparison with traditional statistical and econometric techniques (DA, LA and PA) has confirmed the finding that this new non-parametric approach is indeed an efficient tool that can be used by credit analysts in obtaining credit ris estimates. Of course, the implications of the M.H.DIS method are not only restricted to credit ris assessment; they also involve other financial ris management fields, including among others portfolio selection and management, credit ris assessment and financial distress prediction. Other fields such as mareting, environmental management, medicine are also within the area of possible applications of the M.H.DIS method. Its applicability in these fields is worth further exploration. References Altman, E.I., Financial ratios, discriminant analysis and the prediction of corporate banruptcy. The Journal of Finance 23, Altman, E.I., Saunders, A., Credit ris measurement: Developments over the last 20 years. Journal of Baning and Finance 21, Altman, E.I., Avery, R., Eisenbeis, R., Stiney, J., Application of classification techniques in business, baning and finance. In: Contemporary Studies in Economic and Financial Analysis, vol. 3. JAI Press, Greenwich, CT. Bardos, M., Detecting the ris of company failure at the Banque de France. Journal of Baning and Finance 22, Bergeron, M., Martel, J.M., Twarabimenye, P., The evaluation of corporate loan applications based on the MCDA. Journal of Euro-Asian Management 2 (2), Casey, M., McGee, V., Stiney, C., Discriminating between reorganized and liquidated firms in banruptcy. The Accounting Review, April, pp Courtis, J.K., Modelling a financial ratios categoric framewor. Journal of Business Finance and Accounting 5 (4), Grablowsy, B.J., Talley, W.K., Probit and discriminant factors for classifying credit applicants: A comparison. Journal of Economics and Business 33,