European Journal of Operational Research 138 (2002) 392–412 www.elsevier.com/locate/dsw
Credit risk assessment using a multicriteria hierarchical discrimination approach: A comparative analysis M. Doumpos a, K. Kosmidou a, G. Baourakis b, C. Zopounidis a,* a
Technical University of Crete, Department of Production Engineering and Management, Financial Engineering Laboratory, University Campus, 73100 Chania, Greece b Mediterranean Agronomic Institute of Chania, Department of Economic and Management Sciences, 73100 Chania, Greece
Abstract Corporate credit risk 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 seeking credit into homogeneous groups representing different levels of credit risk. 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 risk assessment model using a large sample of firms derived from the loan portfolio of a leading Greek commercial bank. A total of 1411 firms are considered in both training and holdout samples using financial information through the period 1994–1997. 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 risk assessment. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Credit risk assessment; Multicriteria decision aid; Classification; Case study
1. Introduction Credit risk 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 risk, and finally make 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.: +30-821-37236, 69551; fax: +30-821-69410, 37236. E-mail addresses:
[email protected],
[email protected] (C. Zopounidis).
0377-2217/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 1 ) 0 0 2 5 4 - 5
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of a firm (financial ratios and strategic variables) to its credit risk. According to this relationship the components of credit risk are identified, and decision models are developed to assess credit risk and the corresponding creditworthiness of firms as accurately as possible. Decisions regarding credit risk assessment concern the evaluation of the firms’ financial and non-financial characteristics in order to make ‘‘optimal’’ decisions which incorporate a tradeoff between the potential risk 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 seeking financing from banks 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 risk include the financial characteristics of firms, strategic variables of qualitative nature which affect the general operation of the firm and its relation with the market, and even macroeconomic factors (i.e., inflation, interest rates, etc.). The credit analysts have to identify the most relevant factors for credit risk 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 make a final decision. Usually, factors affecting credit risk assessment lead to conflicting results and decisions. The credit/financial analysts, when performing credit risk 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 risk assessment process has necessitated the construction of credit risk assessment models, based on the sorting approach, which can be used by financial and credit analysts both as evaluation systems of new firms seeking financing as well as screening tools of the firms which are included in the loan portfolio of a bank or a credit institution (Lane, 1972; Altman et al., 1981; Grablowsky and Talley, 1981; Srinivasan and Kim, 1987; Srinivasan and Ruparel, 1990). A comprehensive review of credit risk 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 risk assessment, based on the methodological framework 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 Greek commercial bank. 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 risk 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 risk assessment model through the
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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 a1 ; a2 ; . . . ; an , classified into q ordered classes C1 C2 Cq (C1 is preferred to C2 , C2 is preferred to C3 , etc.) is used for model development (i.e., training sample). The firms are described (evaluated) along a set of m evaluation criteria x ¼ fx1 ; x2 ; . . . ; xm g. The evaluation of a firm aj on criterion xi is denoted as xij . 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 Siskos, 1982), for ranking problems in the UTADIS method (a variant of the UTA method for sorting problems; JacquetLagreze, 1995; Zopounidis and Doumpos, 1999), as well as in the context of the ELECTRE-TRI method (Mousseau and Slowinski, 1998), a well-known outranking relations approach for addressing classification problems (Yu, 1992). The major characteristic of the M.H.DIS method during the development of credit risk 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
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proceeds progressively in the classification of the firms, starting from class C1 (lowest risk group). In the first stage, the firms found to belong to class C1 (correctly or incorrectly) are excluded from further consideration. The objective of the second stage is to identify the firms that belong to class C2 . 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 k of the aforementioned hierarchical discrimination process. The form of these utility functions is the following: 1 Uk ðxÞ ¼
m X
hki uki ðxi Þ
and
U k ðxÞ ¼
i¼1
m X
h ki u ki ðxi Þ:
i¼1
The former utility function Uk ðxÞ characterizes all firms belonging to class Ck whereas the latter utility function U k ðxÞ characterizes all firms belonging to ‘‘lower’’ (worse) classes than Ck at stage k of the hierarchical discrimination process. The corresponding marginal utility functions for each criterion xi are denoted as uki ðxi Þ and between 0 and 1, while the criterion weights hki and h ki Pum ki ðxi Þ which are Pnormalized m sum-up to 1, i.e., h ¼ 1 and h ¼ 1. The marginal utility functions uki ðxi Þ are increasing i¼1 ki i¼1 ki functions on the criterion scale for all criteria xi that are negatively related to credit risk (e.g., profitability ratios, liquidity ratios, etc.) and decreasing functions for all criteria xi that are positively related to credit risk (e.g., solvency ratios, expenses ratios). Similarly, the marginal utility functions u ki ðxi Þ are decreasing functions for all criteria xi that are negatively related to credit risk and increasing functions for all criteria xi that are positively related to credit risk. 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 Uk ðxÞ is higher than the global utility estimated according to the utility function U k ðxÞ, then the firm is assigned to class Ck . Otherwise, if the global utility of a firm according to the utility function U k ðxÞ is higher than the global utility estimated according to the utility function Uk ðxÞ, then the classification decision is not to assign the firm in class Ck . Such a case indicates that the firm should be classified into one of the classes Ckþ1 ; Ckþ2 ; . . . ; Cq (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.
2.2. 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 k 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 Ck and the lower groups Ckþ1 ; Ckþ2 ; . . . ; Cq (henceforth denoted as Ck ). 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 Uk ðxÞ ¼ interval ½0; 1 .
Pm
i¼1
uki ðxi Þ and U k ðxÞ ¼
Pm
i¼1
u ki ðxi Þ if uki ðxi Þ and u ki ðxi Þ are not normalized in the
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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: ! ! 1 X 1 X EC ¼ wk Ikj þ w k I kj ; ð1Þ Nk 8aj 2Ck N k 8aj 2C k where Ikj and I kj are 0–1 variables representing the classification status of each firm belonging to groups Ck and Ck , respectively (0 indicates correct classification, whereas 1 indicates misclassification). Nk represents the number of firms belonging to credit risk group Ck , whereas N k represents the number of firms belonging to the set of groups Ck . The weighting parameters wk and w k should be defined on the basis of the misclassification costs and the a priori default probabilities: wk ¼ pk MCk , w k ¼ p k MC k such that wk þ w k ¼ 1 and wk P 0, w k P 0, where pk and p k are the a priori probabilities that a firm belongs to the credit risk groups Ck and C k , respectively, whereas MCk and MC k are the misclassification costs
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associated with the classification errors of the forms Ck ! C k and C k ! Ck . The definition of wk and w k depends on the decision maker. In the credit risk assessment problem, usually two classes of firms are considered, i.e., the financially sound firms (class C1 ) and the firms that face financial problems (financially distressed firms; class C2 ). 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 > MC1 ). 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., p1 < p 1 ; Theodossiou et al., 1996). Therefore, setting w1 ¼ w 1 ¼ 0:5 is a reasonable choice. This specification is the one used in the application regarding credit risk assessment presented later on, in this paper. The development of a pair of utility functions that minimize the overall misclassification cost (1) (let ECmin 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 EC0 that approximates the overall misclassification cost: ! ! 1 X 1 X 0 EC ¼ wk ekj þ w k e kj : ð2Þ Nk 8aj 2Ck N k 8aj 2C k The error variables ekj and e kj are surrogates of the 0–1 error variables Ikj and I kj in (1). Both these classification errors are positive real numbers representing the magnitude of the violation of the classification rules employed during model development (xj denotes the vector consisting of the performances of the firm aj on all the evaluation criteria): n o ekj ¼ max 0; U k ðxj Þ Uk ðxj Þ ; n o ð3Þ e kj ¼ max 0; Uk ðxj Þ U k ðxj Þ : The minimization of the function EC0 is performed through the solution of the following mathematical programming problem: LP1: Minimization of the overall classification error Min EC0 subject to: m X i¼1 m X i¼1 m X i¼1
uki ðxij Þ
m X
u ki ðxij Þ þ ekj P s
8aj 2 Ck ;
ð4Þ
i¼1
u ki ðxij Þ
m X
uki ðxij Þ þ e kj P s
8aj 2 C k ;
ð5Þ
i¼1
uki ðxi Þ ¼ 0;
m X i¼1
uki ðxi Þ ¼ 1;
m X i¼1
u ki ðxi Þ ¼ 1;
m X i¼1
u ki ðxi Þ ¼ 0;
ð6Þ
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uki ðxi Þ increasing function;
ð7Þ
u ki ðxi Þ decreasing function;
ð8Þ
uki ðxi Þ P 0; u ki ðxi Þ P 0; ekj P 0; e kj 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 eki and e ki . Constraint set (6) is used to normalize the results of global utilities functions in the interval ½0; 1 . In these constraints xi and xi represent the least preferred value and the most preferred one for the criterion xi . Solving LP1 yields an initial pair of utility functions that minimize the total classification error function EC0 (let EC0min denote the minimum total classification error obtained after solving LP1). If these utility functions classify correctly all firms, then the error variables eki and e ki will all be zero. Therefore, EC0min ¼ ECmin ¼ 0. However, this is not always the case. Usually, EC0min 6¼ 0 and consequently ECmin 6¼ 0. In such cases, bearing in mind the fact that EC0 is an approximation of EC, it becomes apparent that the utility functions corresponding to EC0min will not necessarily yield the minimum overall misclassification cost ECmin . For instance, consider that in a sample consisting of four firms classified into two groups C1 and C2 (low-risk and high-risk, respectively) the utility functions obtained after solving LP1 lead to two misclassified firms i (low-risk firm) and j (high-risk firm) with the following classification errors: e1i ¼ 0:2 and e 1j ¼ 0:1. In this case EC0min ¼ 0:075 and EC ¼ 0:5 (assuming w1 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 EC0 ¼ 0:125 > EC0min , but ECmin 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 EC0 P EC0min , 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 0 1 0 1 mis Nkmis N k X X 1 1 Min EC ¼ wk @ mis Ikj A þ w k @ mis I kj A Nk j¼1 N k j¼1 subject to: m X
uki ðxij Þ
i¼1 m X
m X
i¼1 m X
u ki ðxij Þ
i¼1 m X i¼1 m X i¼1 m X
i¼1
aj ¼ 1; 2; . . . ; Nkcor ; ð9Þ
uki ðxij Þ P s;
aj ¼
cor 1; 2; . . . ; N k ;
i¼1
uki ðxij Þ
m X
u ki ðxij Þ þ Ikj P s;
i¼1 m X
u ki ðxij Þ
aj ¼ 1; 2; . . . ; Nkmis ; ð10Þ
uki ðxij Þ þ I kj P s;
mis aj ¼ 1; 2; . . . ; N k ;
i¼1
uki ðxi Þ ¼ 0;
i¼1 m X
u ki ðxij Þ P s;
u ki ðxi Þ ¼ 1;
m X
uki ðxi Þ ¼ 1;
i¼1 m X i¼1
ð11Þ u ki ðxi Þ
¼ 0;
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399
uki ðxi Þ increasing function;
ð12Þ
u ki ðxi Þ decreasing function;
ð13Þ
uki ðxi Þ P 0; Ikj ; I kj integers: 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 Ikj and I kj are not associated to all firms, but only to the ones misclassified by LP1 (constraints (10)). The number of firms actually belonging to group Ck which are misclassified by LP1 is denoted as Nkmis , mis whereas N k denotes the number of firms actually belonging to the set of groups Ck , which are classified cor by LP1 into group Ck . Similarly, Nkcor and N k denote the number of corresponding correct classifications obtained by LP1. All these correct classifications are retained (constraints (9)). Since, in most cases, the mis number of firms misclassified by LP1 ðNkmis þ N k 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 risk assessment. Of course, it is difficult to ensure high predictability during model development. However, utility functions that clearly distinguish firms belonging to different credit risk 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. d ¼ minfd1 ; d2 g; where d1 ¼
min j¼1;2;...;Nkcor
0
0
fUk ðxj Þ U k ðxj Þg and
d2 ¼
min cor j¼1;2;...;N k
0
fU k ðxj Þ Uk ðxj Þg
0
cor (Nkcor and N k denote the number of firms belonging to groups Ck and Ck , 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: m X i¼1 m X i¼1
uki ðxij Þ
m X
u ki ðxij Þ d P s;
i¼1 m X
u ki ðxij Þ
i¼1
0
aj ¼ 1; 2; . . . ; Nkcor ; ð14Þ
uki ðxij Þ d P s;
aj ¼
cor0 1; 2; . . . ; N k ;
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M. Doumpos et al. / European Journal of Operational Research 138 (2002) 392–412 m X i¼1
uki ðxij Þ
m X
u ki ðxij Þ
m X
i¼1
i¼1
0
aj ¼ 1; 2; . . . ; Nkmis ;
i¼1
m X
m X
u ki ðxij Þ 6 0;
ð15Þ uki ðxij Þ 6 0;
aj ¼
mis0 1; 2; . . . ; N k ;
i¼1
uki ðxi Þ ¼ 0;
m X
uki ðxi Þ ¼ 1;
i¼1
m X
u ki ðxi Þ ¼ 1;
i¼1
m X
u ki ðxi Þ ¼ 0;
ð16Þ
i¼1
uki ðxi Þ increasing function;
ð17Þ
u ki ðxi Þ decreasing function;
ð18Þ
uki ðxi Þ P 0; d P 0: 0
0
mis denote the number of LP2 begins with the utility functions obtained after solving MIP. Nkmis and N k firms actually belonging to groups Ck and Ck , respectively, misclassified by MIP. LP2 seeks 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 risk assessment purposes.
2.3. An illustrative example To illustrate the functionality of the M.H.DIS method, consider a simple example consisting of four firms F1 , F2 , F3 and F4 , evaluated along two financial ratios (earnings before interest and taxes/total assets: x1 , current assets/current liabilities: x2 ). The firms are classified into two groups as healthy (group C1 ) and distressed (group C2 ). 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 U1 ðxÞ and U 1 ðxÞ. On the basis of these functions the corresponding global utilities of the firms are expressed as follows: Firm F1 : U1 ðx1 Þ ¼ u11 ð10%Þ þ u12 ð2:97Þ and U 1 ðx1 Þ ¼ u 11 ð10%Þ þ u 12 ð2:97Þ; Firm F2 : U1 ðx2 Þ ¼ u11 ð7:5%Þ þ u12 ð1:05Þ and U 1 ðx2 Þ ¼ u 11 ð7:5%Þ þ u 12 ð1:05Þ;
Table 1 Data of the illustrative example F1 F2 F3 F4
x1
x2
Group
10% 7.5% 8% 3%
2.97 1.05 0.80 1.10
C1 C1 C2 C2
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Firm F3 : U1 ðx3 Þ ¼ u11 ð8%Þ þ u12 ð0:80Þ and U 1 ðx3 Þ ¼ u 11 ð8%Þ þ u 12 ð0:80Þ; Firm F4 : U1 ðx4 Þ ¼ u11 ð3%Þ þ u12 ð1:10Þ and U 1 ðx4 Þ ¼ u 11 ð3%Þ þ u 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: Min EC0 ¼ 0:5 12ðe11 þ e12 Þ þ 0:5 12ðe 13 þ e 14 Þ () Minðe11 þ e12 þ e 13 þ e 14 Þ subject to: Firm F1 : ½u11 ð10%Þ þ u12 ð2:97Þ ½u 11 ð10%Þ þ u 12 ð2:97Þ þ e11 P 0:001; Firm F2 : ½u11 ð7:5%Þ þ u12 ð1:05Þ ½u 11 ð7:5%Þ þ u 12 ð1:05Þ þ e12 P 0:001; Firm F3 : ½u 12 ð8%Þ þ u 12 ð0:80Þ ½u11 ð8%Þ þ u12 ð0:80Þ þ e 13 P 0:001; Firm F4 : ½u 11 ð3%Þ þ u 12 ð1:10Þ ½u11 ð3%Þ þ u12 ð1:10Þ þ e 14 P 0:001; u11 ð10%Þ þ u12 ð2:97Þ ¼ 1; u11 ð3%Þ þ u12 ð0:80Þ ¼ 0; u 11 ð3%Þ þ u 12 ð0:80Þ ¼ 1; u 11 ð10%Þ þ u 12 ð2:97Þ ¼ 0; u11 ð10%Þ P u11 ð8%Þ P u11 ð7:5%Þ P u11 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; u12 ð2:97Þ P u12 ð1:10Þ P u12 ð1:05Þ P u12 ð0:80Þ P 0; u 12 ð0:80Þ P u12 ð1:05Þ P u 12 ð1:10Þ P u 12 ð2:97Þ P 0; e11 ; e12 ; 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 F1 and F2 are assigned to category C1 while firms F3 and F4 are assigned to category C2 . 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 x1 3% 7.5% 8% 10%
Criterion x2 u11
u 11
0.000 0.501 0.501 0.501
0.498 0.498 0.000 0.000
0.80 1.05 1.10 2.97
Global utilities u12
u 12
0.000 0.000 0.499 0.499
0.502 0.002 0.002 0.000
F1 F2 F3 F4
U1
U 1
1.000 0.501 0.500 0.499
0.000 0.500 0.501 0.500
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M. Doumpos et al. / European Journal of Operational Research 138 (2002) 392–412
Max d subject to: Firm F1 : ½u11 ð10%Þ þ u12 ð2:97Þ ½u 11 ð10%Þ þ u 12 ð2:97Þ d P 0:001; Firm F2 : ½u11 ð7:5%Þ þ u12 ð1:05Þ ½u 11 ð7:5%Þ þ u 12 ð1:05Þ d P 0:001; Firm F3 : ½u 12 ð8%Þ þ u 12 ð0:80Þ ½u11 ð8%Þ þ u12 ð0:80Þ d P 0:001; Firm F4 : ½u 11 ð3%Þ þ u 12 ð1:10Þ ½u11 ð3%Þ þ u12 ð1:10Þ d P 0:001; u11 ð10%Þ þ u12 ð2:97Þ ¼ 1; u11 ð3%Þ þ u12 ð0:80Þ ¼ 0; u 11 ð3%Þ þ u 12 ð0:80Þ ¼ 1; u 11 ð10%Þ þ u 12 ð2:97Þ ¼ 0; u11 ð10%Þ P u11 ð8%Þ P u11 ð7:5%Þ P u11 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; u12 ð2:97Þ P u12 ð1:10Þ P u12 ð1:05Þ P u12 ð0:80Þ P 0; u 12 ð0:80Þ P u12 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: U1 ðxÞ ¼ 0:333u11 ðx1 Þ þ 0:667u12 ðx2 Þ; U 1 ðxÞ ¼ u 11 ðx1 Þ: The associated discrimination rule is to assign a firm to category C1 if U1 ðxÞ > U 1 ðxÞ and to category C2 otherwise. Table 3 Solution of the problem LP2 for the data of the illustrative example Criterion x1 3% 7.5% 8% 10%
Criterion x2 u11
u 11
0.000 0.000 0.000 0.333
1.000 0.333 0.333 0.000
0.80 1.05 1.10 2.97
Global utilities u12
u 12
0.000 0.667 0.667 0.667
0.000 0.000 0.000 0.000
F1 F2 F3 F4
U1
U 1
1.000 0.667 0.000 0.667
0.000 0.333 0.333 1.000
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3. Data and preliminary findings 3.1. Data The data used in this article are derived from the loan portfolio of the Commercial Bank of Greece, one of the leading Greek commercial banks. Overall 1411 firms are considered from different business sectors. These firms are included in two data sets. The first one was provided by the bank for model development purposes (training sample). It consists of the financial data of 200 firms over the period 1994–1997. On the basis of the latest information available for these firms (year 1997), the credit officers of the bank assigned half of them as firms of high credit risk. The remaining 100 firms of the training sample were evaluated as firms of low credit risk. Thus, the credit risk 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 risk 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 risk and 118 firms of high credit risk. On the basis of the available financial data of the firms 11 financial ratios are used as adequate measures of corporate credit risk (Table 4). The selection of these ratios has been performed with the collaboration of expert credit risk analysts from the Commercial Bank of Greece in order to consider the credit risk policy of the bank and the financial analysis approach employed in the daily practice of credit risk 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.
3.2. 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/working 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 risk. The financial ratios quick assets/current liabilities (QA/CL) and cash/current liabilities (CASH/CL) involve the liquidity of the Table 4 List of financial ratios Codification
Financial ratio
EBIT/TA NI/NW SALES/TA GP/TA NI/WC TD/TA LTD/(LTD+NW) QA/CL CASH/CL CL/NW TD/WC
Earnings before interest and taxes/total assets Net income/net worth Sales/total assets Gross profit/total assets Net income/working 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/working capital
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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 risk. The ratios total debt/total assets (TD/TA), long-term debt/ (long-term debt + net worth) [LTD/(LTD+NW)], and total debt/working 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
1994
1995
1996
1997
EBIT/TA
Healthy Distressed t-value
0.1622 )0.0274 (4.47)
0.1436 )0.1071 (5.32)
0.1210 )0.2170 (5.49)
0.1323 )1.1269 (1.60)
NI/NW
Healthy Distressed t-value
0.3641 )1.7129 (1.54)
0.4091 )0.3674 (3.82)
0.2918 )0.7507 (4.16)
)0.4748 )6.0223 (1.16)
SALES/TA
Healthy Distressed t-value
1.8241 0.9808 (3.98)
1.7096 0.8724 (4.81)
1.6401 0.7032 (4.84)
1.7316 0.4275 (8.96)
GP/TA
Healthy Distressed t-value
0.3702 0.1825 (3.16)
0.3594 0.0746 (2.74)
0.3483 0.0405 (3.58)
0.3733 )0.0578 (6.69)
NI/WC
Healthy Distressed t-value
)0.2516 )0.9522 (0.88)
)0.1559 )5.0017 (1.37)
0.8086 )0.7804 (2.99)
0.8257 )7.8608 (1.81)
TD/TA
Healthy Distressed t-value
0.4637 0.8849 ()4.32)
0.4661 1.0520 ()5.20)
0.4544 1.2447 ()5.71)
0.4215 1.7475 ()6.76)
LTD/(LTD+NW)
Healthy Distressed t-value
0.0700 0.2264 ()1.99)
0.0717 0.5982 ()1.65)
0.0759 0.4220 ()2.57)
0.09532 0.5434 ()1.97)
QA/CLa
Healthy Distressed t-value
19. 3203 4.6826 (1.17)
5. 6959 5.7405 ()0.01)
4.4356 1.5339 (2.26)
19.2478 2.0176 (2.20)
CASH/CL
Healthy Distressed t-value
15.3058 3.5088 (0.99)
2.6361 1.9544 (0.44)
2.4971 0.6874 (1.84)
7.2732 0.8536 (1.65)
CL/NW
Healthy Distressed t-value
1. 9149 27.7680 ()1.15)
2.1431 4.7687 ()1.59)
2.4161 6.6406 ()1.34)
3.3895 5.7523 ()1.18)
TD/WC
Healthy Distressed t-value
7.9579 5.9755 (0.74)
8. 9572 22.3323 ()1.27)
4.9652 6.2418 ()0.71)
5.7931 18.4063 ()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 Quick assets ¼ Current assets)Inventories.
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to generate more income to meet their obligations and repay their debt. Consequently these ratios are positively related to credit risk. 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
1994
1995
1996
1997
EBIT/TA
Healthy Distressed t-value
0.1203 )0.0599 (6.84)
0.1000 )0.0764 (3.15)
0.1031 )0.2190 (2.82)
0.1076 )0.4497 (4.67)
NI/NW
Healthy Distressed t-value
0.0709 )0.7434 (3.30)
0.2910 1.6317 ()0.59)
0.2548 )0.9098 (4.62)
0.0985 )2.4434 (5.34)
SALES/TA
Healthy Distressed t-value
1.7709 0.8963 (6.39)
1.7375 0.7544 (8.30)
1.7975 1.6273 (0.19)
1.9009 0.7102 (8.22)
GP/TA
Healthy Distressed t-value
0.3344 0.1996 (3.41)
0.3290 0.1224 (4.99)
0.3361 0.1080 (5.95)
0.3522 )0.2173 (4.64)
NI/WC
Healthy Distressed T-value
6.3668 )4.2045 (1.48)
0.0141 )0.4550 (1.29)
0.4206 )1.0242 (1.12)
0.4511 )3.9801 (2.38)
TD/TA
Healthy Distressed t-value
0.4585 0.9890 ()2.85)
0.4617 1.1535 ()3.34)
0.4397 1.4062 ()3.91)
0.4116 1.9054 ()5.07)
LTD/(LTD+NW)
Healthy Distressed t-value
0.0925 2.1945 ()1.03)
0.0960 0.5016 ()1.50)
0.0875 0.1855 ()2.57)
0.1049 0.3347 ()3.10)
QA/CL
Healthy Distressed t-value
24.6852 50.1508 ()0.53)
86.6224 249.5027 ()1.00)
28.6339 37.9408 ()0.42)
41.7127 9.6991 (2.67)
CASH/CL
Healthy Distressed t-value
12.0946 48.9676 ()0.772)
46.6567 235.8084 ()1.18)
21.2828 11.0454 (0.98)
22.1399 5.7096 (2.13)
CL/NW
Healthy Distressed t-value
2.8244 6.0123 ()2.07)
2.6815 23.7707 ()1.16)
2.7088 6.0325 ()1.74)
3.0989 9.7248 ()2.97)
TD/WC
Healthy Distressed t-value
23.3397 37.0955 ()0.49)
8.2275 10.0202 ()0.54)
8.6572 6.7867 (0.93)
5.2319 9.2790 ()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.
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significant throughout all years. EBIT/TA is significant at the 5% level during the years 1994–1996, while the ratios net income/net worth (NI/NW), net income/working capital (NI/WC) and quick 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 alike with the training sample is also significant in the holdout sample with the exception of year 1996. 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 risk 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 EC0 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 risk assessment purposes. The utility function U1 ðxÞ characterizes the firms of low credit risk, whereas the utility function U 1 ðxÞ characterizes the high-risk 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-risk firms and the solid line corresponds to the function developed for the high-risk 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
h1i (%)
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
16.46 1.98 13.91 19.85 7.33 8.07 1.04 8.26 1.96 19.14 1.99
32.48 1.98 20.46 3.35 10.73 1.99 1.04 19.22 1.96 4.80 1.99
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Fig. 3. Marginal utility functions of the financial ratios in the credit risk 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
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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-risk 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-risk 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 risk. On the contrary, although low values of the liquidity ratio QA/CL can be considered a significant indication of high-risk firms, high values do not indicate low-risk (often high liquidity indicates that a firm does not use appropriately its available funds to improve its profitability). The credit risk 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 risk 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 risk into the low-risk group, whereas the type II error corresponds to the classification of low-risk firms into the high-risk 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 Type I error Type II error Total error
Holdout sample
1997 (%)
1996 (%)
1995 (%)
1994 (%)
1997 (%)
1996 (%)
1995 (%)
1994 (%)
0.0 1.0 0.5
15.0 16.0 15.5
28.0 13.0 20.5
32.0 14.0 23.0
3.4 9.9 6.6
20.3 17.2 18.8
22.9 19.3 21.1
28.8 19.2 24.0
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error, the a priori probability that a firm belongs to the low-risk group is considerably lower than the probability that a firm belongs to the high-risk 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 risk 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 risk from the ones of high credit risk.
5. Comparison with discriminant analysis, logit analysis and probit analysis DA can be considered as the first approach to take 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 making problems, including credit risk 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 bXi Þ that a firm belongs to the low-risk group on the basis of its performance on the financial ratios Xi . The developed LA model has the form of the cumulative logistic probability function F ða þ bXi Þ ¼ 1=ð1 þ eðaþbXi Þ Þ. Based on this probability a firm is classified as healthy or financial distressed, using a cut-off probability. Maximum likelihood estimation procedures are employed to determine the parameters a and b. The developed PA model is computed from the standardized normal cumulative distribution function Z aþbXi 1 2 F ða þ bXi Þ ¼ ez =2 dz: 1=2 ð2pÞ 1 The consideration of LA and PA in this comparative study complements the obtained results, since their advantages make them more appealing in credit risk assessment than DA. Furthermore, both methods have been widely used in the past in several applications related to credit risk assessment and financial distress prediction (Ohlson, 1980; Zavgren, 1985; Casey et al., 1986; Keasey et al., 1990; Skogsvik, 1990). DA, LA and PA are applied following the same methodology used for the development of credit risk assessment model through the M.H.DIS method. More specifically, the year 1997 is used for model development purposes. The application of the credit risk 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
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for the LA model and 0.245 for the PA credit risk assessment model. Table 9 presents the credit risk 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 risk 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 risk 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 risk assessment models developed through DA, LA and PA are biased towards higher type I error rates (high-risk firms classified into the low-risk 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 risk assessment model developed through M.H.DIS.
Table 9 Credit risk assessment models developed through DA, LA and PA DA EBIT/TA NI/NW SALES/TA GP/TA NI/WC TD/TA LTD/(LTD+NW) QA/CL CASH/CL CL/NW TD/WC Constant
LA
0.6118 )0.1381 0.7604 0.3609 0.0381 )0.3398 )0.0560 0.0050 0.0006 )0.0596 0.0009 0.2952
PA
43.5548 )0.2557 10.9729 )15.3934 3.8190 )11.5096 1.9235 0.0448 0.3234 )0.0230 0.1365 1.7478
24.8027 )0.1367 6.2990 )8.7871 2.1691 )6.6230 1.1165 0.0246 0.1875 )0.0088 0.0778 0.9466
Table 10 Error rates for the DA, LA and PA models Error type
Training sample 1997 (%)
1996 (%)
1995 (%)
1994 (%)
1997 (%)
1996 (%)
1995 (%)
1994 (%)
DA
Type I Type II Total
7.0 4.0 5.5
29.0 10.0 19.5
47.0 7.0 27.0
49.0 6.0 27.5
18.6 11.1 14.9
39.8 13.0 26.4
39.0 13.4 26.2
48.3 12.9 30.6
LA
Type I Type II Total
2.0 2.0 2.0
22.0 8.0 15.0
37.0 10.0 23.5
35.0 11.0 23.0
6.8 9.6 8.2
28.0 13.6 20.8
32.2 13.2 22.7
39.8 12.6 26.2
PA
Type I Type II Total
2.0 2.0 2.0
22.0 8.0 15.5
37.0 10.0 23.5
36.0 11.0 23.5
8.5 9.4 8.9
28.8 13.5 21.2
34.7 13.0 23.9
40.7 12.7 26.7
Method
Holdout sample
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6. Concluding remarks and future perspectives Credit risk assessment is a complex financial problem that consists of two major aspects: the identification of firms that are likely 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 risk 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 risk 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 risk assessment, since non-financial data such as the management quality of the firms, their organization, their research and development level, the market 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 Greek commercial bank. 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 risk 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 (1994–1997) 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 risk estimates. Of course, the implications of the M.H.DIS method are not only restricted to credit risk assessment; they also involve other financial risk management fields, including among others portfolio selection and management, credit risk assessment and financial distress prediction. Other fields such as marketing, 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., 1968. Financial ratios, discriminant analysis and the prediction of corporate bankruptcy. The Journal of Finance 23, 589–609. Altman, E.I., Saunders, A., 1998. Credit risk measurement: Developments over the last 20 years. Journal of Banking and Finance 21, 1721–1742. Altman, E.I., Avery, R., Eisenbeis, R., Stinkey, J., 1981. Application of classification techniques in business, banking and finance. In: Contemporary Studies in Economic and Financial Analysis, vol. 3. JAI Press, Greenwich, CT. Bardos, M., 1998. Detecting the risk of company failure at the Banque de France. Journal of Banking and Finance 22, 1405–1419. Bergeron, M., Martel, J.M., Twarabimenye, P., 1996. The evaluation of corporate loan applications based on the MCDA. Journal of Euro-Asian Management 2 (2), 16–46. Casey, M., McGee, V., Stinkey, C., 1986. Discriminating between reorganized and liquidated firms in bankruptcy. The Accounting Review, April, pp. 249–262. Courtis, J.K., 1978. Modelling a financial ratios categoric framework. Journal of Business Finance and Accounting 5 (4), 371–386. Grablowsky, B.J., Talley, W.K., 1981. Probit and discriminant factors for classifying credit applicants: A comparison. Journal of Economics and Business 33, 254–261.
412
M. Doumpos et al. / European Journal of Operational Research 138 (2002) 392–412
Jacquet-Lagreze, E., 1995. An application of the UTA discriminant model for the evaluation of R&D projects. In: Pardalos, P.M., Siskos, Y., Zopounidis, C. (Eds.), Advances in Multicriteria Analysis, Kluwer Academic Publishers, Dordrecht, pp. 203–211. Jacquet-Lagreze, E., Siskos, Y., 1982. Assessing a set of additive utility functions for multicriteria decision making: The UTA method. European Journal of Operational Research 10, 151–164. Keasey, K., McGuinness, P., Short, H., 1990. Multilogit approach to predicting corporate failure: Further analysis and the issue of signal consistency. Omega 18 (1), 85–94. Lane, S., 1972. Submarginal credit risk classification. Journal of Financial and Qualitative Analysis 7, 1379–1386. Mousseau, V., Slowinski, R., 1998. Inferring an ELECTRE-TRI model from assignment examples. Journal of Global Optimization 12 (2), 157–174. Ohlson, J.A., 1980. Financial ratios and the probabilistic prediction of bankruptcy. Journal of Accounting Research 18, 109–131. Skogsvik, K., 1990. Current cost accounting ratios as predictors of business failure: The Swedish case. Journal of Business Finance and Accounting 17 (1), 137–160. Srinivasan, V., Kim, Y.H., 1987. Credit granting: A comparative analysis of classification procedures. The Journal of Finance XLII/3, 665–683. Srinivasan, V., Ruparel, B., 1990. CGX: An expert support system for credit granting. European Journal of Operational Research 45, 293–308. Theodossiou, P., Kahya, E., Saidi, R., Philippatos, G., 1996. Financial distress and corporate acquisitions: Further empirical evidence. Journal of Business Finance and Accounting 23 (5–6), 699–719. Yu, W., 1992. ELECTRE TRI: Aspects methodologiques et manuel d’utilization. Document du Lamsade, no 74, Universite de Paris, Dauphine. Zavgren, C.V., 1985. Assessing the vulnerability to failure of American industrial firms: A logistic analysis. Journal of Business Finance and Accounting 12 (1), 19–45. Zopounidis, C., 1987. A multicriteria decision making methodology for the evaluation of the risk of failure and an application. Foundations of Control Engineering 12 (1), 45–67. Zopounidis, C., Doumpos, M., 1999. A multicriteria decision aid methodology for sorting decision problems: The case of financial distress. Computational Economics 14 (3), 197–218. Zopounidis, C., Doumpos, M., 2000. Building additive utilities for multi-group hierarchical discrimination: The M.H.DIS method. Optimization Methods and Software 14 (3), 219–240.
