Firm credit risk evaluation: a series two-stage DEA modeling framework

Abstract

This paper documents a new series two-stage DEA modeling framework for credit risk evaluation in terms of operating performance efficiency and effectiveness that is implemented to a sample of listed Greek firms of basic resources and chemicals sector. In the series stages two types of DEA metrics are used: The first type is based on the range adjusted measure (RAM) whereas the second type is based on a common set of weights (CSW) of RAM. Performance inefficiency is uncovered in both performance dimensions, but the real problem of inefficiency of the sampled firms is a lower level of effectiveness, rather than operating performance efficiency. The operating efficiency is not correlated with effectiveness, and thus it seems that there is not a link between the performance at the operational (cost-oriented) and financial (profit-oriented) spaces of the firm. Therefore, sample firms should give more emphasis on their profit-oriented policies to ensure their success in the industry. The research framework may benefit not only Greek listed firms, but also firms in other countries to quantify their performance and improve their competitive advantages.

Appendices

Appendix 1

DEA efficiency scores of DMUs are derived by using different sets of weights obtained during the optimization process. For finding CSW the following model (3) based on fractional programming is used. This model is formulated to simultaneously maximize the ratio of outputs to inputs for all DMUs. For the use of fractional programming to derive DEA efficiency scores, the interested reader is referred to Cooper et al. (2007a, b) and Tsai et al. (2006).

$$ \begin{gathered} {\text{Max}}\,\left\{ {\frac{{\sum\nolimits_{r = 1}^{k} {\mu_{r} y_{r1} + \omega } }}{{\sum\nolimits_{i = 1}^{m} {v_{i} x_{1} } }}, \ldots,\;\frac{{\sum\nolimits_{r = 1}^{k} {\mu_{r} y_{rk} + \omega } }}{{\sum\nolimits_{i = 1}^{m} {v_{i} x_{im} \,} }}} \right\} \hfill \\ s.t \hfill \\ \frac{{\sum\nolimits_{r = 1}^{k} {\mu_{r} y_{rj} + \omega } }}{{\sum\nolimits_{i = 1}^{m} {v_{i} x_{ij} } }}\, \le \,1,\quad j = 1,2, \ldots ,n \hfill \\ - \mu_{r} \le - 1/R_{r}^{ + } (m + k),\quad r = 1,2, \ldots ,k \hfill \\ - v_{i} \le - 1/R_{i}^{ - } (m + k),\quad i = 1,2, \ldots ,m \hfill \\ \omega \;{\text{free}}\;{\text{on}}\;{\text{sign}} \hfill \\ \end{gathered} $$

(3)

For solving this model, the goal-programming formulation (model 4) based on the L^∞ norm is used. This formulation takes into account the efficiency ratio of all DMUs to calculate and find a CSW so that the efficiency ratio of all DMUs becomes better as the ratio gets larger (Tsai et al. 2006).

$$ \begin{gathered} {\text{Max}}\,\left( {{\text{Min}}\,\left\{ {\frac{{\sum\nolimits_{r = 1}^{k} {\mu_{r} y_{r1} + \omega } }}{{\sum\nolimits_{i = 1}^{m} {v_{i} x_{1} } }}, \ldots ,\;\frac{{\sum\nolimits_{r = 1}^{k} {\mu_{r} y_{rk} + \omega } }}{{\sum\nolimits_{i = 1}^{m} {v_{i} x_{im} } }}} \right\}} \right) \hfill \\ s.t \hfill \\ \frac{{\sum\nolimits_{r = 1}^{k} {\mu_{r} y_{rj} + \omega } }}{{\sum\nolimits_{i = 1}^{m} {v_{i} x_{ij} } }}\, \le 1,\quad j = 1,2, \ldots ,n \hfill \\ - \mu_{r} \le - 1/R_{r}^{ + } (m + k),\quad r = 1,2, \ldots ,k \hfill \\ - v_{i} \le - 1/R_{i}^{ - } (m + k),\quad i = 1,2, \ldots ,m \hfill \\ \omega \;{\text{free}}\;{\text{on}}\;{\text{sign}} \hfill \\ \end{gathered} $$

(4)

By introducing a positive goal achievement variable, z, model (4) is converted to the following model (Jahanshahloo et al. 2005; Tsai et al. 2006):

$$ \begin{gathered} {\text{Max}}\quad z \hfill \\ s.t. \hfill \\ \sum\limits_{r = 1}^{k} {\mu_{r} y_{rj} } \, - \,\sum\limits_{i = 1}^{m} {v_{i} x_{ij} + \omega \le 0,} \quad j = 1,2, \ldots \; ,n \hfill \\ \sum\limits_{r = 1}^{k} {\mu_{r} y_{rj} } \, - \,z\sum\limits_{i = 1}^{m} {v_{i} x_{ij} + \omega \ge 0} ,\quad j = 1,2, \ldots ,n \hfill \\ - \mu_{r} \le - 1/R_{r}^{ + } (m + k),\quad r = 1,2, \ldots ,k \hfill \\ - v_{i} \le - 1/R_{i}^{ - } (m + k),\quad i = 1,2, \ldots ,m \hfill \\ \omega \;{\text{free}}\;{\text{on}}\;{\text{sign}} \hfill \\ \end{gathered} $$

(5)

Model (5) is identical to model (2). A set of (μ _r *, v _i *), i.e., CSW, can be calculated according to Eq. (6) and the efficiency score p _j of a DMU can be calculated with the CSW.

$$ p_{j} = \frac{{\sum\nolimits_{r = 1}^{k} {\mu_{r}^{*} y_{rj} + \omega } }}{{\sum\nolimits_{i = 1}^{m} {v_{i}^{*} x_{ij} } }},\quad j = 1,2, \ldots ,n $$

(6)

In case that the Eq. (6) using CSW does not provide a complete ranking of DMUs, an alternative procedure by omitting the corresponding constraints of efficient DMUs can be employed (Jahanshahloo et al. 2005).

Appendix 2

The sampled listed firms are presented in the following Table 4.

Table 4 Sample firms