**4. Results and Discussion**

During the research process presented in the paper, we created two models. One was developed based on DEA as an ADD model. As the outcome of the ADD model is an extensive set of data, we present selected data as an example in Table 2. In the case of businesses that have a score of zero and all their slacks equal to zero, it can be said that they are likely to go bankrupt. From the Table 3, it can be seen that such a situation applies to e.g., company number 122 and company number 126.


**Table 2.** Selected results of the Additive DEA model (ADD model).



The classification of businesses into individual DEA zones is shown in Table 3. In case of three financial distress zones, the risk of going bankrupt affects 55 companies. With regard to the first zone of the DEA model, it turned out that 17 companies face risk of bankruptcy. With regard to the second zone, 15 companies face such risk, and in case of the third zone, the risk is being faced by another 23 companies.

To determine a model's estimation accuracy, a cut-off point of 0.5 is usually used. However, this cut-off is not appropriate for every given model. Therefore, we were looking for the optimal cut off corresponding to a point in which the sum of sensitivity and specificity is the highest. The optimal cut-off was found at the level of 0.63. The results are shown in Table 4. In this case, 82% estimation accuracy for bankrupt businesses and 56% estimation accuracy for non-bankrupt businesses could be achieved. We can consider this to be an adequate estimation accuracy rate.


**Table 4.** Classification accuracy of the ADD model corresponding to a cut-off point of 0.63.

We also constructed a ROC curve for the ADD model (see Figure 1), where the AUC is at the level of 0.82.

**Figure 1.** ROC curve for ADD model.

We then formulated Logit model to identify businesses that are likely to go bankrupt. We selected 7 financial indicators for the Logit model. They are indicators TRTA, CR, WCTA, EBTA, LLTA, EBIE and CLTA. Table 5 shows the results of Logit model.


**Table 5.** Logit function coefficients.

Based on the Wald confidence intervals, it can be stated, with 95% confidence, that the coefficients of the variables EBTA, LLTA are within the specified limits of the interval and none of the intervals contains a value of 0, which would exclude the variable from the model. Since no Wald statistics parameter is equal to zero, it can be assumed that all explanatory variables can be included in the model. It follows that the tested variables are suitable for the Logit model. At the same time, Wald statistics determines which of the independent variables is more important than the others. A statistically significant relationship determining probability of bankruptcy was confirmed for the indicators EBTA and LLTA. The above results show that the probability of bankruptcy is determined by profit and indebtedness. The resulting Logit function providing the probability of business bankruptcy is:

$$P\_1 = \frac{1}{1 + e^{-(-2.150.49 \times TRTA + 0.01 \times \text{CR} - 1.59 \times \text{WCTA8.53 \times EBTA + 1.31 \times \text{LTAA} - 0.003 \times \text{ERIE} + 0.19 \times \text{CLTAA})}}, \tag{4}$$

To assess the estimation accuracy of the model, we constructed a Receiver Operating Characteristic (ROC) curve (see Figure 2). In our case, the AUC accounts for 79.65%, which we evaluated positively. Therefore, we can state that our model has good estimation accuracy.

**Figure 2.** Receiver Operating Characteristic (ROC) curve for the Logit model.

The Hosmer–Lemeshow test signalized good conformity of the final model with given data. The *p*-value of the test was 0.71. This value is higher than the significance level, so we accepted the null hypothesis—the distribution of predicted and achieved results is the same across all groups of businesses. According to Nagelkerke's R Square, the model explains 26.73% variability of the binary dependent variable. Total estimation accuracy of the Logit model was found to be 86.6%, for non-bankrupt businesses the result was 96%, and estimation accuracy for bankrupt businesses was 30% (see Table 6). The error type I was 70% and error type II was 4%. The model was found to have higher classification accuracy for businesses that are financially sound.

**Table 6.** Classification accuracy of the Logit model corresponding to the same cut-off as the ADD model.


A comparison of the classification ability of the models is given in Table 7 (corresponding to a cut-off point of 0.63). Several researchers compared the results of the ADD model and the Logit model. We already described the research of Premachandra et al. (2009) in the Introduction. The comparison of these two models was also performed by the Slovak researchers Mendelová and Stachová (2016, p. 103) who based on their research concluded that, in general, they cannot say that one method is better than the other one, because the accuracy and suitability of each method depends on the particular data used, its size, and its proportions. The results of the ADD model and Logit model were also compared by Araghi and Makvandi (2012). They found out that DEA is an effective tool for predicting business bankruptcy, but it is not as efficient as the Logit model—DEA achieved a weak performance in identifying bankrupt and non-bankrupt companies.

**Table 7.** Comparison of the estimation accuracy of the ADD and Logit models corresponding to optimal cut-off for the DEA model.


The ADD model was found to have a lower classification accuracy for non-bankrupt businesses corresponding to a cut-off of 0.63 and the Logit model has lower classification accuracy for bankrupt businesses at this cut-off (see Table 7). Therefore, it is necessary to state the optimal cut off for the Logit model, which is 0.16. At this point, the Logit model achieves a higher classification accuracy for bankrupt businesses.

Figure 3 shows the estimation accuracy for bankrupt and non-bankrupt businesses, as well as the percentage of total correct predictions for the DEA model. In the case of non-bankrupt businesses, the highest estimation accuracy was found at a cut-off of 1. Then the estimation accuracy decreases. In the case of bankrupt businesses, the estimation accuracy gradually increases up to 100% at a cut-off point of 0.5 and lower. The overall estimation accuracy decreases with a decreasing cut-off.

**Figure 3.** Percentage of correct predictions using the DEA model. Legend: BR—Estimation Accuracy for Bankrupt Businesses. NBR—Estimation Accuracy for Non-Bankrupt Businesses, O—Overall Estimation Accuracy.

Figure 4 illustrates the estimation accuracy for bankrupt and non-bankrupt businesses, as well as the percentage of total correct predictions for the Logit model.

**Figure 4.** Percentage of correct predictions using the Logit model.

Figure 4 shows that estimation accuracy in the case of the Logit model is different in comparison with the results of the DEA model. In the case of non-bankrupt businesses, the highest estimation accuracy is from a cut-off point of 1 to cut-off point of 0.4; from a cut-off of 0.3, the estimation accuracy decreases rapidly. In the case of bankrupt businesses, the estimation accuracy gradually increases up to 100% at a cut-off point of 0. The overall estimation accuracy slightly increases up to a cut-off of 0.3 and then rapidly decreases. At the end of this discussion, it is necessary to point out the fact that each of the models has its optimal cut-off and the estimation accuracy of the models is given by the selected value of each cut-off.

#### **5. Conclusions**

In this paper, we created specific bankruptcy prediction models for the analysed sample of businesses with the use of the DEA method and Logit model. Inspired by authors who dealt with the causes of bankruptcy and based on the correlation matrix, we selected financial indicators as inputs and outputs for the constructed models. Based on the scientific literature, we also identified the bankruptcy condition of indebtedness, which was necessary to classify businesses into bankrupt and non-bankrupt businesses. This condition was applied in the case of the Logit model and together with the profit, it was verified by this model as a symptom of bankruptcy. Based on the results outlined in the Results and Discussion section, we can say that the ADD model achieved an estimation accuracy for bankrupt businesses of 82%. A similar estimation accuracy rate for businesses threatened with bankruptcy was presented in the work of Premachandra et al. (2009) of 84.89%; Mendelová and Stachová's (2016) accuracy rate was 10–42.86% and Cielen et al.'s (2004) accuracy rate was 74.4–75.7%. We obtained an estimation accuracy of the ADD model for non-bankrupt businesses of 56%, and an overall estimation accuracy of DEA model of 59%. We can compare this result with the outcomes of the above-mentioned authors: Premachandra et al. (2009) obtained 75–77%, Mendelová and Stachová (2016) obtained 88–95%, and Cielen et al. (2004) obtained 85.1–86.4%. The error type I for the ADD model was 18% and the error type II was 44%.

Several studies (Premachandra et al. 2011; Paradi et al. 2014) confirm that the traditional cut-off point of 0.5 may not be appropriate for assessing bankruptcy models' estimation accuracy. This was also confirmed in our research. The optimal cut off for DEA model was found to be 0.63, and we compared the results of the DEA model with the results of theLogit model at this cut-off. Overall estimation accuracy and estimation accuracy for non-bankrupt businesses was higher in the case of the Logit model. On the other hand, DEA has a higher estimation accuracy for bankrupt businesses. Also error type I was lower in the case of the DEA model. The optimal cut-off for Logit model was different, corresponding to 0.16. It should be noted that estimation accuracy of the models depends on their cut-off values. Both models have different optimal cut-offs, so the results cannot be clearly compared. DEA identifies fewer businesses at risk of bankruptcy, but at a higher probability of achieving bankruptcy. Logit identifies more businesses at risk of bankruptcy, but with a lower probability of the identified businesses achieving bankruptcy. This fact may also speak in favour of the application of DEA model in predicting the financial distress of businesses. In this paper, the optimal cut-off was set as the value at which the sum of sensitivity and specificity is the highest. Another way of determining an optimal cut-off is to calculate an index based on two DEA models, one representing the financial health frontier and the other representing the financial distress frontier.

Results of the constructed models can be a starting point to improve financial health, prosperity, and competitiveness of analysed businesses. Based on the achieved results, we can conclude that within our research sample, DEA identifies bankrupt businesses at a higher probability of bankruptcy than the Logit model. The DEA method does not take into account initial bankruptcy conditions but its results are based on the achieved values of financial indicators, so they are independent of any assumptions. A significant benefit of this method is that it allows us to accept the specifics of companies and industry. In contrast to the Logit model, it offers us goal values of indicators, which the Logit model does not offer. Based on the above-mentioned factors, we can conclude that the DEA method is a suitable alternative for predicting the failure of businesses from the analysed sample. In order to increase estimation accuracy of the DEA model and decrease type I and II errors for this model, in our further research we will focus on selecting explanatory variables from a wider range of financial and even non-financial indicators.

A limitation of our research is the sample of businesses used, which consisted of a limited number of companies and insufficient data; therefore, we will improve our sample and data to overcome these shortcomings in the future. However, it is important to note that the research sample consisted of real businesses and it took into account all businesses in the Slovak heat sector. Therefore, we can say that results will be beneficial for that industry.

**Author Contributions:** All authors contributed to all aspects of this work. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** This paper was prepared within the grant scheme VEGA No. 1/0741/20 (The application of variant methods in detecting symptoms of possible bankruptcy of Slovak businesses in order to ensure their sustainable development).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

Achim, Monica Violeta, Codruta Mare, and Sorin Nicolae Borlea. 2012. A statistical model of financial risk bankruptcy applied for Romanian manufacturing industry. *Procedia Economics and Finance* 3: 132–7. [CrossRef]


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