An Analysis of the Robustness of Bankruptcy Prediction Models Industrial Concerns in the Czech Republic in the Years

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1 988 Vision 2020: Sustainable Growth, Economic Development, and Global Competitiveness An Analysis of the Robustness of Bankruptcy Prediction Models Industrial Concerns in the Czech Republic in the Years Michal Karas, Brno University of Technology Faculty of Business and Management, Brno, Czech Republic, Mária Režňáková, Brno University of Technology Faculty of Business and Management, Brno, Czech Abstract The development of a bankruptcy prediction model, i.e. a model capable of identifying in advance companies threatened by bankruptcy with a high degree of accuracy, is a difficult process in view of the absence of data on companies that have gone into bankruptcy. This leads to the use of models that have already been developed, for which their authors have declared a high degree of prediction accuracy. The subject of the research presented in this paper is the testing of the accuracy of such models in a period or environment other than that for which they were designed. The ability of these models to differentiate between companies threatened by bankruptcy and prospering companies and their identification error, i.e. indicating companies that go into bankruptcy as prospering companies and vice versa, were tested. Testing was performed on data on companies in the manufacturing industry operating in the Czech Republic in the years with the use of three models. The first of the models used is the Altman model, which originated in a different environment and in a different period of time to the tested companies. The second model tested was the Neumaier model, which originated in a similar environment, though at a different time. The last of the models tested is a model developed by the authors of this paper, which originated on the basis of data from the same field, though in a different period. The use of these models over a long time period also enabled testing of the predication capability of the individual models in advance, i.e. one to nine years before bankruptcy. The results of this research clearly demonstrate that the accuracy of such models falls markedly when used in a different environment and different time period. The best results were obtained by the use of a model created with data on the same field, though in a different period of time. Keywords: Bankruptcy model, financial ratios, the prediction accuracy of bankruptcy models. Introduction The publications by William Beaver (1966) and Edward Altman (1968) are pioneering works in this field. Many other authors (Deakin, 1972; Altman, 1977; Ohlson, 1980; Zmijewski, 1984; Shumway, 2001, and others) have followed on from their conclusions in subsequent years. The model produced by Altman, in particular, is still considered appropriate for predicting the threat of bankruptcy to this day by a number of authors. The use of bankruptcy models in economic environments or in time periods different from those for which they were designed raises the question as to whether such models can simply be transferred from their original economic environment to a different environment (e.g. at a different stage of economic development, with a differing performance of the economy, etc.) or for how long such models retain their prediction capability. Authors such as Platt and Platt (1990), Grice and Dugan (2001), Niemann et al (2008) and Wu, Gaunt and Gray (2010), have pointed out this problem and indicated that the predication accuracy of bankruptcy models (their ability to differentiate correctly between a company threatened by bankruptcy and a prospering company) falls markedly when they are applied to a different branch, period or economic environment than that from which the data on which they were developed was taken. The aim of this paper is to test the robustness of a model in time and in a different environment over a longer time interval. The robustness of a bankruptcy model is understood here as the possibility of

2 Vision 2020: Sustainable Growth, Economic Development, and Global Competitiveness 989 applying the given model in a moment in time or environment different to that in which it was created without any significant loss of accuracy. The accuracy of a total of three bankruptcy models was tested for this purpose. The first of these models is the Bankruptcy Index model that resulted from our previous piece of research (Karas and Režňáková, 2013). This model was derived in the Czech environment. The second is the IN05 model (see Neumaierová and Neumaier, 2005) which was also derived for the Czech environment, though on the basis of data from a different period to the preceding model. A comparison of these two models allows us to demonstrate robustness in time. The third model is the Altman model, specifically the version designed for unquoted companies (see Altman, 2000). A comparison of the first two models with this model allows us to analyse robustness in time and environment. The subject of this research is also the ability to identify a company threatened with bankruptcy (hereafter a bankrupt company) a number of years before it goes into bankruptcy. All the three given models were created by the discriminant analysis method, with the model parameters being financial indicators on the companies a year before bankruptcy. It is, however, desirable to identify the risk of impending company bankruptcy as early as possible, i.e. several years before bankruptcy, for which reason the accuracy of the models will also be tested a number of time intervals before bankruptcy. Sample and Methods The sample under investigation is comprised of 2,659 companies in the manufacturing industry (NACE rev. 2 main section C: Manufacturing), of which 2,092 financially healthy (active) companies and 567 companies threatened with bankruptcy (bankrupt). The bankrupt companies investigated went into bankruptcy in the years 2008 to 2013, although data on these companies was monitored over 9 time intervals. In the case of the bankrupt companies, the first interval studied is the year before bankruptcy, which is referred to as period t+1. The period studied is then the period between the years 1999 and The data was obtained from the Amadeus database provided to the company Bureau Van Dijk. Bankruptcy Index (BI) This model was developed for companies in the manufacturing industry in the Czech Republic on the basis of data on the years It works by combining linear discriminant analysis and Box- Cox transformation of variables (see Box and Cox, 1964). The model was originally designed for an application in the currency CZK (see Karas and Režňáková, 2013), though its coefficients were later modified for indicators defined in EUR (see Karas and Režňáková, 2014). The currency of the indicators is important for the model because the model contains one absolute variable. The model in its original form (for CZK values), which is tested in this piece of research, is as follows (see Karas and Režňáková, 2013): X X BI X where X 1 is the total assets turnover ratio (ratio of sales to total assets), X 2 is the ratio of quick assets (current assets minus inventories) to sales, X 3 is the value of total assets [CZK]. 3 (1) A company is evaluated by the model as bankrupt if the index < , otherwise it is evaluated as active. The model achieved an average total accuracy of % on the original sample (data from the Czech Republic from the years 2007 to 2010).

3 990 Vision 2020: Sustainable Growth, Economic Development, and Global Competitiveness The IN05 Model The IN05 model represents a combination of a bankruptcy and solvency model, or a model capable of assessing whether a company creates value for its owners. Data on a total of 1,526 industrial companies for the year 2004 was used in its creation. The model can be written in the following form (see Neumaier and Neumaierová, 2005): IN05 = 0.13*TA/TL+0.04*EBIT/I+3.97*EBIT/TA+0.21*OR/TA+0.09*CA/CL (2) where TL/TA represents the ratio total liabilities to total assets, EBIT/I the ratio of earnings before interest and taxes to interest, EBIT/TA the ratio of earnings before interest and taxes to total assets, OR/TA the ratio of operating revenue to total assets, CA/CL the ratio of current assets to current liabilities. Because the indicator EBIT/I often attains extreme values during practical application in view of the low value of the denominator, the authors recommend limiting its value to a maximum of 9 (Neumaier, Neumaierová, 2005). IN05 model values are interpreted as follows. At IN05 values < 0.9 the company does not create value for its owners or destroys value, at IN05 values > 1.6 the company creates new value for its owners, and at values falling within a range of 1.6 > IN05 > 0.9 no definitive conclusive can be determined (the grey zone ). At the time at which the model was created, its authors summarised its predication ability as follows (Neumaier and Neumaierová, 2005): If the index value for a given company falls beneath the lower limit, there is a 97 % probability that the company is headed for bankruptcy and a probability of 76 % that it will not create value. A company in the grey zone has a practically 50 % probability of bankruptcy and a 70 % probability of creating value. A company above the upper limit will have a 92 % probability of not going bankrupt and a 95 % probability of creating value. The Altman Model version for unquoted companies The original version of this model, known as ZETA, comes from Altman later revised the model in the year 2000 in order for it to be useable for companies whose shares are not listed on the capital market (see Altman, 2000). This modification consisted of the substitution of the ratio of the market value of equity to total liabilities with the ratio of the accounting value of equity to total liabilities and the adjustment of coefficients and the boundaries of the grey zone. According to Altman the significance of this modified predictor is slightly lower, though it still represents the model s third most important predictor as in the original version. The revised model takes the following form (see Altman, 2000): Z-score = 0.717*WC/TA+0.847*RE/TA+3.107*EBIT/TA+0.42*E/D+0.998*S/TA (3) where WC/TA is the ratio of net working capital to total assets, RE/TA is the ratio of retained earnings to total assets, EBIT/TA is the ratio of earnings before interest and taxes to total assets, E/D is the ratio of the accounting value of equity to total liabilities, S/TA is the ratio of sales to total assets. Z-score values are interpreted as follows. Companies with a Z-score value < 1.23 are evaluated as being threatened with bankruptcy. Z-score values > 2.9 indicate the company is financially healthy, while the interval of values 1.23 > Z-score > 2.9 means no unambiguous conclusion may be drawn, i.e. the grey zone of inconclusive results.

4 Vision 2020: Sustainable Growth, Economic Development, and Global Competitiveness 991 Altman (2000) states the accuracy of this model as 90.9 % correctly classified bankrupt companies and 97 % correctly classified active companies over a period of one year before bankruptcy. The accuracy of the model falls as the period before bankruptcy increases. The author of the model states that over a period of five years before bankruptcy, its accuracy is 69.8 % correctly classified bankrupt companies and 82.1 % correctly classified active companies. All three tested models are based on the method of linear discriminant analysis, which is essentially a parametric method. The conclusions of parametric methods are influenced by the presence of extreme values in the sample and by deviation from normality. The presence of extreme values in the sample can be identified by the Grubbs test (see Grubbs, 1969), which tests the null hypothesis that there are no outliers in the data sample. The Grubbs test statistic is the largest absolute deviation from the sample mean in units of the sample standard deviation and can be written as follows (Grubbs, 1969): G max Yi Y 1,2 N s i,..., (4) Where Y and s denote the sample mean and the standard deviation, respectively. Fulfilment of the condition of normality can be assessed either by testing, e.g. the Shapiro-Wilk test (see Shapiro and Wilk, 1965), or by the values of skewness and kurtosis. The selective coefficient of skewness (third moment) takes the shape: T 3 1 x t x ˆ 1 (5) T ˆ t 1 The selective coefficient of kurtosis (fourth moment) takes the shape: T 4 1 x ˆ2 t x 3 (6) T t 1 ˆ The variables used have the following meaning: T is the range of the selection, x is the arithmetic mean, ˆ is the selective standard deviation. Results The descriptive statistics of the values of variables used in the individual models quantified for the sample under examination are given in the following table. Variables for active companies are given as (A), variables for bankrupt companies as (B).

5 992 Vision 2020: Sustainable Growth, Economic Development, and Global Competitiveness Table 1, Descriptive statistics of variables of tested models Variable Mean G.Test Std. p-value Median Min. Max. Stat. dev. Skew. Kurt. CA/CL (A) CA/CL (B) E/D (A) E/D (B) EBIT/I (A) EBIT/I (B) EBIT/TA (A) EBIT/TA (B) OR/TA (A) OR/TA (B) QA/S (A) QA/S (B) RE/TA (A) RE/TA (B) S/TA (A) S/TA (B) TA (A) TA (B) TL/TA (A) TL/TA (B) WC/TA (A) WC/TA (B) Source: Own processing of data from the Amadeus database. According to the results of the Grubbs test, all the variables contain at least one extreme value. The influence of these extreme values is also clear from a comparison of the average and median values of indicators. Another characteristic shared by all the indicators is a marked deviation from a normal distribution, particularly in terms of the high degree of kurtosis of the values. This high degree of kurtosis was expressed particularly markedly in the model indicator CA/CL (A), i.e. the ratio of current assets to current liabilities in the sample of active companies, and the indicator QA/S (A), i.e. the ratio of quick assets to sales, again in the sample of active companies. Quick assets represent the difference between current assets and inventories. The IN05 model works with the indicator CA/CL, the BI model with the indicator QA/S. The following table shows the percentage of valid observations on companies for the individual models divided into active and bankrupt companies. A valid observation on a company is understood as an observation for which all the variables of the given model could be quantified. Data on active companies is considerably more readily available than data on bankrupt companies, even after a long time interval (see table 2). Table 2, Number of valid observations on companies t+1 t+2 t+3 t+4 t+5 t+6 t+7 t+8 t+9 BI (A) BI (B) IN05 (A) IN05 (B) Altman (A) Altman (B) Source: Own processing of data from the Amadeus database. There was a pronounced fall in the number of companies that could be included in the test among bankrupt companies. The proportion of tested companies fell to 1.76 %, i.e. the average number of observations is 9, in the period nine years before bankruptcy (the period t+9).

6 Vision 2020: Sustainable Growth, Economic Development, and Global Competitiveness 993 Testing the accuracy of the models From the viewpoint of assessing the accuracy of the models, a company may either be evaluated by a model correctly, the company may fall in the grey zone, or a type one or type two error may occur. A type one error represents a situation in which an active company is evaluated by the model as bankrupt. A type two error represents the opposite situation. In the case of a type one error, the creditor stands to lose potential yield of interest, while in the case of a type two error he stands to lose his entire credit commitment (unpaid debt capital plus agreed interest). For this reason a type two error is between two and twenty times more serious (i.e. costly) than a type one error (see Zhou and Elhag, 2007). Type one or type two error is quantified as the number of erroneously evaluated companies to the total number of active or bankrupt companies. The accuracy of the models (correctly identified active or bankrupt company) was quantified as the ratio of the number of correctly evaluated active or bankrupt companies to the total number of valid observations on active or bankrupt companies. The total accuracy of the models, represented by the balanced average of correct observations on active and bankrupt companies, weighted by the number of valid observations, was also evaluated. Table 3, Results of testing on the Bankruptcy Index model t+1 t+2 t+3 t+4 t+5 t+6 t+7 t+8 t+9 Min. Max. Mean Active Bankrupt Total Grey zone type I error type II error Source: Own processing of data from the Amadeus database. The original version of the BI model does not use a grey zone, so the number of unevaluated companies is therefore zero. The model can accurately identify % of active companies (77.82 % on average) in the sample tested. The number of correctly evaluated companies fell with the increasing length of time from bankruptcy, i.e. the number of years before bankruptcy. On average % of bankrupt companies were erroneously evaluated as active companies and % of active companies evaluated as companies threatened with bankruptcy. This value increased for individual periods with the increasing number of years before bankruptcy. On the other hand, the model manages to correctly identify bankrupt companies in a maximum of % of cases (74.88 % on average), with a maximum of 50 % of bankrupt companies being erroneously evaluated as active companies (25.12 % on average). The overall accuracy of the model ranged in time from to 82.7 %. The second model to be tested was the IN05 model. Table 4, Results of testing on the IN05 model t+1 t+2 t+3 t+4 t+5 t+6 t+7 t+8 t+9 Min. Max. Mean Active Bankrupt Total Grey zone (T) Grey zone (A) Grey zone (B) type I error type II error Source: Own processing of data from the Amadeus database. Note: A active, B bankrupt, T total.

7 994 Vision 2020: Sustainable Growth, Economic Development, and Global Competitiveness The IN05 model can correctly identify an active company in the tested sample in a maximum of 31.6 % of cases (26.9 % on average), with an average of % of active companies remaining unevaluated (a maximum of %) and as many as 40 % of active companies being erroneously evaluated as bankrupt companies (32.68 % on average). Within the sample of bankrupt companies, the model can correctly identify a company in a maximum of % of cases (50.50 % on average), while an average of % of bankrupt companies remain unevaluated (the maximum number of unevaluated companies is %) and as many as 40 % of bankrupt companies are erroneously evaluated as active companies (32.68 %) on average. The values for the prediction accuracy given for bankrupt companies indicate that the accuracy of the model begins to rise following an initial fall (the values in periods t+8, t+9). This apparent increase in model accuracy was caused by the low number of observations. Index values lying in the grey zone are most often shown by active companies on average % of the companies in the grey zone in the IN05 model are active companies. On average, only 4.79 % of companies in the grey zone are bankrupt companies. The total accuracy of the model in time ranges from to %. The total accuracy of the model is considerably reduced by its low accuracy in the sample of active companies, heightened by the number of observations on these companies. The third model tested is the Altman model from the year 2000 (see Altman, 2000). Table 5, Results of testing on the Altman model t+1 t+2 t+3 t+4 t+5 t+6 t+7 t+8 t+9 Min. Max. Mean Active Bankrupt Total Grey zone Grey zone (A) Grey zone (B) type I error type II error Source: Own processing of data from the Amadeus database. Altman s model can, in the tested sample, correctly identify an active company in 53.9 % of cases (51.21 % on average). As many as % of active companies remain unevaluated (40.42 % on average) and as many as % are erroneously evaluated as bankrupt companies (12.35 % on average). In the sample of bankrupt companies, the model can correctly identify a company in as many as % of cases (28.33 % on average), with as many as a third of bankrupt companies remaining unevaluated (the largest proportion of unevaluated companies is %) and as many as % being erroneously evaluated as active companies (12.35 % on average). This value is surprising in relation to the accuracy of identification of bankrupt companies, which is only %. On the other hand, the number of financially healthy companies that were evaluated as bankrupt increased significantly. This value grows unambiguously with an increasing period of time from bankruptcy, i.e. an increasing number of years before bankruptcy. Index values lying in the grey zone are again most frequently shown by active companies (on average % of companies in the grey zone in Altman s model are active companies, with only an average of 7.61 % of companies in the grey zone being bankrupt companies). The total accuracy of the model in time ranges from to %. The total accuracy of the model is, in contrast to the IN05 model, supported by accuracy in the sample of active companies combined with the number of observations on active companies.

8 Vision 2020: Sustainable Growth, Economic Development, and Global Competitiveness 995 Discussion All the models under investigation were tested on data from a period different to that in which they were derived. Altman s model was, what s more, tested in a different environment. The testing performed demonstrated that all the models function with a lower degree of accuracy than in the period and environment in which they were created. On the sample of active companies investigated, the highest level of precision was shown by the BI model, followed by Altman s model and the IN model. On the sample of bankrupt companies, the highest level of precision was shown by the BI model, followed by the IN model and Altman s model. In contrast to the original accuracy values stated by the authors of the model, the Czech BI model operates with a precision % lower on active data and 4.6 % lower on bankrupt data. Only the ability of the other Czech model (IN05) to predict bankruptcy was tested. Its accuracy for active companies (companies not threatened by bankruptcy) was shown to be % lower than its original accuracy and its accuracy with data on bankrupt companies % lower. The application of both Czech models, in the same environment though a different period, led to a significant increase in type one errors, i.e. a reduced ability to identify active companies. The increase in type one errors was smaller. A probable explanation for this is the shift in the boundaries of the grey zone in time. This theory is also supported by the fact that approximately ten times fewer bankrupt companies were found in the grey zone in the IN05 model than stated by the authors (originally 50 %, currently 4.79 %). The majority of the companies in the grey zone (i.e %) were not clearly threatened with bankruptcy, but were active companies. Since the indicator EBIT/I used in the IN05 model often attains extreme values during practical application in view of the low value of the denominator, the authors of the model recommend limiting its value to a maximum of 9 (Neumaier and Neumaierová, 2005). This recommendation was tested during the testing of the model. The purpose of restricting the given indicator is to eliminate distortion of the values of the index as a whole caused by the frequent extreme values of the indicator EBIT/I attained. During the testing of the model, a variant without restriction of the values of this indicator was investigated. The result was an increase in accuracy in the sample of active companies, for example in the year t+1 from a value of % to a value of % correctly identified companies. This remains, however, a significantly lower value for the model s identification ability than the value declared by its authors. A significant loss in accuracy was also seen for the Altman model, which in this piece of research is an example of a model created in a different environment and a different time period, though more detailed analysis reveals differences. The predication accuracy of the model fell by approximately the same percentage in the identification of active companies as for companies threatened by bankruptcy. To be precise, its ability to identify active companies is % lower and its ability to identity bankrupt companies 45.7 % lower. The accuracy of the Altman model does not exceed the level of random selection (i.e. 50 %) either in the group of active companies or in the group of bankrupt companies. In both the IN05 model and the Altman model, the vast majority of companies grouped in the grey zone were active companies. We might note that the common feature of these models is the fact that they were created in a time period different to the test period. The comparison of the accuracy of these models offers the theory that the reduction to their robustness due to the affects of time is reflected, first and foremost, in an increase in type two errors, i.e. a reduction to their ability to correctly identify bankrupt companies. Application in a different environment also results in an increase in type one errors, i.e. a reduction to their ability to identify active companies. The causes of this increase in the error rate can be seen primarily in the shift of the boundaries of the model s grey zone (see Lukáš, 2013). The question, however, remains as to why this shift occurs. According to Altman (2000) it is likely that the profile of failure changes, which leads to the necessity of updating the models.

9 996 Vision 2020: Sustainable Growth, Economic Development, and Global Competitiveness Another factor in lower model accuracy in relation to the tested data may be the characteristics of the tested data. All three tested models were created with the use of a linear discriminant analysis method, which is part of the group of parametric models. Parametric models are usually based on the assumption of a normal distribution of data. As is clear from Table 1, certain model variables show a high degree of kurtosis and extreme values, for which reason it can be justifiably assumed that the prediction accuracy of the models was also influenced by the given properties of these variables. Conclusion The testing of bankruptcy models created in different time periods and environments showed that their accuracy falls markedly when they are applied to a different period or environment. This finding is in accordance with the conclusions reached by other authors. The testing of two models created in the same environment, though in a different period, along with the testing of a model from another environment and branch, makes it possible to expand these conclusions with the finding that the reduction to model robustness as an influence of time is expressed differently to the reduction in robustness as an influence of a different environment. In the case of the use of a model merely in a different period of time, its ability to identify a financially healthy company is diminished, i.e. type one error increases. The influence of a different environment reduces the accuracy of a model in both respects, i.e. in its ability to identify both bankrupt and active (financially healthy) companies. The prediction accuracy of these models was also shown to fall over time. The BI model, which was developed on the basis of data from the same field, though in a different period of time, attains an overall prediction accuracy of 70 % nine years before bankruptcy. The accuracy of prediction of bankrupt companies was significantly diminished, though this may have been caused by the low number of observations. For the sake of completeness, we would also state that the prediction accuracy of Altman s model in the same period was 46 %, and that of the IN05 model only 28 %. References Altman, E. I., (1968) Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy, The Journal of Finance, 23, Altman, E. I., Haldeman, R. G. and Narayanan, P. (1977) ZETA Analysis. A new model to identify bankruptcy risk of corporations, Journal of Banking and Finance, 1, Altman, E. I., (2000) Predicting financial distress of companies: Revisiting the Z-score and Zeta models [online]. [Retrieved April 23, 2013] Beaver, W. H., (1966). Financial Ratios as predictors of Failure, Journal of Accounting Research 4, Box, G. E. P. and Cox, D. R. (1964) An Analysis of Transformations, Journal of the Royal Statistical Society, 26(2), Deakin, E. B. (1972) A Discriminant Analysis of Predictors of Business Failure, Journal of Accounting Research, 10, Grice, J. S. and Dugan, M. T. (2001) The limitations of bankruptcy prediction models: Some cautions for the researchers, Review of Quantitative Finance and Accounting, 17, Grubbs, F. E. (1969). Procedures for Detecting Outlying Observations in Samples, Technometrics,11(1), Karas, M., Režňáková, M. (2013) Bankruptcy prediction model of industrial enterprises in the Czech republic, International journal of mathematical models and methods in applied sciences 7, Karas, M., Režňáková, M. (2014) Možnosti využití bankrotního modelu k měření úvěrového rizika podniku, [Possibilities for the application of a bankruptcy prediction model for measuring credit risk of a company]. Proceedings of the Hradecké ekonomické dny 2014,

10 Vision 2020: Sustainable Growth, Economic Development, and Global Competitiveness 997 Lukáš, L. (2013). Some connections between bankruptcy models and their new possibilities, Proceedings of the Faktory prosperity podniků v lokálním a globálním prostředí optikou roku 2013, Neumaier, I., Neumaierová, I. (2005) Index IN 05, Proceedings of the Evropské finanční systémy Niemann, M., Schmidt, J. H. and Neukirchen, M. (2008) Improving performance of corporate rating prediction models by reducing financial ratio heterogeneity, Journal of Banking & Finance, 32, Ohlson, J. A. (1980) Financial Ratios and the Probabilistic Prediction of Bankruptcy, Journal of Accounting Research, 18, Platt, D. H. and Platt, M. B. (1990) Development of a Class of Stable Predictive Variables: The Case of Bankruptcy Prediction, Journal of Business Finance & Accounting, 17(1), Shapiro, S. S. and Wilk, M. B. (1965) An Analysis of Variance Test for Normality (Complete Samples), Biometrika, 52(3/4), Shumway, T. (2001) Forecasting Bankruptcy More Accurately: A Simple Hazard Model, Journal of Business, 74, Wu, Y., Gaunt, C. and Gray, S. (2010) A comparison of alternative bankruptcy prediction models, Journal of Contemporary Accounting & Economics, 6, Zhou, Y., Elhag, T. M. S. (2007) Apply Logit analysis in Bankruptcy Prediction Proceedings of the 7th WSEAS International Conference on Simulation, Modelling and Optimization, pp Zmijewski, M. E. (1984) Methodological issues related to the estimation of financial distress prediction models, Journal of Accounting Research, 22,