Creating a new bankruptcy prediction model: The

Crafting Global Competitive Economies: 2020 Vision Strategic Planning & Smart Implementation

Creating a new bankruptcy prediction model: The grey zone problem Michal Karas, Brno University of Technology Faculty of Business and Management, Brno, Czech Republic, [email protected] Mária Rež áková, Brno University of Technology Faculty of Business and Management, Brno, Czech Republic,[email protected]

Bankruptcy prediction models based on the method of linear discriminant analysis (“indexes”) represent a widely used tool for analyzing corporate failures or evaluating a company’s financial health. However, such models are ineffective in alternative economic environments or industries. This is one of the reasons for the necessity of adjusting existing models or creating better new models. There are many papers on creating bankruptcy prediction models, but many of these papers deal with the problem of finding a better set of predictors or finding a better classification algorithm, while limited attention is paid to the problem of setting optimal cut-off scores or grey zone borders. As the grey zone problem can be viewed as a trade-off between a model’s accuracy and the number of companies that remain unevaluated, it represents a factor influencing model accuracy and effectiveness. In this paper we aim to suggest a criterion that can be used for deriving a grey zone that on one hand maximizes model accuracy, while on the other hand minimizes the number of unevaluated companies. : Bankruptcy model, financial ratios, the prediction accuracy of bankruptcy models.

Beaver (1966) was the first to show how financial ratios could be used for predicting a company’s failure (i.e. bankruptcy). Altman (1968) followed on from his work and created the first multivariate bankruptcy prediction model. Altman’s model was based on the method of linear discriminant analysis (LDA). In following years, many other researchers created similar models by applying a variety of different approaches. We might mention models based on logit or probit approaches (Martin, 1977; Ohlson, 1980; Zmijewski, 1984), on the Cox’s hazard model (Henerby, 1996; Schumway, 2001) or on nonparametric methods such as neural networks, multi-adaptive regression splines, support vector machines and many others (see Back, Laitinen, Sere, 1996; Kim, Kang, 2010; De Andres et al., 2011; Chen, Hsiao, 2008). When speaking about the methods that can be used, one should be aware that there is a trade-off between method accuracy and the interpretability of its outcomes (see James et al., 2013). The advantage of models based on LDA is that it is easy to interpret their outcomes. This is possibly why the LDA method still remains the most widely used method (see Aziz, Dar, 2006) and why previous models based on LDA (such as Altman’s) are still widely used and discussed. Another frequently discussed question relating to bankruptcy prediction models is whether these models can be effectively applied to different economic environments or time periods from that in which the learning data were observed. 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 industry, period or economic environment than that from which the data on which they were developed was taken. Such arguments motivate efforts aimed at creating new bankruptcy prediction models. In this context, Thomas Ng, Wong and Zhang (2011) point out the need of creating models for branches such as construction, as the existing models are inappropriate

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for this industry. In general, Kapli ski (2008) claims that bankruptcy prediction models should be adjusted to the economic conditions of the given country or even industry. According to Kapli ski (2008), building a new model based on the LDA method (an index) entails: • Choosing a set of indicators most suitable from the viewpoint of the aim of the analysis and the reduction of potential indicators • Defining the weight of particular indicators • Setting up a synthetic indicator (an index) • Defining the critical value of the index on the basis of which it can be predicted whether an assumed occurrence will or will not be present Defining an index’s critical value is a complicated matter in practice, as mentioned by Letza, Kalupa, Kowalski (2003): “it is difficult to establish a precise cut-off value showing a clear distinction between the two samples of companies, failed and non-failed. Often, a range of overlap exists between the two samples and consequently there is some degree of flexibility on exactly where the cut-off value should rest within this grey zone”. In our opinion, limited attention is paid to the last step in creating new indexes (i.e. defining the critical value of the index) in the literature. Moreover, this step influences the overall accuracy of the index created. The aim of this paper is to present a possible approach for setting the grey zone for a new model based on LDA (an index). This suggested approach could be helpful to other authors creating their own models.

The sample under investigation is comprised of 1,236i companies in the manufacturing industry (NACE rev. 2 main section C: Manufacturing), of which 857 (69.3 %) financially healthy (active) companies and 379 (30.7 %) companies threatened with bankruptcy (bankrupt). The bankrupt companies investigated went into bankruptcy in the years 2007 to 2012. In the case of the bankrupt companies, data from a year before bankruptcy are analyzed. The data was obtained from the Amadeus database provided to the company Bureau Van Dijk.

The objective of this method is, according to (Hebák et al., 2004), "to find a linear combination of p monitored predictors, i.e. Y = bTx, where bT = [b1, b2,…, bp] is a vector of parameters that would segregate better than any other linear combination the H groups under consideration, so that its variability within the groups would be minimal and its variability between the groups maximal.” The LDA method produces a discriminatory rule (function) which according to calculated predictors assigns each company to a group of enterprises either threatened or not threatened by bankruptcy. Discriminant analysis works with the assumption of multivariate normal distribution of data. The density of probability of multivariate normal distribution of a variable x can be written as follows (Hebák et al., 2004, p. 108): f k (x ) =

1

(2π )

p/2

Σk

1/ 2

exp −

1 (x − µ k )T Σ −k1 (x − µ k ) 2

(1)

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where x is the vector of independent predictors, where x = (x1, x2, …, xp) µk is the vector of middle values of the quantity x k-th group k is the covariance matrix of the k-th group Linear Discriminant Analysis (LDA) is a special kind of discriminant analysis which adds the assumption of identical covariance matrices ( k). Under these assumptions the discriminant rule, based on the Mahalanobis distance, can be written as follows (Hebák et al., 2004):

x T Σ − 1 ( µ 1 − µ 2 ) > 1/2 ( µ 1 + µ 2 ) T Σ group 1 (e.g. active) x T Σ −1 ( µ 1 − µ 2 ) < 1 / 2 ( µ 1 + µ 2 ) T Σ group 2 (e.g. bankrupt)

−1

−1

(µ1 − µ 2 )

(2)

(µ1 − µ 2 )

(3)

where 1 or 2 is a priori the probability of units belonging to the group corresponding to the range group 1 or 2.

The suggested procedure of grey zone setting will be presented on the following model. The model was derived by using Altman’s variables, as this set of variables is considered a set of effective predictors. This model was derived by using a linear discrimination method or rather its stepwise variant. The data analysed in this paper were used as a learning sample. Details of the derivation of the model (i.e. the significance of the model’s predictors) can be found in (Karas, Rež áková, 2014). The model can be written in the following form: Model CZ = 0.6294*WC/TA+0.7436*RE/TA+6.7840*EBIT/TA-0.1524*S/TA+4.3151

(4)

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 S/TA is the ratio of sales to total assets This form of model has been derived by using a stepwise discrimination analysis. One of the results of the use of this procedure was that the E/D indicator was not included in the model in view of its low significance.

The setting of an index’s grey zone can be viewed as a trade-off between the accuracy of the model and the number of unevaluated companies (i.e. companies in the grey zone). Moreover, according to Thomas Ng, Wong, Zhang (2011): “the cost of type I and type II error rates should be considered when defining the optimal cut-off score” (i.e. the grey zone). A type I error occurs when a bankruptcy-prone company is assessed as financially stable. A type II error describes the opposite situation, i.e. perceiving a financially stable company as facing bankruptcy. According to Zhou, Elhag (2007), a type I error is 2 to 20 times more serious (and thereby more costly) than a type II error. It is important to mention that reducing one type of error means a simultaneous increase in the second type of error.

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An optimally set grey zone should on one hand lead to maximum model accuracy and on the other hand to a minimum number of companies in the grey zone. Moreover, the unequal weight of the two error types should be considered. The above-mentioned requirements may be achieved by maximising the following ratio:

A + w. B G

(5)

where A is the number of correctly evaluated active companies (i.e. financially healthy companies) B is the number of correctly evaluated bankrupt companies G is the number of unevaluated companies (i.e. companies in the grey zone) w is the cost of a type I error relative to the cost of a type II error (e.g. how many times is a type I error more costly than a type II error) In this paper we analyse the use of this criterion for three weight settings, i.e. three values of parameter w. Firstly, for the outer values of interval suggested by Zhou, Elhag (2007), i.e. for w = 2 and w = 20. Secondly, for w = 1 to analyse the impact of equally set weights for both error types. For w = 1 (i.e. a type I error is considered as equally costly as a type II error) the criterion (5) takes the form:

A+ B G

(6)

For w = 2 (i.e. a type I error is considered twice as costly as a type II error) the criterion (5) takes the form:

A + 2.B G

(7)

For w = 20 (i.e. a type I error is considered twenty times as costly as a type II error) the criterion (6) takes the form:

A + 20 . B G

(8)

The given criterion is presented in three different forms. The form (6) represents the criterion with equally weighted type I and type II errors, forms (7) and (8) represent the criterion with a type I error weighted 2 or 20 times more heavily than a type II error.

The following steps should be taken to derive the optimal setting of the index’s grey zone: 1) Analyse the index’s values corresponding to the misclassified companies 2) Divide the values from the previous step into n+1 intervals by using n quantile borders. To avoid the effect of outliers, set the first or last quantile border to a value of the 5 % or 95 % quantile, respectively 3) Use all possible pairs of quantile borders as possible grey zone borders. The number of all possible pairs is:

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n 2

=

n! n ⋅ (n − 1) = 2!⋅(n − 2 )! 2

(9)

where: n – is the number of applied quantile borders 4) Find the grey zone borders that maximise the ratios (6, 7, 8)

The descriptive statistics of values of the index of all companies and misclassified companies respectively are shown in the following Table 1. Table 1. Descriptive statistics of the values of the index – all companies vs. misclassified No. Mean Median Min. Max. Std. Dev. Skew. Kurt. Model CZ all index’s values 1236 1.8269 4.6650 -1856.2 1459.515 81.40784 -10.52 404.22 Model CZ – index’s values 871 0.8835 0.8616 -8.109 3.12654 0.49003 -9.81 155.70 of misclassified companies Source: Own analysis based on data from Amadeus database Without a setting for the grey zone, the model fails in classifying 871 companies. However, the values of the misclassified companies index ranges in a relatively narrow interval compared to the values of all companies indexes. The index values of misclassified companies were, in the following step, divided into six intervals (i.e. n = 6) by using the five quantile borders 5, 23, 41, 59, 77 and 95 %ii. The following table shows the values of the index of misclassified companies corresponding to these quantile borders. Table 2. Values of the index of misclassified companies corresponding to the values of the given quantile borders No. Quantile Corresponding index value 1 5 -1.7995 2 23 4.1855 3 41 4.6664 4 59 4.9950 5 77 5.5182 6 95 6.4487 Source: Own analysis based on data from Amadeus database As there are 6 borders, a set of 15 different pairs could be made (see equation 9), in other words there are 15 possible settings of the grey zone using these borders. The accuracy of the analysed index (see equation 4) was tested by using all of the 15 possible grey zone settings. Moreover, the number of active and bankrupt companies in the grey zone was also tested. The results of this testing are listed in the following Table 3. Settings that maximise the criterion in form (6), (7) or (8) are written in bold letters.

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Table 3. The possible grey zone settings and corresponding model accuracy GZ setting Active Bankrupt G (A) G (B) Form of the criterion No. LB UB No. % No. % No. % No. % Eq. (6) Eq. (7) Eq. (8) 1 1 2 706 82.38 46 12.14 150 17.5 201 53.03 2.14 2.27 4.61 2 2 3 541 63.13 247 65.17 165 19.25 56 14.78 3.57* 4.68 40.16 3 3 4 376 43.87 303 79.95 165 19.25 31 8.18 3.46 5.01* 59.89 4 4 5 211 24.62 334 88.13 165 19.25 20 5.28 2.95 4.75 72.86 5 5 6 46 5.37 354 93.4 165 19.25 15 3.96 2.22 4.19 80.71* 6 1 3 541 63.13 46 12.14 315 36.76 257 67.81 1.03 1.11 2.93 7 2 4 376 43.87 247 65.17 330 38.51 87 22.96 1.49 2.09 21.92 8 3 5 211 24.62 303 79.95 330 38.51 51 13.46 1.35 2.14 31.24 9 4 6 46 5.37 334 88.13 330 38.51 35 9.23 1.04 1.96 37.03 10 1 4 376 43.87 46 12.14 480 56.01 288 75.99 0.55 0.61 2.17 11 2 5 211 24.62 247 65.17 495 57.76 107 28.23 0.76 1.17 15.44 12 3 6 46 5.37 303 79.95 495 57.76 66 17.41 0.62 1.16 21.34 13 1 5 211 24.62 46 12.14 645 75.26 308 81.27 0.27 0.32 1.71 14 2 6 46 5.37 247 65.17 660 77.01 122 32.19 0.37 0.69 11.99 15 1 6 46 5.37 46 12.14 810 94.52 323 85.22 0.08 0.12 1.38 Note: LB – lower border, UB – upper border, G (A) – active companies in grey zone (i.e. unevaluated active companies), G (B) – bankrupt companies in grey zone (i.e. unevaluated bankrupt companies), * – maximal value. Source: Own analysis based on data from Amadeus database Without setting the grey zone, or rather with a grey zone setting of zero length, the model accuracy would be 99.88 % of correctly classified active companies, but only 16.09 % correctly classified bankrupt companies. By applying the grey zone concept to this model we can markedly increase the number of correctly classified bankrupt companies, but on other hand the number of correctly classified active companies will decrease. Now we will present the results of setting the grey zone with the application of the suggested criterion. By evaluating three forms of criterion we obtain three different optimal settings of the grey zone due to the given value of the w parameter (see equation 5). The optimal setting for the criterion in form (6), i.e. with equally weighted error types, is setting number 2. The resulting model accuracy is 63.13 and 65.17 % of correctly evaluated active and bankrupt companies, respectively. The number of unevaluated active and bankrupt companies, respectively, attains a value of 19.25 and 14.28 % of all analysed active and bankrupt companies. By incorporating unequal weights, the resulting optimal grey zone setting differs. If we consider a type I error 2 times more severe than a type II error, i.e. we apply the criterion of form (8), we obtain the following results. The number of correctly evaluated bankrupt companies will increase to 79.95 % and the number of correctly evaluated active companies will fall to 43.87 %. Simultaneously, the number of unevaluated bankrupt companies will fall to 8.18 % and the number of unevaluated active companies will not change. For the purpose of presenting the impact of the application of significantly different error type weights, we can use the last form criterion, i.e. form (8). The criterion in this form takes a type I error to be 20 times more severe than a type II error. Such a change results in a relatively small increase in the number of correctly evaluated bankrupt companies (from 79.95 to 93.4 %), on the other hand there is a significant decrease in the number of correctly evaluated active companies (from 43.87 to 5.37 %). The number of unevaluated bankrupt companies decreases to 3.96 %, while the number of unevaluated active companies remains unchanged. As there are three possible optimal grey zone settings according to the applied form of the criterion, it is necessary to choose the best one. As the criterion in form (6) does not reflect the unequal weight of error types and the application of the criterion in form (8) leads to a significant decrease in the number of correctly evaluated active companies, the criterion in form (7) seems to give the best

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results of all the options tested. The optimal grey zone setting is then setting number 3 (see Table 3). Borders values can be found in Table 2. The final form of the analysed model (i.e. Model CZ, see equation 4) can be written in the following form: Model CZ = 0.6294*WC/TA+0.7436*RE/TA+6.7840*EBIT/TA-0.1524*S/TA (10) Companies with a Z value < 0.3513 are evaluated as being threatened with bankruptcy iii. Model CZ > 0.6799 indicates the company is financially healthy, while the interval of values 0.3513 > Model CZ > 0.6799 means no unambiguous conclusion may be drawn, i.e. the “grey zone” of inconclusive results.

The bankruptcy prediction model or rather the index without a grey zone could classify a company only as financially healthy (active) or as a company threatened with bankruptcy. However, one can argue that there is still a long distance between these two states. The role of the application of a grey zone to the index is to fill this gap. A company in the grey zone could then be viewed on one hand as threatened by bankruptcy, but on the other hand there will probably be efforts made to reverse such a fate in the given company. From this point of view, the incorporation of a grey zone in such models is a necessity. The problem with a grey zone or any given cut-off score is that such a measure is temporally unstable (see Lukáš, 2013), or in other words the grey zone borders shift over time. A possible explanation could be: “the instability of relationships among the variables within the equation over time” (Hayes, Hodge, Hughes, 2010). We can, therefore, say that there is a constant need to adjust the model or, at the least, the need of actualising its grey zone border values. In this paper we present a possible procedure for finding optimal grey zone border values. By using our suggested criterion it is possible to simultaneously maximise the accuracy of the model and minimise the number of companies in the grey zone. Moreover, it is possible to apply different costs of type I and type II errors. The parameter w of the suggested criterion according to Zhou, Elhag (2007) ranges from 2 to 20. For practical application, it is useful to set the value of this parameter more precisely. One of the possible approaches is described in Altman et al. (1977). Altman et al. (1977) viewed the cost of a type I error as a loss of borrowed capital, while the cost of a type II error is the loss of interest from a loan not granted. All three optimal settings found by maximising the criterion have a common length of grey zone (one interval value), regardless of the w parameter setting. It is possible that by using a higher number of quantile values to divide the index’s value of misclassified companies the criterion would prefer narrower settings. An excessively narrow setting could prove to be at the expense of the ability of model generalisation (i.e. model robustness). One can argue that more sophisticated methods are available that can be used for deriving a bankruptcy prediction model, particularly methods of artificial intelligence, but there is a trade-off between method accuracy and the interpretability of its outcomes (see James et al., 2013). The advantage of models based on LDA is that it is easy to interpret theirs outcomes.

Building a new model based on LDA methods involves first choosing suitable variables and defining weights. Many authors focus on searching for a better new set of variables that would have greater power of discrimination than alternative ones or focus on finding a better classification algorithm. But there is another matter of no less importance, which is setting an optimal cut-off score or grey zone borders as there may be a thin border between financially healthy companies and a company that is about to go bankrupt. The aim of this article is to present a possible approach to deriving grey zone borders. We suggest a criterion that reflects the major requirements dealing with model effectiveness. We showed that a grey zone could be set as a compromise between model accuracy and the number of unevaluated companies, while respecting the different costs of both error types.

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i

Number of companies with complete financial statements The length of intervals is given by the following formula: (95-5)/(n-1), in this case (95-5)/5=18. The following intervals of this given length were tested: (0;5>, (5;23>, (23;41>, … , (95;0) iii The final border values of the grey zone were adjusted by using a model constant, i.e. LB = 4.6664 - 4.3151 = 0.3513 or rather UB = 4.9950 - 4.3151 = 0.6799 ii

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