applying backpropagation neural networks to bankruptcy prediction

International Journal of Electronic Business Management, Vol. 3, No. 2, pp. 97-103 (2005)

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APPLYING BACKPROPAGATION NEURAL NETWORKS TO BANKRUPTCY PREDICTION 1

Yi-Chung Hu1* and Fang-Mei Tseng2 Department of Business Administration Chung Yuan Christian University 2 Department of International Business Yuan Ze University Chung-Li (320), Taiwan

ABSTRACT Bankruptcy prediction is an important classification problem for a business, and has become a major concern of managers. In this paper, two well-known backpropagation neural network models serving as data mining tools for classification problems are employed to perform bankruptcy forecasting: one is the backpropagation multi-layer perceptron, and the other is the radial basis function network. In particular, the radial basis function network can be treated as a fuzzy neural network. Through examining their classification generalization abilities, the empirical results from the data resources consisting of bankrupt and nonbankrupt firms in England, demonstrated that the radial basis function network outperforms the other classification methods, including the multi-layer perceptron, the multivariate discriminant analysis, and the probit method. Keywords: Bankruptcy Prediction, Neural Networks, Fuzzy Sets, Discriminant Analysis, Probit Method

1. INTRODUCTION *

Bankruptcy prediction has long been an important classification problem for a business. The empirical literature of bankruptcy prediction has recently gained more attention from financial institutions. Academics and practitioners have realized that the problem of asymmetric information between banks and firms lies at the heart of important market failures such as credit rationing, and that improvement in monitoring techniques represents a valuable alternative to any incomplete contractual arrangement aimed at reducing the borrowers’ moral hazard [3, 26, 27, 35]. The focus of this paper is to employ two data mining tools, namely backpropagation multi-layer perceptron (MLP) and the radial basis function network (RBFN), to quantitative bankruptcy prediction, since the backpropagation neural networks have played an important role [2, 13, 14, 29] for classification problems. Both MLP and RBFN are multi-layer neural networks and can be trained by the backpropagation algorithm [8, 9]. Additionally, MLP and RBFN have been also widely used in other fields, such as function approximations [4, 9, 12, 13, 20, 24] and management sciences [25, 32]. Since under some conditions, a RBFN is functionally equivalent to a *

Corresponding author: [email protected]

zero-order Sugeno fuzzy inference system [13], a RBFN can be treated as a fuzzy neural network. For examining the generalization ability of the above neural networks, the applications forecasting the corporate distress of UK companies are taken into account. Also, five attributes are considered [16, 17]: management inefficiency, capital structure, insolvency, adverse economic effects, and income volatility. In order to compare the performances obtained by the backpropagation neural networks with those obtained by the multivariate techniques, discriminant analysis, and the probit method are taken into account. In fact, discriminant analysis was the dominant method for predicting corporate failure from 1966 until the early part of the 1980s [4, 12, 20]. It gained wide popularity due to its ease of use and interpretation. Later, the probit method [36] and the logit model (logistic regression model) [1, 17, 21] have been increasing in popularity. The probit method, in particular, is commonly used in qualitative response studies. Thus, the above-mentioned four forecasting methods (i.e., MLP, RBFN, discriminant analysis, and probit method) are used to predict the corporate distress of UK companies. It is known that the backpropagation MLP and RBFN can perform nonlinear regressions. The additivity of the interaction among attributes is not assumed for MLP and RBFN, whereas this is the main difference between these two neural networks

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International Journal of Electronic Business Management, Vol. 3, No. 2 (2005)

and the above-mentioned multivariate techniques. Using computer simulations of the financial data sources extracted from DATASTREAM and FT EXTEL Company Research, we show that the RBFN outperforms the other classification methods. By comparing various forecasting methods with additive or non-additive property, the analytical results demonstrate the potential applicability or related methodological development of the RBFN, and even the nonlinear regression tools considering non-additivity of the interaction among attributes, in predicting bankruptcy.

The rest of this paper is organized as follows. The framework of MLP and RBFN are introduced in Section 2. In this section, the relationship between a RBFN and a fuzzy inference system is also described. Thorough descriptions of discriminant analysis and the probit method can be found in [22, 23]. In Section 3, the above-mentioned five forecasting methods are applied to forecasting the corporate distress of UK companies. The generalization ability of these methods is carefully examined by computer simulations. We end this paper with concluding remarks in Section 4.

Figure 1: A three-layer backpropagation MLP

2. BACKPROPAGATION NEURAL NETWORKS The feature of the backpropagation neural networks is that such networks can be trained by the backpropagation algorithm. The objective of the backpropagation algorithm is to minimize a square error measure [9], E: E=

1 m (d j − y j ) 2 ∑ 2 j =1

(1)

where dj and yj are the desired output and the actual output of the jth input training data, respectively, and m is the number of the training data. During the training process, some parameters, such as connection weights, can be adjusted by using individual update rules derived by the error measure (i.e., E) with the gradient descent method. The desired goal of a backpropagation MLP and RBFN is to minimize the error function so as to perform a function approximation. These two well-known backpropagation neural networks are described in Subsections 2.1 and 2.2, respectively. 2.1 Multi-Layer Perceptron There exists distinct relationships between any two subsystems in the real world, although we do not know exactly what these relationships are [6, 10]. In other words, an unknown mapping or relationship does exist in any two subsystems in the real world. It may be not possible to directly implement the above mapping or relationship only by a linear function

because the considered attributes are not always independent of each other. It is known that a regression tool can be used to realize a relationship between input and output variables. In order to realize the above relationship, MLP trained by the backpropagation algorithm using the gradient descent [8, 9], which is usually used as a tool of the approximation of functions like regression [24], can be taken into account. Actually, it has been shown that a backpropagation MLP with a single hidden layer and any fixed continuous sigmoidal function is sufficient to approximate any continuous function [5]. A three-layer architecture is thus taken into account in this paper. Although the numbers of hidden nodes and hidden layers can determine the approximation or classification performance, it is not easy to specify the appropriate numbers of these two parameters prior to training [9]. An example of a three-layer perceptron with n input attributes, (x1, x2, x3,…, xn), and a single output, y, is depicted in Fig. 1. It is clear that this model maps a vector, x (i.e., (x1, x2, x3,…, xn)), to y such that y = f(x). From the above viewpoints, it is feasible to apply MLP to realize an implicit relationship between the selected variables for bankruptcy prediction (i.e., management inefficiency, capital structure, insolvency, adverse economic effects, and income volatility) and bankrupt (e.g., 1.0) or nonbankrupt (e.g., 0.0) indicator for each firm. It is clear that, in our simulation, the numbers of input and output nodes are six and one, respectively. Actually, MLP is trained by the update rules to adjust its connection weights: wij(t) = wij(t−1) + ∆wij(t−1)

(2)

Y. C. Hu, and F. M Tseng: Applying Backpropagation Neural Networks to Bankruptcy Prediction where wij is the connection weight between the ith output (or hidden) node and the jth hidden (or input) node. Furthermore, ∆wij(t−1) = −ηB

∂E ∂wij

+ αB∆wij(t−2)

(3)

It can be seen that two parameters used in the update rule during the training process must be further taken into account: one is the learning rate, say ηB, and the other is the momentum parameter, say αB, which controls the influence of the last weight change on the current weight update. ηB and αB can significantly influence the speed of convergence.

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Usually, the higher the value of ηB, the faster a three-layer network learns. However, the gradient descent can oscillate widely if ηB is too large [9]. In order to make the gradient descent smooth, αB should be taken into account. Furthermore, the neural network converges to a set of weights as long as a termination condition or a stopping criterion is reached. In our simulation, pattern-by-pattern learning [13] is employed to immediately update the connection weights after each input-output pair has been presented. After tB training data have been presented to the MLP, the training procedure is terminated.

Figure 2: A structure of RBFM 2.2 Radial Basis Function Network The RBFN is a structure with locally tuned and overlapping receptive fields [11, 13]. The structure of an RBFN with n receptive field units, s hidden units and a single output, y, is depicted in Fig. 2. We can see that the architecture of an RBFN is similar to a three-layer MLP. In our simulation, there are 5 input nodes and one output node for RBFN. In fact, there are differences between MLP and RBFN. First, in a MLP, each node (i.e., hidden node and output node) has the same transfer function, such as the logistic function. In an RBFN, each hidden node has its own radial basis function, such as a Gaussian function:

Ri(x) = exp(−

x − ui 2 2σi

2

) , i = 1, 2, …, s−1

(4)

where Ri(x) is the radial basis function in the ith receptive field unit, σi is the spread width, and ui is the center of that unit and is a vector with the same dimension as x. Since the sth hidden unit is a bias node, let Rs(x) be equal to 1. Furthermore, the output node performs the weighted sum associated with each receptive field. Second, each connection has its own weight in a MLP, while in RBFN, there are no connection weights between the input layer and the hidden layer. Using backpropagation update rules, a RBFN can be trained by adjusting the aforementioned parameter specifications including ui = (ui1, ui2, ui3,…,

uin), σi, and the connection weights between the hidden layer and the output layer, where 1 ≤ i ≤ s−1 and 1 ≤ j ≤ n. When the tth training epoch is performed, wi, uij, and σi can be determined by the update rules as follows: wi(t) = wi(t−1) + ∆wi(t−1) uij(t) = uij(t−1) + ∆uij(t−1) σi(t) = σi(t−1) + ∆σi(t−1) ∆wi(t−1)=ηR1(d(t−1)−u(t−1))Ri(x)(t−1) +αR1∆wi(t−2)

(5) (6) (7) (8)

where ∆uij(t−1) =ηR2(d(t−1)− u(t−1))wi(t−1)Ri(x)(t−1)

x j (t − 1) − uij (t − 1)

+ αR2∆uij(t−2)

σi

2

∆σi(t−1)

=ηR3(d(t−1)−

x(t − 1) − u j (t − 1)

σi

3

(9)

u(t−1))wi(t−1)Ri(x)(t−1)

2

+ αR3∆σi(t−2)

(10)

Although many effective methods have been proposed, such as the supervised learning algorithms [15, 34] and the sequential training algorithms [18, 19], these methods are not the focus of this paper. In the update rules, the learning rates (i.e., ηR1, ηR2, ηR3) and the momentum parameters (i.e., αR1, αR2, αR3) in the individual update rules must be pre-specified for training an RBFN. After tR training data have been

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presented to the RBFN, the training procedure is terminated. As with MLP, the classification capability of RBFN can be influenced by the number of hidden nodes. But, this parameter is not easy to appropriately specify by the user. Previously, Takagi, Sugeno, and Kang proposed the well-known Sugeno fuzzy model (also known as the TSK fuzzy model) [28, 31] to develop a systematic approach to generating fuzzy rules from numerical data. In the Sugeno fuzzy model, the consequence is a crisp function, say f. When f is a constant, the resulting fuzzy rules and fuzzy inference system are called simplified fuzzy rules and a zero-order Sugeno fuzzy inference system, respectively. Since each radial basis function and each connection weight between the hidden layer and output layer are equal to a multidimensional composite membership function of the premise part and a consequent part of a fuzzy rule, respectively, an RBFN is functionally equivalent to a zero-order Sugeno fuzzy inference system under some conditions [13]. From the above viewpoints, a RBFN can be treated as a fuzzy neural network or an adaptive fuzzy inference system. The number of hidden nodes used in the RBFN is equal to the number of fuzzy

Aspect Theory Based I/O Relationship Statistical Assumption Data Requirement Training Time

if-then rules used in a zero-order Sugeno fuzzy inference system. Furthermore, the actual output obtained from the RBFN can be treated as a result of defuzzification of the above-mentioned fuzzy inference system. In comparison with a backpropagation MLP, the input-output process of RBFN is not a “black box”. Additionally, for a backpropagation MLP and an RBFN, there is no upper or lower limits to the size of a training data set. Moreover, they cannot explicitly interpret which variables are influential factors for the output. For the probit method, this approach can present a crisp relationship between explanatory and response variables of the given data from a statistical viewpoint and does not assume a multivariate normality. In particular, the probit method assumes that the cumulative probability distribution is the standardized normal distribution. The differences between the probit method, the backpropagation MLP and the RBFN are briefly summarized in Table 1. It can be seen that the drawback of the backpropagation MLP and the RBFN is the fact that they take a lot more time generating a nonlinear mapping. Nevertheless, they have the attractive advantage of not requiring any statistical assumptions.

Table 1: Comparisons of three classification models Method Back Propagation RBFN Profit MLP Possibility theory Neural network Neural network Crisp function Block box Zero-order Sugeno fuzzy inference system Yes No No More Less Less None More More

3. EMPIRICAL RESULTS The criteria for selection variables used to explain the failure of a corporate are justified in Subsection 3.1. In Subsection 3.2, the examinations of the classification generalization ability for various forecasting approaches are described in detail. 3.1 Variable Selection In order to demonstrate the appropriateness and effectiveness of the above-mentioned forecasting methods for bankruptcy prediction, the financial data sources DATASTREAM and FT EXTEL Company Research are taken into account. From which, 904 UK public companies in the general industrial sector are selected. Of these, 353 companies either failed or were acquired by another party between March 1985 and March 1994. The corresponding data of failed firms are taken from the financial reports one year prior to failure. In addition, the failed firms are identified by investigating the outcomes of those firms that were dropped from DATASTREAM due to

merger, acquisition, change of name or bankruptcy. Actually, 32 failed firms, whose individual financial data are judged to be sufficient, are selected as the sample data. Furthermore, 45 of the 551 non-bankrupt firms are randomly selected on the basis of having more than nine fiscal years of data [16, 17]. Lin in [16, 17] pointed out that three important financial ratios, Working Capital/Op Expenditure (x1), After-tax Profit/Total Assets (x2) and Change in Net Cash/Total Liabilities (x3), are critical and independent variables that can be used to explain the failure of a corporate. The main reasons that are adopted are poor management and insolvency. They are briefly described below. A direct cause of corporate failure is the inability of a company to meet debt obligations. The choice of a cash-based or working capital-based liquidity ratio is not conclusive, and x3 (i.e., the Change in Net Cash/Total Liabilities) is a surrogate for solvency. Also, the no-credit-interval in a more general form of x1 (i.e., Working Capital/Operating Expenditure) has

Y. C. Hu, and F. M Tseng: Applying Backpropagation Neural Networks to Bankruptcy Prediction been used by Taffler [30] as a powerful indicator of short-term liquidity. As for x2 (i.e., After-tax Profit/Total Assets), in a poor management situation, the allocation of resources is distorted and it fails to

integrate and achieve the corporate goals. The operational costs also increase along with the poor management. Thus, x2 is an appropriate measure of management inefficiency.

Table 2: Classification accuracy rate on average for various methods Method Data Set Back Propagation Discriminant RBFN MLP Analysis 81.64 81.96 77.94 Training Set 83.75 93.75 80.00 Testing Set 3.2 Examining Generalization Ability 3.2.1 Parameter Specifications The parameter specifications used to obtain experimental results by the backpropagation MLP and RBFN are as follows: ηB = ηR1 = ηR2 = ηR3 = 0.05; αB = αR1 = αR2 = αR3 = 0.01; tB = tR = 200000; Number of hidden nodes in the backpropagation MLP: 15; Number of hidden nodes in the RBFN: 15. The above parameters are arbitrarily specified in order to train the backpropagation MLP and RBFN. In addition, both tB and tR are specified as larger values to minimize individual error measures. However, the specifications are reasonable, because the backpropagation MLP and RBFN must be sufficiently trained prior to the training procedure being terminated. Since it is not easy to determine the appropriate values of the learning rates (i.e., ηB, ηR1, ηR2, ηR3) and the momentum parameters (i.e., αB, αR1, αR2, αR3), for simplicity, all learning rates and momentum parameters are specified as 0.05 and 0.01, respectively. To avoid oscillation during the training process, the learning rates and the momentum parameters should be specified as a smaller value. 3.2.2 Implementations The generalization ability of the backpropagation MLP and RBFN are examined by the holdout method [7, 33], which is a common technique for assessing the classifier accuracy. In the holdout method, 80% and 20% of the given data are randomly partitioned into a training set and a testing set, respectively. Subsequently, the training set and the testing set are employed to construct a backpropagation neural network and are used to estimate classifier accuracy, respectively. Using the holdout method, 80% of the 77 selected firms (i.e., 61 firms) and 20% of those (i.e., 16 firms) are randomly partitioned into a training set and a testing set, respectively. Actually, the holdout method is repeated five times by the random subsampling. Thus, the overall estimated accuracy is equal to the average of the accuracies obtained by individual iterations [7]. The above backpropagation MLP and RBFN are implemented in Delphi 5.0 on a Pentium 4 personal computer with a clock rate of 1700 MHz. The discriminant analysis and the probit method are performed by the SAS program.

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Profit 78.36 87.5

3.2.3 Result Analysis In Table 2, we summarize the classification accuracy rates as an average for various forecasting methods. It is easily seen that, not only in the training set and but also in the testing set, the RBFN outperforms the other forecasting methods. The experimental results indicate that, in comparison with the other methods, the RBFN has better fitting ability and generalization ability. In comparison with the backpropagation MLP, the RBFN performs well for the testing data. It seems that the number of hidden nodes is not a serious problem for a RBFN. It is also seen that the backpropagation MLP performs well on the training set, but poorly on the testing set. That is, it seems that the backpropagation MLP suffers from over-fitting. However, it is possible to improve the generalization ability of the backpropagation MLP by using as few hidden nodes as possible [9]. But, as we have mentioned above, the appropriate numbers of hidden nodes are not known in advance. Among the multivariate techniques, the discriminant analysis performs poorly on the testing set. Furthermore, the probit method outperforms the discriminant analysis on the training set and testing set. It is known that the probit method can be employed to determine the relationship between input-output data based on the probability theory. The relationship between inputs and outputs is given by a crisp linear function. In addition, a large amount of input-output data is generally needed to reflect the objectivity of a phenomenon. However, the amount of input-output observations are usually insufficiently large. In other words, 77 selected firms may not be sufficient for the probit method in order to generate best results.

4. DISCUSSIONS AND CONCLUSIONS Bankruptcy prediction is a class of interesting and important problems. A better understanding of the causes will have tremendous financial and managerial consequences. However, they may fail miserably when statistical assumptions are not met. The experimental results are further analyzed as follows:

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1.

In comparison with the discriminant analysis, it seems that the probit method is a promising multivariate technique that should be given considerable consideration when solving real problems like bankruptcy prediction. 2. It is seen that a RBFN is superior to the other methods. This may demonstrate that the RBFN can perform excellent approximations for curve fitting problems [12]. The analytical results may lead to the potential applicability or related methodological development of the nonlinear regression tools considering non-additivity of the interaction among attributes, such as fuzzy rule-based systems, in the bankruptcy prediction. Actually, the backpropagation MLP and a RBFN are powerful tools because of their nonlinear and nonparametric adaptive-learning properties and there being less constraint on the number of observations. Although an advantage of the multivariate techniques is to indicate which variables are influential, due to each method having its unique advantages, decision makers should select an appropriate tool to solve the problems they face. The improvement in generalization ability of the backpropagation MLP will be taken into account. It is known that too few training patterns or too many weights in a network may result in poor generalization ability [9]. Other than these principles, several useful methods proposed to construct a network for improving generalization can be found in [9], and will be considered in the future works.

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ABOUT THE AUTHORS Yi-Chung Hu received the B.S. degree in information and computer engineering from the Chung Yuan Christian University, Chung-Li, Taiwan, the M.S. degree in computer and information science from the National Chiao Tung University, Hsinchu, Taiwan, and the Ph.D. degree in information management from the National Chiao Tung University, Hsinchu, Taiwan, in 1991, 1993, and 2003, respectively. He is currently an Assistant Professor in the Department of Business Administration at the Chung Yuan Christian University, Chung-Li, Taiwan. His research interests include soft computing, multiple criteria decision making, and electronic commerce. Fang-Mei Tseng received the B.S. degree in statistics from the National Chengchi University, Taipei, Taiwan, the M.S. degree in industrial management from the National Taiwan University of Science and Technology, Taipei, Taiwan, and the Ph.D. degree in technology management from the National Chiao Tung University, Hsinchu, Taiwan, in 1988, 1990, and 1998, respectively. She is currently an Associate Professor in the Department of International Business at the Yuan Ze University, Chung-Li, Taiwan. Her research interests include technology management, quantitative methods, and business valuation. (Received July 2004, revised September 2004, accepted November 2004)