A novel method for financial distress prediction based on sparse neural networks with L 1 / 2 regularization.

Corporate financial distress is related to the interests of the enterprise and stakeholders. Therefore, its accurate prediction is of great significance to avoid huge losses from them. Despite significant effort and progress in this field, the existing prediction methods are either limited by the number of input variables or restricted to those financial predictors. To alleviate those issues, both financial variables and non-financial variables are screened out from the existing accounting and finance theory to use as financial distress predictors. In addition, a novel method for financial distress prediction (FDP) based on sparse neural networks is proposed, namely FDP-SNN, in which the weight of the hidden layer is constrained with L 1 / 2 regularization to achieve the sparsity, so as to select relevant and important predictors, improving the predicted accuracy. It also provides support for the interpretability of the model. The results show that non-financial variables, such as investor protection and governance structure, play a key role in financial distress prediction than those financial ones, especially when the forecast period grows longer. By comparing those classic models proposed by predominant researchers in accounting and finance, the proposed model outperforms in terms of accuracy, precision, and AUC performance.

Introduction

Corporate financial distress is one of the important research issues internationally. Both theoretical researchers and practical experience show that failures and bankruptcy filings are a result of financial or economic distress. Even if the firms survive from corporate failures, financial distress can still cause significant direct and indirect costs to them and their stakeholders [3, 5, 25, 35]. Therefore, how to detect and prevent financial distress on a timely basis would offer great attention and significant value to firms, regulators, investors, and other interest-related parties.

Researchers have devoted great efforts to find efficient and effective methods for corporate financial distress prediction, which mainly contains two research streams. One is the expansion and supplement of the predictive factors based on classic statistical methods, e.g., multivariate discriminant analysis (MDA) and logit regression model (Logit) or probit regression models(Probit). In this research stream, various financial and non-financial variables had been explored [2, 7, 11, 19, 52]. However, these classic statistical models were restricted by strict assumptions, such as variables being normally distributed, equal variance covariance matrices across treating and control groups and the absence of multi-collinearity etc. [6, 48]. These assumptions had greatly limited the number of predictors in the models, which made it hard to deal with a large number of predictors and improve accuracy.

The other stream in financial distress prediction is the choice and innovation of the method. To overcome the limitations of classic statistical methods, some researchers started to apply machine learning methods into FDP, among which support vector machine (SVM), decision tree (DT) and neural networks (NN) has been widely used [53, 62, 71]. Compared with classic statistical methods, machine learning methods increase the quantity of variables in the models, enabling features selection and accuracy improvement. Especially, as an efficient method, neural network has been proved to possess the ability to approximate any nonlinear functions, and has been successfully applied to and exhibited excellent performance in FDP [12, 20]. However, existing methods for FDP always utilize only financial variables as predictors, ignoring non-financial variables. Actually, the related researchers have found that financial variables are the reflection of corporate financial situation, while non-financial variables including strategy and governance structure indeed determine the financial situation [38, 44]. Therefore, non-financial variables may be more powerful than financial variables in FDP.

Considering the above issues, this paper adopts more predictors including financial variables, non-financial variables, and also proposes a novel feature selection and prediction method for financial distress using sparse neural networks with regularization, in which the weights connect the input and hidden layer are designed in sparse coding, achieving the purpose of feature selection. Meantime, based on the simplified data, the recognition networks can act out better classification effects. The contributions of this paper are listed as follows:The organization of the paper is described below. Sect. 2 reviews the related work. Section 3 introduces the related technologies and the proposed method. Section 4 exhibits and analyses experiments. Section 5 is conclusion and future work.

A novel prediction method for financial distress is proposed, which adopts the sparse neural networks with regularization to simplify features, and further improves recognition accuracy. Besides, it is extremely beneficial to the interpretability of the model.

This paper considers extensive predictors including financial and non-financial variables, which greatly improves the accuracy of the FDP model. Besides, the results also show that non-financial variables are more important in the FDP.

Related works

In this section, the related works on the financial distress and the neural networks, especially the technique in neural networks and sparsity.

Financial distress prediction

The FDP has been extensively researching areas since the late 1960s. Various statistical and intelligence techniques have been used in this area. However, the most widely used methods are still those classic statistical methods, e.g., MDA, Logit and Probit. Because the most important issue for finance and accounting researchers is to explore new financial distress predictors to build and verify the FDP theory and those methods are qualified for these tasks. Since Altman [2] have innovatively used financial variables to predict financial bankruptcy, researchers started their work on the expansion and supplement of financial predictors [9, 19, 47]. Hereafter, both accounting and finance researchers found that except for financial variables, the non-financial variables, such as government structure [11, 57], information disclosure [34, 52], investor protection [8] and strategy [23] can be used to predict financial distress. During exploring new factors, classic statistical methods had been widely used. However, those statistical models were questioned and criticized by strict assumptions of variation homogeneity of data [48], which makes them being sensitivity to multi-collinearity and limited in the number of predictors. Kumar and Ravi [43] found that the maximum number of significant predictors in those models is 20 variables and more predictors would not improve the prediction performance anymore. That is, limitation for variables have restricted the information content in prediction models. As a result, it has been hard to select key features from a large number of predictors and improve accuracy.

With the development of statistical methods and computer technology, machine learning methods started to be applied in FDP, including support vector machine, decision tree and neural networks. For example, Min and Lee [53] constructed a FDP model based on SVM with 38 financial variables as predictors. Sun and Li [62] design a FDP model based on DT with 35 financial variables as predictors. Chen and Du [12] test a FDP model based on NN with 37 financial variables as predictors. Zhou et al. [71] combined multiple machine learning approaches to select 20 features from 338 financial variables for FDP. Compared with classic statistical methods, the machine learning methods have advantages in their capability of modeling complex relationships between independent and dependent features without strong model assumptions, which makes it possible to put more predictors in a model so as to select features and improve accuracy. However, existing machine learning-based methods for FDP always use only financial variables without considering non-financial variables. Detailed comparison between classic statistical methods and machine learning methods for FDP is shown in Table 1.

Table 1

Comparison between classic statistical methods and machine learning methods for FDP

Streams	Methods	Examples	Achievements	Shortcoming
Classic Statistical Methods	MDA	Altman [2]	New financial distress predictors can be explored to build and verify FDP theory	The maximum number of predictors in model is limited so that it is hard to select key features and improve accuracy
		Deakin [19]
	Logit	Martin [51]
		Ohlson [56]
	Probit	Casey et al. [11]
		Zmijewski [72]
Machine Learning Methods	SVM	Hua et al. [37]	More predictors can be included in the model so that key features can be selected and accuracy can be improved	Only financial variables are considered as predictors and non-financial variables are ignored
		Min and Lee [53]
	DT	Frydman et al. [24]
		Sun and Li [62]
	NN	Altman [2]
		Chen and Du [12]

Neural networks

Machine learning methods, especially neural networks, have been proved can fit linear and nonlinear relationships [31] and have been applied in various fields [1, 13, 30]. Kiran et al. [42] applied artificial neural networks (ANN) to predict the number of students taking make-up examinations. Singh [61] used it to determine the length of intervals in fuzzy time series (FTS) forecasting. Singh et al. [65] adopted the backpropagation neural network (BP-NN) to reconstruct the missing color-channel data. Namasudra, Dhamodharavadhani, and Rathipriya [54] proposed a neural network-based tool to predict the confirmed, recovered, and death of COVID-19. Goel, Murugan, Mirjalili, and Chakrabartty [29] also achieved its automatic diagnosis by using convolutional neural network.

Especially, Chen and Du [12] applied data mining techniques in the form of neural networks to build and test financial distress prediction models. Meatime, they also demonstrated its feasibility and validity. Hereafter, many researchers supported the neural networks approach and found that neural networks performed better in financial distress predicting than decision trees and other alternative approaches such as SVM [26]. Despite its excellent performance, neural networks still face great challenges in dealing with high-dimensional data.

Sparsity regularization

The redundant information in the high-dimensional data [46] will seriously influence the performance of classifiers, especially these methods that have no feature extraction or selection [69]. Sparsity provides an effective method to reduce features and improve performance, and plays an increasingly important role in fields such as machine learning and image processing [55].

The sparsity approach removes a large number of redundant variables and retains only the explanatory variables that are most relevant to the response variables, simplifying the model and effectively solving many problems in modeling high-dimensional datasets [45, 66]. It has better explanatory power and facilitates data visualization, reduced computational effort and transmission storage.

regularization is the first sparse regularization method applied to variable selection and extraction, which can give the optimal variable selection constrained by the number of parameters. However, it needs to solve a difficult combinatorial optimization problem. The regularization proposed by Tibshirani [63] provides a powerful tool that only needs to solve a quadratic programming problem. However, its sparsity is lower than . The regularization between them had proved to have better feature selection ability and compression representation ability than , which has a wide range of value and significance [70]. M. Chen, Mi, He, Deng, and Wei [14] replaced the regularization with regularization in the reconstruction of the CT images, achieving great unbiasedness and acceleration. Liu et al. [50] proved its effectiveness in the variable selection. Wu et al. [68] investigated gene selection in cancer classification using the regularized logistic regression, which outperforms the other sparse methods.

Methodology

In this section, the basic methods and the proposed FDP-SNN will be described. In addition, the basic methods include neural networks and regularization.

Basic methods

Neural networks

The neural networks is a mathematical or computational model that mimics the structure and function of a biological neural network. It consists of a large number of neurons linked together for computation. In most cases, neural networks can change their internal structure on the basis of external information and are adaptive systems [21]. The neural networks are a nonlinear statistical data modeling tool, often used to model complex relationships between inputs and outputs, or to explore patterns in data.

Figure 1 illustrates an example of the neural network. For the input data X, its actual output is Y and dimension is , among which the N is the number of the sample, and the M is the number of the feature. The value of the hidden layer is computed by Eq. (1).where the W1 is the weight connecting the input layer and the hidden layer, and the b1 is the corresponding bias. The is the activation function.

Fig. 1

An example for the neural network with one hidden layer

Based on the value of the hidden layer H, the output of the neural network is computed by Eq. (2).where the W2 is the weight connecting the hidden layer and the output layer, and the b2 is the corresponding bias. Similarly, the is the activation function.

After obtaining the predicted output, its weight and bias are trained by using gradient descent optimization algorithms [59]. For the predicted output Z, the loss function can be established when using the Cross-entropy function, and is represented by Eq. (3).

The gradients of loss function L with respect to W and b are calculated by Eqs. (4) and (5), respectively.Then the weight Wj is updated iteratively by Eq. (6).where is the learning rate, while the bias b is updated iteratively by Eq. (7).During the optimization, an iteration termination condition is set, either by terminating the recursion when the error is less than a certain value, or by setting the number of iterations. When it is finished, the neural network with optimal parameters can be obtained.

regularization

Variable selection and feature extraction are the basic problems when processing the high-dimensional and massive data. If there are redundant variables in the data, identifying the real variables while eliminating the redundant ones is called the sparse problem.

Since the regularization can produce a sparser solution than the regularization and is easier to solve than the regularization, it has been widely applied in sparse problem [45]. For the data , assuming there is a unknown but definite dependencies . Based on the training data, the variable selection aiming at prediction accuracy can be achieved by minimizing the expected risk as Eq. (8).where is the obtained parameter finally.

Since the distribution function of is unknown, the expected risk is replaced by empirical risk and calculated by minimizing empirical risk, as Eq. (9).

Generally, over fitting occurs when solving the Eq. (9). To avoid this issue, it is solved by imposing some constraints on Eq. (9), such as Eq. (10).where is the sparse regularization term, and is its coefficient. When regularization is adopted, the parameter estimates can be calculated using Eq. (11).

FDP-SNN

The framework of the proposed FDP-SNN is shown in Fig. 2. First, the original data are input into the neural network with sparse regularization for optimization. After learning, the neural network can predict whether the company is facing financial distress. Meantime, the importance of various variables can be sorted by the weight in sparse coding, achieving the purpose of variables selection. The details about the method are described in the following subsections.

Fig. 2

The framework of FDP-SNN. In the figure, the weights in dotted line represent their values are less than E-03

Since the financial distress in this work is high-dimensional data including financial variables and non-financial variables, it is necessary to reduce the redundant features information. Aiming at the characteristics of mutual correlation and nonlinearity between the characteristics of financial data, the advantages of neural network in solving nonlinear problems are combined with sparsity norm to solve the problem of feature selection and classification. In addition, the norm is adopted to sparse the weights in the neural network, because it has better sparsity than and norm, and even their combination.

As shown in Fig 2, the original financial distress data are fed into the single hidden neural network with p input nodes, q hidden nodes and 2 output nodes. Its initial weight of each layer are and . The transfer function from hidden layer to output layer is . Particularly, it adopts a sigmoid function as an example. is a defined energy function. Its final predict output Z is calculated by Eq. (12).To achieve the purpose of the feature selection, the weights W1 between the input and hidden layer are restricted by regularization, and the loss function of neural network is modified to Eq. (13).Then in the process of back propagation, its gradients are represented by Eq. (14) and Eq. (15).The weight W1 is updated iteratively by Eq. (16).where the weight W2 is updated iteratively by Eq. (17).

The algorithm runs IteraMax iterations to obtain the optimal classification model. In addtion, the other purpose of the method is to select the influential features to further explained and analyzed the model.

Based on the obtained weights W1 connecting the input layer and the hidden layer, the feature selection process can be implemented. Their absolute values are used as the ranking basis to get the feature ordering for the financial distress, which can be called predictive power weight Wp.

For a variable , its predictive power weight can be computed by Eq. (18).For all variables, a sorted list can be obtained, it is represented by Eq. (19).In which list, the top features can be selected and analyzed to explain the model. The entire processes are described in Algorithm 1.

Experiments and results

In this section, the FDP data, evaluation standard, parameter analysis, model performance and its interpretability will be introduced.

FDP data and evaluation standard

Previous research set many criteria to define and distinguish whether the company is or will be in financial distress, among which firm bankruptcy [2, 32, 52, 56], debt restructuring [4, 11, 27], and debt default [10, 28, 33] are most widely used.1 In China, the sample size of listed firms filing for bankruptcy is extremely small [8, 15], thus we use the other two criteria to define corporate financial distress: (1) whether the firm is experiencing a debt restructuring in a given year (debt restructuring), and (2) whether the firm has a debt default in a given year (debt default). In the China stock market, the listed companies always disclose their financial statements for the last fiscal year around April in a year [71]. To predict the financial distress for a company, we used predictor data obtained 2 and 3 years before the companies met the financial distress criteria. Detailed definitions of financial distress variables are shown in Table 2.

Table 2

Definitions of financial distress

Definitions	Variables	Definitions
Debt	F2-ds	Dummy variable indicating debt restructuring in next 2 years
Restructuring	F3-ds	Dummy variable indicating debt restructuring in next 3 years
Debt	F2-df	Dummy variable indicating debt default in next 2 years
Default	F3-df	Dummy variable indicating debt default in next 3 years

To identify those predictors, we investigate financial distress prediction papers in top accounting and finance journals both at home and abroad, and finally conclude 199 predictor variables. These predictor variables include both financial variables and non-financial variables measuring kinds of aspects of an enterprise, such as capital structure, liquidity, profitability, capacity of corporate governance and strategy. Detailed descriptions of predictor variables are shown in Table 3.

Table 3

Descriptions of predictors

Category	Types	Num	Examples
Financial	Capital structure	29	Accounts payable/assets
			Bank debt/ liabilities
	Cash management	18	Funds for working capital/net flows
			Cash flow from operations/assets
	Development capability	14	Dummy variable indicating whether real growth rate of company is higher than sustainable growth rate
			Sustainable growth rate
	Liquidity	16	Accounts receivable/assets
			Quick assets/assets
	Profitability	26	Internal rate of return to investor in common stock
			Core profit/non-core profit
	Shareholder benefit	14	Daily turnover rate of stock
			Book value/market value
	Size	7	Number of employees per ¥ 10,000 of assets
			The natural logarithm of liabilities
	Turnover	13	Sales/assets
			360 *(Accounts receivable/sales)
	Variability	26	Trend breaks in net income
			Standard deviation of fixed assets/net assets
Non-financial	Governance structure	25	Dummy variable indicating state-owned enterprise
			Dummy variable indicating replacement of chairman or CEO
	Information disclosure	6	Dummy variable indicating whether forecast earnings is larger than actual earnings
			Dummy variable indicating non-standard audit opinions
	Investor protection	3	Dummy variable indicating whether the company registered in the developed provinces:Jiangsu, Zhejiang, Shanghai, Guangdong and Beijing
			Dummy variable indicating whether the company is punished for fraud
	Strategy	2	Dummy variable indicating whether the company is investment-oriented
			(Long-term equity investment in parent company-statement long-term equity investment in consolidated statement)/assets in parent company statement
Total		199

We collect data of financial distress variables and predictor variables from two commonly used database: China Stock Market and Accounting Research Database (CSMAR) and Chinese Research Data Services Platform (CNRDS). The sample period is from 2007 to 2019 and there are 10,731 company-year observations. The financial distress variables whose missing values take more than 10% of the total company-year observations are excluded. After that, there are 163 financial variables measuring capital structure, cash management, development capability, liquidity, profitability, shareholder benefit, size, turnover, variability and 36 non-financial variables measuring governance structure, information disclosure, investor protection and strategy. Table 4 is the sample distribution of financial distress firms (FSMs) by year. We can see that the absolute number of FSMs is increasing over time with a transitory decline in year 2010-2011. However, the proportion of FSMs is decreasing by year before 2015.

Table 4

Distribution of financial distress variables by year

Year	Debt restructuring			Debt default			Total
	Non-FSMs	FSMs	Proportion of FSMs	Non-FSMs	FSMs	Proportion of FSMs
2007	369	82	0.18	347	104	0.23	451
2008	429	87	0.17	402	114	0.22	516
2009	426	81	0.16	402	105	0.21	507
2010	407	66	0.14	385	88	0.19	473
2011	523	68	0.12	492	99	0.17	591
2012	629	75	0.11	598	106	0.15	704
2013	762	97	0.11	737	122	0.14	859
2014	805	104	0.11	778	131	0.14	909
2015	844	109	0.11	827	126	0.13	953
2016	917	129	0.12	905	141	0.13	1046
2017	992	148	0.13	975	165	0.14	1140
2018	1072	166	0.13	1030	208	0.17	1238
2019	1178	166	0.12	1128	216	0.16	1344
Total	9353	1378	0.13	9006	1725	0.16	10,731

Figure 3 describes the distribution of predictor variables. We can see that the number of financial predictor variables is much larger than those non-financial ones, manifesting those previous researchers have focused on financial variables to predict financial distress. Among those prediction variables, capital structure, profitability and variability are the most commonly used financial variables, while governance structure is the most frequently used non-financial variables.

Fig. 3

The distribution of predictor variables by category

In the experiments, the samples are consisted of 10,533 firm-year observations to predict financial distress in the next 2 years and 9189 firm-year observations to predict financial distress in the next 3 years. For each one, 80% of them is randomly selected as the training set, and the remaining 20% as the test set.

In the experiment, the performance of the models is measured in terms of the accuracy and precision, which are calculated by the Eqs. (20) and (21) [18].where the variables, e.g. TP, are listed in the Table 5.

Table 5

The definitions of variables in the Eqs. 20 and 21

Variables	Definitions
TP (true positive)	An instance is positive class and is also judged to be a positive class
FN (false negative)	An instance is originally positive class while is judged to be false class
FP (false positive)	An instance is originally a false class while is judged to be positive one
TN (true negative)	An instance is a false class and is also determined to be a false class

Besides, since financial distress is originally highly imbalanced, we thereby adopt Area under ROC curve (AUC) to measure the performance of the models. ROC graph is a two-dimensional graph in which sensitivity is plotted on the Y axis and 1-specificity is plotted on X axis. An ROC graph depicts relative trade-off between benefits (TP) and costs (FP). AUC is a good performance measure especially for the highly imbalance data [22].

Parameter analysis

In order to capture the optimal performance of the model, the parameters of the FDP-SNN are analyzed and shown in Fig. 4. For the regularization coefficient , a small or large value both are not conducive to the improvement of model performance, and their accuracy is only about 50%. When , the accuracy reaches the highest value 86.48%. With regard to hidden nodes, there are some upward and downward trends in a certain accuracy in a small range. Relatively speaking, more neural nodes can improve the accuracy of the model. Finally, 45 hidden nodes are adopted and its accuracy is 86.48%.

Fig. 4

Parameter analysis of the FDP-SNN when predicting F2-ds

Besides, the iteration is also analyzed. With the increase of the iteration, the training accuracy has been rising, while the test accuracy has decreased sharply when iteration is 50,000. This is because the model with large iteration will be overfit, and further led to a significant reduction of the test accuracy. For the learning rate, despite large fluctuations, its overall trend is also rising first and then dropping. When the learning is set to 0.4, the FDP-SNN has the highest accuracy. Based on the above parameter analysis, the parameter settings of all experiments are determined, which have displayed in Table 6.

Table 6

The parameter settings of all experiments

Instances	Hidden node	λ	η	Iterations
F2-ds	45	0.0001	0.4	40000
F3-ds	40	0.0001	0.4	20000
F2-df	35	0.0001	0.4	30000
F3-df	55	0.0001	0.4	40000

Verification on sparsity

In this paper, regularization is adopted to sparse the weight in neural networks to select the effective features and make it correct decisions, improving performance. To verify its effectiveness, a verification experiment is set up. It compares the accuracy on two predictors (neural networks) that with or without the sparse regularization.

As shown in Fig. 5, the predictor that with sparse regularization accurately takes advantage. Despite different predicted targets, its all accuracy is higher than those without one. Particularly, the maximum promoting value reaches 6.53% on the F3-ds. These phenomena prove that selecting more effective feature using sparse regularization is indeed helpful to improve the ability of model recognition.

Fig. 5

The verification on effectiveness of sparse regularization

Results and analysis

Table 7 compares test accuracy of the proposed sparse neural networks and other intelligent methods on the four indexes. It can be observed that sparse neural networks are superior to other methods in accuracy. Regardless of the four indexes, the effect of naive bayes is the worst because it assumes that attributes are independent of each other. However, both the number and the correlation of attributes in this work are large, which makes classification effect poor. Contrast to other methods, the neural networks successfully improve the classification effect, and their accuracy on the F2-ds has up to 82.20%. Importantly, the values have been improved further when introducing the sparse regularization, where it has up to 87.30% on the F3-ds, which has proved that selecting valid features is helpful for classification.

Table 7

The comparison of the test accuracy on different methods

Methods	Test accuracy(%)
	F2-ds	F2-df	F3-ds	F3-df
Naive bayes	73.25	71.84	75.44	68.53
K-Nearest neighbor	77.92	73.70	78.23	73.41
Support vector machine	78.60	73.20	78.60	73.71
Decision tree	78.63	74.20	79.13	74.20
Decision stump	79.43	76.86	79.68	75.93
Neural networks	82.20	79.92	81.83	81.15
Random forest	84.86	81.40	84.92	81.19
Sparse neural networks	85.23	82.20	87.30	83.30

The significance of bold values represents the best result

Additionally, to verify performance improvement of the proposed model, it is also compared with models proposed by predominant researches, which employ classic statistical methods. The Z-Score proposed by Altman [2] is the first study that uses MDA to predict corporate financial distress. O-Score proposed by Ohlson [56] is the first work that adopted Logit to predict the financial distress of listed companies. Campbell et al. [10] proposed a simplified financial distress prediction model, which combined the traditional financial variables with non-financial variables (the stock market variables), which is also supposed to be more predictive than Z-Score and O-Score. Their comparisons are detailed in Table 8.

Table 8

Comparisons of our model with benchmark models

Model	Method	References
Our	FDP-SNN	199 variables (163 financial variables and  36 non-financial variables)
Altman	MDA	5 financial variables
Ohlson	Logit	9 financial variables
Campbell	Logit	10 financial variables and 5 non-financial variables

As shown in Tables 9 and 10, the proposed model outperforms the benchmark models on all of the performance measures. Among them, the precision in our model is almost twice that of the benchmark model, attributing the success of the improvement to the effectiveness of the variables and the features selection in the sparse neural networks. On the one hand, our model allows the input of more variables. However, due to the limitation of the number of input variables, traditional models only use financial variables, ignoring non-financial variables with more predictive power, so the performance is weaker. On the other hand, sparse neural networks can focus on more effective features, making the model correct decisions, to improve performance. Experimental results proved that the sparse neural networks for feature selection and prediction is effective.

Table 9

The performance of different models predicting financial distress in the next 2 years

Methods	F2-ds			F2-df
	Accuracy	Precision	AUC	Accuracy	Precision	AUC
Altman	0.797	0.207	0.639	0.762	0.271	0.640
Ohlson	0.802	0.227	0.653	0.760	0.263	0.627
Campbell	0.797	0.213	0.618	0.750	0.239	0.613
Our	0.852	0.439	0.690	0.822	0.412	0.695

The significance of bold values represents the best result

Table 10

The performance of different models predicting financial distress in the next 3 years

Methods	F3-ds			F3-df
	Accuracy	Precision	AUC	Accuracy	Precision	AUC
Altman	0.804	0.261	0.639	0.766	0.312	0.639
Ohlson	0.803	0.259	0.654	0.765	0.307	0.637
Campbell	0.819	0.273	0.640	0.766	0.290	0.635
Our	0.873	0.450	0.700	0.833	0.443	0.685

The significance of bold values represents the best result

Interpretability of FDP-SNN

Based on the obtained FDP-SNN, each predictor variable can be given a predictive power weight, which is able to explain the model to some extent. As shown in Table 11, non-financial variables have greater predictive power than those financial variables. Besides, this difference is more significant when our model is predicting financial distress in the next 3 years rather than next 2 years, which indicates that non-financial variables become more important with the forecast period grows longer. The average weights of strategy predictors are instable among the groups of financial distress. They have greater prediction weights for F2-ds and F3-df, but are relatively lower for F3-ds and F2-df. One possible reason is that there are too few variables in the strategy group. Investor protection group, by contrast, is consistently and highly predictive among all financial distress groups. This result consists with the theory of law and finance, which suggests that investor protection is a key factor affecting corporate finance [58, 60].

Table 11

Average predictive power weights of all groups of financial distress variables

Category	Types	Average Weights
		F2-ds	F3-ds	F2-df	F3-df
Financial	Capital structure	2.310	2.802	1.912	3.249
	Cash management	2.661	3.121	1.938	3.911
	Development capability	0.967	1.891	0.495	1.218
	Liquidity	3.387	3.919	2.595	3.606
	Profitability	1.841	2.240	2.504	2.270
	Shareholder benefit	1.661	3.001	1.592	2.871
	Size	1.060	1.369	1.113	2.129
	Turnover	1.611	2.749	1.136	2.479
	Variability	2.765	0.869	0.875	1.532
Subtotal		2.173	2.422	1.666	2.611
Non-financial	Governance structure	2.826	5.487	2.123	7.205
	Information disclosure	1.426	7.864	2.827	3.573
	Investor protection	17.418	20.304	15.037	14.716
	Strategy	20.408	2.735	2.005	35.882
Subtotal		4.786	6.965	3.310	8.819
Total		2.646	3.244	1.963	3.734

Tables 12 and 13 demonstrate the top 10 features with the largest weights extracting from models predicting debt restructuring and debt default. The definitions of these features are summarized in Table 15. Consist with our findings that non-financial variables have greater predictive power, all the features with the largest weight for F2-ds, F3-ds, F2-df and F3-df are non-financial variables. Specifically, the feature Develop a dummy variable indicating whether the company registered in the developed provinces2 has the greatest predictive power for F2-ds and F2-df. This feature was proposed by Hu and Jin [36], who believed that the theory of political tournaments implied local governments had a strong incentive to internalize social burdens in listed companies in their jurisdictions. Thus, it is reasonable to participate that the level of development where a company is located would affect the firms financial positions. The feature SOE, a dummy variable indicating whether the firm is state-owned enterprise or not, has the greatest power to predict F3-ds. Consist with Wu and Wu [67], we believed that state-owned enterprises had strong support from the government and were less likely to run into financial distress. The feature HPAINV, a dummy variable indicating whether the company is investment-oriented or not, has the greatest power to predict F3-df. This feature was put forward by Wang et al. [64], who suggested that compared with operation-oriented, investment-oriented company suffer less financial risk.

Table 12

The top 10 features with the largest weight from models predicting debt restructuring

	Feature	Types	Financial predictor	Weighs
F2-ds	Develop	Investor protection	No	47.397
	HPAINV	Strategy	No	35.725
	TBI	Variability	Yes	34.548
	ARTA	Liquidity	Yes	11.860
	QATA	Liquidity	Yes	9.605
	SD-FANW	Variability	Yes	8.978
	APA	Capital structure	Yes	8.430
	MPNMP	Profitability	Yes	8.274
	OCNF	Cash management	Yes	7.872
	STA	Turnover	Yes	7.402
F3-ds	SOE	Governance structure	No	52.579
	Develop	Investor protection	No	50.813
	Over-predict	Information disclosure	No	33.576
	EGR	Development capability	Yes	15.199
	QATA	Liquidity	Yes	13.720
	CTA	Liquidity	Yes	11.101
	IRRI	Profitability	Yes	10.232
	ARTA	Liquidity	Yes	9.429
	APA	Capital structure	Yes	9.188
	STA	Turnover	Yes	8.548

Table 13

The top 10 features with the largest weight from models predicting debt default

	Feature	Types	Financial predictor	Weighs
F2-df	Develop	Investor protection	No	38.899
	TBQAI	Profitability	Yes	35.030
	APA	Capital structure	Yes	7.509
	QATA	Liquidity	Yes	6.388
	ARTA	Liquidity	Yes	6.359
	BDTL	Capital structure	Yes	6.356
	Fraud	Investor protection	No	5.592
	NS-opinions	Information disclosure	No	5.481
	INNWC	Liquidity	Yes	5.463
	OCFA	Cash management	Yes	5.265
F3-df	HPAINV	Strategy	No	62.902
	Execut-turn	Governance structure	No	37.333
	Develop	Investor protection	No	30.642
	Man-hold	Governance structure	No	12.823
	CM-hold	Governance structure	No	12.795
	SOE	Governance structure	No	12.560
	MPNMP	Profitability	Yes	11.592
	QATA	Liquidity	Yes	11.057
	ARTA	Liquidity	Yes	10.231
	OCFA	Cash management	Yes	9.878

Table 15

Definitions of variables

Variables	Definitions	References
APA	Accounts payable/assets	Jiang and Sun [39]
ARTA	Accounts receivable/assets	Jiang and Sun [39]
BDTL	Bank debt/ liabilities	Gilson et al. [28]
Board-hold	Shareholding ratio of board	Wu and Wu [67]
CTA	Cash/assets	Deakin [19]
Develop-prov	Dummy variable indicating whether the company  registered in the developed provinces: Jiangsu, Zhejiang, Shanghai, Guangdong and Beijing	Hu and Jin [36]
EGR	Dummy variable indicating whether real growth rate of company is higher than sustainable growth rate	Cui and Wang [16]
Execut-turn	Dummy variable indicating replacement of chairman or CEO	Hu and Jin [36]
Fraud	Dummy variable indicating whether the company is punished for fraud	Wu and Wu [67]
HPAINVEST	Dummy variable indicating whether the company is investment-oriented	Wang et al. [64]
INNWC	Inventory/net working capital	Dambolena and Khoury [17]
IRRI	Internal rate of return to investor in common stock	Blum [9]
ITA	Intangible assets/assets	Jiang, Zhang, Lu, and Chen [40]
KFNI	Net income per share excluding non-recurring gains  and losses	Liu and He [49]
Man-hold	Stock option percentage	Casey et al. [11]
MPNMP	Core profit/non-core profit	Wang et al. [64]
NS-opinions	Dummy variable indicating non-standard audit opinions	Hopwood et al. [34]
OCFA	Cash flow from operations/assets	Jones and Hensher [41]
OCNF	Funds for working capital/net flows	Gentry et al. [27]
Over-predict	Dummy variable indicating whether forecast earnings  is larger than actual earnings	Jiang et al. [40]
QATA	Quick assets/assets	Deakin [19]
SD-FANW	Standard deviation of fixed assets/net assets	Dambolena and Khoury [17]
SD-INNWC	Standard deviation of inventory/net working capital	Dambolena and Khoury [17]
SD-LA	Standard deviation of liabilities/assets	Dambolena and Khoury [17]
SD-LDNWC	Standard deviation of funded  liabilities/net working capital	Dambolena and Khoury [17]
SOE	Dummy variable indicating state-owned enterprise	Wu and Wu [67]
STA	Sales/assets	Altman [2]
TBI	Trend breaks in net income	Blum [9]
TBQAI	Trend breaks in (net quick assets/inventory)	Blum [9]

Moreover, non-financial predictor variables are more important with the forecast period grows longer. As shown in Table 12, non-financial variables occupy the top 2/3 features when predicting F2-ds/F3-ds. Most importantly, when we use debt default to proxy financial distress (results are shown in Table 13), there are only 3 non-financial variables in top 10 when predicting F2-df. However, when predicting F3-df, the number of non-financial variables raise up to 6 with top positions.

Based on the comparison between the weights of financial and non-financial predictor variables, we may conclude that non-financial predictor variables are more important in predicting financial distress. However, the number of financial predictor variables is 163, while the number of non-financial predictor variables is 36, which means that the difference of weight may be driven by the difference of variables number. Thus, we carry out a robust test. Specifically, 36 financial indicators with the highest weight are selected and put into the model together with 36 non-financial indicators to re-compare the weights.

As shown in Table 14, except for predicting F2-ds, non-financial indicators have strong predictive ability in predicting other financial distress variables, including higher average weight and higher proportion in the ten variables with the highest weight. This suggests that financial variables are only better at predicting short-term debt restructurings. As the scope of financial distress expands and the predicting period becomes longer, the predictive power of non-financial variables increases.

Table 14

Comparison of predictive power between 36 financial variables and 36 non-financial variables

	Average predictive power weights		Numbers of variables in top 10
	Financial	Non-financial	Financial	Non-financial
F2-ds	2.343	2.112	8	2
F2-df	2.141	3.167	5	5
F3-ds	2.248	3.439	4	6
F3-df	3.273	4.604	3	7

Those results are consistent with the new finding that non-financial variables are more powerful than financial ones in explaining and predicting corporate financial conditions. There are two main reasons. Firstly, non-financial variables are usually the determinants of the corporate financial situation, while financial variables are its reflection. The current financial situation of a company is the result of its past operation and governance, and the future financial situation depends on its current management model ([44]). In summary, financial variables measure what a company “has done,” while non-financial variables measure what a company “is doing.” Thus, non-financial variables, measuring a company’s operation and governance, are more future-orientated.

Secondly, financial variables are generated by the financial accounting information disclosed by the company, which is easily manipulated by the management [38]. It has the following three attributes: (1) the production process is complicated, (2) can be affected by management accounting policies, and (3) often used to evaluate management performance, and thus management has the motivation and ability to manipulate financial accounting information. In contrast, non-financial accounting information does not possess these attributes. To a certain extent, non-financial variables are also more reliable than financial variables.

Conclusion

In this study, a novel prediction method for financial distress is proposed, which is based on sparse neural networks and whose hidden layer with regularization can select the efficient feature, so as to improve the performance on prediction. Based on the existing accounting and finance theory, we identify 163 financial variables and 36 non-financial variables that might affect financial distress and then select the top 10 predictors with the largest weights. The empirical results show that non-financial predictor variables are more important in financial distress prediction especially when the forecast period grows longer. Besides, the performance of FDP-SNN is assessed by comparing it with three benchmark models and find that FDP-SNN outperforms these benchmark models in accuracy, precision, and AUC performance by a large margin.

From this study, we can get the following inspirations: First, sparse neural networks with regularization can be used to select features and build a better model for predicting financial distress. Second, the neural networks model enables us to consider the financial distress predictors from multiple aspects comprehensively without considering the limitation of input variables. Finally, some future-oriented non-financial variables play a key role in the prediction of financial distress, which is consistent with accounting and finance theory.

Future research might also include other variables which have not been mentioned by these existing papers but are important in finance and accounting, such as managers’ personalities, corporate cultures, and organizational identification, etc. Besides, whether the correlations of these predictors are determinant factors in the financial distress prediction will also be our future research.