Discrete Dynamics in Nature and Society

Volume 2015, Article ID 527854, 8 pages

http://dx.doi.org/10.1155/2015/527854

## Complexity Study of the Credit Risk of a Business Group

^{1}College of Management and Economics, University of Electronic Science and Technology of China, Chengdu 610054, China^{2}School of Economics, Sichuan University, Chengdu 610064, China

Received 25 March 2015; Accepted 31 May 2015

Academic Editor: Taher Lotfi

Copyright © 2015 Yang Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A business group is a complex system; thus it is much more difficult to predict its credit risk than that of an individual company. This study proposes an iterative model, which describes the internal interactions and dynamic credit risk of a business group. The proposed model was analyzed from a complex dynamics perspective. The simulation results based on this model show that chaos will emerge in the credit risk of a business group due to the dynamic decision-making processes of its subsidiaries, even if the interactions in the business group are fairly simple. The results of this study might explain some economic phenomena, and they also provide insights into the credit risk of a business group.

#### 1. Introduction

With the continuous development of economic globalization, business groups play increasingly important roles in economic activities and are prevalent around the world (e.g., [1, 2]). Because of the scale advantages of business groups, attempts by banks to attract business groups by providing favorable rates and flexible policies have been a tremendous success in the past two decades. Many banks have gained sizable benefits from their business group customers (e.g., [3, 4]). However, the worldwide financial crisis in 2008, and subsequent bankruptcy of numerous business groups, demonstrate that banks can never afford to ignore the importance of the credit risk of business groups. Thus, the need for credit risk control by banks with business group customers has never been as urgent as it is at present. Compared with individual companies, the credit risk of business groups is always much more complex and often unpredictable in reality. But why should this be the case? Why is it difficult to predict credit risk for business groups? What is the mechanism that underlies the evolution of the credit risk of a business group? This study attempts to answer these questions by constructing a mathematical model, conducting numerical simulations, and performing a dynamic system analysis. We argue that chaos will occur in the credit risk of a business group as time passes.

In recent years, increasingly close attention has been paid to the credit risk of business groups. Siegel and Choudhury [5] argued that business groups have a relatively low credit risk compared with individual companies. Subsidiaries with limited access to intermediate funds in business groups can benefit from the support of the rest of group when they suffer negative cash-flow shocks. Gopalan et al. [6] and Khanna and Yafeh [7] found that the internal capital market in a business group can decrease the level of credit risk for its subsidiaries. These conclusions are also consistent with the view of Masulis et al. [8] who stated that the financing advantages of business groups outweigh their disadvantages. By contrast, some researchers have stated the opposite opinion, where they suggest that the credit risk of business groups is often higher than that of individual companies because of tunneling (e.g., [9, 10]). In addition, other studies have demonstrated that whether the credit risk of subsidiaries affiliated with business groups is lower than that of individual companies depends on a variety of parameters (e.g., [11, 12]). In general, the inconsistent conclusions of previous studies suggest that the credit risk of business groups is complex. Chen and Zhou [13] developed a method to measure the credit risk of business groups based on a structural model. Many studies have analyzed the default correlation using copula functions (e.g., [14–16]). Although these studies are significant, they still fail to explain the complexity of the credit risk of business groups.

In addition, many major economic events, such as the collapse of Barings Bank, the failure of China Aviation Oil (Singapore), the Enron scandal, and the bankruptcy petition of General Motors, serve as reminders that the credit risk of business groups is sensitive, and a butterfly effect may exist in the credit risk of a business group. We, therefore, constructed a newsvendor model, which is a fairly simple method, to describe the interactions among the subsidiaries of a business group. Next, we analyzed the complexity of the credit risk of the business group based on numerical simulations and a dynamic systems analysis of the model. The results of this study show that the credit risk of business groups leads to chaos even if the interactions within the business group are very simple. Our results also provide new insights into credit risk management for business groups.

The remainder of this paper is organized as follows. In the next section, we construct a model using discrete dynamics to describe the credit risk of a business group and find the Nash equilibrium point of credit risk with a given set of parameters. The emergence of chaos in the credit risk of a business group is demonstrated in Section 3 based on a numerical simulation and stability analysis. Section 4 provides the conclusions of this study.

#### 2. Model Construction and Analysis

The model we constructed in this study was as concise as possible. This iterative model shows that credit risk for business groups can be complex and unpredictable, even if the interaction among their subsidiaries is described in a very simple manner.

##### 2.1. Assumptions and Nomenclature

Following Jarrow and Turnbull [17], Ha and Tong [18], Chen and Bell [19], and Wang and Ma [20], the specific assumptions and notations of the proposed model are summarized as follows.(1)The business group in our model is assumed to comprise subsidiaries, which are independent decision makers with the goal of profit maximization.(2)The products of all the subsidiaries are perishable. Each subsidiary faces a newsvendor-type demand market. We assume that the demand of any subsidiary , follows the uniform distribution , where represents the effort level selected by subsidiary in period .(3)To describe the collaboration and interactions among subsidiaries in the business group, we simply assume that the market price , which depends on the total effort level, is the same for all the subsidiaries: where .(4)Each subsidiary has a nonlinear cost function, , for their effort level. The subsidiaries always make the optimal effort level decision to obtain the maximum margin of profit in every period:(5)During each period, the business group has a fixed financial cost , which is used to repay its external debt. If the total profit of the business group is lower than in any period, the business group will default. In this study, the probability of default is used to describe the credit risk of the business group.

The assumptions given above imply three intuitive conclusions. First, increasing the effort level can improve the demand market and increase the expected demand. Thus, the expected demand will be higher when the level of effort selected by the subsidiary is also higher. However, a high effort level can lead to a higher cost. Furthermore, a very high level of effort may also decrease the profit of other subsidiaries, because of the relatively low market price. Second, the market price is decided jointly by all the subsidiaries of the business group, which can affect the profits of all the subsidiaries. Third, the uncertainty in our model only originates from market demand. The volatility of demand directly affects the credit risk of the business group. The pattern of the business group we consider is similar to a modularized monopoly structure, which has the shape of horizontal integration and the character of competition. However, if we interpret the variable as the production decision making and interpret the formula (1) as the nonlinear inverse demand function of subsidiary in period , we can probably explain these assumptions in the context of the supply chain.

To simplify the model, the influence on the subsidiaries due to the actual controller and the internal tunneling problem of the business group are not considered. Each subsidiary pursues its own profit maximization strategy and does not care about the overall welfare of the business group, even though the decisions of each subsidiary affect the other subsidiaries in the business group.

##### 2.2. Model Construction

We define as the profit of subsidiary in period :

According to the assumptions in the previous section, the subsidiaries of the business group are independent and pursue their own profit maximization strategies. When the subsidiaries decide their effort level, they tend to increase their effort until the expected marginal revenue is equal to the marginal cost. Therefore,

The optimal effort level might not be achieved in every period because of the information asymmetry among the subsidiaries and the bounded rationality of decision makers. The subsidiaries should adjust their effort level in period according to their performance in period . A strategy that the subsidiaries are highly likely to adopt in the dynamic decision-making process is marginal adjustment, to achieve the aim of profit maximization. This strategy could be represented by the following iterative equation:where is the adjustment coefficient for subsidiary . In our model, this coefficient reflects the flexibility of the subsidiaries. The subsidiaries with higher adjustment coefficients may respond more vigorously to market changes compared with the subsidiaries with relatively lower adjustment coefficients. The coefficient can also be regarded as a measure of the individual characteristics of different decision makers. The subsidiaries that are controlled by radical decision makers tend to have higher adjustment coefficients, whereas the subsidiaries that are controlled by conservative decision makers tend to have lower adjustment coefficients. Since the adjustment coefficient vector is closer to the optimal decision when the information is more comprehensive and transparent, it may also represent some features of the information structure of the business group.

According to (4) and (5), the dynamic decision-making process of subsidiary can be written as follows:

Hence, the dynamic decision-making process of the business group can be represented by an -dimensional nonlinear iterative system:

The profit of subsidiary is ; thus the total profit of the business group in period can be written as follows:

We also assume that are independent for different subsidiaries and we define as follows:

Consequently, follows the uniform distribution .

In this study, the probability of default is used to measure the credit risk of the business group. According to the fifth assumption, the default will only occur in a period when the total profit of the business group is less than its fixed financial cost. Therefore, the business group defaults if and only if the following inequality is satisfied:

Hence, the probability of default can be written as follows:

Since follows the uniform distribution, the probability of default in period can ultimately be written as follows:where .

Specifically, the financial cost of the business group may be due to the liabilities of its subsidiaries, and the constant measures the total financial cost of all the subsidiaries in each period. In reality, the business group often moves money to subsidiaries that face the risk of default from other subsidiaries based on its internal capital market; thus rob Peter to pay Paul. Therefore, we do not need to consider the different debt structures among the subsidiaries, and we only assume a fixed total financial cost, to simplify the model in this study.

##### 2.3. Model Analysis

Without loss of generality, we consider the special case of for simplicity. Then, the dynamic decision-making process model can be constructed as the following 3-dimensional iterative system:where , , , , , and .

Therefore, the fixed points in our model satisfy the following algebraic equations:

In the dynamic decision-making process, parameters are relatively fixed whereas the adjustment coefficients , which depend on the specific characteristics of the subsidiaries, are not. In order to make the study convenient, we fix the parameters as follows:

Then, the algebraic equations (14) can be rewritten as follows:

There are six fixed points: , , , , , and . According to the algebra, we can easily show that the equilibrium and the adjustment coefficients are independent. In our model, the nonpositive equilibrium solutions are meaningless. Thus, only is the Nash equilibrium of the dynamic decision-making process. At the equilibrium, the probability of default is 41.02% for the business group.

In order to find the stable region of the Nash equilibrium point, we put into the Jacobian matrix and obtain the following:

Therefore, the characteristic equation of the Jacobian matrix can be written as follows: where

According to the Routh-Hurwitz criterion, the necessary and sufficient condition for the asymptotic stability of the equilibrium point can be written as follows:

Figure 1 shows the stability region of the Nash equilibrium point represented by algebraic equations (20). If the vector of the adjustment coefficients is in this three-dimensional region, the Nash equilibrium point will be reached after a certain number of iterations.