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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.

With the continuous development of economic globalization, business groups play increasingly important roles in economic activities and are prevalent around the world (e.g., [

In recent years, increasingly close attention has been paid to the credit risk of business groups. Siegel and Choudhury [

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

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.

Following Jarrow and Turnbull [

The business group in our model is assumed to comprise

The products of all the subsidiaries are perishable. Each subsidiary faces a newsvendor-type demand market. We assume that the demand of any subsidiary

To describe the collaboration and interactions among subsidiaries in the business group, we simply assume that the market price

where

Each subsidiary

During each period, the business group has a fixed financial cost

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

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.

We define

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 (

Hence, the dynamic decision-making process of the business group can be represented by an

The profit of subsidiary

We also assume that

Consequently,

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

Specifically, the financial cost of the business group may be due to the liabilities of its subsidiaries, and the constant

Without loss of generality, we consider the special case of

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

In the dynamic decision-making process, parameters

Then, the algebraic equations (

There are six fixed points:

In order to find the stable region of the Nash equilibrium point, we put

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

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

The stability region of the Nash equilibrium.

We conducted numerical simulations to understand the evolution of the credit risk in more depth. The model we constructed shows that the evolution of the credit risk is driven by the dynamic decision-making processes of the subsidiaries, which are described by algebraic equations (

The Lyapunov characteristic exponent of a dynamical system is a commonly used quantity, which characterizes the rate of separation of infinitesimally close trajectories. The dynamic decision-making process in our model is a three-dimensional dynamic system. The Lyapunov characteristic exponent in each dimension of the dynamic decision-making process is shown in Figure

Lyapunov characteristic exponent of the dynamic decision-making process.

Figure

To obtain a better understanding of the complex credit risk, the bifurcation shown in Figure

Bifurcations of the decision-making process and the credit risk.

We also show the chaotic attractors of the dynamic system in Figure

Chaotic attractors of the dynamic system.

Figures

Sequence diagram showing the changes in the credit risk of the business group.

Next, we illustrate the butterfly effect of the credit risk. The chaotic system is sensitive to its initial conditions; thus we can expect that trivial differences in the initial conditions of the subsidiaries will result in departures from the evolutionary trajectories. We simulated the evolution of the credit risk of the business group with the initial conditions:

Sensitivity of the credit risk to the initial conditions.

In fact, if we define

Credit risk of a business group is complicated and difficult to predict compared with that of an individual company. The available literature has pointed out some of the main reasons for the complexity of credit risk of a business group. However, previous studies have not produced a formal framework that explains the mechanism behind this phenomenon. In this paper, we proposed an iterative model to describe the internal interactions and dynamic decision-making process of a business group. The credit risk of the business group was then characterized in each iteration by default probability. The subsequent stability analysis and Lyapunov characteristic exponents derived by numerical simulations have at last revealed the complexity of the credit risk of a business group. First, the credit risk of a business group might be led into chaos as time passes, even if the internal interactions of the business group were described in a fairly simple manner. Second, the flexibility of the subsidiaries, characteristics of decision makers, and information structure in a business group can all affect the Nash equilibrium of its credit risk. Lastly, the credit risk of a business group appeared to be acutely sensitive to initial conditions, and the butterfly effect was found in the evolution of the credit risk of a business group.

In this study, we have two particular contributions. First, we proposed an iterative model, which was constructed to be as concise as possible, to capture the main characteristics of the credit risk of a business group. The analysis and simulations of this model essentially revealed the complexity of the credit risk of a business group and, to some extent, explained the observed phenomenon that the credit risk of a business group is more difficult to predict than that of an individual company. Second, our results provided important theoretical insights for understanding the complexity of the credit risk of a business group, which can benefit further research in this area. In addition, this study also provided practical implications for risk control of a business group. It suggested that an appropriate internal control mechanism, information sharing, and decision process management can help to decrease the credit risk of a business group. Although the numerical simulation results in this study are based on the special case of a business group with three subsidiaries, they can easily be extended to a general circumstance involving a business group with more subsidiaries.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was funded by the National Natural Science Foundation of China (Approval no. 71271043 and no. 71401116) and the Specialized Research Fund for Doctoral Program of Higher Education (Approval no. 20110185110021).