Complexity

Volume 2018, Article ID 2929157, 11 pages

https://doi.org/10.1155/2018/2929157

## Credit Risk Contagion Based on Asymmetric Information Association

^{1}Glorious Sun School of Business and Management, Donghua University, Shanghai 200051, China^{2}Jiangsu Key Laboratory of Big Data Analysis Technology, Nanjing University of Information Science and Technology, Nanjing 210044, China

Correspondence should be addressed to Hong Fan; nc.ude.uhd@nafgnoh

Received 8 February 2018; Revised 5 May 2018; Accepted 3 June 2018; Published 11 July 2018

Academic Editor: Andreas Flache

Copyright © 2018 Shanshan Jiang 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

The study of the contagion law of credit risk is very important for financial market supervision. The existing credit risk contagion models based on complex network theory assume that the information between individuals in the network is symmetrical and analyze the proportion of the individuals infected by the credit risk from a macro perspective. However, how individuals are infected from a microscopic perspective is not clear, besides the level of the infection of the individuals is characterized by only two states: completely infected or not infected, which is not realistic. In this paper, a credit risk contagion model based on asymmetric information association is proposed. The model can effectively describe the correlation among individuals with credit risk. The model can analyze how the risk individuals are infected in the network and can effectively reflect the risk contagion degree of the individual. This paper further analyzes the influence of network structure, information association, individual risk attitude, financial market supervision intensity, and individual risk resisting ability on individual risk contagion. The correctness of the model is verified by theoretical deduction and numerical simulation.

#### 1. Introduction

Credit risk refers to the risk of economic loss caused by the failure of the counterparty to fulfill the obligations stipulated in the contract, and it is the main type of financial risk. In recent years, the contagion effects of credit risk occur frequently in financial markets, which have caused severe impacts on the financial market and economic development of almost all market economy countries. The subprime mortgage crisis, which originated in the United States in 2008, eventually became a global financial crisis and shocked the global capital market. The global financial risks caused by the subprime mortgage crisis in the United States are directly reflected in the credit risks and contagion effects in the financial market. Credit is the cornerstone of the market economy, and the risk of the capital market is largely from the credit crisis. The contagion of credit risk will increase the complexity of credit risk in the capital market and reduce the transparency of credit risk in the capital market. At present, the introduction and rapid development of CRT (credit risk transfer) market make credit risk management more difficult, and the credit risk contagion is more extensive [1–4]. Therefore, the study of the contagion law of credit risk in the financial market has attracted much attention of researchers.

At present, the research on the contagion model of credit risk in the financial market mainly includes the following three categories: the simplified model, the structured model, and the complex network evolution model. The stochastic theory-based simplified model and structured model are used to describe the impact and contagion effects on the creditor under different circumstances of credit default strength and default loss rate of the debtor [5–10]. The method of structural model assumes that the dynamic process of corporate assets depends on a set of common state variables, and that the interfirm default correlation arises from the dynamic evolution of the firm’s asset value [9]. The simplified model directly models the process of corporate default intensity, and the default correlation is determined by the intensity of the default process, without considering the relationship between the default and the company value. Comparing with other models, the simplified model is easier to calculate the default intensity and becomes the main framework for the study of the contagion model of credit risk [10]. The simplified model and the structured model mainly assume that credit default is exogenous. The influence of endogenous factors such as psychological behavior, correlation mechanism, and network structure on the behavior of the model is not considered in the modeling of credit risk contagion for the simplified model and the structured model, while the process of credit risk contagion is the result of many endogenous factors and exogenous factors [11]. The latest research in behavioral finance believes that the psychological characteristics of investors seriously affect people’s psychological expectations and decision-making behavior [12] and increase market risk and investors’ attitudes and emotions can be transmitted in the market [13]. Therefore, in the study of credit risk contagion, the influence and function of economic subject’s psychological and behavioral factors cannot be ignored [14, 15]. Many researches indicate that the network structure of credit risk holders also has an important impact on the spread of risk [16]. In addition, empirical research shows that market supervision has a strong inhibitory effect on the infection of credit risk [17]. Therefore, credit risk is not a simple credit default dependency contagion but mixed with endogenous and exogenous factors: psychological and behavioral factors, network structure of credit risk holders, and the market regulator.

The method of complex network is paid more and more attention in the research of financial risk contagion in recent years. The financial risk contagion system is a self-organized social system [18]. Complex networks can visually describe the complex relationship between credit risk individuals and can effectively analyze endogenous and exogenous factors that affect risk transmission [19, 20]. Cimini and Serri [21] defined a systemic risk metric that estimated the potential amplification of losses in interbank markets accounting for both credit and liquidity contagion channels. This work indicated that losses reverberate among banks and eventually were amplified because of the complex structure of interbank exposures, which lead to the occurrence of the financial crisis. Bardoscia et al. [22] proposed a dynamical “microscopic” theory of instability by iterating balance sheet identities of individual banks and by assuming that transfer of shocks from borrowers to lenders. Allen and Gale [23] pioneered a study of risk contagion in the interbank market. They believed that the transmission of financial risks mainly depended on the internal relations of the financial system, such as the structure of lending relationship. This work indicated that sparse networks were more likely to infect risks, and the reason was that the tight network dispersed the impact of single bank failures on the overall system, which was similar to a complex social system [24]. But the opposite view was that the tight network reduced the risk of a single bank but increased the correlation between banks, thereby increasing the risk of contagion [25, 26]. The work [27] formalized an extension of a financial network model originally proposed by Nier et al. [25]. Acemoglu’s work [28] showed that the network structure is not a monotonic linear relationship with contagion effects. When the negative impact was less than a certain threshold, the tight network was more stable; while the negative impact was greater than a particular threshold, the weak link network was more stable. Upper [29] summarized the simulation methods of the spread of network risk in the interbank market, discussed the assumptions and applications of various simulation methods, and pointed out that infectious default could not be completely eliminated. Gai and Kapadia [30] pointed out that the high connectivity of the financial network can reduce the probability of infection but also increase the risk of infection when the problem occurs. Li’s work [31] indicated that the increased connectivity between banks reduced contagion effects, but will lead to liquidity problems, causing the risk infection. Heise and Kühn [32] studied dynamic risk contagion in the financial network and pointed out that the derivative securities risk exposure was an additional channel of contagion, which could reduce losses but did not rule out very large tail risks, and that risk contagion and loss may be increased in stressful situations. Filiz et al. [33] used algebraic geometry technique and maximum likelihood estimation method to study the problem of bank related default in simple graphs. Mastromatteo et al. [34] used the information transfer method and the maximum entropy theory to study the systemic risk of financial network structure, which highlighted the sparsity and heterogeneity of financial networks. Glasserman and Young [35] used the complex network theory to investigate the bankruptcy costs and mark-to-market losses resulting from credit quality deterioration or a loss of confidence. Bardoscia’s work [36] indicated that the origin of instability resided in the presence of specific types of cyclical structures. Tonzer [37] analyzed whether international linkages in interbank markets affected the stability of interconnected banking systems. Li and Sui [38] investigated contagion risk in an endogenous financial network. Deng’s work [39] investigated how systemic risk was affected by the structure of the banking system.

There is something in common between the financial system and the ecosystem. The contagion of financial risks is very similar to the spread of epidemics [40]. In recent years, epidemic models have been introduced into the field of economics and finance to study the diffusion effects of economic and financial risks [41]. Garas et al. [42] introduced the epidemic contagion mechanism into the actual financial network model. This work used the SIR epidemic model to simulate the contagion of the crisis in the global economic network combining with ecology, epidemiology, and complex network theory. Haldane [43] studied the relationship among network complexity, diversity, and financial vulnerability and explained the reasons for the vulnerability of the network structure. Chen and He [44] constructed a network model of credit risk contagion with related factors of credit principal behavior and revealed some relations among credit subjects in social networks. This work also studied the risk attitude of credit subject and the ability to resist credit risk. All above works show the advantages of complex network theory in the application of risk contagion. However, there are still some points needed to be improved: (i) Above models basically analyzed the characteristics of network risk contagion from a macro perspective, mainly analyzed the proportion of individuals infected; however, there is no analysis of how individuals are infected in the network; (ii) The connection relation of network nodes is not well stated, and the relation among network nodes in most studies is symmetrical; (iii) In the above models, individuals are infected at two levels, that is, they are either completely infected or not infected, while in real financial networks, individuals are infected to varying degrees. Toivanen [45] used an epidemiologic SIR model to model the spreading of the contagion in the interbank network and analyzed the importance of individual bank-specific factors on financial stability. Brandi and Clemente [46] developed an Exposed-Distressed-Bankrupted model based on SIR model for the dynamics of liquidity shocked reverberation between banks. The above two works effectively analyzed the process of individual credit risk contagion from the perspective of liquidity, and banks were shown to be in three discrete states: exposed, distressed, and bankrupted. Moreover, the asymmetric risk contagion probability was established based on loan correlation.

Based on the above analysis, based on the existing correlation theory and the complex network theory, this paper establishes an asymmetric information association model. Considering micro behavior of investors, such as the risk attitude, the ability to resist risks, and the monitoring behavior of financial market supervisors, this paper studies the contagion behavior of individuals and its evolution mechanism. In this paper, the association between risk holders is asymmetric, which can analyze the evolution process of individual risk contagion. And the degree of individual infection is ranged from 0 to 1 instead of two levels: completely infected or not infected. Compared with the work of Toivanen and Brandi and Clemente [45, 46], the proposed model considers more factors and uses different infectious model, the definition of interbank relationship function is different, and banks are infected to varying degrees.

#### 2. Credit Risk Contagion Model

In the financial market, the contagion of credit risk is a complex process related to social psychology, economic behavior, and information communication. In this process, the credit risk holder propagates the risk to other individuals through information association or interest association. Risk contagion is a game process of various factors. The strength of financial market supervision, individual ability to resist risks, and individual attitude to risk plays an important role in the process of risk transmission.

##### 2.1. The Assumptions and Notation

In this paper, we assume that the network structure of credit risk propagation in the financial market remains unchanged. In the financial market, the probability of the individual being infected by the credit risk is related to four factors, such as the relationship among the credit risk holders, the risk attitude, the ability to resist risks, and the monitoring strength of the financial market regulators. At the same time, we assume that the number of individuals in the network is , which is limited. All parameters used in this paper are defined as follows. (i) is the average contagion rate of credit risk in the financial network, and .(ii) is the monitoring strength of the financial market regulators, and .(iii) is the effect strength of credit event, and .(iv) are individual attitudes and emotions to credit risk contagion, which can characterize the impact of credit events on individual behavior in financial markets. And , , and , which indicate that the influence of credit events has an increasing marginal impact on individual risk aversion.(v) are the ability or resilience of individuals to resist credit risk contagion in financial markets, and .(vi) are the information association between individuals with market risk holders, and . In the actual risk propagation network, the relation between individuals is bidirectional and asymmetrical. Thus, is an asymmetric matrix. In fact, in this work, can be considered as bilateral exposures in a real financial network. This article focuses on the point of physical dynamics, so the expression “information association” is used. The lending relationship between financial institutions can be regarded as information association, that is to say, is the liability matrix. In real financial networks, can be obtained through maximum entropy [29], exponential random graphs [47, 48], or minimum density [49] based on the balance sheet of financial institutions.(vii) is the degree to which the credit risk is transmitted, and . Previous studies had only two states: infected or not infected, ignoring the degree of infection. In this paper, the degree of infection is taken into consideration.(viii) is the speed that individuals restore to the health status after being infected credit risk.

##### 2.2. The Credit Risk Contagion Model

The contagion mechanism of credit risk in the financial system is similar to the physical phenomenon of network flow. In financial markets, individuals who are strongly associated with individuals who have been infected by credit risk are more likely to be infected. In this work, the average intensity of infection for individual by other infected individuals is defined as

From the (1), the intensity of contagion monotonically increases with . For a fixed structure risk contagion network, the contagion process of credit risk can be regarded as a Markov process. For any individual , the degree of infection satisfies the differential equation as follows: where is the monotonically increasing convex function. In this work, is used. The first item in the right of (2) indicates that the intensity increases in which individuals are infected by infected individuals, and the second item is the recovery of individuals who are infected by credit risk. This work models credit risk contagion drawing on virus infection model. On the one hand, individuals with a large degree are easy to be infected by credit risks. On the other hand, individuals who are highly associated with infected individuals are also susceptible to infection. Previous works [25, 27] indicated that this credit risk contagion mechanism in the financial system is similar to the physical phenomenon of network flow. represents the effect of other related financial institutions to node . reflects the connection strength of node and other nodes, and the greater the association with other nodes, the easier the node will be infected. is the average infection degree of the nodes connected to node . The bigger is, the more likely the node is to be infected. is the average contagion rate of credit risk in the financial network, and the bigger is, the greater the node will be infected. represents individuals’ attitudes and emotions towards credit risk contagion, depicting the impact of credit event influence on individual behavior in financial markets. The bigger , the bigger , the greater the credit risk will affect the individual, which makes the individual more susceptible to be infected. is the strength of market regulation, the stronger the regulation is, the more stable the financial market is, and the lower the degree of individual credit risk infection. is the ability or resilience of individual to resist credit risk contagion. In the real financial system, which can be regarded as the fundraising capability and management capability of financial institutions in times of crisis. When , the contagion intensity of node in the equilibrium state of credit risk contagion system can be obtained as where is risk transfer rate of credit risk contagion. From (3), it can be obtained that , , , , , , , and . Obviously, the contagion intensity of credit risk is a monotonic increasing convex function of individual risk attitude and the influence of credit events, and the contagion intensity of credit risk is a monotonically decreasing concave function of financial market supervision intensity and individual risk resisting ability. Thus, risk aversion can increase the infection and impact of credit risk, and individual risk resistance and financial market regulation will reduce the contagion of credit risk. Then, we plug (3) into (1) and can get an autonomous equation: where . Equation (4) describes the influence strength of infected nodes on node when the credit risk contagion system reaches equilibrium. This equation also describes the conditions and the factors that need to be satisfied when the credit risk contagion system reaches equilibrium. Obviously, is the trivial solution of (4). The trivial solution indicates that there is no risk contagion in the network. However, nontrivial solutions are not the concern of contagion, and nonzero nontrivial solutions are important for risk contagion networks.

Theorem 1. *In the incomplete market, when the credit risk system is in equilibrium status, there is only a unique equilibrium for at most in the credit risk contagion system.*

* Proof. *Let

It obviously that the intersection point of (5) and (6) is the solution of (4). Solving first and two order derivatives of (6), we get

From (4), it is easy to find that and . Thus, , and . Equations (7) and (8) indicate that (6) is monotonic increasing convex function. Due to , we get

Equation (9) indicates that there are at most two fixed points of (4) in the interval shown in Figure 1, in which , when is a fixed point. If (4) has nontrivial solutions , the following conditions must be satisfied: