Advances in Mathematical Physics

Volume 2016 (2016), Article ID 1297832, 8 pages

http://dx.doi.org/10.1155/2016/1297832

## The Stability of Interbank Market Network: A Perspective on Contagion and Risk Sharing

^{1}College of Business Administration, Hunan University, Changsha 410082, China^{2}Center of Finance and Investment Management, Hunan University, Changsha 410082, China^{3}Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215, USA

Received 16 October 2015; Accepted 24 January 2016

Academic Editor: Kiseop Lee

Copyright © 2016 Chi Xie 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

As an important part of the financial system, interbank market provides banks with liquidity and credit lending and also is the main channel for risk contagion. In this paper, we test the existence of systematic risk contagion within the Chinese interbank market. By building the networks of the Chinese interbank market for each year and using the measure of mutual information, we quantitatively detect the changes of interbank market networks and observe that the correlations between banks become increasingly tighter in recent years. With the bilateral risk exposure among Chinese listed commercial banks, we find that the possibility of systemic risk contagion in Chinese interbank market is fairly small. But of great concern on each individual bank, the matter is different. Our simulation shows that the failures of three special banks (i.e., Agricultural Bank of China and Bank of China and Industrial and Commercial Bank of China) most likely lead to systemic risk contagion. Furthermore, we test the antirisk ability of the Chinese interbank market from the perspective of risk sharing and discover that the interbank market is stable when the loss scale is lower than forty percent of banks’ total core capital.

#### 1. Introduction

Interbank market plays a prominent role in the situation that the globalized financial activities promote the economic development. And the stability of the interbank market guarantees the health and sustainability of the economy. Interbank lending connects banks as a network and easily triggers systemic risks in the interbank market. A huge amount of relevant research shows that the failure of one bank will triggers a chain reaction causing other banks to fail through interbank lending. For example, a small liquidity preference shock in one bank can spread by contagion through the interbank lending [1] and the breakdown of a single bank can lead to risk contagion in a banking system [2].

In this paper, we test the existence of systematic risk contagion within the Chinese interbank market and the antirisk ability of the Chinese interbank market from the perspective of risk sharing. Unfortunately, we cannot observe how the actual structure of bilateral exposures affects the danger of contagion because banks do not have to disclose their counterparts [2]. But a single bank’s interbank annual loans and deposits are available. Yet, on the basis of interbank lending, it is difficult to conduct an effective method to evaluate risk contagion effects between banks. Therefore, we measure the interbank exposure matrix by using maximizing entropy method and construct network of the Chinese interbank market. We also introduce a measure of mutual information to observe the changes of interbank market networks in a quantitative way. Then we estimate the possibility of risk contagion in the Chinese interbank market and analyze the ability to insure against risks from the perspective of risk sharing.

The remainder of this paper is organized as follows. Section 2 reviews the literature on financial contagion and risk sharing on the interbank market. Section 3 constructs the network of Chinese interbank market. Section 4 uses mutual information to observe the changes of interbank market networks in a quantitative way and deals with the risk contagion in the Chinese interbank market and then investigates the effects of risk sharing to the stability of the banking system. Section 5 concludes the paper.

#### 2. Literature Review

A great deal of research, until recently, has recognized the importance of systemic risk in banking system. However, there is no consensus on an accepted definition of financial systemic risk. More recently Schweitzer et al. [3] explain that systemic risk can be generally described as the risk where a failing agent causes the failure of other agents. This definition differs from other notions of risk. For example, Embrechts et al. [4] treat the default of individual agents as an extreme event. They calculate the probability with the exclusion of the interaction among economic agents. In this sense, Tasca et al. [5] provide a lower bound for systemic risk. Billo et al. [6] define systemic risk as the probability of correlated defaults among financial institutions, even just in a short time, which in turn trigger a widespread liquidity shortage and loss of confidence in the financial system as a whole. Zheng et al. [7] address this definition as any set of circumstances that threatens the stability of the financial system, which may potentially spur financial crisis.

There are three primary approaches of measuring systemic risk: simulation method, network analysis, and matrix method. The first common method is simulation method. It measures the change of assets and liabilities of banks in each set period of time by determining specific distribution pattern of bank assets termly and includes KMV and credit metrics model. Angelini et al. [8] analyze the systemic risk of Italian interbank payment system with simulation method. Câmara et al. [9] estimate the implied probability of default from stock and options in the US by using KMV. In Galos and Soramaki’s [10] study, an isolated, sudden, and unexpected failure of a bank can generate a widespread of consequences in alternative pan-European large-value payment system designs. They think in the risk of a particular systemic event, a sudden and unexpected failure of a bank where contagion is contained to the payment system, that is, which can be rather low. Yet, the simulation method does not apply to the interbank market in developing countries.

Another important method is the network approach, which analyzes the interaction between banks by building a credit network of banks with certain specification. Müller [11] employs network analysis to calculate risk contagion of banking system. In a related paper by Boss et al. [12], the network structure of the Austrian interbank market is of great concern, and the vertex between individual banks is linearly related to their contagion impact. Minoiu and Reyes [13] also demonstrate the global banking based on the network analysis. Unfortunately, the network approach is only valid when there is one or more center on the interbank market.

The third method is matrix analysis. Matrix method, averting from the data limitation, measures the risk contagion with risk exposure. Meng et al. [14] investigate the systemic risk of the US housing market at the state level based on the Random Matrix Theory. Upper and Worms [2] estimate a matrix of bilateral credit relationships for the German banking system. Similarly, Mistrulli [15] observe risk contagion within the Italian interbank market by using the same method. These works all find that the breakdown of a single bank can lead to contagion.

Based on the above-mentioned three methods, many contributions have devoted to prove the existence of risk sharing. In a work by Campos [16], the risk sharing can generate feedback between the reallocation of wealth and the aggregate size of a shock. Similarly, Ortigueira and Siassi [17] reveal that intrahousehold risk sharing behavior exerts fairly large quantitative effects on all the margins detected. Ladley [18] indicates that interbank lending boosts stability through risk sharing or provides a channel to spread failure. These papers suggest that risk contagions coexist with risk sharing in the interbank market. Accordingly, risk sharing may support banking system stability.

Contagion and risk sharing in the interbank networks have been a hot research subject, but few are related to China. We herein select 16 representative commercial banks to detect contagion and risk sharing behavior in the Chinese interbank market. Taking no account of foreign and domestic nonbank transactions data, we build a risk exposure matrix of the Chinese interbank market by using the maximizing entropy method and then construct the network structure of the Chinese interbank market. After this, to quantify the changes of interbank market networks in an efficient way, we consider the mutual information of links between two successive networks. To better observe the dynamics of risk contagion, especially the changing behavior after the financial crisis, we simulate the risk contagion of the interbank market from 2005 to 2013. Besides we also discuss the stability of interbank market from risk sharing with a unique data set in 2013.

#### 3. The Model

##### 3.1. Interbank Matrix

Related studies have valuable insights into the matrix approach [2, 19]. We use matrix analysis in the interbank market by following these studies. Suppose that there are banks in the interbank market, and their lending relationships can be represented by the following matrix:where is the exposure of bank vis-*à*-vis bank . The sum of each row and column, respectively, is total assets of bank on other banks and total liabilities of bank on other banks in the interbank market. Though there is no directly observed bilateral exposures information , each bank’s interbank annual assets and liabilities are acquirable. Consider

However, we cannot calculate all elements of matrix only according to (2) and still have unknown elements. In order to estimate the bilateral exposures, we first use the information entropy.

Information entropy represents the statistics of the probability distribution state of random events in the whole system. The smaller the entropy value is, the lower the randomness of the events is. It is easier to predict which event will occur in the system. Assuming that an event exhibit two scenarios: the occurrence probability of scenario A is 1 and probability of scenario B is 0. The consequence of this event will be up to the expression of entropy ( is the probability that event occurred). For two different events and , the probability matrix () joint entropy is defined as follows:where is the joint probability of . Thus matrix information entropy is . To simplify the problem of maximum entropy of the distribution of interbank exposures, we normalize the interbank assets and liabilities to unity as suggested by Wells [20] (i.e., ) and calculate the key elements by the solution of the following linear programming:

The Lagrangian function to this problem is given byand according to the first-order conditions the solution is given by

We substitute (6) into the adding-up constraints of (4) and obtain the following equations:

Because of the normalization (i.e., ), based on (6), we obtain the expression

By combining (8) with (7), we acquire the formulas

By substituting (9) into the solution (6), we obtain the result of (4):In accordance with the fact that banks cannot lend to themselves, the main diagonal elements must be zero. A new matrix is set to enhance accuracy of the assumption and satisfy

The problem does not necessarily disappear as the number of banks increases if interbank lending or borrowing is relatively concentrated. This means that we have to minimize the entropy of with respect to a matrix [2]:

Consequently, we assume that , if and only if , and .

##### 3.2. The Network Structure of the Chinese Interbank Market

In this paper we secure matrix of bilateral exposures by programming. is the element of which represents the exposure of bank vis-*à*-vis bank . But what we present here is a complete network structure which is clearly inconsistent with the facts. When the lending ratio between two banks is less than a predetermined threshold value , the edges between two banks are not contained in the network structure. In addition, we suppose , and is the amended matrix.

If the matrix is multiplied by the total liabilities and then divided by the bank ’s interbank liabilities, we can obtain the matrix , which is a matrix of interbank debt ratio. is the ratio between the asset of bank held by bank and the total amount of interbank liabilities of bank .

#### 4. Empirical Researches

##### 4.1. Data

We focus on the representative banks of the Chinese interbank market, that is, 4 state-owned commercial banks and 12 joint-stock commercial banks by considering the difficulty of getting access to the data and these banks carry on most transactions in the Chinese interbank market. The empirical analysis is based on the annual reports of each bank from 2005 to 2013. In order to improve the precision of results, we only consider the lending relationships among domestic banks. Table 1 presents the name of primary Chinese banks in the control sample, where each symbol represents the bank in the empirical process. Figure 1 shows the size of total lending (the sum of each bank’s interbank lending) in the Chinese interbank market, which has been growing quickly from 2007 to 2012. For observing the specific connecting way in the Chinese interbank market network, we demonstrate the network structure of 2013 in the scenario of (Figure 2) as a sample.