Discrete Dynamics in Nature and Society

Volume 2017, Article ID 9732678, 13 pages

https://doi.org/10.1155/2017/9732678

## A Regulation Model for the Solvency of Banking System: Based on the Pinning Control Theory of Complex Network

School of Economics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

Correspondence should be addressed to Hongbing Ouyang; moc.621@bhgnayuo

Received 3 May 2017; Revised 28 July 2017; Accepted 20 August 2017; Published 20 December 2017

Academic Editor: Ricardo López-Ruiz

Copyright © 2017 Xiang Gao 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 dynamic model is proposed based on the pinning control theory of complex network in order to simulate government bailouts against financial crisis and then is applied to a stress test of China’s interbank borrowing and lending network from 2007 to 2014. The proposed model takes many cases into account, so it is able to simulate bailout effects with different parameters, capture temporal and individual differences of banks’ spillovers effects, and reflect their sensitivity to government bailouts indirectly. This paper offers an innovative model to identify the systemic-important banks in financial crisis and construct a macroprudential regulation system based on network theory.

#### 1. Introduction

The characteristic of financial network is one of explanations about systemic risk and contagion. Recently, governments [1] take some measures such as macroprudential regulation to solve these problems. As a result, the macroprudential regulation framework and methodology become important subjects of researches and discussions. It was Allen and Gale [2] who first applied the network theory to study the financial system and later a lot of researchers analyzed and expanded the financial systems worldwide based on the network theory and drew different conclusions. The theory network offers a comprehensive and systematic analysis perspective and leads people to have a better understanding about the formation of systemic risks [3, 4] and contagion modes [5–7] as well as effective bailout measures [8, 9].

In general, a wealth of literature on the studies of financial system using the network theory is concentrated on the description of financial systems. Huang and Jia [10] validated the network model using the data of large value payment system, described China’s bank network from multiple dimensions, and found important bank nodes. With Australia’s interbank market data, Sokolov et al. [11] constructed the interbank network and studied its topological properties. Drehmann and Tarashev [12] measured the systemic importance of interrelated banks. Some other literature mainly discusses the modeling methods. Mistrulli [13], Degryse and Nguyen [14], and Ma et al. [15] applied the maximum entropy method to study the structure of bank networks in different countries. However, Mistrulli [13] found that the risk contagion effect was underestimated in the bank networks constructed with the maximum entropy method and thus added a constraint totally with the actual situation. Later, Wang et al. [16] built the interbank market network based on a constraint of interbank borrowing and lending preference and measured the systemic risks. Anand et al. [17] combined the minimum density solution and the maximum entropy method to determine the boundary of contagion effect. Above all, most literature just takes advantage of the network theory to study the macroscopic characteristics of bank system, identify important banks, or measure the factors of systemic risks [18–20] but hardly involves the controllability of network.

It is obvious that we could understand financial system more clearly and deeply after studying the controllability of financial network and thus provide some bailouts strategies and supervision model accordingly. In this paper, we pay more attention to the optimization of policy and supervision; for more details, we present a model on bailouts for the solvency of banking system based on the pinning control theory of complex network [21–25] and then apply it to a stress test of China’s interbank network, which generates a methodology about financial regulation.

The pinning control theory mentioned above has been used to study network synchronization thoroughly and also widely applied to opinion evolution of social network, multimode laser system, and so on [26, 27]. The basic thinking of pinning control is to selectively exert control on a few nodes in the network so that the whole network shows desired behaviors [28]. Rong et al. [29] and Tang et al. [30] compared selective strategies of pinning control. Besides, Turci and Macau [31] found that a max-degree pinning scheme is better than a random pinning one in a disassortative network. Moreover, Zou and Chen [32] illustrated that pinning the big nodes is, in fact, always better than pinning the small ones in normalized weighted scale-free networks. This paper extends pinning control to financial network and deems the government’s bailout against risks as pinning control or an exogenous policy restraint for certain banks, so that the solvency of the whole bank network changes in line with the government’s expectations and then the goal of bailout is attained. The comparative result of different bailout strategies offers a basis of judgment for bailout behaviors [33].

The rest of this paper is organized as follows. Section 2 constructs a model of financial network based on interbank borrowing and lending and further builds the dynamic evolution model of solvency based on the pinning control theory of complex network; Section 3 analyzes China’s bank system using the aforementioned model, compares the results of government bailouts in different cases, and gives suggestions for policy-making; Section 4 summarizes the whole paper, discusses the significance of the dynamic evolution model of solvency in macroprudential regulation as well as its role in the process of government’s decision-making as a new dynamic stress test method.

#### 2. Modeling

##### 2.1. Financial Network Model Based on Interbank Borrowing and Lending

Since it is difficult to obtain complete information of both parties in interbank borrowing and lending, the maximum entropy method [34] is adopted to estimate the structure of interbank borrowing and lending in China and establish a directed network composed of nodes showing one-to-one correspondence with banks and weighted edges^{1} according to the common practice in studies on bank systems.

To compare the interrelation degrees revealed in interbank borrowing and lending in the same dimension, the concrete numerical values are subject to normalization to obtain a relative weight matrix . As the network changes with time, there is a relative weight matrix at any time , which is given byin which the weight of edge connecting nodes such as represents the percentage of the lending amount from bank to bank in the total lending amount of bank at time and the diagonal element is zero because the borrowing and lending amount of a bank to itself are zero.Then, turns to :The elements in can be solved by And if , ; if , , , and , which are the constraint conditions.

Therefore, a larger weight of edge connecting bank and bank indicates that bank has a closer relation with bank and simultaneously reflects the relative importance of bank among all banks holding a lending relation with bank . Similarly, represents the relative importance of bank according to borrowing relation. Based on such a directed network of weighted edges, an evolution model of solvency which is affected by the interrelation among banks is constructed to study the bailouts of the lender of last resort against systemic risk.

##### 2.2. Dynamic Evolution Model of Solvency Based on Pinning Control

The bailouts of “the lender of last resort” to certain banks in the bank system can be interpreted as the pinning control on the solvency of individuals in the network which enables the individuals to reach the desired values first and then brings the whole network to a normal level. In complex financial network, it will be a highly efficient and cost-saving method to maintain the stability of the whole financial system by exerting control on a few key financial institutions, and in particular, prompt government bailouts can stimulate the recovery of economic vigor when the systemic risk occurs to the financial system. This paper develop an dynamic evolution model, which takes solvency as a basic parameter, banks are mutually affected in terms of solvency due to their interrelation, government bailouts are deemed as exogenous control on the network and brought into the influential factors for solvency to judge bank differences and bailouts effect through differential treatment, and also the result clarifies how to identify “important financial institutions.”

###### 2.2.1. Hypotheses

*Hypothesis 1. *There are participants endowed with solvency in a system (). 0 suggests that a bank suffers from serious solvency crisis and faces with bankruptcy; 1 indicates that a bank is in the state of normal operation and shows high solvency.

*Hypothesis 2. *When an adverse exogenous shock acts on bank system, all will suffer from solvency crisis, and the solvency of each node in the network deviates from its nominal value and fluctuates in the range of small value. Considering that the initial value of solvency does not affect the stability of network^{2}, the initial values of solvency of banks are taken from the range of randomly and satisfy the requirement of uniform distribution. The central bank, as “the lender of last resort,” duly and moderately injects liquidity into banks so that the whole market receives bailouts and remains stable at a high level (economic revitalization) which is taken as the desired value () for model. Figures 1(a), 1(b), 1(c), and 1(d) display random distribution of banks in the range of , and the conclusions in this paper are verified by these different random distribution of initial solvency. And the final solvency is shown in Figure 1(e), and it is reasonable and factual that some banks’ solvency is close to the standard but not up to it.