We develop an equilibrium model of credit network and trust network in the interbank market. We consider two kinds of decision makers including banks with liquidity surplus and banks with liquidity shortage. We model the behavior of the decision makers, derive the equilibrium conditions, and establish the variational inequality formulation for interbank credit network and trust network. We then utilize the variational inequality formulation to obtain qualitative properties of the equilibrium pattern in terms of existence and uniqueness.
Interbank markets are among the most important in financial systems. They allow exchanges among financial institutions, facilitating the allocation of the liquidity surplus to banks with liquidity shortage. A particular feature in interbank markets is the threat of systemic risk, where the failure of one bank spreads to other banks, through interbank connections. Notwithstanding, the global financial crisis, which burst in August 2007, has shown the dark side of interbank connections. In recent years, interbank connections have received significant attention in the literature. However, the literature mainly focuses on network structure, based on which the effect of systemic risk is simulated.
The intricate structure of interbank connections can be captured by using a network representation, and this network is called
The above relevant literature makes it clear that interbank credit network structures play an important role in the resilience of banking systems to systemic risk. Therefore, analyzing credit lending behavior in the interbank market is very useful in understanding the formation of interbank credit networks. It is also important for monetary authorities, since the interbank market lies at the heart of monetary policy. Actually, interbank lending behavior is a multiobjective decision making problem among banks. Trust is ubiquitous in social and economic activity, and credit markets exemplify the way in which trust lies at the very foundation of modern financial systems [
In this section, we construct the equilibrium model of credit network and trust network in the interbank market. Assuming that there are
Notations for the equilibrium model.
Notations | Description |
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The number of banks in the interbank market |
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Trust network in the interbank market |
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Credit network in the interbank market |
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A bank with liquidity surplus, |
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A bank with liquidity shortage, |
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The trust level of bank |
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The value of the trust level |
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The scale of liquidity surplus of bank |
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The scale of liquidity shortage of bank |
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The scale of credit lending of bank |
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The lending rate of bank |
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The borrowing rate of bank |
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The interest rate for the credit lending |
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The opportunity cost of bank |
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The risk faced by bank |
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The borrowing cost of bank |
We now describe the behavior of banks in the interbank market and their optimality conditions. For bank
The optimization problem of bank
In this paper, we assume that the liquidity in the interbank market is sufficient and can meet the demands of all banks with liquidity shortage. In a similar way to the above analysis, we can know that the optimization problem of bank
In equilibrium, the optimality conditions for all banks with liquidity surplus and the optimality conditions for all banks with liquidity shortage must be simultaneously satisfied. We now formally state the equilibrium condition of interbank credit network and trust network as following.
The equilibrium state of interbank credit network and trust network is one where interbank credit scales and trust levels satisfy the sum of conditions in (
The equilibrium condition governing interbank credit network and trust network is equivalent to the solution to the variational inequality problem given by
After algebraic simplification, the summation of (
Assume that the functions
Let
Interbank credit lending behavior actually is a multiobjective decision making problem among banks based on interbank trust relationships. In this paper, we developed a framework for the formulation and qualitative analysis of solutions to the equilibrium problem of credit network and trust network in the interbank market.
We described the behavior of the decision makers concluding banks with liquidity surplus and banks with liquidity shortage, where banks with surplus faced a multicriteria decision making problem consisting of revenue maximization, opportunity cost minimization, and risk minimization, and banks with liquidity shortage faced bicriteria decision making problem consisting of borrowing cost minimization and trust value maximization. We derived the optimality conditions for banks with liquidity surplus as well as banks with liquidity shortage, under suitable assumptions of the underlying functions, along with the equilibrium conditions. Moreover, we derived the variational inequality formulation of the governing equilibrium condition of interbank credit network and trust network. The variational inequality was then utilized to obtain the existence of the equilibrium credit scale and trust level pattern as well as uniqueness.
Hence, under the interaction of interbank credit network and trust network, using the variational inequality formulation, we can obtain the equilibrium credit scale and trust level pattern to make the optimality conditions for all banks be simultaneously satisfied. At the moment, the interbank market is under the equilibrium state, and the whole interest of the interbank market is the largest, which is useful to enhance the resilience of the interbank market to systemic risk. If the equilibrium condition is not fulfilled, banks with liquidity surplus will not obtain the maximum profit, and banks with liquidity shortage will obtain liquidity at relatively larger cost or their liquidity needs are not met, which may trigger default risk and contagion risk. In order to maintain the equilibrium of the bank network system, the central bank should keep the liquidity in the interbank market sufficient, and banks should enhance their credit level and strengthen their cooperation with other banks.
This research is supported by NSFC (no. 71071034, no. 71201023, and no. 71273048), NBRR (no. 2010CB328104-02), and Humanities and Social Science Youth Foundation of the Ministry of Education of China (no. 12YJC630101).