Network model of credit risk contagion in the interbank market by considering bank runs and the fire sale of external assets

Journal Pre-proof Network model of credit risk contagion in the interbank market by considering bank runs and the fire sale of external assets Tingqiang Chen, Yutong Wang, Qianru Zeng, Jun Luo

PII: DOI: Reference:

S0378-4371(19)31698-X https://doi.org/10.1016/j.physa.2019.123006 PHYSA 123006

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Physica A

Received date : 13 January 2019 Revised date : 25 March 2019 Please cite this article as: T. Chen, Y. Wang, Q. Zeng et al., Network model of credit risk contagion in the interbank market by considering bank runs and the fire sale of external assets, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123006. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Network model of credit risk contagion in the interbank market by considering bank runs and the fire sale of external assets Tingqiang Chen1, Yutong Wang1, Qianru Zeng1, Jun Luo2

（1 School of Economic and Management, Nanjing Tech University, Nanjing 211816, China；

Pr e-

p ro

of

2 School of Health Economics and Management, Nanjing University of Chinese Medicine, Nanjing 210023, China) Abstract: We construct a network model of credit risk contagion in the interbank lending market based on time series. Such network model is based on imperfect information considering that the initial credit default of the debt bank will cause the creditor bank to run on its debt bank and the debt bank’s fire sale of external assets held to repay the debt or control the leverage ratio that reduce asset prices and create spillovers. By the theoretical deduction and simulation method, we study how the contagion effects of credit risk accumulate and spread in the interbank market network. In addition, we study the evolution characteristics of credit risk contagion caused by the initial default of debt banks in the interbank market. The following are the results of the study. First, when banks sell external assets at a fixed price, the fire sale of external assets can effectively reduce the probability of credit default. However, when the price of external assets decrease with the increase of fire sales volume, banks' fire sale of external assets will lead to more serious contagion spillover consequences than that of none of the banks' selling strategies. Second, the panic of bank decision makers in the banking system has a nonlinear evolution characteristic of increasing and decreasing credit risk contagion. Third, the higher the proportion of banks' liquid assets, the stronger their ability to resist credit risks. Last, the higher the proportion of banks' interbank assets, the stronger the contagion effect of credit risk on the entire banking system. Keywords: interbank market; credit lending network; interbank asset; credit risk; contagion model

1. Introduction

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Credit risk contagion in the interbank market is manifested in the impact and spillover effect of debt banks’ credit default on other banks in the interbank credit lending network. When a creditor bank fails to receive the payment of debt as scheduled, it will transfer the default status to the bank associated with the debt (Rochet and Tirole, 1996; Eisenberg and Noe, 2001; Nier et al., 2007; Greenwood et al., 2015). Such transference causes a series of credit default events that spreads and harms the entire banking system. The 2008 subprime mortgage crisis in the United States and the subsequent European sovereign debt crisis are typical cases showing the devastating impact of contagion and spillover effects of credit default risk. Credit default risk can be transmitted and spread among banks through various ways (Shin, 2008; Brunnermeier and Pederson, 2009; Barro and Basso, 2010; Upper, 2011; Provenzaro, 2012; Glasserman and Young, 2015; Chen et al., 2015; Chen et al., 2017). According to existing literature, three main channels of credit risk contagion and diffusion emerge: direct credit connection between banks, run on banks, and asset sales by banks. Direct credit connection happens when two parties establish a credit lending relationship and the debt bank defaults and fails to repay the debt on schedule, causing the creditor bank to lack the funds needed to fulfill the debt to a third party (Allen and Babus, 2009; Duffie, 2011; Kallestrup et al., 2011; Diebold and Yilmaz, 2011; Gorton and Metrick, 2012; Giglio, 2013). For example, Provenzaro (2012) uses the agent based model to explore contagion from one institution to another that can stem from the existence of financial contracts. Caballero and Simsek (2013) presents a model of financial crises that stem from endogenous complexity. They conceptualize complexity as banks’ uncertainty about the financial network exposures. Second, when a credit default event occurs in the banking system, banks are afraid of being run by their creditor banks, leading them to repay their debts in advance and lack liquidity assets. Consequently, they run on their creditor banks to obtain liquidity and cope with the possible liquidity shock, thereby manifesting a run on banks (Kodres and Pritsker, 2002; Diamond and Rajan, 2005; Acharya et al., 2008). Acharya and Yorulmazer (2008) corroborated that the information acquired by agents about the asset quality of debtor banks affects creditor banks’ decisions on their runs, thereby generating credit default risk. The third channel, involving asset sales by banks, is considered to be an important driving factor of credit risk in modern financial markets based on a wide range of theoretical literature on asset selling. Greenwood et al. (2015) asserted that, when banks and other financial institutions are affected by financial events, they often choose to sell illiquid assets to return the leverage ratio to the   Corresponding

author：[email protected] (T. Chen) 

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normal level. The resulting drop in asset prices will affect other financial institutions holding the same assets, who could now turn to selling other assets to optimize their balance sheets. This contagion channel of spreading credit risk may occur between seemingly unrelated assets and financial institutions. The inter-bank market has formed a relatively complex credit network structure through credit lending business, which provides a business connection channel for credit risk contagion. Such structure’s features significantly impact credit risk contagion in the inter-bank market. Georg et al. (2013) established a dynamic agency model with a central bank. Banks can make subjective decisions. Money center network is affirmed to be more stable than the random graph network by comparing the structure of different inter-bank networks. Ladley (2013) studied the influence of the increase of inter-bank loans on the stability of the financial system in the case of different inter-bank market structures. He confirmed that no inter-bank market structure could maximize the stability under all conditions. Lux (2016) constructed the bank asset scale subject to the Pareto distribution and set the probability of inter-bank credit loan connection based on the bank asset scale, thereby establishing the inter-bank credit market network. Silva (2016) studied the impact of the core peripheral structure on the efficiency of banks and proved that it has cost benefits that will encourage them to participate in the financial network. However, this core-peripheral inter-bank market structure has a higher level of systemic risk. Aldasoro et al. (2017) analyzed the utility function of banks and the clearing equilibrium in the credit market. In analyzing the mechanism of credit risk contagion in inter-bank market, Nier et al. (2007) described the network structure of inter-bank credit market by random network. They studied the contagion mechanism of bank credit default in the inter-bank market, that is, the inter-bank loan exposure and the selling externality, and briefly discuss the influence mechanism of the network structure of the inter-bank market on the bank credit default contagion. Battiston et al. (2012) regarded the interbank market network as a uniform network and explored the relationship between the bank default probability caused by the contagion of credit default risk and the average connectivity of the interbank market network. Glasserman and Young (2015), on the basis of the liability matrix constructed by Eisenberg and Noe (2001), studied the evolution mechanism of inter-bank credit default contagion effects on bank network parameters. Aldasoro et al. (2016) believe that banks are usually faced with the joint action of liquidity shortage risk and solvency risk, and based on this, they build an inter-bank market network model of asset-to-liability contagion on the balance sheet. They also used the model to fit European data and study the effect of liquidity coverage on financial system stability. Li and Li (2016) investigates the impact of social network structures of depositors on bank runs. They find that the average degree of depositor networks has a significant impact on bank runs, but this impact is related to the proportion of impatient depositors and the confidence levels of depositors in banks. González-Avella et al. (2016) explore the characteristics of financial contagion in interbank networks whose distribution of links approaches a power law, and the results suggest that more connected networks that have a high concentration of credit are more resilient to contagion than other types of networks analyzed. As discussed above, the existing research on interbank credit lending network structure and credit risk contagion channels is relatively rich. However, research for credit default caused by short-term lending between banks run, illiquid and falling asset prices, and a series of negative externalities and its spillover effect is relatively scarce. On the basis of credit rating, we construct an inter-bank market network and a time-serial-based credit risk contagion model. The contagion effects of bank runs and asset sales are analyzed by using the data in the bank balance sheet. The model constructed in this paper considers the condition of the banks’ asset holdings, the adverse impact taken by the bank balance sheet adjustment decisions, and the effects of these decisions on asset prices. This study also explores the interbank market in the process of credit risk contagion between banks, bank of assets accounted for, and liquidity assets accounted factors such as investment sentiment for the evolution of credit risk contagion effects between banks. The structure of this article is arranged as follows. The second section gives the construction of the interbank market, specifically, including the network topology of the interbank market and the bank balance sheet. The third section elucidates the contagion and amplification mechanism of credit default risk in the interbank market in a mathematical way. The fourth section establishes the time series model of credit risk contagion according to the mechanism in the third section. The fifth part carries on the computation experiment and the simulation to the model established on the fourth part. The sixth part concludes the entire paper.

2. Interbank Network

The inter-bank market in this paper refers to the capital lending market formed by banks as trading subjects, that is, the inter-bank lending market. In the inter-bank market, the formation of inter-bank credit lending relationship is due to the establishment of the credit lending relationship between banks to meet the demand of liquidity, which leads to the formation of a complex capital credit lending network. Complex network theory is an important tool to study the interaction relationship in complex systems; it records

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of

the main body of interaction in complex systems as nodes and the interaction between nodes as edges. We regard the inter-bank market as a complex system. Each bank in the inter-bank market is a node in the complex network, and the inter-bank credit lending relationship is the connection between network nodes. Given the relatively small number of banks with large assets in the inter-bank market, banks with large assets generally have a high degree of credit, whereas banks with small assets have a relatively small degree of credit. Therefore, the credit degree of banks in the inter-bank market network is assumed to obey the power law distribution on the interval (0, 1). When determining the inter-bank credit lending relationship, it is jointly determined by the banks with excess liquidity and those with shortage liquidity and is determined by the interaction of inter-bank credit. On the basis of the above analysis and hypothesis, the network structure of the inter-bank market is constructed by the following algorithm.

2.1 Topology of interbank network

probability density function is as follows:

i

by

ci , where ci is selected from a power law distribution; its

Pr e-

1 as well. Thus, we denote the credit rating of bank

p ro

We assume that the number of banks in the interbank market is n and describe the interbank network by directed graph G  (V , E ) , where V is the set of banks in the interbank market and E is the set of edge links between banks in the interbank network. In the interbank market, each bank has its own credit rating due to its assets size, profitability, and solvency. Evidence on the right-hand skewness of the firm size distribution dates back at least to the 1950s and has recently been confirmed in a comprehensive data set for the U.S. economy (Axtell, 2001). Axtell finds that the entire population of tax-paying firms is very close to a Zipf distribution, i.e. a power-law distribution with exponent 1. A recent study of the distribution of bank sizes (Bremus et al., 2013) confirms that banks are no exception to this regularity. For a sample of more than 10,000 banks from 83 countries they report power-law exponents mostly around

f (ci ) 

l  ci   1 l 1  ( ) h

I{l ci h}

（1）

where l represents the minimum value of the generated node bank credit, h represents the maximum value of the generated node bank credit, and  is the power law exponent of the truncated power law distribution. l , h , and  jointly determine the distribution of bank credit. The larger the  , the fewer banks with higher credit, that is, the lower the overall credit level of the entire market.

is the eigenfunction of set

{l  ci  h} , and its meaning

al

is as follows:

I{l ci h}

1 if l  ci  h I{l ci h}   0 otherwise

（2）

urn

According to the above hypothesis, in the inter-bank market, credit rating is the basis for the inter-bank credit lending relationship without guarantee and mortgage. Moreover, in the inter-bank credit lending relationship, creditor banks have the right of future recovery value, and debtor banks have the obligation of future repayment value. Therefore, when the bank establishes the credit borrowing relationship based on the credit degree, it should mainly consider the credit degree of the debt bank. On the basis of the above analysis, we define the interaction between bank and bank credit as follows:

cij  ci c j

（3）

Jo

in which   1    0 。 In the process of building the interbank market network model, the key to building the interbank market network is to determine the connection relationship between the network nodes. We use the threshold method to determine the interbank credit lending relationship, that is, when the credit interaction

cij

between bank

i

and bank j is greater

than a certain threshold  , bank i and bank j have a credit lending relationship. Therefore, the credit lending relationship between bank i and bank j is expressed by the adjacency matrix of the figure as follows:

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1 cij   xij    0 cij  

（4）

the adjacency matrix X

 (xij )nn .

p ro

of

where    lh , for i  1, 2,...n , j  1, 2,...n . l and h are the lower and upper limits of truncated power law distribution, respectively, and  is a positive real number.  measures the difficulty of establishing a credit connection between banks. The greater  is, the more difficult it is for banks to establish a credit connection. Given the credit distribution of banks,  only depends on constant  ; thus the constant  measures the banks' expectations of the future economic situation in the interbank market network. When the economic situation is expected to be better in the future, establishing credit lending relationships is easier for banks. When the economic situation is expected to be poor in the future, establishing credit lending relationships will be difficult for banks. According to the above analysis, the topological structure of the interbank market network can be described by Given that the interbank credit lending relationship must distinguish between

creditor and debtor banks, the interbank market network is a directed network, and matrix. The number of creditor banks of a bank

i

X  (xij )nn

is an asymmetric

is called the out-degree of nodes i , denoted as

dout (i) ,

the

i is called the in-degree of nodes i , denoted as din (i) , and then the expressions of out-degree and in-degree of nodes i can be obtained according to the adjacency matrix X :

Pr e-

number of debtor banks of a bank

n

d out (i )   xij

（5）

j 1

n

din (i)   x ji

（6）

j 1

On the basis of the above algorithm, the credit degree of banks in the interbank market network c is determined by (l , h,  ) . The topology of the interbank market network is determined by ( c ,  ,  ,  , n ) . Therefore, nodes V

al

and edges E of the interbank market network represented by the directed graph G  (V , E ) can be determined by the above parameters, that is, G  G ((l , h ,  ),  ,  ,  , n ) can be used to express the structure model of the interbank market network.

2.2 Balanced sheet

Ai

For the convenience of research, we simplify the balance sheet in reality and assume that the total assets

i

are composed of three parts:

AiN

represents the short-term interbank loans of the bank i ,

urn

the bank

AiM

the cash or liquid assets held by the bank i , and

LiN ,

AiC represents

represents the off-bank assets held by the bank i . We assume

that the assets outside the bank are the mortgages that the bank makes outside. Total liabilities composed of two parts:

of

the short-term inter-bank borrowing of bank

i

and deposits

Li

LDi that

of a bank

i

are

the bank absorbs

Jo

from outside the interbank market. Therefore, the balance sheet of node banks in the interbank market network satisfies the following relationship:

where the difference

Ei

Ai  AiN  AiC  AiM Li  L  L N i

between

Ai

and

Li

D i

（7） （8）

is called net asset or holder's equity:

Ei  Ai  Li  AiN  AiC  AiM  LNi  LDi

（9）

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Table 1. Balanced sheet of bank Asset

Ai

i

Liability

Short-term interbank loans

Short-term inter-bank

N i

A

Borrowing

Off-bank assets

AiC

LiN

Deposits

AiM

LDi

of

Liquid assets

Li

Eisenberg and Noe (2001) were used for reference to express the matrix of bank network structure, and the bank

following expression, we define

ANji

p ro

L  ( LijN )nn was defined to represent the liabilities of banks to banks. For the convenience of the

liability matrix

LijN . Therefore, it can be obtained that: n

LNi   LijN j 1

n

（10）

n

AiN   AijN   LNji

（11）

j 1

Pr e-

j 1

On the basis of the balance sheet structure of node banks in the interbank market network constructed above, the following paragraphs further define the values of each variable in the balance sheet in the following manner. We assume that the initial total assets of the bank n

i

are

Ai

where i  1, 2, ..., n so that the total assets of all banks are

 A . Second, according to the analysis above, variables in the balance sheet of banks i in the interbank market are i 1

i

defined and calculated according to the following algorithm:

LDi . We ignore the heterogeneity of depositors' preferences for Banks and assume that their choice of

1) Deposit

bank to deposit money is random. As a result of expectations, the ratio of deposits to total assets is the same for each

L   Ai . D i

AiN

and interbank borrowing

urn

2) Interbank lending

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bank. For any bank i , suppose the proportion of depositors' deposits in the total assets

Ai

of the bank is

,

i.e.,

LiN . If the sum of interbank assets accounts for a proportion

 of the total assets of each bank, then the total amount of interbank assets is

n

n

i 1

i 1

 AiN    Ai . According to the

above, it can be obtained that:

n

n

n

n

 L   A i 1 j 1

N ji

i 1 j 1

N ij

n

n

i 1

i 1

  AiN    Ai

（12）

the elements parameter

Jo

Boss et al. (2004) contended that the interbank credit lending scale obeys the power-law distribution. Therefore,

LijN

of liability matrix

L is assumed to obey the power-law distribution with the power-law distribution

 . and satisfy the requirements

borrowings

N i

L

n

n

L i 1 j 1

N ji

n

   Ai . Therefore, interbank loans Ai and interbank N

i 1

can be calculated through (10) and (11).

3) Banks' liquid assets

AiC . The proportion of liquid assets is a measure of the ability of Banks to cope with

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withdrawals. In order to meet the demand of depositors, commercial Banks have to keep part of their funds in cash and other assets that can be quickly converted into cash. On the one hand, if the proportion of cash assets is too high, it will form idle funds and reduce the overall rate of return. On the other hand, if the cash stock is too small, then the reserves to be paid for withdrawal may be insufficient, and the capital turnover may be difficult, which reflects the unsound operation of the bank. To reasonable arrangement of liquidity and the necessary control can effectively prevent Banks, regulators have the specified minimum liquidity assets proportion is usually can satisfy the requirements of the bank's deal with escrow, as a result, we assume that the commercial Banks in order to achieve the highest possible yields, will keep regulators to the specified minimum liquidity assets accounted for. We use  to

AiC   Ai Therefore, the total liquid assets in the interbank market are as follows: n

n

i 1

i 1

（13）

p ro

AiC  Ai

of

represent the proportion of the bank i 's liquid assets in the bank i 's total assets, then:

（14）

According to Equations (7), (12), and (14), the total external asset size of the inter-bank market is as follows: n

(1 ) Ai

（15）

i 1

AiM . Nier et al. (2007) put forward the restriction condition for banks' external assets, that is,

Pr e-

4) External assets

banks' credit lending only exists to solve short-term liquidity shortage of banks rather than become a means of longterm financing. For the enhanced investigation of the contagion and impact of credit risk in the inter-bank market, ensuring that banks at all nodes in the inter-bank market network are in normal state when the initial shock arrives is necessary. Therefore, we assume that the difference between inter-bank loans and inter-bank loans is less than the investment amount of the bank, that is, meet the following requirements:

AiM  LiN  AiN . Therefore, we make the external investment of banks to AiM  (LNi  AiN ) 

n 1 (1  ) Ai N i 1

（16）

al

The difference in the proportion of external investment in total assets presented in Equation (16) actually reflects the heterogeneous belief of banks in decision-making. 5) Equity. According to steps one to four, the total assets

urn

as follows:

Ai

Ai

and total liabilities

Li

AiN  AiM  AiC

Li LiN  LDi

of a bank

i

can be expressed （17） （18）

Therefore, the net value of node banks in the interbank market will be calculated according to steps 1–5 above. On the basis of the above analysis, we present the balance sheet structure of the credit relationship between banks at each node in the inter-bank market network. For the convenience of the following calculation, the initial total assets of the bank

i

are assumed to be a normal number, that is,

Ai  A . Therefore, with a certain network structure of the

Jo

inter-bank market, the balance sheet of a bank is jointly determined by the initial total assets A , the proportion of depositors' deposits  , the proportion of inter-bank assets  , the power-law distribution parameter  of matrix L , and the proportion of liquid assets  ; thus, we call

(G, A, ,,,) a banking system.

3. Contagion Channel

For the convenience of the later research, liability shock is defined as the pressure on banks to reduce their liabilities, such as the need to repay debts due to a run by creditors. Asset shock refers to the reduced pressure on banks' assets, for instance, assets held by banks in various forms cannot be repaid on time.

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On the basis of the above analysis, the necessary conditions for banks are assumed to be the following:

i

to fall into credit default at any time

it 0

i

（19）

at the time t . In reality, when i is less than a certain positive value

(we call it the minimum equity ratio and write it down as 0 ), bank

i

is required by regulators to sell their long-term

assets to get the equity ratio i back above the minimum equity ratio

i

Ai i Ai , it will cause bank i

is small, bank

i

Ai

will lose its assets

to fall into default, which is equivalent to

measure a bank

i 's

if a financial event occurs; when

Ai  i . The value at risk (VaR) of a Ai

p ro

“stability.” When the value of

0 . Equity ratios i

of

where it is the equity ratio of the bank

t

bank i determines to what extent the bank may be affected by the reduction of assets. When making investment decisions, banks will be cautious about the VaR, and its control is required by the regulatory authorities. The bigger

i

is, the more capital it can withstand without falling into default. Therefore, equity ratio is an important indicator to

Pr e-

measure the “stability” of a bank. For convenience of expression, the term “stability” will be used later to refer to the size of the equity ratio. For the convenience of mathematical processing and without loss of generality, this study assumes that the minimum equity ratio is 0, that is, 0  0 .

urn

al

For the debtor of a company, the low equity ratio is a dangerous signal, which means that once the operator of the company decides to go into liquidation, the interests of creditors are difficult to be guaranteed. Therefore, the low equity ratio will increase the probability that creditors will put pressure on the operator of the company or choose to run on the bank. At the same time, the low equity ratio also reflects that the operator of the enterprise has insufficient protection of the interests of creditors, which makes the reputation of the enterprise decline and makes it difficult to raise funds through debt channels in the later operation process. There are therefore good reasons for Banks to keep their equity ratios at a healthy level, even by selling assets. In the inter-bank market network, short-term bank creditors will make decisions based on the information available to them. For instance, if short-term bank creditors can be sure that their debtors are in good economic condition, short-term bank creditors do not need to withdraw this deposit in advance. On the contrary, if short-term inter-bank creditors know that their creditors have been greatly impacted by financial events and will find repaying their debts in the future difficult, they will choose to stop redepositing to reduce their possible losses. If short-term creditors in the interbank market do not have access to any information about counterparties, no reason at any time for them to change their current decisions. In fact, as the decision-making subject, the type of information banks can obtain from the market directly affects their decisions and further affects the operation of the entire market.

3.1. Interbank credit default contagion If the debt of bank

i:

i

and j defaults at time t , then this will have an impact on the interbank assets

Ai N

AiN (t)  AiN (t 1)  aAijN tj (t)

of the bank

AiN (t) represents the long-term assets of the bank i i

Jo

where

at time t ,

（20）

AiN (t 1)

represents the long-term assets

at time t  1 , and a represents the credit default loss rate, which is assumed a  1 .

the part of the inter-bank assets held by the bank j ,

of bank

AijN

represents

 tj (t ) is the characteristic function of the bank j 's default at

time t , namely:

1 0

 tj (t )  

j default at time t otherwise

（21）

When the initial credit default occurs, no new exogenous default is assumed to occur during the spread of credit

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default, and all new credit defaults are caused by the initial credit default contagion. Although this assumption is somewhat different from the fact that financial risks are spread in reality, the generation of exogenous defaults can be obtained by the superposition of the results without exogenous shocks in the contagion process described by our model. As such, as the creditor bank i of a credit default bank, the necessary conditions for bank i to fall into a credit default at time t are as follows:

Ai (t 1)  Li (t 1)  aAijN tj (t)  0

of

3.2 Bank run

（22）

p ro

Bank run refers to the behavior that short-term creditors require their agents to repay debts in advance. In fact, it is the result of the decisions taken by depositors as the decision-making subjects to maximize their utility, which often leads to greater and non-negligible losses suffered by banks. We will focus on bank runs associated with defaults and analyze their risk contagion mechanisms. Various reasons for bank runs emerge. For instance, short-term creditors receive some information indicating that their agents will be greatly impacted in the future. Short-term creditors are inclined to run on their agents to reduce their losses or may choose to run on their agents because they are impacted by liabilities and need liquid assets to repay outstanding debts. Therefore, given that the bank has its own information set, assuming that the creditor bank will not be able to know the exact robustness level of its counterparty, it will only be able to estimate the robustness level of its counterparty based on the information available to it. In our model, creditor banks i are assumed to

Pr e-

determine whether to run on their debt banks based on the following three variables: the robustness level

i

of

creditor banks i , the proportion of the number of defaulting banks in the entire financial system in the total number of n banks   d , and the proportion of the number of banks in the transaction pair of bank i that have credit default n

i 

nid , where nd represents the number of credit default banks in the entire interbank market network, ni ni

represents the total number of counterparties of the bank, and nid represents the number of credit default occurred in the counterparties of the bank's transactions. In the above three variables, the bank

i 's

own level of robustness

i

measures the bank

al

Higher robustness level i means that the bank i 's financial situation will be better, thus bank

i

i 's

own finances.

has a reason that its

urn

creditor banks will not run themselves, maintain a relatively low level of liquidity, and put more assets into interestbearing assets to achieve their own profit maximization. Hence, the probability of bank i running its bank debt is small. In addition, when the robustness level of bank i is low, bank i is worried that after its robustness level is known by its creditor banks, it will bear the adverse impact caused by a run on its creditor banks. Hence bank i tends to hoard liquidity in advance in order to prevent potential adverse shocks. Therefore, bank i runs on its counterparties to obtain liquidity with a high probability. The proportion of banks that default in the banking system to the total number of banks  is common knowledge

Jo

and provides measure on the current economic situation. When  is maintained at a low level, bank i has a reason to believe that the current economic situation is good and that the probability of financial crisis in the future is low; hence, bank i has a low probability of running on its agent. On the contrary, when  is large, bank i believes that the current economic situation is relatively poor. Even if the agent of the bank i has sufficient solvency, it is more likely to fall into default in the future. To reduce the possible losses caused by the default of the counterparty, bank i is more likely to run on its agent. The proportion of banks that have credit defaults among banks

i

measures the bank

i 's evaluation of the establishment of credit lending relationship. Smaller i means less default on bank i 's debts. Even if the default rate of the banking system is high at this time, bank i still has a reason to think that banks that

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establish credit lending relationship with itself may not be affected by default in the short term. A large

i

indicates

that the bank i establishing credit lending relationship with the bank has a large range of credit defaults, which will make bank i believe that it has followed some wrong principles in the process of establishing credit lending; hence, bank i has a high probability of running on its debt bank. As such, we assume that the probability of a creditor bank running on its short-term debtors P (i ) is a linear function of its own robustness level i , the proportion of credit default banks to the total number of banks in the

P (i )  min{(1i  2i i  3ii )  , 0}

1i

is the sensitivity of bank

i

to macroeconomic situation,

2i

（23）

is the sensitivity of bank

p ro

where

i :

of

entire financial system  , and the proportion of credit default banks to the banks’ debt banks

i

to

counterparty solvency, and 3i is the sensitivity of bank i to their own robustness level. For the convenience of the following analysis, we assume that all banks are equally sensitive to the three categories, that is, for any i  1, 2,...n , we have 1i 1, 2i  2 and 3i  3 . Banks' sensitivity to macroeconomic conditions, to counterparty solvency, and to their own levels of soundness

Pr e-

actually measure their panic. For the same macroeconomic situation  , the more sensitive 1 the macroeconomic situation is, the higher the probability that a bank runs on its debt banks P (i ) , which means the more pessimistic the interbank market is. For the same state of counterparty default

2 , the greater the probability that bank i in the interbank market. When bank

i

i , the greater the sensitivity of counterparty solvency

runs on its debt banks P (i ) , which indicates that more pessimism emerges

takes back the short-term interbank loan

AijN

to bank j , it is bound to cause bank j to be

impacted by the reduction of short-term interbank liabilities. Bank j must use an equal amount of liquid assets to repay bank i 's short-term debt. When bank j 's liquid assets are insufficient to repay, it is:

al

ACj  LNji

（24）

At this point, bank j is at risk of illiquidity. Hence, the changes of illiquidity risk reflected in the balance sheet are as follows: the interbank assets of the run party decrease

AiN , the liquid assets increase AiN , and the interbank

urn

AiN simultaneously. According to the above analysis, bank i only runs change on the form of assets held by debtors to bank i , that is, from short-term interbank assets to liquid assets. Therefore, a bank run on its debtor will not change bank i 's robustness level but will increase bank i 's liquidity level. The effect of a bank i run on its debtor bank j is that interbank liabilities and liquid assets are N reduced Ai at the same time, which does not change the absolute value of the bank j 's equity. Therefore, when the liabilities and liquid assets of the run party decrease

Jo

robustness level of bank j is positive, this will enhance the robustness of the bank:

( A j   AiN )  ( L j   AiN )

Ai  Li A  Li （25）  i N A j  A A j   Ai Aj Therefore, given that part of the decrease in bank j ’s assets is liquid assets, the liquidity level of bank j will decline. The following paragraphs continue to discuss the illiquidity and asset selling behaviors that may be triggered by the run on creditor banks. N i



3.3 Illiquidity and fire sale The analysis of 3.1 and 3.2 shows that as the decision-making subjects in the interbank market, banks will face

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assets, rather than involving changes in debt. Therefore, even if bank

i

of

the asset impact of the reduction of interbank assets caused by the credit default of their debt banks and the liability impact caused by the run of short-term creditors at the same time. Among them, the credit default of the debt bank may make the equity ratio of the bank lower than the minimum equity ratio required by the regulation. At this time, the regulatory authority will require the bank to sell some long-term assets to return the equity level to the normal level. However, the liability impact brought by the run of short-term creditors will cause banks to use part of their liquid assets to repay debts. Moreover, when the inter-bank run brings a big impact on the liabilities and even the liquidity assets of the bank cannot pay off its liabilities, the debt banks run will fall into illiquidity, and, at this time, the banks will be forced to choose to sell the long-term assets they hold. The bank's fire sale is essentially a move by banks to exchange long-term assets held at market prices for liquid still has a positive equity ratio i , it may fall

into illiquidity. If the bank will give priority to the sale of mortgage debt

AiM

in the face of illiquidity and all

Pr e-

p ro

mortgage debt is of the same type of asset, then the selling price of mortgage debt will be determined by its supply and demand. When the supply and demand of mortgage creditor's rights assets are roughly equal, banks can obtain the book value of mortgage creditor's rights assets without loss and rebalance their liquidity situation. When demand exceeds supply in the mortgage debt asset market, the price of the mortgage debt asset will rise until the price makes the demand and supply reach equilibrium again. Similarly, when the supply in the market is greater than the demand, the price of mortgage debt falls until the supply and demand in the market for mortgage debt assets are back in equilibrium. Thus, when banks are hit by counterparty credit default and short-term creditor run, they will tend to sell mortgage debt assets to maintain equity level and obtain liquidity. We assume that the demand function of asset price changes caused by supply and demand action is an exponential function: （26） P ( x )  exp(   x )

al

Among them, x is the number of mortgage creditor's rights assets being sold in the current market, and  is the sensitivity of price supply, reflecting the change range of asset price with the sale amount of mortgage creditor's rights assets.  makes the price of mortgage creditor's rights assets change to the extent that the market clears in each period. Therefore, when a bank is impacted by assets or liabilities and chooses to sell part of the illiquid assets in a certain period to maintain the equity level or pay off debts, the bank may also face losses caused by the decline in the price of the illiquid assets.  measures the sentiment of investors in the external asset market. The larger  is, the more serious the investor panic is; hence, the price drops sharply. The smaller  is, the better the investor's expectation of the future is.  

4. Time Series Model of Interbank Network Credit Risk Contagion

Jo

urn

The risk contagion mechanism and its impact caused by various credit defaults, runs and asset sales and their interactions are analyzed above. Bank i 's debt bank credit default will cause bank i to be impacted by the decrease of interbank assets. In the interbank market, the short-term creditors of bank i will decide whether to run on its debtor bank j based on the information obtained, and when deciding to run on bank j , it will face the liability shock of needing to repay the short-term interbank credit lending. When a bank is hit by an asset, it is likely to fall into a credit default. When a bank is hit by debt, it can run into illiquidity. To avoid falling into illiquidity, banks can choose to sell certain mortgage debt assets to obtain liquid assets. If more mortgage debts were to be sold at the same time, then prices would fall sharply. At this point, all banks holding mortgage debt assets will bear losses caused by the decline in the asset price. In the present study, with regard to the time series of credit risk contagion in the model, each issue can be due to bank assets resulting from counterparty credit default, or paper losses caused by selling into a credit default state. In a credit default, the current state of the bank can enter a new phase of asset impact on the interbank market network. Therefore, the credit risk of the interbank market network will be infected by the financial events in each phase. When t  1, an exogenous given initial shock s appears in the interbank market network. The set of banks that are caught in credit default due to the initial shock is denoted as  | S1 |  nd (1) .

s

is denoted as

S1 , and the number of elements in the set of credit default banks

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The initial shock s must be determined to study the contagion of credit risk in the interbank market network. We draw lessons from the assumption of Grasserman et al. (2015) on initial shocks. Initial shocks are usually positively correlated with each other. Frey and McNeil (2003) used several different Copula to fit the default correlation in reality and found expressing the initial shock distribution difficult when the default correlation was expressed in a simple mathematical form. We assume that initial shock s comes from external assets, namely, the default of the mortgage loan, and that the proportion of initial shock received by each bank in its external assets is independently and identically distributed, represented by Beta distribution:

y p 1 (1  y )q 1 I{0 y 1} B ( p, q )

p  1 , q  1 , the probability density function

hp,q ( y)

p

and q, so that

y







hp,q ( y)dy  1. When

p ro

where p , q  1 , B ( p , q ) is a normalized parameter that depends on

（27）

of

hp , q ( y ) 

is about the decline of

, that is, the bank is impacted with

a relatively small probability. Furthermore, beta distribution can be used to fit more cases. For instance, when q  1 ,

p  1 , the probability density function

hp,q ( y) is increasing, that is, the “fat-tail distribution”, which means that the

loss is greater with a larger probability and smaller with a smaller probability. In the case that the distribution of initial shock is determined, the following paragraphs mainly build the time series contagion model of interbank network credit risk contagion when t  1, as follows:

Pr e-

St 1 generated in the previous period will be impacted

Step 1. The creditor banks of the credit default bank set by assets of the size of

AiN 



jSt 1

A

jSt 1

AijN , or AiN 



jSt 1

N ij

. Therefore, if the interbank assets of bank

i

are updated from

AiN

to

LNji , then the time series model of the interbank assets of bank i is as follows:



AiN (t )  AiN (t  1)  Bank

i

jSt 1

LNji ， i  1, 2,...n

（28）

will reassess its level of robustness i after it is hit by a reduction in interbank assets as a result of a

al

AiN (t )  AiM (t  1)  AiC (t  1)  LNi (t )  LDi (t ) credit default. Hence, when i (t )   0 , bank i is going to default AiN (t )  AiM (t  1)  AiC (t  1) t 1

urn

on its credit. Deposit the newly generated credit default bank number of the current period into

St .

That is, if

i (t)  0 and i   Sm , then i St . m0

t

i V \  Sm , according to the analysis in Section 3.2, bank i will determine whether to run on its

Step 2. For

m 0

short-term debtor based on its own robustness level i , current economic situation  and the default state of its

Jo

counterparty i for fear of further losses caused by credit default. Let their debtors in the current period, then debtor j is

L

iRt

N ji

t

Rt

be the set of banks that decide to run on t

Rt  V \  Sm . For each j V \  Sm , the liability impact of the run on m0

. Therefore, the total interbank liability

m0

N j

L

of bank j is updated to LNj 

L

i Rt

N ji

. Accordingly,

debtor j will use equal amounts of liquid assets to pay off debts, and debtor j 's liquid assets will be updated from

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A Cj

to ACj 

L

iRt

N ji

. Therefore, the time series model of total interbank liabilities and liquid assets of bank j can be

obtained as follows:

LNj (t )  LNj (t  1)   LNji

（29）

ACj (t )  ACj (t  1)   LNji

（30）

iRt

When

L

iRt

N ji

is large,

of

iRt

ACj (t) is likely to be negative.

p ro

Step 3. The above discussion discusses the asset impact brought by credit default and the liability impact brought by a bank run, among which the asset impact brought by credit default will reduce the interbank assets of creditor banks, while the liability impact brought by a bank run will reduce the interbank liabilities and liquid assets of debtors at the same time. Banks hit by assets can re-evaluate the proportion of their liquid assets

AiC (t ) i (t )  N and use it as a measure of their liquidity. Basel III requires that the ratio of liquid Ai (t )  AiM (t )  AiC (t )

assets with the current market value mortgage creditor's rights assets sold

AiM ,

AiM

Pr e-

assets should not fall below a lower boundary  . When the proportion of bank i 's liquid assets is lower than the proportion of the regulatory requirements  , bank i will be required to sell long-term assets to obtain liquidity, so as to meet the regulatory requirements of the regulatory authorities on liquidity. When the liquidity of the bank is lower than the proportion of the liquid assets required by the regulation, the bank can sell mortgage creditor's rights that is, when

AiC (t )  i (t )   , the amount of AiN (t )  AiM (t )  AiC (t )

meets:

AiC (t )  AiM  AiN (t )  AiM (t )  AiC (t )

（31）

Without considering the short-sale mechanism, considering whether the balance of mortgage debt assets held by banks reaches this amount is also necessary. Therefore,

al

 ( A N (t )  AiM (t )  AiC (t ))(  i )  AiM (t )   M i  Ai (t )

 AiM  AiM (t )  AiM  AiM (t )

（32） （33）

AiC (t)  AiC (t) AiM (t)

（34）

urn

AiM (t)  AiM (t 1) AiM (t)

According to Section 3.3, when supply in the market of mortgage creditor's rights and assets increases and demand decreases, the price of mortgage creditor's rights and assets will decline. According to the above assumption, the proportion of price decline is the following: （35） P ( x (t ))  exp(   x ( t )) n

x(t )   AiM (t ) represents the total value of all mortgage debt assets that banks intend to sell in the

Jo

where

i 1

current market. When the price of a mortgage asset falls, any bank that holds a mortgage asset suffers a paper loss. Therefore, if the value of mortgage creditor's rights assets is updated from

AiM

i becomes the following: A (t)  ( AiM (t 1) AiM (t))P(x(t))

to

AiM P(x) ,

then the value of

mortgage creditor's rights assets held by bank M i

（36）

Falling mortgage asset prices can lead certain banks to default on their credit. Therefore, after the decline of

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mortgage debt assets, calculating the robustness level of each bank again is necessary:

i (t ) 

AiN (t )  AiM (t )  AiC (t )  LNi (t )  LDi (t ) AiN (t )  AiM (t )  AiC (t )

Step 4. At the end of time t , update

St

（37）

according to the robustness level i of Equation (37). If the credit risk

of

contagion in period t leads to new credit default, then change t into t  1 , enter into the next time, and continue iteration from Step 1. When the spread of credit risk in the t -phase does not lead to a new credit default, the spread of credit default risk stops during this period. The following further proves that the time series model of inter-bank network credit risk contagion constructed above is convergent. To this end, we point out the fact that a bank entering a default will not return to a non-default

through any step in the iterative algorithm, it will not lead to i

p ro

state in the subsequent liquidation process. A stronger conclusion than this fact is that: When i



Ai  Li  0 goes Ai

0. We can verify the impact of each event on i

turn: Step 1 will reduce the amount of interbank assets of the affected bank

i

by a certain amount

in

AiN , therefore, i

Pr e-

AiN  i  0 ; Step 2 does not change the robustness of the bank run party and causes the Ai

will be updated to i 

stability of the bank impacted by liabilities to change to

Ai  AiN  ( Li  AiN ) A  Li A  Li  i  i  i  0 ; N N Ai  Ai Ai  Ai Ai

Step 3 negative externalities brought about by unliquidated selling, that is, the decline in asset prices, have no impact on the stability of banks; Step 4 will cause bank i 's external assets

i

will be changed into

AiM

to reduce

AiM ; hence, bank i 's stability

Ai  AiM  Li Ai  Li    i  0 . At this point, it shows that the bank that has defaulted Ai  AiM Ai

cannot alleviate itself of the default in the subsequent credit risk contagion process. Combined with the above conclusions, the necessary condition of iteration to the period k is: the number of

nd  k

(for any k  1 ). One reason is that the iteration does not stop requiring that, for each

al

banks in default state

time from 1 to k , new defaults occur. At least one new default is generated in each time. Combined with the conclusion proved in the previous paragraph, these defaulting banks will not return to the non-defaulting state in the

urn

subsequent credit risk contagion process. Therefore, the number of defaulting banks in time k must satisfy

nd  k .

Given that the network size of the interbank market is n , at most after n periods, either the iteration has stopped or all banks have fallen into default. Therefore, we prove that the above time series model of interbank network credit risk contagion stops in a finite number of steps. Given credit risk contagion of the interbank market through the network of time series analysis, we can see that the credit risk in the interbank market in the network (G , A,  ,  ,  ,  ) transmission depends on the initial economic

2 ,

Jo

impact on the interbank market network s , macroeconomic situation sensitivity its sensitivity robustness level

3 ,

1 , counterparty solvency sensitivity

regulation allowed minimum levels of liquidity, demand

sensitivity of mutual interaction , where, the initial shock and q, i.e., s  s ( p , q ) . For the convenience of expression, risk contagion process of the interbank market network.

s

,

and price

is determined by the parameters of beta distribution

p

(( p, q),1,2 ,3, , ) is called the time series credit

5. Simulation Analysis The regularity characteristics of credit risk contagion of the interbank market network are analyzed in order to

Journal Pre-proof

test the validity and scientific basis of the credit risk contagion model of the interbank market network. Simulation analysis is carried out through computational experiments. First, the relevant parameters of the interbank market network's bank credit degree were assumed to truncate the minimum value l  0.3 and maximum value h  1 of the power law distribution and the power law exponential   1.25 of the power law distribution.   0.1 ,   3 ,

Pr e-

p ro

of

  0.3 and interbank market network size N  1000 were generated for the topological structure of the interbank market network. According to the requirements of the Basel III agreement on the composition of assets, set the parameters in the bank balance sheet as follow: initial total assets of the bank A  1000 , proportion of depositors' deposits   0.55 , total interbank assets of all banks   0.4 , the power-law distribution parameter   0.9 , and proportion of liquidity assets   0.04 . We will carry out computational experiments under the following three different scenarios and describe the influence mechanism and evolution of interbank network credit risk contagion in accordance with the construction process of the time series contagion model of interbank network credit risk contagion in Section 4. Scenario 1. When the initial shock arrives and causes some banks to default on credit derivatives, all banks will become vigilant and withdraw liquidity, and there is a certain probability of a run on their debt banks. When some banks suffer a run, they may not have enough liquidity to pay off their debts. Therefore, they will sell external assets to obtain liquidity. When the number of similar external assets sold in the market increases, the price of such assets will decrease. Given that the mark-to-market characteristics of the balance sheet, all banks holding such assets will suffer book losses. Therefore, scenario 1 considers interbank asset losses caused by the initial shock, bank runs caused by asset losses, and bank asset sales caused by the run on the debt banks, thereby resulting in book losses caused by the decline in asset prices. Scenario 2. On the basis of scenario 1, the decline in the price of external assets caused by the bank selling external assets in each phase is not considered, that is, the bank can always sell external assets at the initial price of external assets. Scenario 2 considers a bank run and asset selling, but it does not take into account book loss of asset prices due to asset selling. Scenario 3. Bank runs are not considered and banks are not allowed to sell external assets to obtain liquidity, which means that all the credit defaults in scenario 3 are caused by the initial shock contagion and that initial credit default is not amplified by shortage of liquidity. 5.1 Structure characteristics of interbank market network

urn

al

On the basis of the above parameter setting, this section mainly explores the influence of the lower limit l of bank credit, the threshold  of interbank credit, and the size N of interbank market network on the average edge connection probability P of the interbank market network to analyze the evolutionary characteristics of the topology structure of the interbank market network. The calculation experiment process was repeated for 100 times to reduce the accidental influence of random factors, and the mean of the average edge connection probability P of the interbank market network for 100 times was calculated (Figure 1). 1

1

1 0.9

0.9

0.9

0.8

0.8

0.8

0.7

0.7

0.7

0.6

0.6

0.5 0.4

0.5

0.6

0.3

0.4

0.5

0.2

Jo

0.4 0.1

0.2

0.3

0.3

0.4

0.5

0.6

Lower limitation of credit l

0.7

0.8

0.9

0.2 0.1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Inter-bank credit threshold

0.8

0.9

1

0

0

100

200

300

400

500

600

The number of banks n

700

800

900

1000

(b) P depends on  (c) P depends on n (a) P depends on l  Fig. 1. Effects of the lower limit l , interbank credit threshold , and the size N of the interbank market on the average connection probability P . Figure 1 (a), demonstrates that the average edge connection probability P of the interbank market network is positively correlated with the lowest credit rating in the entire interbank market. The greater the credit of bank i and

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bank j in the interbank market network, the greater the probability that the credit interaction

cij

between bank

i

and

 lh . Moreover, when the lower limit l of credit rating is greater than 0.6, the average connection probability P  1 , that is, the interbank market network evolves into a fully connected network. Figure 1(b) depicts that the average connection probability P of the interbank market network is negatively

bank j exceeds the credit threshold

of

correlated with constant  , which illustrates that, when banks in the interbank network market expect economic development in the future to be relatively sluggish, it will be more difficult for them to establish credit-lending relationships. Figure 1(c) illustrates that the size of the interbank market network N has a weak impact on the average connection probability P of the interbank market network.

p ro

The in-degree distribution Pin(d) and out-degree Pout (d) distribution of the connectivity degree of the interbank market network (for directed network, the connectivity degree can distinguish the in-degree from the out-degree) are important indicators to describe the network structure of the interbank market. The average edge connectivity probability P and the degree distribution satisfy the relationship:  1  P  ( dPin (d )   dPout (d )) 2 d 1 d 1

Therefore, the probability distribution of the out-degree

and in-degree din of the interbank market network

al

Probability P in (d)

Probability P out (d)

Pr e-

is shown in Figure 2.

dout

（38）

5.2 Initial shock

urn

(a) In-degree distribution (b) Out-degree distribution Fig. 2. Plots of in-degree distribution of the interbank market network in (a) and out-degree distribution of the interbank network in (b). Figure 2(a) illustrates that the number of debt banks of each bank is distributed in an extreme way, with a small number of one or two debt banks and a large number of almost all banks in the network being their debt banks. This is consistent with the way of using the threshold method to construct the network of the interbank market in the present study. A bank with a high credit rating can become the debt of most other banks, while a bank with a low credit rating can only become the debt of a bank with a high credit rating. Figure 2(b) exhibits that the number of creditor banks of each bank is almost equal. According to the interaction of credit rating, most banks can establish credit lending relationship with several banks with a high credit rating and become their debt banks.

Jo

Given the analysis in Section 3.4, we assume that the proportion of initial shock to the external assets of bank i obeys beta distribution. We make 200 initial shock calculation experiments for each value of beta distribution index q from 1 to 100 with p  1 . On the basis of this computational experiment, the evolution of credit risk contagion in the interbank network under three scenarios was analyzed.

p ro

of

Bank credit difault probability

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Pr e-

Fig. 3. Effects of Beta distribution parameter q on bank credit default probability in three different scenarios. According to Figure 3, by comparing the evolution characteristics of bank credit default probability under scenarios 2 and 3, no matter what the Beta distribution index of the initial shock obeys, there will always be

p2  p3 .

Given the situation where there is a difference between scenarios 2 and 3, in situation 2 banks can be sold at the market value of constant external assets to obtain liquidity assets, and Figure 3 shows that, although the banks face creditor banks runs between bank debt to reduce impact, as long as the sale of external assets will not lead to declines in asset prices, the risk aversion of banks to sell assets can effectively weaken their own debt repayment of liquidity shocks and thus avoid giving itself a credit default. In Figures 1, 2 and 3, by comparing the scenarios of bank credit default probability it can be found, when considering a bank run, that banks sell external assets to cause a decline in asset prices, paper losses are brought about by situation (1) scenarios, bank credit default probability and the tendency of the Beta distribution parameters q is not monotonous, but there is a rise after an initial falling process. Given that the only difference between scenarios 1 and 2 is whether the book loss caused by the decline of external asset prices is

p1  p2 of bank credit default in scenario 1 is higher than that in scenario 2, which is

al

considered, the probability

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completely caused by the decline of external asset prices. Therefore, when q  [20, 35] , with the increase of q, the probability of bank credit default in scenario 1 shows an upward trend, which is caused by the sharp increase of liquidity demand caused by the increase of default. When q  [20, 35] , with increased q, although the initial impact was monotonously reduced, as the interbank run became increasingly serious, all banks needed a higher level of liquidity, which led to excessive selling of external assets. Thus, the price of external assets fell sharply and the bank's book loss rose sharply, leading to a rise in the probability of bank credit default instead of a decline. Considering the bank run, leading to sale of assets, causing the fall of asset prices and paper losses under the condition of scenario 1, the bank's credit default probability

p1 is significantly higher than when it does not consider

bank run behavior, and does not allow banks to sell external assets to obtain liquidity situation (scenario 3) bank credit

p3 . This scenario reveals that each bank hopes to mitigate the liquidity impact caused by the run

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default probability

on its creditor banks by selling external assets to avoid risks. However, the selling of external assets also leads to decline in the price of external assets, resulting in book losses. When deciding whether to run on their debtors or not, banks are trapped in a “prisoner’s dilemma”, that is, a non-pareto optimal Nash equilibrium. When more initial defaults occur, for banks, whether other banks run on their debt banks or not, it is the best strategy for bank i to choose to run on their debt banks, so as to avoid falling into credit default to the greatest extent.

5.3 Sentiment According to Figure 2, liquidity shortage caused by interbank runs has a significant impact on credit risk

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contagion. As noted earlier, bank runs are linked to the mood of bank policymakers. In the bank run model, and

1 , 2

3 measure the investment sentiment of bank decision makers and determine the probability of bank decision

makers running on their debt banks. When panic is severe in the overall economic environment, the probability of bank runs is higher, which corresponds to the larger

1 , 2 and smaller 3 . At this time, market participants tend to

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think that there will be a series of serious credit defaults in the future, which can have a serious impact on their own robustness. Creditor banks will be more likely to run on their debt banks in order to avoid liquidity shortage.

(a) scenario 1

(b) scenario 2

1 and 2 on bank default probability

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Fig. 4. Effects of the interaction of bank decision-makers' sentiment

(c) scenario 3

under different scenarios, in which (a)(b)(c) is for scenarios 1, 2, and 3.

(b) scenario 2

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(a) scenario 1

Fig. 5. Effects of the interaction of bank decision-makers' sentiment

(c) scenario 3

1 and 3 on bank default probability

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under different scenarios, in which (a)(b)(c) is for scenarios 1, 2, and 3.

(a) scenario 1

(b) scenario 2

Fig. 6. Effects of the interaction of bank decision-makers' sentiment

(c) scenario 3

2 and 3 on bank default probability

under different scenarios, in which (a)(b)(c) is for scenarios 1, 2, and 3. As illustrated in Figures 4, 5, 6, and 7, the probability of bank credit default caused by credit risk contagion changes significantly with parameters

1 , 2 and 3 under three different scenarios. As depicted in Figure 4(a),

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1 and 2 , the probability of bank credit default p1 first increases and then decreases. This is because when 1 and 2 are small and the increase of 1 and 2 will lead to credit default in the whole banking with the increase of

system, the phenomenon of run will slowly increase and liquidity shortage will gradually become serious. Therefore, the probability of bank credit default in the entire interbank market network will slowly increase. When

1 and 2

situation under the bank credit default probability

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are large, it means that banks are in a state of extreme panic. When a bank's credit default occurs in the banking system, all banks will run on their debt banks, and each bank has almost the same amount of interbank loans. Therefore, when all banks choose to run on their debt banks at the same time, any bank can use the liquidity assets obtained from the run on its debt banks to repay the run it faces. Therefore, the liquidity impact brought by the run will be reduced, and the consequence of interbank credit risk contagion will be reduced. As demonstrated in Figure 4(b), the probability that the bank will fall into default due to credit risk contagion is related to the gradual decrease of panic when the price of external assets decreases due to the bank's selling of external assets not being considered. When all banks are able to keep a balance between banks and can turn external assets into liquid assets unconditionally, banks have sufficient liquidity to defend against the claims of a bank run liquidity shortage, so the

p2 is less than 2 situation under 1 bank credit default probability

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p1 . As denoted in Figure 4(c), when neither bank runs are allowed nor banks are allowed to sell external assets (scenario 3), the probability of bank credit default p3 is a quantity independent of the changes in 1 and 2 , and p3  p1, indicating that the bank's selling of external assets actually intensifies the contagion effect of credit risk. The

(a)

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above conclusions are further confirmed in Figures 5 and 6.

Fig. 3. Effects of the sentiment parameter

(b)

(c)

1 , 2 and 3 on bank credit default probability in three different

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scenarios. As presented in Figure 7, in scenario 1, which is closest to the real situation, increased sensitivity of banks to

3 can reduce the contagion effect of interbank credit risk. In the case of scenario 2, the increase of 3 only aggravates the contagion effect of credit risk. In scenario 3, 3 has a relatively weak impact on credit risk

contagion.

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their own stability

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Credit default probability

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Fig. 8. Effects of price sensitivity  on bank credit default probability in three different scenarios. As illustrated in Figure 8, during a bank run, when banks sold external assets to cause a decline in asset prices and bring the book loss under the situation of the bank credit default probability

p1 increased with the increase of

asset price sensitivity , and is sensitive to the change of . This illustrates that when the asset price sale assets fell on the interbank market liquidity risk transmission network, there was a more significant effect. In scenarios 2 and 3, external asset prices do not change, so the probability of bank credit default is not affected by 's change in asset price sensitivity.

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Credit default probability

5.4 Banking regulatory indicators

Fig. 9. Effects of the initial liquidity level  on bank credit default probability in three different scenarios. As illustrated in Figure 9, the initial liquid level of  can more significantly reduce during a bank run when banks sell external assets to cause a decline in asset prices and bring the book loss under the situation of the bank credit default probability

p1 , but declines in asset prices that are not considered to bring the book loss under the

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p2 , does not consider the behavior of a bank run, and does not allow banks to sell external assets to obtain a liquidity situation under the influence of bank credit default probability p3 , situation of the bank credit default probability

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p ro

Credit default probability

of

which is relatively weak.

Fig. 10. Effects of the initial interbank asset ratio  on bank credit default probability in three different scenarios. As demonstrated in Figure 10, the balance between banks accounted for   0 (  0 is a threshold , which is

default probability

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approximately 0.34 in our simulation result), and almost does not lead to credit risk contagion bank credit default, but when   [ 0 , 1 ] ( 1 is approximately 0.44 according to our simulation result), interbank assets accounted for at the results of the interbank market credit risk contagion effect are remarkable, the higher the initial bank assets proportion between infection caused by the bank credit risk, the higher the default probability. By comparing scenarios 1 and 2 of

p1 and p2 , it can be concluded that the banks that sell external asset prices falling paper losses on

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interbank credit risk contagion caused relatively significant impact, but between bank asset accounts for the proportion of total assets  is 0.44 or higher, the influence of paper losses differences gradually disappears. With the increase of  , the proportion of banks holding external assets will decrease correspondingly, and the impact of falling external asset prices will gradually weaken. In addition, by comparing bank default probability

p2 and p3 obtained

from scenarios 2 and 3, without considering the decline of external asset prices (scenario 2), selling external assets can effectively reduce the default probability of banks compared with taking no measures.

6. Conclusion

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In the interbank market, due to the asymmetry of information, the initial credit default will cause bank institutions to worry about their future liquidity asset level, and then run on their debt banks. Moreover, to repay debts or control their leverage ratio, debt banks often choose to sell illiquid assets, which will in turn affect other banks holding the same type of assets and generate negative externalities. We consider that the initial credit default of a debt bank will cause the creditor bank to worry about the future liquidity level, and then run on its debt bank. Debt at the same time, considering the bank for the repayment of the debt or leverage, caused when external assets held by the sale price is lower, then affects other holders .The interbank lending market network model is based on credit, and on this basis to build a based-on-time series of interbank credit default risks transmission model, the accumulation of credit default risk in the banking system is studied and the contagion effects, as well as the banking system in the initial default leads to credit risk contagion evolution characteristics.

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Competing interests The authors declare that they have no competing interests.

Acknowledgments

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In this study, the impact mechanism of initial default shock, bank emotion factor, and bank supervision index on bank default probability is simulated and analyzed by means of computational experiment. Results corroborate the following: (1) When banks are able to convert non-liquid assets into liquid assets without limit, the selling behavior of banks will effectively reduce the probability of credit default. In reality, when asset prices fall as sales increase, the negative externalities of bank selling lead to a more severe contagion effect than if banks did not sell at all. (2) Panic of bank decision makers in the banking system has a non-monotonous impact on results of the contagion effect of credit risk. With the increase of bank runs, the probability of bank credit default first increases and then decreases. (3) The probability of bank credit default decreases linearly with respect to the proportion of banks' liquid assets. (4) When the proportion of interbank assets is less than a certain threshold, the proportion of interbank assets has no significant impact on the probability of bank credit default. However, when the proportion of interbank assets is larger than the threshold, the probability of bank default increases in proportion of interbank assets in a cliff-like growth. The above conclusions are of great theoretical and practical significance to the monitoring of credit risk contagion in the interbank market. However, this study did not use real data to estimate and fit model parameters, which is one of the possible development directions of this study in the future.  

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We wish to express our gratitude to the referees for their invaluable comments. This work was supported by the National Natural Science Foundation of China (Nos. 71871115, 71501094, 71771116, 71501131), the Natural Science Foundation of Jiangsu Province of China (Nos. BK20150961, BK20161398), the Key Project of Philosophy and Social Science Research in Colleges and Universities in Jiangsu Province (No. 2017ZDIXM074), the innovation team Project of philosophy and social sciences in Colleges and Universities in Jiangsu Province (No. 2017ZSTD005), Education ministry science and technology innovation platform cultivation project（2017PT27）.  

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Highlights

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Design/methodology/approach – This study constructs a network model of credit risk contagion in the interbank lending market based on time series. Such network model is based on imperfect information considering that the initial credit default of the debt bank will cause the creditor bank to run on its debt bank and the debt bank’s fire sale of external assets held to repay the debt or control the leverage ratio that reduce asset prices and create spillovers. By the theoretical deduction and simulation method, we study how the contagion effects of credit risk accumulate and spread in the interbank market network. In addition, we study the evolution characteristics of credit risk contagion caused by the initial default of debt banks in the interbank market. Findings – Results show that (1) When banks are able to convert non-liquid assets into liquid assets without limit, the selling behavior of banks will effectively reduce the probability of credit default. In reality, when asset prices fall as sales increase, the negative externalities of bank selling lead to a more severe contagion effect than if banks did not sell at all. (2) Panic of bank decision makers in the banking system has a non-monotonous impact on results of the contagion effect of credit risk. With the increase of bank runs, the probability of bank credit default first increases and then decreases. (3) The probability of bank credit default decreases linearly with respect to the proportion of banks' liquid assets. (4) When the proportion of interbank assets is less than a certain threshold, the proportion of interbank assets has no significant impact on the probability of bank credit default. However, when the proportion of interbank assets is larger than the threshold, the probability of bank default increases in proportion of interbank assets in a cliff-like growth. Originality/value – This paper are of great theoretical and practical significance to the monitoring of credit risk contagion in the interbank market. Financial regulators can adjust fit model parameters to manage and control the credit risk contagion in the interbank market.