Chain bankruptcy size in interbank networks: the effects of asset price volatility and the network structure
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Abstract
One bankruptcy of a certain bank can make another bank go bankrupt. This phenomenon is called chain bankruptcy. Chain bankruptcy is a kind of “systemic risk,” a topic that has received a great deal of attention from researchers, recently. Here, we analyzed the effect of the asset price fluctuation and the interbank lending and borrowing network on chain bankruptcy by using an agentbased simulation. We found that: (1) as the rate of change in asset price grows, the total number of bankruptcies increases. On the other hand, when the rate of change in asset prices exceeds a certain value, the total number of bankruptcies became unvarying; (2) as the density of links increases, the total number of bankruptcies decreases, except when a certain situation occurs in core–periphery networks. These results suggest that factors causing bankruptcy are asset price fluctuations and the network structure of the interbank network.
Keywords
Systemic risk Chain bankruptcy Interbank networksIntroduction
The 2008 global financial crisis derived from a much smaller incident that occurred in an American company’s housing loan department [2]. Moreover, the European sovereign debt crisis, which had a significant negative impact on Spain and Italy, began with a sharp downgrade of Greece’s debt status by creditrating companies [8].
The risk posed to the whole financial system caused by the bankruptcy of a single financial institution is called “systemic risk,” and in recent years, the studies that promote systemic stability have attracted a great deal of attention.
Previous studies
Based on the above background, we examined the factors affecting a kind of systemic risk called chain bankruptcy in our study evaluating financial systemic stability.
Imakubo [3] and Hashimoto and Kurahashi [1, 6] have researched systemic risk. Imakubo [3] focused on shortterm financing and did not deal with chain bankruptcy. Hashimoto and Kurahashi [6] explained the causal dependence between bankruptcy and financial transaction networks and devised a means of forecasting the progress of chain bankruptcies. Those work, however, treated the price of risk assets such as stocks and bonds on the balance sheet as exogenous and did not consider the interaction between price fluctuation and systemic risk.
 1.
the volatility of marketable assets that banks hold and
 2.
the type of interbank lending and borrowing network.
Model
Brief description of model
In our model, a certain bank is made to go bankrupt forcibly at the first moment of each trial. Under this condition, we count how many banks go bankrupt at the last moment. In this way, we examine situations in which the number of bankruptcies increases. We denote the bank made to go bankrupt at the beginning as the “start bank.”
 1.
Initialize the settings.
 2.
Buy or sell risk assets.
 3.
Update the price of risk assets.
 4.
Update the balance sheet of each bank.
 5.
Judge whether banks become bankrupt or not, and deal with bankrupt banks.
 6.
Repeat steps 2–5.
Interbank network
Each bank borrows money from another bank and lends money to another bank. A network in which each bank corresponds to a node and an account is a link is called an interbank network. In the simulation, the interbank network is a directed graph to indicate whether a bank borrows or lends money. There is a 50% chance of a bank borrowing and a 50% chance of a bank lending. The three networks used in this model are described below.
 1.
core–periphery,
 2.
scale free,
 3.
random.
Core–periphery networks
 1.
a twolayer structure consisting of a core and periphery,
 2.
all nodes in the core connect to every other node in the core,
 3.
core is a hub to the periphery,
 4.
the periphery consists of clusters.
 1.
Connect the nodes of the ten large banks to each other.
 2.
Divide the 90 smaller banks into ten groups.
 3.
Each large bank and each group of smaller banks made in step 2 are made to correspond to each other in a onetoone fashion, and the large bank connects to each smaller bank in the group.
 4.
In each group of smaller banks, four smaller banks connect to each other and form a cluster.
 5.
Each bank connects to other banks which it does not unite in random order.
Scalefree networks

Growth: networks get larger over time as nodes are added to them.

Preferential selection: nodes that connect more to other nodes have a higher probability of connecting to new nodes.
These characteristics are called scale free and are observed in realworld networks such as the Web [7].
 1.
Connect all nodes of the ten large banks to each other.
 2.
Add nodes as smaller banks to the network made in step 1.
LinkSum is doubled in the denominator of Eq. 1, because a link that connects bank A and bank B is counted twice (one link is considered to be a link of A connected to B and a link of B connected to A).
Equation 1 means that the more links a bank has, the higher is the probability of it connecting to an added bank. As a result, a lot of links are concentrated on a small number of nodes, and the scalefree property is realized. This addition is repeated 90 times. Each node is connected by roulette selection.
Random networks
Random networks are those in which links are connected in random order. In this study, a Monte Carlo method is used to make random networks.
Risk assets
Brief description of market
Algorithm to buy and sell bank assets
 1.
Fundamental term
In this term, the price is predicted by considering the difference between the logical price and market price.In Eq. 7, \(F^{i,s}_{t}\) is the fundamental term of a stock s which a bank i holds, \(p^{s*}_{t}\) is the logical price at a time point t, \(p^{s}_{t}\) is the market price at a time point t, and \(\tau ^{s*}\) is the average regression speed, which is the speed at which the market price approaches the logical price when the market price is largely different from the logical price.$$\begin{aligned} F^{i,s}_{t} = \frac{1}{\tau ^{s*}}\ln \left( \frac{p^{s*}_{t}}{p^{s}_{t}}\right) . \end{aligned}$$(7)  2.
Chart term
This term predicts prices by considering the tendency of the market price before the current step.In Eq. 8, \(C^{i,s}_{t}\) is a chart term of a stock s which a bank i holds, \(/tau^{i}\) is the length for which the tendency of the market price is dealt, and \(p^{s}_{t}\) is the market price at time point t.$$\begin{aligned} C^{i,s}_{t} = \frac{1}{\tau ^{i}}\sum ^{\tau ^{i}}_{j=1}\ln \frac{p^{s}_{tj}}{p^{s}_{tj1}}. \end{aligned}$$(8)  3.
Noise term
The noise term of a stock s a bank i holds, \(N^{i,s}_{t}\), follows a normal distribution of \(N(0,\sigma _{\epsilon })\).
Bank bankruptcy
Judging whether banks are bankrupt
VaR stands for value at risk, which is a quantitative index to measure the risk of a certain stock. The denominator of Eq. 9 is the product of the risk of the stock and the number of stocks; i.e., it is the total risk. When VaR falls below 0.04, the bank goes bankrupt.
Dealing with bankrupt banks
When a certain bank goes bankrupt, other banks which have lent to it can no longer collect debts from it. On the other hand, the bankrupt bank does not have to clear its debts. For the sake of brevity, the marketable assets of the bankrupt bank are not sold to the market. Namely, each bank continues to hold marketable assets after it goes bankrupt, but it can no longer buy and sell them.
Results and discussion
 1.
the volatility of marketable assets that banks hold and
 2.
the type of interbank lending and borrowing network.
Effect on fluctuation of marketable assets
Values of coefficient of price fluctuation \(\alpha \)
\(\alpha \)  0  0.25  0.5  0.75  1.0  2.0 
The results in Fig. 2 indicate that as the coefficient of price fluctuation grows, the number of bankrupt banks increases.
Comparing the cases in which \(\alpha \) is 0 and \(\alpha \) is 1.0 or 2.0, we can see that as soon as the price fluctuates, the number of bankrupt banks increases. This is because the fluctuation of marketable assets enlarges the risk that banks hold. In particular, the number of bankrupt banks rapidly increases between the case in which \(\alpha \) is 0 and the case in which \(\alpha \) is 0.5. However, the average number of bankrupt banks does not change much between the cases in which \(\alpha \) is 0.5 and \(\alpha \) is 2.0. This is because when \(\alpha \) exceeds a certain limit, almost all banks holding marketable assets over a certain amount go bankrupt.
Effect on interbank lending and borrowing networks
The results for each network are described below.
Core–periphery networks
In the core–periphery networks, the number of links determined in step 5 in Sect. 2.2 was set as the parameter.
Value of parameters in core–periphery networks
Number of links of large banks  20  30  50  
Number of links of smaller banks  5  8  10  20 
Patterns of parameters in core–periphery networks
Large banks  Smaller banks  Large banks  Smaller banks  

Pattern 1  20  5  Pattern 7  20  10 
Pattern 2  30  5  Pattern 8  30  10 
Pattern 3  50  5  Pattern 9  50  10 
Pattern 4  20  8  Pattern 10  20  20 
Pattern 5  30  8  Pattern 11  30  20 
Pattern 6  50  8  Pattern 12  50  20 
As the number of links increases, the average number of bankrupt banks decreases, except for certain patterns. The exceptions are patterns 7–9 and 10–12, where the numbers of links are 10 or 20. The reason for this is explained in the next section.
Scalefree networks
Patterns of parameters in scalefree networks
Number of links of added banks  2  3  5  8  10 
The number of bankrupt banks decreased in a monotone manner as the number of links of added banks and the number of links in the whole network increased.
Random networks
Parameter patterns in random networks
Large banks  Smaller banks  Large banks  Smaller banks  

Pattern 1  6  3  Pattern 3  20  10 
Pattern 2  10  5  Pattern 4  50  20 
The results in Fig. 5 show that as the number of links increases, the number of bankrupt banks decreases.
Figure 6 shows a schematic view of the loan amounts. The length of the bar at the bottom represents the loan amount; it is almost constant in patterns I and II. The bar is partitioned; each square corresponds to the amount of money which bank X lent to each bank.
In this situation, the debts which bank X can not collect varies depending on patterns I and II. The debts that bank X cannot collect in pattern I is larger than in pattern II. Therefore, when the number of links is small and a client goes bankrupt, the damage is larger. This is the reason for the result that the number of bankrupt banks decreases as the number of links increases.
Moreover, as the number of links increases, the number of bankrupt banks decreases, but in some cases of the core–periphery networks, the number of bankrupt banks did not decrease in a monotone manner. This happens when the number of links of smaller banks is 10 and 20. Let us consider the behavior in that case. When the number of links of the smaller banks is large, the change in the number of links of the large banks does not affect the number of bankrupt banks. On the other hand, when the number of links of the large banks is large, as the number of links of the smaller banks increases, the number of bankrupt banks decreases.
 1.
links which connect large banks to large banks,
 2.
links which connect large banks to smaller banks,
 3.
links which connect smaller banks to smaller banks.
With all of the above things considered, when the number of links is large in core–periphery networks, the change in the number of type2 links does not affect the number of bankrupt banks, whereas an increase in the number of type3 links makes the interbank networks stable.
Conclusion
 1.
the volatility of risk assets that banks hold, and
 2.
the type of interbank lending and borrowing network.
 1.
As the coefficient of price fluctuation \(\alpha \) increases, the number of bankrupt banks increases.
 2.
As the number of links increases, the number of bankrupt banks decreases.
Regarding conclusion 2, it can be said that when banks lend almost the same amount of money, they should divide up that amount and lend those portions to more banks so as to disperse the risk.
Future plans
By adding a supply chain to an interbank network, we can consider banks loaning to companies and price fluctuations due to company activities. We want to make a framework in which “banks”, “companies”, and “markets” are integrated and thereby observe the effect of a bankruptcy of a certain company on the whole economic system.
As a first step, we will examine the supply chain and combine it with the interbank network. Regarding supply chains, there are studies of Miura [5]. In Miura’s research [5], a growth model of the enterprise’ s trading network is set up, and features of the supply chain are explained. Regarding risk asset markets, we also plan to build an artificial market based on Torii’s research [10].
Notes
Acknowledgements
This research was supported by the Japan Society for the Promotion of Science (KAKENHI grant no. 15H02745) and the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan, via the Exploratory Challenges on PostK computer study on multilayered multiscale spacetime simulations for social and economic phenomena.
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