Chain Bankruptcy Size in Inter-bank Networks:
the Eﬀects of Asset Price Volatility
and the Network Structure
Ryo Hamawaki1 , Kiyoshi Izumi1 , Hiroki Sakaji1 ,Takashi Shimada1 , and Hiroyasu Matsushima1
1. School of Engineering, The University of Tokyo, Tokyo, Japan

Abstract. One bankrupt of a certain bank can make another bank gobankrupt. 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 theasset price fluctuation and the inter-bank lending and borrowing network on chain bankruptcy by using an agent-based simulation. We found
that: (1) As the rate of change in asset price grows, the total numberof 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 theinter-bank network.
Keywords: Systemic risk · Chain bankruptcy · Inter-bank networks.

1

Introduction

1.1

Introduction

The 2008 global financial crisis derived from a much smaller incident that occurred in an American company’s housing loan department [2]. Moreover, theEuropean sovereign debt crisis, which had a significant negative impact on Spainand Italy, began with a sharp downgrade of Greece’s debt status by credit-ratingcompanies[8].
The risk posed to the whole financial system caused by the bankruptcy ofa single financial institution is called “systemic risk,” and in recent years, thestudies to promote systemic stability have attracted a great deal of attention.
1.2

Previous studies

Based on the above background, we examined the factors aﬀecting a kind ofsystemic risk called chain bankruptcy in our study evaluating financial systemicstability.

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R.Hamawaki, K.Izumi, H.Sakaji, T.Shimada, H .Matsushima

Imakubo[3] and Hashimoto and Kurahashi[1][6] have researched systemicrisk. Imakubo[3] focused on short-term 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 offorecasting the progress of chain bankruptcies. Those work, however, treated theprice 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.
This study used agent simulations to examine the eﬀect on chain bankruptcy
of varying
1. the volatility of marketable assets that banks hold and2. the type of inter-bank lending and borrowing network.

2

Model

2.1

Brief description of modelIn our model, a certain bank is made to go bankrupt forcibly at the first momentof 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 numberof bankruptcies increases. We denote the bank made to go bankrupt at the
beginning as the “start bank.”
The procedure of each trial is as follows:
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 to 5
In each trial of the simulation, Steps 2 to 5 are repeated 100 times. The
model includes 100 banks. As city (large) banks account for 10% to 20% of thewhole, the number of large banks is 10, while the number of smaller banks is 90.
The inter-bank network is described in section 2.2. A large bank has more links
in the inter-bank network than a smaller bank does. Moreover, the amount ofassets that each bank holds rises as the number of links increases, as well[4].
2.2Inter-bank 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 linkis called an inter-bank network. In the simulation, the inter-bank network is adirected 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.

Chain Bankruptcy Size in Inter-bank Networks

3

The structure of inter-bank networks diﬀers from country to country and fromperiod to period, but most inter-bank networks have the same characteristics as
the following three kinds of network.
1. core-periphery
2. scale-free
3. random
The outline of these networks and how to make them are described below.
Core-periphery networks
Characteristics of core-periphery networks According to Imakubo[3], the interbank networks of the short-term inter-bank market have the following characteristics.
1. a two-layer 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
Graph 1 is a conceptual diagram of a core-periphery network.

Fig. 1. Conceptual diagram of core-periphery network

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R.Hamawaki, K.Izumi, H.Sakaji, T.Shimada, H .Matsushima

How to make a core-periphery network To reproduce the features of core-peripherynetworks in our simulation, we perform the following steps:
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 tocorrespond to each other in a one-to-one 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 otherand form a cluster.
5. Each bank connects to other banks which it does not unite in random order.
Step 1 corresponds to feature 2 above, while steps 2 and 3 correspond tofeatures 1 and 3, and step 4 to feature 4.
Scale-free networks
Characteristics of scale-free networks Scale-free networks have two importantcharacteristics[7].
– 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.
According to above features, the number of links that each node has is variable.
A small minority of nodes in the network, the hub, have a lot of links. On theother hand, most nodes have only a few links.
These characteristics are called scale-free and are observed in real world networks such as the Web [7].
How to make scale-free networks To create of scale-free networks and reproduce
their features, we perform the following steps:
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.
The procedure for adding nodes is as follows. Added nodes have a fixed
number of links. These nodes connect to the existing network, and the probabilityof connecting to certain bank i,pi , is determined by Eq. 1.
Linki
(1)
2 ∗ LinkSum
In this equation, Linki means the total number of links connected to banki, and LinkSum means the total number of links in the networks.
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).
pi =Chain Bankruptcy Size in Inter-bank Networks

5

Equation 1 means that the more links a bank has, the higher the probabilityof it connecting to an added bank is. As a result, a lot of links are concentrated
on a small number of nodes, and the scale-free 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 connectedin random order. In this study, a Monte Carlo method is used to make random
networks.
2.3

Risk assets

Brief description of market Each bank holds risk assets such as stocks andtheir price at a time point t, pt , as follows:(nb,t−1 − ns,t−1 ),(2)Nwhere nb,t is the number of risk assets bought at t, ns,t is the number of risk
assets sold at t, N is the total number of stocks, and α is a coeﬃcient of pricefluctuation.
In this situation, when the coeﬃcient of price fluctuation, α, is zero, eq. 3 issatisfied.
pt = pt−1(3)
pt = pt−1 + αpt−1

This equation means that the price of marketable assets is fixed at any time.
In this study, a single kind of marketable asset is used.
Algorithm to buy and sell bank assets The algorithm to buy and sellmarketable assets is based on the local trader of Torii[9]. The logical price ofmarketable assets at a time point t, ps∗t , is exogenously given by Eq. 4.
s∗
s∗
s∗
s∗
ps∗
t = pt−1 + µs∗ pt−1 + σs∗ pt−1 ∆Wt−1(4)

In eq. 4, µs∗ means the average logical price, σs∗ means the standard deviations∗of the logical price, and ∆Wt−1
means a Wiener process.
Local traders determine the expected return by using Eq. 5, in which theweighted average of the next three items is calculated.
r̂ti,s =

(
)
1
i,s
i,s
i,s
i
i
i
w
F
+
w
C
+
w
N
F t
C t
N t
i + wiwFi + wC
N

(5)

i
i
In Eq. 5, wFi , wC
and wNare the weights of the “fundamental term”, “chartterm”, and “noise term.” Each weight is determined for each bank by using arandom number so that each will have its own trading strategy.
The expected price at a time point, t + τ i , p̂i,st+τ i is determined using eq. 6i,sand r̂t , which is calculated in eq. 5.
)
(
i,s i
s
(6)
p̂i,s
=
p
exp
r̂
τ
i
t
t
t+τ

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R.Hamawaki, K.Izumi, H.Sakaji, T.Shimada, H .Matsushima

1. Fundamental term
In this term, the price is predicted by considering the diﬀerence between the
logical price and market price.
( s∗ )
pt1i,s
Ft = s∗ ln
(7)
τ
pst
In Eq. 7, Fti,s is the fundamental term of a stock s which a bank i holds, ps∗t
is the logical price at a time point t, pst is the market price at a time pointt, and τ s∗ is the average regression speed, which is the speed at which themarket price approaches the logical price when the market price is largely
diﬀerent from the logical price.
2. Chart termThis term predicts prices by considering the tendency of the market pricebefore the current step.
τ
pst−j
1 ∑
ln
τ i j=1 pst−j−1i

Cti,s =

(8)

In eq. 8, Cti,s is a chart term of a stock s which a bank i holds, /taui isthe length for which the tendency of the market price is dealt, and pst is themarket price at time point t.
3. Noise term
The noise term of a stock s a bank i holds, Nti,s , follows a normal distributionof N (0, σϵ ).
The expected price, which is calculated in eq. 6, p̂i,st+τ i is compared to thes,thebank buys a stock, andislargerthanpmarket price at t, pst . When p̂i,s
t
t+τ i
i,s
s
when p̂t+τ i is smaller than pt , the bank sells a stock. Each bank performs atransaction, following the above algorithm.
2.4

Bank Bankruptcy

Judging whether banks are bankrupt Whether banks go bankrupt or isdetermined by eq. 9.
CARt =

N etwortht
V aR ∗ StockN umt(9)

In this equation, CARt is the capital adequacy ratio at t, N etwortht means
net worth at t, and StockN umt is the number of stocks the bank holds.
V aR stands for value at risk, which is a quantitative index to measure therisk of a certain stock. The denominator of Eq. 9 is the product of the risk of thestock and the number of stocks; i.e., it is the total risk. When V aR falls below
0.04, the bank goes bankrupt.

Chain Bankruptcy Size in Inter-bank Networks

7

Dealing with bankrupt banks When a certain bank goes bankrupt, otherbanks 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 canno longer buy and sell them.

3

Results and Discussion

As mentioned in section 1.2, we studied chain bankruptcy by using an agent
simulation in which two parameters were varied:

1. the volatility of marketable assets that banks hold and2. the type of inter-bank lending and borrowing network.

The experiments were conducted 100 times for each parameter, and the average number of bankrupt banks was calculated.

3.1

Eﬀect on fluctuation of marketable assets

The coeﬃcient of price fluctuation α, defined in eq. 2, was set as the parameter.
Table 1 shows the six patterns of α tried in the simulations.

Table 1. Values of coeﬃcient of price fluctuation α

α 0 0.25 0.5 0.75 1.0 2.0

We used core-periphery networks in which the large banks had 30 links andthe smaller banks had ten links.

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R.Hamawaki, K.Izumi, H.Sakaji, T.Shimada, H .Matsushima

Fig. 2. Relationship between coeﬃcient of price fluctuation and the number ofbankrupt banks
The length of error bars corresponds to each sample variance.

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 α is 0 and α is 1.0 or 2.0, we can see that assoon as the price fluctuates, the number of bankrupt banks increases. This isbecause the fluctuation of marketable assets enlarges the risk that banks hold.
In particular, the number of bankrupt banks rapidly increases between the casein which α is 0 and the case in which α is 0.5. However, the average number ofbankrupt banks does not change much between the cases in which α is 0.5 and α
is 2.0. This is because when α exceeds a certain limit, almost all banks holding
marketable assets over a certain amount go bankrupt.

Chain Bankruptcy Size in Inter-bank Networks

3.2

9Eﬀect on inter-bank 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 section 2.2 was set as the parameter.
Table 2 shows the number of links of large banks and smaller banks.

Table 2. 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

We conducted a round-robin on the number of large banks versus the numberof smaller banks, as shown in Table 3.

Table 3. Patterns of parameters in core-periphery networks

pattern 1
pattern 2
pattern 3
pattern 4
pattern 5
pattern 6large banks smaller banks
large banks smaller banks
20
5
pattern 7
20
10
30
5
pattern 8
30
10
50
5
pattern 9
50
10
20
8
pattern 10
20
20
30
8
pattern 11
30
20
50
8
pattern 12
50
20

Figure 3 shows the result.

10

R.Hamawaki, K.Izumi, H.Sakaji, T.Shimada, H .Matsushima

Fig. 3. Relationship between the number of links and the number of bankrupt banks
in core-periphery networks
The length of error bars corresponds to each sample variance.

As the number of links increases, the average number of bankrupt banks
decreases, except for certain patterns. The exceptions are patterns 7 to 9 and 10
to 12, where the numbers of links are 10 or 20. The reason for this is explainedin the next section.

Scale-free networks In scale-free networks, the number of links of the added
banks in step 2 in section 2.2 was set as the parameter. Table 4 shows the numberof links of added banks.

Table 4. Patterns of parameters in scale-free networks

number of links of added banks 2 3 5 8 10

Figure 4 shows the result.

Chain Bankruptcy Size in Inter-bank Networks

11

Fig. 4. Relationship between the number of links and the number of bankrupt banks
in scale-free networks
The length of error bars corresponds to each sample variance.

The number of bankrupt banks decreased in a monotone manner as thenumber of links of added banks and the number of links in the whole network
increase.

Random networks In random networks, the number of links that large banks
and smaller banks have is set as the parameter. Our purpose for studying random
networks was to know the basic behavior of the number of bankruptcy banks as
the number of links increased, so we examined four diﬀerent parameters shownin Table 5.

Table 5. Parameter patterns in random networks

pattern 1
pattern 2large banks smaller banks
large banks smaller banks
6
3
pattern 3
20
10
10
5
pattern 4
50
20

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R.Hamawaki, K.Izumi, H.Sakaji, T.Shimada, H .Matsushima

Fig. 5. Relationship between the number of links and the number of bankrupt banks
in random networks
The length of error bars corresponds to each sample variance.

The results in Fig. 5 show that as the number of links increases, the numberof bankrupt banks decreases.

To see the reason for this result, we note that as the number of links increases,
the number of bankrupt banks decreases in the random networks and scale-free
networks. In these simulations, the sum of all the assets that each bank holds is
set to a certain range, so no matter how large or small the number of links ofeach bank is, the loan amount is almost the same. For example, let us consider
cases in which the number of links of a certain bank X is 3 (pattern I) or 20(pattern II).

Chain Bankruptcy Size in Inter-bank Networks

13

Fig. 6. Schematic view of loans

Figure 6 shows a schematic view of the loan amounts. The length of the barat 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.
Here, let us suppose a bank to which bank X lent money goes bankrupt (Fig.
7).

Fig. 7. Schematic view of loans when a certain bank goes bankrupt

In this situation, the debts which bank X can not collect varies dependingon 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 ofbankrupt banks did not decrease in a monotone manner. This happens when

14

R.Hamawaki, K.Izumi, H.Sakaji, T.Shimada, H .Matsushima

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.
Here, we can distinguish three types of link.
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
In this network model, because every large bank is connected each other, type1 does not increase even if the number of links changes. So, when the number oflinks of the large banks increases, only type 2 increases. Also, when the numberof links of the smaller banks increases, type 2 and 3 increase.
With all of the above things considered, when the number of links is large incore-periphery networks, the change in the number of type-2 links does not aﬀect
the number of bankrupt banks, whereas an increase in the number of type-3 links
makes the inter-bank networks stable.

4

Conclusion

4.1

Conclusion

We studied the eﬀect on chain bankruptcy of varying
1. the volatility of risk assets that banks hold, and2. the type of inter-bank lending and borrowing network.
The conclusions are as follows:
1. As the coeﬃcient of price fluctuation α increases, the number of bankrupt
banks increases.
2. As the number of links increases, the number of bankrupt banks decreases.
Regarding conclusion 1, while the number of bankruptcies rapidly increases
between the case in which α is 0 and the case in which α is 0.5, it is almost
constant when α is larger than 0.5.
Regarding conclusion 2, it can be said that when banks lend the almost sameamount of money, they should divide up that amount and lend those portions
to more banks so as to disperse the risk.

Chain Bankruptcy Size in Inter-bank Networks

4.2

15

Future plans

We should consider the eﬀect of connecting the inter-bank network to a supply
chain, i.e., a network of companies.

Fig. 8. Framework of future plans

By adding a supply chain to an inter-bank network, we can consider banks
loaning to companies and price fluctuations due to company activities. We wantto 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 theinter-bank 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].

4.3

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 Post-K computer study on multilayered multiscale space-time simulations forsocial and economic phenomena.

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R.Hamawaki, K.Izumi, H.Sakaji, T.Shimada, H .Matsushima

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