Contagion in Chinese banking system: A comparative study of maximum entropy method and transfer entropy method
Changzhi Lianga,b, Yanzhen Yaoc, Xiaoqian Zhua,b, Jianping Lia,1 a b c
Institution of Policy and Management, Chinese Academy of Sciences, Beijing 100190, China School of Management, University of Chinese Academy of Sciences, Beijing 100190, China
School of Mathematics and Statistics, Central China Normal University, Beijing 430079, China
Abstract: Maximum entropy method (MEM) is the traditional approach to estimate interbank exposure matrix, which is the key to assess contagion effect in banking system. Recently, a new transfer entropy method (TEM) is proposed to estimate interbank exposure matrix. This paper employs the two approaches to estimate interbank exposure matrix of Chinese banking system, and then simulate the contagion process given the initial failure of a bank in the system. The comparison of the results indicate that MEM is consistent with TEM when it comes to general and qualitative features of Chinese banking system; while the differences begin to emerge when it comes to the exact and quantitative features of Chinese banking system. Key words：maximum entropy method, transfer entropy method, interbank exposure matrix
1. Introduction
With the deregulation policy and financial innovation, financial market has been much more complex in the last decade. The inter-correlation between different financial instruments, different financial institutions, different financial sectors and even different countries has also been strengthened significantly. Similar to butterfly effect in meteorology, for the excessively complex
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financial system, a perturbation can cause a profound change to the state of the system. Latest examples include the subprime crisis and European sovereign debt crisis. Considering the devastating effect of financial crisis, both practitioners and researchers are beginning to study how a negative event can spread risk to the whole financial system and produce systematic consequences. Some scholars regard the banking system as a network with specific structure, and investigate its behavior in terms of contagion; their results show that structure of interbank market plays a critical role in determining the consequences of contagion [1-7]. Meanwhile, some other authors analyze the contagion effect in real banking system, such as German, UK, Italy, Switzerland, and Dutch [8-13]. In these empirical studies, authors employ maximum entropy method (MEM) to estimate interbank exposure matrix, and use the matrix as a description of the interbank market structure. MEM is thought to be an unbiased estimation method under inadequate information [14], however, it is also criticized as to deviate from reality. Recently, a transfer entropy method(TEM) is proposed to estimate interbank exposure matrix [15], and reveal a different contagion process compared with MEM. This paper employs both MEM and TEM to analyze contagion effect of Chinese banking system in 2011 and 2012. Then results of the two methods are compared in terms of similarities and differences. The comparison indicates that
2. MEM and TEM For a banking system consisted of N banks，interbank exposure matrix can be represented by the following N × N matrix： x11 x X = 21 ... xN 1
x12 x22 ... xN 2
... x1N ... x2 N ... ... ... xNN
The element xij , located in row i and column j , represents the credit exposure of bank i to bank j , it can be interpreted as bank j ' s liability to bank i as well. By adding up all elements in row i , we get bank i ' s total interbank assets ai ; and by adding up elements in column j , we get bank j ' s total interbank liabilities l j . If we denote the summation of the
matrix as L , then we have the following equation: N
N
N
N
a ∑ = l ∑∑= x ∑=
=i 1
i
=j 1
j
=i 1 =j 1
ij
L
Generally speaking, ai and l j can be obtained from financial statements of the bank, while how ai and l j is allocated among xij is unknown, which is why we need to estimate interbank exposure matrix X .
2.1
Estimate interbank exposure matrix with MEM
According to quite a few previous studies, MEM is described as an unbiased approach to estimate the interbank exposure matrix. It turns the estimation into an optimization problem whose object is to maximize the entropy of the interbank exposure matrix. The entropy of the matrix is N
N
defined as −∑∑ xij ln xij , thus the optimization problem can be summarized as: =i 1 =j 1
N
N
min ∑∑ xij ln xij =i 1 =j 1
N
N
s= .t. ∑ xij a= i， ∑ xij l j
=j 1 =i 1
= xii 0,
for i ≠ j , xii > 0
Solution of the problem can be calculated in two steps: 1.
Calculate the approximate solution. In this step, we first derive the initial solution, with the method of Lagrange multiplier, which is xij = ai l j . Then the initial solution is modified according to the second constraint, thus we have the approximate solution, written as:
0, i = j xij = ai l j , i ≠ j Obviously, the approximate solution doesn’t meet the requirement of the first constraint; therefore it will be adjusted in step 2. 2.
Adjust the approximate solution with RSA algorithm, which can be summarized as the following steps:
I.
Row adjustment: xij → xij ρi , ρi =
ai
∑
∀j / xij > 0
II.
xij lj
Column adjustment: x ji → x jiσ j , σ j =
∑
∀i / xij > 0
III.
xij
Return to step I
When ρi − 1 , σ j − 1 < 10−3 , the iteration is terminated and the interbank exposure matrix is obtained.
2.2
Estimate interbank exposure matrix with TEM
For two time series I and J , the transfer entropy from J to I is defined as:
= TJ → I
∑ p (i
t +1
, itk , jtl ) log p ( it +1 | itk , jtl ) − ∑ p ( it +1 , itk ) log p ( it +1 | itk )
= ∑ p ( it +1 , i , j ) log k t
l t
p ( it +1 | itk , jtl ) p ( it +1 | itk )
Where, it +1 represents the state of I at t + 1 , itk represents the sequence it , , it − k +1 ， and jtl represents the sequence jt , , jt −l . According to the definition of transfer entropy [16], TJ → I measures the information flowing from J to I and vice versa.
Li et al. [15] suggest that the transfer entropy matrix T , whose element Tij is the transfer entropy from stock price of bank i to stock price of bank j , can be used to approximate the structure of interbank exposure matrix. Thus the approximate solution for interbank exposure matrix is xij = Tij . Since Tii = 0, ∀i ∈ {1, N } , the approximate solution is consistent with the constraint that N
N
xii = 0, ∀i ∈ {1, N } , yet the constraint = ∑ xij a= ∑ xij l j is not satisfied. Hence we adjust i, =j 1 =i 1
the approximate solution with RSA algorithm to obtain the exact solution as well.
2.3
Furfine’s risk contagion mechanism
Once a bank defaults, how will other banks response and how is the risk spreading through interbank exposure matrix depend on the contagion mechanism. Furfine [12] regards contagion as
a sequential default process. Denote the equity capital of bank i as Ci , loss rate given default as
θ , external equity capital loss as α Ci , then Furfine’s mechanism can be generalized as: Step 1: An initial bank i defaults. Step 2: Since bank j lends X ji to bank i , it suffers a capital loss of θ X ji . The condition
θ X ji +α C j >C j is satisfied and bank j defaults because of insolvency. Step 3: Since bank k lends X ki to bank i and X kj to bank j , it suffers a capital loss of θ ( X ki + X kj ) , reaching the critical point that θ ( X ki +X kj ) +α Ck >Ck , as a result bank k defaults. Step 4: Continue step 3 until no further bank defaults. In the empirical study, the contagion process of Chinese banking system is simulated with Furfine’s mechanism.
3. Empirical study on Chinese banking system
In this section, we adopt MEM and TEM to estimate the exposure matrix of Chinese banking system in 2011 and 2012, and then simulate the contagion process. According to China Banking Regulatory Commission, the total assets of Chinese banking industry amount to 133.6 trillion Yuan, in which the 16 listed banks contribute 85.9 trillion Yuan, accounting for a percentage of 64.3%. Other banks are much smaller and can be merged as a single bank called bank A, which along with other 16 listed banks constitute Chinese banking system. Table 1 gives the names of the 16 listed banks. Table 1 The 16 listed banks Names of the banks ChinaMinsheng Bank(CMBC)
Bank of Beijing(BCCB)
Spd Bank(SPDB)
Bank of Ningbo(BONB)
Industrial Bank(IB)
China Construction Bank(CCB)
China Merchants Bank(CMB)
ChinaEverbright Bank(CEB)
Bank of Communications(BOCOM)
Bank of Nanjing(BONJ)
Agricultural Bank of China(ABC)
Bank of China(BOC)
Huaxia Bank(HXB)
ChinaCitic Bank(CITIC)
Industrial And Commercial Bank of China(ICBC)
Pingan Bank(PAB)
3.1 Data description The data come from several sources. The interbank market data, including deposits in other banks, due from banks, financial assets purchased under resale, deposits from other banks, interbank borrowing, repurchase agreements, and equity capital, are extracted from the annual reports of the 16 listed banks in 2011 and 2012. Stock prices of the 16 listed banks are daily data from Wind database, and the sampling period is from 2011/1/4 to 2012/12/31, with 487 records. 16
For the merged bank, its equity capital C A = ∑ Ci
16
∑T C
=i 1 =i 1
16
i
i
, the aggregated interbank assets
16
16
a A = ∑ ai 16 , and the aggregated interbank liabilities l A + ∑ li =a A + ∑ ai . Such arrangement =i 1 =i 1
i =1
ensures that the merged bank along with other 16 banks form an enclosed system.
3.2 Simulation results with MEM We derive the interbank exposure matrix of Chinese banking system in 2011 and 2012 with MEM. Considering the size of the matrix, only bilateral exposures of five biggest banks, namely ICBC, CCB, BOC, ABC, BOCOM, are presented, as shown in Table 2 and Table 3. Table 2 Bilateral exposures of five biggest banks with MEM in 2011
billion
ICBC
CCB
BOC
ABC
BOCOM
ICBC CCB BOC ABC BOCOM
0.0 104.3 221.9 152.0 77.1
103.5 0.0 146.4 100.3 50.9
173.2 115.2 0.0 167.9 85.1
82.5 54.9 116.7 0.0 40.5
82.4 54.8 116.5 79.8 0.0
Table 3 Bilateral exposures of five biggest banks with MEM in 2012
billion
ICBC
CCB
BOC
ABC
BOCOM
ICBC CCB BOC ABC BOCOM
0.0 153.9 184.0 191.6 76.4
116.8 0.0 115.2 120.0 47.8
201.5 166.3 0.0 207.1 82.6
102.1 84.3 100.8 0.0 41.8
95.8 79.1 94.6 98.5 0.0
According to Furfine’s mechanism, both loss rate given default θ and capital loss rate α affect interbank contagion process, thus we set scenarios with different combinations of α and θ ,
in which α takes the value among 0, 0.3, 0.5, and θ increases from 0.1 to 1 with the step size of 0.1. Table 4 and Table 5 present results of the simulated contagion process. Column 1 represents the initial defaulted bank, while column 2 to 11 represents the total number of defaulted banks at different α and θ . For example, 1 means only the initial bank defaults, and no other banks are defaulted, 17 means all other 16 banks default in the contagion process triggered by the initial defaulted bank. For banks not listed in the table, they won’t cause other banks to default for any combination of α and θ .
Table 4 Chinese banking contagion process in 2011 with MEM
θ
0.1
ICBC CCB BOC ABC BOCOM CIB A
1 1 1 1 1 1 1
ICBC CCB BOC ICBC BOC
1 1 1 1 1
0.2 1 1 1 1 1 1 1 1 1 1 1 1
0.3 1 1 1 1 1 1 1 1 1 1 1 1
0.4 1 1 10 1 1 1 1 1 1 1 1 1
0.5 α = 0.5 11 1 11 1 1 1 1
α = 0.3 1 1 1
α=0 1 1
0.6
0.7
0.8
0.9
1
16 1 14 1 1 1 1
17 17 17 1 1 1 1
17 17 17 17 17 1 1
17 17 17 17 17 17 1
17 17 17 17 17 17 17
10 1 11
11 1 11
16 1 14
16 17 15
17 17 17
1 1
1 1
1 10
11 11
11 11
Table 5 Chinese banking contagion process in 2012 with MEM
θ
0.1
ICBC CCB BOC ABC BOCOM CMBC CIB A
1 1 1 1 1 1 1 1
0.2 1 1 1 1 1 1 1 1
0.3 1 1 1 1 1 1 1 1
0.4 1 1 1 1 1 1 1 1
0.5 α = 0.5 17 1 15 1 1 1 1 1
α = 0.3
0.6
0.7
0.8
0.9
1
17 1 17 1 1 1 1 1
17 1 17 1 1 1 1 1
17 17 17 1 1 1 17 1
17 17 17 17 1 1 17 1
17 17 17 17 17 17 17 17
ICBC BOC ICBC BOC
1 1
1 1
1 1
1 1
1 1
1 1
1 1 1 1
1 1 α=0 1 1
1 1
17 15
17 17
17 17
17 17
1 1
1 1
1 1
1 14
17 15
3.3 Simulation results with TEM The first step of TEM is to calculate the transfer entropy matrix of stock price sequences. The sequence is divided into two parts, the first part is from 2011/1/4 to 2011/12/31, and the second part is from 2012/1/4 to 2012/12/31. With TEM, we obtain interbank exposure matrix in 2011 and 2012, as shown in Table 6 and Table 7.
Table 6 Bilateral exposures of five biggest banks in with TEM 2011
billion
ICBC
CCB
BOC
ABC
BOCOM
ICBC CCB BOC ABC BOCOM
0 128.2 183.6 151 44
130.9 0 49.8 72 24.9
103.1 62.9 0 72.1 77.6
51.8 34.6 64.3 0 19.8
36.4 29.9 51.7 30.6 0
Table 7 Bilateral exposures of five biggest banks in with TEM 2012
billion
ICBC
CCB
BOC
ABC
BOCOM
ICBC CCB BOC ABC BOCOM
0 191.5 304.9 171.5 65.7
127.8 0 89 74.2 37.4
253.5 251.6 0 96.7 131.1
88.8 65.6 27.5 0 19.5
21.6 28.1 43.3 227 0
The contagious process is simulated under the same scenarios as in section 3.2. Table 8 and Table 9 give the results.
Table 8 Chinese banking contagion process in 2011 with TEM
θ
0.1
ICBC BOC CIB SPDB
1 1 1 1
0.2 2 1 1 1
0.3 2 1 1 1
0.4 9 1 1 1
0.5 α = 0.5 10 1 1 1
0.6
0.7
0.8
0.9
1
10 1 1 1
10 1 2 1
17 12 9 1
17 17 10 1
17 17 10 10
CITIC
1
ICBC CIB
1 1
ICBC
1
1
1
1 1
1
2 1
1
2 1
1
2
1 α = 0.3 2 1
α=0 2
1
1
1
1
10
10 1
10 1
10 1
10 2
10 3
2
2
9
10
10
Table 9 Chinese banking contagion process in 2012 with TEM
θ
0.1
ICBC BOC CIB BOCOM
1 1 1 1
ICBC BOC
1 1
ICBC
1
0.2
0.3
1 1 1 1
1 1 1 1
1 1
1 1 1 1
1 1
1
0.4
1 1
1
1
0.5 α = 0.5 11 1 1 1
α = 0.3 1 1
α=0 1
0.6
0.7
0.8
0.9
1
11 11 1 1
17 17 1 1
17 17 10 1
17 17 10 1
17 17 17 17
1 1
11 1
11 11
17 14
17 17
1
1
1
1
11
4. Comparison of the results
To better understand the simulation results in Table 4, Table 5, Table 8 and Table 9, we choose the scenario when α =0.5 , and plot the number of defaulted banks against loss rate given default θ for both MEM and TEM, as illustrated in Figure 1 and Figure 2.
Defaulted banks triggered by ICBC
Defaulted banks triggered by BOC
20
20
15
15
ND 10 5
T-ICBC M-ICBC
0 0
0.5
θ
1
1.5
ND 10
T-BOC
5
M-BOC
0 0
0.5
Figure 1 Number of defaulted banks versus loss rate given default in 2011
θ
1
1.5
Defaulted banks triggered by BOC
Defaulted banks triggered by ICBC 20
20
15
15
ND 10
T-ICBC
5
M-ICBC
0 0
0.5
θ
1
1.5
ND
10
T-BOC
5
M-BOC
0 -5 0
0.5
θ
1
Defaulted banks triggered by BOCOM 20 15 T-BOCOM
ND 10
M-BOCOM
5 0 -5 0
0.5
θ
1
1.5
Figure 2 Number of defaulted banks versus loss rate given default in 2012
4.1 Features of Chinese banking system revealed by results of both methods Combining Table 4, Table 5, Table 8 and Table 9, as well as Figure 1 and Figure 2, the results of both methods reveal the following features of Chinese banking system. 1.
Chinese banking system is more stable in 2012 than in 2011. Comparing the results in 2011 and 2012, we find that for the same initial bank and α , the starting value of θ to trigger a contagion process is higher in 2012 than in 2011, which means the system is more resilient to risk contagion in 2012.
2.
The seriousness of contagion is positively and nonlinearly correlated with both θ and α . On one hand, greater α means the bank has less capital to cover interbank loss, which makes it more vulnerable to interbank risk contagion. On the other hand, greater θ means
1.5
greater interbank loss, thus a bank’s default will cause greater damage to other banks and increase the possibility of interbank contagion. 3.
Given certain α , there exists a threshold value θ µ for loss rate given default θ . If θ < θ µ , then the default of any single bank will not trigger further defaults. If θ > θ µ , then the default will trigger massive further defaults. A reasonable inference is that in normal condition the interbank market provides liquidity to the banking system, which helps diversify the risk and avoid the occurrence of default. However, in extreme condition, the liquidity shortage caused by a bank’s default rapidly spreads through the interbank borrowing and lending chain, which has a knock on effect on the banking system and leads to massive defaults.
4.2 Differences between results of the two methods Apart from general features of Chinese banking system, the results also indicate some significant differences between maximum entropy method and transfer entropy method. 1.
For the same initial default, α and θ , the two methods give different number of total defaults, corresponding to different system stability. In Figure 1 and Figure 2, the number of defaults given by maximum entropy method is equal to or slightly lower than the number of defaults given by transfer entropy method when θ is small. With θ above the threshold value θ µ , the number of defaults given by maximum entropy method increases rapidly and soon exceeds the number of
defaults given by transfer entropy method, and the gap
between them increases with θ . 2.
For the same initial default and α , MEM and TEM give different threshold value θ µ . Generally speaking, MEM gives a smaller threshold value θ µ than TEM.
5. Conclusion
In this paper, we conduct a comparative study on the traditional MEM and the newly proposed TEM, by applying both methods to analyze contagion effect of Chinese banking system
in 2011 and 2012. At first, interbank exposure matrix is estimated with MEM and TEM respectively. Then contagion process in different scenarios with certain α and θ is simulated. The results show that MEM is consistent with TEM when it comes to general and qualitative features of Chinese banking system. Specifically, results of both methods indicate: banking system is more stable in 2012 than in 2011; their exists positive-nonlinear relationship between seriousness of contagion and θ as well as α ; there exists a threshold value θ µ for loss rate given default θ . However, when it comes to exact and quantitative features, the differences begin to emerge, such as the different number of defaulted banks given identical scenario and different threshold value. Generally speaking, MEM gives a relatively more evenly distributed interbank exposure matrix, thus makes the number of total defaults more sensitive to loss rate given default. As a preliminary research, this paper compares MEM and TEM mainly from an empirical perspective, and need improvement in terms of convincible theoretical demonstration. In future study, theoretical basis of MEM and TEM shall be thoroughly investigated and the two methods shall be evaluated in a more comprehensive way.
Acknowledgements
This research is supported by National Nature Science Foundation of China (No.71071148, No.71003091), Key Research Program of Institute of Policy and Management, Chinese Academy of Sciences (Y201171Z05), the Youth Innovation Promotion Association of Chinese Academy of Sciences.
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