Measuring systemic importance of banks considering risk interactions: An ANOVA-like decomposition method

Journal Pre-proof Measuring Systemic Importance of Banks Considering Risk Interactions: An ANOVALike Decomposition Method Chunbing Bao, Dengsheng Wu, Jianping Li PII:

S2096-2320(19)30098-8

DOI:

https://doi.org/10.1016/j.jmse.2019.12.001

Reference:

JMSE 15

To appear in:

Journal of Management Science and Engineering

Please cite this article as: Bao C., Wu D. & Li J., Measuring Systemic Importance of Banks Considering Risk Interactions: An ANOVA-Like Decomposition Method, Journal of Management Science and Engineering, https://doi.org/10.1016/j.jmse.2019.12.001. 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. © [Copyright year] Production and Hosting by Elsevier B.V. on behalf of China Science Publishing & Media Ltd.

Measuring Systemic Importance of Banks Considering Risk Interactions: An ANOVA-Like Decomposition Method Chunbing Bao a, Dengsheng Wu b, c, Jianping Li b, c,

*

Corresponding author. E-mail address: [email protected] (J. Li).

*

a

School of Management, Shandong University, Jinan, Shandong, China.

b

Institute of Science and Development, Chinese Academy of Sciences, Beijing, China.

c

University of Chinese Academy of Sciences, Beijing, China.

Abstract： ：The systemic importance of a bank is usually measured by its effect on the banking system, conditional on the insolvency of the bank and solvency of other banks. However, banks encounter different kinds of shocks simultaneously in reality. So that, the conditional results give biased estimates of banks’ systemic importance when potential risks are ignored. Researchers like Tarashev et al. proposed the Shapley value method to deal with risk interactions, but it suffers heavy computational costs. This paper proposes an ANOVA-like decomposition method to measure the systemic importance of banks in more complicated and realistic environments, which considers both interactions and individual effects of multiple shocks and provides a more exact estimation of systemic importance. It is found that the method proposed in this paper fits well in the network models. And meanwhile, a discussion between the method proposed in this paper and the Shapley value method is made based on the numerical example, which aims to demonstrate it’s the advantages. The Shapley value method requires 2 n subsystems, while the ANOVA-like decomposition method requires only n+1 model runs. In the application part, the proposed method is adopted to measure the systemic importance of 16 Chinese listed banks. With low computational costs, the model outputs the individual effect, interaction, and total effect of each bank. The results confirm that interactions of different shocks play a significant role in the systemic importance of a bank; thus, the total effect considering interactions should be adopted.

KEYWORDS:

Systemic risk; Interbank network; Risk interaction; Systemically important bank;

ANOVA-like decomposition method

1. Introduction Against the backdrop of rapid integration of the global financial market, banks have become more interconnected via a sophisticated network of multilateral exposures. As a result, insolvency of one bank may trigger an amplified effect on the entire banking system, which is termed as systemic risk. Systemic risk in the banking system has received increasing attention from researchers, regulators, and supranational agencies (Elsinger et al., 2006; Sui et al., 2014; Sui & Wang, 2015; Demange, 2016; Tarashev et al., 2016; Yao et al., 2017; Asongu & Odhiambo, 2019). It is important to identify important banks systemically, which contribute the most to the banking system or have more systemic risk than others have. Generally, two strands of methods are adopted when systemic risk is studied. One focuses on capturing the potential spillover effect of a bank relying on market data, such as first getting the system-wide loss distribution and then measuring the expected loss of the banking system conditional on the adverse event of a bank (Gauthier et al., 2012; Drehmann & Tarashev, 2013; Girardi & Tolga Ergün, 2013; López-Espinosa et al., 2015; Laeven et al., 2016; Nucera et al., 2016; Sedunov, 2016; Tarashev et al., 2016). The other strand of research uses balance-sheet data to construct a network based on bilateral exposures of the banks in the interbank system (Elsinger et al., 2006; Upper, 2011; Raffestin, 2014; Glasserman & Young, 2015; Chen et al., 2016; Roukny et al., 2018). Then, different kinds of mechanisms are proposed to characterize the risk contagion in the banking system, such as Eisenberg and Noe’s interbank clearing algorithm (Eisenberg & Noe, 2001), Furfine’s sequential default algorithm (Furfine, 2003), and several improved algorithms (Nier et al., 2007; Rogers & Veraart 2013). Both strands of methods have their respective advantages and challenges, and one may refer to Vanhoose and Bisias et al. (VanHoose, 2011; Bisias et al., 2012) for a detailed comparison. In this paper, the network models are adopted since they give a causal explanation of the generation of systemic risk instead of a correlational one, compared with the first stream of methods. Although Furfine’s sequential default algorithm has been popular in recent literature (Degryse & Nguyen, 2007; Toivanen, 2009), a major challenge for this algorithm is that it

relieves the losses from later-default banks to banks that had defaulted previously. However, such reduced losses amplify the severity of the next round of contagion effects, which can be treated as a resilient effect in the system. Therefore, Furfine’s contagion mechanism underestimates the systemic risk. To provide one or more shocks to the banking system, Eisenberg and Noe’s (2001) interbank clearing algorithm is adopted to characterize the risk contagion process. Given the initial shocks to the banking system, one can get the interbank payments after clearing, which is an equilibrium considering all possible resilient effects. When assessing the systemic importance of a bank using network models, researchers usually measure the corresponding systemic risk conditional on the insolvency of the bank (Furfine, 2003; Cont et al., 2010; Martínez-Jaramillo et al., 2010; Upper, 2011; Li et al., 2013). It is intuitive to measure the domino effect of the distress of a bank. However, such measures ignore the fact that besides the initial external shock on the particular bank, additional shocks to other banks in the domino path are possible. In fact, all banks in the system may encounter different shocks simultaneously through (i) bilateral exposures (Eisenberg & Noe, 2001; Elsinger et al., 2006); (ii) correlated exposures under a common risk (Huang et al., 2012); (iii) feedback effects from fire-sale of assets (Nier et al., 2007); and (iv) informational contagion (Hasman & Samartín, 2008; Nikitin & Smith, 2008). Thus, the traditional measure considering only the shock to the particular bank is likely to yield a biased estimation to the systemic importance. The purpose of this paper is to discover systemically important banks using a credible measure, under an increasingly complicated environment. Obviously, when the banking system suffers multiple shocks, the shocks not only have individual effect but also affect the network and thus the total effect gets amplified (Frank et al., 2009). For example, systemic loss resulting from simultaneous shocks to bank A and B may be larger than the sum of systemic loss resulting from sequential shocks to bank A and B. In this case, the traditional systemic importance measure of a bank, which refers to the change of systemic risk when a shock (usually an extreme shock resulting in bankruptcy) to the bank materializes, will be inaccurate since the shock to the bank experiences shocks to other banks. Besides, there are higher-order interactions, like the interactions of a triplet.

Some scholars have recognized the importance of simultaneous exogenous shocks to the banking system and have proposed overall systemic risk measures to the banking system (Elsinger et al., 2006). However, they have not given a specific method to determine important banks systemically against such a comprehensive background. Drehmann and Tarashev (Drehmann & Tarashev, 2013; Tarashev et al., 2016) measured systemic importance of banks according to the risk attribution of different banks based on the Shapley value. They treated the banks as players in a game, and the Shapley value is the solution to allocate the players’ contributions in the game. The Shapley value measures the contribution of a player according to the average of the incremental contributions of all combinations of the players. As a result, interactions of different banks are considered if the Shapley value is adopted to measure systemic importance. However, an obvious flaw of the Shapley value method is that the total calculation is 2n times since all subsystems are considered ( Cn + Cn + L + Cn ), 0

which means when

1

n

n is quite large, it is impossible to get the Shapley values.

In this paper, we use an ANOVA-like method (analysis of variance) to identify systemically important banks. Different from the Shapley value method based on game theory, we classify banks systemically importance according to sensitivity to external shocks. If a bank is more sensitive to external shock, it means greater systemic risk given the same external shock. Thus, it is our opinion that banks more sensitive to external shocks should be more systemically important. The most important advantage of the method proposed in this paper is the low computational cost, which is n + 1 . It is found that the proposed method works well in network models. First, we define a systemic risk measure in the network. The measure is based on clearing payments (Eisenberg & Noe, 2001) and has strong properties like monotonicity and concavity, which lead to some interesting conclusions with regards to interaction. Furthermore, an ANOVA-like decomposition method is proposed to study the interactions in the presence of some shocks. We present the banks’ interactions of any order, and show that the sum of the interactions (calculation of which is 2 n ) can be calculated simply (calculation of which is n + 1 ). Non-negativity of interactions is also found through this model, further proving the necessity

of adding interactions into the overall measure of a bank’s systemic importance. What’s more, we prove that superadditivity of systemic importance exists in the banking system, which means systemic loss due to several shocks will be larger than the sum of systemic loss due to each individual shock. Thus, the managerial insight is that we should reduce the possibility of large shocks attacking several banks simultaneously. Another advantage of the model proposed in this paper is that not only the systemic importance of a bank can be estimated, but also that of the banking group dose. Sometimes we care about systemic importance from the point of banking groups. For example, the important banks may be measured and ranked at the national level among all the global systemically important banks. Meanwhile, the model is illustrated by a numerical example through which the foregone Shapley value-based method and the model proposed in this paper are compared and the essential difference between them is explained. Finally, the model is applied to the Chinese banking system. It is showed that widely held banks with large asset sizes are relatively more systemically important according to the method proposed in this paper, which accounts for its applicability when studying systemic importance. Besides, it is found that interactions play a central role in the total effect and, thus, rankings of a bank are different according to the traditional measure and the measure proposed in this paper. The remainder of this paper is organized as follows. First, how to measure systemic risk based on the interbank network is explained. Then a new method to measure systemic importance is proposed, and some properties of systemic importance are discussed. A comparison between the Shapley value method and the proposed method is proposed. Lastly, the results of empirical analysis of systemically important banks in the Chinese banking system are presented.

2. Systemic risk indicator from interbank network Interbank exposures have played an important role in the propagation of the financial crisis of 2007-2008. The interbank exposure structure reflects the trade-off between

risk-taking and risk-spreading through cross-liabilities by banks in the system (Demange, 2016). If one denotes involved banks with a finite set N = {1,L , n} , the structure of the lending relationships among these banks can be represented as the n´ n interbank liability matrix

L , as shown in Equation (1), where l ij represents the obligation of bank i to bank j . Instead of specific elements in the matrix, one can usually get the total interbank assets and total interbank liabilities of a bank, which are the sums of rows and columns ( a i and li ). Therefore, the elements of the credit exposure matrix must be estimated first.

∑j  l11 M  L =  li1  M ln1  ∑i

L l1 j L l1n  l1 O M N M  M L lij L lin  li  N M O M M L lnj L lnn  ln a1 L a j L an

(1)

There are many methods to estimate liability matrices, such as the maximum entropy method (Upper & Worms 2004), the transfer entropy method (Li et al., 2013), and others (Anand et al., 2014, Gandy & Veraart, 2016). Among these methods, the most frequently used is the maximum entropy method combined with a standard iterative algorithm (RAS), due to its easy operation and low demand for data (Sheldon & Maurer, 1998; Eisenberg & Noe, 2001; Upper, 2011; Fang et al., 1997). Maximum entropy method optimizes problems from an information-theoretic perspective, which can be seen as minimizing the total uncertainty subject to all the constraints. Maximum entropy method has two notable advantages. Firstly, very little information, only an individual bank’s total interbank assets and liabilities, is needed. Secondly, entropy optimization is straightforward to implement, using RAS, which can be generalized to handle additional constraints (Anand et al., 2014). After getting the interbank liability matrix, Eisenberg and Noe’s interbank clearing algorithm is employed to describe the contagion mechanism. A formal presentation of this algorithm is given as follows. In the liability matrix L , we denote the total obligation of

bank i to other banks by pi =

å

l . Thus, the vector of total obligations of all banks is

j Î N ij

p = ( p1 , p2 , L , pn ) .

To describe the structure of a network, let ìï lij ïï P ij = ïí pi ïï ïïî 0

if pi > 0 otherwise

(2) ,

and the corresponding P represent the relative liabilities matrix. Additionally, each bank is characterized by an exogenous operating cash flow e i , and one can describe a system by a triplet (P , p, e) , where e is the related vector of e i .

e is the operating cash flow in Eisenberg and Noe (Eisenberg & Noe, 2001), since they only consider the interbank system. In the banking system, insolvency of interbank business does not mean failure of a bank, since non-interbank assets and liabilities are not considered. In this paper, we consider e as the net assets outside the interbank system, which equal to non-interbank assets less the sum of non-interbank liabilities and equity. Under this setting, the condition that bank i defaults becomes n

å

P ji p j + ei - pi < 0.

(3)

j= 1

Given multiple shocks, for a particular bank, if condition (3) is satisfied, failure of bank

i is regarded as a fundamental default. Through the network, a bank’s default may trigger the default of other bank(s). Thus, this extra default is called the contagious default satisfying the following conditions: n

å

n

P ji p j + ei - pi ³ 0 and

j= 1

å

P ji p*j + ei - pi £ 0

(4)

j= 1

where p*j is the total obligation of bank j after clearing, defined as follows. * Definition 1. p is the clearing vector of the banking system (P , p, e) if and only if the

following condition holds:

n

pi* = min[ei +

å

P ji p*j , pi ] " i Î N

(5)

j= 1

The existence and uniqueness of the clearing vector has been proven by Eisenberg and Noe under defined regularity conditions (Eisenberg & Noe, 2001). Besides, a fictitious default algorithm was proposed to find the fixed point at which the clearing vector reaches equilibrium. This algorithm mainly contains the following steps: Step 1: Initialize pi equaling to the initial value. Calculate the net value of each bank, n

å

n

P ji p j + ei - pi . If

å

j= 1

* P ji p j + ei - pi ³ 0 , no bank defaults, and pi equals the initial

j= 1

value and the algorithm terminates; otherwise, go to Step 2. n

Step

2:

Banks

with

å

P ji p j + ei - pi < 0

default.

Record

the

ratio

j= 1

n

qi = (å P ji p j + ei ) / pi < 1 . Replace lij by qi lij , and new P ij , lij , and pi are obtained. j= 1

Repeat Step 2 until no bank defaults. And the final pi can be treated as the approximate

pi* . Given the clearing vector, we use the loss ratio of total assets of the banking system as the systemic risk measure, instead of the number of defaulting banks, as the researchers have usually used (Upper & Worms, 2004, Mistrulli, 2011, Chen et al., 2016), since the former measure considers asset size and is thus more accurate. Combining the fundamental and contagion default loss, we define the endogenous systemic risk metric as follows. Definition 2. Systemic risk due to shocks to the bank i1, i2 ,L , is is defined as loss ratio of the total assets:

å lossratioi1 ,i2 ,L ,is (e ') =

iÎ i1 , i2 ,L ,is

D ei +

å

pi -

iÎ N

å

å iÎ N

pi

pi* (e ') (6)

iÎ N

where

å iÎ i1 ,i2 ,L ,is

D ei denotes total loss of net asset outside the interbank system, i.e.,

fundamental loss; e ' denotes the net asset after absorbing shocks;

pi* (e ') / å pi denotes

å iÎ N

the system recovery rate, and thus

å iÎ N

å

pi -

å

pi* (e ') is seen as a contagious loss where

iÎ N

pi* (e ') is the total assets after clearing and

iÎ N

iÎ N

å

pi is the nominal total assets. Both

iÎ N

fundamental loss and contagious loss are important to systemic risk, which can be seen as external and internal dimensions of the total loss. Obviously, lossratioi1 ,i2 ,L ,is can be used to measure systemic risk of any combination of banks. One may care for the case where there is no contagious loss. If so, there will be only fundamental loss in the systemic importance measure, namely,

å iÎ i1 ,i2 ,L ,is

D ei / å pi . As a iÎ N

result, systemic importance of a bank is determined by the corresponding external loss. Generally, under the same macro environment, or given the same loss ratio, a bank with a larger asset size usually suffers a greater loss; therefore, it has higher systemic importance. This is consistent with some existing literature (Drehmann & Tarashev, 2011), which proves the rationality of the measure. Besides, in the empirical analysis part in Section 4, we show that if a contagious loss occurs, systemic importance measure by Equation (6) is in accordance with the common sense. One may find that we do not hold a distributive perspective. As argued by Brunnermeier and Oehmke, systemic risk usually builds up in the background in a low volatility environment during the run-up phase, because regulations based on risk measures that rely mostly on contemporaneous volatility are not useful (Brunnermeier & Oehmke, 2013). Therefore, the uncertainty of our risk measurement indicator is derived from the occurrence probability of large shocks. What the investigation is that the effect when the large shocks occur. That is to say, we quantify the loss generated from the shocks to different banks in a macroeconomic stress scenario. In section 4, it shows that the indicator is suitable for studying the interactions because it is able to measure the conditional risks of any shock group. As it has been found by Eisenberg and Noe, if the vector p * is a convex, increasing the function of operating cash flow e (Eisenberg & Noe, 2001), each element of the vector p *

and the sum of these elements are also convex and increase the function of e . Given that total assets equal total liabilities, we conclude that the total assets after clearing are convex, increasing the function of e ' . Therefore, the following proposition holds:

Proposition 1. Systemic risk measuring loss ratioi1 ,i2 ,L ,is is concave, decreasing the function of capital e . Proposition 1 states that although the process through which the systemic risk is derived is universally seen as a black box, this function presents some special characteristics that play a crucial role in analyzing interactions. This will be discussed further in the next section. Usually, interbank liabilities matrix L is assumed to be constant when the clearing vector is employed (Elsinger et al., 2006; Rogers & Veraart, 2013; Glasserman & Young, 2015). In this paper, we set e as the sole driver of systemic risk. Therefore, we investigate the interactions of shocks to the banking system, which shows the capacity of banks to absorb shocks.

3. A framework for measuring systemic importance of banks Knowing whether the failure of a particular bank could trigger the failure of others is important not only for crisis management but also for crisis prevention. As we have argued before, simultaneous shocks to different banks may occur for a variety of reasons. It would be useful to extend the simulation by relaxing the assumption that the defaulting bank is randomly selected. Once the notion that simultaneous shocks to different banks are possible is accepted, analysis from the perspective of a hypothetical defaulting bank is inadequate. As proven by Myers and Montgomery (Myers & Montgomery, 1995), if a model is not an additive function of the inputs, the sum of outputs due to individual effects does not equal to the total of outputs. In other words, the total effect resulting from a shock to one bank is not accurately characterized by systemic risk measured in the scenario where only one bank is affected. The difference is due to the shock interacting with other shocks. Thus, in the remainder of this

section we first present an ANOVA-like decomposition method to introduce interactions, then calculate interactions involved in systemic risk, and lastly examine some special properties of systemic risk which are measured by the indicator defined in Equation (6).

3.1 ANOVA-like decomposition method to measure risk interactions Like many problems in engineering, the black box between the inputs and the outputs of the banking system is a particular function which creates systemic risk without supplying a specific mathematical expression. The nominal liability matrix, capital, external shocks and many other factors can be treated as the inputs. Besides, the contagion mechanism works as the function, and systemic risk is the output. When studying risk interactions, we first consider all outputs of different subsets (interactions of factor pairs, interactions of factor triplets and so on). In a mathematical language, if inputs are measurable by a measure µ (such as probability measure or others), the function f ( x ) can be decomposed as follows (Takemura, 1983; Rabitz & Aliş, 1999): n

f ( x ) = f 0 + ∑ f i ( xi ) + ∑ f i , j ( xi , x j ) + L + f i1 ,i2 Kin ( xi1 , xi2 ,L , xin )

(7)

 f0 = E µ [ f ] = L f (x)d µ , ∫ ∫   f ( x ) = E [ f x ] − f = L f (x)∏ d µ − f , i 0 µ ∫ ∫ k ≠i k 0  i i   f (x , x ) = E [ f x , x ] − f (x ) − f (x ) − f , 0 i j i i j j µ  i, j i j L.

(8)

i =1

i< j

In (8), f0 denotes the expected value under measure m and fi ( xi ) denotes the change from expected value if xi is known, and so on. In Equation (7), 2n orthogonal terms correspond to all the subsets. When inputs vary, that is to say, inputs turn from nominal value x 0 to a mutative value x1 , change of output f is n

∆f = f (x1 ) − f (x 0 ) = ∑ ∆fi + ∑ ∆fi , j + L + ∆fi1 ,i2Kin . i =1

(9)

i< j

Such a process is treated as sensitivity analysis, and x1 presents a sensitivity case (Borgonovo & Smith, 2011). In particular, when µ is the Dirac − δ measure, a general

value ∆fi1,i2Kis equals to the value of

f

1 1 1 obtained with xi1 , xi2 ,L, xis at x i1 , x i2 ,L, x is . The

remaining parameters are at the base case value (Sobol, 2003; Borgonovo, 2010). Thus,

∆fi1,i2Kin becomes  ∆ i f = f ( xi1 , x (0− i ) ) − f ( x 0 ),  1 1 0 0  ∆ i , j f = f ( xi , x j , x ( − i , j ) ) − ∆ i f − ∆ j f − f ( x ),  L , where ( xi , x( −i ) ) = ( x1 , x2 ,L, xi −1, xi , xi +1,L, xn ) denotes the input vector with 1

0

0

0

0

1

0

0

(10)

xi

fixed

at xi1 and the remaining x0 , and so on. Generally, indicators in Equation (10) are equal to the values with factors of one set fixed by subtracting the indicators of the subsets. These indicators reflect the residual effects of factors in a set and can thus be treated as the interaction index (Borgonovo & Smith, 2011). In the next subsection, we define the interactions in systemic risk.

3.2 Measure systemic importance by ANOVA-like decomposition method When measuring interactions in systemic risk, the output is systemic risk metrics, such as loss ratio i , i 1

2

,L , is

defined in Equation (6), and inputs are risk factors used in the risk

measuring process, such as external shocks, internal structure, and so on. In previous studies, failure is imposed on a particular bank, and a conditional loss is obtained. Such a loss is caused by the contribution of an individual factor, i.e., ei . Next, it is posited that the conventional conditional loss of a hypothetical vulnerable bank is the same as the first indicator in Equation (10), which is named the individual effect of negative shock.

Definition 3. The individual effect of a shock to a bank i is the change of systemic risk resulting from bank i ’s net assets outside the interbank system suffering a loss. The individual effect at differentiable points when the shock is infinitesimal can be written as follows (Borgonovo, 2010):

λi1 = f (ei1 , e(0−i ) ) − f (e0 )  fi′(e0 )dei ,

(11)

0 1 where ei denotes the capital of bank i in the changed environment, e( −i ) denotes the

nominal capital vector except e i . We see from Equation (11) that individual effect refers to the effect of bank i to the banking system, where only bank i suffers a loss to net assets outside the interbank system, and other banks do not suffer any loss. The individual effect is a kind of spillover effect, which is widely used by many scholars studying systemic risk (Furfine, 2003; Upper, 2011). Different from the traditional assumption, we consider a setting where simultaneous shocks occur. Particularly, when two banks are affected simultaneously, interaction between the effects on the two banks inevitably contributes to the total loss. Such contributions of factor pairs are called 2-order interactions of the shock to bank i or j and are denoted as follows, given infinitesimal shock changes:

λi, j = f (ei1 , e1j , e(0−i, j ) ) − ∆i f − ∆ j f − f (e0 )  fi,′′j (e0 )dei de j .

(12)

s banks are affected concurrently, an s -order effect

Generally, if

∆ f i1 ,i 2 K in to bank i

is defined as follows.

Definition 4. An s -order interaction of shock(s) to the bank i is the residual contribution to systemic risk, which equals to systemic loss ratio i ,i ,i 1

2

,L , i s - 1

subtracting all the lossratios of

all subsets of {i, i1,L , is- 1} . Therefore, the definition of total effect is below.

Definition 5. The total effect of a shock to the bank i , and is used to indicate the systemic importance, is the additive effect of individual effects and any order interactions associated with ei . We write total effect as n

λiT := ∑

∑

s =1 i1 < i2
λi ,i 1

n

2

,K , is

= λi1 + ∑

∑

s = 2 i1 < i2
λi ,i 1

2

,K , is

.

When shocks are small enough and f is differentiable, the total effect equals to

(13)

λiT  fi′(e0 )dei +

n

∑

j =1, j ≠i

fi′′, j (e0 )dei dej +L+ f xn1 ,x2 ,L, xn (e0 )d e1de2 Lden .

By summing all the terms associated with

ei

(14)

, a simple expression of λ iT

is

(Borgonovo 2010) λ iT = f ( e 1 ) − f ( ei0 , e 1( − i ) ). ,

where

(15)

(ei0 , e1(−i) ) = (e11, e12 ,L, ei1−1, ei0 , ei1+1,L, en1 ) . That is to say that the total effect of ei

is

equal to the output when all parameters are in the sensitivity case, subtracting the output when fixing

ei

at its base value and all other parameters in the sensitivity case. Therefore, each

bank’s total effect of the shock to the bank can be derived at the computational cost of n + 1 times. According to Equation (13), total interactions (second term on the right of the equation) contain 2 n - C n1 - C n0 items, which means a high computational cost. However, if we move individual effects to the left of the equation from the right, there will be

λiI := λiT − λi1. In Equation (16)‚ λ i I

(16)

represents the contribution of

ei

to ∆f due to interactions. By

Equation (11) and Equation (15), we have

λiI = λiT − λi1 = f (e1 ) − f (ei0 , e1(−i ) ) − f (ei1, e0(−i) ) + f (e0 ).

(17)

Equation (17) indicates that interactions can be derived by 2 n + 2 model runs, which means lower computational cost. This alludes to the feasibility of using such a method to measure interactions in systemic risk and other high dimensional problems. In the realistic networks, it is interesting to study the total effect containing interactions of shocks to bank groups, for example, in the global banking system, i.e. whether European banks have a higher systemic importance than Asian banks. It is obvious that if the function is not additive, the sum of total effects of inputs does not equal to the effects of the case where 0 these factors change together. A simple example is that f (x1, x2 ) = x1x2 , x = (1,2) and

T T T x1 = (2,3) , thus λ1 + λ2 = (6 −1× 3) + (6 − 2 × 2) ≠ (λ1,2 = 6 − 2) . As a result, when

investigating the systemic risk of factor groups, we cannot simply add total effects of all individual factors. Hence, next we introduce how to measure the systemic importance of bank groups. We consider a general case where banks in the system are partitioned into

m

groups,

and the corresponding impact factor e is partitioned as: e1 e2 L es1 es1 +1 es1 + 2 L es2 K esm−1 +1 esm−1 + 2 L en . 1424 3 14 4244 3 1442443 g1

g2

(18)

gm

We write the change of ∆ f due to the changes in parameters by groups: n

m

i =1

i< j

∆f = f (g1 ) − f (g0 ) = ∑∆gi f + ∑∆gi , g j f +L+ ∆gi , gi ,L,gi f , 1

2

m

(19)

where  ∆ gi f = f ( gi1 , g 0( − i ) ) − f (g 0 ),  1 1 0 0  ∆ gi , g j f = f ( gi , g j , g ( − i , j ) ) − ∆ gi f − ∆ g j f − f (g ),  L.

(20)

Then individual effect, total effect, and interaction effects can be similarly written as:

λg1 = f ( gi1, g(0−i) ) − f (g0 ),

(21)

λgT = f (g1 ) − f ( gi0 , g1(−i) ),

(22)

λgI = λ gT − λg1 .

(23)

3.3 Properties of systemic importance of banks We start from the notion of interactions between multiple shocks. Obviously, if none of the banks are interconnected, interactions will not happen. In other words, the contagion process of multiple shocks is the base for the existence of interactions. As a result, the fundamental loss should have no effect on the interactions, which forms the following proposition. Proposition 2. Interactions are the same when one measures the systemic risk from a contagion loss perspective or a total loss perspective. Proof of Proposition 2. If interactions are measured by the total loss as given in Definition 2, interactions are expressed in Equation (17):

λiI = λiT − λi1 = f (e1 ) − f (ei0 , e1(−i ) ) − f (ei1 , e(0−i) ) + f (e0 ) . Substituting f ( x ) with loss ratioi1 ,i2 ,L ,is , we have

λ

I i

∑ ∆e + ∑ p − ∑ p (e ) ∑ ∆e = − ∑p k

k∈N

k ∈N

k

k∈N

k

k ∈N

∑ p − ∑ p (e , e ∑p k

k ∈N

− ∆ei +

1

k

k ∈N

∆ei +

* k

k ∈N

k ∈N

* k

0 i

1 ( −i )

)

k

k∈N

k ∈N

k ∈N

* k

k∈N

k∈N

1

k∈N

k

∑ p − ∑ p (e , e ∑p

k ∈N

k

k ∈N

k∈N

k ∈N

k ∈N

* k

k

0 i

1 ( −i )

)

k

k ∈N

* k

0 i

1 ( −i )

) −

k

∑ p − ∑ p (e ) + ∑p k

* k

0

k

k∈N

k

k∈N

* k

∑ p − ∑ p (e ) ∑ p − ∑ p (e , e = − ∑p ∑p k

∑ p − ∑ p (e , e ∑p

0 i

k

1 ( −i )

) −

k

∑ p − ∑ p (e ) + . ∑p k ∈N

k

* k

k ∈N

k∈N

0

k

After simplification, the items on the right of the equation are the same as the result calculated by the only contagious loss. W It seems whether interactions are the same with or without fundamental loss could be treated as a criterion for determining rationality of a systemic importance measure. Proposition 1 is in accordance with the intuition, as stated before, and proves the rationality of choosing the sum of fundamental and contagious loss as the systemic risk measure. Thus, some measures, such as the number of defaulting banks, seem defective. However, it does not mean fundamental loss is not relative to interactions since it is the incentive for further contagion. When interactions occur, one managerial insight of interest is whether the interactions between effects of multiple shocks have a positive or negative impact on the systemic risk. Proposition 3 in the following answers the question. Proposition 3. If systemic risk is measured by the loss rate defined in Definition 2, the interactions of shocks to individual banks are nonnegative. Proof of Proposition 3. Since lossratioi1 ,i2 ,L ,is (we denote it by f in the proof) is decreasing with respect to e , the following equation must hold:

λi1 = f (ei − ∆ei , e(−i ) ) − f (e) ≥ 0 ,

λiT = f (ei − ∆ei , e(−i ) − ∆e(−i ) ) − f (ei , e(−i ) − ∆e(−i ) ) ≥ 0 . What remains is the need to prove the non-negativity of the interactions, which is to prove

λiI = λiT − λi1 =f (ei −∆ei ,e(−i) −∆e(−i) ) − f (ei ,e(−i) −∆e(−i) ) − f (ei −∆ei ,e(−i) ) + f (e) ≥ 0 . Define the two assistant functions:

f1 (ei ) = f (ei , e(−i) −∆e(−i) ) and f2 (ei )=f (ei , e(−i) ) . Then the upper equation can be rewritten as:

f1 (ei ) − f2 (ei ) ≤ f1(ei −∆ei ) − f2 (ei −∆ei ) Letting g (ei ) be g(ei )=f1 (ei ) − f2 (ei ) , we next need to prove that g (ei ) is decreasing. In fact, we have

g′(ei )=f1′(ei ) − f2′(ei ) . Since f is a convex function, we have f ′′ ≥ 0 , and the decreasing monotonicity of g (ei ) holds:

g ′(ei )=f1′(ei ) − f 2′(ei )=f ′(ei , e(− i ) − ∆e( −i ) ) − f ′(ei , e( −i ) ) ≤ 0 and thus g(ei ) is decreasing. W Proposition 3 states that when multiple shocks occur, interactions positively contribute to the total changes. This suggests the non-ignorable role of interactions. And as a result, the individual effect of a shock to a bank is an underestimated assessment of the bank’s systemic importance. In addition, apparently when we study interactions of shock groups, the property as shown in Proposition 3 still holds, namely: Corollary 1. Interactions of shock groups are nonnegative when systemic risk is measured by

lossratioi1 ,i2 ,L ,is . Proof of Corollary 1. We denote shock group gi with a set {e1 , e2 ,L , es } , thus interactions of shock group gi can be written as:

λgI = λgT − λg1 =f (ei − ∆ei , ei − ∆ei ,L , ei − ∆ei , e( − g ) − ∆e( − g ) ) − i

i

i

1

1

2

2

s

s

i

i

f (ei1 − ∆ei1 , ei2 − ∆ei2 ,L , eis − ∆eis , e( − gi ) − ∆e ( − gi ) ) − f (ei1 − ∆ei1 , ei2 − ∆ei2 ,L , eis − ∆eis , e( − i ) ) + f (e) Substitute ei in proof of Proposition 3 with (e1 , e2 ,L , es ) , and the process to prove

I λgI ≥ 0 is the same as the process to prove λi ≥ 0 in proof of Proposition 3. W i

Corollary 1 states a generalized property of interactions if shocks are partitioned in groups, which shows that interactions of a shock or of shock groups are always non-negative and are significant in the total effect. Among the interactions, the 2-order interactions are the most attractive from the simplicity perspective. For example, one may care if two banks suffer outside shocks simultaneously, whether the total effect is larger than the sum of the effects when the two banks suffer shocks individually, namely, whether f (A+B) > f (A) + f (B). This is related to the concept of super-additivity. Therefore, we present the properties of 2-order interactions (interactions of factor pairs). Proposition 4. If systemic risk is measured by the loss rate defined in Definition 2, 2-order interactions are non-negative. Proof of Proposition 4. 2-order interactions can be written as follows:

λi , j = f (ei1 , e1j , e 0( − i , j ) ) − ∆ i f − ∆ j f − f (e 0 ) = f (ei1 , e1j , e 0( − i , j ) ) − f (ei1 , e 0( − i ) ) − f (e1j , e 0( − j ) ) + f (e 0 ) =f (ei − ∆ei , e j − ∆e j , e ( − i , j ) ) − f (ei − ∆ei , e ( − i ) ) − f (e j − ∆e j , e ( − j ) ) + f (e)

Let f1 (e j ) = f (ei - D ei , e j , e(- i , j ) ) and f 2 (e j ) = f (ei , e j , e(- i , j ) ) . In addition, let g (e j ) = f1 (e j ) - f 2 (e j ) = f (ei - D ei , e j , e(- i , j ) ) - f (ei , e j , e(- i , j ) ) . We need to prove that g (e j - D e j ) ³ g (e j ) . Notice that f is a decreasing and convex function with respect to e , we have f ¢¢(e) ³ 0 , and thus f ¢(e) is increasing with respect to e , so the following equation holds: g ¢(e j ) = f ¢(ei - D ei , e j , e (- i , j ) ) - f ¢(ei , e j , e (- i , j ) ) £ 0.

So g (e j ) is decreasing with respect to e , and we have

g (e j - D e j ) ³ g (e j ). W “Bank regulation and bank monitoring are mostly implemented at the level of the individual bank” (Elsinger et al., 2006). Proposition 3 suggests that simultaneous shocks to two banks trigger a larger loss than two independent shocks because of the non-negativity of

the interaction of the two banks. More generally, super-additivity of systemic loss exists, which is summed as follows. Corollary2. If systemic risk is measured by the loss rate defined in Definition 2, superadditivity exists in the 2-order interaction if the nominal loss rate is null or small enough to be ignored: f (ei1 , e1j , e(0− i , j ) ) ≥ f (ei1 , e 0( −i ) ) + f (e1j , e0( − j ) )

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0 Proof of Corollary 2. Generally, let output ( f (x ) ) of the nominal case be null or small

enough. Then the second order interaction is

λi , j = f ( xi1 , x1j , x 0( − i , j ) ) − ∆ i f − ∆ j f − f ( x 0 )=f ( xi1 , x1j , x (0− i , j ) ) − f ( xi1 , x (0− i ) ) − f ( x1j , x (0− j ) ). Since λi , j ≥ 0 , f ( xi1 , x1j , x 0( − i , j ) ) − f ( xi1 , x 0( − i ) ) − f ( x1j , x 0( − j ) ) ≥ 0 . W We show that the condition “the nominal loss rate is null” is reasonable. Consider a case where the banking system has been cleared initially, after which there is no insolvent bank. Thus, if we focus on the latter clearings based on the initial clearing, the nominal losses are always null. What’s more, usually small shocks will not result in the contagious loss, and the total loss will thus be minor. “One potential caveat of all macroprudential capital requirements is that capital allocations can be negative” (Gauthier et al., 2012). In fact, if a bank has a negative correlation with another bank, the simultaneous occurrence of shocks to the two banks reduces the risk of the system, i.e. f ( s, t ) £ f ( s) + f (t ) . When one measures the bank’s systemic importance with some systemic measures and allocates capital according to the systemic importance, one may get a negative capital requirement for the two banks since simultaneous insolvency of the two banks leads to less loss. With the measure proposed in this paper, such a problem is resolved since super-additivity exists. More generally, in a multi-dimensional problem, the following proposition holds. Proposition 5. If systemic risk is measured by the loss rate defined in Definition 2, loss of factor group is not less than the sum of loss of individual factors:

f (ei11 , ei12 ,L , ei1g , e(0- i1 ,i2 ,L ,ig ) ) ³ f (ei11 , e(0- i1 ) ) + f (ei12 , e(0- i2 ) ) + L f (ei1g , e(0- ig ) ) .

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Proof of Proposition 5. Similar to proof of Corollary 2, if we study the interactions of factor groups, we can prove that 2-order interactions are non-negative. We divide the factors ei1 , ei2 ,L , eig into two groups: ei1 , ei2 ,L , eis and eis+ 1 , eis+ 2 , L , eig . If the nominal loss is null or

small enough, we have

f (ei11 , ei12 ,L , ei1g , e(0- i1 ,i2 ,L ,ig ) ) ³ f (ei1 , ei2 ,L , eis , e(0- i1 ,i2 ,L ,is ) ) + f (eis+ 1 , eis+ 2 ,L , eig , e0(- is+ 1 ,is+ 2 ,L ,ig ) ) . Similarly, we can divide ei1 , ei2 ,L , eis into any two groups and divide eis+ 1 , eis+ 2 ,L , eig into two other groups and so on. Thus, there will be

f (ei11 , ei12 ,L , ei1g , e(0- i1 ,i2 ,L ,ig ) ) ³ f (ei1 , ei2 ,L , eis , e0(- i1 ,i2 ,L ,is ) ) + f (eis+ 1 , eis+ 2 ,L , eig , e0(- is+ 1 ,is+ 2 ,L ,ig ) ) ³ L ³ f (ei11 , e(0- i1 ) ) + f (ei12 , e(0- i2 ) ) + L f (ei1g , e(0- ig ) ).

W

Proposition 5 is an extension of Corollary 2. Systemic risk is underestimated by summing all the individual contributions. In other words, decreasing the probability of concurrent large shocks to banks is more important than focusing on the safety of individual banks.

3.4 A numerical example: comparison between Shapley value and the ANOVA-like decomposition method This section introduces how the proposed model is applied to measure systemic importance through a numerical example. As stated earlier, the Shapley value is another method of measuring the systemic importance of banks while considering interactions. We will compare Shapley value and the model proposed in this paper. Consider an interbank system with the interbank liability matrix L, where

0 2 L =  1  1

3 3  . The total assets of the interbank system are 22. Besides, the initial net 0 1  2 0

1 2 0 1 2 3

T assets outside the interbank system are e = (2,1,1,1) .

Assume all banks suffer different degrees of shock and e becomes null ultimately,

1 T namely, e = (0,0,0,0) . After clearing, the interbank liability matrix L becomes L′ , where

 0 1.8 1 0.93  0.62 0 2 2.78  . And the total assets after clearing are 18.7. As a result, the L′ =  1.24 0.9 0 1.86    0  1.87 2.7 1 total loss is 8.3 (=22-18.7+5). Therefore, the overall systemic risk is 0.377 (=8.3/22). Similar to the analysis above, we can derive different magnitudes of systemic risk of different subsets of all the shocks. Here we report the results as shown in Table 1. Table 1. Systemic risk of different subsets of all the shocks Different order 0 1

2

3 4

Systemic risk

f (e0 ) =0. f (e11 , e0( −1) ) =0.1818, f (e12 , e (0−2) ) =0.0455, f (e31 , e 0( −3) ) =0.0455, f (e14 , e (0−4) ) =0.0455. f (e11 , e12 , e 0( −1,2) ) =0.2486, f ( e11 , e31 , e 0( −1,3) ) =0.2273, f (e11 , e14 , e 0( −1,4) ) =0.2273, f (e12 , e31 , e 0( −2,3) ) =0.0909, f (e12 , e41 , e (0−2,4) ) =0.0909, f (e31 , e14 , e 0( −3,4 ) =0.0909.

f (e11 , e12 , e31 , e40 ) =0.2941, f (e11 , e12 , e14 , e30 ) =0.3314, f (e11 , e31 , e14 , e20 ) =0.2727, f (e12 , e31 , e14 , e10 ) =0.1364. f (e11 , e12 , e31 , e14 ) =0.3769.

Correspondingly, individual effects and all high orders of interactions are reported in Table 2. Table 2. Interactions of different orders Different order

Systemic risk

0

f (e0 ) =0.

1

λ11 =0.1818, λ21 =0.0455, λ31 =0.0455, λ41 =0.0455.

2

3

4

λ1,2 = f (e11 , e12 , e 0( −1,2) ) - λ11 - λ21 - f (e0 ) =0.0213, λ1,3 =0, λ1,4 =0, λ2,3 =0, λ2,4 =0, λ3,4 =0. λ1,2,3 = f (e11 , e12 , e31 , e40 ) - λ1,2 - λ1,3 - λ2,3 - λ11 - λ21 - λ31 - f (e0 ) =0, λ1,2,4 =0.0373, λ1,3,4 =0, λ2,3,4 =0 λ1,2,3,4 = f (e11 , e12 , e31 , e14 ) - λ1,2,3 - λ1,2,4 - λ1,3,4 - λ2,3,4 - λ1,2 - λ1,3 - λ1,4 - λ2,3 - λ2,4 λ3,4 - λ11 - λ21 - λ31 - λ41 =0.

According to our model, the total systemic risk of shock to bank 1 is

λ1T = λ11 + λ1,2 + λ1,3 + λ1,4 + λ1,2,3 + λ1,2,4 + λ1,3,4 + λ1,2,3,4 = 0.2405 , which can be computed simply, namely,

λ1T = f (e1 ) − f (e10 , e1( −1) ) = 0.3769 − 0.1364 = 0.2405. T T Other total systemic risks of shocks to other banks are: λ2 = 0.1042 , λ3 = 0.0455 , and

λ4T = 0.0828 . In addition, some results, such as non- negativity of all effects, f (e11 , e12 , e 0( −1,2) ) > 1

1

1

0

f (e11 , e0( −1) ) + f (e12 , e (0−2) ) , f (e1 , e2 , e3 , e4 ) > f (e11 , e0( −1) ) + f (e12 , e (0−2) ) + f (e31 , e 0( −3) ) and so on, are

in accordance with conclusions stated in Section 3.3. Let’s return to the Shapley value measure. Let the total asset loss ratio defined in Equation (7) be the characteristic function of the Shapley value, namely, µ = lossratio . According to the Shapley value, as defined in Equation (26),

(n − k − 1)!k ! ∑ (µiK − µK ) . n! k =0 K ⊂ N \i n −1

ShVi ( N , µ ) = ∑

(26)

K =k

The Shapley value of shock to bank 1 is

1 1 ShV1 = ( f (e11 , e0( −1) − f (e0 )) + ( f (e11 , e12 , e(0−1,2) ) − f (e12 , e(0−2) ) + f (e11 , e31 , e(0−1,3) ) − f (e31 , e(0−3) ) 12 4 1 + f (e11 , e14 , e(0−1,4) ) − f (e41 , e(0−4) )) + ( f (e11 , e12 , e31 , e40 ) − f (e12 , e31 , e0( −2,3) ) + f (e11 , e12 , e14 , e30 ) 12 − f (e12 , e14 , e0( −2,4) ) + f (e11 , e31 , e14 , e20 ) − f (e31 , e14 , e(0−3,4 )) 1 + ( f (e11 , e12 , e31 , e41 ) − f (e12 , e31 , e14 , e10 )) = 0.205. 4 Other the Shapley values are ShV2 = 0.0686 , ShV3 = 0.0455 , ShV4 = 0.058 . We now focus on the connections and differences between the two methods to measure the systemic importance of a bank. (1) Essentially, the Shapley value measures systemic importance from the perspective of the game, and the model proposed in this paper is from the perspective of sensitivity to external shocks. Both explain well why they can be used to measure systemic importance. The Shapley value measures the average incremental contributions of all subsystems and banks which contribute more should be systemically more important. The ANOVA-like decomposition method measures the overall sensitivity of any order, and banks which are more sensitive to external shocks should be paid more attention

since they cause more of systemic risk given the same shock. (2) Although both methods consider interactions, they are different. The difference can be seen in their computational formulas. The ANOVA-like decomposition method subdivides the interactions into any order, and a higher-order interaction does not contain interaction of a lower-order. The Shapley value, however, considers the incremental contribution, instead of subdividing the incremental contribution into any detailed parts. (3) The Shapley value measures the contribution of a player in a static state, during the ANOVA-like decomposition method in a dynamic state. For example, we can measure the systemic importance of a bank at the initial state or the state of shock with the Shapely values. When using the ANOVA-like decomposition method, we need to consider both states, i.e. before and after the shock. As a result, we posit the ANOVA-like decomposition method is not suitable in cases where risk measures contain banks’ data of any state, such as value at risk or similar measures like expected loss (ES). (4) The most important difference between the two methods is computation. As we show, the computation of Shapley value is 2 n and, thus, when the number of banks increases significantly, it results in heavy computational cost. However, the model proposed in this paper gives the results at a much lower computational cost, which is n + 1 . Through the above numerical example, we show the feasibility of the proposed model. In the next section, we apply our method to study the interactions in China’s banking system.

4. Empirical analysis: 16 listed commercial banks in China 4.1 Data description and processing Considering the availability of data, we use data from listed Chinese commercial banks’ balance sheets. There are only 16 listed banks in China, accounting for 62.86% of assets of the banking industry. The 16 listed banks and their total assets, as well as interbank assets and liabilities, are shown in Table 3. Besides, because most banks were listed in 2007, annual reports of these banks from 2007 to 2017 were collected.

Table 3. Data of the 16 Chinese listed banks Abbr.

Full name of bank

Total assets

Total liabilities

Interbank assets

Interbank liabilities

ICBC

Industrial and Commercial Bank of China

26,087,043

23,945,987

1,657,783

2,257,033

Net assets outside interbank system 2,740,306

CCB

China Construction Bank

22,124,383

20,328,556

708,598

1,794,913

2,882,142

BOC

Bank of China

19,467,424

17,890,745

1,037,165

2,013,692

2,553,206

ABC

Agriculture Bank of China

21,053,382

19,623,985

1,175,900

1,574,580

1,828,077

BOCOM

Bank of Communications

9,038,254

8,361,983

806,998

590,408

459,681

CMB

China Merchants Bank

6,297,638

5,814,246

461,164

736,661

758,889

IB

Industrial Bank

6,254,567

5,851,068

193,354

1,758,087

1,968,232

SPDB

Shanghai Pudong Development Bank

6,035,033

5,614,909

186,555

1,607,958

1,841,527

CMBC

China Minsheng Banking Corp. Ltd.

5,693,896

5,321,706

243,709

1,432,022

1,560,503

CITIC

China International Trust and Investment Corp. Bank

5,677,691

5,265,258

351,045

1,010,102

1,071,490

CEB

China Everbright Bank

4,088,243

3,782,807

285,011

729,826

750,251

PAB

Pingan Bank

3,248,474

3,026,420

231,157

465,287

456,184

HXB

Huaxia Bank

2,455,401

2,288,700

111,250

324,853

380,304

BOBJ

Bank of Beijing

2,304,456

2,130,340

214,006

378,801

338,911

BONJ

Bank of Nanjing

1,131,357

1,064,570

71,273

102,574

98,088

BONB

Bank of Ningbo

1,032,042

974,836

93,334

167,886

131,758

Note: Data is from financial statement in 2017; unit: million RMB.

The main difficulty mentioned in the literature is that for one bank we were only able to obtain the total interbank assets and liabilities instead of all bilateral liabilities. To overcome this, the maximum entropy method was adopted to assess the nominal liabilities matrix L . We present the interbank liability structure in 2017 using the “pie chart” matrix (Fig. 1). The more the pie is filled, the bigger the corresponding liability is. Besides, the magnitude of liability is denoted by the color of the pie. Thus, through the “pie chart” matrix, a general cognition about the interbank liability structure is formed; for example, ABC played an important role in the system in 2017. We assume that the nominal liabilities matrix keeps constant in a year.

Fig 1. Interbank liability matrix in 2017.

A hypothetical bank plays the role of the other banks, except for the 16 that are listed. Assets of the hypothetical bank are calculated as the difference between the total interbank assets of the Chinese banking industry and those of the 16 listed banks. And the interbank liabilities of the hypothetical bank are calculated using total interbank assets of the 17 banks less interbank liabilities of the 16 listed banks. Therefore, the 17 banks (contains the hypothetical bank) constitute the Chinese banking system, where total assets of all banks are

equal to their total liabilities. However, when investigating the contagion process, the hypothetical bank is not included in the system for two main reasons: (i) interbank data of non-listed banks are not available for these banks do not need to release financial reports; (ii) the 16 listed banks’ total assets account for more than 60% of all, and thus play a main part in the banking system. Therefore, this paper is done in the subsystem with most important banks. Choice of a case of sensitivity will greatly affect the magnitude of systemic risk, since

D f changes as x 1 varies. To be exact, the choice should be combined with quintiles of the loss distribution of net assets outside the interbank system, for example, VaR0.95 (li ) and

VaR0.05 (li ) of the loss distribution can be used as the base case and the sensitivity case under shock, respectively. Contagion is a low probability but high impact event (Elsinger et al., 2006), because data used are usually in the background during periods of low volatility. This may be why no accurate prediction is proposed, based on evidence prior to 2007. Since the 95th percentile may not lead to failure of a bank, this raises the question as to which percentile should be chosen. Such a problem seems more severe in the Chinese banking system, since no banks have failed. Moreover, since simultaneous shocks occur, a joint distribution is needed, which calls for more data. Scarcity of data due to the short listing time (mostly shorter than 10 years) further stops us from adopting a distribution-based case. As has been pointed out earlier, when investigating bank contagion, scholars usually do not “analyze macro-factors that may be behind contagious defaults” (Elsinger et al., 2006). These macro-factors include GDP and its components, inflation, interest rates, monetary aggregates, asset price indices, public debt, and so on (Reinhart & Rogoff, 2008; Alessi & Detken, 2011). Therefore, in this paper, since all banks exist in the same macro environment, it is assumed that all banks suffer a loss of the same proportion to the net assets outside the interbank system. Intuitively, different loss proportions result in different insolvent banks, and as a result, the corresponding systemic risks vary. However, it is showed in Table 4 that no matter which proportion between 0 and 1 is chosen, the banks rank nearly the same, which is confirmed by Spearman’s rank correlation coefficient.

Table 4. Spearman’s rank correlation coefficients of banks under different loss proportion Loss proportion

Coefficients 0.1 0.2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1

1

0.9971

0.9882

0.9471

0.9618

0.9647

0.9794

0.9971

1

0.9971

0.9882

0.9471

0.9618

0.9647

0.9794

0.9971

0.9971

0.9882

0.9471

0.9618

0.9647

0.9794

0.9971

0.9941

0.9588

0.9706

0.9735

0.9853

0.9912

0.9706

0.9794

0.9824

0.9912

0.9853

0.9794

0.9706

0.9706

0.9441

0.9971

0.9912

0.9647

0.9941

0.9676

0.3 0.4 Loss proportion

0.5 0.6 0.7 0.8 0.9

0.9824

And for explicitly exploring how risks propagate in the interbank network under large shocks, the sensitivity of the interbank system is tested to have a heavy stress that the net assets outside the interbank system are swept away, namely, ∆ei = ei . Since small shocks do not trigger bank failures, there are no interactions of risks; and the choice of different loss proportions does not change the ranking. Therefore, we posit that the extreme shocks amplify the tiny effect of small risks, which clarifies the systemic risk in the Chinese banking system.

4.2 Systemic importance of 16 listed banks As mentioned, it is not accurate to describe the contribution of a bank to systemic risk with individual effects when banks face simultaneous shocks. In this section, we present the total effects consisting of both individual effects and interactions with the method mentioned in Section 3 based on the data described in the last subsection. Results for 2017 are shown in Table 5. Such results are obtained after 34 ( 2 ´ 16+2 ) model runs for each year. Table 5. The individual effects, total effects, and interactions in 2017 Bank list

Total effect

Individual effect

Interactions

Ranking by total effect

Ranking by individual effect

Ranking by interactions

ICBC

0.3208

0.1869

0.1338

1

4

1

Bank list

Total effect

Individual effect

Interactions

Ranking by total effect

Ranking by individual effect

Ranking by interactions

CCB

0.3136

0.2223

0.0913

2

1

4

BOC

0.3056

0.1977

0.1080

3

3

2

ABC

0.2229

0.1246

0.0983

6

7

3

BOCOM

0.0662

0.0258

0.0405

11

14

9

CMB

0.1033

0.0579

0.0454

10

10

8

IB

0.2741

0.1980

0.0761

4

2

5

SPDB

0.2529

0.1828

0.0701

5

5

6

CMBC

0.2188

0.1540

0.0648

7

6

7

CITIC

0.1489

0.0969

0.0520

8

8

10

CEB

0.1061

0.0669

0.0391

9

9

11

PAB

0.0657

0.0387

0.0270

12

11

12

HXB

0.0505

0.0333

0.0172

13

12

14

BOBJ

0.0512

0.0282

0.0230

14

13

13

BONJ

0.0141

0.0072

0.0068

16

16

15

BONB

0.0201

0.0116

0.0085

15

15

16

Note: Values in the table are loss ratio.

As discussed, traditional methods measure systemic risk according to individual effects. However, we see from Table 5 that when simultaneous shocks occur, individual effects underestimate the systemic risk. The sum of total effects is 2.5349, sum of individual effects is 1.6328, and sum of interactions is 0.9021. As a result, interactions play an important role in the sensitivity case. We rank the banks according to total effects, individual, and interactions in the fifth, sixth, and seventh columns, respectively, of Table 5. Although some banks have the same rankings no matter which measure is adopted, two-thirds of banks have different rankings according to different measures. As we explained, the individual effect is essentially the traditional “one-stains-the-whole” measure, which has not taken shocks to other banks into consideration. Besides, both fundamental loss and contagious loss are contained in the individual effect. However, the fundamental loss is not included in the interactions (one may refer to Proposition 2), and thus, interactions only measure the additional effect that

fundamental loss brings to the interbank system. Three measures as follows can be summarized: (1) Individual effects measure systemic importance, assuming that only the target bank suffers a loss. However, in the case where simultaneous shocks happen, the target bank may interact with other banks and the interaction is non-negative. Thus, individual effects underestimate the systemic risk of the target bank. (2) Interactions contain the co-effects of bank combinations in the case where multiple shocks occur. However, this measure cannot bring the fundamental loss into the final system risk, and it excludes the individual effect of the target bank. As a result, interactions reflect the activity of a bank’s interplay with others. (3) Total effect contains both fundamental loss and contagious loss as well as individual effects and interactions. Therefore, it provides a more accurate estimation of the systemic importance of a bank under a more complicated environment. The data in Table 5 support the notion that big banks, such as ICBC, CCB, BOC, ABC, are more systemically important. Obviously, the fundamental loss is highly related to e, namely, the net assets outside the interbank system, which leads to the lower rankings of CMB and BOCOM (rank higher according to total assets), since they have smaller e. IB and SPDB, CMBC have much large interbank liabilities, which means they are more active in the interbank system. Thus, the three banks have higher rankings than BOCOM, which is one of the Big Five Banks. As discussed, interactions reflect the activity of a bank’s interplay with others. A bank with higher interaction means it is more sensitive to outside shocks. Therefore, when studying the interplay of the banks, interactions should be employed. Next we explore how systemic risk changes during different periods. To see the variation of systemic risk, we present the interactions of the Big Five Banks, namely, ICBC, CCB, BOC, BOCOM, and ABC, as time varies from 2008 to 2017 in Fig. 2 (ABC listed in 2007, and thus we report the results from 2008). Fig. 2 shows that from 2008 to 2010, the banks had higher systemic risks than from 2011 to 2017. As we stated before, larger total effects mean higher sensitivity to risks. Therefore, according to Fig. 2, it can be concluded that the banks

tend to adjust their financial structure to better shield against external risks after the global financial crisis in 2008. ICBC

CCB

BOC

ABC

BOCOM

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2008

2009

2010

2011

2012

2013

2014

2015

2016

2017

Fig 2. Total effects of Big Five Banks as time varies. Since total effects reflect the sensitivity of banks to external shocks, this subsection also focuses on finding the correlated factors which affect total effects. Many researchers have claimed before that banks with large asset sizes are usually systemically important. We agree with the claim partly, because according to Fig. 3, it does not necessarily mean that a bank with a larger asset base has a higher systemic importance. For example, ABC has more assets than BOCOM but has a lower ranking.

0.4

0.35

Total effect

0.3

0.25

0.2

0.15

0.1

0.05

0

0

0.5

1

1.5

Total assets

2

2.5 7

x 10

Fig 3. The relation between total effect and total assets in 2017. Combined with the network model, intuitively, total effects relate to external shocks

(corresponding to fundamental loss) and interbank liability matrix (base of contagion). As

shown in Fig. 4, the bank suffering a larger shock (or heavier fundamental loss) triggers a larger systemic risk. Besides, according to the clearing mechanism, the interbank liability matrix consisting of interbank assets and liabilities affects the output in an implicit way, which can be explained by neither interbank assets nor interbank liabilities alone (Fig. 4). As a result, net assets outside the interbank system, which absorb external shocks, and interbank liability matrix, which determines the contagion once a contagious loss occurs, make up the financial structure of a bank. A higher systemic importance of a bank means that the allocation of the bank’s assets and liabilities is sensitive to external risks and needs to be rearranged to ensure a more stable state. Generally speaking, a bank, which is more connected with other banks (means these banks have more assets exposed to the interbank system), suffers large systemic risk and thus has a higher systemic importance. 0.4 0.35

Total effect

0.3

0.25 0.2 0.15

0.1 0.05

0

0

0.5

1

1.5

Shock

2

2.5

3 6

x 10

0.4

0.35

Total effect

0.3

0.25

0.2

0.15

0.1

0.05

0

0

5

10

15 5

Interbank assets

x 10

0.4 0.35

Total effect

0.3

0.25 0.2 0.15

0.1 0.05

0

0

0.5

1

1.5

Interbank liabilities

2

2.5 6

x 10

Fig 4. The relation between total effect and external shock & interbank assets & liabilities in 2017.

4.3 Systemic importance of bank groups In this subsection, we apply our method to investigate the systemic importance of banks at the group level. In China, there are three kinds of listed banks, i.e., Big Five banks, joint-stock banks, and city banks. Among the three groups, members of Big Five banks are all nationalized banks which play a central role in the national economy. In fact, assets of the Big Five banks account for 71% of the 16 listed banks’ total assets and 46% of the banking industry’s assets. City banks mainly serve the local economies, and assets of the three city banks thus only account for 2.4% of the total assets of the 16 listed banks. By contrast,

joint-stock banks have more assets than city banks and work more actively in economic development. We list the individual effects, total effects, and interactions as time varies from 2008 to 2017 in Table 6. Table 6. Interactions at group level in China banking system from 2007 to 2017 Big Five banks year

Eight joint-stock banks

Three city banks

Individual effect

Total effect

Interactions

Individual effect

Total effect

Interactions

Individual effect

Total effect

Interactions

2017

0.8029

1.0531

0.2501

0.4047

0.6539

0.2491

0.0472

0.0847

0.0376

2015

0.7107

0.9970

0.2863

0.3846

0.6721

0.2875

0.0335

0.0659

0.0324

2015

0.7204

1.0000

0.2796

0.2919

0.5691

0.2773

0.0318

0.0582

0.0264

2014

0.7676

1.1373

0.3697

0.4616

0.8230

0.3614

0.0431

0.0933

0.0502

2013

0.4842

0.5761

0.0920

0.2278

0.3182

0.0904

0.0304

0.0519

0.0215

2012

0.4703

0.5397

0.0694

0.1039

0.1694

0.0655

0.0241

0.0400

0.0159

2011

0.6952

0.7980

0.1029

0.0784

0.1684

0.0900

0.0156

0.0386

0.0229

2010

1.1239

1.3446

0.2207

0.0941

0.2966

0.2026

0.0100

0.0333

0.0232

2009

0.9818

1.2257

0.2439

0.1145

0.3457

0.2312

0.0109

0.0286

0.0178

2008

1.0756

1.3163

0.2408

0.1588

0.3948

0.2359

0.0080

0.0231

0.0151

Note: Entries of the first sub-column are interactions; Big Five banks include ICBC, ABC, CCB, BOC, and BOCOM; Eight joint-stock banks include CMB, SPDB, CMBC, IB, PAB, CITIC, CEB, and HXB; Three city banks include BOBJ, BONJ, and BONB.

In each year, systemic importance of Big Five banks is much larger than other groups. However, the difference becomes smaller as time varies, which means these relatively smaller banks developed quickly in recent years and have become more connected with other banks, improving their participation in the national economy. According to Table 6 (marked with bold font), interactions of shocks to the 8 joint-stock banks are almost the same as Big Five banks in each year. Of course, when considering only individual or total effects, Big Five banks group is much more important than the 8 joint-stock banks since their assets are 2.5 times larger than those of the latter. However, when we focus on the effect of a bank co-working with other banks, namely, the sum of interactions of any order, Big Five Banks have the same systemic importance as that of the 8 joint-stock banks.

As explained, it means that the group of the 8 joint-stock banks participates as actively as the group of the Big Five Banks. There are many kinds of interactions, namely, interactions of shock pairs, interactions of shock triplets, and so on. In this paper, we only report the interaction of shock pairs since it is usually kept and others omitted when interactions are studied. We intend to find (1) what the percentage of 2-order interactions is in the overall interactions; and (2) which bank pair will trigger the largest systemic risk. The results are presented in Table 7.

Table 7. 2-order interaction of shocks to different banks in 2017 Banks

ICBC

CCB

BOC

ABC

BOCOM

CMB

IB

SPDB

CITIC

CMBC

CEB

PAB

HXB

BONJ

BOBJ

BONB

ICBC

0

0.0080

0.0084

0.0052

0

0.0026

0.0083

0.0072

0.0091

0.0018

0.0003

0.0006

0.0045

0.0031

0.0014

0.0014

CCB

0.0080

0

0.0064

0.0053

0

0.0023

0.0039

0.0036

0.0043

0.0013

0.0003

0.0005

0.0028

0.0020

0.0011

0.0009

BOC

0.0084

0.0053

0

0.0056

0

0.0025

0.0055

0.0049

0.0060

0.0015

0.0003

0.0005

0.0034

0.0024

0.0012

0.0011

ABC

0.0052

0

0

0

0

0.0017

0.0056

0.0048

0.0061

0.0012

0.0002

0.0004

0.0030

0.0021

0.0009

0.0009

BOCOM

0

0.0023

0.0025

0.0017

0

0

0

0

0

0

0

0

0

0

0

0

CMB

0.0026

0.0039

0.0055

0.0056

0

0

0.0021

0.0019

0.0024

0.0005

0.0001

0.0002

0.0012

0.0009

0.0004

0.0004

IB

0.0083

0.0036

0.0049

0.0048

0

0.0019

0

0.0018

0.0018

0.0011

0.0003

0.0004

0.0019

0.0015

0.0010

0.0006

SPDB

0.0072

0.0043

0.0060

0.0061

0

0.0024

0.0018

0

0.0019

0.0010

0.0003

0.0004

0.0018

0.0014

0.0009

0.0006

CITIC

0.0091

0.0013

0.0015

0.0012

0

0.0005

0.0011

0.0010

0

0.0012

0.0003

0.0005

0.0021

0.0016

0.0011

0.0006

CMBC

0.0018

0.0003

0.0003

0.0002

0

0.0001

0.0003

0.0003

0.0003

0

0.0001

0.0001

0.0007

0.0005

0.0003

0.0002

CEB

0.0003

0.0005

0.0005

0.0004

0

0.0002

0.0004

0.0004

0.0005

0.0001

0

0.0000

0.0002

0.0001

0.0001

0.0001

PAB

0.0006

0.0028

0.0034

0.0030

0

0.0012

0.0019

0.0018

0.0021

0.0007

0.0002

0

0.0003

0.0002

0.0001

0.0001

HXB

0.0045

0.0020

0.0024

0.0021

0

0.0009

0.0015

0.0014

0.0016

0.0005

0.0001

0.0002

0

0.0011

0.0006

0.0004

BONJ

0.0031

0.0011

0.0012

0.0009

0

0.0004

0.0010

0.0009

0.0011

0.0003

0.0001

0.0001

0.0006

0

0.0004

0.0003

BOBJ

0.0014

0.0009

0.0011

0.0009

0

0.0004

0.0006

0.0006

0.0006

0.0002

0.0001

0.0001

0.0004

0.0003

0

0.0002

BONB

0.0014

0.0009

0.0011

0.0009

0

0.0004

0.0006

0.0006

0.0006

0.0002

0.0001

0.0001

0.0004

0.0003

0.0002

0

Total

Total 2-order interaction of bank i

0.0618

0.0371

0.0452

0.0439

0

0.0173

0.0346

0.0310

0.0385

0.0117

0.0027

0.0041

0.0234

0.0175

0.0096

0.0078

0.3862

Percentage of overall interactions (%)

46.14%

40.70%

41.87%

44.63%

0

38.04%

49.43%

47.74%

50.61%

43.46%

39.77%

47.73%

44.97%

44.77%

41.74%

45.22%

42.82%

Table 7 shows that the 2-order interactions of BOCOM is 0, which means BOCOM has no effect on the interactions. This is because the net assets of the interbank system of BOCOM is positive and, as a result, contagious default will not happen to ABC. The result is in accordance with Demange’s study on the network of 10 European Union countries (Demange, 2016). Table 7 shows that the percentage of 2-order interactions is only 42.82%, which is less than half of the overall interactions. Hence, in this setting, the high-order interactions cannot be omitted. Besides, similar with the total effects, banks with larger interbank assets have larger 2-order interactions, which means interbank lending and borrowing of big banks should be paid more attention to avoid the simultaneous large shocks to these big banks.

5. Conclusion In this paper, a method to measure the systemic importance of banks in a more complicated environment is explored where simultaneous shocks to different banks are possible. In this environment, the traditional measures conditional on the insolvency of one bank and solvency of any other banks will underestimate the systemic importance, since interactions of different banks play an important role in the process. The Shapley value is a method for studying the problem that considers the interactions, but an apparent flaw is high computational cost, when the number of banks in a system becomes larger. To solve the problem, the ANOVA-like decomposition method is proposed in this paper. Specifically, a systemic risk indicator is defined by systemic assets loss based on the clearing payment vector proposed by Eisenberg and Noe. The indicator consists of exogenous fundamental loss and endogenous contagion loss, which is concave, a decreasing function of bank capital. We then developed the ANOVA-like decomposition method to study the interactions of extreme shocks to banks in systemic risk and to study its properties. Based on the mathematical expression, risk individual effect and risk interactions effect are defined and are proved that the individual effect is the same as that when assuming idiosyncratic failure of one bank. Since systemic assets loss rate is a decreasing and convex function of the initial

capital, risk interactions effects are uncovered. It is posited that all individual effects, interactions effects, and total effects are non-negative. In addition, super-additivity exists in the systemic risk, which indicates that simultaneous shocks to a group of banks will trigger a higher systemic loss than shocks that come one after another. Meanwhile, a numerical example is used to identify the connections and differences between the Shapley value and the proposed method. The primary advantage of the ANOVA-like decomposition method is its low computational cost. In the empirical study, the risk interaction of systemic risk is investigated in the Chinese banking system. At the individual banking level, it can be found that the interactions play an important role in the measure. Thus, rankings of systemic importance with and without considering interactions are different. At the bank group level, it is found that the interactions of Big Five banks are almost as big as 8 joint-stock banks, while the city banks played a smaller role in the banking system. Besides that, the 2-order interaction only accounts for a small portion of the whole, which means higher order interactions cannot be omitted. Systemic risk is a very import issue in China. Our model can be used to identify to which banks should be paid more attention. Besides, with our model, the government can supervise the banks with bank groups. For example, if bank group A and B suffer large shock, it will trigger a much larger loss to the banking system. In this case, A and B should be taken as a group to be supervised. This paper aims to contribute to the literature on systemic risk. When measuring systemic importance, if simultaneous shocks occur, it is more reasonable to adopt the total effect consisting of individual effect and interactions. In the future, to obtain a more accurate estimate of systemic importance of banks, a more reasonable sensitivity case should be given. Not only interbank data, but also macroeconomic data are needed.

Acknowledgement This research was supported by the National Natural Science Foundation of China under Grants 71425002, 71571179. We acknowledge Shandong Key Laboratory of Social Supernetwork Computation and Decision Simulation in Universities (Shandong University).

Conflicts of interest The authors are not aware of any conflicts of interest concerning this paper.

References Alessi, L., & Detken, C. (2011). Quasi real time early warning indicators for costly asset price boom/bust cycles: A role for global liquidity. European Journal of Political Economy, 27(3), 520-533. Anand, K., Craig, B., & Peter, G. V. (2014). Filling in the Blanks: Network Structure and Interbank Contagion. Quantitative Finance, 15(4), 625-636. Asongu, S. A., & Odhiambo, N. M. (2019). Size, efficiency, market power, and economies of scale in the African banking sector. Financial Innovation, 5(1), 4. Bisias, D., Flood, M., Lo, A. W., & Valavanis, S. (2012). A Survey of Systemic Risk Analytics.

Annual Review of Financial Economics, 4(76), 255-296. Borgonovo, E. (2010). Sensitivity analysis with finite changes: An application to modified EOQ models. European Journal of Operational Research, 200(1), 127-138. Borgonovo, E., & Smith, C. L. (2011). A Study of Interactions in the Risk Assessment of Complex Engineering Systems: An Application to Space PSA. Operations Research, 59(6), 1461-1476. Brunnermeier, M. K., & Oehmke, M. (2013). Bubbles, Financial Crises, and Systemic Risk.

Handbook of the Economics of Finance, 2, 1221–1288. Chen, N., Liu X., & Yao, D. D. (2016). An optimization view of financial systemic risk modeling: Network effect and market liquidity effect. Operations research, 64(5), 1089-1108. Cont, R., Moussa, A., & Santos, E. B. E. (2010). Network Structure and Systemic Risk in Banking Systems. SSRN Electronic Journal, 326-367. Degryse, H., & Nguyen, G. (2007). Interbank exposures: An empirical examination of contagion risk in the Belgian banking system. International Journal of Central Banking, 3(2), 123-171. Demange, G. (2016). Contagion in Financial Networks: A Threat Index. Center of Economic

Studio & Ifo Institute Working Paper, 5307. Drehmann, M., & Tarashev, N. (2013). Measuring the systemic importance of interconnected banks. Journal of Financial Intermediation, 22(4), 586-607. Drehmann, M., & Tarashev, N. (2011). Systemic importance: some simple indicators. Bis

Quarterly Review, 25-37.

Eisenberg, L., & Noe, T. H. (2001). Systemic Risk in Financial Systems. Management Science,

47(2), 236-249. Elsinger, H., Lehar, A., & Summer, M. (2006). Risk assessment for banking systems. Management

science, 52(9), 1301-1314. Fang, S.-C., Rajasekera, J. R., & Tsao, H.-S. J. (1997). Entropy optimization and mathematical

programming, New York: Springer Science & Business Media. Frank, S., Giorgio, F., Didier, S., Fernando, V. R., Alessandro, V., & White, D. R. (2009). Economic Networks: The New Challenges. Science, 325(5939), 422-425. Furfine, C. (2003). Interbank exposures: Quantifying the risk of contagion. Journal of Money,

Credit, and Banking, 35(1), 111-128. Gandy, A., & Veraart, L. A. (2016). A Bayesian methodology for systemic risk assessment in financial networks. Management Science, 63(12): 4428-4446. Gauthier, C., Lehar , A., & Souissi, M. (2012). Macroprudential capital requirements and systemic risk. Journal of Financial Intermediation, 21(4), 594-618. Girardi, G., & Tolga Ergün, A. (2013). Systemic risk measurement: Multivariate GARCH estimation of CoVaR. Journal of Banking & Finance, 37(8), 3169-3180. Glasserman, P., & Young, H. P. (2015). How likely is contagion in financial networks? Journal of

Banking & Finance, 50, 383-399. Hasman, A., & Samartín, M. (2008). Information acquisition and financial contagion. Journal of

Banking & Finance, 32(10), 2136-2147. Huang, X., Zhou, H., & Zhu, H. (2012). Systemic risk contributions. Journal of financial services

research, 42(1-2), 55-83. Laeven, L., Ratnovski, L., & Tong, H. (2016). Bank size, capital, and systemic risk: Some international evidence. Journal of Banking & Finance, 69, S25-S34. Li, J., Liang, C., Zhu, X., Sun, X., & Wu, D. (2013). Risk contagion in Chinese banking industry: A Transfer Entropy-based analysis. Entropy, 15(12), 5549-5564. López-Espinosa, G., Moreno, A., Rubia A., & Valderrama, L. (2015). Systemic risk and asymmetric responses in the financial industry. Journal of Banking & Finance, 58, 471-485. Martínez-Jaramillo, S., Pérez, O. P., Embriz, F. A., & Dey, F. L. G. (2010). Systemic risk, financial contagion and financial fragility. Journal of Economic Dynamics & Control, 34(11), 2358-2374.

Mistrulli, P. E. (2011). Assessing financial contagion in the interbank market: Maximum entropy versus observed interbank lending patterns. Journal of Banking & Finance, 35(5), 1114-1127. Myers, R. H., & Montgomery, D. C. (1995). Response Surface Methodology: Process and Product

in Optimization Using Designed Experiments. New York: Wiley-Interscience. Nier, E., Yang, J., Yorulmazer, T., & Alentorn, A. (2007). Network models and financial stability.

Journal of Economic Dynamics and Control, 31(6), 2033-2060. Nikitin, M., & Smith, R. T. (2008). Information acquisition, coordination, and fundamentals in a financial crisis. Journal of Banking & Finance, 32(6), 907-914. Nucera, F., Schwaab, B., Koopman, S. J., & Lucas, A. (2016). The information in systemic risk rankings. Journal of Empirical Finance, 38, 461-475. Rabitz, H., & Aliş, Ö. F. (1999). General foundations of high‐dimensional model representations.

Journal of Mathematical Chemistry, 25(2-3), 197-233. Raffestin, L. (2014). Diversification and systemic risk. Journal of Banking & Finance, 46, 85-106. Reinhart, C. M. & Rogoff, K. S. (2008). This time is different: A panoramic view of eight centuries of financial crises, National Bureau of Economic Research Working Paper. 13882. Rogers, L. C. G., & Veraart, L. A. M. (2013). Failure and Rescue in an Interbank Network.

Management Science, 59(4), 882-898. Roukny, T., Battiston, S., & Stiglitz, J. E. (2018). Interconnectedness as a Source of Uncertainty in Systemic Risk. Journal of Financial Stability, 35, 93-106. Sedunov, J. (2016). What is the systemic risk exposure of financial institutions? Journal of

Financial Stability, 24, 71-87. Sheldon, G., & Maurer, M. (1998). Interbank lending and systemic risk: An empirical analysis for switzerland. Revue Suisse D'Economie Politique Et de Statistique, 134, 685-704. Sobol, I. M. (2003). Theorems and examples on high dimensional model representation.

Reliability Engineering & System Safety, 79(2), 187-193. Sui, C., Chi, G., & Wang, Z. (2014). Network structure and systemic risk by banks. Journal of

Management Science and Engineering, 17(04), 57-70. (In Chinese) Sui, C., & Wang, Z. (2015). Interbank network scale-free characteristics. Journal of Management

Science and Engineering, 18(2), 18-26. (In Chinese) Takemura, A. (1983). Tensor analysis of ANOVA decomposition. Journal of the American

Statistical Association, 78(384), 894-900. Tarashev, N., Tsatsaronis, K., & Borio, C. (2016). Risk Attribution Using the Shapley Value: Methodology and Policy Applications. Review of Finance, 20, 1189-1213. Toivanen, M. (2009). Financial interlinkages and risk of contagion in the Finnish interbank market.

Bank of Finland Research, Discussion Paper, 6, 1-17. Upper, C. (2011). Simulation methods to assess the danger of contagion in interbank markets.

Journal of Financial Stability, 7(3), 111-125. Upper, C., & Worms, A. (2004). Estimating bilateral exposures in the German interbank market: Is there a danger of contagion? European economic review, 48(4), 827-849. VanHoose, D. D. (2011). Systemic risks and macroprudential bank regulation: a critical appraisal.

Nfi Policy Briefs, 33, 45-60. Yao, Y., Li, J., Zhu, X., & Wei, L. (2017). Expected default based score for identifying systemically important banks. Economic Modelling, 64, 589-600.