Novel three-bank model for measuring the systemic importance of commercial banks

Economic Modelling 43 (2014) 238–246

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Novel three-bank model for measuring the systemic importance of commercial banks Jing Lu a,⁎, Xiaohong Hu b a b

Department of Finance, School of Economics and Business Administration, Chongqing University, Chongqing, China Chongqing Branch, Agricultural Bank of China, Chongqing, China

a r t i c l e

i n f o

Article history: Accepted 15 August 2014 Available online xxxx Keywords: Systemic importance of banks Systemic risk Size of bank Multivariate extreme theory

a b s t r a c t Relaxing the hypothesis on the scale level of a bank, the present paper develops an improved three-bank model for analyzing the relationship between the size and the systemic importance of a bank. The proposed model is more general and more operational compared with other models. By introducing the L function based on the multivariate extreme theory and the systemically important index, the effect of the size on the systemic importance of a bank is analyzed. The size is found to be a necessary but insufficient condition for measuring the systemic importance of a bank. The size of a bank plays a critical role in evaluating systemic importance, but when the size reaches a certain threshold, its effect is weakened. The current study has theoretical and practical significance for the recognition and supervision of the systemic importance of banks. © 2014 Elsevier B.V. All rights reserved.

1. Introduction In summarizing the lessons of the financial crisis in 2008, the International Monetary Fund (IMF) has cited that one of the roots of the crisis is the insufficient attention provided by the regulatory authorities to risk concentration and systemic risk accumulation resulting from financial innovation (IMF, 2009). Therefore, a reform to enhance systemic risk monitoring and coordinate the international efforts to execute macroprudential supervision should be implemented (BIS et al., 2009). Moreover, a large number of scholars agree that the global financial crisis reflects the defects of microprudential supervision, which only emphasizes a partial equilibrium in the entire financial system without seeking an overall equilibrium (Kashyap et al., 2008; Brunnermeier et al., 2009; French et al., 2010). As Bernanke (2008) stated: “Under our current system of safety-and-soundness regulation, supervisors often focus on the financial conditions of individual institutions in isolation. An alternative approach, which has been called systemwide or macroprudential oversight, would broaden the mandate of regulators and supervisors to encompass consideration of potential systemic risks and weaknesses as well.” Hence, the Basel Committee for Banking Supervision (BCBS) has placed a higher capital requirement for systemically important banks, which are asked to hold at least 1% more capital than others through

⁎ Corresponding author. Tel.: +86 23 65106325. E-mail address: [email protected] (J. Lu).

http://dx.doi.org/10.1016/j.econmod.2014.08.007 0264-9993/© 2014 Elsevier B.V. All rights reserved.

the BCBS-issued Basel III on December 16, 2010 (BCBS, 2010). According to Weistroffer (2011), the systemic importance of a financial institution lies in the fact that a total disorder or a serious crisis will ensue when it falls into a liquidity crisis, encounters insolvency, or withdraws from the financial market because of bankruptcy. After the subprime crisis, both the academia and regulatory authorities began to study the systemic importance of financial institutions, and proposed different methods to measure the systemic importance of banks. For example, Zhou (2010) develops a three-bank model for analyzing the relationship between the size and the systemic importance of a bank by adding another bank, as opposed to the two-bank model proposed by De Vries (2005), which is aimed at gauging systemic risk. However, the three-bank model by Zhou simply comprises one large bank and two comparably smaller banks. The three-bank model is actually similar to the two-bank model in its consideration of bank size. Moreover, there may be a non-linear relationship between the size and systemic importance of banks, the validation of which requires further study. Based on this concept, the present study constructs an improved three-bank model by resetting the scale levels as large, medium, and small for improved generalizability. Combining the L function in the Extreme Value Theory and the PAO index (the probability that at least one bank becomes distressed) proposed by Segoviano and Goodhart (2009), the relationship is further assessed, and the size of a bank is found to be a necessary but insufficient condition for determining its systemic importance. This result satisfies the role of the size of the bank in evaluating its systemic importance, which was set by the BCBS as well. The current study indicates that size is not the only factor affecting the systemic importance of a bank, thus extending and enriching the research in this area.

J. Lu, X. Hu / Economic Modelling 43 (2014) 238–246

2. Literature review 2.1. Systemically important banks (SIBs) Systemically important banks are main parts of systemically important financial institutions (SIFIs) because banks are major financial intermediations in most countries. G-20 members agree that a financial institution can be considered systemically important if its failure or malfunction may cause widespread distress, either as a direct impact or as a trigger for broader contagion (BIS et al., 2009). BIS et al. (2011) classified SIFIs into two categories, namely, global systemically important financial institutions (G-SIFIs) and domestic systemically important financial institutions (D-SIFIs). BCBS (2012) proposed global systemically important banks (G-SIBs) and domestic systemically important banks (D-SIBs) for banking. Undoubtedly, SIFIs exhibit several negative characteristics. 2.1.1. Huge negative externalities Externalities (spillover) are impacts of the economic activities of an entity on another without paying any cost or getting any compensation. Externalities can be considered positive if these activities are beneficial; and negative otherwise. BCBS believes that negative externality is an essential characteristic of SIFIs, which means that the business behavior of SIFIs could adversely impact the financial system and other financial institutions during a financial crisis. (a) Too big to fail (TBTF) means too big to be allowed to fail in a fashion that includes options other than the entire bank rescue, which imposes losses on people other than the equity providers (Turner, 2009). The impact on society cannot be estimated as soon as SIFIs go bankrupt. The bankruptcy of SIFIs could sometimes result into a recession for the entire economy, which the government would want to avoid. Therefore, the government will choose to rescue these financial institutions in many cases, which is precisely one of the painful lessons of the financial crisis. (b) Too connected to fail (TCTF) is the financial institution that is most active in the market. This institution is the net center of financial system and is closely connected to other financial institutions. In the present financial crisis, the larger banks did not commit the first mistakes and intensified the crisis. By contrast, the real “culprits” were the banks that were not large, but are strongly associated with other financial institutions. SIFIs have the ability to spread the financial and economic system of financial stress based on the size of their business, important market functions, and relationship with other financial institutions and markets. In view of this, several scholars have paid attention to the correlation between SIFIs and other financial institutions (Drehmann and Tarashev, 2011). (c) Too-Important-to-Fail (TITF) was first proposed in the International Monetary Fund (ÖtkerRobe et al., 2011) policy report in May 2011. The market believes that SIFIs could obtain some invisible guarantee, which allows them to conduct risky business because of the TITF status of SIFIs. This situation finally resulted in moral hazard and dilemmas for the government. Meanwhile, the abuse of the TITF status by SIFIs may lead to financial market distortions and unfair competition and profit use with regulatory gaps. (d) Too-Similar-to-Fail (TSTF) is another SIFI feature that was highlighted in the present crisis. As important nodes of highly interconnected financial network, SIFIs form common exposure through equity investment, issued and held financial bonds, and reported other security derivatives. Thus, most or the entire part of the financial system simultaneously face bankruptcy as soon as any institution holding the same or similar risk assets and positions goes into bankruptcy or malfunction. 2.1.2. Moral hazard (a) Regulatory tolerance policy. The government or the central bank normally bails out SIFIs either through fiscal assistance or through refinancing, which is called regulatory tolerance policy. This approach is employed to distinguish other regulatory policies for non-SIFIs because of the huge negative externalities when SIFIs go bankrupt. The

239

moral hazard of SIFIs results in the final relief from the government or central bank. The huge relief funds from the government can be extensively turned into private benefits of executives and shareholders. The emergence of adverse selection is even worse, which makes SIFIs more risky and more likely to engage in high-risk operations. (b) Emergence of forced and shift mechanism. SIFIs make full use of the features of TBTF or TCTF to pursue high-risk businesses, which leaves the costs to the government or the taxpayers and forces the government to relieve them. These performances contribute disorder to the financial system. (c) Vicious circle. Regulatory authorities paid attention to moral hazard at the beginning of the present financial crisis. Thus, the US Ministry of Finance and Fed did not rescue the Lehman Brothers when it went bankrupt. However, the US government did not expect the mass-market volatility caused by the bankruptcy of the Lehman Brothers. The volatility compelled the government to give up moral hazard and to relieve AIG. However, the latter issued up to $165 million worth of executives bonuses in 2008. 2.1.3. Unfairness of competition The appearance of SIFIs results in distorted competitive markets. On the one hand, the implicit guarantee of the government results in a slow and indistinct market, which twists the fair competitive mechanism and accumulates universal risk. On the other hand, the psychological expectations of participants cause the market failure to correct the malfunction mechanism. The negative effects of regulatory tolerance policy provides the market with a psychological expectation that SIFIs will never fail, which does not let market investors vote with their feet or their hands and restricts the high-risk business of SIFIs. 2.1.4. Asymmetric costs and benefits The high-risk activities of SIFIs could let them obtain high profit while they leave the risk to the entire financial system and society. The government and taxpayers paid for systemic crisis from SIFIs. Undoubtedly, the difficulty of securing fragile credit during the crisis is attributed to this system. Thus, the rules of economic operation are weakened and the responsibilities, functions, and credibility of financial institutions are reduced. 2.2. Measures of SIBs Two categories of measures are used to distinguish SIBs. One of the methods is called indicator methodology, which directly provides the indicators and evaluation value based on the main features of SIBs. The other measure is called market method, which uses related market volatility data among financial institutions to measure the risk contribution of SIBs to the financial system. The main difference between the two methods is the different perspectives in understanding the meaning of SIBs. The indicator methodology reflects the accumulated experiences of international financial regulatory authorities and monetary authorities, which is a more intuitive understanding and judgment to SIFIs. The market method is based on the risk management model of financial markets and calculates the contribution of every participant to the financial system. 2.2.1. Indicator method International financial regulatory authorities were the first to use the indicator method. The advantages of this method include increased transparency and a quick and simple way of identifying SIFIs. The disadvantages of this method are its empirical and arbitrary features, which result in its inability to distinguish the contribution of a financial institution to systemic risk with risky events. In 2013, BCBS announced an updated assessment methodology for global systemically important banks (BCBS, 2013). The selected indicators include size, interconnectedness, substitutes or financial institution infrastructure, cross-jurisdictional activity, and complexity. Every indicator has a uniform weight of 20% to calculate the systemic important score of a bank. However, with the

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exception of size, the other four indicators are further divided into two or more individual indicators (sub-indicators). Interconnectedness is further divided into intra-financial system assets, intra-financial system liabilities, and securities outstanding. Substitutes or financial institution infrastructure is divided into assets under custody, payments activity, underwritten transactions in debt, and equity markets. Crossjurisdictional activity is divided into cross-jurisdictional claims and cross-jurisdictional liabilities. Complexity is divided into notional amount of over-the-counter (OTC) derivatives, Level 3 assets, trading, and available-for-sale securities. BCBS uses different buckets to transfer scores into systemically important levels. The banks with similar systemic importance fall into the same bucket. Banks that fall into different buckets have different systemic importance, whereas banks in the same bucket have the same systemic importance. The indicator method can quickly, transparently, and dynamically select SIFIs. Selected SIFIs should face public pressure monitoring. Weistroffer (2011) believes that this method is useful in reducing moral hazard and systemic risk, which facilitates the positive reaction of SIFIs to regulation. In other words, this method can encourage financial institutions to withdraw from SIFI list instead of joining it. 2.2.2. Market method Market method suggests that systemic risk originates from three channels, namely, risk of volatility of asset prices, risk contagion arising from investor panic and psychological factors, and impact on its counterparts arising from excessive association portfolio or bankruptcy of financial institution. This method includes Marginal Expected Shortfall (MES), Shapley Value, CoVaR, CDS spread, and Extreme Value Model. MES represents the expected equity loss of an institution. Acharya (2009) and Acharya et al. (2010) use SES to identify SIFIs. Brownlees and Engle (2011) use public market information and the basic idea of the stress test to combine leverage and MES into a Systemic Risk Index (SRISK). Shapley Value is an important game theory instrument and can calculate the importance of every participant to the whole system and their expected rewards from the cooperation, and based on this, Shapley Value can estimate the contribution of every financial institution to the systemic risk (Gauthier et al., 2011; Drehmann and Tarashev, 2011). CoVaR is the Value at Risk (VaR) of the financial system conditional on institutions being in distress. The contribution of an institution to systemic risk is defined as the difference between the CoVaR conditional on the institution during distress and the CoVaR in the median state of the institution (Adrian and Brunnermeier, 2011). Huang et al. (2009, 2012) think that CDS spread can be thought as systemic risk because it grasps the market volatility of asset prices and market expectations. Zhou (2008) uses the Extreme Value Theory to assess the systemic importance. Combined with CoVaR and Segoviano and Goodhart's (2009) PAO and Zhou (2010) proposes two other indices to identify SIFIs, namely, Systemic Impact Index (SII) and Vulnerability Index (VI). Regulatory authorities seldom use market methods although they are considered as a theoretical approach because several market values are difficult to obtain and these values vary heavily. 2.3. Size and SIBs Several methods have been introduced to calculate the systemic importance of banks, but the relationship between size and its systemic importance is unclear. Several studies agree that the size of financial institutions is related to their systemic importance. Gauthier et al. (2010) use Canadian financial institutions as samples and compare alternative mechanisms for allocating the overall risk of a banking system to its member banks. They find that systemic importance is related to size and increases with interbank assets. However, their study emphasized on how to meet regulatory capital without the spillover risk of a bank. Tarashev

et al. (2009) use Shapley values to measure the systemic importance of banks. They find that the systemic importance of banks increases with bank size. Moore and Zhou (2012) point out that larger banks that hold more diversified portfolios are typically recognized as more stable, whereas when one fails, the probability that this shock will also affect other banks is higher. By contrast, smaller banks hold a smaller set of assets and are more isolated from the rest of the system. Therefore, size may be positively associated with systemic importance. Furthermore, Moore and Zhou (2012) find that the relationship between size and systemic importance can be highly non-linear, which could be dependent on other macro-economic conditions. The marginal contribution of individual banks to the systemic risk is mostly determined by its size when considering the risk spillover effect of an individual financial institution (Huang et al., 2012). Zhou (2010) measures systemic importance under a multivariate extreme-value-theory framework and finds that size should not be considered as a proxy for systemic importance. Komarkova et al. (2011/ 2012) confirm that size is not a deciding factor for determining systemic importance but does play an important role. According to Tarashev et al. (2010), size is not the sole deciding factor, but the probabilities of default and exposures to common risk factors interact in a non-linear fashion to determine the contribution of financial institutions to a system-wide risk. Gravelle and Li (2013) present a set of marketbased measures on the systemic importance of a financial institution or a group of financial institutions and find that the size of a financial institution should not be considered as a proxy of systemic importance. Most of the above studies are empirical (Tarashev et al., 2009; Komarkova et al., 2011/2012; Huang et al., 2012). Our theoretical study is relatively similar to the studies of Gravelle and Li (2013) and Zhou (2010). Also using Extreme Value Theory to calculate the systemic importance of a bank, Gravelle and Li (2013) build an extreme model and use 34 international active banks to find empirical evidence that the size of a financial institution should not be considered as a proxy of systemic importance. The studies of Gravelle and Li (2013) have the following differences: (a) Gravelle and Li (2013) measure the systemic risk conditional on the crash of a group of banks by calculating the simultaneous collapses of at least r (r ≥ 1) banks that is conditional on the collapses of a subset of banks, whereas we use the L function to capture extreme co-movements; (b) Gravelle and Li (2013) measure the systemic importance by measuring the increase in the systemic risk conditional on the crash of a group of banks. Therefore, evaluating the contribution of size to systemic importance is difficult. The present study expands Zhou's method and thoroughly discusses the relationship of size and systemic importance at the individual bank level. 3. Description of systemic importance A financial system comprising n banks is assumed to be indicated by (X1, X2, ⋯, Xn), and Xi is the risk column vector of bank i. If an inequality Xi N VaRi(p) with extremely low probability p exists, taking the Value at Risk (VaR) as a high risk threshold, the bank i is considered to be suffering a crisis. Xi N VaRi(p) is therefore an extreme event, such as bankruptcy of bank i. If the bank is listed company, Xi denotes its stock return. Otherwise, Xi denotes net equity, return of assets, return of equity or other financial indicators. In order to define all used formulae in terms of upper tail value, we adopt the convention to take the negative of these indicators. The multivariate Extreme Value Theory (multivariate EVT), which is an extension of the univariate EVT (Embrechts et al., 1997), is adopted to measure the co-movements among, Xi, i = 1, 2, ⋯ n, which can be considered as risk spillover of banks in distress. For any x1, x2, ⋯ xn N 0, when p → 0, we suppose that P ðX 1 NVaR1 ðx1 pÞ or ⋯ or X n NVaRn ðxn pÞÞ →Lðx1 ; x2 ; ⋯ xn Þ: p

ð1Þ

J. Lu, X. Hu / Economic Modelling 43 (2014) 238–246

Regarding the property of the L function, see De Haan and Ferreira (2006). At a special point (1, 1, ⋯ 1), Eq. (1) can be rewritten as Lð1; 1; ⋯1Þ ¼ lim p→0

P ðX 1 NVaR1 ðpÞ or ⋯ or X n NVaRn ðpÞÞ : p

ð2Þ

Moreover, the L function has the tailed character of a copula function. Presuming F(x1, x2, ⋯, xn) as the joint distribution function of (X1, X2, ⋯, Xn) and Fi(xi), i = 1, 2, ⋯ n as the marginal distribution function, then only one distribution function C(x1, x2, ⋯, xn) exists on [0, 1]d. Hence, F ðx1 ; x2 ; ⋯; xn Þ ¼ C ð F 1 ðx1 Þ; F 2 ðx2 Þ; ⋯; F n ðxn ÞÞ:

ð3Þ

Given Eq. (3), (1) is then equal to the relation as follows:For any x1, x2, ⋯ xn N 0, when p → 0, 1−C ð1−px1 ; ⋯1−pxn Þ →Lðx1 ; x2 ; ⋯; xn Þ: p

ð4Þ

At the inflection point (1, 1, ⋯ 1) ∈ [0, 1]d, we have 1−C ð1−p; ⋯1−pÞ →Lð1; 1; ⋯; 1Þ: p

ð5Þ

As Zhou (2010) points out that L function is connected to the copula and does not contain any marginal information therefore, in modeling the linkage of banking distress, the L function is irrelevant to the risk profile of the individual bank and less restrictive. The L function grasps the tailed behavior of the copula function and deals well with the extreme co-movements. Segoviano and Goodhart (2009) introduced the PAO index to measure the systemic importance of banks, which is a

241

probability problem defined as the conditional probability of having at least one extra bank being a failure, given that a particular bank fails. The model is expressed as PAOi ðpÞ :¼ P

n o   ∃ j ≠ i; s:t: X j N VaR j ðpÞ X i N VaRi ðpÞ :

ð6Þ

Under the multivariate EVT, the extreme value of the PAO can be expressed by the L function. Hence, the result of the PAO can be obtained approximately if the L function is known. Suppose (X1, X2 ⋯, Xn) meets the multivariate EVT. Based on the definition of PAO in (6), we have PAOi :¼ lim PAOi ðpÞ ¼ L≠i ð1; 1; ⋯; 1Þ þ 1−Lð1; 1; ⋯; 1Þ p→0 n o  P ∃j≠i; s:t:X j N VaR j ðpÞ ∩fX i N VaRi ðpÞg PAOi ¼ P ðX i N VaRi ðpÞÞ  1  ¼ P ∃ j≠i; s:t:X j N VaR j ðpÞ p  o 1 n þ1− P ∃ j≠i; s:t:X j N VaR j ðpÞ ∪fX i N VaRi ðpÞg p  1  ¼ P ∃ j≠i; s:t:X j N VaR j ðpÞ p o 1 n :¼ I1 þ 1−I 2 þ1− P ∃ j; s:t:X j N VaR j ðpÞ p

ð7Þ

where p → 0, and L≠i(1, 1, ⋯, 1) is denoted as I1 and L(1, 1, ⋯, 1) as I2. Eq. (7) suggests that with a very small p, PAOi(p) is close to lim PAOi p→0

ðpÞ. Moreover, PAO can be calculated easily and directly as long as the L function is known.

4. Improved three-bank model Based on the three-bank model by Zhou (Zhou, 2010), the size conditions are relaxed, and the three banks in the financial system are defined as big, medium, and small. The three-bank model allows the relationship between the bank size and its systemic importance to be studied. The improved three-bank model is more general, more operational, and has a more universal significance compared with the previous models. 4.1. Model setup Given the three banks (X1, X2, X3) that have a portfolio investment on three independent projects (A, B, C), respectively, assume that X1 holds three unit capitals for investment, whereas X2 and X3 hold two unit capitals and one unit capital,1 respectively. Meanwhile, project A will receive three unit investment capitals in total, and projects B and C will receive two and one unit of investment each. Therefore, the market clearing equations are 8 < X 1 ¼ ð3−3λÞA þ 2λB þ λC X ¼ 2λA þ ð2−2λ−ηÞB þ ηC : : 2 X 3 ¼ λA þ ηB þ ð1−λ−ηÞC

ð8Þ

In this case, 0 b λ b 1, 0 b η b 1, and λ + η b 1. The investment scale is viewed as the bank size. Obviously, X1 is the biggest, representing the large banks. X2 represents the medium-sized banks, and X3 represents the small banks. The parameters λ and η refer to the weight of the investment of the banks. Provided that A, B, and C follow the fat-tailed distribution, and (X1, X2, X3) meet the multivariate EVT, an analysis of the size and systemic importance of the bank can be provided. For simplicity, following the method by Zhou, not all λ are discussed, but three cases will be given focus: λ is close to 1, λ is equal to 1/2, and λ is close to 0.

1

In Zhou's three-bank model (Zhou, 2010), X1 holds two unit capitals for investment, whereas X2 and X3 hold one unit capital each.

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4.2. Solution of the model According to the definition of PAO in Eq. (7), we have 8 < PAO1 −PAO2 ¼ Lð0; 1; 1Þ−Lð1; 0; 1Þ PAO2 −PAO3 ¼ Lð1; 0; 1Þ−Lð1; 1; 0Þ : : PAO1 −PAO3 ¼ Lð0; 1; 1Þ−Lð1; 1; 0Þ

ð9Þ

The comparison among PAOi, i = 1, 2, 3, depends completely on L(1, 1, 0), L(1, 0, 1), and L(0, 1, 1). Hence, the focus is on calculating the L function. Before proceeding with the solution, the two lemmas proposed by Zhou (2010) should be introduced. Lemma 1. Supposing two positive independent random variables U and V exist, with both having a fat-tailed distribution with the same tail index α. Therefore, when s → ∞, P ðU þ VNsÞ  P ðUNsÞ þ P ðVNsÞ: Lemma 2. If the assumptions of Lemma 1 are met, for any positive constant mij, 1 ≤ i, j ≤ 2, the distribution functions of (m11U1 + m12U2, m21U1 + m22U2) and (m11U1 ∨ m12U2, m21U1 ∨ m22U2) are tail-equivalent. The individual risks taken by the three banks are then distinguished. Under the improved three-bank model, as p → 0, VaR2 ðpÞ  c1 VaR1 ðpÞ; VaR3 ðpÞ  c2 VaR1 ðpÞ

ð10Þ



In this case;

−α1 ð3−3λÞα þ ð2λÞα þ λα ; α α α ð2λÞ þ ð2−2λ−ηÞ þ η 1  α α α ð3−3λÞ þ ð2λÞ þ λ −α : c2 :¼ α α α λ þ ð1−λ−ηÞ þ η

c1 :¼

Given Lemma 1, as s → ∞, P ðX 1 NsÞ  P ðð3−3λÞANsÞ þ P ð2λBNs Þ þ P ðλCNsÞ α α α  −α ¼ ð3−3λÞ þ ð2λÞ þ λ s K ðsÞ

ð11Þ

 α α α  −α P ðX 2 NsÞ  ð2λÞ þ ð2−2λ−ηÞ þ η s K ðsÞ

ð12Þ

 α α α  −α P ðX 3 NsÞ  λ þ ð1−λ−ηÞ þ η s K ðsÞ:

ð13Þ

By comparing (11) and (12) and (11) and (13), both c1 and c2 are less than 1, which is consistent with the fact that the large bank X1 can take more risks than the other two banks. L(1, 1, 0), L(1, 0, 1), and L(0, 1, 1) can then be calculated, as denoted by v(p) := VaR1(p). From Lemma 2, as p → 0 P ðX 1 NVaR1 ðpÞ or X 2 NVaR2 ðpÞÞ  P ðð3−3λÞA∨2λB∨λC N vðpÞ or 2λA∨ð2−2λ−ηÞB∨ηCNc1 vðpÞÞ 1 0 B ¼ PB @AN 0

vðpÞ 2λ ð3−3λÞ∨ c1

vðpÞ vðpÞ C C or C N ηA 2−2λ−η λ∨ 2λ∨ c1 c1 1 0 0

or BN 1

1

C C C B B vðpÞ C þ P BBN C þ P BC N vðpÞ C @ @ ηA 2λA 2−2λ−ηA λ∨ ð3−3λÞ∨ 2λ∨ c1 c1 c1  α  α  α

2λ 2−2λ−η η  ð3−3λÞ∨ P ðA NvðpÞÞ þ 2λ∨ þ λ∨ c1 c1 c1 B  PB @AN

vðpÞ

From (11), p ~ ((3 − 3λ)α + (2λ)α + λα)P(A N v(p)) is obtained as p → 0. Combining (14) and the L function, we derive P ðX 1 NVaR1 ðpÞ or X 2 NVaR2 ðpÞÞ Lð1; 1; 0Þ ¼ lim p→0 p h α  α  α i 2λ P ðANvðpÞÞ ð3−3λÞ∨ c þ 2λ∨ 2−2λ−η þ λ∨ cη c1 1 1 ¼ ðð3−3λÞα þ ð2λÞα þ λα ÞP ðANvðpÞÞ  α  α  α ð3−3λÞ∨ 2λ þ 2λ∨ 2−2λ−η þ λ∨ cη c1 c1 1 ¼ ð3−3λÞα þ ð2λÞα þ λα

ð14Þ

J. Lu, X. Hu / Economic Modelling 43 (2014) 238–246

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Also,  α  α  α ð3−3λÞ∨ cλ þ 2λ∨ cη þ γ∨ 1−λ−η c2 P ðX 1 NVaR1 ðpÞ or X 3 NVaR3 ðpÞÞ 2 2 Lð1; 0; 1Þ ¼ lim : ¼ p→0 ð3−3λÞα þ ð2λÞα þ λα p 

2λ λ c1 ∨ c2

P ðX 2 NVaR2 ðpÞ or X 3 NVaR3 ðpÞÞ Lð0; 1; 1Þ ¼ lim ¼ p→0 p

α

þ





α 2−2λ−η ∨ cη þ c1 2 α α





η 1−λ−η α c1 ∨ c2 α

ð3−3λÞ þ ð2λÞ þ λ

With the aforementioned conditions, Theorem 1 is obtained. Theorem 1. Considering the improved three-bank model and market clearing conditions in (8), and assuming that the losses of the projects A, B, and C are fat-tailed with α N 1, the three cases are as follows: Case 1. If 3/4 ≤ λ b 1, then PAO2 N PAO1 or PAO3. η Case 2. If λ = 1/2, then: (i) η ∈ (0, 1/4) then PAO1 N PAO2 or PAO3; (ii) η ∈ (1/3, 1/2), then PAO1 ≥ PAO2 and PAO1 N PAO3, especially 1−2η 2c2 N c1 then PAO1 = PAO2 N PAO3.

  Case 3. If 0 b λ ≤ 1/3, then: (i) η∈ð0; λ∪ 23 ð1−λÞ; 1−λ ; if 3−3λ≤ cλ2, then PAO1 N PAO2 or PAO3. If not, no clear relationship exists among them;  2  ≤ cη2 , and if 2−2λ−η N cη2 , no clear relationship among them (ii) η∈ λ; 3 ð1−λÞ , if 3−3λN cλ2 , then PAO2 N PAO1 or PAO3 with the condition that 2−2λ−η c1 c1 exists. If 3−3λ ≤ cλ2 , then PAO1 ≥ PAO2 and PAO1 N PAO3, especially 2−2λ−η ≤ cη2 then PAO1 = PAO2 N PAO3. c1     , and λN max cη1 ; 1−λ−η . Proof of Case 1. When λ ≥ 3/4, then λ/c2 N 2λ/c1 N 3 − 3λ, 2λN max cη2 ; 2−2λ−η c1 c2  PAO1 −PAO2 ¼ Lð0; 1; 1Þ−Lð1; 0; 1Þ ¼  PAO2 −PAO3 ¼ Lð1; 0; 1Þ−Lð1; 1; 0Þ ¼

2λ λ c1 ∨ c 2

α

þ





α 2−2λ−η ∨ cη c1 2

þ



η c1

α  α  α  α ∨ 1−λ−η − ð3−3λÞ∨ cλ − 2λ∨ cη − λ∨ 1−λ−η c c 2

2

2

2

ð3−3λÞα þ ð2λÞα þ λα  α      α 2−2λ−η η α η 1−λ−η α λ þ ∨c þ c ∨ c − cλ −ð2λÞα −λα c c

b0 ð3−3λÞα þ ð2λÞα þ λα  α  α  α  α  α  α ð3−3λÞ∨ cλ þ 2λ∨ cη þ λ∨ 1−λ−η − ð3−3λÞ∨ 2λ − 2λ∨ 2−2λ−η − λ∨ cη c c c 2

1

2

2

1

2

2

2

2

1

1

1

ð3−3λÞα þ ð2λÞα þ λα  α α α λ þ ð2λÞ þ λ − 2λ −ð2λÞα −λα c2 c1  N0 ð3−3λÞα þ ð2λÞα þ λα  α  α  α  α  α  α 2λ λ þ 2−2λ−η ∨ cη þ cη ∨ 1−λ−η − ð3−3λÞ∨ 2λ − 2λ∨ 2−2λ−η − λ∨ cη c1 ∨ c2 c1 c1 c2 c1 2 1 1 PAO1 −PAO3 ¼ Lð0; 1; 1Þ−Lð1; 1; 0Þ ¼ ð3−3λÞα þ ð2λÞα þ λα  α      α 2−2λ−η η α η 1−λ−η α λ þ ∨c þ c ∨ c − 2λ −ð2λÞα −λα c2 c1 c1 2 1 2  α α α ð3−3λÞ þ ð2λÞ þ λ  α

Therefore, PAO2 N PAO1 or PAO3. Proof of Case 2. When λ = 1/2, then 1/2 b c1 b 2/3, 1/4 b c2 b 1/3, and 3/2 b 1/c1 b 1/2c2.

 Lð1; 1; 0Þ ¼

3 1 2 ∨ c1

α

 α  α þ 1∨ 1−η þ 12 ∨ cη c 1

1

ð3−3λÞα þ ð2λÞα þ λα α  α  α 3 1 þ 1∨ cη þ 12 ∨ 1−2η 2 ∨ 2c 2c

 Lð1; 0; 1Þ ¼

2

2

2

ð3−3λÞα þ ð2λÞα þ λα α    α η α 1 1 þ 1−η þ cη ∨ 1−2η c ∨ 2c c ∨c 2c

 Lð0; 1; 1Þ ¼

1

(i) 0 b η b 1=4

2

1

2

1

ð3−3λÞα þ ð2λÞα þ λα

2

244

J. Lu, X. Hu / Economic Modelling 43 (2014) 238–246

In this condition, cη b 1

1 2

b

1−2η 2c2

and cη b 1 b 2

1−η c1 .

Thus,



1 2c2

PAO1 −PAO2 ¼ Lð0; 1; 1Þ−Lð1; 0; 1Þ ¼

α

þ



 1−η α c1

þ



  α  α 1−2η α − 2c1 −1− 1−2η 2c2 2c2 2 α α α

N0 ð3−3λÞ þ ð2λÞ þ λ α  α  α  α   1 þ 1−η þ 1−2η − c1 − 1−η − 12 α 2c2 c1 2c2 c1 1 PAO1 −PAO3 ¼ Lð0; 1; 1Þ−Lð1; 1; 0Þ ¼ N0 α α α ð3−3λÞ þ ð2λÞ þ λ  α  α  α  α   1 þ 1 þ 1−2η − c1 − 1−η − 12 α 2c2 2c2 c1 1 PAO2 −PAO3 ¼ Lð1; 0; 1Þ−Lð1; 1; 0Þ ¼ : α α α ð3−3λÞ þ ð2λÞ þ λ 

α



Hence, PAO1 N PAO2 or PAO3.   (ii) 1/3 b η b 1/2, cη N 12 and cη N max 1−η c ; 1 . Thus, 1

2

1



η c1

PAO1 −PAO2 ¼ Lð0; 1; 1Þ−Lð1; 0; 1Þ ¼

α  α ∨ 1−2η − 12 ∨ 1−2η 2c 2c 2

2

ð3−3λÞα þ ð2λÞα þ λα 

PAO1 −PAO3 ¼ Lð0; 1; 1Þ−Lð1; 1; 0Þ ¼

1 2c2

α

þ

 α η c2

þ



η c1

∨ 1−2η 2c

α

2

 α  α  α − c1 − 1∨ 1−η − cη c 1

1

1

ð3−3λÞα þ ð2λÞα þ λα 

PAO2 −PAO3 ¼ Lð1; 0; 1Þ−Lð1; 1; 0Þ ¼

1 2c2

α

þ

 α η c2



þ

α

1−2η 1 2 ∨ 2c2 α

 α  α  α − c1 − 1∨ 1−η − cη c 1

1

1

ð3−3λÞ þ ð2λÞα þ λα

N0

:

η If 1−2η 2c N c , PAO1 = PAO2 N PAO3 is considered. If not, PAO1 N PAO2 or PAO3. 2

1

Proof of Case 3. (i)

2 3 ð1−λÞ b η b 1−λ,

    then cη1 N max λ; 1−λ−η , and cη2 N max 2λ; 2−2λ−η . c2 c1  α

PAO1 −PAO2 ¼ Lð0; 1; 1Þ−Lð1; 0; 1Þ ¼

λ c2

PAO2 −PAO3 ¼ Lð1; 0; 1Þ−Lð1; 1; 0Þ ¼

 α η c1

 α  α − ð3−3λÞ∨ cλ − λ∨ 1−λ−η c 2

2

ð3−3λÞα þ ð2λÞα þ λα  α

PAO1 −PAO3 ¼ Lð0; 1; 1Þ−Lð1; 1; 0Þ ¼

þ

λ c2

þ

 α η c2

 α  α − ð3−3λÞ∨ 2λ − 2λ∨ 2−2λ−η c c 1

1

ð3−3λÞα þ ð2λÞα þ λα  α  α  α  α  α  α ð3−3λÞ∨ cλ þ cη þ λ∨ 1−λ−η − ð3−3λÞ∨ 2λ − 2λ∨ 2−2λ−η − cη c c c 2

2

2

1

1

1

ð3−3λÞα þ ð2λÞα þ λα

If 3−3λ≤ cλ , then PAO1 N PAO2 or PAO considered. If not, no  3 is   clear  relationship exists among them. 2 N max 2λ; cη , and 1−λ−η N max λ; cη . Thus, When 0 b η ≤ λ, then 2−2λ−η c c 1

2

2

 α PAO1 −PAO2 ¼ Lð0; 1; 1Þ−Lð1; 0; 1Þ ¼

λ c2

PAO2 −PAO3 ¼ Lð1; 0; 1Þ−Lð1; 1; 0Þ ¼

þ

α

2−2λ−η c1

 α  α − ð3−3λÞ∨ cλ − 2λ∨ cη 2

2

ð3−3λÞα þ ð2λÞα þ λα  α

PAO1 −PAO3 ¼ Lð0; 1; 1Þ−Lð1; 1; 0Þ ¼

1



λ c2

þ





 α  α ð3−3λÞ∨ 2λ − λ∨ cη c

1−λ−η α − c2 α

1

1

ð3−3λÞ þ ð2λÞα þ λα

 α  α  α  α  α  α ð3−3λÞ∨ cλ þ 2λ∨ cη þ 1−λ−η − ð3−3λÞ∨ 2λ − 2−2λ−η − γ∨ cη c c c 2

2

2

ð3−3λÞα þ ð2λÞα þ λα

If 3−3λ≤ cλ , PAO1 N PAO2 or PAO If not, no clear relationship exists among them.  3 is considered.  2 ; cη , and λb cη b 1−λ−η . (ii) λ b η ≤ 23 ð1−λÞ, then 2λb min 2−2λ−η c c 1

2

1

 α PAO1 −PAO2 ¼ Lð0; 1; 1Þ−Lð1; 0; 1Þ ¼

λ c2

þ

2



α  α  α 2−2λ−η ∨ cη − ð3−3λÞ∨ cλ − cη c1 2 2 2 α α α ð3−3λÞ þ ð2λÞ þ λ

1

1

1

:

J. Lu, X. Hu / Economic Modelling 43 (2014) 238–246

 α λ c2

PAO1 −PAO3 ¼ Lð0; 1; 1Þ−Lð1; 1; 0Þ ¼





α 2−2λ−η ∨ cη c1 2

þ





1−λ−η α − c2 α

 α  α  α ð3−3λÞ∨ 2λ − 2−2λ−η − cη c c 1

1

1

ð3−3λÞ þ ð2λÞα þ λα 

PAO2 −PAO3 ¼ Lð1; 0; 1Þ−Lð1; 1; 0Þ ¼

þ

245

ð3−3λÞ∨ cλ

2

α

þ

 α η c2

þ





 α  α  α ð3−3λÞ∨ 2η − 2−2λ−η − cη c c

1−λ−η α − c2 α

1

1

1

ð3−3λÞ þ ð2λÞα þ λα

If 3−3λN cλ2 , then PAO2 N PAO1 or PAO3 is considered, with the condition that 2−2λ−η ≤ cη2 . Without the same condition, no clear c1 ≤ cη2 , then relationship exists among them. Moreover, if 3−3λ≤ cλ2 , then PAO 1 ≥ PAO 2 and PAO 1 N PAO 3 are considered. When 2−2λ−η c1 PAO1 = PAO2 N PAO3.

5. Discussion on Theorem 1

6. Conclusion

When λ is close to 1, most of the capital of the large bank X1 is invested into projects B and C. However, the capital of medium X2 and small X3 are invested into project A. A huge difference is determined between the portfolio of X1 and that of X2 or X3. Hence, the large bank X1 may have a less systemic effect compared with the other two banks. As in the results of Case 1, the PAO1 of X1 is lower than that of X2, implying that larger banks have less systemic importance. As for X2 and X3 having a similar interconnection, the bigger bank has more systemic importance. Therefore, the relationship between bank size and systemic importance is not a simple linear correlation. Moreover, the systemic importance of a bank may also depend on other factors, such as the interconnection among banks. When λ = 1/2, the portfolio of the large bank X1 is similar to that of the medium-sized bank X2 and the small bank X3. In this case, the relationship between the size and systemic importance of the bank are: (i) η ∈ (0, 1/4) and PAO1 N PAO2 or PAO3; (ii) η ∈ (1/3, 1/2), PAO1 ≥ PAO2, and PAO1 N PAO3. For most values of η, PAO2 and PAO3 are lower than PAO1. Hence, the systemic importance of the large bank X1 is greater than those of the other two banks. If the same interconnections are found, large banks generally have more systemic importance without comparing PAO2 and PAO3. When λ belongs to (0, 1/3], the large bank X1 focuses on project A, whereas X2 and X3 primarily focus on projects B and C. Thus, the following results are obtained:   (i) for η∈ð0; λ∪ 23 ð1−λÞ; 1−λ , if 3−3λ ≤ cλ2 , then PAO1 N PAO2 or PAO3 is considered.

The size of a bank, as one of the five indicators proposed by the BCBS, plays a critical role in its systemic importance. In the current study, based on the three-bank model (Zhou, 2010), an improved three-bank model is introduced to analyze the relationship between the size and systemic importance of a bank by resetting the scale level so as to obtain a more general result. Combining the PAO index with the L function under the multivariate EVT, the systemic importance of the bank is compared among different bank sizes. Size has been shown to play an important role in the systemic importance of a bank, but this function is not a dominant factor. Although the small banks mostly have less systemic importance, once the size reaches a certain threshold, the effect would weaken. No proof exists to support that the systemic importance must follow the size of a bank. Additionally, the systemic importance of a bank will be affected by the interconnection among banks. Therefore, bank size is merely a necessary but insufficient condition for measuring the systemic importance of a bank, which is in compliance with the definition of bank size set by the BCBS.

Regardless of the comparison between PAO2 and PAO3, large banks   have greater systemic importance. (ii) for η∈ λ; 23 ð1−λÞ , if 3−3λN cλ2 , then PAO2 N PAO1 or PAO3 is considered on the condition that 2−2λ−η N c1

η c2. This condition suggests

that large banks do not have greater systemic

importance, and the relationship between size and systemic importance remains unclear. If 3−3λ ≤ cλ2 , PAO1 ≥ PAO2 and PAO1 N PAO3 are considered, especially when 2−2λ−η ≤ cη2 , where PAO1 = PAO2 N PAO3. c1 This condition suggests that large banks have greater systemic importance. Therefore, as λ changes, the relationship between the size and the systemic importance of a bank is not fixed. Generally, no linear relationship is found between the size and the systemic importance of a bank, and large banks do not always have a huge systemic importance. In a way, systemic importance depends on the interconnection among banks. However, the small banks are always found to have less systemic importance in general. Therefore, based on the improved three-bank model, the assumption is that when a bank is small, its systemic importance is always lower. However, once the size reaches a certain threshold, this assumption does not apply, and other factors, such as interconnection among banks, may have an effect on the relationship between the size and systemic importance of a bank.

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