Financial contagion in interbank networks and real economy Asako Chiba∗ December 2016

Abstract Understanding of financial market as networks is essential in designing regulations which prevent system-wide breakdown. As a stylized representation of the interbank lending market, this paper construct a model in which banks’ liabilities form a core-periphery network. The model also includes endogenously reacting asset prices as contagion proceeds. The main contributions are as follows. First, the simulation shows that when the asset price is endogenized in the contagious insolvency, the degree of contagion increases significantly compared with that in exogenous price model. Second, the analytical study of core-periphery network shows the non-motonotic expansion of insolvency in the strength of the links. Given the exposures between core banks and peripheral banks, the likelihood of contagious insolvency is high if the exposures between core banks are in intermediate range. If the exposures between core banks are very low, the contagion between core banks are not likely. On the other hand, if they are very high, each bank has small exposure to the asset price, hence the devaluation of asset does not have enough influence to damage market value of banks.

∗

Graduate School of Economics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan: [email protected].

1

1

Introduction

Interdependencies among financial institutions generate chain reaction bankruptcy. Understanding of financial market as networks is essential in designing regulations which prevent system-wide breakdown. As a stylized representation of the interbank lending market, this paper construct a model in which banks’ liabilities form a core-periphery network. The model also includes endogenously reacting asset prices as contagion proceeds. The main contributions are as follows. First, the simulation shows that when the asset price is endogenized in the contagious insolvency, the degree of contagion increases significantly compared with that in exogenous price model. Second, the analytical study of core-periphery network shows the non-motonotic expansion of insolvency in the strength of the links. Given the exposures between core banks and peripheral banks, the likelihood of contagious insolvency is high if the exposures between core banks are in intermediate range. If the exposures between core banks are very low, the contagion between core banks are not likely. On the other hand, if they are very high, each bank has small exposure to the asset price, hence the devaluation of asset does not have enough influence to damage market value of banks. The model borrows much from Elliott, Golub, and Jackson (2014). They study the expansion of default from one bank to another bank connected through a given network of liabilities. The cascades of default in their model can be summarized as follows: If a bank’s value falls below a failure threshold, it discontinuously loses its full value. This imposes losses on other banks, and these losses then propagate to others, even those who are not directly connected to the banks that initially failed. After this contagion, other banks may hit failure thresholds and also lose their values. An idiosyncratic shock that is relatively small can be amplified in this way. Focusing mainly on random networks, their main results show how the probability and the degree of cascades are affected by two feature values of network: integration and diversification. Integration refers to how much of a bank is held by final investors, and how much is held by other banks. Diversification refers to how many others a bank is held by. Each of these two feature values has non-monotonic effect on the amplification of failure. Financial system is most vulnerable to widespread cascades when both integration and diversification are intermediate. If there is no integration, then clearly there is no contagion. As integration increases, the exposure of banks to each other increases and so contagions become more likely. Thus, higher level of integration leads to increase in the probability and impact of contagions. The opposing effect is that a bank is less dependent on its own primitive assets as it becomes more integrated. Therefore, although 2

integration can empower the amplification, it also decreases the likelihood of the first failure. Diversification has also trade-offs. Here the exposure of organizations is fixed but the number of organizations cross-held varies. If the diversification is low, banks are very sensitive to particular counterparts, but the network of interconnections is weak. As diversification increases, banks have enough of their ownerships that are concentrated in particular others so that a cascade take places, and yet the cross-holdings is strong enough for the contagion to be far-reaching. Finally, as diversification is extremely high, banks’ portfolios are diversified so that they become less susceptible to any particular bank’s failure. Putting these results together, a financial system is most likely to collapse by widespread cascades when both integration and diversification are intermediate. There are two differences between their work and this paper. First, this model includes feedback effect between advance of failure and reduction in asset price. Specifically, this model describes the price of each proprietary asset decreases as the number of failed banks increases. This extension reflects the frequently observed fact that asset prices fall together rapidly when the bubble is bursting. This paper provides results on the systemic behaviour when each bank’s individual asset price reacts against the contagion of failure. Second, this model mainly focus on core-periphery network. As several empirical studies show, interbank networks consists of two types of banks: local banks and global banks, or, retail banks and wholesale banks. The former depends relatively high on real sectors rather than financial sectors, whereas the latter is engaged heavily on interbank borrowing and lending. In other words, global banks are linked to each other and play roles as hub nodes, with local banks linked to some small number of global banks. As a well description of the reality, this paper distinguishes global banks and local banks and models as core-periphery networks. As Brunnermeier and Oehmke (2012) review, the recent financial crisis had a wide-spread effect triggered by a single event for the study of financial economics. The view that financial interlinkages amplify the shock has been gaining traction among many economists. A large body of work has focused on the role of interconnections among banks as amplifiers of shocks. Most of this literature takes interbank structure as exogenously given and analyzes the likelihood of contagion by comparative statistics. This strand of literature dates back to the works of Allen and Gale (2013), Freixas, Parigi, and Rochet (2000) and Eisenberg and Noe (2001). Using a network with four banks, Allen and Gale (2013) illustrate that the network structure among banks determines whether contagion occurs or not. When all banks are completely connected to each other, the impact of a shock is so small that there is no contagion, since the shock is shared among banks. 3

However, if each bank has links to only a fraction of others, those neighbors of the bank initially hit incur substantial losses to liquidate long-term assets, thus bring other banks into the contagion. Freixas et al. (2000) show that networks may also be fragile with money-center banks and the peripheral banks only linked to the center. Eisenberg and Noe (2001) show that the set of payments of banks that satisfies their obligations uniquely exists under mild regularity conditions, and provide an algorithm which corresponds to a process of sequential default. Various algorithms to cauculate steady state are organized and presented by Upper (2011). Cifuentes, Ferrucci, and Shin (2005) incorporate fire sales into the setting of Eisenberg and Noe (2001); their approach is further extended by Gai and Kapadia (2010) and Feinstein (2015). Gai and Kapadia (2010), using a standard network model of epidemics characterized by its degree distribution, suggest that financial systems are robust-yet-fragile tendency: the effects can be extremely widespread depending on the structure of the network and on the location of the target node. Feinstein (2015) provides a framework for modeling the financial system with multiple illiquid assets, where the model of Cifuentes et al. (2005) incorporate a single asset. Some of the more recent studies, including Elliott et al. (2014), provide a comparative analysis of the extent of financial contagion as a function of the structure of the linkages among financial institutions. Awiszus and Weber (2015) incorporate direct liabilities, cross-holdings and fire sales and characterize the equilibrium as the vector of clearing payments and the price of the common illiquid asset exposed to price effects. In contrast, some of the recent literature focus on the endogenous formation of linkages and study the implications. Zawadowski (2013), Babus (2014), Farboodi (2014) and Acemgolu and Tahbaz-Salehi (2015) explicitly model interbank liabilities to investigate the relationship between counterparty risk and the equilibrium interest rates. Farboodi (2014) studies a model of endogenous intermediation among debt financed banks. She shows that if some banks have access to investment, equilibrium networks have core-periphery structures and may not be efficient. Acemgolu and Tahbaz-Salehi (2015) also consider the endogenous formation of financial linkages, but interest rate is endogenously determined in their model, where Farboodi’s model takes the face value of the contracts and the allocation rule of intermediation rents as exogenously given. This strand of literature points out that banks come into contract voluntarily, and that the structures of the networks should be equilibrium objects endogenously determined. In other words, regulations based on the network analysis that do not regard the network structure as the equilibrium are subject to the critique suggested by Lucas (1976). Although these criticisms are reasonable, it is still meaningful to examine the resulting cascades with the network fixed. The 4

concept of network structure exogenously given is valid under the assumption of short-term story. If one is to model the resulting contagion just after the triggering event of crisis, it suffices to fix network structure, because there is no room for the agents to optimize linkages. In this sense, my work is to study of shortterm behavior of banks on networks, especially when asset prices are treated as endogenous factor.

2 2.1

Model Banks and interbank networks

There are n banks and M assets. The price of asset k is denoted pk . Banks hold assets and liabilities to other banks. Asset holdings are described as follows. Let Dik be the share of the value of asset k held by bank i and let "asset-holdings matrix" D ∈ RN ×M denote the matrix whose (i, k)th entry is equal to Dik . Holdings of liabilities are described as follows. For any bank i and j, let Cij denote the fraction of bank j’s value of assets held by bank i. Cii = 0 for each i. Let "cross-holdings matrix" C ∈ RN ×N denote a network where there is a directed link from bank i to bank j if i holds a positive fraction of bank j’s assets, so that Cij > 0. The remainder of the fractions after all these interbank cross-holding areP accounted for ˆ is capital, or shares held by other investors than banks. Cii = 1 − j Cji denote this capital of bank i, which is assumed to be positive.

2.2

Value of banks

The accounting and the key valuation equations are set as follows. Let Vi denote the total value of bank i’s asset. Vi is the sum of the value of bank i’s individual assets and the value of its liabilities to other banks: X X Vi = Dik pk + Cij Vj (1) j

k

In the matrix form, equation (1) is written as

and solved to yield

V = Dp + CV,

(2)

V = (I − C)−1 Dp

(3)

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, where V ∈ RN ×1 denotes the vector of banks’ asset values. Let vi denote the market value of bank i, which is defined as the value of assets which belong to final investors of bank i. vi = Cˆii Vi (4) Therefore,

ˆ = C(I ˆ − C)−1 Dp = ADp v = CV

(5)

, where v ∈ RN ×1 denotes the vector of banks’ market values. Equation (5) shows that the value of a bank equals the sum of the values of its final claims on individual assets, with bank i having a fraction Aij of j’s direct holdings of individual assets. ˆ which capture The dependency matrix A is substantially different from C and C, the direct claims, because A takes into account all indirect holdings as well as direct holdings.

2.3

Insolvency

A bank lose its value discontinuously if its market value falls below the threshold. If vi , the value of bank i, falls below some threshold level vi , bank i becomes insolvent and incurs insolvency cost βi (p). This discontinuity brings nonlinearities into the financial system. When this insolvency cost is taken into account, each bank’s balance sheet is directly affected. The equity value of bank i becomes: X X (6) Dik pk − βi Ivi
k

where Ivi
(7)

where bi (v, p) = βi (p)Ivi
(8)

An entry Aij of the dependency matrix denotes the proportion of j’s failure costs that i incurs when j fails as well as i’s claims on the individual assets that j directly holds. If bank j fails and incurs insolvency costs of βj , then i’s value decreases by Aij βj . 6

2.4

Asset price

This paper introduces the price of each bank’s individual asset endogenously reacting to the expansion of insolvency. The basic economic idea is that, if the contagious default expands on the financial networks and more banks fail, asset price is pushed down. Assuming the individual asset of each bank is tradable in some market, it faces decreasing demand just after the realization of triggering event of crisis. Thus, asset price declines. Awiszus and Weber (2015) assume all the banks hold the same asset and express the price of this single asset as an inverse demand function of the number of unit of the asset which has been sold as of then. Although each bank in this model is assumed to hold different asset from one another, same argument holds and thus asset prices can be modeled in similar manner. To integrate this idea into the baseline network model of contagious defaults, I assume that price of each asset is expressed an exogenously given positive continuous function. In particular, using parametric exponential function, the price of individual asset held by bank i is written as follows. pi = exp(−γx)

(9)

, where x denotes the total number of assets sold-off at that time. This setting implicitly incorporates the mutual effect between financial system and real economy, abstracting away from describing details of the whole mechanism, which is complicated in reality.

2.5

Equilibrium

The equilibrium consists of {p, v} which satisfies equation (8) and (9). The time horizon in this model is as follows. In the first period, cross-holdings is determined, given feature values. Each bank’s asset price is initially 1. Then one of the peripheral bank’s asset price falls to 0. In the second period, each bank’s market value is recalculated. A bank with market value below the threshold becomes insolvent and its asset price falls to 0. Each bank’s asset price reacts negatively to the number of insolvent banks. In the next period, each bank’s market value is recalculated and some banks newly become insolvent. This process is continued until the set of insolvent banks does not change before and after the period.

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2.6

Description of core-periphery networks

This paper focus on a specific type of network, consisting of global banks playing as hub nodes and local banks connected to limited number of global banks. I call it "core-periphery" network and describe the details below. There are two types of banks: Nc of them, called core banks, have mutually directed liabilities to each other, and the rest N − Nc of them, called peripheral banks, have mutually directed liabilities with one of the core banks. The cross-holdings matrix C is characterized by the strength of links to each other: how much of bank’s liability is held by other banks, and how much by outside investors. Let ccc denote the fraction of a core bank’s asset value held by other core banks and ccp denote the fraction of a core bank’s asset value held by its neighboring peripheral banks. In other words, a core bank ’s exposure to other core banks is ccc and the exposure to the peripheral banks is ccp . In addition, let the banks have different assets with each other. Then, M = N and D = I. This setting gives a simple approximation of the interbank market in the real world, where regional banks borrow from and lend to a particular large bank and those large banks are connected to each other in the global financial market. Figure 1 shows some examples with N = 12 and Nc = 3.

Figure 1: Examples of core-periphery network

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3 3.1

Simulation and analysis simulation

This section shows the result of simulation and detailed analysis. Figure 2 illustrates how the number of insolvent banks changes as the level of core exposures ccc , peripheral exposures ccp and the price sensitivity γ change. Here, Nc = 10, N = 100 and θ = 0.98. The left figure shows the number of insolvent banks with γ = 0, which corresponds to the case where the asset price is fixed through the expansion of insolvency. The middle and the right figures show the result with γ = 5 × 10−6 and γ = 5 × 10−5 . In each figure, the vertical axis is ccc , integration between core banks, and the horisontal axis is cpc , integration between a core bank and a peripheral bank. These result exhibit two noticeable things. First, in the right figure, the number of insolvent banks are significantly high compared with that in the left figure. This shows that the effect of endogenous asset price has substantial effect on the expansion of insolvency, even if its sensitivity is very small. Second, given the peripheral exposure ccp , the number is nonlinear in the core exposure ccc . Starting from ccc = 0, the degree of contagion increases as ccc increases, but from a certain point, it turns into decrease. The degree of contagion is low when the strength of links between core banks are either very low or very high, which is consistent with the experiment in Elliott et al. (2014). The next section gives the formal explanation to this phenomena.

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Figure 2: The number of failed banks on core-periphery networks

3.2

Analysis

The phenomenon observed in the previous section can be analytically verified. The key parameter is the dependency matrix A. A bank is judged to be insolvent based on its market value, which reflects the set of insolvent banks before that period. Since D = I in (5), (i, j)the entry in A measures the change in bank iś market value when bank jś asset fall by 1, which exactly means bank j has become insolvent in the case of exogenous asset price. In other words, if Aij is large, bank i is likely to become insolvent following bank j. For this reason, this section gives a closer to look at the matrix A. The cross-holdings matrix C can be decomposed in four blocks.   Cc Cp C≡ (10) Cp0 O where Cc ∈ RNc ×Nc denotes the links among core banks and Cp ∈ RNc ×(N −Nc ) denotes those between core banks and peripheral banks. Since peripheral banks do not have links with each other, the bottom-right block which represents the links among peripheral banks is O ∈ R(N −Nc )×(N −Nc ) . These blocks in C are characterized by the strength of links to each other: how much of bank’s liability 10

is held by other banks, and how much by outside investors. Let ccc denote the fraction of a core bank’s asset value held by other core banks and ccp denote the fraction of a core bank’s asset value held by its neighboring peripheral banks. In other words, a core bank ’s exposure to other core banks is ccc and the exposure to the peripheral banks is ccp . Each block of cross-holdings matrix is then, ccc (110 − I) ∈ RNc ×Nc Nc − 1 Nc Cp = ccp L ∈ RNc ×(N −Nc ) N − Nc Cc =

, where L is a Nc × (N − Nc ) matrix such that Lij equals 1 if bank i and bank j are linked, and 0 otherwise. 110 is the Nc × Nc matrix of ones. With ccc and cpc , fraction of each bank’s value held by outside investors is denoted as follows.   c I O cc Cˆ = O ccp I Hereafter, this paper focus on the specific core-periphery network where (i) each core bank has links with exactly the same number of peripheral banks, and (ii) each peripheral bank has links with only one of the core banks. , without loss of generality. The network in this model contains seven types of connection. Each entry in A represents the type of connection and they are interpreted as follows: 1. Core bank’s exposure against its own proprietary asset: Acc_ii , diagonal elements of Acc 2. Core bank’s exposure against another core bank’s proprietary asset: Acc_ij , off-diagonal elements of Acc 3. Peripheral bank’s exposure against its own proprietary asset: App_ii , diagonal elements of App 4. Peripheral bank’s exposure against another peripheral bank which shares the same core bank: App_ij_linked 11

5. Peripheral bank’s exposure against another peripheral bank which does not share the same core bank: App_ij_nonlinked 6. Core bank’s exposure against a linked peripheral bank: Acp_linked 7. Core bank’s exposure against a non-linked peripheral bank: Acp_nonlinked Figure 3 and 4 show the example with N = 12, Nc = 3, ccc = 0.5 and ccp = 0.3.

Figure 3: Example of core-peripheryl network

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Figure 4: Elements in Matrix A In this setting, each entries can be explicitly derived as a function of ccc and ccp . Given the set of N and Nc in a certain range, Acc_ij and Acp_linked are significantly high compared to others, apart from Acc_ii and App_ii . In other words, each core bank’s market value is highly dependent on the price of asset held by its directly linking banks. By focusing on these two entries, following propositions are obtained. Proposition 1 Let Aij_cc > 0 denote the change in core bank i’s market value per unit change in core bank j ’s asset price. Fixing ccp , limccc →0

∂ Aij_cc > 0 ∂ccc

, limccc →1−ccp , and

∂ Aij_cc < 0 ∂ccc

∂2 Aij_cc < 0 ∂c2cc 13

Proposition 2 Let Aij−cp_linked > 0 denote the change in core bank i’s market value per unit change in its linking peripheral bank j’s asset price. Fixing ccp , ∂ Aij_cp_linked < 0 ∂ccc Proposition 1 is interpreted as follows. Given the exposures between a core bank and a peripheral bank which cross-hold liabilities to each other, a core bank is exposed to relatively high risk of insolvency after another core bank if the exposures between core banks are in the intermediate range. The intuition is obtained by assuming the opposite case, where the exposures between two core banks are extremely low or high. If these core exposures are very low, links between core banks have relatively low importance in this financial network, so any core bank is less likely to be affected by another core bank. On the other hand, if they are very high, the links of cross-holdings between core banks is strong. The shock generated by one core bank’s insolvency is more likely to be shared across solvent core banks. Therefore, every single core bank’s market value is not so damaged as to fall below the threshold. Figure 5 and figure 6 illustrate this.

Figure 5: Contagion from a core bank to another core bank.

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Figure 6: Decrease in core bank i’s market value[%] caused by 1-unit fall in core bank j’s asset price. ccp = 0.25, Nc = 10 and N = 100. Proposition 2 means that given the exposures between a core bank and a peripheral bank which cross-hold liabilities to each other, a core bank is exposed to relatively high risk of insolvency after its neighboring peripheral bank if the exposures between these two are very low. The intuition is similar to proposition 1. Figure 7 and figure 8 illustrate this.

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Figure 7: Contagion from a peripheral bank to its neighboring core bank.

Figure 8: Decrease in core bank i’s market value[%] caused by 1-unit fall in its linking peripheral bank j’s asset price. ccp = 0.25, Nc = 10 and N = 100. Taking these two effects together into account, one can see that if the core 16

exposures are very low, global contagions are unlikely, and if core-exposures are very high, both global contagions and local contagions are unlikely (figure 9). Otherwise, i.e. if the core exposures are in intermediate range, neither global prevention nor local prevention functions, hence the system-wide contagions are likely.

Figure 9: Decrease in a core bank’s market value[%] caused by 1-unit fall in its linking banks’ asset price. ccp = 0.25, Nc = 10 and N = 100.

4

Conclusion

This paper construct a model in which banks’ liabilities form a core-periphery network with endogenously reacting asset prices as contagion proceeds, as a stylized representation of the interbank lending market. There are two main findings. First, when the asset price is endogenized in the contagious insolvency, 17

the degree of contagion increases significantly compared with that in exogenous price model. Second, given the exposures between core banks and peripheral banks, the likelihood of contagious insolvency is high if the exposures between core banks are in intermediate range. If the exposures between core banks are very low, the contagion between core banks are not likely. On the other hand, if they are very high, each bank has small exposure to the asset price, so the devaluation of asset does not have enough influence to damage market value of banks.

References Acemgolu, D. a. A. O., & Tahbaz-Salehi, A. (2015). Systemic risk and stability in financial networks. American Economic Review, 105. Allen, F., & Gale, D. (2013). Financial contagion. Journal of Political Economy, 108, 1-33. Awiszus, K., & Weber, S. (2015). The joint impact of bankruptcy costs, crossholdings and fire sales on systemic risk in financial networks. Manuscript sumbitted for publication. Babus, A. (2014). The formation of financial networks. Discussion Paper, Tinbergen Institute, 06-093. Brunnermeier, M. K., & Oehmke, M. (2012). Bubbles, financial crises, and systemic risk. National Bureau of Economic Research(No. w18398). Cifuentes, R., Ferrucci, G., & Shin, H. S. (2005). Liquidity risk and contagion. Journal of the European Economic Association, 3, 556-66. Eisenberg, L., & Noe, T. H. (2001). Systemic risk in financial systems. Management Science, 47, 236-249. Elliott, M., Golub, B., & Jackson, M. O. (2014). Financial networks and contagion. American Economic Review, 104, 3115-3153. Farboodi, M. (2014). Intermediation and voluntary exposure to counterparty risk. Working Paper. Feinstein, Z. (2015). Financial contagion and asset liquidation strategies. Manuscript sumbitted for publication. Freixas, X., Parigi, B. M., & Rochet, J.-C. (2000). Systemic risk, interbank relations, and liquidity provision by the central bank. Journa of Money, Credit, and Banking, 32, 611-638. Gai, P., & Kapadia, S. (2010). Contagion in financial networks. Bank of England Working Papers, 383.

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Lucas, R. E. (1976). Ągeconometric policy evaluation: A critique. In CarnegieRochester Conference Series on Public Policy (Karl Brunner and Alan Meltzer, eds.), 1, 19-46, Elsevier. Upper, C. (2011). Simulation methods to assess the danger of contagion in interbank markets. Journal of Financial Stability, 7(3), 111-25. Zawadowski, A. (2013). Entangled financial systems. The Review of Financial Studies, 26, 1291-1323.

A

Proof of propositions

I − C can be decomposed as follows:   I − Cc −Cp I −C = −Cp0 I Its inverse matrix is then, (I − C)

−1

 =

Z ZCp 0 Cp Z I + Cp0 ZCp



(11)

where Z = (I − Cc − Cp Cp0 )−1 = αI − β110 α=1+

Using Z −1 = α−1 I +

(I − C)−1 =

Nc ccc − c2 Nc − 1 (N − Nc ) cp ccc β= Nc − 1

β 110 , α(α−Nc β)

β β α−1 I + α(α−N 110 (α−1 I + α(α−N 110 )Cp c β) c β) β β Cp0 (α−1 I + α(α−N 110 ) I + Cp0 (α−1 I + α(α−N 110 )Cp c β) c β)

Cˆ =



(1 − ccc − ccp )I 0 0 (1 − ccp )I

By (10), (11) and (12), 19



!

(12)

 A=

   Acc Acp (1 − ccc − ccp )Z (1 − ccc − ccp )ZCp ≡ A0cp App (1 − ccp )(I + Cp0 ZCp ) (1 − ccp )Cp0 Z

Two propositions are derived by checking the partial derivatives of ln Acc_ij and ln Acp_linked with respect to ccc .

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