Systemic risk through contagion in a coreperiphery structured banking network
Abstract
We contribute to the understanding of how systemic risk arises in a network of creditinterlinked agents. Motivated by empirical studies we formulate a network model which, despite its simplicity, depicts the nature of interbank markets better than a homogeneous model. The components of a vector OrnsteinUhlenbeck process living on the vertices of the network describe the financial robustnesses of the agents. For this system, we prove a LLN for growing network size leading to a propagation of chaos result. We state properties, which arise from such a structure, and examine the effect of inhomogeneity on several risk management issues and the possibility of contagion.
AMS 2010 Subject Classifications:  60K35 ; 60H30 ; 91B30 . 
JEL Classification:  G18 ; G21 ; G23 . 
Keywords: coreperiphery bank model, financial contagion, inhomogeneous graph, interacting particles, systemic risk
1 Introduction
Interbank lending patterns and financial contagion have been in the focus of central banks and regulators already before, but predominantly since the financial crisis has started in 2007, succeeded by the government debt crisis in Europe. Consequently, there is a number of empirical studies performed by central banks dealing with issues of contagion; see e.g. [7, 9, 14, 15, 17, 20, 23]. However, as indicated in Mistrulli [14], data limitations  especially not or only partly available bilateral exposures between agents  often enforced the use of the socalled maximumentropy method. This method actually rules out structural information about the market, while assuming that each bank lends to all others, possibly leading to over or underestimation of contagion (cf. Mistrulli [14] for a discussion and data analysis). With respect to structural properties of interbank markets, the empirical studies [4] for the Austrian market, [18] for the Fedwire interbank payment network, [5] for the Brazilian market, and [6] for the German market share the same finding: There is a small number of highly connected big banks acting as financial intermediaries for a large number of smaller banks, which mostly do not interact directly. Our approach weakens the homogeneity assumptions by suggesting a more realistic model allowing for toptier and lowertier banks.
Several studies model the financial market as a random graph. Financial contagion in the market is then treated by investigating and simulating a discrete bankruptcy cascade, which is initiated by some triggering mechanism like a first passage event; see e.g. [1, 3, 10, 13]. Another approach is based on mean field models of interacting systems of diffusions as used in physics to model the evolution of particles. This yields a homogeneous financial market; see e.g. [8, 11, 12]. Our approach extends such models twofold. Firstly, we replace the driving Brownian motion (BM) by a Lévy process, which does not require new techniques. Secondly and more important, we modify the model away from homogeneity to the above mentioned twotier market structure, and derive for this a new limit theorem. We thus enter a new line of research that allows for more flexibility in modeling the robustness of the financial market and its agents.
Our paper is organised as follows. We introduce the twotier financial market model in Section 2 and present subsequently the robustness process for all agents in the market. In Section 3 we prove a LLN for the new financial system, which may be interpreted as a propagation of chaos result. The limit model is further studied and used for the sake of systemic risk assessment in Section 4. We investigate the systemic risk of the market in terms of the standard deviation risk and the inverse first passage time risk. We examine in particular the effect of individual risk management decisions on the risk of contagion. The paper concludes with an outlook on future research in Section 5.
2 The contagion model
2.1 Market model
We model a credit interbank market as a weighted directed graph with a finite number of vertices . This is a common approach, see e.g. [1, 3]. While the set of vertices represents the agents in the market, information about their bilateral credit relationships is encoded in the set of edges. In the interbank market each agent manages a credit portfolio , where holds if and only if agent is a debtor of agent . We agree upon that no agent lends to herself, i.e. for all and denote by the set of debtors of agent . Correspondingly, indicates the outdegree of agent being the number of issued credits by agent in our context. The set endows every edge with an individual weight , that is, the credit issued from agent to agent corresponds to % of the total credit amount agent has issued to the overall interbank market. Consequently,
In the homogeneous graph of [3] each agent issues credits to exactly other agents from the same market so that for all . Moreover, the credit weights are assumed to be uniformly for all and 0 otherwise.
We extend this model to an inhomogeneous graph in the sense that we allow for two types of agents, the core banks and the periphery banks. There is empirical evidence for a twotiered structure of interbank markets: toptier banks and lowertier banks, cf. [4, 5, 18, 20] and in particular [6], which develops a more specific coreperiphery network model. In a simplified purely tiered network toptier (core) banks can potentially lend to and borrow from any bank in the network, while lowertier (periphery) banks exclusively interact with toptier banks but not with banks from their own tier.
We underlay this coreperiphery interbank market with the following specifying assumptions:

The interbank market is partitioned into a set of core banks and a set of periphery banks ; i.e., .

The set of debtor banks of a core bank can be partitioned into the two subsets
Analogously, for a periphery bank we set

The banks (nodes) have the following outdegree structure:

Each agent acts as a creditor and issues credits to other agents from the same market:
The adjacency matrix indicates the bilateral credit relationships between the agents of the network by entries of ones and zeros, more precisely,
(2.1) 
In view of the twotiered structure of the market, a block model can be employed, which is a common approach in social network analysis; cf. [22]. In our case the adjacency matrix is a block matrix composed of 4 blocks corresponding to the core and periphery decomposition of :
(2.2) 
The block having dimension lists the credit relationships among the core banks, the block provides the information about the relationships among the periphery banks and the blocks and cover the exchange of credits between core and periphery, respectively.
The weighted adjacency matrix is defined through
Example 2.1.
[Craig and von Peter [6]] Here the blocks are specified as follows:

is a matrix of ones exceptional the zero diagonal: all core banks issue credits to all other core banks;

is a matrix of zeros: periphery banks issue no credits among each others;

is row regular, that is, each row has at least one 1: each core bank issues credits to at least one periphery bank;

is column regular, that is, each column is covered by at least one 1: at least one periphery bank issues a credit to one of the core banks.
2.2 Financial robustness
Following [3, 8], we endow each agent in our network by a measure called financial robustness which quantifies an agent’s financial constitution over time. In the following we specify this measure as a continuoustime stochastic process, where all stochastic quantities will be defined on a probability space . In our specification we suppose that the behavior of the financial robustness is related to two sources: On the one hand, an agent’s robustness depends on the robustness of its debtors. If the debtors’ robustness is low, an agent has to face higher counterparty risk and, thus, its robustness will suffer as well. On the other hand, the robustness will also be affected by any noninterbank market investment. We model this by a vector OrnsteinUhlenbeck process given as solution of the vector stochastic differential equation (SDE)
(2.3) 
where denotes an dimensional mean 0 Lévy process with finite variance. The component models the interdependence resulting from the agents’ interbank market activity, whereas the Lévy process covers the impact from external market sources. By incorporating the robustness process is explicitly addressing the network structure. For our purposes this network structure is kept constant over time, which is in line with the findings of [6] about the structural stability of the German interbank market.
The following result gives the solution of the SDE (2.3) and the second order moment structure.
Proposition 2.2.
For the SDE (2.3) with and initial vector the following assertions hold.
(a)
The SDE has a unique explicit solution given by
(2.4) 
with the matrix exponential , and is the unit matrix.
(b) The mean of the process is given by
(c) For every the covariance matrix function is given by
where is the diagonal variance matrix of .
3 Financial robustness in large networks
If we pick out one row of Eq. (2.3), then the financial robustness of agent follows the dynamic
(3.1) 
As described above the drift term adjusts the process towards the mean robustness of agent ’s debtors. Note that the mean is calculated over with , hence this ensemble mean is independent of the driving process .
When all weights are chosen to be equal and the driving process is a Brownian motion, then this is a classical example in physics for interacting particle systems going back to McKean. We extend McKean’s mean field example of interacting diffusions to the inhomogeneous system (2.4) driven by independent Lévy processes.
We choose the weights based on the following market assumption, which are in line with with a perfectly tiered interbank market as considered in [6]. All core banks interact with each other and every periphery bank is creditor and debtor to every core bank. For the periphery banks, any credit relationship among them is excluded. Then the SDE (3.1) becomes
(3.2)  
(3.3) 
where all Lévy processes are independent with mean , and standardized second moment for . The constants model the standard deviations of the core and periphery banks respectively. The Lévy processes are for all identically distributed, as well as the for all . Moreover, we assume the following simple scenario for the weights. For all we assume that for and for some , and also that for , so that . For all we assume that for all and for all . Then (3.2) and (3.3) read as
(3.4)  
(3.5) 
This is a coupled system, where the robustness of the core banks is influenced by the mean robustness of all other core banks and the mean robustness of all periphery banks. The robustness of the periphery banks, on the other hand, is influenced by that of the core banks only. We prove a LLN for the empirical distributions given by the weighted sums when the system becomes large; i.e. for .
Theorem 3.1.
Assume the coreperiphery model (3.4) and (3.5) with independent driving Lévy processes, which are identically distributed for all and all , respectively. Define the limit system by the dynamics
(3.6)  
(3.7) 
where and ; i.e. is the distribution of for all and that of for all . Take the same driving Lévy processes as above and the same initial conditions for , independent of all Lévy processes. Denote . Then for every , , and a constant independent of ,
(3.8) 
Proof.
For the proof we adapt the arguments of the proof of Theorem 1.4 of Sznitman [19] to the inhomogenous system. First note that for
Summing this equality over all , and using the fact that and for are equally distributed, respectively, we obtain for and some ( always denotes some positive constant, whose value may vary from line to line) by taking the modulus under the Lebesgue integral
Now we estimate for using the triangular inequality
Hence, the structure of this inequality is of the form ready to apply Gronwall’s Lemma, which yields
(3.9) 
Next note that for
We take all terms under the integral corresponding to the core banks and obtain (we dropped the factor )
Then we take all terms under the integral corresponding to the periphery banks (dropped the factor ) and obtain
Now we estimate for , taking the modulus under the Lebesgue integral and use the triangular inequality
Summing the previous inequality over all , which are identically distributed, as well as all for ,
To estimate the last integral we use the fact that for all expectations are equal, and take under the integral the supremum over all . This gives for arbitrary
where the last inequality follows from the bound in (3.9). Now we take this term back under the common integral, recall that all our bounds depend on and call again simply . Then adding and subtracting in the above bound, and using the triangular inequality,
This implies
(3.10)  
hence, by Gronwall’s Lemma,
(3.11)  
Now we have for , since all are independent,
so that by the CauchySchwarz inequality, for all ,
(3.12) 
where does not depend on . The same argument applies for the sum over , so that we obtain from (3.11)
(3.13) 
For we go back to (3.9) and, invoking (3) and (3.13), we find
This implies the result. ∎
Remark 3.2.
(1) Note that in the limit system (3.6) and (3.7) all processes are independent, so that we have propagation of chaos, meaning that for the system size getting large, all robustness processes become independent.
(2) From the result (3.8) we see that all banks are mean reverted to a mean process provided that the number of core banks gets large, and the number of core banks and periphery banks satisfy a certain growth condition. In a real market we would think of many more periphery banks than core banks, so that as would seem realistic.
4 Risk management in the coreperiphery market
In order to study certain diversification effects in the coreperiphery bank model we introduce, similar to [8, 11, 12], friction parameters for the core banks and the periphery banks, respectively. Hence, the model (3.4) and (3.5) is extended to
(4.1)  
(4.2) 
Corollary 4.1.
We consider and as parameters emphasizing how strong the corresponding agent is weighting interbank activity in its investment strategy; i.e., a higher value indicates a larger investment into interbank credits. This higher value will increase the effect of the mean reversion term in the OrnsteinUhlenbeck dynamic.
We start the discussion by a simulation study based on Eq. (2.3) for the finite network with the specific structure assumed in Theorem 3.1 and the additionally introduced friction parameters; that is
(4.5) 
where and are diagonal matrices with diagonals and , respectively, for positive constants and . For our simulation we choose as network size with and . For all our simulations the robustness processes start in 1 and are driven by Brownian motions. Based on data in [6] we take .
4.1 Hedging changes in the market volatility
We examine the consequences of changes in the market volatilities to either core or periphery banks, based on the paths of the agents’ robustnesses. We consider different scenarios.
Initially core and periphery will face the same economic environment and we choose and , respectively. For this choice of parameters Figure 1 shows in the upper left plot sample paths of the robustness for all five core banks and five (out of 50) periphery banks. In a next step we suppose higher volatility in the market. Whereas core banks can keep the volatility of their noninterbank assets constant due to sophisticated hedging strategies; i.e., the same value holds, periphery banks do not have the resources and expertise for such methods. Hence, the standard deviation of their noninterbank assets increases to . If periphery banks do not undertake a shift of their assets but keep their investment strategy unchanged, their robustness will show a higher variation, which is confirmed by the upper right plot in Figure 1. The two lower plots in the same figure highlight that an increase of (by increased investment into interbank credits) can reduce variation.
A further analysis of the consequences of such a change in the market volatility is summarized in Table 4.1, where is the estimated robustness of one periphery bank at time based on 100 simulation runs for varying parameters of . The standard errors in brackets indicate that an increase of can indeed reduce the variation in the robustness of a periphery bank.
So far, our model suggests that periphery banks can reduce uncertainty in their robustness, resulting from higher volatility in noninterbank assets, by higher investment into interbank assets. This strategy has certain drawbacks, when a shock hits all core banks’ robustness at the same instant of time. For pointing out the resulting effect we redo the simulations and assume a reduction of all core banks’ robustness by 0.3 at and investigate the market at (immediately after the shock) and at . Such a shock, being restricted to the core, means that for a short term the robustness of core banks and periphery banks will diverge. However, due to the mean reversion in Eq. (4.5) the mean of core and periphery banks’ robustnesses will again revert to a common value in the long run (cf. Corollary 4.3 below). Table 4.1 illustrates that in a shock scenario an increased still reduces variation, but a shocked core will affect the periphery more intensive for higher values of . For smaller values of the robustness of the periphery banks exhibits a lower sensitivity with respect to the shock on the core. In this case core banks can in turn benefit from their interbank activity with more robust periphery banks. This becomes apparent in the estimates of the robustness at , which show approximately the new common robustness of core and periphery in the postshock regime. Apparently, for the core banks can at least recover partially from the shock, which is, however, not the case any more, if becomes too large.
Overall, we conclude that periphery banks can have an incentive to invest more into the interbank market in order to hedge their volatility, however, the increase of interbank investment makes them more vulnerable for contagion resulting from a corewide shock. The whole network will also suffer, if the periphery invests too much into the core as the increased sensitivity of the periphery with respect to the core’s constitution will have negative feedback effects on the core itself and its ability to recover from past shock events.

without shock ()  with shock ()  with shock ()  

0.99 (0.34)  0.98 (0.14)  0.96 (0.34)  0.70 (0.14)  0.77 (0.33)  0.75 (0.15)  
1.00 (0.22)  0.98 (0.14)  0.92 (0.22)  0.69 (0.14)  0.71 (0.24)  0.71 (0.15)  
1.00 (0.16)  0.98 (0.14)  0.86 (0.16)  0.69 (0.14)  0.70 (0.19)  0.70 (0.15)  
1.00 (0.14)  0.98 (0.14)  0.81 (0.14)  0.69 (0.14)  0.69 (0.16)  0.69 (0.16)  
1.00 (0.13)  0.98 (0.14)  0.77 (0.13)  0.69 (0.14)  0.69 (0.15)  0.69 (0.16)  
1.00 (0.12)  0.98 (0.14)  0.74 (0.12)  0.69 (0.14)  0.69 (0.14)  0.68 (0.16)  
1.00 (0.11)  0.98 (0.14)  0.72 (0.11)  0.69 (0.14)  0.69 (0.14)  0.68 (0.16) 
4.2 Risk management of structural breaks in the market
In this section we want to shed light on the outcome of the previous simulations in a more concrete way by relying on first passage times.
The following corollary presents the first and second moments of the limit processes.