Abstract
This paper presents a new approach to randomly generate interbank networks while overcoming shortcomings in the availability of bank-by-bank bilateral exposures. Our model can be used to simulate and assess interbank contagion effects on banking sector soundness and resilience. We find a strongly non-linear pattern across the distribution of simulated networks, whereby only for a small percentage of networks the impact of interbank contagion will substantially reducoe average solvency of the system. In the vast majority of the simulated networks the system-wide contagion effects are largely negligible. The approach furthermore enables to form a view about the most systemic banks in the system in terms of the banks whose failure would have the most detrimental contagion effects on the system as a whole. Finally, as the simulation of the network structures is computationally very costly, we also propose a simplified measure—a so-called Systemic Probability Index—that also captures the likelihood of contagion from the failure of a given bank to honour its interbank payment obligations but at the same time is less costly to compute. We find that the SPI is broadly consistent with the results from the simulated network structures.
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Notes
Brusco and Castiglionesi (2007) in contrast highlight that in the presence of moral hazard among banks, in the sense that liquidity coinsurance via the interbank market entails higher risk-taking, more complete networks may in fact prove to be more, not less, contagious.
For a few representative country-specific studies using real-time overnight transactions data or large exposure data as well as entropy approaches, see e.g. Furfine (2003), Upper and Worms (2004), Boss et al. (2004), van Lelyveld and Liedorp (2006), Soramaki et al. (2007) and Degryse and Nguyen (2007).
See also Karas et al. (2008).
The bank level exposure data were downloaded from the EBA website: http://www.eba.europa.eu.
Further interesting reading about the application of network measures can be found in von Goetz (2007).
A more elaborate discussion included into the working paper version of the article can be provided on request.
Anyway, we conservatively assume that \(R_i=0\) for all banks.
See Article 111 of Directive 2006/48/EC that introduces the limits.
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Acknowledgments
The authors are indebted to J. Henry, I. Alves, M. Groß, G. Šimkus and S. Tavolaro and an anonymous referee who provided valuable comments. We are grateful for some inspiring e-mail discussions with A. Barvinok.
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Appendix
Appendix
1.1 Proof of theorem 3.1
We focus on the triggering bank \(I\). Let us define a mapping \(\Psi _I:[0,1]^N\rightarrow [0,1]^N\) as
where \(A(z)\) is an isotone, positive mapping and \(B(z)\) a given mapping (in \(\mathbb R \)).
Suppose that \(z_1\in [0,1]^N\) is such that \(z_{1i}\ge \gamma \) and \(z_1\succ z_2\). Then, \(\Psi _I(z_1)\succ \Psi _I(z_2)\). It follows from the fact that \(A(\cdot )\) is isotone and positive. In fact, \(A(\cdot )\) and \(B(\cdot )\) both depend on \(j\) but we drop the index for brevity. Let us notice, that for \(e\) being a unit vector (e.g. \(e^{(k)}:=[\underbrace{0\dots 0}_{k}\ 1\ \underbrace{0\dots \ 0}_{N-k-1}]\)), \(e^{(k)}\preceq \Psi _I(e^{(k)})\), since by definition \(\Phi _I\) is bounded by 0 and 1. If \(\Psi _Ij(e^{(k)})\ge \gamma \), then the sequence \(\Psi _Ij(e^{(k)}),\ \Psi _Ij\circ \Psi _Ij(e^{(k)}),\dots ,\ \Psi _Ij\circ \dots \circ \Psi _Ij(e^{(k)}),\dots \) is non-decreasing and, since is bounded by 1, it converges. It is, then, sufficient to prove the theorem by showing that \(\Psi _I\) is isotone if \(A(z)\) is replaced by \([z_1L_1l_1,\dots ,z_NL_Nl_N]^{\top }\). But trivially, \(A_j(z)\) is increasing in every \(z_i\). This completes the proof.
Remark 6.1
Why \((P_{Ij}^{(k)})\) may not be globally convergent? Set \(b:\!=\![z_1L_1l_1\,\dots \,z_N\) \(L_Nl_N]^{\top }\). Let \(B(z)\) be replaced by
Let us represent \(B\) in the following way (we slightly abuse the notation introducing \(z\) to power \(n^{\text{ th }}\), i.e. \(z^n:=[z_1^n,\dots ,x_N^n]^{\top }\)):
where
for positive vectors \(Q^{(21)}, Q^{(22)}, Q^{(23)}\) and \(Q^{(24)}\). We determine a region where \(B\) is increasing. Namely, differentiating \(B^1\) with respect to \(z_i\) (in the set \(\{z|\mathbf P ((\pi ^{\top })_I\cdot A(z)>C_j)<\gamma \}\)), one observes that it is increasing if
It happens for \(z\) bounded from \(0^N\), i.e. for all \(i\in \{1,\dots ,N\}\) satisfying
In case of \(B^2\) the differentiation with respect to \(z_i\) brings us to the following inequality
that translates into increasing \(B^2\). The sufficient condition for the inequality to hold is \(C_j-Q^{(23)}\cdot z>0\).
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Hałaj, G., Kok, C. Assessing interbank contagion using simulated networks. Comput Manag Sci 10, 157–186 (2013). https://doi.org/10.1007/s10287-013-0168-4
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DOI: https://doi.org/10.1007/s10287-013-0168-4