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Assessing interbank contagion using simulated networks

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

  1. See Rochet and Tirole (1996), Allen and Gale (2000) and Freixas et al. (2000) for some early prominent examples.

  2. See e.g. Aharony and Swary (1983), Peavy and Hempel (1988), Docking et al. (1997), Slovin et al. (1999), Cooperman et al. (1992), Smirlock and Kaufold (1987), Musumeci and Sinkey (1990), Wall and Peterson (1990) and Kho et al. (2000).

  3. See e.g. Longin and Solnik (2001), Hartmann et al. (2004, 2005), Gropp et al. (2009).

  4. See e.g. Cappiello et al. (2005), Engle and Manganelli (2004), White et al. (2010) and Adrian and Brunnermeier (2011).

  5. For example, interbank connections may produce co-insurance against liquidity shocks and may enhance peer monitoring; see e.g. Bhattacharya and Gale (1987), Flannery (1996), Rochet and Tirole (1996) and Freixas et al. (2000).

  6. See also Battiston et al. (2012a, b) and Gai et al. (2011). Nier et al. (2007) and Allen and Babus (2009) provides surveys of the recent literature.

  7. 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.

  8. See e.g. Flannery (1996), Ferguson et al. (2007), Heider et al. (2009) and Morris and Shin (2012).

  9. See e.g. Albert et al. (2000), Barabási and Albert (1999), Doyle et al. (2005) and Nagurney and Qiang (2008).

  10. See also Schaefer and Graham (2002) and Kossinets (2006) for some applications to social networks.

  11. 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).

  12. See also Karas et al. (2008).

  13. Gabrieli (2011) as an exception also provides a network analysis covering at least part of the euro area money market. Likewise, Garratt et al. (2011) presents a global interbank network in their paper.

  14. The bank level exposure data were downloaded from the EBA website: http://www.eba.europa.eu.

  15. Further interesting reading about the application of network measures can be found in von Goetz (2007).

  16. Convergence follows the Tarski’s fixed point theorem for the isotone mappings on a complete lattice (see also Cifuentes et al. 2005 or Hałaj 2012).

  17. A more elaborate discussion included into the working paper version of the article can be provided on request.

  18. Anyway, we conservatively assume that \(R_i=0\) for all banks.

  19. See, for example, Adrian and Shin (2010), Geanakoplos (2009) and Brunnermeier (2009).

  20. 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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Correspondence to Grzegorz Hałaj.

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The views expressed in the paper are those of the authors and do not necessarily reflect those of the ECB.

Appendix

Appendix

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

$$\begin{aligned} \Psi _{Ij}(z)=\left\{ \begin{array}{ll} \mathbf P ((\pi ^{\top })_I\cdot A(z)>C_j),&{}\mathbf P ((\pi ^{\top })_I\cdot A(z)>C_j)>\gamma \\ \min \left\{ \gamma ,B(z)+\mathbf P ((\pi ^{\top })_I\cdot A(z)>C_j)\right\} ,&{}\mathbf P ((\pi ^{\top })_I\cdot A(z)>C_j)\le \gamma \\ \end{array}\right. \end{aligned}$$

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

$$\begin{aligned} \left[ \begin{array}{cc} \omega (P^{geo}_{\cdot 1},d_{\cdot 1},b)\frac{K_1}{1+\left| \frac{C_1-b\cdot \mathbf E [\pi ^G_{\cdot 1}]}{\sqrt{b^2\cdot D^2[\pi ^G_{\cdot 1}]}}\right| ^3}\\ \vdots \\ \omega (P^{geo}_{\cdot N},d_{\cdot N},b)\frac{K_1}{1+\left| \frac{C_N-b\cdot \mathbf E [\pi ^G_{\cdot N}]}{\sqrt{(b)^2\cdot D^2[\pi ^G_{\cdot N}]}}\right| ^3}\\ \end{array}\right] . \end{aligned}$$

Let us represent \(B\) in the following way (we slightly abuse the notation introducing \(z\) to power \(n^{\text{ th }}\), i.e. \(z^n:=[z_1^n,\dots ,x_N^n]^{\top }\)):

$$\begin{aligned} B(z)=B^1(z)B^2(z) \end{aligned}$$

where

$$\begin{aligned} B^1(z)&= \frac{Q^{(21)}\cdot z^3}{\sqrt{Q^{(22)}\cdot z^2}^3}\\ B^2(z)&= \frac{1}{1+\frac{C_j-Q^{(23)}\cdot z}{\sqrt{Q^{(24)}\cdot z^2}}} \end{aligned}$$

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

$$\begin{aligned} 3Q^{(21)}_iz_i^2\sqrt{Q^{(22)}\cdot z^2}^3-3Q^{(21)}\cdot z^3\sqrt{Q^{(22)}\cdot z^2}Q^{(22)}_iz_i>0. \end{aligned}$$

It happens for \(z\) bounded from \(0^N\), i.e. for all \(i\in \{1,\dots ,N\}\) satisfying

$$\begin{aligned} z_i>\frac{Q^{(22)}_i\sum _{m\ne i}Q^{(21)}_mz_m^3}{Q^{(21)}_i\sum _{m\ne i}Q^{(22)}_mz_m^2}. \end{aligned}$$

In case of \(B^2\) the differentiation with respect to \(z_i\) brings us to the following inequality

$$\begin{aligned} Q^{(23)}_iz_i\sqrt{Q^{(24)}\cdot z^2}+(C_j-Q^{(23)}\cdot z)\frac{Q^{(24)}_iz_i}{\sqrt{Q^{(24)}\cdot z^2}}>0 \end{aligned}$$

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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Keywords

  • Network theory
  • Interbank contagion
  • Systemic risk
  • Banking
  • Stress-testing