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Assessing interbank contagion using simulated networks


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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  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:

  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.


  • Adrian T, Brunnermeier M (2011) CoVaR. Working Paper 17454, NBER

  • Adrian T, Shin HS (2010) Financial intermediaries and monetary economics. In: Friedman B, Woodford M (eds) Handbook of Monetary Economics. North-Holland, New York

    Google Scholar 

  • Aharony J, Swary V (1983) Contagion effects of bank failures: evidence from capital markets. J Bus 56(3):305–317

    Article  Google Scholar 

  • Albert R, Jeong H, Barabási A-L (2000) Error and attach tolerance of complex networks. Nature 406: 378–382

    Google Scholar 

  • Allen F, Babus A (2009) Networks in finance. In: Kleindorfer P, Wind J (eds) The network challenge: strategy, profit, and risk in an interlinked world. Wharton School Publishing, Philadelphia

    Google Scholar 

  • Allen F, Gale D (2000) Financial contagion. J Polit Econ 108(1):1–33

    Article  Google Scholar 

  • Barabási A-L, Albert R (1999) Emergence of scaling in random networks. Science 268:509–512

    Google Scholar 

  • Battiston S, Gatti D, Gallegat M, Greenwald B, Stiglitz J (2012a) Liaisons dangereuses: increasing connectivity, risk sharing, and systemic risk. J Econ Dyn Control 36(8):1121–1141

    Google Scholar 

  • Battiston S, Delli Gatti D, Gallegati M, Greenwald B, Stiglitz JE (2012b) Default cascades: when does risk diversification increase stability? J Financial Stab 8(3):138–149

    Google Scholar 

  • Bhattacharya S, Gale D (1987) Preference shocks, liquidity and central bank policy. In: Barnett W, Singleton K (eds) New approaches to monetary economics. Cambridge University Press, New York

    Google Scholar 

  • Boss M, Elsinger H, Thurner S, Summer M (2004) Network topology of the interbank market. Quant Finance 4:1–8

    Article  Google Scholar 

  • Bradley D, Gupta R (2002) On the distribution of the sum of n non-identically distributed uniform random variables. Ann Inst Stat Math 54(3):689–700

    Article  Google Scholar 

  • Brunnermeier M (2009) Deciphering the liquidity and credit crunch 2007–8. J Econ Perspect 23(1):77–100

    Article  Google Scholar 

  • Brusco S, Castiglionesi F (2007) Liquidity coinsurancemoral hazard and financial contagion. J Finance 65(5):2275–2302

    Article  Google Scholar 

  • Cappiello L, Gerard B, Manganelli S (2005) Measuring comovements by regression quantiles. Working paper 501, ECB

  • Cifuentes R, Ferrucci G, Shin HS (2005) Liquidity risk and contagion. J Eur Econ Assoc 3(2/3):556–566

    Google Scholar 

  • Cooperman E, Lee W, Wolfe G (1992) The 1985 Ohio thrift crisis, FSLIC’s solvency, and rate contagion for retail CDs. J Finance 47(3):919–941

    Article  Google Scholar 

  • Degryse H, Nguyen G (2007) Interbank exposures: an empirical examination of systemic risk in the Belgian banking system. Int J Central Bank 3(2):123–171

    Google Scholar 

  • Diebold FX, Yilmaz K (2011) On the network topology of variance decompositions: measuring the connectedness of financial firms. Working paper.

  • Docking D, Hirschey M, Jones V (1997) Information and contagion effects of bank loan-loss reserve announcements. J Financial Econ 43(2):219–239

    Article  Google Scholar 

  • Doyle JC, Alderson D, Li L, Low SH, Roughan M, Shalunov S, Tanaka R, Willinger W (2005) The “robust yet fragile” nature of the Internet. Proc Natl Acad Sci 102(40):14123–14475

    Article  Google Scholar 

  • ECB. evaluating interconnectedness in the financial system on the basis of actual and simulated networks. Financial stability review special feature, European Central Bank, June 2012

  • Eisenberg L, Noe TH (2001) Systemic risk in financial systems. Manag Sci 47(2):236–249

    Article  Google Scholar 

  • Engle RF, Manganelli S (2004) Caviar: conditional autoregressive value at risk by regression quantile. J Bus Econ Stat 22(4):367–381

    Article  Google Scholar 

  • Ferguson R, Hartmann P, Panetta F, Portes R (2007) International financial stability. Geneva report on the world economy, vol 9, CEPR

  • Flannery M (1996) Financial crises, payment system problems, and discount window lending. J Money Credit Bank 28(4):804–824

    Google Scholar 

  • Freixas X, Parigi BM, Rochet J-C (2000) Systemic risk, interbank relations, and liquidity provisions. J Money Credit Bank 32(3):611–638

    Article  Google Scholar 

  • Furfine C (2003) Interbank exposures: quantifying the risk of contagion. J Money Credit Bank 35(1): 111–638

    Google Scholar 

  • Gabrieli S (2011) The microstructure of the money market before and after the financial crisis: a network perspective. Research paper 181, CEIS.

  • Gai P, Haldane A, Kapadia S (2011) Complexity, concentration and contagion. J Monetary Econ 58(5): 453–470

    Google Scholar 

  • Garratt RJ, Mahadeva L, Svirydzenka K (2011) Mapping systemic risk in the international banking network. Working paper 413, Bank of England.

  • Geanakoplos J (2009) The leverage cycle. Cowles Foundation for Research in Economics, Yale University. Cowles Foundation Discussion Papers no. 1715

  • Georg C-P (2011) The effect of the interbank network structure on contagion and common shocks. Discussion paper series 2: banking and financial studies 2011, 12. Deutsche Bundesbank, Research Centre

  • Gropp R, Lo Duca M, Vesala J (2009) Cross-border contagion risk in Europe. In: Shin H, Gropp R (eds) Banking, development and structural change

  • Hałaj G (2012) Systemic bank valuation and interbank contagion. In: Kranakis E (ed) Advances in network analysis and applications. Proceedings of MITACS workshops, Springer series on mathematics in industry

  • Hałaj G, Kok Ch (2013) Assessing interbank contagion using simulated networks. Working paper series 1506, European Central Bank

  • Hartmann P, Straetmans S, de Vries C (2004) Asset market linkages in crisis periods. Rev Econ Stat 86(1):313–326

    Article  Google Scholar 

  • Hartmann P, Straetmans S, de Vries C (2005) Banking system stability: a cross-Atlantic perspective. Working paper 11698, NBER

  • Heider F, Hoerova M, Holthausen C (2009) Liquidity hoarding and interbank market spreads. Working paper 1107, ECB

  • Holló D, Kremer M, Lo Duca M (2012) CISS—a composite indicator of systemic stress in the financial system. Working paper series 1426, European Central Bank

  • Iori G, De Masi G, Precup OV, Gabbi G, Caldarelli G (2008) A network analysis of the Italian overnight money market. J Econ Dyn Control 32(1):259–278

    Article  Google Scholar 

  • Karas A, Schoors KJL, Lanine G (2008) Liquidity matters: evidence from the Russian interbank market. Working paper, Ghent University

  • Kho B, Lee D, Stulz R (2000) US banks, crises and bailouts: from Mexico to LTCM. Am Econ Rev 90(2):28–31

    Article  Google Scholar 

  • Kodres LE, Pritsker M (2002) A rational expectations model of financial contagion. J Finance 57(2):769–799

    Article  Google Scholar 

  • Kossinets G (2006) Effects of missing data in social networks. Soc Netw 28:247–268

    Article  Google Scholar 

  • Longin F, Solnik B (2001) Extreme correlation of international equity markets. J Finance 56(2):649–676

    Article  Google Scholar 

  • Lu L, Zhou T (2010) Link prediction in complex networks: a survey. CoRR abs/1010.0725.

  • Mistrulli PE (2011) Assessing financial contagion in the interbank market: maximum entropy versus observed interbank lending patterns. J Bank Finance 35:1114–1127

    Article  Google Scholar 

  • Morris S, Shin HS (2012) Contagious adverse selection. Am Econ J Macroecon 4(1):1–21

    Article  Google Scholar 

  • Musumeci J, Sinkey JF Jr (1990) The international debt crisis, investor contagion, and bank security returns in 1987. J Money Credit Bank 22:209–220

    Article  Google Scholar 

  • Nagurney A, Qiang Q (2008) An efficiency measure for dynamic networks modeled as evolutionary variational inequalities with application to the internet and vulnerability analysis. Netnomics 9(1):1–20

    Article  Google Scholar 

  • Newman MEJ (2005) Random graphs as models of networks. In: Bornholdt S, Schuster HG (eds) Handbook of graphs and networks: from the genome to the internet. Wiley Publishing, New York

    Google Scholar 

  • Nier E, Yang J, Yorulmazer T, Alentorn A (2007) Network models and financial stability. J Econ Dyn Control 31(6):2033–2060

    Article  Google Scholar 

  • Peavy JW, Hempel GH (1988) The Penn Square bank failure: effect on commercial bank security returns—a note. J Bank Finance 12:141–150

    Article  Google Scholar 

  • Polson NG, Scott JG (2011) Explosive volatility: a model of financial contagion. Working paper

  • Rochet J-C, Tirole J (1996) Interbank lending and systemic risk. J Money Credit Bank 28(4):733–762

    Article  Google Scholar 

  • Schaefer JL, Graham JW (2002) Missing data: our view of the state of the art. Psychol Methods 7(2):147–177

    Article  Google Scholar 

  • Slovin M, Sushka ME, Polonchek J (1999) An analysis of contagion and competitive effects at commercial banks. J Financial Econ 54(2):197–225

    Article  Google Scholar 

  • Smirlock M, Kaufold H (1987) Bank foreign lending, mandatory disclosure rules, and the reaction of bank stock prices to the Mexican debt crisis. J Bus 60:347–364

    Article  Google Scholar 

  • Soramaki K, Bech ML, Arnold J, Glass RJ, Beyeler WE (2007) The topology of interbank payment flows. Physica A 379:317–333

    Article  Google Scholar 

  • Upper C, Worms A (2004) Estimating bilateral exposures in the German interbank market: is there a danger of contagion? Eur Econ Rev 48(4):827–849

    Article  Google Scholar 

  • van Lelyveld I, Liedorp F (2006) Interbank contagion in the Dutch banking sector: a sensitivity analysis. Int J Central Bank 2(2):99–133

    Google Scholar 

  • von Goetz P (2007) International banking centres: a network perspective. BIS Q Rev 33–45

  • Wall LD, Peterson DR (1990) The effect of Continental Illinois’ failure on the financial performance of other banks. J Monetary Econ 26(1):77–99

    Google Scholar 

  • White H, Kim T-H, Manganelli S (2010) VAR for VaR: measuring systemic risk using multivariate regression quantiles. Working paper

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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.



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. \(z^n:=[z_1^n,\dots ,x_N^n]^{\top }\)):

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


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

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).

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