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On the Computational Complexity of Measuring Global Stability of Banking Networks

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Abstract

Threats on the stability of a financial system may severely affect the functioning of the entire economy, and thus considerable emphasis is placed on the analyzing the cause and effect of such threats. The financial crisis in the current and past decade has shown that one important cause of instability in global markets is the so-called financial contagion, namely the spreadings of instabilities or failures of individual components of the network to other, perhaps healthier, components. This leads to a natural question of whether the regulatory authorities could have predicted and perhaps mitigated the current economic crisis by effective computations of some stability measure of the banking networks. Motivated by such observations, we consider the problem of defining and evaluating stabilities of both homogeneous and heterogeneous banking networks against propagation of synchronous idiosyncratic shocks given to a subset of banks. We formalize the homogeneous banking network model of Nier et al. (J. Econ. Dyn. Control 31:2033–2060, 2007) and its corresponding heterogeneous version, formalize the synchronous shock propagation procedures outlined in (Nier et al. J. Econ. Dyn. Control 31:2033–2060, 2007; M. Eboli Mimeo, 2004), define two appropriate stability measures and investigate the computational complexities of evaluating these measures for various network topologies and parameters of interest. Our results and proofs also shed some light on the properties of topologies and parameters of the network that may lead to higher or lower stabilities.

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Notes

  1. This model assumes that all the depositors are insured for their deposits, e.g., in United States the Federal Deposit Insurance Corporation provides such an insurance up to a maximum level. Thus, we will omit the parameters d v for all v in the rest of the paper when using the model. Similarly, ℓ v quantities (which depend on the d v ’s) are also only necessary in writing the balance sheet equation and will not be used subsequently.

  2. If |c v (t 0)|>b v then the depositors incur a loss of b v −|c v (t 0)|, but as already mentioned before this model assumes that all the depositors are insured for their deposits.

  3. Intuitively, a value of ∞ signifies that the corresponding quantity is undefined.

  4. However, this exact construction will not work in the proof of Theorem 8.1 since the entire network needs to fail in that proof.

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Acknowledgements

The authors would like to thank the organizers of the Industrial-Academic Workshop on Optimization in Finance and Risk Management at the Fields Institute in Toronto (Canada) for an opportunity to discuss some of the results in this paper and receive valuable feedbacks.

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Authors and Affiliations

  1. Department of Computer Science & Engineering, Pennsylvania State University, University Park, PA, 16802, USA

    Piotr Berman

  2. Department of Computer Science, University of Illinois at Chicago, Chicago, IL, 60607-7053, USA

    Bhaskar DasGupta & Lakshmi Kaligounder

  3. Department of Computer Science, University of Bonn, 53117, Bonn, Germany

    Marek Karpinski

Authors
  1. Piotr Berman
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  2. Bhaskar DasGupta
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  3. Lakshmi Kaligounder
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  4. Marek Karpinski
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Corresponding author

Correspondence to Bhaskar DasGupta.

Additional information

Talks based on these results were given or will be given at the \(4^{\rm th}\) annual New York Computer Science and Economics Day, New York University, September 16, 2011, at the Industrial-Academic Workshop on Optimization in Finance and Risk Management, October 3–4, 2011, Fields Institute, Toronto, Canada, and at the Mathematical Finance theme, 2012 Annual Meeting of the Canadian Applied and Industrial Mathematics Society, July 24–28, 2012.

Appendix

Appendix

For the benefit of the reader, we provide explanations for a few finance terminologies frequently used in this paper.

External asset::

refers to the case of financial institutions borrowing from investors and similar outside entities.

Interbank exposure::

refers to the case of financial institutions borrowing from other financial institutions.

Net worth or equity::

a fixed proportion of the total asset of a bank. In general, higher equity imply better stability for an individual bank.

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Berman, P., DasGupta, B., Kaligounder, L. et al. On the Computational Complexity of Measuring Global Stability of Banking Networks. Algorithmica 70, 595–647 (2014). https://doi.org/10.1007/s00453-013-9769-0

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