Mapping Systemic Risk: Critical Degree and Failures Distribution in Financial Networks
July
Mapping Systemic Risk: Critical Degree and Failures Distribution in Financial Networks
Matteo Smerlak 0 1 2
Brady Stoll 0 1 2
Agam Gupta 0 1 2
James S. Magdanz 0 1 2
0 1 Perimeter Institute for Theoretical Physics , 31 Caroline Street North, N2L 2Y5 Waterloo ON , Canada , 2 Department of Mechanical Engineering, University of Texas at Austin , 204 E. Dean Keaton, C2200, Austin, TX 78712 , United States of America, 3 Indian Institute of Management Calcutta , Diamond Harbour Road, Joka, Kolkata, West Bengal 700104 , India , 4 Resilience and Adaptation Program, University of Alaska Fairbanks , PO Box 757000, Fairbanks, AK 99775-7000 , United States of America
1 Funding: Research at the Perimeter Institute is supported in part by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript
2 Editor: Naoki Masuda, University of Bristol , UNITED KINGDOM
The financial crisis illustrated the need for a functional understanding of systemic risk in strongly interconnected financial structures. Dynamic processes on complex networks being intrinsically difficult to model analytically, most recent studies of this problem have relied on numerical simulations. Here we report analytical results in a network model of interbank lending based on directly relevant financial parameters, such as interest rates and leverage ratios. We obtain a closed-form formula for the “critical degree” (the number of creditors per bank below which an individual shock can propagate throughout the network), and relate failures distributions to network topologies, in particular scalefree ones. Our criterion for the onset of contagion turns out to be isomorphic to the condition for cooperation to evolve on graphs and social networks, as recently formulated in evolutionary game theory. This remarkable connection supports recent calls for a methodological rapprochement between finance and ecology.
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Competing Interests: The authors have declared
that no competing interests exist.
In the financial sector, shock propagation mechanisms are at the core of systemic risk [1, 2],
and banks play the most important role [3]. An important and potentially vulnerable arena for
financial contagion is the interbank loan market, which allows banks to rapidly exchange large
amounts of capital for short durations to accommodate temporary liquidity fluctuations [4–6].
In a seminal work that laid down a framework for exploring systemic risk, Eisenberg and Noe
[7] developed a clearing payment algorithm for the interbank loan market and, subsequently,
interbank loan networks have been of particular interest [8–10].
In recent years, random network theory [11, 12] has provided a useful framework to study
contagion effects in interconnected structures [13]. Applied to the financial sector [14],
network approaches have clarified the role of connectivity [15, 16], bank size [17], shock size [10]
and overlapping portfolios [18] in systemic risks. Increased understanding of contagion in
finance [19, 20] has led to an increased interest by regulators and central bankers [21, 22] in
using network measures to evaluate systemic risk. Network centrality measures have been
widely used by researchers to identify important nodes in financial networks. Battiston et al.
[23] introduced DebtRank, a measure based on feedback centrality to identify systemically
important nodes. Markose et al. [24] used eigenvector centrality to design a super-spreader tax
to make financial systems more robust.
An essential insight of Allen and Gale [25], confirmed in [8] and deepened (with different
assumptions) in [16], was that increasing network connectivity—measured by its mean degree
z—can have opposite effects depending on the baseline value of z. On the one hand, when the
network is sparsely connected, increasing z will open new channels for contagion and weaken
the network. On the other hand, when z is sufficiently large, increasing z further will dilute the
effect of a localized shock and strengthen the network. From this perspective, the key question
is not if, but when, enhanced connectivity helps secure network robustness. Battison et. al. [26,
27], Stiglitz [28], and Roukney [29] incorporated the effects of illiquid assets and potential
amplifications of failures due to human behavior in crises. These findings suggested that
intermediate-degree networks would be most stable. Additionally, Battiston et. al [26] suggested in
case of non-amplification that increasing degree may have an ambiguous relationship with
contagion depending on the parameter values of initial robustness. However, these studies
assumed a normal distribution of initial robustness. We focus on the initial failures of the
system as studied by Allen and Gale—not amplifications—and study more realistic financial
networks where (...truncated)