Controlling systemic risk: Network structures that minimize it and node properties to calculate it

Sebastian M. Krause, Hrvoje Štefančić, Guido Caldarelli, and Vinko Zlatić

Phys. Rev. E 103, 042304 – Published 5 April 2021

Abstract

Evaluation of systemic risk in networks of financial institutions in general requires information of interinstitution financial exposures. In the framework of the DebtRank algorithm, we introduce an approximate method of systemic risk evaluation which requires only node properties, such as total assets and liabilities, as inputs. We demonstrate that this approximation captures a large portion of systemic risk measured by DebtRank. Furthermore, using Monte Carlo simulations, we investigate network structures that can amplify systemic risk. Indeed, while no topology in general sense is a priori more stable if the market is liquid (i.e., the price of transaction creation is small) [T. Roukny et al., Sci. Rep. 3, 2759 (2013)], a larger complexity is detrimental for the overall stability [M. Bardoscia et al., Nat. Commun. 8, 14416 (2017)]. Here we find that the measure of scalar assortativity correlates well with level of systemic risk. In particular, network structures with high systemic risk are scalar assortative, meaning that risky banks are mostly exposed to other risky banks. Network structures with low systemic risk are scalar disassortative, with interactions of risky banks with stable banks.

Received 5 December 2019
Accepted 4 March 2021

DOI:https://doi.org/10.1103/PhysRevE.103.042304

Physics Subject Headings (PhySH)

Complex systems Econophysics Network optimization Network resilience

NetworksInterdisciplinary PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

Sebastian M. Krause^1,2, Hrvoje Štefančić³, Guido Caldarelli^4,5,6, and Vinko Zlatić ¹

¹Division of Theoretical Physics, Rudjer Bošković Institute, 10000 Zagreb, Croatia
²Faculty of Physics, University of Duisburg-Essen, 47057 Dusiburg, Germany
³Catholic University of Croatia, Ilica 242, 10000 Zagreb, Croatia
⁴DSMN, University of Venice Ca'Foscari, Via Torino 155, 30172, Venezia Mestre, Italy and ECLT Ca'Bottacin Dorsoduro 3911, Calle Crosera 30123 Venice, Italy
⁵London Institute for Mathematical Sciences, Royal Institution, 21 Albemarle Street, London W1S 4BS, United Kingdom
⁶IMT Piazza San Francesco 19, 55100 Lucca, Italy

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Vol. 103, Iss. 4 — April 2021

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Images

Figure 1
(a) In the upper panel, the shock multiplier $Ψ$ is shown during maximization (red line) and minimization (green line), where $n$ denotes sweeps (with $N^{2}$ update trials). This is for an artificial example with $N = 30$ banks, equities from a Pareto distribution with exponent three, and interbank leverages $0.32 < A_{i} / E_{i} = L_{i} / E_{i} < 0.96$ from a uniform distribution. In the lower panel, a scalar assortativity measure with respect to interbank leverage is shown for the same optimization runs. Minimal systemic risk is connected to disassortative networks [see also (b)], while maximal systemic risk is connected to assortative networks [see also (c)]. The networks in (b) and (c) encode the total assets $A_{i}$ of a bank $i$ as node size, and the interbank leverage $A_{i} / E_{i}$ as node color from pink (light shade - low values) to cyan (darker shade - high values).
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Figure 2
(a) Liability network with $N = 53$ banks in the Italian market in 1999. (d) For this network, source and target properties are uncorrelated. (e) After minimizing systemic risk, the network becomes disassortative, with anticorrelations among different scalar properties for source and target node. For $A_{i} = L_{i}$ , these correlations can be described with the simpler measure of scalar disassortativity as shown in Fig. 1 on the bottom. The gray band indicates the area between first and third quartiles for $A_{i} / E_{i}$ , so half of the values $A_{i} / E_{i}$ around the median lie within the gray area. We see that for small systemic risk, a bank $i$ with high $L_{i} / E_{i}$ should lend to a bank $j$ with low $A_{j} / E_{j}$ . Interpretation: A bank $i$ with high $L_{i} / E_{i}$ has high impact on its lenders, while a bank $j$ with low $A_{j} / E_{j}$ has only small exposure to its borrowers, thus shocks are dampened. (b) The network with minimal risk. Total assets $A_{i}$ of a bank $i$ shown as node size, and the interbank $L_{i} / E_{i}$ as edge color at the edge source, $A_{j} / E_{j}$ at edge target, node color from pink (light shade - low values) to cyan (darker shade - high values). (c) Network with maximal risk. (f) Maximal systemic risk is connected to assortativity.
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Figure 3
Finite size effects for unrestricted optimization (circles) and restricted optimization, where small degree and asymmetric investment matrix is forced (diamonds). (a) With finite size scaling we find that for large $N, Ψ$ after minimization approaches $Ψ_{\min}^{\infty} \approx 2.0947 \pm 5 \times 10^{- 4}$ , with a finite size deviation about $\propto N^{- 1.15}$ . Numerical results are shown with circles (unrestricted) and diamonds (restricted optimization). The dashed line indicates a power law with exponent $- 1.15$ . (b) Results of (a) are repeated with linear scale (green symbols and lower dashed line), and compared to results of maximization (red symbols and upper dashed line indicating results of a finite size scaling). The dotted lines indicate $Ψ_{\min}^{\infty}$ and $Ψ_{\max}^{\infty} \approx 2.315 \pm 0.01$ . (c) Assortativity after restricted optimization for minimization (green diamonds) and maximization (red diamonds). Results indicate that, independent of the network size, least risky networks are strongly disassortative, while most risky networks are strongly assortative with respect to leverage.
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Figure 4
Rescaling single bank leverage $A_{i} / E_{i} \to c \times A_{i} / E_{i}$ , stress propagation can switch from dampened to exponentially growing. We see this for an example case with $c = 2$ . While the largest eigenvalue $λ$ of the stress propagation matrix $Λ$ increases from below one (solid lines, minimization green, maximization red) to above one (dashed lines), the optimization procedure has a similar outcome with respect to assortativity in both cases.
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