Sensitivity of the Eisenberg--Noe Clearing Vector to Individual Interbank Liabilities

Abstract

We quantify the sensitivity of the Eisenberg-Noe clearing vector to estimation errors in the bilateral liabilities of a financial system. The interbank liabilities matrix is a crucial input to the computation of the clearing vector. However, in practice central bankers and regulators must often estimate this matrix because complete information on bilateral liabilities is rarely available. As a result, the clearing vector may suffer from estimation errors in the liabilities matrix. We quantify the clearing vector's sensitivity to such estimation errors and show that its directional derivatives are, like the clearing vector itself, solutions of fixed point equations. We describe estimation errors utilizing a basis for the space of matrices representing permissible perturbations and derive analytical solutions to the maximal deviations of the Eisenberg-Noe clearing vector. This allows us to compute upper bounds for the worst case perturbations of the clearing vector. Moreover, we quantify the probability of observing clearing vector deviations of a certain magnitude, for uniformly or normally distributed errors in the relative liability matrix.Applying our methodology to a dataset of European banks, we find that perturbations to the relative liabilities can result in economically sizeable differences that could lead to an underestimation of the risk of contagion. Our results are a first step towards allowing regulators to quantify errors in their simulations.

A Continuous-Time Model of Financial Clearing

“…They considered an environment, in which all participants (henceforth, "banks") default within a single clearing mechanism, and demonstrated that there always exist a "clearing payment vector" that satisfy some natural requirements. The Eisenberg-Noe approach has been successfully extended to incorporate liquidity spillovers (Cifuentes, Ferrucci and Shin, 2005;Shin, 2008), outside liabilities (Elsinger, 2009;Glasserman and Young, 2015), costs of default (Rogers and Veraart, 2013), liabilities of different seniority (Kusnetsov and Veraart, 2019), mandatory disclosures (Alvarez and Barlevy, 2015), and other financial instruments, and has become a cornerstone in analysis of systemic financial risk (see, e.g., Hurd, 2016;Feinstein et al, 2018;Kabanov, Mokbel and El Bitar, 2018, for recent surveys).…”

“…Liu and Staum (2010) provided a formula for sensitivity analysis of Eisenberg and Noe's one-period model of contagion via direct bilateral links. Feinstein et al (2018) quantified the Eisenberg-Noe clearing vector's sensitivity.…”

Systemic illiquidity in the interbank network

“…For example, Cifuentes, Ferrucci & Shin (2005) allow for liquidity considerations and Rogers & Veraart (2013) introduce costs of default. More recently, Feinstein, Pang, Rudloff, Schaanning, Sturm & Wildman (2018) test the sensitivity of the Eisenberg-Noe clearing vector to estimation errors in bilateral interbank liabilities. There are also studies on alternative clearing processes.…”

Network valuation in financial systems

“…Often, even regulators have only a partial view of the detailed structure of interbank assets. Indeed, a separate strand of literature is focused specifically on trying to "reconstruct" interbank assets from publicly available information (Anand, Craig, & Von Peter, 2015;Cimini, Squartini, Garlaschelli, & Gabrielli, 2015;Gandy & Veraart, 2016;Squartini, Cimini, Gabrielli, & Garlaschelli, 2017;Squartini, Caldarelli, Cimini, Gabrielli, & Garlaschelli, 2018) or to assess the impact of their misestimation (Feinstein et al, 2018).…”