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Characterization of Invariance, Periodic Solutions and Optimization of Dynamic Financial Networks

Leonardo Stella, Dario Bauso, Franco Blanchini, Patrizio Colaneri

TL;DR

This work addresses the propagation of financial contagion in networks with cross-holdings and asset holdings by modeling the system as a discrete-time monotone dynamical process. It derives conditions for regional attraction and invariance across orthants, demonstrates that periodic solutions longer than two can occur in discrete time, and proposes a minimal-cash-injection strategy to steer the system to the maximal healthy invariant region. The results yield a tractable framework for assessing stability margins under uncertain cross-holdings and for designing interventions to contain contagion. Collectively, the paper provides theoretical tools for systemic-risk management and practical optimization-based strategies to mitigate cascading defaults.

Abstract

Cascading failures, such as bankruptcies and defaults, pose a serious threat for the resilience of the global financial system. Indeed, because of the complex investment and cross-holding relations within the system, failures can occur as a result of the propagation of a financial collapse from one organization to another. While this problem has been studied in depth from a static angle, namely, when the system is at an equilibrium, we take a different perspective and study the corresponding dynamical system. The contribution of this paper is threefold. First, we carry out a systematic analysis of the regions of attraction and invariance of the system orthants, defined by the positive and negative values of the organizations' equity. Second, we investigate periodic solutions and show through a counterexample that there could exist periodic solutions of period greater than 2. Finally, we study the problem of finding the smallest cash injection that would bring the system to the maximal invariant region of the positive orthant.

Characterization of Invariance, Periodic Solutions and Optimization of Dynamic Financial Networks

TL;DR

This work addresses the propagation of financial contagion in networks with cross-holdings and asset holdings by modeling the system as a discrete-time monotone dynamical process. It derives conditions for regional attraction and invariance across orthants, demonstrates that periodic solutions longer than two can occur in discrete time, and proposes a minimal-cash-injection strategy to steer the system to the maximal healthy invariant region. The results yield a tractable framework for assessing stability margins under uncertain cross-holdings and for designing interventions to contain contagion. Collectively, the paper provides theoretical tools for systemic-risk management and practical optimization-based strategies to mitigate cascading defaults.

Abstract

Cascading failures, such as bankruptcies and defaults, pose a serious threat for the resilience of the global financial system. Indeed, because of the complex investment and cross-holding relations within the system, failures can occur as a result of the propagation of a financial collapse from one organization to another. While this problem has been studied in depth from a static angle, namely, when the system is at an equilibrium, we take a different perspective and study the corresponding dynamical system. The contribution of this paper is threefold. First, we carry out a systematic analysis of the regions of attraction and invariance of the system orthants, defined by the positive and negative values of the organizations' equity. Second, we investigate periodic solutions and show through a counterexample that there could exist periodic solutions of period greater than 2. Finally, we study the problem of finding the smallest cash injection that would bring the system to the maximal invariant region of the positive orthant.
Paper Structure (8 sections, 11 theorems, 52 equations, 3 figures, 1 algorithm)

This paper contains 8 sections, 11 theorems, 52 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Preliminary Results
  4. Region of Attraction and Invariance
  5. The Uncertain Case
  6. Periodic Solutions
  7. Minimal Cash Injection for a Healthy System
  8. Conclusions

Key Result

Lemma 1

Consider system (eq:model). $V(t)\ge 0, \forall t\ge 0$ and $V(0) \ge \mathbb 0_n$ if and only if $\square$

Figures (3)

  • Figure 1: Phase portrait in the $V_1$-$V_2$ plane. The equilibria are indicated by red circles and the thick black rectangles denote the invariant sets in quadrants 2 and 4.
  • Figure 2: The trajectory of system $x(t)$, which is a periodic movement of period $8$. Each color represents a different organization.
  • Figure 3: Starting from an initial condition where some of the companies have failed, we find the minimal cash investment that over time brings the system to the maximal positive invariant region $\mathcal{M}^+$.

Theorems & Definitions (16)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Example 1
  • Theorem 6
  • ...and 6 more