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Chaos and Synchronization in Financial Leverages Dynamics: Modeling Systemic Risk with Coupled Unimodal Maps

Marco Ioffredi, Stefano Marmi, Matteo Tanzi

Abstract

Systemic financial risk refers to the simultaneous failure or destabilization of multiple financial institutions, often triggered by contagion mechanisms or common exposures to shocks. In this paper, we present a dynamical model of bank leverage (the ratio of asset holdings to equity) a quantity that both reflects and drives risk dynamics. We model how banks, constrained by Value-at-Risk (VaR) regulations, adjust their leverage in response to changes in the price of a single asset, assumed to be held in fixed proportion across banks. This leverage-targeting behavior introduces a procyclical feedback loop between asset prices and leverage. In the dynamics, this can manifest as logistic-like behavior with a rich bifurcation structure across model parameters. By analyzing these coupled dynamics in both isolated and interconnected bank models, we outline a framework for understanding how systemic risk can emerge from seemingly rational micro-level behavior.

Chaos and Synchronization in Financial Leverages Dynamics: Modeling Systemic Risk with Coupled Unimodal Maps

Abstract

Systemic financial risk refers to the simultaneous failure or destabilization of multiple financial institutions, often triggered by contagion mechanisms or common exposures to shocks. In this paper, we present a dynamical model of bank leverage (the ratio of asset holdings to equity) a quantity that both reflects and drives risk dynamics. We model how banks, constrained by Value-at-Risk (VaR) regulations, adjust their leverage in response to changes in the price of a single asset, assumed to be held in fixed proportion across banks. This leverage-targeting behavior introduces a procyclical feedback loop between asset prices and leverage. In the dynamics, this can manifest as logistic-like behavior with a rich bifurcation structure across model parameters. By analyzing these coupled dynamics in both isolated and interconnected bank models, we outline a framework for understanding how systemic risk can emerge from seemingly rational micro-level behavior.
Paper Structure (7 sections, 6 theorems, 31 equations, 9 figures)

This paper contains 7 sections, 6 theorems, 31 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Unimodal Evolution of the Leverage for One Bank Trading One Asset
  3. Coupled Unimodal Maps Model for the Leverage Evolution of Banks Trading a Common Asset
  4. The Model
  5. Homogeneous case: sychronization
  6. Heterogeneous case
  7. 'Big vs small' bank: the skew-product case

Key Result

Theorem 1

(Synchronization in the Homogenous Case) If $\omega_1=\omega_2$, $\forall (\lambda_{1,0},\lambda_{2,0})\in\mathcal{V}$

Figures (9)

  • Figure 1: Map $T$ for $\omega=0.3$ (Blue), $\omega=0.5$ (Orange), $\omega=0.8$ (Green). Here $\alpha=1.64$, $\Sigma_\epsilon=0.0015^2$, $\gamma=100$
  • Figure 2: Bifurcation diagrams for $T$ (obtained discarding the first 1000 values and considering the next 800) for $\Sigma_\epsilon=0.0015^2$, $\gamma=100$, $\alpha=1.64$.
  • Figure 3: Lyapunov exponent for the map $T$ (calculated over 100 time steps) for $\Sigma_\epsilon=0.0015^2$, $\gamma=100$, $\alpha=1.64$.
  • Figure 4: Leverages synchronization for $\omega_1=\omega_2=0.8, 0.6, 0.3$ (top to bottom) and $\pi_1=0.5$
  • Figure 5: Example of asymptotic orbits for the two banks (green for Bank 1 and blue for Bank 2) as $\pi_1$ varies and for different choices of $\omega_1, \omega_2$. Here $\omega_1=0.8,\omega_2=0.3$. In doing the plot, the first 1000 values have been discarded and the next 500 plotted.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6