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

TL;DR

This paper models systemic financial risk using coupled dynamical systems of bank leverage, revealing how procyclical feedback loops can lead to complex behaviors and potential systemic failures.

Contribution

It introduces a novel dynamical model of bank leverage responding to asset prices, highlighting bifurcation structures and systemic risk emergence from micro-level behaviors.

Findings

Leverage dynamics exhibit logistic-like behavior with rich bifurcation structures.

Coupled bank models can produce systemic risk through feedback loops.

Procyclical leverage adjustments can lead to financial instability.

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.