Analysis of bank leverage via dynamical systems and deep neural networks

TL;DR

This paper models bank leverage dynamics using a slow-fast dynamical system with noise, analyzes its stability and chaos, and employs deep neural networks to estimate parameters from real bank data, revealing chaotic leverage behavior especially in large banks.

Contribution

It introduces a novel dynamical systems model for leverage with noise and demonstrates how deep neural networks can estimate model parameters from financial data.

Findings

Leverage dynamics can exhibit chaotic behavior.

Large banks' leverage time series tend to be more chaotic.

The model's parameters can be estimated accurately using neural networks.

Abstract

We consider a model of a simple financial system consisting of a leveraged investor that invests in a risky asset and manages risk by using Value-at-Risk (VaR). The VaR is estimated by using past data via an adaptive expectation scheme. We show that the leverage dynamics can be described by a dynamical system of slow-fast type associated with a unimodal map on [0,1] with an additive heteroscedastic noise whose variance is related to the portfolio rebalancing frequency to target leverage. In absence of noise the model is purely deterministic and the parameter space splits in two regions: (i) a region with a globally attracting fixed point or a 2-cycle; (ii) a dynamical core region, where the map could exhibit chaotic behavior. Whenever the model is randomly perturbed, we prove the existence of a unique stationary density with bounded variation, the stochastic stability of the process and the almost certain existence and continuity of the Lyapunov exponent for the stationary measure. We then use deep neural networks to estimate map parameters from a short time series. Using this method, we estimate the model in a large dataset of US commercial banks over the period 2001-2014. We find that the parameters of a substantial fraction of banks lie in the dynamical core, and their leverage time series are consistent with a chaotic behavior. We also present evidence that the time series of the leverage of large banks tend to exhibit chaoticity more frequently than those of small banks.