  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.
