  We present an analytical model to study the role of expectation feedbacks and
overlapping portfolios on systemic stability of financial systems. Building on
[Corsi et al., 2016], we model a set of financial institutions having Value at
Risk capital requirements and investing in a portfolio of risky assets, whose
prices evolve stochastically in time and are endogenously driven by the trading
decisions of financial institutions. Assuming that they use adaptive
expectations of risk, we show that the evolution of the system is described by
a slow-fast random dynamical system, which can be studied analytically in some
regimes. The model shows how the risk expectations play a central role in
determining the systemic stability of the financial system and how wrong risk
expectations may create panic-induced reduction or over-optimistic expansion of
balance sheets. Specifically, when investors are myopic in estimating the risk,
the fixed point equilibrium of the system breaks into leverage cycles and
financial variables display a bifurcation cascade eventually leading to chaos.
We discuss the role of financial policy and the effects of some market
frictions, as the cost of diversification and financial transaction taxes, in
determining the stability of the system in the presence of adaptive
expectations of risk.
