  We present a simple agent-based model of a financial system composed of
leveraged investors such as banks that invest in stocks and manage their risk
using a Value-at-Risk constraint, based on historical observations of asset
prices. The Value-at-Risk constraint implies that when perceived risk is low,
leverage is high and vice versa, a phenomenon that has been dubbed pro-cyclical
leverage. We show that this leads to endogenous irregular oscillations, in
which gradual increases in stock prices and leverage are followed by drastic
market collapses, i.e. a leverage cycle. This phenomenon is studied using
simplified models that give a deeper understanding of the dynamics and the
nature of the feedback loops and instabilities underlying the leverage cycle.
We introduce a flexible leverage regulation policy in which it is possible to
continuously tune from pro-cyclical to countercyclical leverage. When the
policy is sufficiently countercyclical and bank risk is sufficiently low the
endogenous oscillation disappears and prices go to a fixed point. While there
is always a leverage ceiling above which the dynamics are unstable,
countercyclical leverage can be used to raise the ceiling. We also study the
impact on leverage cycles of direct, temporal control of the bank's riskiness
via the bank's required Value-at-Risk quantile. Under such a rule the regulator
relaxes the Value-at-Risk quantile following a negative stock price shock and
tightens it following a positive shock. While such a policy rule can reduce the
amplitude of leverage cycles, its effectiveness is highly dependent on the
choice of parameters. Finally, we investigate fixed limits on leverage and show
how they can control the leverage cycle.
