  Effective risk control must make a tradeoff between the microprudential risk
of exogenous shocks to individual institutions and the macroprudential risks
caused by their systemic interactions. We investigate a simple dynamical model
for understanding this tradeoff, consisting of a bank with a leverage target
and an unleveraged fundamental investor subject to exogenous noise with
clustered volatility. The parameter space has three regions: (i) a stable
region, where the system always reaches a fixed point equilibrium; (ii) a
locally unstable region, characterized by cycles and chaotic behavior; and
(iii) a globally unstable region. A crude calibration of parameters to data
puts the model in region (ii). In this region there is a slowly building price
bubble, resembling a "Great Moderation", followed by a crash, with a period of
approximately 10-15 years, which we dub the "Basel leverage cycle". We propose
a criterion for rating macroprudential policies based on their ability to
minimize risk for a given average leverage. We construct a one parameter family
of leverage policies that allows us to vary from the procyclical policies of
Basel II or III, in which leverage decreases when volatility increases, to
countercyclical policies in which leverage increases when volatility increases.
We find the best policy depends critically on three parameters: The average
leverage used by the bank; the relative size of the bank and the
fundamentalist, and the amplitude of the exogenous noise. Basel II is optimal
when the exogenous noise is high, the bank is small and leverage is low; in the
opposite limit where the bank is large or leverage is high the optimal policy
is closer to constant leverage. We also find that systemic risk can be
dramatically decreased by lowering the leverage target adjustment speed of the
banks.
