Endogenous Leverage: VaR and Beyond

We study endogenous leverage in a general equilibrium model with incomplete markets. We prove that in any binary tree leverage emerges in equilibrium at the maximum level such that VaR = 0, so there is no default in equilibrium, provided that agents get no utility from holding the collateral. When the collateral does affect utility (as with housing) or when agents have sufficiently heterogenous beliefs over three or more states, VaR = 0 fails to hold in equilibrium. We study commonly used examples: an economy in which investors have heterogenous beliefs and a CAPM economy consisting of investors with different risk aversion. We find two main departures from VaR = 0. First, both examples show that with enough heterogeneity among the investors, equilibrium default is normal. Second, we find that more than one contract is actively traded in equilibrium on the same collateral, that is, the same asset is bought at different margin requirements by different agents. Finally, we study the relationship between leverage and asset prices. We provide an example that shows that as the regulatory authority gradually relaxes leverage restrictions from low levels and permits leverage to rise, asset prices start to rise, but eventually increased leverage paradoxically tends to reduce asset prices because the risky bonds become substitutes for the asset used as collateral.