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.

A recent literature shows how an increase in volatility reduces leverage. However, in order to explain pro-cyclical leverage it assumes that bad news increases volatility, that is, it assumes an inverse relationship between first and second moments of asset returns. This paper suggests a reason why bad news is more often than not associated with higher future volatility. We show that, in a model with endogenous leverage and heterogeneous beliefs, agents have the incentive to invest mostly in technologies that become more volatile in bad times. Agents choose these technologies because they can be leveraged more during normal times. Together with the existing literature this explains procyclical leverage. The result also gives a rationale to the pattern of volatility smiles observed in the stock options since 1987. Finally, the paper presents for the first time a dynamic model in which an asset is endogenously traded simultaneously at different margin requirements in equilibrium.

We show that binomial economies with financial assets are an informative and tractable model to study endogenous leverage and collateral equilibrium: endogenous leverage can be highly volatile, but it is always easy to compute. The possibility of default can have a dramatic effect on equilibrium, if collateral is scarce, yet we prove the No-Default Theorem asserting that, without loss of generality, there is no default in equilibrium. Thus potential default has a dramatic effect on equilibrium, but actual default does not. This result is valid with arbitrary preferences, contingent promises, many assets and consumption goods, production, and multiple periods. On the other hand, we show that the theorem fails in trinomial models. For example, in a CAPM model, we find that default is robust. In a model with heterogeneous beliefs, we find that different agents might borrow on the same asset with different LT V s.