The Size of Background Risk and the Theory of Risk Bearing

Abstract

We establish a necessary and sufficient condition for the risk aversion of an agent’s derived utility function to increase with independent, zero-mean background risk. This condition is weaker than standard risk aversion. For small risks, the condition is that the ratio of the third to the first derivative of the utility function is decreasing in income. In a market with state-contingent marketable claims, an increase in background risk, which raises the agent’s derived risk aversion, reduces the slope of the agent’s optimal sharing rule. Under a weak aggregation condition, an increase of background risk for many agents in the economy raises the prices of marketable claims in states with a low level of marketable aggregate income relative to the prices in states with a higher level of such income.