Prudence with Multiplicative Risk

In this paper we analyze the conditions under which the presence of a multiplicative background risk induces a more “prudent” behavior. We show that the results from Kimball (1990) concerning the convexity of the marginal utility are no longer sufficient with multiplicative risk. An agent is multiplicative risk prudent when the coefficient of relative prudence is greater than two. We introduce the concept of quintessence in order to guarantee the decrease of relative temperance. Both decreasing relative prudence and decreasing relative temperance are sufficient to guarantee more “prudent” behavior with a multiplicative risk-prudent agent.