In this paper we analyze the conditions under which the presence of a multiplicative background risk induces a more “prudent” behavior.
Professor, EDHEC Business School
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. Nevertheless, the convexity of the relative prudence combined with the decrease (respectively increase) of the relative prudence implies more prudent behavior when the agent is multiplicative risk prudent (respectively risk imprudent). Surprisingly, the derived utility function inherits some properties of the direct utility function and the presence of the multiplicative background risk preserves comparative riskprudent behavior under certain conditions. Our results apply to savings and portfolio selection. For example, a risk-prudent agent will lower his or her willingness to hold risky assets when he or she faces a multiplicative background risk if his or her relative risk aversion is convex and decreases.
Prudence with Multiplicative Risk...
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