Multiplicative Risk Prudence, Inflation Uncertainty and the Real Risk-Free Rate Puzzle

Abstract

We examine the optimal saving decision of individuals who face a multiplicative risk. An individual is defined to be multiplicative risk prudent if multiplying a pure risk to her future wealth raises her optimal savings. We show that convex marginal utility is not sufficient to induce multiplicative risk prudent. Instead, an individual is multiplicative risk prudent if and only if her relative prudence of future consumption uniformly exceeds two. Our results provide an explanation to the real risk-free rate puzzle. Intuitively, the presence of (or an increase in) inflation uncertainty in an economy should stengthen the aggregate precautionary motive to save, leading to a reduction in the equilibrium real risk-free rate. Thus neglecting inflation uncertainty may result in an overestimation of real risk-free rate. We also study jointly the impact of correlated additive and multiplicative risks on the optimal savings decision and demonstrate that the concept of multiplicative risk prudence is stronger than that of additive-multiplicative risk prudence. Our findings suggest one should take the condition of multiplicative risk prudence as a natural restriction on preference.