Time-Consistency of Optimal Investment under Smooth Ambiguity

Abstract

We study portfolio choice in a Black-Scholes world under drift uncertainty. Preferences towards risk and ambiguity are modeled using the smooth ambiguity approach under a double power utility assumption and a normal distribution assumption on the unknown drift. Optimal investment in this setting is time-inconsistent: While utility is maximized by a precommitment strategy resembling the classical Merton solution, the investor’s future selves prefer to constantly increase the riskiness of the strategy. In contrast, the optimal dynamically consistent investment strategy accounts for variations in the perceived severity of drift uncertainty, thus increasing the riskiness of the strategy gradually over time. We provide a detailed comparative analysis of the mechanics and interplay of ambiguity, myopia and optimal decisions in this setting. We show that an investor who pre-commits will regret that decision from some time point onwards, wishing that she had followed the dynamically consistent strategy. This “point of regret” always lies near the middle of the investment horizon.