Robust Mean–Variance Portfolio Optimization: Mean–Variance–Variance Criterion Versus Mean–Variance–Standard Deviation Criterion

ABSTRACT We study a dynamic portfolio optimization problem under the mean–variance–variance (M‐V‐V) criterion proposed by Maccheroni et al. It is an analogue of the Arrow–Pratt approximation to the well‐known smooth ambiguity model. Under the standard Black–Scholes framework, we derive fully explicit equilibrium investment strategies in which a DM's level of ambiguity, risk aversion, and ambiguity aversion are transparently captured. We find that the time horizon appears inconsistently in the objective function of the M‐V‐V criterion, in turn causing the equilibrium strategies to be nonmonotonic with respect to risk aversion. In response, we introduce a new mean–variance–standard deviation (M‐V‐SD) criterion to address this issue. Equilibrium strategies under the M‐V‐SD criterion exhibit an appealing feature of limited stock market participation which provides a theoretical justification to this phenomenon.