On time consistency for mean-variance portfolio selection

This paper addresses a comparison between different approaches to time inconsistency for the mean-variance portfolio selection problem. We define a suitable intertemporal preferences-driven reward and use it to compare three common approaches to time inconsistency for the mean-variance portfolio selection problem over $[t_0,T]$: precommitment approach, consistent planning or game theoretical approach, and dynamically optimal approach. We prove that, while the precommitment strategy beats the other two strategies (that is a well-known obvious result), the consistent planning strategy dominates the dynamically optimal strategy until a time point $t^*in(t_0,T)$ and is dominated by the dynamically optimal strategy from $t^*$ onwards. Existence and uniqueness of the break even point $t^*$ is proven.