# Multi-period portfolio allocation: Time-inconsistent approach

Consider a multi-period mean-variance portfolio optimization so that at time $$t$$ I find the strategy that maximizes my expected terminal wealth $$X_T$$, subject to a constraint on risk, \begin{align*} \Pi_t = \mathbb{E}_t[X_T]-Var_t[X_T]. \end{align*}

Presumably I can do the same tomorrow, but it turns out that the strategy set in motion today will be sub-optimal for me tomorrow, so I will deviate from it. In other words, the strategy set in motion today will never be realized.

There does exist a solution concept that deals with this time-inconsistency and takes future behavior into account (subgame-perfect solution). However, the approach described above seems to be widely used, and I my question is whether it can be rationalized? That is, can it be rational today to decide a strategy that will be sub-optimal tomorrow and thus not carried out?

In any case, I would not expect that portfolio weights would predictably change significantly. What makes a stock attractive today, should make it attractive tomorrow. Also, in practice, I would not expect the effect of approaching the horizon will have a large impact on allocation. If you follow the rule of thumb: stock allocation is $$100\% - \textrm{age}$$, then the change is not even a basis point per day.