# 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?

This is a difficult question per se, but people in the literature have tried different ways of dealing with the time-inconsistency of the mean variance problem.

Basak and Chabakauri (2010) is one of the seminal references.From their paper:

"In this article, we solve the dynamic asset allocation problem of a meanvariance optimizer in an incomplete-market setting and provide a simple, tractable solution for the risky stock holdings. To our knowledge, ours is the first to obtain within a general environment a fully analytical characterization of the dynamically optimal mean-variance policies, from which the investor has no incentive to deviate, namely, the time-consistent policies. "

Also, I have to take a little detour and say that on an optimal life-cycle model the rule highlighted by Bob Jansen though very appealing brings large welfare losses. Take a look at table X from Cocco, Gomes and Maenhout (2005):

Such a rule brings reduction on consumption equivalent units between 0.5% and 1.6% per anuum.

If for the optimal policy transaction costs are ignored (to simplify the problem) it might be beneficial to choose a policy that appears sub-optimal now and keep that initial portfolio to the end of the investment horizon.

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.

If you allow more time between periods so that the above arguments do not hold. It would be nice to make some more general statements about this regarding transactions, updating expectations and certainty around the current set of expectations...