Time consistent policy of multi-period mean-variance problem in stochastic markets

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Due to the non-separability of the variance operator, the optimal investment policy of the multi-period mean-variance model in Markovian markets doesn't satisfy the time consistency. We propose a new weak time consistency in stochastic markets and show that the pre-commitment optimal policy satisfies the weak time consistency at any intermediate period as long as the investor's wealth is no more than a specific threshold. When the investor's wealth exceeds the threshold, the weak time consistency no longer holds. In this case, by modifying the pre-commitment optimal policy, we derive a wealth interval, from which we determine a more efficient revised policy. The terminal wealth obtained under this revised policy can achieve the same mean as, but not greater variance than those of the terminal wealth obtained under the pre-commitment optimal policy; a series of superior investment policies can be obtained depending on the degree the investor wants the conditional variance to decrease. It is shown that, in the above revising process, a positive cash flow can be taken out of the market. Finally, an empirical example illustrates our theoretical results. Our results generalize existing conclusions for the multi-period mean-variance model in deterministic markets.