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The approach to portfolio construction proposed in this paper is based on recent results on stochastic reachability. It is assumed that investors' preferences are expressed in terms of target sets to be reached at each time period over a specified finite horizon. A portfolio is defined optimal if it maximizes the probability of its value to belong to the target sets. A case study drawn from the US market shows the interest and applicability of the approach. The optimal solution we obtain exhibits a contrarian attitude, whereby risky exposures are enhanced in case of negative performances and reduced in case of positive performances. A comparison with the constant proportion portfolio insurance method highlights advantages and drawbacks of the proposed approach.