We develop and test multistage portfolio selection models maximizing expected end-of-horizon wealth while minimizing one-sided deviation from a target wealth level. The trade-off between two objectives is controlled by means of a non-negative parameter as in Markowitz Mean-Variance portfolio theory. We use a piecewise-linear penalty function, leading to linear programming models and ensuring optimality of subsequent stage decisions. We adopt a simulated market model to randomly generate scenarios approximating the market stochasticity. We report results of rolling horizon simulation with two variants of the proposed models depending on the inclusion of transaction costs, and under different simulated stock market conditions. We compare our results with the usual stochastic programming models maximizing expected end-of-horizon portfolio value. The results indicate that the robust investment policies are indeed quite stable in the face of market risk while ensuring expected wealth levels quite similar to the competing expected value maximizing stochastic programming model at the expense of solving larger linear programs.