We present a robust optimization approach to portfolio management under uncertainty when randomness is modeled using uncertainty sets for the continuously compounded rates of return, which empirical research argues are the true drivers of uncertainty, but the parameters needed to define the uncertainty sets, such as the drift and standard deviation, are not known precisely. Instead, a finite set of scenarios is available for the input data, obtained either using different time horizons or assumptions in the estimation process. Our objective is to maximize the worst-case portfolio value (over a set of allowable deviations of the uncertain parameters from their nominal values, using the worst-case nominal values among the possible scenarios) at the end of the time horizon in a one-period setting. Short sales are not allowed. We consider both the independent and correlated assets models. For the independent assets case, we derive a convex reformulation, albeit involving functions with singular Hessians. Because this slows computation times, we also provide lower and upper linear approximation problems and devise an algorithm that gives the decision maker a solution within a desired tolerance from optimality. For the correlated assets case, we suggest a tractable heuristic that uses insights derived in the independent assets case.