Abstract The paper investigates quadratic convex reformulations (QCR) for the portfolio selection problem with Value-at-Risk (VaR) constraint (PS-VaR). Problem (PS-VaR) is in fact equivalent to a chance constrained problem. With an assumption of discrete distribution, problem (PS-VaR) can be generally formulated as a standard mixed-integer problem (MIP). By means of copy constraints, we first introduce semicontinuous variables for problem (PS-VaR). By exploiting the special property of semicontinuous variables, we then propose a class of quadratic convex reformulations for problem (PS-VaR), which have continuous relaxation bounds at least as tight as that of the standard MIP reformulation. We further discuss how to find the QCR with the best continuous relaxation bound among this class via strong duality and lifting techniques. Computational experiments are then conducted to illustrate the effectiveness of the proposed approach in contrast to the standard MIP formulation.