We propose a novel approach to the benchmark replication problem that uses a minimum tracking error variance as an objective subject to a target expected outperformance. When no budget constraint is imposed on the replicating portfolio, the solution is a two-fund portfolio involving the standard hedge portfolio and the tangent portfolio constructed using the replicating securities. In the presence of a budget constraint, the solution is a three-fund portfolio, which includes, in addition, the minimum variance portfolio constructed using the replicating securities. We implement our theoretical results using recent data for three widely followed US stock indices with very good out-of-sample performance.