We develop methods for determining whether relaxing sparsity constraints on portfolios improves the investment opportunity set for risk-averse investors. We formulate a new estimation procedure for sparse second-order stochastic spanning based on a greedy algorithm and Linear Programming. We show the optimal recovery asymptotically whether spanning holds or not. From large equity datasets, we estimate the expected utility loss due to under-diversification. There is no benefit, statistically or economically, from expanding beyond 45 assets. The opti- mal portfolio cuts tail risk vis-à-vis a sparse mean-variance portfolio. On rolling windows, the number of assets shrinks to 25 assets in crisis periods.

Keywords : Nonparametric estimation, stochastic dominance, spanning, under-diversification, greedy algorithm, Linear Programming.

JEL : C13, C14, C44, C58, C61, D81, G11.