We develop and implement methods for determining whether relaxing sparsity con-

straints 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

of the sparse solution asymptotically whether spanning holds or not. From large equity

datasets, we estimate the expected utility loss due to possible under-diversification,

and find that there is no benefit from expanding a sparse opportunity set beyond 45

assets. The optimal sparse portfolio invests in 10 industry sectors and cuts tail risk

when compared to a sparse mean-variance portfolio. On a rolling-window basis, the

number of assets shrinks to 25 assets in crisis periods, while standard factor models

cannot explain the performance of the sparse portfolios