  We develop and implement 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 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.
