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.

翻译：我们开发并实施了用于判断放松投资组合稀疏性约束是否能够改善风险厌恶型投资者的投资机会集的方法。基于贪心算法和线性规划，我们提出了一种新的稀疏二阶随机跨度估计程序。我们证明了无论跨度是否成立，稀疏解都能渐近地实现最优恢复。利用大型股票数据集，我们估算了由于可能出现的分散不足导致的预期效用损失，并发现将稀疏机会集扩展到超过45只资产并无益处。最优稀疏投资组合投资于10个行业板块，与稀疏均值-方差投资组合相比，能够削减尾部风险。在滚动窗口基础上，危机时期资产数量缩减至25只，而标准因子模型无法解释稀疏投资组合的表现。