  The existing approaches to sparse wealth allocations (1) are limited to
low-dimensional setup when the number of assets is less than the sample size;
(2) lack theoretical analysis of sparse wealth allocations and their impact on
portfolio exposure; (3) are suboptimal due to the bias induced by an
$\ell_1$-penalty. We address these shortcomings and develop an approach to
construct sparse portfolios in high dimensions. Our contribution is twofold:
from the theoretical perspective, we establish the oracle bounds of sparse
weight estimators and provide guidance regarding their distribution. From the
empirical perspective, we examine the merit of sparse portfolios during
different market scenarios. We find that in contrast to non-sparse
counterparts, our strategy is robust to recessions and can be used as a hedging
vehicle during such times.
