Sparse Weighted Norm Minimum Variance Portfolio

Abstract:     
We propose to impose a weighted L1 and squared L2 norm penalty on the portfolio weights to improve performances of portfolio optimization when number of assets N becomes large. We show that under certain conditions, the realized risk of the portfolio obtained from this strategy will asymptotically be less than that of some benchmark portfolios with high probability. An intuitive interpretation for why including a fewer number of assets may be benefcial in the high dimensional situation is built on a constraint between sparsity of the optimal weight vector and the realized risk. The theoretical results also imply that the penalty parameters for the weighted norm penalty can be specified as a function of the number of assets N and sample size n for estimating parameters used in the portfolio optimization. We then propose a coordinate-wise descent algorithm to solve the penalized weighted norm portfolio optimization problem. Wefind performances of the weighted norm strategy dominate other benchmarks for the case of Fama-French 100 size and book to market ratio portfolios, but are mixed for the case of individual stocks. We also propose several novel alternative norm penalties and show that their performances are comparable to the weighted norm strategy.

Keywords: Sparsity and Diversity, Weighted Norm Portfolio, Coordinate-Wise Descent Algorithm

JEL Classification: C40, C61, G11

working papers series