Risk reduction and efficiency increase in large portfolios: Leverage and shrinkage

Two basic solutions have been proposed to fix the well-documented incompatibility of the sample covariance matrix with Markowitz mean-variance portfolio optimization: first, restrict leverage so much that no short sales are allowed; or, second, linearly shrink the sample covariance matrix towards a parsimonious target. Mathematically, there is a deep connection between the two approaches, and empirically they display similar performances. Recent developments have turned the choice between no-short-sales and linear shrinkage into a false "either-or" dichotomy. What if, instead of 0% leverage we considered fully-invested, long-short 130/30 portfolios, or even 150/50, given that prime brokers, fund regulators and investors have started to allow it? And instead of linearly shrinking the unconditional covariance matrix, what if we allowed for each of the eigenvalues of the sample covariance matrix to have its own shrinkage intensity, optimally determined under large-dimensional asymptotics, while also incorporating Multivariate GARCH effects? Our empirical evidence finds that, indeed, these new developments enable us to have "the best of both worlds" by combining some appropriate leverage constraint with a judiciously chosen shrinkage method. The overall winner is a 150/50 investment strategy where the covariance matrix estimator is a combination of DCC (Dynamic Conditional Correlation - a well-known Multivariate GARCH model) - with NL (Non-Linear shrinkage, a substantial upgrade upon linear shrinkage technology); although 130/30 DCC-NL comes a close second. This is true both in the "pure" case of estimating the Global Minimum Variance portfolio, and also for textbook-style construction of Markowitz mean-variance efficient portfolio.