Large Scale Mean--Variance Investing Via Nuclear Hedging

We propose a new approach for constructing global minimum–variance and mean–variance efficient portfolios in large asset markets. Instead of estimating the mean vector and precision matrix of returns separately, we express both objects through the coefficients and residual variances of a system of multi–response hedging regressions, in which each asset is optimally hedged using all others. We show that, under general (possibly approximate) factor structures, the matrix of hedging portfolio returns is low-rank: a small number of common hedging portfolios spans most of the systematic comovement in returns. This result motivates a penalized reduced–rank hedging estimator that (i) enforces a low–dimensional hedging structure, (ii) shrinks excessive hedging coefficients promoting diversified portfolios, and (iii) regularizes mean estimates toward economically plausible targets. This approach delivers a dense but structured precision matrix that reflects realistic factor exposures without imposing hard sparsity or orthogonality. In simulations and empirical applications with both $N<T$ and $N>T$, the resulting portfolios are stable, well diversified, and exhibit substantially lower realized volatility and higher out–of–sample Sharpe ratios than leading benchmark methods.