Analytic approach to variance optimization under an ℓ1 constraint

The optimization of the variance of a portfolio of N independent but not identically distributed assets, supplemented by a budget constraint and an asymmetric ℓ1 regularizer, is carried out analytically by the replica method borrowed from the theory of disordered systems. The asymmetric regularizer allows us to penalize short and long positions differently, so the present treatment includes the no-short-constrained portfolio optimization problem as a special case. Results are presented for the out-of-sample and the in-sample estimator of the regularized variance, the relative estimation error, the density of the assets eliminated from the portfolio by the regularizer, and the distribution of the optimal portfolio weights. We have studied the dependence of these quantities on the ratio r of the portfolio’s dimension N to the sample size T, and on the strength of the regularizer. We have checked the analytic results by numerical simulations, and found general agreement. Regularization extends the interval where the optimization can be carried out, and suppresses the large sample fluctuations, but the performance of ℓ1 regularization is rather disappointing: if the sample size is large relative to the dimension, i.e. r is small, the regularizer does not play any role, while for r’s where the regularizer starts to be felt the estimation error is already so large as to make the whole optimization exercise pointless. We find that the ℓ1 regularization can eliminate at most half the assets from the portfolio (by setting their weights to exactly zero), corresponding to this there is a critical ratio r = 2 beyond which the ℓ1 regularized variance cannot be optimized: the regularized variance becomes constant over the simplex. These facts do not seem to have been noticed in the literature. Graphical abstract