We consider the variance portfolio optimization problem with a ban on short selling. We provide an analytical solution by means of the replica method for the case of a portfolio of independent, but not identically distributed, assets. We study the behavior of the solution as a function of the ratio r between the number N of assets and the length T of the time series of returns used to estimate risk. The no-short-selling constraint acts as an asymmetric  regularizer, setting some of the portfolio weights to zero and keeping the out-of-sample estimator for the variance bounded, avoiding the divergence present in the non-regularized case.

However, the ban on short positions does not prevent the phase transition in the optimization problem, only shifts the critical point from its non-regularized value of  to 2, and changes its character: at  the out-of-sample estimator for the portfolio variance stays finite and the estimated in-sample variance vanishes, while another critical parameter, related to the estimated portfolio weights and the condensate density, diverges at the critical value . We also calculate the distribution of the optimal weights over the random samples and show that the regularizer preferentially removes the assets with large variances, in accord with one’s natural expectation.

I. Kondor, G. Papp, F. Caccioli, Analytic solution to variance optimization with no short positions, J. Stat. Mech. (2017) 123402