# Analytic solution to variance optimization with no short-selling

Imre Kondor, G\'abor Papp and Fabio Caccioli

Papers from arXiv.org

Abstract: A large portfolio of independent returns is optimized under the variance risk measure with a ban on short positions. The no-short selling constraint acts as an asymmetric $\ell_1$ 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 susceptibility, i.e. the sensitivity of the optimal portfolio weights to changes in the returns, diverges at a critical value $r=2$. This means that a ban on short positions does not prevent the phase transition in the optimization problem, it merely shifts the critical point from its non-regularized value of $r=1$ to $2$. At $r=2$ the out of sample estimator for the portfolio variance stays finite and the estimated in-sample variance vanishes. We have performed numerical simulations to support the analytic results and found perfect agreement for \$N/T

New Economics Papers: this item is included in nep-rmg
Date: 2016-12, Revised 2017-01
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