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Analytic solution to variance optimization with no short positions

Kondor, I; Papp, G; Caccioli, F; (2017) Analytic solution to variance optimization with no short positions. Journal of Statistical Mechanics: Theory and Experiment , Article 123402. 10.1088/1742-5468/aa9684.

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Abstract

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 `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 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 r = 1 to 2, and changes its character: at r = 2 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 r = 2. 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

Type: Article
Title: Analytic solution to variance optimization with no short positions
DOI: 10.1088/1742-5468/aa9684
Publisher version: http://dx.doi.org/10.1088/1742-5468/aa9684
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Science & Technology, Technology, Physical Sciences, Mechanics, Physics, Mathematical, Physics, cavity and replica method, quantitative finance, risk measure and management, Dimensional Covariance Matrices, Portfolio Optimization, Noise Sensitivity, Risk Measures, Selection, Constraints, Estimator, Shrinkage
UCL classification: UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science
URI: https://discovery.ucl.ac.uk/id/eprint/10044428
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