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Replica approach to mean-variance portfolio optimization

Varga-Haszonits, I; Caccioli, F; Kondor, I; (2016) Replica approach to mean-variance portfolio optimization. Journal of Statistical Mechanics: Theory and Experiment , 2016 , Article 123. 10.1088/1742-5468/aa4f9c. Green open access

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Abstract

We consider the problem of mean-variance portfolio optimization for a generic covariance matrix subject to the budget constraint and the constraint for the expected return, with the application of the replica method borrowed from the statistical physics of disordered systems. We find that the replica symmetry of the solution does not need to be assumed, but emerges as the unique solution of the optimization problem. We also check the stability of this solution and find that the eigenvalues of the Hessian are positive for r  =  N/T  <  1, where N is the dimension of the portfolio and T the length of the time series used to estimate the covariance matrix. At the critical point r  =  1 a phase transition is taking place. The out of sample estimation error blows up at this point as 1/(1  −  r), independently of the covariance matrix or the expected return, displaying the universality not only of the critical exponent, but also the critical point. As a conspicuous illustration of the dangers of in-sample estimates, the optimal in-sample variance is found to vanish at the critical point inversely proportional to the divergent estimation error.

Type: Article
Title: Replica approach to mean-variance portfolio optimization
Open access status: An open access version is available from UCL Discovery
DOI: 10.1088/1742-5468/aa4f9c
Publisher version: http://dx.doi.org/10.1088/1742-5468/aa4f9c
Language: English
Additional information: © 2016 IOP Publishing Ltd and SISSA Medialab srl. This is an author-created, un-copyedited version of an article accepted for publication/published in Journal of Statistical Mechanics: Theory and Experiment. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at 10.1088/1742-5468/aa4f9c.
Keywords: Cavity and replica method, quantitative finance, risk measure and management, Correlation-matrices, Expected Shortfall, Noise Sensitivity, Risk Measures, Selection, Constraints, Uncertainty, Estimator, Parameter, Models
UCL classification: UCL
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/1502721
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