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Replica approach to mean-variance portfolio optimization

Varga-Haszonits, Istvan, Caccioli, Fabio and Kondor, Imre (2016) Replica approach to mean-variance portfolio optimization. Journal of Statistical Mechanics: Theory and Experiment, 2016 (Dec.). ISSN 1742-5468

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Identification Number: 10.1088/1742-5468/aa4f9c


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.

Item Type: Article
Official URL:
Additional Information: © 2016 IOP Publishing Ltd and SISSA Medialab srl
Divisions: Systemic Risk Centre
Subjects: H Social Sciences > HA Statistics
Date Deposited: 20 Jan 2017 17:10
Last Modified: 20 Oct 2021 00:32
Projects: ES/K002309/1
Funders: Economic and Social Research Council

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