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Mean-Variance Portfolio Optimization: Eigendecomposition-Based Methods
Linköping University, Department of Mathematics. Linköping University, Faculty of Science & Engineering.
2015 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

Modern portfolio theory is about determining how to distribute capital among available securities such that, for a given level of risk, the expected return is maximized, or for a given level of return, the associated risk is minimized. In the pioneering work of Markowitz in 1952, variance was used as a measure of risk, which gave rise to the wellknown mean-variance portfolio optimization model. Although other mean-risk models have been proposed in the literature, the mean-variance model continues to be the backbone of modern portfolio theory and it is still commonly applied. The scope of this thesis is a solution technique for the mean-variance model in which eigendecomposition of the covariance matrix is performed.

The first part of the thesis is a review of the mean-risk models that have been suggested in the literature. For each of them, the properties of the model are discussed and the solution methods are presented, as well as some insight into possible areas of future research.

The second part of the thesis is two research papers. In the first of these, a solution technique for solving the mean-variance problem is proposed. This technique involves making an eigendecomposition of the covariance matrix and solving an approximate problem that includes only relatively few eigenvalues and corresponding eigenvectors. The method gives strong bounds on the exact solution in a reasonable amount of computing time, and can thus be used to solve large-scale mean-variance problems.

The second paper studies the mean-variance model with cardinality constraints, that is, with a restricted number of securities included in the portfolio, and the solution technique from the first paper is extended to solve such problems. Near-optimal solutions to large-scale cardinality constrained mean-variance portfolio optimization problems are obtained within a reasonable amount of computing time, compared to the time required by a commercial general-purpose solver.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2015. , 51 p.
Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1717
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-118362DOI: 10.3384/lic.diva.118362ISBN: 978-91-7519-038-9 (print)OAI: oai:DiVA.org:liu-118362DiVA: diva2:814636
Presentation
2015-06-09, ACAS, A-huset, ingång 17, Campus Valla, Linköpings universitet, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2015-05-27 Created: 2015-05-27 Last updated: 2015-05-27Bibliographically approved
List of papers
1. Eigendecomposition of the mean-variance portfolio optimization model
Open this publication in new window or tab >>Eigendecomposition of the mean-variance portfolio optimization model
2015 (English)In: Optimization, control, and applications in the information age / [ed] Athanasios Migdalas and Athanasia Karakitsiou, Springer, 2015, 209-232 p.Chapter in book (Refereed)
Abstract [en]

We provide new insights into the mean-variance portfolio optimization problem, based on performing eigendecomposition of the covariance matrix. The result of this decomposition can be given an interpretation in terms of uncorrelated eigenportfolios. When only some of the eigenvalues and eigenvectors are used, the resulting mean-variance problem is an approximation of the original one. A solution to the approximation yields lower and upper bounds on the original mean-variance problem; these bounds are tight if sufficiently many eigenvalues and eigenvectors are used in the approximation. Even tighter bounds are obtained through the use of a linearized error term of the unused eigenvalues and eigenvectors.

We provide theoretical results for the upper bounding quality of the approximate problem and the cardinality of the portfolio obtained, and also numerical illustrations of these results. Finally, we propose an ad hoc linear transformation of the mean-variance problem, which in practice significantly strengthens the bounds obtained from the approximate mean-variance problem.

Place, publisher, year, edition, pages
Springer, 2015
Series
Springer Proceedings in Mathematics & Statistics, ISSN 2194-1009 ; Vol. 130
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-118358 (URN)10.1007/978-3-319-18567-5_11 (DOI)000380540400011 ()9783319185668 (ISBN)9783319185675 (ISBN)
Available from: 2015-05-27 Created: 2015-05-27 Last updated: 2016-08-19Bibliographically approved
2. Tight Upper Bounds on the Cardinality Constrained Mean-Variance Portfolio Optimization Problem Using Truncated Eigendecomposition
Open this publication in new window or tab >>Tight Upper Bounds on the Cardinality Constrained Mean-Variance Portfolio Optimization Problem Using Truncated Eigendecomposition
2016 (English)In: Operations Research Proceedings 2014: Selected Papers of the Annual International Conference of the German Operations Research Society (GOR), RWTH Aachen University, Germany, September 2-5, 2014 / [ed] Marco Lübbecke, Arie Koster, Peter Letmathe, Reinhard Madlener, Britta Peis and Grit Walther, Springer, 2016, 385-392 p.Conference paper, Published paper (Refereed)
Abstract [en]

The mean-variance problem introduced by Markowitz in 1952 is a fundamental model in portfolio optimization up to date. When cardinality and bound constraints are included, the problem becomes NP-hard and the existing optimizing solution methods for this problem take a large amount of time.

We introduce a core problem based method for obtaining upper bounds to the meanvariance portfolio optimization problem with cardinality and bound constraints. The method involves performing eigendecomposition on the covariance matrix and then using only few of the eigenvalues and eigenvectors to obtain an approximation of the original problem. A solution of this approximate problem has a relatively low cardinality and is used to construct a core problem. When solved, the core problem provides an upper bound. We test the method on large scale problems of up to 1000 assets. The obtained upper bounds are of high quality and the time required to obtain them is much less than what state-of-the-art mixed integer softwares use, which makes it practically useful.

Place, publisher, year, edition, pages
Springer, 2016
Series
Operations Research Proceedings, ISSN 0721-5924
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-118359 (URN)10.1007/978-3-319-28697-6_54 (DOI)9783319286952 (ISBN)9783319286976 (ISBN)
Conference
Annual International Conference of the German Operations Research Society (GOR), RWTH Aachen University, Germany, September 2-5, 2014
Available from: 2015-05-27 Created: 2015-05-27 Last updated: 2016-11-18Bibliographically approved

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