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Eigendecomposition of the mean-variance portfolio optimization model
Department of Mathematics, Makerere University, Kampala, Uganda.
Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0003-2094-7376
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. 209-232 p.
Series
, Springer Proceedings in Mathematics & Statistics, ISSN 2194-1009 ; Vol. 130
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-118358DOI: 10.1007/978-3-319-18567-5_11ISI: 000380540400011ISBN: 9783319185668 (hardcover)ISBN: 9783319185675 (ebook)OAI: oai:DiVA.org:liu-118358DiVA: diva2:814620
Available from: 2015-05-27 Created: 2015-05-27 Last updated: 2016-08-19Bibliographically approved
In thesis
1. Mean-Variance Portfolio Optimization: Eigendecomposition-Based Methods
Open this publication in new window or tab >>Mean-Variance Portfolio Optimization: Eigendecomposition-Based Methods
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:nbn:se:liu:diva-118362 (URN)10.3384/lic.diva.118362 (DOI)978-91-7519-038-9 (print) (ISBN)
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

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