An Iterative Approach to Ill-Conditioned Optimal Portfolio Selection
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An Iterative Approach to Ill-Conditioned Optimal Portfolio Selection Mårten Gulliksson1 · Stepan Mazur2 Accepted: 29 October 2019 © The Author(s) 2019
Abstract Covariance matrix of the asset returns plays an important role in the portfolio selection. A number of papers is focused on the case when the covariance matrix is positive definite. In this paper, we consider portfolio selection with a singular covariance matrix. We describe an iterative method based on a second order damped dynamical systems that solves the linear rank-deficient problem approximately. Since the solution is not unique, we suggest one numerical solution that can be chosen from the iterates that balances the size of portfolio and the risk. The numerical study confirms that the method has good convergence properties and gives a solution as good as or better than the solutions that are based on constrained least norm Moore–Penrose, Lasso, and naive equal-weighted approaches. Finally, we complement our result with an empirical study where we analyze a portfolio with actual returns listed in S&P 500 index. Keywords Mean–variance portfolio · Singular covariance matrix · Linear ill-posed problems · Second order damped dynamical systems
1 Introduction Modern portfolio theory has drawn much attention in the academic literature starting from 1952 when Harry Max Markowitz published his seminal paper about portfolio selection (see Markowitz 1952). He proposed efficient way of portfolio allocation that guarantees the lowest risk for a given level of the expected return. A number of papers are devoted to questions like, e.g., how can an optimal portfolio be constructed, monitored, and/or estimated by using historical data (see, e.g., Alexander and Baptista 2004; Golosnoy and Okhrin 2009; Bodnar 2009; Bodnar et al.
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Stepan Mazur [email protected]
1
School of Science and Technology, Örebro University, 70182 Örebro, Sweden
2
School of Busniess, Örebro University, 70182 Örebro, Sweden
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M. Gulliksson, S. Mazur
2017a; Bauder et al. 2018), what is the influence of parameter uncertainty on the portfolio performance (cf., Okhrin and Schmid 2006; Bodnar and Schmid 2008; Javed et al. 2017), how do the asset returns influence the portfolio choice (see, e.g., Jondeau and Rockinger 2006; Mencia and Sentana 2009; Adcock 2010; Harvey et al. 2010; Amenguala and Sentana 2010), how is it possible to estimate the characteristics of the distribution of the asset returns (see, e.g., Jorion 1986; Wang 2005; Frahm and Memmel 2010), how can the structure of optimal portfolio be statistically justified (Gibbons et al. 1989; Britten-Jones 1999; Bodnar and Schmid 2009). Björk et al. (2014) studied the mean–variance portfolio optimization in continuous time, whereas Liesiö and Salo (2012) developed a portfolio selection framework which uses the set inclusion to capture incomplete information about scenario probabilities and utility functions. Chiarawongse et al. (2012) formulated a mean–variance portfolio selection problem that accommodates qualitative input about ex
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