Abstract
As the research for portfolio selection evolves, traditional models and models with one quadratic objective and multiple linear objectives are being solved. In this paper, I propose models with multiple quadratic and multiple linear objectives. Due to the difficulty involved, I study the new models by lines in decision space, analyze the criterion vectors of the lines by projection, and approximate the nondominated sets by the criterion vectors. As an illustration, I extend Merton’s portfolio selection model, propose algorithms to approximate the nondominated sets by the criterion vectors of portfolios with cardinality 3 and then K, and demonstrate the number of the criterion vectors.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The main-stream style in finance (e.g., Huang and Litzenberger (1988)’s method in Chapter 3) is an e-constraint approach, but the result is only an approximation of nondominated sets.
In this paper, bold symbols (e.g., \(\mathbf x \)) denote vectors or matrices, while normal symbols (e.g., t) denote scalars.
A covariance matrix can be defined as symmetric and nonnegative definite, as documented by Brockwell and Davis (1987) (pp. 33 & 35).
Smooth is defined by many textbooks, e.g., that of Thomas and Finney (1998) (p. 744).
Data source: The website of Dow Jones Industrial Average, http://www.djindexes.com/, and the website of Wharton Research Data Services (WRDS), https://wrds-web.wharton.upenn.edu/wrds/, October 2, 2014.
References
Bana e Costa CA, Soares JO (2001) Multicriteria approaches for portfolio selection: an overview. Rev Financ Mark 4(1):19–26
Best MJ (1996) An algorithm for the solution of the parametric quadratic programming problem. In: Fischer H, Riedmüller B, Schäffler S (eds) Applied mathematics and parallel computing: festschrift for klaus ritter. Physica-Verlag, Heidelberg, pp 57–76
Brockwell PJ, Davis RA (1987) Time series: theory and methods, 1st edn. Springer, Berlin
Chow G (1995) Portfolio selection based on return, risk, and relative performance. Financ Anal J 51(2):54–60
Dorfleitner G, Leidl M, Reeder J (2012) Theory of social returns in portfolio choice with application to microfinance. J Asset Manag 13(6):384–400
Ehrgott M (2005) Multicriteria optimization, volume 491 of lecture notes in economics and mathematical systems, 2nd edn. Springer, Berlin
Ehrgott M, Klamroth K, Schwehm C (2004) An MCDM approach to portfolio optimization. Eur J Oper Res 155(3):752–770
Hirschberger M, Qi Y, Steuer RE (2010) Large-scale MV efficient frontier computation via a procedure of parametric quadratic programming. Eur J Oper Res 204(3):581–588
Hirschberger M, Steuer RE, Utz S, Wimmer M, Qi Y (2013) Computing the nondominated surface in tri-criterion portfolio selection. Oper Res 61(1):169–183
Huang C, Litzenberger RH (1988) Foundations for financial economics. Prentice Hall, Englewood Cliffs
Korhonen P, Yu GY (1997) A reference direction approach to multiple objective quadratic-linear programming. Eur J Oper Res 102(3):601–610
Korhonen P, Yu GY (1998) On computing objective function values in multiple objective quadratic-linear programming. Eur J Oper Res 106(1):184–190
Markowitz HM (1956) The optimization of a quadratic function subject to linear constraints. Nav Res Logist Q 3(1–2):111–133
Markowitz HM (1959) Portfolio selection: efficient diversification in investments, 1st edn. Wiley, New York
Markowitz HM (1991) Foundations of portfolio selection. J Financ 46(2):469–477
Markowitz HM (2013) How to represent mark-to-market possibilities with the general portfolio selection model. J Portf Manag 39(4):1–3
Merton RC (1972) An analytical derivation of the efficient portfolio frontier. J Financ Quant Anal 7(4):1851–1872
Niedermayer A, Niedermayer D (2010) Applying Markowitz’s critical line algorithm. In: Guerard JB Jr (ed) Handbook of portfolio construction: contemporary applications of Markowitz techniques. Springer, Berlin, pp 383–400
Qi Y, Hirschberger M, Steuer RE (2009) Dotted representations of mean–variance efficient frontiers and their computation. Inf Syst Oper Res 47(1):15–22
Spronk J, Hallerbach WG (1997) Financial modelling: Where to go? With an illustration for portfolio management. Eur J Oper Res 99(1):113–127
Stein M, Branke J, Schmeck H (2008) Efficient implementation of an active set algorithm for large-scale portfolio selection. Comput Oper Res 35(12):3945–3961
Steuer RE (1986) Multiple criteria optimization: theory, computation and application. Wiley, New York
Steuer RE, Qi Y, Hirschberger M (2007) Suitable-portfolio investors, nondominated frontier sensitivity, and the effect of multiple objectives on standard portfolio selection. Ann Oper Res 152(1):297–317
Steuer RE, Wimmer M, Hirschberger M (2013) Overviewing the transition of Markowitz bi-criterion portfolio selection to tri-criterion portfolio selection. J Bus Econ 83(1):61–85
Thomas GB, Finney RL (1998) Calculus and analytic geometry, 9th edn. Addison Wesley, Reading
Xidonas P, Mavrotas G, Krintas T, Psarras J (2012) Multicriteria portfolio management. Springer, Berlin
Xu J, Li J (2002) A class of stochastic optimization problems with one quadratic & several linear objective functions and extended portfolio selection model. J Comput Appl Math 146(1):99–113
Acknowledgments
The help from Dr. Ralph E. Steuer in the Department of Finance at the University of Georgia is highly appreciated. Also acknowledged is the support from the Ministry of Education of China (Grant No. 14JJD630007); the National Natural Science Foundation of China (Grant No. 71132001); and the Program for Changjiang Scholars and Innovative Research Team in University, IRT0926.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interet
The author has no further conflict of interest.
Human rights
For research involving human participants, the author doesn’t invite any human participants; and
Informed consent
For informed consent, the author doesn’t utilize any animals.
Rights and permissions
About this article
Cite this article
Qi, Y. On the criterion vectors of lines of portfolio selection with multiple quadratic and multiple linear objectives. Cent Eur J Oper Res 25, 145–158 (2017). https://doi.org/10.1007/s10100-015-0431-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10100-015-0431-6