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On the Equivalence of Quadratic Optimization Problems Commonly Used in Portfolio Theory

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  • Taras Bodnar
  • Nestor Parolya
  • Wolfgang Schmid

Abstract

In the paper, we consider three quadratic optimization problems which are frequently applied in portfolio theory, i.e, the Markowitz mean-variance problem as well as the problems based on the mean-variance utility function and the quadratic utility.Conditions are derived under which the solutions of these three optimization procedures coincide and are lying on the efficient frontier, the set of mean-variance optimal portfolios. It is shown that the solutions of the Markowitz optimization problem and the quadratic utility problem are not always mean-variance efficient. The conditions for the mean-variance efficiency of the solutions depend on the unknown parameters of the asset returns. We deal with the problem of parameter uncertainty in detail and derive the probabilities that the estimated solutions of the Markowitz problem and the quadratic utility problem are mean-variance efficient. Because these probabilities deviate from one the above mentioned quadratic optimization problems are not stochastically equivalent. The obtained results are illustrated by an empirical study.

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  • Taras Bodnar & Nestor Parolya & Wolfgang Schmid, 2012. "On the Equivalence of Quadratic Optimization Problems Commonly Used in Portfolio Theory," Papers 1207.1029, arXiv.org, revised Apr 2013.
  • Handle: RePEc:arx:papers:1207.1029
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    References listed on IDEAS

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    Cited by:

    1. Yam, Sheung Chi Phillip & Yang, Hailiang & Yuen, Fei Lung, 2016. "Optimal asset allocation: Risk and information uncertainty," European Journal of Operational Research, Elsevier, vol. 251(2), pages 554-561.
    2. Taras Bodnar & Yarema Okhrin & Nestor Parolya, 2016. "Optimal shrinkage-based portfolio selection in high dimensions," Papers 1611.01958, arXiv.org, revised May 2017.
    3. Bodnar, Taras & Mazur, Stepan & Okhrin, Yarema, 2017. "Bayesian estimation of the global minimum variance portfolio," European Journal of Operational Research, Elsevier, vol. 256(1), pages 292-307.
    4. Bodnar, Taras & Parolya, Nestor & Schmid, Wolfgang, 2015. "On the exact solution of the multi-period portfolio choice problem for an exponential utility under return predictability," European Journal of Operational Research, Elsevier, vol. 246(2), pages 528-542.

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