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Econometrical analysis of the sample efficient frontier


  • Taras Bodnar
  • Wolfgang Schmid


The efficient frontier is a parabola in the mean-variance space which is uniquely determined by three characteristics. Assuming that the portfolio asset returns are independent and multivariate normally distributed, we derive tests and confidence sets for all possible arrangements of these characteristics. Note that all of our results are based on the exact distributions for a finite sample size. Moreover, we determine a confidence region of the whole efficient frontier in the mean-variance space. It is shown that this set is bordered by five parabolas.

Suggested Citation

  • Taras Bodnar & Wolfgang Schmid, 2009. "Econometrical analysis of the sample efficient frontier," The European Journal of Finance, Taylor & Francis Journals, vol. 15(3), pages 317-335.
  • Handle: RePEc:taf:eurjfi:v:15:y:2009:i:3:p:317-335
    DOI: 10.1080/13518470802423478

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    References listed on IDEAS

    1. John H. Cochrane, 1999. "Portfolio advice of a multifactor world," Economic Perspectives, Federal Reserve Bank of Chicago, issue Q III, pages 59-78.
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    1. repec:eee:ejores:v:266:y:2018:i:1:p:371-390 is not listed on IDEAS
    2. Taras Bodnar & Yarema Okhrin & Nestor Parolya, 2016. "Optimal shrinkage-based portfolio selection in high dimensions," Papers 1611.01958,, revised May 2017.
    3. Bodnar, Taras & Parolya, Nestor & Schmid, Wolfgang, 2018. "Estimation of the global minimum variance portfolio in high dimensions," European Journal of Operational Research, Elsevier, vol. 266(1), pages 371-390.
    4. Bodnar, Taras & Parolya, Nestor & Schmid, Wolfgang, 2013. "On the equivalence of quadratic optimization problems commonly used in portfolio theory," European Journal of Operational Research, Elsevier, vol. 229(3), pages 637-644.
    5. Bodnar Taras & Schmid Wolfgang, 2011. "On the exact distribution of the estimated expected utility portfolio weights: Theory and applications," Statistics & Risk Modeling, De Gruyter, vol. 28(4), pages 319-342, December.
    6. Bodnar, Taras & Mazur, Stepan & Podgórski, Krzysztof, 2016. "Singular inverse Wishart distribution and its application to portfolio theory," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 314-326.
    7. Begoña Font, 2016. "Bootstrap estimation of the efficient frontier," Computational Management Science, Springer, vol. 13(4), pages 541-570, October.
    8. Muhammad Najib Razali, 2011. "Portfolio Optimisation Model for Malaysian Property Market," ERES eres2011_131, European Real Estate Society (ERES).
    9. Taras Bodnar & Wolfgang Schmid & Taras Zabolotskyy, 2013. "Asymptotic behavior of the estimated weights and of the estimated performance measures of the minimum VaR and the minimum CVaR optimal portfolios for dependent data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(8), pages 1105-1134, November.
    10. David Bauder & Taras Bodnar & Nestor Parolya & Wolfgang Schmid, 2018. "Bayesian mean-variance analysis: Optimal portfolio selection under parameter uncertainty," Papers 1803.03573,
    11. Bodnar Taras & Schmid Wolfgang & Zabolotskyy Tara, 2012. "Minimum VaR and minimum CVaR optimal portfolios: Estimators, confidence regions, and tests," Statistics & Risk Modeling, De Gruyter, vol. 29(4), pages 281-314, November.
    12. Olha Bodnar & Taras Bodnar, 2009. "Statistical inference procedure for the mean–variance efficient frontier with estimated parameters," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 93(3), pages 295-306, September.


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