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A Note on Portfolio Selection under Various Risk Measures

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  • Enrico De Giorgi

Abstract

This work gives a brief overview of the portfolio selection problem following the mean-risk approach first proposed by Markowitz (1952). We consider various risk measures, i.e. variance, value-at-risk and expected-shortfall and we study the efficient frontiers obtained by solving the portfolio selection problem under these measures. We show that under the assumption that returns are normally distributed, the efficient frontiers obtained by taking value-at-risk or expected-shortfall are subsets of the mean-variance efficient frontier. We generalize this result for all risk measures that can be written as a particular combination of mean and variance and we show that for these measures Tobin separation holds under some restrictions.

Suggested Citation

  • Enrico De Giorgi, "undated". "A Note on Portfolio Selection under Various Risk Measures," IEW - Working Papers 122, Institute for Empirical Research in Economics - University of Zurich.
  • Handle: RePEc:zur:iewwpx:122
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    References listed on IDEAS

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    1. De Giorgi, Enrico, 2005. "Reward-risk portfolio selection and stochastic dominance," Journal of Banking & Finance, Elsevier, vol. 29(4), pages 895-926, April.
    2. Alexander, Gordon J. & Baptista, Alexandre M., 2002. "Economic implications of using a mean-VaR model for portfolio selection: A comparison with mean-variance analysis," Journal of Economic Dynamics and Control, Elsevier, vol. 26(7-8), pages 1159-1193, July.
    3. Carlo Acerbi & Dirk Tasche, 2002. "Expected Shortfall: A Natural Coherent Alternative to Value at Risk," Economic Notes, Banca Monte dei Paschi di Siena SpA, vol. 31(2), pages 379-388, July.
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    Citations

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    Cited by:

    1. De Giorgi, Enrico, 2005. "Reward-risk portfolio selection and stochastic dominance," Journal of Banking & Finance, Elsevier, vol. 29(4), pages 895-926, April.
    2. Bruno S. Frey & Alois Stutzer, "undated". "Direct Democracy: Designing a Living Constitution," IEW - Working Papers 167, Institute for Empirical Research in Economics - University of Zurich.
    3. Wozabal, Nancy, 2009. "Uniform limit theorems for functions of order statistics," Statistics & Probability Letters, Elsevier, vol. 79(12), pages 1450-1455, June.
    4. Enrico De Giorgi & Stefan Reimann, "undated". "The ?-Beauty Contest: Choosing Numbers, Thinking Intervals," IEW - Working Papers 183, Institute for Empirical Research in Economics - University of Zurich.
    5. Solange M. Berstein & Rómulo A. Chumacero, 2012. "VaR limits for pension funds: an evaluation," Quantitative Finance, Taylor & Francis Journals, vol. 12(9), pages 1315-1324, May.
    6. Alejandro Reveiz & Carlos León, 2010. "Efficient Portfolio Optimization in the Wealth Creation and Maximum Drawdown Space," Palgrave Macmillan Books, in: Arjan B. Berkelaar & Joachim Coche & Ken Nyholm (ed.), Interest Rate Models, Asset Allocation and Quantitative Techniques for Central Banks and Sovereign Wealth Funds, chapter 7, pages 134-157, Palgrave Macmillan.
    7. Mario Brandtner, 2016. "Spektrale Risikomaße: Konzeption, betriebswirtschaftliche Anwendungen und Fallstricke," Management Review Quarterly, Springer, vol. 66(2), pages 75-115, April.
    8. Georg Pflug & Nancy Wozabal, 2010. "Asymptotic distribution of law-invariant risk functionals," Finance and Stochastics, Springer, vol. 14(3), pages 397-418, September.
    9. Babaei, Sadra & Sepehri, Mohammad Mehdi & Babaei, Edris, 2015. "Multi-objective portfolio optimization considering the dependence structure of asset returns," European Journal of Operational Research, Elsevier, vol. 244(2), pages 525-539.
    10. Winker, Peter & Maringer, Dietmar, 2004. "The Hidden Risks of Optimizing Bond Portfolios under VaR," Research Notes 13, Deutsche Bank Research.
    11. Brandtner, Mario, 2013. "Conditional Value-at-Risk, spectral risk measures and (non-)diversification in portfolio selection problems – A comparison with mean–variance analysis," Journal of Banking & Finance, Elsevier, vol. 37(12), pages 5526-5537.

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    More about this item

    Keywords

    decision under risk; mean-risk models; portfolio optimization; value-at-risk; expected shortfall; efficient frontier;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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