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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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    File URL: https://www.zora.uzh.ch/id/eprint/52022/1/iewwp122.pdf
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Paul Embrechts & Andrea Höing & Alessandro Juri, 2003. "Using copulae to bound the Value-at-Risk for functions of dependent risks," Finance and Stochastics, Springer, vol. 7(2), pages 145-167.
    4. De Giorgi, Enrico, 2005. "Reward-risk portfolio selection and stochastic dominance," Journal of Banking & Finance, Elsevier, vol. 29(4), pages 895-926, April.
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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. Martin Herdegen & Nazem Khan, 2020. "Mean-$\rho$ portfolio selection and $\rho$-arbitrage for coherent risk measures," Papers 2009.05498, arXiv.org, revised Jul 2021.
    4. Wozabal, Nancy, 2009. "Uniform limit theorems for functions of order statistics," Statistics & Probability Letters, Elsevier, vol. 79(12), pages 1450-1455, June.
    5. 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.
    6. 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.
    7. 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.
    8. Mario Brandtner, 2016. "Spektrale Risikomaße: Konzeption, betriebswirtschaftliche Anwendungen und Fallstricke," Management Review Quarterly, Springer, vol. 66(2), pages 75-115, April.
    9. Georg Pflug & Nancy Wozabal, 2010. "Asymptotic distribution of law-invariant risk functionals," Finance and Stochastics, Springer, vol. 14(3), pages 397-418, September.
    10. Martin Herdegen & Nazem Khan, 2022. "Mean‐ρ$\rho$ portfolio selection and ρ$\rho$‐arbitrage for coherent risk measures," Mathematical Finance, Wiley Blackwell, vol. 32(1), pages 226-272, January.
    11. Gilles Boevi Koumou, 2020. "Diversification and portfolio theory: a review," Financial Markets and Portfolio Management, Springer;Swiss Society for Financial Market Research, vol. 34(3), pages 267-312, September.
    12. Anthony Hatherley & Jamie Alcock, 2007. "Portfolio construction incorporating asymmetric dependence structures: a user's guide," Accounting and Finance, Accounting and Finance Association of Australia and New Zealand, vol. 47(3), pages 447-472, September.
    13. 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.
    14. Winker, Peter & Maringer, Dietmar, 2004. "The Hidden Risks of Optimizing Bond Portfolios under VaR," Research Notes 13, Deutsche Bank Research.
    15. Brandtner, Mario & Kürsten, Wolfgang & Rischau, Robert, 2020. "Beyond expected utility: Subjective risk aversion and optimal portfolio choice under convex shortfall risk measures," European Journal of Operational Research, Elsevier, vol. 285(3), pages 1114-1126.
    16. 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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