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Construction of uncertainty sets for portfolio selection problems

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  • Wiechers, Christof

Abstract

While modern portfolio theory grounds on the trade-off between portfolio return and portfolio variance to determine the optimal investment decision, postmodern portfolio theory uses downside risk measures instead of the variance. Prominent examples are given by the risk measures Value-at-Risk and its coherent extension, Conditional Value-at-Risk. When avoiding distributional assumptions on the process that generates the risky assets' returns, historical return data or expert knowledge remain the only data available to the investor. His problem is then to maximize the return of his portfolio given the risk constraint that his portfolio does not fall short of some threshold return. For the Conditional Value-at-Risk, the solution is known to be achievable by a linear program. This paper extends the solution to the investor's problem whenever his risk preferences are given by any coherent distortion risk measure. More precisely, it is shown that whenever the risk constraint is given by a coherent distortion risk measure, a linear program leads to the solution. A geometric interpretation of this solution is immediate, which is related to the non-parametric description of data by socalled weighted-mean trimmed regions. The connections of the solution to robust optimization and decision theory are illustrated.

Suggested Citation

  • Wiechers, Christof, 2011. "Construction of uncertainty sets for portfolio selection problems," Discussion Papers in Econometrics and Statistics 4/11, University of Cologne, Institute of Econometrics and Statistics.
  • Handle: RePEc:zbw:ucdpse:411
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    References listed on IDEAS

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    1. Gordon J. Alexander & Alexandre M. Baptista, 2004. "A Comparison of VaR and CVaR Constraints on Portfolio Selection with the Mean-Variance Model," Management Science, INFORMS, vol. 50(9), pages 1261-1273, September.
    2. Bazovkin, Pavel & Mosler, Karl, 2012. "An Exact Algorithm for Weighted-Mean Trimmed Regions in Any Dimension," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 47(i13).
    3. Ogryczak, Wlodzimierz & Ruszczynski, Andrzej, 1999. "From stochastic dominance to mean-risk models: Semideviations as risk measures," European Journal of Operational Research, Elsevier, vol. 116(1), pages 33-50, July.
    4. Mosler, Karl & Lange, Tatjana & Bazovkin, Pavel, 2009. "Computing zonoid trimmed regions of dimension d>2," Computational Statistics & Data Analysis, Elsevier, vol. 53(7), pages 2500-2510, May.
    5. Dyckerhoff, Rainer & Mosler, Karl, 2012. "Weighted-mean regions of a probability distribution," Statistics & Probability Letters, Elsevier, vol. 82(2), pages 318-325.
    6. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
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    More about this item

    Keywords

    Portfolio Optimization; Risk Constraints; Coherent Distortion Risk Measures; Uncertainty Sets;

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C18 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Methodolical Issues: General
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill

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