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Portfolio optimization with pw-robustness

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EURO Journal on Computational Optimization

Abstract

This paper investigates a portfolio optimization problem under uncertainty on the stock returns, where the manager seeks to achieve an appropriate trade-off between the expected portfolio return and the risk of loss. The uncertainty set consists of a finite set of scenarios occurring with equal probability. We introduce a new robustness criterion, called pw-robustness, which seeks to maximize the portfolio return in a proportion p of scenarios and guarantees a minimum return over all scenarios. We model this optimization problem as a mixed-integer programming problem. Through extensive numerical experiments, we identify the instances that can be solved to optimality in an acceptable time using off-the-shelf software. For the instances that cannot be solved to optimality within the time frame, we propose and test a heuristic that exhibits excellent practical performance in terms of computation time and solution quality for the problems we consider. This new criterion and our heuristic methods therefore exhibit great promise to tackle robustness problems when the uncertainty set consists of a large number of scenarios.

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References

  • Belles-Sampera J, Merigo J, Guillén M, Santolino M (2013) The connection between distortion risk measures and ordered weighted averaging operators. Insur Math Econ 52:411–420

    Article  Google Scholar 

  • Benati S (2015) Using medians in portfolio optimization. J Oper Res Soc 66:720–731

    Article  Google Scholar 

  • Benati S, Rizzi R (2007) A mixed integer linear programming formulation of the optimal mean value at risk portfolio problem. Eur J Oper Res 176(1):423–434

    Article  Google Scholar 

  • Bertsimas D, Lauprete GJ, Samarov A (2004) Shortfall as a risk measure: properties, optimization and applications. J Econ Dyn Control 28(7):1353–1381

    Article  Google Scholar 

  • Calafiore GC (2010) Random convex programs. SIAM J Optim 20:3427–3464

    Article  Google Scholar 

  • Calafiore GC (2013) Direct data-driven portfolio optimization with guaranteed shortfall probability. Automatica 49:370–380

    Article  Google Scholar 

  • Calafiore GC, Campi MC (2006) The scenario approach to robust control design. IEEE Trans Autom Control 51:742–753

    Article  Google Scholar 

  • Calafiore GC, Garatti S (2008) The exact feasibility of randomized solutions of uncertain convex programs. SIAM J Optim 19:1211–1230

    Article  Google Scholar 

  • Cetinkaya E, Thiele A (2015) Data-driven portfolio management with quantile constraints. OR Spectr 37:761–786

    Article  Google Scholar 

  • Cornuejols G, Tütüncu R (2007) Optimization methods in finance. Cambridge University Press, New York

    Google Scholar 

  • Feng M, Wächter A, Staum J (2015) Practical algorithms for value-at-risk portfolio optimization problems. Quant Finance Lett 3:1–9

    Article  Google Scholar 

  • Gabrel V, Murat C, Wu L (2011) New models for the robust shortest path problem: complexity, resolution and generalization. Ann Oper Res. https://doi.org/10.1007/s10479-011-1004-2

    Google Scholar 

  • Gabrel Virginie, Murat Cécile, Thiele Aurélie (2014) Recent advances in robust optimization: an overview. Eur J Oper Res 235(3):471–483

    Article  Google Scholar 

  • Iyengar G, Ma AKC (2013) Fast gradient descent method for mean-CVaR optimization. Ann Oper Res 205:203–212

    Article  Google Scholar 

  • Jorion P (2006) Value at risk: the new benchmark for managing financial risk, 3rd edn. McGraw-Hill Education, New York

    Google Scholar 

  • Kandasamy H (2008) Portfolio selection under various risk measures. Ph.D. thesis, Clemson University

  • Kolm PN, Tütüncü R, Fabozzi FJ (2014) 60 years of portfolio optimization: practical challenges and current trends. Eur J Oper Res 234:356–371

    Article  Google Scholar 

  • Kouvelis P, Yu G (1997) Robust discrete optimization and its applications. Kluwer Academic Publishers, Boston

    Book  Google Scholar 

  • Laengle S, Loyola G, Merigo J (2017) Mean–variance portfolio selection with the ordered weighted average. IEEE Trans Fuzzy Syst 25:350–362

    Article  Google Scholar 

  • Lin C (2009) Comments on a mixed integer linear programming formulation of the optimal mean value at risk portfolio problem. Eur J Oper Res 194(1):339–341

    Article  Google Scholar 

  • Mansini R, Ogryczak W, Speranza MG (2003) LP solvable models for portfolio optimization: a classification and computational comparison. IMA J Manag Math 14(3):187–220

    Article  Google Scholar 

  • Mansini R, Ogryczak W, Speranza MG (2014) Twenty years of linear programming based portfolio optimization. Eur J Oper Res 234:518–535

    Article  Google Scholar 

  • Mansini R, Ogryczak W, Speranza MG (2015) Linear models for portfolio optimization. Springer, Berlin, pp 19–45

    Google Scholar 

  • Markowitz H (1952) Portfolio selection. J Finance 7(1):77–91

    Google Scholar 

  • Markowitz H (1959) Portfolio selection: efficient diversification of investments. Number 16. Yale University Press, New Haven

    Google Scholar 

  • Ogryczak W (2000) Multiple criteria linear programming model for portfolio selection. Ann Oper Res 97:143–162

    Article  Google Scholar 

  • Ogryczak W, Sliwinski T (2003) On solving linear programs with the ordered weighted averaging objective. Eur J Oper Res 148(1):80–91

    Article  Google Scholar 

  • Perny P, Spanjaard O, Storme L-X (2006) A decision-theoretic approach to robust optimization in multivalued graphs. Ann OR 147(1):317–341

    Article  Google Scholar 

  • Polak GG, Rogers DF, Sweeney DJ (2010) Risk management strategies via minimax portfolio optimization. Eur J Oper Res 207:409–419

    Article  Google Scholar 

  • Rockafellar RT, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2:21–42

    Article  Google Scholar 

  • Romanko O, Mausser H (2016) Robust scenario-based value-at-risk optimization. Ann Oper Res 237:203–218

    Article  Google Scholar 

  • Roy B (2010) Robustness in operational research and decision aiding: a multi-faceted issue. Eur J Oper Res 200(3):629–638

    Article  Google Scholar 

  • Young MR (1998) A minimax portfolio selection rule with linear programming solution. Manag Sci 44(5):673–683

    Article  Google Scholar 

Download references

Acknowledgements

We would like to dedicate our work to Professor Bernard Roy (1934–2017) who passed away as we were revising this paper. Professor Roy, who founded the LAMSADE group (Laboratoire d’Analyse et de Modélisation des Systèmes pour l’Aide à la Décision), was instrumental in getting us started in this area of research and provided valuable feedback on our paper. We are deeply honored to have benefited from the guidance of this pioneer of operations research. We would also like to thank two anonymous referees, whose comments have significantly improved the paper.

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Authors and Affiliations

  1. Université Paris-Dauphine, PSL Research University, CNRS, LAMSADE, 75016, Paris, France

    Virginie Gabrel & Cécile Murat

  2. Department of Engineering Management, Information and Systems, Southern Methodist University, Dallas, TX, 75205, USA

    Aurélie Thiele

Authors
  1. Virginie Gabrel
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  2. Cécile Murat
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  3. Aurélie Thiele
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Correspondence to Virginie Gabrel.

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Gabrel, V., Murat, C. & Thiele, A. Portfolio optimization with pw-robustness. EURO J Comput Optim 6, 267–290 (2018). https://doi.org/10.1007/s13675-018-0096-8

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  • DOI: https://doi.org/10.1007/s13675-018-0096-8

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