Portfolio Choice Models Based on Second-Order Stochastic Dominance Measures: An Overview and a Computational Study

  • Csaba I. FábiánEmail author
  • Gautam Mitra
  • Diana Roman
  • Victor Zverovich
  • Tibor Vajnai
  • Edit Csizmás
  • Olga Papp
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 163)


In this chapter we present an overview of second-order stochastic dominance-based models with a focus on those using dominance measures. In terms of portfolio policy, the aim is to find a portfolio whose return distribution dominates the index distribution to the largest possible extent. We compare two approaches, the unscaled model of Roman et al. (Mathematical Programming Series B 108: 541–569, 2006) and the scaled model of Fabian et al. (Quantitative Finance 2010). We constructed optimal portfolios using representations of the future asset returns given by historical data on the one hand, and scenarios generated by geometric Brownian motion on the other hand. In the latter case, the parameters of the GBM were obtained from the historical data. Our test data consisted of stock returns from the FTSE 100 basket, together with the index returns. Part of the data were reserved for out-of-sample tests. We examined the return distributions belonging to the respective optimal portfolios of the unscaled and the scaled problems. The unscaled model focuses on the worst cases and hence enhances safety. We found that the performance of the unscaled model is improved by using scenario generators. On the other hand, the scaled model replicates the shape of the index distribution. Scenario generation had little effect on the scaled model. We also compared the shapes of the histograms belonging to corresponding pairs of in-sample and out-of-sample tests and observed a remarkable robustness in both models. We think these features make these dominance measures good alternatives for classic risk measures in certain applications, including certain multistage ones. We mention two candidate applications.


Second-order stochastic dominance Portfolio optimization Scenario generation 


  1. AitSahlia, F., C-J. Wang, V.E. Cabrera, S. Uryasev, and C.W. Fraisse (2009). Optimal crop planting schedules and financial hedging strategies under ENSO-based climate forecasts. Annals of Operations Research,  published online, DOI:10.1007/s10479-009-0551-2Google Scholar
  2. Artzner, Ph., F. Delbaen, J.-M. Eber, and D. Heath (1999). Coherent measures of risk. Mathematical Finance  9, 203–227.CrossRefGoogle Scholar
  3. Carr, P., H. Geman, and D. Madan (2001). Pricing and hedging in incomplete markets. Journal of Financial Economics  62, 131–167.CrossRefGoogle Scholar
  4. Cherubini, U., E. Luciano, and W. Vecchiato (2006). Copula Methods in Finance.  Wiley, New York, NY.Google Scholar
  5. Consigli, G. and M.A.H. Dempster (1998). The CALM stochastic programming model for dynamic asset-liability management. In Worldwide Asset and Liability Modeling,  edited by W.T. Ziemba and J.M. Mulvey, pp 464–500. Cambridge University Press, Cambridge.Google Scholar
  6. Deák, I. (1990). Random Number Generators and Simulation.  Akadémiai Kiadó, Budapest.Google Scholar
  7. Delbaen, F. (2002). Coherent risk measures on general probability spaces. Essays in Honour of Dieter Sondermann.  Springer, Berlin, Germany.Google Scholar
  8. Dempster, M.A.H. and R.R. Merkovsky (1995). A practical geometrically convergent cutting plane algorithm. SIAM Journal on Numerical Analysis  32, 631–644.CrossRefGoogle Scholar
  9. Dentcheva, D. and A. Ruszczyński (2003). Optimization with stochastic dominance constraints. SIAM Journal on Optimization  14, 548–566.CrossRefGoogle Scholar
  10. Dentcheva, D. and A. Ruszczyński (2006). Portfolio optimization with stochastic dominance constraints. Journal of Banking & Finance  30, 433–451.CrossRefGoogle Scholar
  11. Ellison, E.F.D., M. Hajian, H. Jones, R. Levkovitz, I. Maros, G. Mitra, and D. Sayers (2008). FortMP Manual.  Brunel University, London and Numerical Algorithms Group, Oxford.
  12. Fábián, C.I., G. Mitra, and D. Roman (2009). Processing Second-Order Stochastic Dominance models using cutting-plane representations. Mathematical Programming, Ser A.  DOI:10.1007/s10107-009-0326-1.Google Scholar
  13. Fábián, C.I., G. Mitra, D. Roman, and V. Zverovich (2010). An enhanced model for portfolio choice with SSD criteria: a constructive approach. Quantitative Finance. DOI: 10.1080/14697680903493607.Google Scholar
  14. Föllmer, H. and A. Schied (2002). Convex measures of risk and trading constraints. Finance and Stochastics  6, 429–447.CrossRefGoogle Scholar
  15. Fourer, R., D. M. Gay, and B. Kernighan. (2002). AMPL: A Modeling Language for Mathe-matical Programming. Brooks/Cole Publishing Company/Cengage Learning.Google Scholar
  16. Hadar, J. and W. Russel (1969). Rules for ordering uncertain prospects. The American Economic Review  59, 25–34.Google Scholar
  17. Heath, D. (2000). Back to the future. Plenary Lecture at the First World Congress of the Bachelier Society,  Paris, June 2000.Google Scholar
  18. Joe, H. (1997). Multivariate Models and Dependence Concepts.  Chapman & Hall, London.Google Scholar
  19. Klein Haneveld, W.K. (1986). Duality in stochastic linear and dynamic programming. Lecture Notes in Economics and Math. Systems  274. Springer, New York, NY.Google Scholar
  20. Klein Haneveld, W.K. and M.H. van der Vlerk (2006). Integrated chance constraints: reduced forms and an algorithm. Computational Management Science  3, 245–269.CrossRefGoogle Scholar
  21. Kroll, Y. and H. Levy (1980). Stochastic dominance: A review and some new evidence. Research in Finance  2, 163–227.Google Scholar
  22. Künzi-Bay, A. and J. Mayer (2006). Computational aspects of minimizing conditional value-at-risk. Computational Management Science  3, 3–27.CrossRefGoogle Scholar
  23. Luedtke, J. (2008). New formulations for optimization under stochastic dominance constraints. SIAM Journal on Optimization  19, 1433–1450.CrossRefGoogle Scholar
  24. McNeil, A.J., R. Frey, and P. Embrechts (2005). Quantitative Risk Management. Princeton University Press, Princeton, NJ.Google Scholar
  25. Ogryczak, W. (2000). Multiple criteria linear programming model for portfolio selection. Annals of Operations Research  97, 143–162.CrossRefGoogle Scholar
  26. Ogryczak, W. (2002). Multiple criteria optimization and decisions under risk. Control and Cybernetics  31, 975–1003.Google Scholar
  27. Ogryczak, W. and A. Ruszczyński (2001). On consistency of stochastic dominance and mean- semideviations models. Mathematical Programming  89, 217–232.CrossRefGoogle Scholar
  28. Ogryczak, W. and A. Ruszczyński (2002). Dual stochastic dominance and related mean-risk models. SIAM Journal on Optimization  13, 60–78.CrossRefGoogle Scholar
  29. Pflug, G. (2000). Some remarks on the value-at-risk and the conditional value-at-risk. In Probabilistic Constrained Optimization:Methodology and Applications, edited by S. Uryasev, pp. 272–281. Kluwer, Norwell, MA.Google Scholar
  30. Rockafellar, R.T. (2007). Coherent approaches to risk in optimization under uncertainty. Tutorials in Operations Research  INFORMS 2007, 38–61.Google Scholar
  31. Rockafellar, R.T. and S. Uryasev (2000). Optimization of conditional value-at-risk. Journal of Risk  2, 21–41.Google Scholar
  32. Rockafellar, R.T. and S. Uryasev (2002). Conditional value-at-risk for general loss distributions. Journal of Banking & Finance  26, 1443–1471.CrossRefGoogle Scholar
  33. Rockafellar, R. T., S. Uryasev, and M. Zabarankin (2002). Deviation measures in risk analysis and optimization. Research Report 2002-7, Department of Industrial and Systems Engineering, University of Florida.Google Scholar
  34. Rockafellar, R. T., S. Uryasev, and M. Zabarankin (2006). Generalised deviations in risk analysis. Finance and Stochastics  10, 51–74.CrossRefGoogle Scholar
  35. Roman, D., K. Darby-Dowman, and G. Mitra (2006). Portfolio construction based on stochastic dominance and target return distributions. Mathematical Programming  Series B  108, 541–569.CrossRefGoogle Scholar
  36. Ross, S.M. (2002). An Elementary Introduction to Mathematical Finance.  Cambridge University Press, Cambridge.Google Scholar
  37. Rudolf, G. and A. Ruszczyński (2008). Optimization problems with second order stochastic dominance constraints: duality, compact formulations, and cut generation methods. SIAM Journal on Optimization  19, 1326–1343.CrossRefGoogle Scholar
  38. Ruszczyński, A. (1986). A Regularized Decomposition Method for Minimizing the Sum of Polyhedral Functions. Mathematical Programming  35, 309–333.CrossRefGoogle Scholar
  39. Ruszczyński, A. and A. Shapiro (2006). Optimization of convex risk functions. Mathematics of Operations Research  31, 433–452.CrossRefGoogle Scholar
  40. Topaloglou, N., H. Vladimirou, and S. Zenios (2008). A dynamic stochastic programming model for international portfolio management. European Journal of Operational Research  185, 1501–1524.CrossRefGoogle Scholar
  41. Sklar A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institut de Statistique de l’Universit de Paris  1959/8, 229–231.Google Scholar
  42. Whitmore, G.A. and M.C. Findlay (1978). Stochastic Dominance: An Approach to Decision- Making Under Risk.  D.C.Heath, Lexington, MA.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Csaba I. Fábián
    • 1
    • 2
    Email author
  • Gautam Mitra
    • 3
    • 4
  • Diana Roman
    • 3
  • Victor Zverovich
    • 3
    • 4
  • Tibor Vajnai
    • 1
  • Edit Csizmás
    • 1
  • Olga Papp
    • 1
    • 5
  1. 1.Institute of Informatics, Kecskemét CollegeKecskemétHungary
  2. 2.Department of OREötvös Loránd UniversityBudapestHungary
  3. 3.School of Information Systems, Computing and Mathematics, The Centre for the Analysis of Risk and Optimisation Modelling ApplicationsBrunel UniversityUxbridgeUK
  4. 4.OptiRisk SystemsUxbridgeUK
  5. 5.Doctoral School in Applied Mathematics, Eötvös Loránd UniversityBudapestHungary

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