Portfolio Optimization with Other Real Features

  • Renata Mansini
  • Włodzimierz Ogryczak
  • M. Grazia Speranza
Part of the EURO Advanced Tutorials on Operational Research book series (EUROATOR)


Real features are all the additional characteristics an investor is interested to consider when selecting a real portfolio, because they reflect preferences or information not captured by the model otherwise, or he/she is obliged to include as restrictions imposed by market conditions. Besides transaction costs, real features include, transaction lots, thresholds on investment, restriction on the number of assets (cardinality constraint), and decision dependency constraints among assets or classes of assets. In this chapter, we focus on all real features different from transaction costs. We analyze their practical relevance and show how to model them in a portfolio optimization problem. The modeling of some real features is possible by using as decision variables the asset shares. In several cases, however, the introduction of real features implies the need of variables that represent the amount of capital invested in each asset. When discussing each real feature, and the way to treat it in a portfolio optimization model, we specify if the use of amounts instead of shares is required. Finally, we show how the modeling of some real features may require the introduction of binary or integer variables giving rise to mixed integer linear programming problems.


Real Effort Portfolio Optimization Model Transaction Lots Cardinality Constraints Real Portfolio 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Acerbi, C. 2002. Spectral measures of risk: A coherent representation of subjective risk aversion. Journal of Banking & Finance 26(7): 1505–1518.Google Scholar
  2. Adcock, C., and N. Meade. 1994. A simple algorithm to incorporate transactions costs in quadratic optimisation. European Journal of Operational Research 79(1): 85–94.Google Scholar
  3. Andersson, F., H. Mausser, D. Rosen, and S. Uryasev. 2001. Credit risk optimization with conditional value-at-risk criterion. Mathematical Programming 89(2): 273–291.Google Scholar
  4. Angelelli, E., R. Mansini, and M.G. Speranza. 2008. A comparison of MAD and CVaR models with real features. Journal of Banking & Finance 32(7): 1188–1197.Google Scholar
  5. Angelelli, E., R. Mansini, and M.G. Speranza. 2010. Kernel search: A general heuristic for the multi-dimensional knapsack problem. Computers & Operations Research 37(11): 2017–2026. Metaheuristics for Logistics and Vehicle Routing.Google Scholar
  6. Angelelli, E., R. Mansini, and M.G. Speranza. 2012. Kernel search: A new heuristic framework for portfolio selection. Computational Optimization and Applications 51(1): 345–361.Google Scholar
  7. Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath. 1999. Coherent measures of risk. Mathematical Finance 9(3): 203–228.Google Scholar
  8. Baumann, P., and N. Trautmann. 2013. Portfolio-optimization models for small investors. Mathematical Methods of Operations Research 77(3): 345–356.Google Scholar
  9. Baumol, W.J. 1964. An expected gain-confidence limit criterion for portfolio selection. Management Science 10: 174–182.Google Scholar
  10. Bawa, V.S. 1975. Optimal rules for ordering uncertain prospects. Journal of Financial Economics 2(1): 95–121.Google Scholar
  11. Beasley, J.E., N. Meade, and T.-J. Chang. 2003. An evolutionary heuristic for the index tracking problem. European Journal of Operational Research 148(3): 621–643.Google Scholar
  12. Bienstock, D. 1996. Computational study of a family of mixed-integer quadratic programming problems. Mathematical Programming 74(2): 121–140.Google Scholar
  13. Canakgoz, N., and J. Beasley. 2009. Mixed-integer programming approaches for index tracking and enhanced indexation. European Journal of Operational Research 196(1): 384–399.Google Scholar
  14. Cariño, D.R., T. Kent, D.H. Myers, C. Stacy, M. Sylvanus, A.L. Turner, K. Watanabe, and W.T. Ziemba. 1994. The Russell-Yasuda Kasai model: An asset/liability model for a Japanese insurance company using multistage stochastic programming. Interfaces 24(1): 29–49.Google Scholar
  15. Cariño, D.R., D.H. Myers, and W.T. Ziemba. 1998. Concepts, technical issues, and uses of the Russell-Yasuda Kasai financial planning model. Operations Research 46(4): 450–462.Google Scholar
  16. Chang, T.-J., N. Meade, J. Beasley, and Y. Sharaiha. 2000. Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research 27(13): 1271–1302.Google Scholar
  17. Chekhlov, A., S. Uryasev, and M. Zabarankin. 2005. Drawdown measure in portfolio optimization. International Journal of Theoretical and Applied Finance 8(1): 13–58.Google Scholar
  18. Chen, A.H., F.J. Fabozzi, and D. Huang. 2010. Models for portfolio revision with transaction costs in the mean–variance framework. In Handbook of portfolio construction, ed. John B. Guerard, 133–151. New York/London: Springer.Google Scholar
  19. Chen, A.H., F.C. Jen, and S. Zionts. 1971. The optimal portfolio revision policy. Journal of Business 44(1): 51–61.Google Scholar
  20. Chiodi, L., R. Mansini, and M.G. Speranza. 2003. Semi-absolute deviation rule for mutual funds portfolio selection. Annals of Operations Research 124(1–4): 245–265.Google Scholar
  21. Elton, E., and M. Gruber. 1995. Modern portfolio theory and investment analysis, Portfolio management series. New York: Wiley.Google Scholar
  22. Elton, E., M. Gruber, S. Brown, and W. Goetzmann. 2003. Modern portfolio theory and investment analysis. New York: Wiley.Google Scholar
  23. Embrechts, P., C. Klüppelberg, and T. Mikosch. 1997. Modelling extremal events: For insurance and finance, Applications of mathematics. New York: Springer.Google Scholar
  24. Espinoza, D., and E. Moreno. 2014. A primal-dual aggregation algorithm for minimizing conditional value-at-risk in linear programs. Computational Optimization and Applications 59(3): 617–638.Google Scholar
  25. Fabian, C.I., G. Mitra, and D. Roman. 2011. Processing second-order stochastic dominance models using cutting-plane representations. Mathematical Programming 130(1): 33–57.Google Scholar
  26. Feinstein, C.D., and M.N. Thapa. 1993. A reformulation of a mean–absolute deviation portfolio optimization model. Management Science 39: 1552–1553.Google Scholar
  27. Fieldsend, J.E., J. Matatko, and M. Peng. 2004. Cardinality constrained portfolio optimisation. In IDEAL, Exeter, vol. 3177, ed. Z.R. Yang, R.M. Everson, and H. Yin. Lecture Notes in Computer Science, 788–793. Springer.Google Scholar
  28. Fishburn, P.C. 1976. Continua of stochastic dominance relations for bounded probability distributions. Journal of Mathematical Economics 3(3): 295–311.Google Scholar
  29. Fishburn, P.C. 1977. Mean-risk analysis with risk associated with below target returns. American Economic Revue 67: 116–126.Google Scholar
  30. Fourer, R. 2013. Linear programming software survey. OR/MS Today 40(3): 40–53.Google Scholar
  31. Gastwirth, J.L. 1971. A general definition of the lorenz curve. Econometrica 39(6): 1037–1039.Google Scholar
  32. Guastaroba, G., R. Mansini, W. Ogryczak, and M. Speranza. 2014. Linear programming models based on Omega ratio for the enhanced index tracking problem. Tech. Rep. 2014–33, Institute of Control and Computation Engineering, Warsaw University of Technology.Google Scholar
  33. Guastaroba, G., R. Mansini, and M.G. Speranza. 2009a. Models and simulations for portfolio rebalancing. Computational Economics 33(3): 237–262.Google Scholar
  34. Guastaroba, G., R. Mansini, and M.G. Speranza. 2009b. On the effectiveness of scenario generation techniques in single-period portfolio optimization. European Journal of Operational Research 192(2): 500–511.Google Scholar
  35. Guastaroba, G., and M.G. Speranza. 2012. Kernel search: An application to the index tracking problem. European Journal of Operational Research 217(1): 54–68.Google Scholar
  36. Hadar, J., and W.R. Russell. 1969. Rules for ordering uncertain prospects. American Economic Review 59(1): 25–34.Google Scholar
  37. Hanoch, G., and H. Levy. 1969. The efficiency analysis of choices involving risk. Review of Economic Studies 36(107): 335–346.Google Scholar
  38. Hardy, G.H., J.E. Littlewood, and G. Pólya. 1934. Inequalities. London: Cambridge University Press.Google Scholar
  39. Jobst, N., M. Horniman, C. Lucas, and G. Mitra. 2001. Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative Finance 1(5): 489–501.Google Scholar
  40. Jorion, P. 2006. Value at risk: The new benchmark for managing financial risk, 3rd edn. New York: Mcgraw-Hill.Google Scholar
  41. Kahneman, D., and A. Tversky. 1979. Prospect theory: An analysis of decision under risk. Econometrica 47(2): 263–291.Google Scholar
  42. Kellerer, H., R. Mansini, and M. Speranza. 2000. Selecting portfolios with fixed costs and minimum transaction lots. Annals of Operations Research 99(1–4): 287–304.Google Scholar
  43. Konno, H., K. Akishino, and R. Yamamoto. 2005. Optimization of a long-short portfolio under nonconvex transaction cost. Computational Optimization and Applications 32(1–2): 115–132.Google Scholar
  44. Konno, H., and A. Wijayanayake. 2001. Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Mathematical Programming 89(2): 233–250.Google Scholar
  45. Konno, H., and R. Yamamoto. 2005. Global optimization versus integer programming in portfolio optimization under nonconvex transaction costs. Journal of Global Optimization 32(2): 207–219.Google Scholar
  46. Konno, H., and H. Yamazaki. 1991. Mean–absolute deviation portfolio optimization model and its application to tokyo stock market. Management Science 37: 519–531.Google Scholar
  47. Koshizuka, T., H. Konno, and R. Yamamoto. 2009. Index-plus-alpha tracking subject to correlation constraint. International Journal of Optimization: Theory, Methods and Applications 1: 215–224.Google Scholar
  48. Kouwenberg, R., and S. Zenios. 2001. Stochastic programming models for asset liability management. In Handbook of asset and liability management, ed. S. Zenios, W. Ziemba, 253–299. Amsterdam: North-Holland.Google Scholar
  49. Krejić, N., M. Kumaresan, and A. Rožnjik. 2011. VaR optimal portfolio with transaction costs. Applied Mathematics and Computation 218(8): 4626–4637.Google Scholar
  50. Krokhmal, P., J. Palmquist, and S. Uryasev. 2002. Portfolio optimization with conditional value-at-risk objective and constraints. Journal of Risk 4(2): 11–27.Google Scholar
  51. Krzemienowski, A. 2009. Risk preference modeling with conditional average: Anapplication to portfolio optimization. Annals of Operations Research 165(1): 67–95.Google Scholar
  52. Krzemienowski, A., and W. Ogryczak. 2005. On extending the LP computable risk measures to account downside risk. Computational Optimization and Applications 32(1–2): 133–160.Google Scholar
  53. Kumar, R., G. Mitra, and D. Roman. 2010. Long-short portfolio optimization in the presence of discrete asset choice constraints and two risk measures. Journal of Risk 13(2): 71–100.Google Scholar
  54. Le Thi, H.A., M. Moeini, and T.P. Dinh. 2009. DC programming approach for portfolio optimization under step increasing transaction costs. Optimization 58(3): 267–289.Google Scholar
  55. Lee, E.K., and J.E. Mitchell. 2000. Computational experience of an interior-point SQP algorithm in a parallel branch-and-bound framework. In High performance optimization, vol. 33, ed. H. Frenk, K. Roos, T. Terlaky, S. Zhang. Applied Optimization, 329–347. Boston: Springer US.Google Scholar
  56. Levy, H. 2006. Stochastic dominance: Investment decision making under uncertainty, 2nd edn. New York: Springer.Google Scholar
  57. Levy, H., and Y. Kroll. 1978. Ordering uncertain options with borrowing and lending. Journal of Finance 33(2): 553–574.Google Scholar
  58. Li, D., X. Sun, and J. Wang. 2006. Optimal lot solution to cardinality constrained mean–variance formulation for portfolio selection. Mathematical Finance 16(1): 83–101.Google Scholar
  59. Lim, C., H.D. Sherali, and S. Uryasev. 2010. Portfolio optimization by minimizing conditional value-at-risk via nondifferentiable optimization. Computational Optimization and Applications 46(3): 391–415.Google Scholar
  60. Lintner, J. 1965. The valuation of risky assets and the selection of risky investments in stock portfolios and capital budget. Review of Economics and Statistics 47: 13–37.Google Scholar
  61. Liu, S., and D. Stefek. 1995. A genetic algorithm for the asset paring problem in portfolio optimization. In Proceedings of the first international symposium on operations research and its application (ISORA), Beijing, 441–450.Google Scholar
  62. Lobo, M., M. Fazel, and S. Boyd. 2007. Portfolio optimization with linear and fixed transaction costs. Annals of Operations Research 152: 341–365.Google Scholar
  63. Mansini, R., W. Ogryczak, and M.G. Speranza. 2003a. LP solvable models for portfolio optimization: A classification and computational comparison. IMA Journal of Management Mathematics 14: 187–220.Google Scholar
  64. Mansini, R., W. Ogryczak, and M.G. Speranza. 2003b. On LP solvable models for portfolio optimization. Informatica 14: 37–62.Google Scholar
  65. Mansini, R., W. Ogryczak, and M.G. Speranza. 2007. Conditional value at risk and related linear programming models for portfolio optimization. Annals of Operations Research 152: 227–256.Google Scholar
  66. Mansini, R., W. Ogryczak, and M.G. Speranza. 2014. Twenty years of linear programming based portfolio optimization. European Journal of Operational Research 234(2): 518–535.Google Scholar
  67. Mansini, R., W. Ogryczak, and M.G. Speranza. 2015. Portfolio optimization and transaction costs. In Quantitative financial risk management: Theory and practice, ed. C. Zopounidis and E. Galariotis, 212–241. Oxford: Wiley.Google Scholar
  68. Mansini, R., and M.G. Speranza. 1999. Heuristic algorithms for the portfolio selection problem with minimum transaction lots. European Journal of Operational Research 114(2): 219–233.Google Scholar
  69. Mansini, R., and M.G. Speranza. 2005. An exact approach for portfolio selection with transaction costs and rounds. IIE Transactions 37(10): 919–929.Google Scholar
  70. Markowitz, H.M. 1952. Portfolio selection. Journal of Finance 7: 77–91.Google Scholar
  71. Markowitz, H.M. 1959. Portfolio selection: Efficient diversification of investments. New York: Wiley.Google Scholar
  72. Mausser, H., D. Saunders, and L. Seco. 2006. Optimising omega. Risk Magazine 19(11): 88–92.Google Scholar
  73. Meade, N., and J.E. Beasley. 2011. Detection of momentum effects using an index out-performance strategy. Quantitative Finance 11(2): 313–326.Google Scholar
  74. Michalowski, W., and W. Ogryczak. 2001. Extending the MAD portfolio optimization model to incorporate downside risk aversion. Naval Research Logistics 48(3): 185–200.Google Scholar
  75. Mossin, J. 1966. Equilibrium in a capital asset market. Econometrica 34: 768–783.Google Scholar
  76. Müller, A., and D. Stoyan. 2002. Comparison methods for stochastic models and risks. New York: Wiley.Google Scholar
  77. Nawrocki, D.N. 1992. The characteristics of portfolios selected by n-degree lower partial moment. International Review of Financial Analysis 1(3): 195–209.Google Scholar
  78. Neumann, J.V., and O. Morgenstern. 1947. Theory of games and economic behavior, 2nd edn. Princeton: Princeton University Press.Google Scholar
  79. Ogryczak, W. 1999. Stochastic dominance relation and linear risk measures. In Financial modelling – Proceedings of the 23rd meeting EURO WG financial modelling, 1998, Cracow, ed. A.M. Skulimowski, 191–212. Progress & Business Publisher.Google Scholar
  80. Ogryczak, W. 2000. Multiple criteria linear programming model for portfolio selection. Annals of Operations Research 97(1–4): 143–162.Google Scholar
  81. Ogryczak, W., and A. Ruszczyński. 1999. From stochastic dominance to mean-risk models: Semideviations as risk measures. European Journal of Operational Research 116(1): 33–50.Google Scholar
  82. Ogryczak, W., and A. Ruszczyński. 2001. On consistency of stochastic dominance and mean-semideviation models. Mathematical Programming 89(2): 217–232.Google Scholar
  83. Ogryczak, W., and A. Ruszczyński. 2002a. Dual stochastic dominance and quantile risk measures. International Transactions in Operational Research 9(5): 661–680.Google Scholar
  84. Ogryczak, W., and A. Ruszczyński. 2002b. Dual stochastic dominance and related mean-risk models. SIAM Journal on Optimization 13(1): 60–78.Google Scholar
  85. Ogryczak, W., and T. Śliwiński. 2011a. On dual approaches to efficient optimization of LP computable risk measures for portfolio selection. Asia-Pacific Journal of Operational Research 28(1): 41–63.Google Scholar
  86. Ogryczak, W., and T. Śliwiński. 2011b. On solving the dual for portfolio selection by optimizing conditional value at risk. Computational Optimization and Applications 50(3): 591–595.Google Scholar
  87. Pflug, G.C. 2000. Some remarks on the value-at-risk and the conditional value-at-risk. In Probabilistic constrained optimization: Methodology and applications, ed. S. Uryasev, 272–281. Boston: Kluwer.Google Scholar
  88. Pflug, G.C. 2001. Scenario tree generation for multiperiod financial optimization by optimal discretization. Mathematical Programming 89(2): 251–271.Google Scholar
  89. Pogue, G.A. 1970. An extension of the Markowtiz portfolio selection model to include variable transaction costs, short sales, leverage policies and taxes. Journal of Finance 25(5): 1005–1027.Google Scholar
  90. Quiggin, J. 1982. A theory of anticipated utility. Journal of Economic Behavior & Organization 3(4): 323–343.Google Scholar
  91. Quirk, J.P., and R. Saposnik. 1962. The efficiency analysis of choices involving risk. Review of Economic Studies 29(2): 140–146.Google Scholar
  92. Rockafellar, R., S. Uryasev, and M. Zabarankin. 2006. Generalized deviations in risk analysis. Finance and Stochastics 10(1): 51–74.Google Scholar
  93. Rockafellar, R.T., and S. Uryasev. 2000. Optimization of conditional value-at-risk. Journal of Risk 2: 21–41.Google Scholar
  94. Roman, D., K. Darby-Dowman, and G. Mitra. 2007. Mean-risk models using two risk measures: A multi-objective approach. Quantitative Finance 7(4): 443–458.Google Scholar
  95. Rothschild, M., and J.E. Stiglitz. 1970. Increasing risk: I. A definition. Journal of Economic Theory 2(3): 225–243.Google Scholar
  96. Roy, A. 1952. Safety-first and the holding of assets. Econometrica 20: 431–449.Google Scholar
  97. Shadwick, W., and C. Keating. 2002. A universal performance measure. Journal of Portfolio Measurement 6(3 Spring): 59–84.Google Scholar
  98. Sharpe, W.F. 1964. Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance 19: 425–442.Google Scholar
  99. Sharpe, W.F. 1971a. A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis 6: 1263–1275.Google Scholar
  100. Sharpe, W.F. 1971b. Mean-absolute deviation characteristic lines for securities and portfolios. Management Science 18: B1–B13.Google Scholar
  101. Shorrocks, A.F. 1983. Ranking income distributions. Economica 50(197): 3–17.Google Scholar
  102. Smith, K.V. 1967. A transition model for portfolio revision. Journal of Finance 22(3): 425–439Google Scholar
  103. Speranza, M.G. 1993. Linear programming models for portfolio optimization. Finance 14: 107–123.Google Scholar
  104. Speranza, M.G. 1996. A heuristic algorithm for a portfolio optimization model applied to the Milan stock market. Computers & Operations Research 23(5): 433–441.Google Scholar
  105. Stone, B.K. 1973. A linear programming formulation of the general portfolio selection problem. Journal of Financial and Quantitative Analysis 8: 621–636.Google Scholar
  106. Tobin, J. 1958. Liquidity preference as behavior towards risk. Review of Economic Studies 25(2): 65–86.Google Scholar
  107. Topaloglou, N., H. Vladimirou, and S.A. Zenios. 2002. CVaR models with selective hedging for international asset allocation. Journal of Banking & Finance 26(7): 1535–1561.Google Scholar
  108. Valle, C., N. Meade, and J. Beasley. 2014a. Absolute return portfolios. Omega 45: 20–41.Google Scholar
  109. Valle, C., N. Meade, and J. Beasley. 2014b. Market neutral portfolios. Optimization Letters 8: 1961–1984.Google Scholar
  110. Whitmore, G.A. 1970. Third-degree stochastic dominance. American Economic Review 60(3): 457–459.Google Scholar
  111. Woodside-Oriakhi, M., C. Lucas, and J. Beasley. 2013. Portfolio rebalancing with an investment horizon and transaction costs. Omega 41(2): 406–420.Google Scholar
  112. Xidonas, P., G. Mavrotas, and J. Psarras. 2010. Portfolio construction on the Athens Stock Exchange: A multiobjective optimization approach. Optimization 59(8): 1211–1229.Google Scholar
  113. Yaari, M.E. 1987. The dual theory of choice under risk. Econometrica 55(1): 95–115.Google Scholar
  114. Yitzhaki, S. 1982. Stochastic dominance, mean variance, and Gini’s mean difference. American Economic Review 72: 178–185.Google Scholar
  115. Young, M.R. 1998. A minimax portfolio selection rule with linear programming solution. Management Science 44(5): 673–683.Google Scholar
  116. Zenios, S., and P. Kang. 1993. Mean-absolute deviation portfolio optimization for mortgage-backed securities. Annals of Operations Research 45(1): 433–450.Google Scholar
  117. Zhu, S., and M. Fukushima. 2009. Worst-case conditional value-at-risk with application to robust portfolio management. Operations Research 57(5): 1155–1168.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Renata Mansini
    • 1
  • Włodzimierz Ogryczak
    • 2
  • M. Grazia Speranza
    • 3
  1. 1.Department of Information EngineeringUniversity of BresciaBresciaItaly
  2. 2.Institute of Control and Computation EngineeringWarsaw University of TechnologyWarsawPoland
  3. 3.Department of Economics and ManagementUniversity of BresciaBresciaItaly

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