ABSTRACT We propose a computational procedure to find the efficient frontier for the standard Markowitz mean-variance model with discrete variables. The integer constraints limit on the one hand the portfolio to contain a predetermined number of assets and, on the other hand, the proportion of the portfolio held in a given asset. We adapt the multiobjective algorithm NSGA for solving the problem. The algorithm ranks the solutions of each generation in layers based on Pareto non-domination. We have applied the procedure in sixty assets of ATHEX. We have also compared the algorithm with a single genetic algorithm. The computational results indicate that the procedure is promising for this class of problems.
Cite this paper
nullK. ANAGNOSTOPOULOS and G. MAMANIS, "Finding the Efficient Frontier for a Mixed Integer Portfolio Choice Problem Using a Multiobjective Algorithm," iBusiness, Vol. 1 No. 2, 2009, pp. 99-105. doi: 10.4236/ib.2009.12013.
 Markowitz, H. M., “Portfolio selection,” Journal of Finance, No. 7, pp. 77–91, 1952.
Markowitz, H. M., “Portfolio selection, efficient diversi- fication of investments,” Blackwell, Cambridge MA and Oxford UK, 1990.
Benati, S. and Rizzi, R., “A mixed integer linear program- ming formulation of the optimal mean/value-at-risk portfolio problem,” European Journal of Operational Research, Vol. 176, No. 1, pp. 423–434, 2007.
Gilli, M. and K?llezi, E., “A global optimization heuristic for portfolio choice with VaR and expected shortfall,” in Kontoghiorghes, E. J., Rustem, B., and Siokos, S., Eds., Computational Methods in Decision-Making, Economics and Finance, Applied Optimization Series, Kluwer Aca- demic Publishers, pp. 167–183, 2002.
Mitra, G., “A review of portfolio planning: Models and systems,” in Satchell, S., Scowcroft, A., Eds., Advances in Portfolio Construction and Implementation, Butter- worth-Heinemann, Amsterdam, pp. 1–39, 2003.
Chang, T. J., Meade, N., Beasley, J. E., and Sharaiha, Y. M., “Heuristics for cardinality constrained portfolio opti- mization,” Computers & Operations Research, 27, pp. 1271–1302, 2000.
Crama, Y. and Schyns, M., “Simulated annealing for complex portfolio selection problems,” European Journal of Operational Research, No. 150, pp. 546–571, 2003.
Jobst, N. J., Horniman, M. D., Lucas, C. A., and Mitra, G., “Computational aspects of alternative portfolio selection models in the presence of discrete asset choice cons- traints,” Quantitative Finance, Vol. 1, pp. 1–13, 2001.
Blum, C. and Roli, A., “Metaheuristics in combinatorial optimization: Overview and conceptual comparison,” ACM Computing Surveys, Vol. 35, No 3, pp. 268–308, 2003.
Mansini, R. and Speranza, M. G., “Heuristic algorithms for the portfolio selection problem with minimum transa- ction lots,” European Journal of Operational Research, Vol. 114, No. 2, pp. 219–233, 1999.
Anagnostopoulos, K. P., Chatzoglou, P. D., and Katsavounis, S., “A reactive greedy randomized adaptive search procedure for a mixed integer portfolio optimi- zation problem,” in P.D. Chatzoglou, Ed., Proceedings of the 2nd International Conference on Accounting and Finance in Transition (CD), 2004.
Coello, C. A., “An updated survey of GA-based multi- objective optimization techniques,” ACM Computing Surveys, Vol. 32, No 2, pp. 109–143, 2000.
Deb, K., “Multiobjective genetic algorithms: Problem difficulties and construction of test problems,” Evolu- tionary Computation, Vol. 7, No. 3, pp. 205–230, 1999.
Schaffer, J. D., “Multiple objective optimization with vector evaluated genetic algorithms,” in Grefenstette, J. J., Ed., Proceedings of the First International Conference on Genetic Algorithms and Their Applications, Lawrence Erlbaum, Hillsdale, New Jersey, pp. 93–100, 1985.
Fonseca, C. M., and Fleming, P. J., “Genetic algorithms for multiobjective optimization: Formulation, discussion, and generalization,” in Forrest, S., Ed., Proceedings of the Fifth International Conference on Genetic Algorithms, Morgan Kaufmann, San Mateo, California, pp. 416–423, 1993.
Horn, J., Nafpliotis, N., and Goldberg, G. E., “A niched Pareto genetic algorithm for multiobjective optimization, Piscataway,” IEEE Service Center, NJ, pp. 82–87, 1994.
Srinivas, N. and Deb, K., “Multiobjective optimization using nondominated sorting in genetic algorithms,” Evolutionary Computation, Vol. 2, No. 3, pp. 221–248, 1994.
Jaszkiewicz, A., “On the computational effectiveness of multiple objective metaheuristics,” in Proceedings of the Fourth International Conference on Multi-Objective Programming and Goal Programming MOPGP'00, Theo- ry & Applications, Berlin-Heidelberg, Springer–Verlag, May 29–June 1, 2000.
Maringer, D., “Portfolio management with heuristic optimization,” Springer, New York, 2005.