# American Institute of Mathematical Sciences

July  2014, 10(3): 905-927. doi: 10.3934/jimo.2014.10.905

## A hybrid approach for index tracking with practical constraints

 1 Institute of Systems Science, Chinese Academy of Science, Beijing 100190, China, China 2 Department of Finance and Investment, Sun Yat-Sen University, Guangzhou 510275, China 3 School of Mathematical Sciences, South China Normal University, Guangzhou, 510631

Received  March 2012 Revised  August 2013 Published  November 2013

Index tracking is a popular way for passive fund management, which aims to reproduce the performance of a market index by investing in a subset of the constituents of the index. The formulation of index tracking with some realistic constraints always leads to an optimization problem that is very hard to solve. In this paper, we propose an approximate formulation to the original optimization problem and analyze the approximation error bound. It is shown that the approximation can be reasonably close to the original problem. We consider both cases where the mean absolute error and mean square error are used as the tracking error measurements. The mean absolute error measurement results in a mixed-integer linear programming problem and can be solved by some standard solvers, such as CPLEX. The mean square error measurement leads to a mixed-integer quadratic programming problem. An efficient hybrid heuristic method is given to solve this problem. We do some numerical experiments by the use of five data sets from OR-Library. The results are promising.
Citation: Yingjie Li, Xiaoguang Yang, Shushang Zhu, Dong-Hui Li. A hybrid approach for index tracking with practical constraints. Journal of Industrial and Management Optimization, 2014, 10 (3) : 905-927. doi: 10.3934/jimo.2014.10.905
##### References:
 [1] E. Aarts and J. Korst, Selected topics in simulated annealing, in Essays and Surveys in Metaheuristics (eds. C. C. Ribeiro and P. Hansen) (Angra dos Reis, 1999), Oper. Res./Comput. Sci. Interfaces Ser., 15, Kluwer Academic Publishers, Boston, MA, 2002, 1-37. doi: 10.1007/978-1-4615-1507-4_1. [2] J. E. Beasley, OR-Library: Distributing test problems by electronic mail, Journal of the Operational Research Society, 41 (1990), 1069-1072. [3] J. E. Beasley, N. Meade and T.-J. Chang, An evolutionary heuristic for the index tracking problem, European Journal of Operation Research, 148 (2003), 621-643. doi: 10.1016/S0377-2217(02)00425-3. [4] S. Browne, Beating a moving target: Optimal portfolio strategies for outperforming a stochastic benchmark, Finance and Stochastics, 3 (1999), 275-294. doi: 10.1007/s007800050063. [5] T.-J. Chang, N. Mead, J. E. Beasley and Y. M. Sharaiha, Heuristics for cardinality constrained portfolio optimisation, Computers and Operations Research, 27 (2000), 1271-1302. [6] N. A. Canakgoz and J. E. Beasley, Mixed-integer programming approaches for index tracking and enhanced indexation, European Journal of Operational Research, 196 (2009), 384-399. doi: 10.1016/j.ejor.2008.03.015. [7] E. Çinlar, Introduction to Stochastic Processes, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1975. [8] R. Flethcer, Ageneral quadratic programming algorithm, J. Inst. Math. Appl., 7 (1971), 76-91. [9] M. Gill and E. Këllezi, Threshold Accepting for Index Tracking, Computing in Economics and Finance Series, 72, Society for Computational Economics, 2001. [10] J. H. Holland, Adaption in Natural and Artificial Systems. An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence, University of Michigan Press, Ann Arbor, Mich., 1975. [11] R. Horst, A general class of branch-and-bound methods in global optimization with some new approaches for concave minimization, Journal of Optimization Theory and Applications, 51 (1986), 271-291. doi: 10.1007/BF00939825. [12] L. Ingber, Simulated annealing: Practice versus theory, Mathematical and Computer Modelling, 18 (1993), 29-57. doi: 10.1016/0895-7177(93)90204-C. [13] S. Kirkpatrick, C. D. Gelatt, Jr. and M. P. Vecchi, Optimization by simulated annealing, Science, 220 (1983), 671-680. doi: 10.1126/science.220.4598.671. [14] P. J. Laarhoven and E. H. Aarts, Simulated Annealing: Theory and Applications, Mathematics and its Applications, 37, Springer, 1997. [15] H. Markowitz, Mean-Variance Analysis in Portfolio Choice and Captial Markets, Basil Blackwell, Oxford, 1987. [16] R. Moral-Escudero, R. Ruiz-Torrubiano and A. Suárez, Selection of optimal investment portfolio with cardinality constraints, in Proceedings of the IEEE Congress on Evolutionary Computation, 2006, 2382-2388. [17] I. H. Osman and J. P. Kelly, eds., Meta-Heuristics: Theory & Applications, Papers from the 1995 International Conference (MIC) held in Breckenridge, Colorado, July 22–26, 1995, Kluwer Academic Publishers, Boston, MA, 1996. [18] A. F. Perold, C. D. Gelatt and M. P. Vecchi, Dynamic strategies for asset allocation, Financial Analysis Journal, 44 (1988), 17-27. [19] R. Ruiz-Torrubiano and A. Suárez, A hybrid optimization approach to index tracking, Anneals of Operation Research, 166 (2009), 57-71. doi: 10.1007/s10479-008-0404-4. [20] J. Shapcott, Index Tracking: Genetic Algorithms for Investment Portfolio Selection, Technical Report EPCC-SS92-24, Edinburgh, Parallel Computing Center, 1992. [21] C. M. S. Sutcliffe, Stock Index Futures: Theories and International Evidence, 2nd edition, International Thompson Business Press, 1997.

show all references

##### References:
 [1] E. Aarts and J. Korst, Selected topics in simulated annealing, in Essays and Surveys in Metaheuristics (eds. C. C. Ribeiro and P. Hansen) (Angra dos Reis, 1999), Oper. Res./Comput. Sci. Interfaces Ser., 15, Kluwer Academic Publishers, Boston, MA, 2002, 1-37. doi: 10.1007/978-1-4615-1507-4_1. [2] J. E. Beasley, OR-Library: Distributing test problems by electronic mail, Journal of the Operational Research Society, 41 (1990), 1069-1072. [3] J. E. Beasley, N. Meade and T.-J. Chang, An evolutionary heuristic for the index tracking problem, European Journal of Operation Research, 148 (2003), 621-643. doi: 10.1016/S0377-2217(02)00425-3. [4] S. Browne, Beating a moving target: Optimal portfolio strategies for outperforming a stochastic benchmark, Finance and Stochastics, 3 (1999), 275-294. doi: 10.1007/s007800050063. [5] T.-J. Chang, N. Mead, J. E. Beasley and Y. M. Sharaiha, Heuristics for cardinality constrained portfolio optimisation, Computers and Operations Research, 27 (2000), 1271-1302. [6] N. A. Canakgoz and J. E. Beasley, Mixed-integer programming approaches for index tracking and enhanced indexation, European Journal of Operational Research, 196 (2009), 384-399. doi: 10.1016/j.ejor.2008.03.015. [7] E. Çinlar, Introduction to Stochastic Processes, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1975. [8] R. Flethcer, Ageneral quadratic programming algorithm, J. Inst. Math. Appl., 7 (1971), 76-91. [9] M. Gill and E. Këllezi, Threshold Accepting for Index Tracking, Computing in Economics and Finance Series, 72, Society for Computational Economics, 2001. [10] J. H. Holland, Adaption in Natural and Artificial Systems. An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence, University of Michigan Press, Ann Arbor, Mich., 1975. [11] R. Horst, A general class of branch-and-bound methods in global optimization with some new approaches for concave minimization, Journal of Optimization Theory and Applications, 51 (1986), 271-291. doi: 10.1007/BF00939825. [12] L. Ingber, Simulated annealing: Practice versus theory, Mathematical and Computer Modelling, 18 (1993), 29-57. doi: 10.1016/0895-7177(93)90204-C. [13] S. Kirkpatrick, C. D. Gelatt, Jr. and M. P. Vecchi, Optimization by simulated annealing, Science, 220 (1983), 671-680. doi: 10.1126/science.220.4598.671. [14] P. J. Laarhoven and E. H. Aarts, Simulated Annealing: Theory and Applications, Mathematics and its Applications, 37, Springer, 1997. [15] H. Markowitz, Mean-Variance Analysis in Portfolio Choice and Captial Markets, Basil Blackwell, Oxford, 1987. [16] R. Moral-Escudero, R. Ruiz-Torrubiano and A. Suárez, Selection of optimal investment portfolio with cardinality constraints, in Proceedings of the IEEE Congress on Evolutionary Computation, 2006, 2382-2388. [17] I. H. Osman and J. P. Kelly, eds., Meta-Heuristics: Theory & Applications, Papers from the 1995 International Conference (MIC) held in Breckenridge, Colorado, July 22–26, 1995, Kluwer Academic Publishers, Boston, MA, 1996. [18] A. F. Perold, C. D. Gelatt and M. P. Vecchi, Dynamic strategies for asset allocation, Financial Analysis Journal, 44 (1988), 17-27. [19] R. Ruiz-Torrubiano and A. Suárez, A hybrid optimization approach to index tracking, Anneals of Operation Research, 166 (2009), 57-71. doi: 10.1007/s10479-008-0404-4. [20] J. Shapcott, Index Tracking: Genetic Algorithms for Investment Portfolio Selection, Technical Report EPCC-SS92-24, Edinburgh, Parallel Computing Center, 1992. [21] C. M. S. Sutcliffe, Stock Index Futures: Theories and International Evidence, 2nd edition, International Thompson Business Press, 1997.
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