American Institute of Mathematical Sciences

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

A hybrid approach for index tracking with practical constraints

Yingjie Li 1, , Xiaoguang Yang 1, , Shushang Zhu 2, and  Dong-Hui Li 3,

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.
Keywords: mixed-integer linear programming, Simulated annealing, mixed-integer quadratic programming., index tracking.
Mathematics Subject Classification: Primary: 65K15; Secondary: 90C1.
Citation: Yingjie Li, Xiaoguang Yang, Shushang Zhu, Dong-Hui Li. A hybrid approach for index tracking with practical constraints. Journal of Industrial & 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.  Google Scholar

[2]

J. E. Beasley, OR-Library: Distributing test problems by electronic mail, Journal of the Operational Research Society, 41 (1990), 1069-1072. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[7]

E. Çinlar, Introduction to Stochastic Processes, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1975.  Google Scholar

[8]

R. Flethcer, Ageneral quadratic programming algorithm, J. Inst. Math. Appl., 7 (1971), 76-91.  Google Scholar

[9]

M. Gill and E. Këllezi, Threshold Accepting for Index Tracking, Computing in Economics and Finance Series, 72, Society for Computational Economics, 2001. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

L. Ingber, Simulated annealing: Practice versus theory, Mathematical and Computer Modelling, 18 (1993), 29-57. doi: 10.1016/0895-7177(93)90204-C.  Google Scholar

[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.  Google Scholar

[14]

P. J. Laarhoven and E. H. Aarts, Simulated Annealing: Theory and Applications, Mathematics and its Applications, 37, Springer, 1997. Google Scholar

[15]

H. Markowitz, Mean-Variance Analysis in Portfolio Choice and Captial Markets, Basil Blackwell, Oxford, 1987.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[18]

A. F. Perold, C. D. Gelatt and M. P. Vecchi, Dynamic strategies for asset allocation, Financial Analysis Journal, 44 (1988), 17-27. Google Scholar

[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.  Google Scholar

[20]

J. Shapcott, Index Tracking: Genetic Algorithms for Investment Portfolio Selection, Technical Report EPCC-SS92-24, Edinburgh, Parallel Computing Center, 1992. Google Scholar

[21]

C. M. S. Sutcliffe, Stock Index Futures: Theories and International Evidence, 2nd edition, International Thompson Business Press, 1997. Google Scholar

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.  Google Scholar

[2]

J. E. Beasley, OR-Library: Distributing test problems by electronic mail, Journal of the Operational Research Society, 41 (1990), 1069-1072. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[7]

E. Çinlar, Introduction to Stochastic Processes, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1975.  Google Scholar

[8]

R. Flethcer, Ageneral quadratic programming algorithm, J. Inst. Math. Appl., 7 (1971), 76-91.  Google Scholar

[9]

M. Gill and E. Këllezi, Threshold Accepting for Index Tracking, Computing in Economics and Finance Series, 72, Society for Computational Economics, 2001. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

L. Ingber, Simulated annealing: Practice versus theory, Mathematical and Computer Modelling, 18 (1993), 29-57. doi: 10.1016/0895-7177(93)90204-C.  Google Scholar

[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.  Google Scholar

[14]

P. J. Laarhoven and E. H. Aarts, Simulated Annealing: Theory and Applications, Mathematics and its Applications, 37, Springer, 1997. Google Scholar

[15]

H. Markowitz, Mean-Variance Analysis in Portfolio Choice and Captial Markets, Basil Blackwell, Oxford, 1987.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[18]

A. F. Perold, C. D. Gelatt and M. P. Vecchi, Dynamic strategies for asset allocation, Financial Analysis Journal, 44 (1988), 17-27. Google Scholar

[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.  Google Scholar

[20]

J. Shapcott, Index Tracking: Genetic Algorithms for Investment Portfolio Selection, Technical Report EPCC-SS92-24, Edinburgh, Parallel Computing Center, 1992. Google Scholar

[21]

C. M. S. Sutcliffe, Stock Index Futures: Theories and International Evidence, 2nd edition, International Thompson Business Press, 1997. Google Scholar

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