Gaustaroba, Gianfranco and Mansini, Renata and Ogryczak, Wlodzimierz and Speranza, M. Grazia (2014): Linear Programming Models based on Omega Ratio for the Enhanced Index Tracking Problem.

PDF
MPRA_paper_67097.pdf Download (1MB)  Preview 
Abstract
Modern performance measures differ from the classical ones since they assess the performance against a benchmark and usually account for asymmetry in return distributions. The Omega ratio is one of these measures. Until recently, limited research has addressed the optimization of the Omega ratio since it has been thought to be computationally intractable. The Enhanced Index Tracking Problem (EITP) is the problem of selecting a portfolio of securities able to outperform a market index while bearing a limited additional risk. In this paper, we propose two novel mathematical formulations for the EITP based on the Omega ratio. The first formulation applies a standard definition of the Omega ratio where it is computed with respect to a given value, whereas the second formulation considers the Omega ratio with respect to a random target. We show how each formulation, nonlinear in nature, can be transformed into a Linear Programming model. We further extend the models to include real features, such as a cardinality constraint and buyin thresholds on the investments, obtaining Mixed Integer Linear Programming problems. Computational results conducted on a large set of benchmark instances show that the portfolios selected by the model assuming a standard definition of the Omega ratio are consistently outperformed, in terms of outofsample performance, by those obtained solving the model that considers a random target. Furthermore, in most of the instances the portfolios optimized with the latter model mimic very closely the behavior of the benchmark over the outofsample period, while yielding, sometimes, significantly larger returns.
Item Type:  MPRA Paper 

Original Title:  Linear Programming Models based on Omega Ratio for the Enhanced Index Tracking Problem 
Language:  English 
Keywords:  Enhanced Index Tracking, Omega Ratio, Portfolio Optimization, Linear Program ming, Mixed Integer Linear Programming. 
Subjects:  C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61  Optimization Techniques ; Programming Models ; Dynamic Analysis C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63  Computational Techniques ; Simulation Modeling G  Financial Economics > G1  General Financial Markets G  Financial Economics > G1  General Financial Markets > G11  Portfolio Choice ; Investment Decisions 
Item ID:  67097 
Depositing User:  prof. Wlodzimierz Ogryczak 
Date Deposited:  07 Oct 2015 13:21 
Last Modified:  28 Sep 2019 17:14 
References:  [1] J.E. Beasley. Portfolio optimisation: Models and solution approaches. In Topaloglu H., editor, Tutorials in Operations Research, Vol. 10, pages 201221. INFORMS, 2013. [2] J.E. Beasley, N. Meade, and T.J. Chang. An evolutionary heuristic for the index tracking problem. European Journal of Operational Research, 148(3):621643, 2003. [3] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, 2004. [4] N.A. Canakgoz and J.E. Beasley. Mixedinteger programming approaches for index tracking and enhanced indexation. European Journal of Operational Research, 196(1):384399, 2009. [5] T.F. Coleman, Y. Li, and J. Henniger. Minimizing tracking error while restricting the number of assets. Journal of Risk, 8(4):3355, 2006. [6] C. Filippi, G. Guastaroba, and M.G. Speranza. A heuristic framework for the biobjective enhanced index tracking problem. Submitted, 2014. [7] A. Frino, D.R. Gallagher, and T.N. Oetomo. The index tracking strategies of passive and enhanced index equity funds. Australian Journal of Management, 30(1):2355, 2005. [8] M. Gilli and E. Schumann. Distributed optimisation of a portfolio’s Omega. Parallel Computing, 36(7):381389, 2010. [9] M. Gilli, E. Schumann, G. di Tollo, and G. Cabej. Constructing 130/30portfolios with the Omega ratio. Journal of Asset Management, 12(2):94108, 2011. [10] M.J. Gruber. Another puzzle: The growth in actively managed mutual funds. The Journal of Finance, 1(3):783810, 1996. [11] G. Guastaroba, R. Mansini, and M.G. Speranza. On the effectiveness of scenario generation techniques in singleperiod portfolio optimization. European Journal of Operational Research, 192(2):500511, 2009. [12] G. Guastaroba and M.G. Speranza. Kernel search: An application to the index tracking problem. European Journal of Operational Research, 217(1):5468, 2012. [13] P. Jorion. Enhanced index funds and tracking error optimization. Unpublished Paper, Graduate School of Management, University of California at Irvine, 2002. Currently available at http://merage.uci.edu/~jorion/papers/enh.pdf. [14] S.J. Kane, M.C. BartholomewBiggs, M. Cross, and M. Dewar. Optimizing Omega. Journal of Global Optimization, 45(1):153167, 2009. [15] M. Kapsos, N. Christofides, and B. Rustem. Worstcase robust Omega ratio. European Journal of Operational Research, 234(2):499507, 2014. [16] M. Kapsos, N. Christofides, B. Rustem, and S. Zymler. Optimizing the Omega ratio using linear programming. The Journal of Computational Finance, Forthcoming. [17] C. Keating and W.F. Shadwick. A universal performance measure. The Journal of Performance Measurement, (3):5984, 2002. [18] T. Koshizuka, H. Konno, and R. Yamamoto. Indexplusalpha tracking subject to correlation constraint. International Journal of Optimization: Theory, Methods and Applications, 1(2):215224, 2009. [19] M.A. Lejeune. Game theoretical approach for reliable enhanced indexation. Decision Analysis, 9(2):146155, 2012. [20] M.A. Lejeune and G. SamathPac. Construction of riskaverse enhanced index funds. INFORMS Journal on Computing, 25(4):701719, 2013. [21] Q. Li, L. Sun, and L. Bao. Enhanced index tracking based on multiobjective immune algorithm. Expert Systems with Applications, 38(5):61016106, 2011. [22] R. Mansini, W. Ogryczak, and M.G. Speranza. On LP solvable models for portfolio selection. Informatica, 14(1):3762, 2003. [23] R. Mansini, W. Ogryczak, and M.G. Speranza. Twenty years of linear programming based portfolio optimization. European Journal of Operational Research, 234(2):518535, 2014. [24] H. Mausser, D. Saunders, and L. Seco. Optimizing Omega. Investment Management, 40(10):24182428, 2013. [25] N. Meade and J.E. Beasley. Detection of momentum effects using an index outperformance strategy. Quantitative Finance, 11(2):313326, 2011. [26] H. Mezali and J.E. Beasley. Quantile regression for index tracking and enhanced indexation. Journal of the Operational Research Society, 64(11):16761692, 2013. [27] G. Mitra, T. Kyriakis, C. Lucas, and M. Pirbhai. A review of portfolio planning: Models and systems. In S. Satchell and A. Scowcroft, editors, Advances in Portfolio Construction and Implementation, pages 139. ButterworthHeinemann, 2003. [28] W. Ogryczak and A. Ruszczynski. From stochastic dominance to meanrisk models: Semideviations as risk measures. European Journal of Operational Research, 116(1):3350, 1999. [29] A. Passow. Omega portfolio construction with Johnson distributions. Risk, April:8590, 2005. [30] J.L. Prigent. Portfolio Optimization and Performance Analysis. Chapman & Hall CRC Financial Mathematics Series, 2007. [31] D. Roman, G. Mitra, and V. Zverovich. Enhanced indexation based on secondorder stochastic dominance. European Journal of Operational Research, 228(1):273281, 2013. [32] A. Scowcroft and J. Sefton. Enhanced indexation. In S. Satchell and A. Scowcroft, editors, Advances in Portfolio Construction and Implementation, pages 95124. Butterworth Heinemann, 2003. [33] W.F. Sharpe. Mutual fund performance. Journal of Business, 39(1):119138, 1966. [34] F.A. Sortino and L.N. Price. Performance measurement in a downside risk framework. The Journal of Investing, 3(3):5964, 1994. [35] F.A. Sortino, R. van der Meer, and Plantinga A. The Dutch triangle. The Journal of Portfolio Management, Fall:5058, 1999. [36] C.A. Valle, N. Meade, and J.E. Beasley. Absolute return portfolios. Omega, 45(0):2041, 2014. [37] T. Wilding. Using genetic algorithms to construct portfolios. In S. Satchell and A. Scowcroft, editors, Advances in Portfolio Construction and Implementation, pages 135160. ButterworthHeinemann, 2003. [38] H.P. Williams. Model Building in Mathematical Programming. John Wiley & Sons, 5th edition, 2013. [39] L.C. Wu, S.C. Chou, C.C. Yang, and C.S. Ong. Enhanced index investing based on goal programming. The Journal of Portfolio Management, 33(3):4956, 2007. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/67097 