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Robust tracking error portfolio selection with worst-case downside risk measures. (English) Zbl 1402.91713

Summary: This paper proposes downside risk measure models in portfolio selection that captures uncertainties both in distribution and in parameters. The worst-case distribution with given information on the mean value and the covariance matrix is used, together with ellipsoidal and polytopic uncertainty sets, to build-up this type of downside risk model. As an application of the models, the tracking error portfolio selection problem is considered. By lifting the vector variables to positive semidefinite matrix variables, we obtain semidefinite programming formulations of the robust tracking portfolio models. Numerical results are presented in tracking SSE50 of the Shanghai Stock Exchange. Compared with the tracking error variance portfolio model and the equally weighted strategy, the proposed models are more stable, have better accumulated wealth and have much better Sharpe ratio in the investment period for the majority of observed instances.

MSC:

91G10 Portfolio theory
90C22 Semidefinite programming
91G80 Financial applications of other theories
90C20 Quadratic programming

Software:

SDPT3
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Full Text: DOI

References:

[1] Alexander, G. J.; Baptista, A. M., Active portfolio management with benchmarkingadding a value-at-risk constraint, J. Econ. Dyn. Control, 32, 779-820, (2008) · Zbl 1181.91288
[2] Bawa, V. S., Optimal rules for ordering uncertain prospects, J. Financ. Econ., 2, 1, 95-121, (1975)
[3] Bawa, V. S.; Lindenberg, E. B., Capital market equilibrium in a mean-lower partial moment framework, J. Financ. Econ., 5, 189-200, (1977)
[4] Bazaraa, M. S.; Sherali, H. D.; Shetty, C. M., Nonlinear programmingtheory and algorithms, (1993), John Wiley & Sons New York · Zbl 0774.90075
[5] Ben-Tal, A.; Nemirovski, A., Robust convex optimization, Math. Oper. Res., 23, 769-805, (1998) · Zbl 0977.90052
[6] Black, F., Litterman, R., 1992. Global portfolio optimization. Financ. Anal. J. 48 (September/October), 28-43.
[7] Brown, D. B.; De Giorgi, E.; Sim, M., Aspirational preferences and their representation by risk measures, Manag. Sci., 58, 11, 2095-2113, (2012)
[8] Chen, H.-H.; Tsai, H.-T.; Lin, D., Optimal mean-variance portfolio selection using Cauchy-Schwarz maximization, Appl. Econ., 43, 21, 2795-2801, (2011)
[9] Chen, L.; He, S.; Zhang, S., Tight bounds for some risk measures, with applications to robust portfolio selection, Oper. Res., 59, 4, 847-865, (2011) · Zbl 1233.91236
[10] Chiu, M. C.; Wong, H. Y., Mean-ariance portfolio selection of cointegrated assets, J. Econ. Dyn. Control, 35, 8, 1369-1385, (2011) · Zbl 1217.91166
[11] Costa, O. L.V.; Paiva, A. C., Robust portfolio selection using linear-matrix inequalities, J. Econ. Dyn. Control, 26, 889-909, (2002) · Zbl 0996.91061
[12] DeMiguel, V.; Garlappi, L.; Uppal, R., Optimal versus naive diversificationhow inefficient is the 1/n portfolio strategy?, Rev. Financ. Stud., 22, 5, 1915-1953, (2009)
[13] El Ghaoui, L., Oks, M., Oustry, F., 2003. Worst-case value-at-risk and robust portfolio optimization: a conic programming approach. Oper. Res. 51, 543-556. · Zbl 1165.91397
[14] Erdoğgany, E., Goldfarb, D., Iyengar, G., 2004. Robust active portfolio management. CORC Technical Report TR-2004-11, Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027-6699, USA. Available at 〈http://www.corc.ieor.columbia.edu/reports/techreports/tr-2004-11.pdf〉.
[15] Fabozzi, F. J.; Huang, D.-H.; Zhou, G.-F., Robust portfolioscontributions from operations research and finance, Ann. Oper. Res., 176, 1, 191-220, (2010) · Zbl 1233.91243
[16] Fishburn, P. C., Mean-risk analysis with risk associated with below-target returns, Am. Econ. Rev., 67, 2, 116-126, (1977)
[17] Glabadanidis, P., 2010. Robust and efficient strategies to track and outperform a benchmark. Working paper. Available at 〈http://papers.ssrn.com/sol3/papers.cfm?abstract-id=1571250〉(February 3, 2010).
[18] Goldfarb, D.; Iyengar, G., Robust portfolio selection problems, Math. Oper. Res., 28, 1-38, (2003) · Zbl 1082.90082
[19] Grootveld, H.; Hallerbach, W., Variance vs downside riskis there really that much difference?, Eur. J. Oper. Res., 114, 304-319, (1999) · Zbl 0935.91021
[20] Guedj, I., Huang, J., 2009. Are ETFs replacing index mutual funds? Available at 〈http://papers.ssrn.com/sol3/papers.cfm?abstract-id=1108728〉(March 2009).
[21] Grinold, R. C.; Kahn, R. N., Active portfolio management, (1999), McGraw-Hill Professional Publishing
[22] Glode, V., Why mutual funds “underperform”, J. Financ. Econ., 99, 3, 546-559, (2011)
[23] Harlow, W. V., Asset allocation in a downside-risk framework, Financ. Anal. J., 47, 5, 28-40, (1991)
[24] Huang, L.; Liu, H., Rational inattention and portfolio selection, J. Finance, 62, 1999-2040, (2007)
[25] Huang, D.; Zhu, S.; Fabozzi, F. J.; Fukushima, M., Portfolio selection with uncertain exit timea robust CVaR approach, J. Econ. Dyn. Control, 32, 2, 594-623, (2008) · Zbl 1181.91292
[26] Isii, K., The extrema of probability determined by generalized moments (I) bounded random variables, Ann. Inst. Stat. Math., 12, 2, 119-134, (1960)
[27] Jorion, P., Portfolio optimization with tracking-error constraints, Financ. Anal. J., 59, 1, 70-82, (2003)
[28] Kahneman, D., Attention and effort, (1973), Prentice Hall New Jersey
[29] Lim, A. E.B.; Watewai, T., Robust asset allocation with benchmark objectives, Math. Finance, 21, 4, 643-679, (2011) · Zbl 1239.91151
[30] Ling, A.-F.; Xu, C.-X., Robust portfolio selection involving options under a marginal + joint ellipsoidal uncertainty set, J. Comput. Appl. Math., 236, 3373-3393, (2012) · Zbl 1239.91152
[31] Liu, H. N., Dynamic portfolio choice under ambiguity and regime switching mean returns, J. Econ. Dyn. Control, 35, 4, 623-640, (2011) · Zbl 1209.91150
[32] Liu, S., Xu, R., 2010. The effects of risk aversion on optimization. 2010 MSCI Barra Research. Available at 〈http://www.mscibarra.com/research/articles/2010/〉.
[33] Lu, Z., Robust portfolio selection based on a joint ellipsoidal uncertainty set, Optim. Methods Softw., 26, 1, 89-104, (2011) · Zbl 1222.91053
[34] Lobo, M. S.; Vandenberghe, L.; Boyd, S.; Lebret, H., Applications of second-order cone programming, Linear Algebra Appl., 284, 1, 193-228, (1998) · Zbl 0946.90050
[35] Miao, J.; Wang, N., Risk, uncertainty, and option exercise, J. Econ. Dyn. Control, 35, 4, 442-461, (2011) · Zbl 1211.91096
[36] Markowitz, H. M., Porfolio selectionefficient diversification of investment, (1959), John Wiley New York
[37] Polik, I.; Terlaky, T., A survey of the S-lemma, SIAM Rev., 49, 3, 371-481, (2007) · Zbl 1128.90046
[38] Pashler, H.; Johnston, J., Attentional limitations in dual-task performance, (Pashler, H., Attention, (1998), Psychology Press)
[39] Rockafellar, R. T.; Uryasev, S., Optimization of conditional value-at-risk, J. Risk, 2, 21-41, (2000)
[40] Roll, R., A mean-variance analysis of the tracking error, J. Portf. Manag., 18, 1, 13-22, (1992)
[41] Rudolf, M.; Wolter, H.-J.; Zimmermann, H., A linear model for tracking error minimization, J. Bank. Finance, 23, 1, 85-103, (1999)
[42] Shapiro, A., 2001. On duality theory of conic linear problems. In: Goberna, M.A., Lopez, M.A. (Eds.) Semi-Infinite Programming: Recent Advances. Kluwer Academic Publishers. · Zbl 1055.90088
[43] Sims, C., Implications of rational inattention, J. Monet. Econ., 50, 665-690, (2003)
[44] Toh, K.-C.; Todd, M. J.; Tutuncu, R. H., SDPT3 version 4.0 (beta), a MATLAB software for semidefinite-quadratic-linear programming, Math. Program. Ser. B, 95, 189-217, (2003) · Zbl 1030.90082
[45] Yao, D.; Zhang, S.; Zhou, X., Tracking a financial benchmark using a few assets, Oper. Res., 54, 232-246, (2006) · Zbl 1167.91409
[46] Yu, L., Zhang, S., Zhou, X., 2006. A downside risk analysis based on financial index tracking models. Stochastic Finance, Part I, 213-236, doi: http://dx.doi.org/10.1007/0-387-28359-5-8 · Zbl 1143.91348
[47] Zhao, Y., A dynamic model of active portfolio management with benchmark orientation, J. Bank. Finance, 31, 11, 3336-3356, (2007)
[48] Zhu, S.; Fukushima, M., Worst-case conditional value-at-risk with application to robust portfolio management, Oper. Res., 57, 1155-1168, (2009) · Zbl 1233.91254
[49] Zhu, S.; Li, D.; Wang, S., Robust portfolio selection under downside risk measures, Quant. Finance, 7, 869-885, (2009) · Zbl 1180.91280
[50] Zymler, S.; Kuhn, D.; Rustem, B., Worst-case value-at-risk of non-linear portfolios, Manag. Sci., 59, 1, 172-188, (2013)
[51] Zymler, S.; Rustem, B.; Kuhn, D., Robust portfolio optimization with derivative insurance guarantees, Eur. J. Oper. Res., 210, 2, 410-424, (2011) · Zbl 1210.91128
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