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Index tracking and enhanced indexing using mixed conditional value-at-risk. (English) Zbl 1408.91238

Summary: Index tracking (IT) and enhanced indexing (EI) are two forms of investment strategies which revolve around the movements of the benchmark index. While IT aims to match the performance of the benchmark index, EI intents to outperform the same. In this paper, we seek to design portfolios for IT and EI problems using mixed conditional value at risk (MCVaR). We propose to use the two tail MCVaR (TMCVaR) measure to track the index. Optimizing TMCVaR is a linear program which minimizes the upper deviation and the downside deviation from the benchmark index and hence meets the objective of IT. On the other hand, we propose a two step procedure for EI problem. In step one, we design a discrete Markov chain model to filter a few stocks on the basis of their high probability of gain over the benchmark index. In step two, we assign optimal weights to the filtered assets through maximizing any of the two variants of the STARR ratio with MCVaR or the STARR ratio with deviation MCVaR (DMCVaR). Maximizing the STARR ratio either with MCVaR or DMCVaR is a linear program and hence tractable. We analyze the empirical performance of the proposed models over 17 world-wide indices using the rolling window approach. We consider two IT and four EI models from the literature for a comparative analysis. It is exhibited that the proposed IT model outperforms the other two IT models over several performance measures including higher correlation value with the benchmark index and lower tracking error, and the two proposed EI models outperform the other four EI models in achieving higher excess mean returns from the benchmark index.

MSC:

91G70 Statistical methods; risk measures
91G10 Portfolio theory
62P05 Applications of statistics to actuarial sciences and financial mathematics
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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[1] Martin, R. Douglas; Rachev, Svetlozar Zari; Siboulet, Frederic, Phi-alpha optimal portfolios and extreme risk management, (Wilmott Magazine of Finance November, Vol. 1 (2003)), 70-83
[2] Canakgoz, Nilgun A.; Beasley, John E., Mixed-integer programming approaches for index tracking and enhanced indexation, European J. Oper. Res., 196, 1, 384-399 (2009) · Zbl 1159.91464
[3] Bruni, Renato; Cesarone, Francesco; Scozzari, Andrea; Tardella, Fabio, On exact and approximate stochastic dominance strategies for portfolio selection, European J. Oper. Res., 259, 1, 322-329 (2017) · Zbl 1395.91396
[4] Treynor, Jack L.; Black, Fischer, How to use security analysis to improve portfolio selection, J. Bus., 46, 1, 66-86 (1973)
[5] Rudd, Andrew, Optimal selection of passive portfolios, Financ. Manag., 57-66 (1980)
[6] Meade, Nigel; Salkin, Gerald R., Index funds-construction and performance measurement, J. Oper. Res. Soc., 40, 10, 871-879 (1989)
[7] Buckley, I. R.C.; Korn, R., Optimal index tracking under transaction costs and impulse control, Int. J. Theor. Appl. Finance, 1, 03, 315-330 (1998) · Zbl 0909.90020
[8] Connor, Gregory; Leland, Hayne, Cash management for index tracking, Financ. Anal. J., 75-80 (1995)
[9] Jansen, Roel; Van Dijk, Ronald, Optimal benchmark tracking with small portfolios, J. Portf. Manag., 28, 2, 33-39 (2002)
[10] Roll, Richard, A mean/variance analysis of tracking error, J. Portf. Manag., 18, 4, 13-22 (1992)
[11] Beasley, John E.; Meade, Nigel; Chang, T.-J., An evolutionary heuristic for the index tracking problem, European J. Oper. Res., 148, 3, 621-643 (2003) · Zbl 1037.90038
[12] Dose, Christian; Cincotti, Silvano, Clustering of financial time series with application to index and enhanced index tracking portfolio, Physica A, 355, 1, 145-151 (2005)
[13] García, Fernando; Guijarro, Francisco; Oliver, Javier, Index tracking optimization with cardinality constraint: a performance comparison of genetic algorithms and tabu search heuristics, Neural Computing and Applications, 1-17 (2017)
[14] Guastaroba, Gianfranco; Speranza, Maria Grazia, Kernel search: An application to the index tracking problem, European J. Oper. Res., 217, 1, 54-68 (2012) · Zbl 1244.91109
[15] Ruiz-Torrubiano, Rubén; Suárez, Alberto, A hybrid optimization approach to index tracking, Ann. Oper. Res., 166, 1, 57-71 (2009) · Zbl 1163.91421
[16] Xu, Fengmin; Lu, Zhaosong; Xu, Zongben, An efficient optimization approach for a cardinality-constrained index tracking problem, Optim. Methods Softw., 31, 2, 258-271 (2016) · Zbl 1382.90117
[17] Alexander, Carol; Dimitriu, Anca, Indexing and statistical arbitrage, J. Portf. Manag., 31, 2, 50-63 (2005)
[18] SantAnna, Leonardo R.; Filomena, Tiago P.; Caldeira, João F., Index tracking and enhanced indexing using cointegration and correlation with endogenous portfolio selection, Quart. Rev. Econ. Finance (2016)
[19] Zenios, Stavros A., Practical Financial Optimization (2008), Wiley-Blackwell: Wiley-Blackwell New York, NY · Zbl 1142.91008
[20] Consiglio, Andrea; Zenios, Stavros A., Integrated simulation and optimization models for tracking international fixed income indices, Math. Program., 89, 2, 311-339 (2001) · Zbl 1013.91098
[21] Dan DiBartolomeo, The enhanced index fund as an alternative to indexed equity management.; Dan DiBartolomeo, The enhanced index fund as an alternative to indexed equity management.
[22] Riepe, Mark W.; Werner, Matthew D., Are enhanced index mutual funds worthy of their name?, J. Invest., 7, 2, 6-15 (1998)
[23] Ahmed, Parvez; Nanda, Sudhir, Performance of enhanced index and quantitative equity funds, Financ. Rev., 40, 4, 459-479 (2005)
[24] Koshizuka, Tomoyuki; Konno, Hiroshi; Yamamoto, Rei, Index-plus-alpha tracking subject to correlation constraint, Int. J. Optim.: Theory Methods Appl., 1, 2, 215-224 (2009) · Zbl 1209.91148
[25] Weng, Yin-Che; Wang, Rui, Do enhanced index funds truly have enhanced performance? evidence from the Chinese market, Emerg. Mark. Finance Trade, 53, 4, 819-834 (2017)
[26] Wu, Liang-Chuan; Chou, Seng-Cho; Yang, Chau-Chen; Ong, Chorng-Shyong, Enhanced index investing based on goal programming, J. Portf. Manag., 33, 3, 49-56 (2007)
[27] Li, Qian; Sun, Linyan; Bao, Liang, Enhanced index tracking based on multi-objective immune algorithm, Expert Syst. Appl., 38, 5, 6101-6106 (2011)
[28] Lejeune, Miguel A., Game theoretical approach for reliable enhanced indexation, Decis. Anal., 9, 2, 146-155 (2012) · Zbl 1398.91531
[29] Filippi, C.; Guastaroba, G.; Speranza, M. G., A heuristic framework for the bi-objective enhanced index tracking problem, Omega, 65, 122-137 (2016)
[30] Bruni, Renato; Cesarone, Francesco; Scozzari, Andrea; Tardella, Fabio, A linear risk-return model for enhanced indexation in portfolio optimization, OR Spectrum, 37, 3, 735-759 (2015) · Zbl 1318.91175
[31] Rudolf, Markus; Wolter, Hans-Jürgen; Zimmermann, Heinz, A linear model for tracking error minimization, J. Banking Finance, 23, 1, 85-103 (1999)
[32] de Paulo, Wanderlei Lima; de Oliveira, Estela Mara; do Valle Costa, Oswaldo Luiz, Enhanced index tracking optimal portfolio selection, Finance Res. Lett., 16, 93-102 (2016)
[33] Guastaroba, Gianfranco; Mansini, Renata; Ogryczak, Wlodzimierz; Speranza, Maria Grazia, Linear programming models based on omega ratio for the enhanced index tracking problem, European J. Oper. Res., 251, 3, 938-956 (2016) · Zbl 1346.91208
[34] Keating, Con; Shadwick, William F., A universal performance measure, J. Perform. Meas., 6, 3, 59-84 (2002)
[35] W. Ogryczak, G. Guastaroba, R. Mansini, M.G. Speranza, Enhanced Index Tracking with CVaR-Based Measures. Technical report, Report of the Institute of Control and Computation Engineering, Warsaw University of Technology, 2016.; W. Ogryczak, G. Guastaroba, R. Mansini, M.G. Speranza, Enhanced Index Tracking with CVaR-Based Measures. Technical report, Report of the Institute of Control and Computation Engineering, Warsaw University of Technology, 2016. · Zbl 1346.91208
[36] Roman, Diana; Mitra, Gautam; Zverovich, Victor, Enhanced indexation based on second-order stochastic dominance, European J. Oper. Res., 228, 1, 273-281 (2013) · Zbl 1332.91104
[37] Sharma, Amita; Agrawal, Shubhada; Mehra, Aparna, Enhanced indexing for risk averse investors using relaxed second order stochastic dominance, Optim. Eng., 18, 2, 407-442 (2017) · Zbl 1371.91167
[38] Dentcheva, Darinka; Ruszczynski, Andrzej, Optimization with stochastic dominance constraints, SIAM J. Optim., 14, 2, 548-566 (2003) · Zbl 1055.90055
[39] Markowitz, Harry, Portfolio selection, J. Finance, 7, 1, 77-91 (1952)
[40] Thomas J. Linsmeier, Neil D. Pearson, Risk measurement: an introduction to value-at-risk. Technical report, OFOR, University of Illinois, Urbana-Champaign, 1996.; Thomas J. Linsmeier, Neil D. Pearson, Risk measurement: an introduction to value-at-risk. Technical report, OFOR, University of Illinois, Urbana-Champaign, 1996.
[41] Group of Thirty. Derivatives: practices and principles. Technical report, Group of Thirty, Washington, D. C., 1993.; Group of Thirty. Derivatives: practices and principles. Technical report, Group of Thirty, Washington, D. C., 1993.
[42] Acerbi, Carlo; Tasche, Dirk, On the coherence of expected shortfall, J. Banking Finance, 26, 7, 1487-1503 (2002)
[43] Rockafellar, R. Tyrrell; Uryasev, Stanislav, Conditional value-at-risk for general loss distributions, J. Banking Finance, 26, 7, 1443-1471 (2002)
[44] Artzner, Philippe; Delbaen, Freddy; Eber, Jean-Marc; Heath, David, Coherent measures of risk, Math. Finance, 9, 3, 203-228 (1999) · Zbl 0980.91042
[45] Pflug, Georg Ch., Some remarks on the value-at-risk and the conditional value-at-risk, (Probabilistic Constrained Optimization (2000), Springer), 272-281 · Zbl 0994.91031
[46] Rockafellar, R. Tyrrell; Uryasev, Stanislav, Optimization of conditional value-at-risk, J. Risk, 2, 21-42 (2000) · Zbl 0989.91052
[47] Ogryczak, Włodzimierz; Ruszczynski, Andrzej, Dual stochastic dominance and related mean-risk models, SIAM J. Optim., 13, 1, 60-78 (2002) · Zbl 1022.91017
[48] Sergey Sarykalin, Gaia Serraino, Stan Uryasev, Value-at-risk vs. conditional value-at-risk in risk management and optimization, in: State-of-the-Art Decision-Making Tools in the Information-Intensive Age, Informs, 2008, pp. 270-294.; Sergey Sarykalin, Gaia Serraino, Stan Uryasev, Value-at-risk vs. conditional value-at-risk in risk management and optimization, in: State-of-the-Art Decision-Making Tools in the Information-Intensive Age, Informs, 2008, pp. 270-294.
[49] Lim, Andrew E. B.; Shanthikumar, J. George; Vahn, Gah-Yi, Conditional value-at-risk in portfolio optimization: Coherent but fragile, Oper. Res. Lett., 39, 3, 163-171 (2011) · Zbl 1219.91130
[50] Andersson, Fredrik; Mausser, Helmut; Rosen, Dan; Uryasev, Stanislav, Credit risk optimization with conditional value-at-risk criterion, Math. Program., 89, 2, 273-291 (2001) · Zbl 0994.91028
[51] Krokhmal, Pavlo; Palmquist, Jonas; Uryasev, Stanislav, Portfolio optimization with conditional value-at-risk objective and constraints, J. Risk, 4, 43-68 (2002)
[52] Roman, Diana; Darby-Dowman, Kenneth; Mitra, Gautam, Mean-risk models using two risk measures: a multi-objective approach, Quant. Finance, 7, 4, 443-458 (2007) · Zbl 1190.91139
[53] Sharma, Amita; Utz, Sebastian; Mehra, Aparna, Omega-cvar portfolio optimization and its worst case analysis, OR Spectrum, 39, 2, 505-539 (2017) · Zbl 1367.91203
[54] Zhu, Shushang; Fukushima, Masao, Worst-case conditional value-at-risk with application to robust portfolio management, Oper. Res., 57, 5, 1155-1168 (2009) · Zbl 1233.91254
[55] Ogryczak, Włodzimierz, Multiple criteria linear programming model for portfolio selection, Ann. Oper. Res., 97, 1, 143-162 (2000) · Zbl 0961.91021
[56] Chekhlov, Alexei; Uryasev, Stanislav; Zabarankin, Michael, Drawdown measure in portfolio optimization, Int. J. Theor. Appl. Finance, 8, 01, 13-58 (2005) · Zbl 1100.91040
[57] R. Tyrrell Rockafellar, Stanislav P. Uryasev, Michael Zabarankin, Deviation measures in risk analysis and optimization, Research Report 2002-7. ISE Dept., University of Florida, 2002.; R. Tyrrell Rockafellar, Stanislav P. Uryasev, Michael Zabarankin, Deviation measures in risk analysis and optimization, Research Report 2002-7. ISE Dept., University of Florida, 2002.
[58] Uryasev, Stan; Zabarankin, Michael, Risk tuning with generalized linear regression, Math. Oper. Res., 33, 3, 712-729 (2008) · Zbl 1218.90158
[59] Sharpe, William F., The sharpe ratio, J. Portf. Manag., 21, 1, 49-58 (1994)
[60] Sortino, Frank A.; Price, Lee N., Performance measurement in a downside risk framework, J. Invest., 3, 3, 59-64 (1994)
[61] Biglova, Almira; Ortobelli, Sergio; Rachev, Svetlozar T.; Stoyanov, Stoyan, Different approaches to risk estimation in portfolio theory, J. Portf. Manag., 31, 1, 103-112 (2004)
[62] Rachev, Svetlozar; Ortobelli, Sergio; Stoyanov, Stoyan; Fabozzi, Frank J.; Biglova, Almira, Desirable properties of an ideal risk measure in portfolio theory, Int. J. Theor. Appl. Finance, 11, 01, 19-54 (2008) · Zbl 1153.91557
[63] Stoyanov, Stoyan V.; Rachev, Svetlozar T.; Fabozzi, Frank J., Optimal financial portfolios, Appl. Math. Finance, 14, 5, 401-436 (2007) · Zbl 1151.91542
[64] Mansini, Renata; Ogryczak, Włodzimierz; Speranza, M. Grazia, Conditional value at risk and related linear programming models for portfolio optimization, Ann. Oper. Res., 152, 1, 227-256 (2007) · Zbl 1132.91497
[65] Yitzhaki, Shlomo, Stochastic dominance, mean variance, and Gini’s mean difference, Amer. Econ. Rev., 72, 1, 178-185 (1982)
[66] Charnes, Abraham; Cooper, William W., Programming with linear fractional functionals, Naval Res. Logist. (NRL), 9, 3-4, 181-186 (1962) · Zbl 0127.36901
[67] Konno, Hiroshi; Yamazaki, Hiroaki, Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Manage. Sci., 37, 5, 519-531 (1991)
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