Enhanced Index Tracking via Omega-CVaR Optimization: A Downside Risk Perspective

Document Type : Original Article

Authors

1 Financial Engineering, Faculty of Accounting and Finance, College of Management, University of Tehran, Tehran, Iran

2 Faculty of Economics, University of Tehran, Tehran, Iran

Abstract

Introduction: This study aims to develop and evaluate a novel portfolio optimization framework for enhanced index tracking. Enhanced index tracking is an intermediate strategy between active and passive portfolio management, in which the goal is to construct a portfolio from the constituents of a benchmark index so as to closely follow the index while achieving returns above the benchmark. The main objective is to jointly pursue “return enhancement” and “strict control of tail risk” in financial markets where returns may exhibit skewness, excess kurtosis, and extreme events. In such environments, relying solely on conventional variance-based risk measures may underestimate downside risks and thereby expose portfolio-weighting decisions to substantial losses. The proposed framework employs the Omega ratio as a distribution-based performance measure and Conditional Value at Risk (CVaR) as a downside risk control metric, with the aim of generating excess returns over the benchmark while enhancing the portfolio’s resilience to severe losses.Methods: The proposed framework is formulated as an Omega–CVaR optimization problem. The objective function maximizes the Omega ratio of the tracking portfolio in order to improve the ratio of returns above a specified threshold to losses below that threshold. Simultaneously, CVaR is imposed as a constraint to control downside risk by limiting the mean of large losses in the tail of the return distribution. Operational constraints include full investment, minimum/maximum weight bounds of 0% and 50% to prevent excessive concentration, and asset selection restricted to the constituents of the benchmark index. The empirical assessment is conducted using 30 rolling time windows; in each iteration, 52 weeks of in-sample data are used for estimation and 12 weeks of out-of-sample data are used for performance evaluation. The study period spans approximately eight years (from late January 2018 to late December 2025), and the results are compared with those of the Tehran Exchange Price Index (TEPIX) and a competing model based on conventional constraints/objectives.Finding: The results indicate that the Omega–CVaR framework delivers a substantial long-term advantage in terms of cumulative returns relative to both the TEPIX and the competing model. The cumulative return of the proposed portfolio is reported at 2,546%, compared with 1,350% for the TEPIX. Analyzing the time path of performance across rolling windows shows that the primary advantage of the model does not necessarily stem from “persistent weekly outperformance,” but rather from two complementary mechanisms. First, the CVaR constraint, by limiting the average losses beyond the confidence level, reduces the depth of drawdowns during bearish phases. Second, maximizing the Omega ratio leads to a weight allocation that increases the share of desirable returns relative to undesirable losses, enabling the portfolio to better exploit the “compounding effect” during recovery periods following downturns. Nonetheless, statistical tests reveal that at weekly horizons, the model’s outperformance relative to the TEPIX and competing models is not consistently and significantly confirmed. This pattern is consistent with the defensive nature of the portfolio, as evidenced by an average beta of about 0.43, indicating lower sensitivity to market fluctuations than the TEPIX.Conclusions: The findings suggest that the Omega–CVaR framework is an effective tool for enhanced index tracking in volatile markets characterized by extreme risks. Although statistically significant short-term outperformance is not observed, effective control of severe losses and reinforcement of the compounding effect can ultimately lead to higher cumulative returns over the long run. This strategy is particularly recommended for investors seeking lower risk and greater stability in their portfolios, especially in markets with non-normal return distributions.

Keywords

References

منابع
Ahmed, P., & Nanda, S. (2005). Performance of enhanced index and quantitative equity funds. Financial Review, 40(4), 459–479. https://doi.org/10.1111/j.1540-6288.2005.00119.x
Alexander, C., & Dimitriu, A. (2005). Indexing and Statistical Arbitrage. The Journal of Portfolio Management, 31(2), 50–63. https://doi.org/10.3905/jpm.2005.470578
Ansari, H. , Behzadi, A. and Tondnevis, F. (2019). Enhanced Index Tracking with a Two-Stage Mixed Integer Programing Model and Pattern Search Algorithm. Financial Management Strategy, 7(4), 1-22. doi: 10.22051/jfm.2019.24888.1998 (in persian)
Artzner, P., Delbaen, F., Eber, J., & Heath, D. (1999). Coherent Measures of Risk. Mathematical Finance, 9(3), 203–228. https://doi.org/10.1111/1467-9965.00068
Beasley, J. E., Meade, N., & Chang, T.-J. (2003). An evolutionary heuristic for the index tracking problem. European Journal of Operational Research, 148(3), 621–643. https://doi.org/10.1016/S0377-2217(02)00425-3
Biglova, A., Ortobelli, S., Rachev, S., & Stoyanov, S. (2004). Different approaches to risk estimation in portfolio theory. In Journal of Portfolio Management (Vol. 31, Issue 1, pp. 103–112). https://doi.org/10.3905/jpm.2004.443328
Brinson, G. P., Hood, L. R., & Beebower, G. L. (1995). Determinants of Portfolio Performance. Financial Analysts Journal, 42(1), 39–44. https://doi.org/10.2469/faj.v42.n4.39
Brooks, V. L., & Reid, I. A. (1983). Effects of blockade of brain angiotensin II receptors in conscious, sodium-deprived dogs. The American Journal of Physiology, 245(6), R881-7. https://doi.org/10.1002/9781119196679
Bruni, R., Cesarone, F., Scozzari, A., & Tardella, F. (2015). A linear risk-return model for enhanced indexation in portfolio optimization. OR Spectrum, 37(3), 735–759. https://doi.org/10.1007/s00291-014-0383-6
Bruni, R., Cesarone, F., Scozzari, A., & Tardella, F. (2017). On exact and approximate stochastic dominance strategies for portfolio selection. European Journal of Operational Research, 259(1), 322–329. https://doi.org/10.1016/j.ejor.2016.10.006
Buckley, I. R. C., & Korn, R. (1998). Optimal Index Tracking Under Transaction Costs and Impulse Control. International Journal of Theoretical and Applied Finance, 01(03), 315–330. https://doi.org/10.1142/S0219024998000187
Canakgoz, N. A., & Beasley, J. E. (2009). Mixed-integer programming approaches for index tracking and enhanced indexation. European Journal of Operational Research, 196(1), 384–399. https://doi.org/10.1016/j.ejor.2008.03.015
Charnes, A., & Cooper, W. W. (1962). Programming with linear fractional functionals. Naval Research Logistics Quarterly, 9(3–4), 181–186. https://doi.org/10.1002/nav.3800090303
de Paulo, W. L., de Oliveira, E. M., & do Valle Costa, O. L. (2016). Enhanced index tracking optimal portfolio selection. Finance Research Letters, 16, 93–102. https://doi.org/10.1016/j.frl.2015.10.005
Dose, C., & Cincotti, S. (2005). Clustering of financial time series with application to index and enhanced index tracking portfolio. Physica A: Statistical Mechanics and Its Applications, 355(1), 145–151. https://doi.org/10.1016/j.physa.2005.02.078
Dueck, G., & Scheuer, T. (1990). Threshold accepting: A general purpose optimization algorithm appearing superior to simulated annealing. Journal of Computational Physics, 90(1), 161–175. https://doi.org/10.1016/0021-9991(90)90201-B
Eyvazloo, R., Fallahpour, S., & Dehghani Ashkezari, M. (2022). Index tracking using Two-tail Mixed Conditional Value-at-risk in Tehran Stock Exchange. Financial Research Journal, 23(4), 545–563. https://doi.org/10.22059/frj.2020.289344.1006927 (in persian)
Eyvazloo, R. , Shafizadeh, M. and Ghahramani, A. (2017). Index Tracking and Enhanced Indexing Using Co-integration and Correlation Approaches. Financial Research Journal, 19(3), 457-474. doi: 10.22059/jfr.2018.245816.1006551 (in persian)
Filippi, C., Guastaroba, G., & Speranza, M. G. (2016). A heuristic framework for the bi-objective enhanced index tracking problem. Omega, 65, 122–137. https://doi.org/10.1016/j.omega.2016.01.004
Goel, A., Sharma, A., & Mehra, A. (2018). Index tracking and enhanced indexing using mixed conditional value-at-risk. Journal of Computational and Applied Mathematics, 335, 361–380. https://doi.org/10.1016/j.cam.2017.12.015
Guastaroba, G., Mansini, R., Ogryczak, W., & Speranza, M. G. (2016). Linear programming models based on Omega ratio for the Enhanced Index Tracking Problem. European Journal of Operational Research, 251(3), 938–956. https://doi.org/10.1016/j.ejor.2015.11.037
Guastaroba, G., Mansini, R., Ogryczak, W., & Speranza, M. G. (2020). Enhanced index tracking with CVaR-based ratio measures. Annals of Operations Research, 292(2), 883–931. https://doi.org/10.1007/s10479-020-03518-7
Guastaroba, G., & Speranza, M. G. (2012). Kernel Search: An application to the index tracking problem. European Journal of Operational Research, 217(1), 54–68. https://doi.org/10.1016/j.ejor.2011.09.004
Hanifi, F., Bahrololum, M. M., and Javadi, B. (2009). Design and comparative analysis of metaheuristic algorithms for implementing index-based investment in the Tehran Stock Exchange. Financial Management Outlook, Issue 32, pp. 89–108. (in persian)
Huyer, W., & Neumaier, A. (1999). Global Optimization by Multilevel Coordinate Search. Journal of Global Optimization, 14(4), 331–355. https://doi.org/10.1023/A:1008382309369
Jansen, R., & van Dijk, R. (2002). Optimal Benchmark Tracking with Small Portfolios. The Journal of Portfolio Management, 28(2), 33–39. https://doi.org/10.3905/jpm.2002.319830
Keating, C., & Shadwick, W. F. (2002). A Universal Performance Measure. Journal of Perfor-Mance Measurement, 6(1), 59–84.
Konno, H., & Yamazaki, H. (1991). Mean-Absolute Deviation Portfolio Optimization Model and Its Applications to Tokyo Stock Market. Management Science, 37(5), 519–531. https://doi.org/10.1287/mnsc.37.5.519
Koshizuka, T., Konno, H., & Yamamoto, R. (2009). Index-plus-Alpha Tracking Subject to Correlation Constraint. International Journal of Optimization: Theory, Methods and Applications, 1, 215–224.
Lejeune, M. A. (2012). Game Theoretical Approach for Reliable Enhanced Indexation. Decision Analysis, 9(2), 146–155. https://doi.org/10.1287/deca.1120.0239
Li, Q., Sun, L., & Bao, L. (2011). Enhanced index tracking based on multi-objective immune algorithm. Expert Systems with Applications, 38(5), 6101–6106. https://doi.org/10.1016/j.eswa.2010.11.001
Mansini, R. (2003). LP solvable models for portfolio optimization: a classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187–220. https://doi.org/10.1093/imaman/14.3.187
Mansini, R., Ogryczak, W., & Speranza, M. G. (2003). On LP Solvable Models for Portfolio Selection. Informatica, 14(1), 37–62. https://doi.org/10.15388/informatica.2003.003
Markowitz, H. (1952). PORTFOLIO SELECTION*. The Journal of Finance, 7(1), 77–91. https://doi.org/10.1111/j.1540-6261.1952.tb01525.x
Martin, R. D., Rachev, S., & Siboulet, F. (2005). PHI-ALPHA OPTIMAL PORTFOLIOS & EXTREME RISK MANAGEMENT.
Mausser, H., Saunders, D., & Seco, L. (2006). Optimizing Omega. RISK-LONDON-RISK MAGAZINE LIMITED.
Meade, N., & Beasley, J. E. (2011). Detection of momentum effects using an index out-performance strategy. Quantitative Finance, 11(2), 313–326. https://doi.org/10.1080/14697680903460135
Meade, N., & Salkin, G. R. (1989). Index Funds—Construction and Performance Measurement. Journal of the Operational Research Society, 40(10), 871–879. https://doi.org/10.1057/jors.1989.155
Nabizade, A. , Gharehbaghi, H. and Behzadi, A. (2017). Index Tracking Optimization under down Side Beta and Evolutionary Based Algorithms. Financial Research Journal, 19(2), 319-340. doi: 10.22059/jfr.2017.226501.1006374 (in persian)
Rahmani, A., & Dehghani Ashkezari, M. (2021). Enhanced indexing using a discrete Markov chain model and mixed conditional value-at-risk. International Journal of Finance & Managerial Accounting, 6(22), 69–80. https://ijfma.srbiau.ac.ir/article_17475.html
Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. The Journal of Risk, 2(3), 21–41. https://doi.org/10.21314/JOR.2000.038
Rockafellar, R. T., & Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, 26(7), 1443–1471. https://doi.org/10.1016/S0378-4266(02)00271-6
Roll, R. (1992). A Mean/Variance Analysis of Tracking Error. The Journal of Portfolio Management, 18(4), 13–22. https://doi.org/10.3905/jpm.1992.701922
Rudd, A. (1980). Optimal Selection of Passive Portfolios. Financial Management, 9(1), 57. https://doi.org/10.2307/3665314
Sant’Anna, L. R., Filomena, T. P., & Caldeira, J. F. (2017). Index tracking and enhanced indexing using cointegration and correlation with endogenous portfolio selection. Quarterly Review of Economics and Finance, 65, 146–157. https://doi.org/10.1016/j.qref.2016.08.008
Schoenfeld, S. A. (2004). Active Index Investing: maximizing portfolio performance and minimizing risk through global index strategies.
Sehgal, R., Sharma, A., & Mansini, R. (2023). Worst-case analysis of Omega-VaR ratio optimization model. Omega, 114, 102730. https://doi.org/10.1016/j.omega.2022.102730
Sharma, A., Utz, S., & Mehra, A. (2017). Omega-CVaR portfolio optimization and its worst case analysis. OR Spectrum, 39(2), 505–539. https://doi.org/10.1007/s00291-016-0462-y
Sharpe, W. F. (1966). Mutual Fund Performance, Part 2: Supplement on Security Prices. The Journal of Business,The University of Chicago, 39(1), 119–138. https://www.jstor.org/stable/2351741
Sharpe, W. F. (1971). Mean-Absolute-Deviation Characteristic Lines for Securities and Portfolios. Management Science, 18(2), B-1-B-13. https://doi.org/10.1287/mnsc.18.2.b1
Sortino, F. A., Meer, R. Van Der, & Plantinga, A. (1999). The Dutch Triangle. The Journal of Portfolio Management, 26(1), 50–57. https://doi.org/10.3905/jpm.1999.319775
Sortino, F. A., & Price, L. N. (1994). Performance Measurement in a Downside Risk Framework. The Journal of Investing, 3(3), 59–64. https://doi.org/10.3905/joi.3.3.59
Treynor, J. L., & Black, F. (1973). How to Use Security Analysis to Improve Portfolio Selection. The Journal of Business, 46(1), 66–86. http://www.jstor.org/stable/2351280
Varsei, M., and Shams, N. (2010). A Creative Approach to Solving the Index Tracking Portfolio Problem and Its Implementation for the First Time in the Tehran Stock Exchange TEPIX, 8th International Management Conference, Tehran, https://civilica.com/doc/99509 (in persian)
Weng, Y.-C., & Wang, R. (2017). Do Enhanced Index Funds Truly Have Enhanced Performance? Evidence from the Chinese Market. Emerging Markets Finance and Trade, 53(4), 819–834. https://doi.org/10.1080/1540496X.2015.1105637
Wu, L.-C., Chou, S.-C., Yang, C.-C., & Ong, C.-S. (2007). Enhanced Index Investing Based on Goal Programming. The Journal of Portfolio Management, 33(3), 49–56. https://doi.org/10.3905/jpm.2007.684753
Yitzhaki, S. (1982). Stochastic Dominance, Mean Variance, and Gini’s Mean Difference. The American Economic Review, 72(1), 178–185. http://www.jstor.org/stable/1808584

Files

Share

How to cite

Statistics