Abstract
We propose an enhanced indexing portfolio optimization model that not only seeks to maximize the excess returns over and above the benchmark index but simultaneously control the risk by introducing a constraint on the weighted conditional value at risk (WCVaR) of the portfolio. The constraint in the proposed model can be seen as hedging the risk described by WCVaR of the portfolio. To carry out a comparative analysis of the proposed model, we also suggest an enhanced indexing CVaR model. We analyze the performance of the proposed model at various risk levels on eight publicly available financial data sets from Beasley OR library, and S&P 500, S&P BSE 500, NASDAQ composite, FTSE 100 index, and their constituents, for average returns, Sharpe ratio, and upside potential ratio. Empirical analysis exhibits superior performance of the portfolios from the proposed WCVaR model over the respective benchmark indices and additionally the optimal portfolios obtained from various other enhanced indexing models that exist in the literature. Furthermore, we present evidence of better performance of WCVaR model over the CVaR model for long-term investment horizons.
Similar content being viewed by others
Notes
\(L_0=0\) is selected only for computational convenience. However, an investor can select any value of \(L \in [L_{min},L_{max}]\).
References
Ahmed, P., & Nanda, S. (2005). Performance of enhanced index and quantitative equity funds. Financial Review, 40(4), 459–479.
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.
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.
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.
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.
DiBartolomeo, D. (2000). The enhanced index fund as an alternative to indexed equity management. Boston: Northfield information services. http://www.northinfo.com/documents/70.pdf. Accessed 16 Jan 2019.
Edirisinghe, N. (2013). Index-tracking optimal portfolio selection. Quantitative Finance Letters, 1(1), 16–20.
Gilli, M., & Këllezi, E. (2002). The threshold accepting heuristic for index tracking. In P. Pardalos & V. K. Tsitsiringos (Eds.), Applied optimization series: Financial engineering, e-commerce and supply chain (pp. 1–18). Boston: Kluwer Academic.
Guastaroba, G., & Speranza, M. G. (2012). Kernel search: An application to the index tracking problem. European Journal of Operational Research, 217(1), 54–68.
Guastaroba, G., Mansini, R., Ogryczak, W., & Speranza, M. G. (2016a). Linear programming models based on omega ratio for the enhanced index tracking problem. European Journal of Operational Research, 251(3), 938–956.
Guastaroba, G., Mansini, R., Speranza, M. G., Ogryczak, W. (2016b). Enhanced index tracking with cvar-based measures. https://www.academia.edu/32399307/Enhanced_Index_Tracking_with_CVaR-Based_Measures. Accessed 16 Jan 2019.
Jeurissen, R., & Van den Berg, J. (2005). Index tracking using a hybrid genetic algorithm. In Proceedings of 2005 ICSC congress on computational intelligence methods and applications, Istanbul. https://doi.org/10.1109/CIMA. 2005.1662364.
Ji, R., Lejeune, M. A., & Prasad, S. Y. (2017). Properties, formulations, and algorithms for portfolio optimization using mean-gini criteria. Annals of Operations Research, 248(1–2), 305–343.
Keating, C., & Shadwick, W. F. (2002). A universal performance measure. Journal of Performance Measurement, 6(3), 59–84.
Koshizuka, T., Konno, H., & Yamamoto, R. (2009). Index-plus-alpha tracking subject to correlation constraint. International Journal of Optimization: Theory, Methods and Applications, 1(2), 215–224.
Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of Risk, 4, 43–68.
Li, Q., Sun, L., & Bao, L. (2011). Enhanced index tracking based on multi-objective immune algorithm. Expert Systems with Applications, 38(5), 6101–6106.
Linsmeier, T. J., Pearson, N. D., et al. (1996). Risk measurement: An introduction to value at risk. In Technical report 96-04. OFOR, University of Illinois, Urbana-Champaign.
Lizyayev, A., & Ruszczyński, A. (2012). Tractable almost stochastic dominance. European Journal of Operational Research, 218(2), 448–455.
Mansini, R., Ogryczak, W., & Speranza, M. G. (2007a). Conditional value at risk and related linear programming models for portfolio optimization. Annals of Operations Research, 152(1), 227–256.
Mansini, R., Ogryczak, W., & Speranza, M. G. (2007b). Tail Gini’s risk measures and related linear programming models for portfolio optimization. In HERCMA conference proceedings, CD, LEA Publishers, Athens.
Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1), 77–91.
Meade, N., & Beasley, J. E. (2011). Detection of momentum effects using an index out-performance strategy. Quantitative Finance, 11(2), 313–326.
Ogryczak, W., & Ruszczyński, A. (2002a). Dual stochastic dominance and quantile risk measures. International Transactions in Operational Research, 9(5), 661–680.
Ogryczak, W., & Ruszczynski, A. (2002b). Dual stochastic dominance and related mean-risk models. SIAM Journal on Optimization, 13(1), 60–78.
Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2, 21–42.
Rockafellar, R. T., & Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, 26(7), 1443–1471.
Roll, R. (1992). A mean/variance analysis of tracking error. The Journal of Portfolio Management, 18(4), 13–22.
Roman, D., Mitra, G., & Zverovich, V. (2013). Enhanced indexation based on second-order stochastic dominance. European Journal of Operational Research, 228(1), 273–281.
Rudolf, M., Wolter, H. J., & Zimmermann, H. (1999). A linear model for tracking error minimization. Journal of Banking & Finance, 23(1), 85–103.
Sant’Anna, L. R., Filomena, T. P., & Caldeira, J. F. (2017). Index tracking and enhanced indexing using cointegration and correlation with endogenous portfolio selection. The Quarterly Review of Economics and Finance, 65, 146–157.
Scowcroft, A., & Sefton, J. (2003). Enhanced indexation. In S. Satchell & A. Scowcroft (Eds.), Advances in portfolio construction and implementation (pp. 95–124). Butterworth-Heinemann Finance.
Sharma, A., Agrawal, S., & Mehra, A. (2017). Enhanced indexing for risk averse investors using relaxed second order stochastic dominance. Optimization and Engineering, 18(2), 407–442.
Sharpe, W. F. (1994). The sharpe ratio. The Journal of Portfolio Management, 21(1), 49–58.
Sortino, F. A., Van Der Meer, R., & Plantinga, A. (1999). The dutch triangle. The Journal of Portfolio Management, 26(1), 50–57.
Wang, M., Xu, C., Xu, F., & Xue, H. (2012). A mixed 0–1 LP for index tracking problem with cvar risk constraints. Annals of Operations Research, 196(1), 591–609.
Yitzhaki, S. (1982). Stochastic dominance, mean variance, and Gini’s mean difference. The American Economic Review, 72(1), 178–185.
Acknowledgements
The authors are profoundly thankful to the Editor-in-Chief and the esteemed referees for their valuable comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sehgal, R., Mehra, A. Enhanced indexing using weighted conditional value at risk. Ann Oper Res 280, 211–240 (2019). https://doi.org/10.1007/s10479-019-03132-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-019-03132-2