A Robust Portfolio Construction Using the Bootstrap Method to Extract Multidimensional Uncertainty Sets: An Application on BIST100 Stocks

Abstract

Asset allocation is the main problem of portfolio management. The goal is to maximize the portfolio's expected return while minimizing investment risk by allocating assets optimally. However, it is not possible to exclude all investment risks due to prediction errors, optimization of incorrect models, uncertainties in parameters, etc. The classical models used in portfolio theory disperse model-based risks but ignore the uncertainties of the predicted parameters. Besides, uncertainty-based models, such as robust optimization help to eliminate uncertainty risks in addition to model-based risks. Robust optimization constructs portfolios by considering worst-case realizations of asset returns within uncertainty sets. In this way, the model's solution remains optimal with high probability, while investors are protected from model-based risks. In this paper, we develop a robust optimization formulation based on Bertsimas and Sim (2004) and combine the model with the bootstrap method to create optimal portfolios. The results show that the expected return of the portfolios decreases as the uncertainty of the robust model increases. The expected return of the robust portfolio is as good as that of the classical portfolio for a moderate level of uncertainty. Out of the sample, the robust portfolios outperform the equally weighted portfolio and the target index

Keywords

• Robust optimization
• bootstrap method
• asset allocation
• BIST100
• uncertainity sets

References

1. Agra, A., Christiansen M., Hvattum L. M., and Rodrigues F. (2018). Robust Optimization for a Maritime Inventory Routing Problem. Transportation Science 52(3): 509-525.
2. Alem, D. J., and Morabito R. (2012). Production Planning in Furniture Settings via Robust Optimization. Computers and Operations Research 39(2): 139-150.
3. Azimli, A. (2020). Pricing the Common Stocks in an Emerging Capital Market: Comparison of the Factor Models. Borsa Istanbul Review 20(4): 334-346.
4. Bakar, N. A., and Rosbi, S. (2019). Robust Statistical Portfolio Investment in Modern Portfolio Theory: A Case Study of Two Stocks Combination in Kuala Lumpur Stock Exchange. International Journal of Engineering and Advanced Technology (IJEAT), 8: 214-221.
5. Bandi, C., Bertsimas D. (2012). Tractable Stochastic Analysis in High Dimensions Via Robust Optimization. Mathematical Programming 134(1): 23-70.
6. Beck, A., and Ben-Tal, A. (2009). Duality in Robust Optimization: Primal Worst Equals Dual Best. Operations Research Letters, 37(1):1-6.
7. Ben-Tal, A., and Nemirovski A. (1999). Robust Solutions of Uncertain Linear Programs. Operations Research Letters 25(1): 1-13.
8. Ben-Tal, A., and Nemirovski A. (2000). Robust Solutions of Linear Programming Problems Contaminated with Uncertain Data. Mathematical Programming 88(3): 411-424.

9. Ben-Tal, A., and Nemirovski, A. (2002). Robust Optimization–Methodology and Applications. Mathematical Programming, 92: 453-480.
10. Ben-Tal, A., Nemirovski.A. (1998). Robust Convex Optimization. Mathematics of Operations Research 23(4): 769-805.
11. Bertsimas, D., and Brown D. B. (2009). Constructing Uncertainty Sets for Robust Linear Optimization. Operations Research 57(6): 1483-1495.
12. Bertsimas, D., and Thiele A. (2004). “A Robust Optimization Approach to Supply Chain Management”. In International Conference on Integer Programming and Combinatorial Optimization, Berlin, Heidelberg, June 88-100.
13. Bertsimas, D., Brown, D. B., and Caramanis, C. (2011). Theory and Applications of Robust Optimization. SIAM review, 53(3): 464-501.
14. Bertsimas, D., Gupta V., and Kallus N. (2018). Data-Driven Robust Optimization. Mathematical Programming 167(2): 235-292.
15. Bertsimas, D., Pachamanova, D., and Sim, M. (2004). Robust Linear Optimization Under General Norms. Operations Research Letters, 32(6): 510-516.
16. Bertsimas, D., Sim M. (2004). The Price of Robustness. Operations Research 52(1): 35-53.
17. Bhansali, V. (2008). Tail Risk Management. The Journal of Portfolio Management 34(4): 68-75.
18. Birge, J. R., and Louveaux, F. (2011). Introduction to stochastic programming. Springer Science and Business Media.
19. Campbell, R. A., Forbes C. S., Koedijk K. G., and Kofman P. (2008). Increasing Correlations or Just Fat Tails?. Journal of Empirical Finance 15(2): 287-309.
20. Chopra, V. and Ziemba, W. T., (1993). The Effects of Errors in Means, Variances, And Covariances on Optimal Portfolio Choice. J. Portfolio Manage,19(2): 6–11
21. Dai, Z., Kang J. (2021). Some New Efficient Mean–Variance Portfolio Selection Models. International Journal of Finance & Economics 26(1).
22. Dai, Z., Wang F. (2019). Sparse and Robust Mean–Variance Portfolio Optimization Problems. Physica A: Statistical Mechanics and Its Applications 523: 1371-1378.
23. Daneshvari, H., and Shafaei R. (2021). A New Correlated Polyhedral Uncertainty Set for Robust Optimization. Computers and Industrial Engineering 157: 107346.
24. Deng G, Dulaney T., McCann C., and Wang O. (2013). Robust Portfolio Optimization with Value-At-Risk-Adjusted Sharpe Ratios. Journal of Asset Management 14(5): 293–305.
25. Diwekar, U. M., and Diwekar, U. M. (2020). Optimization Under Uncertainty. Introduction to Applied Optimization, 151-215.
26. Efron, B., and Tibshirani R. (1985). The Bootstrap Method for Assessing Statistical Accuracy. Behaviormetrika 12(17): 1-35.
27. Eom, C. (2020). Risk Characteristic on Fat-Tails of Return Distribution: An Evidence of the Korean Stock Market. Asia-Pacific Journal of Business 11(4): 37-48.
28. Eom, C., Kaizoji, T., and Scalas, E. (2019). Fat tails in Financial Return Distributions Revisited: Evidence from the Korean Stock Market. Physica A: Statistical Mechanics and its Applications, 526: 121055.
29. Fabozzi, F. J., Kolm P. N., Pachamanova D. A., and Focardi S. M. (2007). Robust Portfolio Optimization. The Journal of Portfolio Management 33(3): 40-48.
30. Fama, E. F. (1965). Portfolio Analysis in a Stable Paretian Market. Management Science 11(3): 404-419.
31. Fama, E. F., French K. R. (1992). The Cross‐Section of Expected Stock Returns. The Journal of Finance, 47(2): 427-465.
32. Fama, E. F., French K. R. (1993). Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics 33(1): 3-56.
33. Gabrel, V., Murat, C., and Thiele, A. (2014). Recent Advances in Robust Optimization: An Overview. European Journal of Operational Research, 235(3): 471-483.
34. Georgantas, A., Doumpos M., and Zopounidis C. (2021). Robust optimization approaches for portfolio selection: a comparative analysis. Annals of Operations Research 301(1): 1-17.
35. Gero, J. S., and Dudnik, E. E. (1978). Uncertainty and the Design of Building Subsystems—a Dynamic Programming Approach. Building and Environment, 13(3): 147-152.
36. Ghaoui, L. E., Oks, M., and Oustry, F. (2003). Worst-Case Value-At-Risk and Robust Portfolio Optimization: A Conic Programming Approach. Operations Research, 51(4): 543-556.
37. Goldfarb, D., and Iyengar, G. (2003). Robust Portfolio Selection Problems. Mathematics of Operations Research, 28(1): 1-38.
38. Golsefidi, A. H., Jokar M. R. A. (2020). A Robust Optimization Approach for the Production-Inventory-Routing Problem with Simultaneous Pickup and Delivery. Computers and Industrial Engineering 143: 106388.
39. Gregory, C., Darby-Dowman K., and Mitra G. (2011). Robust Optimization and Portfolio Selection: The Cost of Robustness. European Journal of Operational Research 212(2): 417-428.
40. Guan, Y., Wang J. (2013). Uncertainty Sets for Robust Unit Commitment. IEEE Transactions on Power Systems 29(3): 1439-1440.
41. Gülpınar, N., and Hu, Z. (2016). Robust Optimization Approaches to Single Period Portfolio Allocation Problem. Robustness Analysis in Decision Aiding, Optimization, and Analytics, 265-283.
42. Haas, M., and Pigorsch, C. (2009). Financial Economics, Fat-Tailed Distributions. Encyclopedia of Complexity and Systems Science, 4(1): 3404-3435.
43. Hahn, G. J., and Kuhn H. (2012). Value-Based Performance and Risk Management in Supply Chains: A Robust Optimization Approach. International Journal of Production Economics 139(1): 135-144.
44. Huang, D., Zhu, S., Fabozzi, F. J., and Fukushima, M. (2010). Portfolio Selection Under Distributional Uncertainty: A relative Robust CVaR Approach. European Journal of Operational Research, 203(1): 185-194.
45. Hubert, M., Debruyne M., and Rousseeuw P. J. (2018). Minimum Covariance Determinant and Extensions. Wiley Interdisciplinary Reviews: Computational Statistics 10(3): e1421.
46. Kallberg, J. G., and Ziemba, W. T. (1984). Mis-Specifications in Portfolio Selection Problems. In Risk and Capital: Proceedings of the 2nd Summer Workshop on Risk and Capital Held at the University of Ulm, West Germany June 20–24, 1983 (pp. 74-87). Berlin, Heidelberg: Springer Berlin Heidelberg.
47. Kapsos M, Christofides N., and Rustem B. (2014). Worst-Case Robust Omega Ratio. European Journal of Operational Research 234(2): 499–507.
48. Kaszuba, B. (2012). Applications of Robust Statistics in the Portfolio Theory. Mathematical Economics, 8: 63-82.
49. Kolm, P. N., Tütüncü R., and Fabozzi F. J.. (2014). 60 Years of Portfolio Optimization: Practical Challenges and Current Trends. European Journal of Operational Research 234(2): 356-371.
50. Koumou, G. B. (2020). Diversification and Portfolio Theory: A Review. Financial Markets and Portfolio Management 34(3): 267-312.
51. Lauprete, G. J., Samarov, A. M., and Welsch, R. E. (2003). Robust Portfolio Optimization. In Developments in Robust Statistics: International Conference on Robust Statistics 2001 (pp. 235-245). Physica-Verlag HD.
52. Lee, Y., Kim, M. J., Kim, J. H., Jang, J. R., and Chang Kim, W. (2020). Sparse and Robust Portfolio Selection Via Semi-Definite Relaxation. Journal of the Operational Research Society, 71(5): 687-699.
53. Lhabitant, François-Serge. (2017). Portfolio Diversification. London: Oxford.
54. Li, X., Hong, J., and Wang, B. (2015, April). The Application of Robust Statistics to Stock Portfolio Problem. In 3rd International Conference on Mechatronics, Robotics and Automation (pp. 1285-1289). Atlantis Press.
55. Mandelbrot, B. Benoit. (1997). Fractals and Scaling in Finance: The Variation of Certain Speculative Prices. New York, NY, USA.
56. Markowitz, H. M. (1952). Portfolio Selection. The Journal of Finance 7(1): 77-91
57. Marshall, C. M. (2015). Isolating the Systematic and Unsystematic Components of a Single Stock’s (or Portfolio’s) Standard Deviation. Applied Economics, 47(1): 1-11.
58. Moret, S.,Babonneau F., Bierlaire M., and Maréchal F. (2020). Decision Support for Strategic Energy Planning: A Robust Optimization Framework. European Journal of Operational Research 280(2): 539-554.
59. Pilbeam, K. (2018). Finance and Financial Markets. Bloomsbury Publishing.
60. Pishvaee, M. S., Rabbani M., and Torabi S. A. (2011). A Robust Optimization Approach to Closed-Loop Supply Chain Network Design Under Uncertainty. Applied Mathematical Modelling 35(2): 637-649.
61. Qiu, H., Han, F., Liu, H., and Caffo, B. (2015). Robust Portfolio Optimization. Advances in Neural Information Processing Systems, 28.
62. Quaranta, A. G., and Zaffaroni A. (2008). Robust Optimization of Conditional Value at Risk and Portfolio Selection. Journal of Banking and Finance 32(10): 2046-2056.
63. Reyna, F. R., Júnior A. M. D., Mendes B. V., and Porto O. (2005). Optimal Portfolio Structuring in Emerging Stock Markets Using Robust Statistics. Brazilian Review of Econometrics 25(2): 139-157.
64. Roald, L. A., Pozo, D., Papavasiliou, A., Molzahn, D. K., Kazempour, J., and Conejo, A. (2023). Power Systems Optimization Under Uncertainty: A Review of Methods and Applications. Electric Power Systems Research, 214: 108725.
65. Rodrigues, F., Agra A., Christiansen M., Hvattum L. M., and Requejo C. (2019). Comparing Techniques for Modelling Uncertainty in A Maritime Inventory Routing Problem. European Journal of Operational Research 277(3): 831-845.
66. Scutella, M. G., and Recchia R. (2013). Robust Portfolio Asset Allocation and Risk Measures. Annals of Operations Research 204(1): 145-169.
67. Sengupta, R. N., Kumar R. (2017). Robust and Reliable Portfolio Optimization Formulation of a Chance Constrained Problem. Foundations of Computing and Decision Sciences 42(1): 83-117.
68. Shapiro, A., and Philpott, A. (2007). A Tutorial on Stochastic Programming. Manuscript. Available at www2. isye. gatech. edu/ashapiro/publications. html, 17.
69. Sheikh, A. Z., Qiao H. (2009). Non-normality of Market Returns: A Framework for Asset Allocation Decision Making. The Journal of Alternative Investments 12(3): 8-35.
70. Shen, F., Zhao L., Du W., Zheng W., and Qian F. (2020). Large-Scale Industrial Energy Systems Optimization Under Uncertainty: A Data-Driven Robust Optimization Approach. Applied Energy 259: 114199.
71. Sheng, L., Zhu, Y., and Wang, K. (2020). Analysis of a Class of Dynamic Programming Models for Multi-Stage Uncertain Systems. Applied Mathematical Modelling, 86: 446-459.
72. Solares, E., Coello C. A. C., Fernandez E., and Navarro J. (2019). Handling Uncertainty Through Confidence Intervals in Portfolio Optimization. Swarm and Evolutionary Computation 44: 774-787.
73. Soyster, A. L. (1973). Convex Programming with Set-Inclusive Constraints and Applications to Inexact Linear Programming. Operations research 21(5): 1154-1157.
74. Stoyanov, S. V., Rachev S. T., Racheva-Yotova B., and Fabozzi F. J. (2011). Fat-tailed Models for Risk Estimation. The Journal of Portfolio Management, 37(2): 107-117.
75. Tütüncü, R. H., Koenig M. (2004). Robust Asset Allocation. Annals of Operations Research 132(1): 157-187.
76. Wang, L., Cheng X. (2016). Robust Portfolio Selection under Norm Uncertainty. Journal of Inequalities and Applications 2016(1): 1-10.
77. Wehrens, R., Putter, H., and Buydens, L. M. (2000). The Bootstrap: A Tutorial. Chemometrics And İntelligent Laboratory Systems, 54(1): 35-52.
78. Xidonas, P., Steuer, R., and Hassapis, C. (2020). Robust Portfolio Optimization: a Categorized Bibliographic Review. Annals of Operations Research, 292(1): 533-552.
79. Xidonas, P., Steuer, R., and Hassapis, C. (2020). Robust Portfolio Optimization: A Categorized Bibliographic Review. Annals of Operations Research, 292(1): 533-552.
80. Yam, S. C. P., Yang, H., and Yuen, F. L. (2016). Optimal Asset Allocation: Risk and İnformation Uncertainty. European Journal of Operational Research, 251(2): 554-561.
81. Yang, L., Couillet R., and McKay M. R. (2015). A Robust Statistics Approach to Minimum Variance Portfolio Optimization. IEEE Transactions on Signal Processing, 63(24): 6684-6697.
82. Yanıkoğlu, İ., Gorissen, B. L., and den Hertog, D. (2019). A Survey of Adjustable Robust Optimization. European Journal of Operational Research, 277(3): 799-813.
83. Zaimovic, A., Omanovic, A., and Arnaut-Berilo, A. (2021). How Many Stocks Are Sufficient For Equity Portfolio Diversification? A Review of the Literature. Journal of Risk and Financial Management, 14(11): 551.
84. Zakaria, A., Ismail, F. B., Lipu, M. H., and Hannan, M. A. (2020). Uncertainty Models for Stochastic Optimization in Renewable Energy Applications. Renewable Energy, 145: 1543-1571.
85. Zeferino, J. A., Cunha M. C., and Antunes A. P. (2012). Robust Optimization Approach to Regional Wastewater System Planning. Journal of Environmental Management 109: 113-122.
86. Zhang, B., Li Q., Wang L., and Feng W. (2018). Robust Optimization for Energy Transactions in Multi-Microgrids Under Uncertainty. Applied Energy 217: 346-360.
87. Zhao, P., Gu C., Hu D., Shen Y., and Hernando-Gil I. (2019). Two-stage Distributionally Robust Optimization for Energy Hub Systems. IEEE Transactions on Industrial Informatics 16(5): 3460-3469.
88. Zhu, X., Zeng B., Dong H., and Liu J. (2020). An Interval-Prediction Based Robust Optimization Approach for Energy-Hub Operation Scheduling Considering Flexible Ramping Products. Energy 194: 116821.
89. Zymler, S., Rustem, B., and Kuhn, D. (2011). Robust Portfolio Optimization With Derivative İnsurance Guarantees. European Journal of Operational Research, 210(2): 410-424.