Stable and Cost-Efficient Tracking of the Tehran Stock Exchange Index through Robust Optimization and a Heuristic Algorithm

Objective
The rising popularity of passive management in recent years is largely due to its advantages, such as lower management fees and reduced transaction costs. A key component of passive management is index tracking, which aims to replicate the performance of a specific index using a smaller set of assets. This paper introduces a novel robust linear optimization approach for tracking indices that is not only more reliable than existing models but also exhibits superior performance on out-of-sample data, effectively tracking indices over extended periods with minimal deviation and without requiring frequent rebalancing. The proposed robust model is highly adaptable, allowing decision-makers to account for a wide range of potential future scenarios, including both the most certain outcomes and the worst-case possibilities. The importance of this model lies in its ability to adapt more effectively to market fluctuations and help investors get closer to their return objectives.
 
 
Methods
Given the NP-hard nature of index tracking problems with cardinality constraints, solving them in polynomial time is challenging. To address this computational burden, a novel heuristic approach was designed. This heuristic integrates the exploration capabilities of genetic algorithms with the focused search capabilities of local search techniques and is designed to solve the problem efficiently. Daily data from the Tehran Stock Exchange and OR-library were utilized to validate the proposed model and heuristic. For comparison purposes, a commercial solver was employed to solve both the proposed robust model and the conventional linear model commonly used in the literature. This approach was adopted instead of the developed algorithm because, although the proposed algorithm demonstrates superior performance compared to the commercial solver, it is inherently stochastic in nature. Therefore, to eliminate the influence of random and probabilistic factors on the results, a deterministic and precise commercial solver was utilized.
 
Results
The results demonstrate that the proposed heuristic not only converges to optimal solutions for moderately sized problems but also produces portfolios that outperform those generated by commercial solvers in terms of both in-sample and out-of-sample data, all within a shorter time frame. Furthermore, given the structure of the Tehran Stock Exchange, the impact of including or excluding the ten largest stocks by market capitalization in the selected portfolio on the final results of the proposed tracking portfolio has been examined. This analysis indicates that the presence of these stocks in the portfolio can influence the performance of the tracking portfolio.
 
Conclusion
This study aimed to develop an effective approach for index tracking by proposing a robust linear mathematical model alongside a novel heuristic algorithm. The performance of the proposed approach was evaluated against a benchmark linear model using five quantitative indicators: correlation, mean absolute deviation (MAD), root mean squared error (RMSE), standard deviation, and beta. The proposed model outperformed the benchmark model across all indicators in the out-of-sample period. In addition to these indicators, the objective function value was used to assess the performance of the proposed heuristic. The algorithm outperformed the CPLEX solver in 77 out of 88 comparisons, 11 indicators under four configurations for the Tehran Stock Exchange and the Hang Seng Index. Furthermore, an alternative tracking portfolio with a novel weighting system was introduced, which effectively reflects the status of the largest industries represented in the Tehran Stock Exchange. The performance of this alternative portfolio was compared with that of the proposed portfolio in terms of return, risk, and tracking error. The tracking error of the proposed portfolio was found to be nearly three times lower than that of the alternative approach.