Index Tracking

Non-Convex Portfolio Optimization via Energy-Based Models: A Comparative Analysis Using the Thermodynamic HypergRaphical Model Library (THRML) for Index Tracking ArXiv ID: 2601.07792 “View on arXiv” Authors: Javier Mancilla, Theodoros D. Bouloumis, Frederic Goguikian Abstract Portfolio optimization under cardinality constraints transforms the classical Markowitz mean-variance problem from a convex quadratic problem into an NP-hard combinatorial optimization problem. This paper introduces a novel approach using THRML (Thermodynamic HypergRaphical Model Library), a JAX-based library for building and sampling probabilistic graphical models that reformulates index tracking as probabilistic inference on an Ising Hamiltonian. Unlike traditional methods that seek a single optimal solution, THRML samples from the Boltzmann distribution of high-quality portfolios using GPU-accelerated block Gibbs sampling, providing natural regularization against overfitting. We implement three key innovations: (1) dynamic coupling strength that scales inversely with market volatility (VIX), adapting diversification pressure to market regimes; (2) rebalanced bias weights prioritizing tracking quality over momentum for index replication; and (3) sector-aware post-processing ensuring institutional-grade diversification. Backtesting on a 100-stock S and P 500 universe from 2023 to 2025 demonstrates that THRML achieves 4.31 percent annualized tracking error versus 5.66 to 6.30 percent for baselines, while simultaneously generating 128.63 percent total return against the index total return of 79.61 percent. The Diebold-Mariano test confirms statistical significance with p less than 0.0001 across all comparisons. These results position energy-based models as a promising paradigm for portfolio construction, bridging statistical mechanics and quantitative finance. ...

A comprehensive review and analysis of different modeling approaches for financial index tracking problem ArXiv ID: 2601.03927 “View on arXiv” Authors: Vrinda Dhingra, Amita Sharma, Anubha Goel Abstract Index tracking, also known as passive investing, has gained significant traction in financial markets due to its cost-effective and efficient approach to replicating the performance of a specific market index. This review paper provides a comprehensive overview of the various modeling approaches and strategies developed for index tracking, highlighting the strengths and limitations of each approach. We categorize the index tracking models into three broad frameworks: optimization-based models, statistical-based models and machine learning based data-driven approach. A comprehensive empirical study conducted on the S&P 500 dataset demonstrates that the tracking error volatility model under the optimization-based framework delivers the most precise index tracking, the convex co-integration model, under the statistical-based framework achieves the strongest return-risk balance, and the deep neural network with fixed noise model within the data-driven framework provides a competitive performance with notably low turnover and high computational efficiency. By combining a critical review of the existing literature with comparative empirical analysis, this paper aims to provide insights into the evolving landscape of index tracking and its practical implications for investors and fund managers. ...

Index-Tracking Portfolio Construction and Rebalancing under Bayesian Sparse Modelling and Uncertainty Quantification ArXiv ID: 2512.22109 “View on arXiv” Authors: Dimitrios Roxanas Abstract We study the construction and rebalancing of sparse index-tracking portfolios from an operational research perspective, with explicit emphasis on uncertainty quantification and implementability. The decision variables are portfolio weights constrained to sum to one; the aims are to track a reference index closely while controlling the number of names and the turnover induced by rebalancing. We cast index tracking as a high-dimensional linear regression of index returns on constituent returns, and employ a sparsity-inducing Laplace prior on the weights. A single global shrinkage parameter controls the trade-off between tracking error and sparsity, and is calibrated by an empirical-Bayes stochastic approximation scheme. Conditional on this calibration, we approximate the posterior distribution of the portfolio weights using proximal Langevin-type Markov chain Monte Carlo algorithms tailored to the budget constraint. This yields posterior uncertainty on tracking error, portfolio composition and prospective rebalancing moves. Building on these posterior samples, we propose rules for rebalancing that gate trades through magnitude-based thresholds and posterior activation probabilities, thereby trading off expected tracking error against turnover and portfolio size. A case study on tracking the S&P~500 index is carried out to showcase how our tools shape the decision process from portfolio construction to rebalancing. ...

Asset pre-selection for a cardinality constrained index tracking portfolio with optional enhancement ArXiv ID: 2503.18609 “View on arXiv” Authors: Unknown Abstract An index tracker is a passive investment reproducing the return and risk of a market index, an enhanced index tracker offers a return greater than the index. We consider the selection of a portfolio of given cardinality to track an index, both without and with enhancement. We divide the problem into two steps - (1) pre-selection of assets; (2) estimation of weights on the assets chosen. The eight pre-selection procedures considered use: forward selection (FS) or backward elimination (BE); implemented using ordinary least squares (OLS) or least absolute deviation (LAD) regression; with a regression constant (c) or without (n). The two-step approach avoids the NP-hard problem arising when asset selection and asset weight computation are combined, leading to the selection of a cardinality constrained index tracking portfolio by computer intensive heuristic procedures with many examples in the literature solving for portfolios of 10 or fewer assets. Avoiding these restrictions, we show that out-of-sample tracking errors are roughly proportional to 1/sqrt(cardinality). We find OLS more effective than LAD; BE marginally more effective than FS; (n) marginally more effective than (c). For index tracking, both without and with enhancement, we use BE-OLS(n) in sensitivity analyses on the periods used for selection and evaluation. For a S&P 500 index tracker, we find that out-of-sample tracking error, transaction volume and return-risk ratios all improve as cardinality increases. By contrast for enhanced returns, cardinalities of the order 10 to 20 are most effective. The S&P 500 data used from 3/1/2005 to 29/12/2023 is available to researchers. ...

Sparse Portfolio Selection via Topological Data Analysis based Clustering ArXiv ID: 2401.16920 “View on arXiv” Authors: Unknown Abstract This paper uses topological data analysis (TDA) tools and introduces a data-driven clustering-based stock selection strategy tailored for sparse portfolio construction. Our asset selection strategy exploits the topological features of stock price movements to select a subset of topologically similar (different) assets for a sparse index tracking (Markowitz) portfolio. We introduce new distance measures, which serve as an input to the clustering algorithm, on the space of persistence diagrams and landscapes that consider the time component of a time series. We conduct an empirical analysis on the S&P index from 2009 to 2022, including a study on the COVID-19 data to validate the robustness of our methodology. Our strategy to integrate TDA with the clustering algorithm significantly enhanced the performance of sparse portfolios across various performance measures in diverse market scenarios. ...

Sparse Index Tracking via Topological Learning ArXiv ID: 2310.09578 “View on arXiv” Authors: Unknown Abstract In this research, we introduce a novel methodology for the index tracking problem with sparse portfolios by leveraging topological data analysis (TDA). Utilizing persistence homology to measure the riskiness of assets, we introduce a topological method for data-driven learning of the parameters for regularization terms. Specifically, the Vietoris-Rips filtration method is utilized to capture the intricate topological features of asset movements, providing a robust framework for portfolio tracking. Our approach has the advantage of accommodating both $\ell_1$ and $\ell_2$ penalty terms without the requirement for expensive estimation procedures. We empirically validate the performance of our methodology against state-of-the-art sparse index tracking techniques, such as Elastic-Net and SLOPE, using a dataset that covers 23 years of S&P500 index and its constituent data. Our out-of-sample results show that this computationally efficient technique surpasses conventional methods across risk metrics, risk-adjusted performance, and trading expenses in varied market conditions. Furthermore, in turbulent markets, it not only maintains but also enhances tracking performance. ...

Reinforcement Learning for Financial Index Tracking ArXiv ID: 2308.02820 “View on arXiv” Authors: Unknown Abstract We propose the first discrete-time infinite-horizon dynamic formulation of the financial index tracking problem under both return-based tracking error and value-based tracking error. The formulation overcomes the limitations of existing models by incorporating the intertemporal dynamics of market information variables not limited to prices, allowing exact calculation of transaction costs, accounting for the tradeoff between overall tracking error and transaction costs, allowing effective use of data in a long time period, etc. The formulation also allows novel decision variables of cash injection or withdraw. We propose to solve the portfolio rebalancing equation using a Banach fixed point iteration, which allows to accurately calculate the transaction costs specified as nonlinear functions of trading volumes in practice. We propose an extension of deep reinforcement learning (RL) method to solve the dynamic formulation. Our RL method resolves the issue of data limitation resulting from the availability of a single sample path of financial data by a novel training scheme. A comprehensive empirical study based on a 17-year-long testing set demonstrates that the proposed method outperforms a benchmark method in terms of tracking accuracy and has the potential for earning extra profit through cash withdraw strategy. ...

A systematic literature review on solution approaches for the index tracking problem in the last decade ArXiv ID: 2306.01660 “View on arXiv” Authors: Unknown Abstract The passive management approach offers conservative investors a way to reduce risk concerning the market. This investment strategy aims at replicating a specific index, such as the NASDAQ Composite or the FTSE100 index. The problem is that buying all the index’s assets incurs high rebalancing costs, and this harms future returns. The index tracking problem concerns building a portfolio that follows a specific benchmark with fewer transaction costs. Since a subset of assets is required to solve the index problem this class of problems is NP-hard, and in the past years, researchers have been studying solution approaches to obtain tracking portfolios more practically. This work brings an analysis, spanning the last decade, of the advances in mathematical approaches for index tracking. The systematic literature review covered important issues, such as the most relevant research areas, solution methods, and model structures. Special attention was given to the exploration and analysis of metaheuristics applied to the index tracking problem. ...