Index-Tracking Portfolio Construction and Rebalancing under Bayesian Sparse Modelling and Uncertainty Quantification

Paper: arXiv 2512.22109

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.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 6.0/10 (capped from 8.5: transaction costs not modelled; see penalty flags)
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced mathematics including proximal MCMC, Laplace priors, and stochastic approximation, while also demonstrating strong empirical rigor with a detailed S&P 500 case study, posterior uncertainty quantification, and practical rebalancing rules.

Research Flowchart

  flowchart TD
  A["<b>Research Goal</b><br>Build sparse, low-turnover<br>index-tracking portfolios<br>with uncertainty quantification"] --> B["<b>Methodology</b><br>Bayesian Sparse Modeling<br>Laplace Prior + Shrinkage<br>Proximal Langevin MCMC"]

  B --> C["<b>Computational Process</b><br>1. Calibrate shrinkage<br>parameter via Empirical Bayes<br>2. Sample posterior<br>portfolio weights<br>3. Quantify uncertainty"]

  C --> D["<b>Decision Process</b><br>Gate trades via<br>magnitude thresholds<br>& posterior probabilities"]

  D --> E["<b>Outcomes</b><br>1. Sparse portfolios<br>2. Controlled turnover<br>3. Quantified uncertainty<br>4. S&P 500 case study"]

Cited References (39)

1. Optimization Methods for Financial Index Tracking: From Theory to Practice (2018) — link
2. Sparse Portfolios for High-Dimensional Financial Index Tracking (2018) — link
3. An Evolutionary Heuristic for the Index Tracking Problem (2003) — link
4. Mixed-Integer Programming Approaches for Index Tracking and Enhanced Indexation (2009) — link
5. A Systematic Literature Review on Solution Approaches for the Index Tracking Problem (2024) — link
6. Index Tracking in Financial Markets: A Comprehensive Review of Optimization, Statistical, and Data-Driven Modeling Approaches (2026) — link
7. Minimal Cost Index Tracking under Nonlinear Transaction Costs and Minimal Transaction Unit Constraints (2001) — link
8. Index Tracking with Fixed and Variable Transaction Costs (2014) — link

0 of 39 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.