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.

Keywords: index tracking, portfolio optimization, uncertainty quantification, Markov chain Monte Carlo, sparse portfolios

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 8.5/10
• Quadrant: Holy Grail
• Why: 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.

  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"]