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

TL;DR

This paper develops a Bayesian sparse modeling approach for index-tracking portfolios, enabling uncertainty quantification and optimized rebalancing strategies to closely follow a reference index while controlling turnover and sparsity.

Contribution

It introduces a Bayesian framework with Laplace priors and MCMC algorithms for sparse index tracking, incorporating uncertainty quantification into portfolio rebalancing decisions.

Findings

Effective control of tracking error and portfolio sparsity.

Rebalancing rules based on posterior probabilities reduce turnover.

Case study demonstrates practical applicability on S&P 500.

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.