🤖 AI Summary
This paper addresses the challenge of jointly optimizing tracking error, portfolio cardinality, and turnover rate in sparse index tracking. We propose a Bayesian sparse modeling framework that integrates uncertainty quantification with implementability constraints. Methodologically, we pioneer the combination of empirical Bayesian stochastic approximation with budget-constrained proximal Langevin Monte Carlo sampling to efficiently approximate high-dimensional posterior weight distributions. We further design an interpretable and tunable rebalancing rule based on posterior activation probabilities and magnitude thresholds. Our contribution lies in unifying sparsity enforcement, dynamic portfolio adjustment, and probabilistic uncertainty characterization—enabling joint quantification of constituent count, tracking deviation, and rebalancing risk. Empirical evaluation on S&P 500 tracking demonstrates significant reductions in both tracking error and turnover, precise control over holding count, and support for decision reliability verification.
📝 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.