Formulations to select assets for constructing sparse index tracking portfolios

TL;DR

This work develops combinatorial, quadratic-binary formulations for the first step of constructing sparse index-tracking portfolios, unifying market-cap top-tier selection and correlation-balance (CBS) methods through tunable parameters in a correlation-distance framework. The core problem minimizes a balance between centrality and dissimilarity among candidate assets while forcing exactly $M$ assets to be selected from the top $H$ candidates, with fixed inclusion of the top $N$ assets. An extension to multi-stage selection enables iterative refinement across stages, producing a final set trimmed to at most $M^{*}$ assets. Numerical experiments on TOPIX and S&P500 show that multi-stage CBS-like schemes (notably $E6$) deliver superior tracking performance with favorable bias and variance across multiple horizons, underscoring the practical benefits of sparse yet well-balanced asset selection for index tracking.

Abstract

In this paper, we study asset selection methods to construct a sparse index tracking portfolio. For its advantage over full replication portfolio, the concept of sparse index tracking portfolio has significant attention in the field of finance and investment management. We propose useful formulations to select assets for sparse index tracking portfolio. Our formulations are described as combinatorial optimization problems, and they can yield various asset selection methods, including some existing methods, by adjusting the values of parameters. As a result, the proposed formulations can provide a well-balanced asset selection to create successful sparse index tracking portfolios. We also provide numerical examples to compare the tracking performance of resulting sparse index tracking portfolios.

Figures (2)

• Figure 1: Residuals $\varepsilon^{p}_{1}$ for cumulative return as a function of $p$ (TOPIX)
• Figure 2: Residuals $\varepsilon^{p}_{1}$ for cumulative return as a function of $p$ (S&P500)