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Sparse Index Tracking: Simultaneous Asset Selection and Capital Allocation via $\ell_0$-Constrained Portfolio

Eisuke Yamagata, Shunsuke Ono

TL;DR

This paper proposes a new problem formulation of sparse index tracking using an inline-formula-norm constraint that enables easy control of the upper bound on the number of assets in the portfolio and develops an efficient algorithm for solving this problem based on a primal-dual splitting method.

Abstract

Sparse index tracking is a prominent passive portfolio management strategy that constructs a sparse portfolio to track a financial index. A sparse portfolio is preferable to a full portfolio in terms of reducing transaction costs and avoiding illiquid assets. To achieve portfolio sparsity, conventional studies have utilized $\ell_p$-norm regularizations as a continuous surrogate of the $\ell_0$-norm regularization. Although these formulations can construct sparse portfolios, their practical application is challenging due to the intricate and time-consuming process of tuning parameters to define the precise upper limit of assets in the portfolio. In this paper, we propose a new problem formulation of sparse index tracking using an $\ell_0$-norm constraint that enables easy control of the upper bound on the number of assets in the portfolio. Moreover, our approach offers a choice between constraints on portfolio and turnover sparsity, further reducing transaction costs by limiting asset updates at each rebalancing interval. Furthermore, we develop an efficient algorithm for solving this problem based on a primal-dual splitting method. Finally, we illustrate the effectiveness of the proposed method through experiments on the S&P500 and Russell3000 index datasets.

Sparse Index Tracking: Simultaneous Asset Selection and Capital Allocation via $\ell_0$-Constrained Portfolio

TL;DR

This paper proposes a new problem formulation of sparse index tracking using an inline-formula-norm constraint that enables easy control of the upper bound on the number of assets in the portfolio and develops an efficient algorithm for solving this problem based on a primal-dual splitting method.

Abstract

Sparse index tracking is a prominent passive portfolio management strategy that constructs a sparse portfolio to track a financial index. A sparse portfolio is preferable to a full portfolio in terms of reducing transaction costs and avoiding illiquid assets. To achieve portfolio sparsity, conventional studies have utilized -norm regularizations as a continuous surrogate of the -norm regularization. Although these formulations can construct sparse portfolios, their practical application is challenging due to the intricate and time-consuming process of tuning parameters to define the precise upper limit of assets in the portfolio. In this paper, we propose a new problem formulation of sparse index tracking using an -norm constraint that enables easy control of the upper bound on the number of assets in the portfolio. Moreover, our approach offers a choice between constraints on portfolio and turnover sparsity, further reducing transaction costs by limiting asset updates at each rebalancing interval. Furthermore, we develop an efficient algorithm for solving this problem based on a primal-dual splitting method. Finally, we illustrate the effectiveness of the proposed method through experiments on the S&P500 and Russell3000 index datasets.
Paper Structure (22 sections, 27 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 22 sections, 27 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions and Paper Organization
  4. Preliminary
  5. Tracking Error Measure
  6. Transaction Costs
  7. Proximal Tools
  8. Primal-Dual Splitting Method
  9. Proposed Method
  10. Problem Formulation
  11. Algorithm
  12. Simultaneous Asset Selection and Capital Allocation
  13. Computational Complexity
  14. Experiments
  15. Dataset and Settings
  16. ...and 7 more sections

Figures (3)

  • Figure 1: The graph of $\mathop{\mathrm{MDTE}}\nolimits [\mathrm{bps}]$ across different sparsity on S&P500, 2012 - 2017. The vertical axis indicates $\mathop{\mathrm{MDTE}}\nolimits [\mathrm{bps}]$ and the horizontal axis indicates the sparsity. To avoid parameter tuning on $\lambda$ (which controls sparsity in LAIT), we fixed the value of $\lambda$ per data point. Therefore, the exact sparsity of per training period differs. The sparsity of Proposed[P, ETE] ($K_1$), NNOMP-PGD and $\ell_0$-ADMM is adjusted to be the same as that of LAIT. As for Proposed[T, ETE], $K_2 = K_1$. Parameter $K$ in the graph indicates the sparsity of the portfolio at $n = 10$.
  • Figure 2: The graph of the investment simulation on S&P500, 2012 - 2017. The vertical axis indicates the normalized accumulated return and the horizontal axis indicates the sparsity of the portfolio ($K_1,K=40$ and $K_2 = K_1/3$). The graph indicates an investment simulation based on one of the data points of Fig. \ref{['fig:MDTE']}. The initial capital is $\$10000$.
  • Figure 3: The graph of the proposed algorithm's ([P,ETE]) convergence behavior on the S&P500 (2012 - 2017) dataset ($K_1 = 40$, Init. A). The vertical axis indicates $\frac{\|{\mathbf w}^{(k)} - {\mathbf w}^{(k-1)}\|_2}{\|{\mathbf w}^{(k-1)}\|_2}$ and the horizontal axis indicates the number of iterations.