\( \newcommand{\bm}[1]{\boldsymbol{#1}} \newcommand{\textm}[1]{\textsf{#1}} \def\T{{\mkern-2mu\raise-1mu\mathsf{T}}} \newcommand{\R}{\mathbb{R}} % real numbers \newcommand{\E}{{\rm I\kern-.2em E}} \newcommand{\w}{\bm{w}} % bold w \newcommand{\bmu}{\bm{\mu}} % bold mu \newcommand{\bSigma}{\bm{\Sigma}} % bold mu \newcommand{\bigO}{O} %\mathcal{O} \renewcommand{\d}[1]{\operatorname{d}\!{#1}} \)

13.6 Summary

Active and passive strategies play a major role in the financial investment arena, both having theoretical and practical justifications albeit stemming from opposite views of the markets. Some key takeaways include:

  • Passive investing methods seek to avoid the fees and limited performance that may occur with frequent active trading.

  • Index tracking is the mainstream approach for passive investment and simply tries to mimic an index, based on the assumption that the market is efficient and cannot be beaten.

  • In the current markets, there are thousands of financial indices that cover a wide range of asset classes, sectors, and regions (e.g., the S&P 500). There are even more ETFs that precisely track any given index and investors can directly trade them (e.g., there are hundreds of ETFs that track the S&P 500 index).

  • Sparse index tracking is closely related to a fundamental problem in statistics called sparse regression. Its goal is to approximate an index but using a small number of active assets. The mathematical problem formulation requires a tracking error measure and a mechanism to control the sparsity level.

  • A variety of tracking error measures can be used for index tracking, such as the \(\ell_2\)-norm tracking error (13.11), the downside risk (13.12), the \(\ell_1\)-norm version (13.13), and the Huberized robust version (13.14), among others.

  • Many algorithms have been proposed for index tracking capable of controlling the sparsity or cardinality level. The iterative reweighted \(\ell_1\)-norm method in Algorithm 13.1 provides the best combination of tracking error while controlling the sparsity at a low computational cost.

  • In practice, deciding the sparsity level is typically done by trial and error while tuning some hyper-parameter in a laborious and computationally demanding way. The recently proposed FDR-controlling index tracking method is able to automatically determine the sparsity based on statistically sound hypothesis testing techniques.