13.6 Summary

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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 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.