Improved Tracking-Error Management for Active and Passive Investing

Challenge

While tracking-error management is rarely addressed in academic literature, it plays a crucial role in the real-world practices of portfolio management. Most portfolio managers are benchmarked against indices such as the S&P 500, Russell 2000, or MSCI World. For some, the goal is to closely track these benchmarks—a task that can be challenging, especially when the benchmarks include assets that are difficult or expensive to trade. In this scenario, the primary objective is to minimize tracking error. Other managers aim to maintain an active tilt while not deviating too far from their benchmark, making the control of tracking error their key focus. In both cases, an accurate estimation of the covariance matrix of numerous (excess) returns is essential. However, this task is complicated by in-sample optimism and the curse of dimensionality, where the number of stocks continues to grow while the available time-series data remains limited.

Solution

This study tackles various tracking-error challenges faced by portfolio managers, including the passive challenge of tracking or mimicking a benchmark and the active manager’s challenge of developing strategies with attractive risk-return properties while limiting deviation from a specified benchmark. The research demonstrates that utilizing advanced shrinkage estimators for the covariance matrix, particularly when combined with multivariate GARCH models, offers substantial improvements in tracking-error management compared to traditional sample covariance matrices and naive methods that avoid covariance matrix estimation altogether. Among the models evaluated, the dynamic DCC-NL model by Engle et al. (2019) emerges as the most effective overall.

Key Findings

To illustrate the study's main findings, Figures 1-3 present results from monthly rebalanced minimum-tracking-error portfolios, which include the 800 largest stocks from the benchmark. These portfolios are based on the value-weighted (VW) benchmark of the 1,000 largest CRSP stocks. The first step in computing the minimum-tracking-error portfolio is estimating the ex-ante tracking error implied by a given covariance matrix estimator. The following estimators are analyzed:

·       Sample: the sample covariance matrix

·       L: the linear shrinkage estimator of De Nard (2022)

·       NL: the nonlinear shrinkage estimator of Ledoit and Wolf (2022)

·       DCC-NL: the multivariate GARCH model of Engle et al. (2019)

·       VW-E: the value-weighted portfolio of the eligible universe (800 largest stocks), based on market cap

Figure 1 graphically illustrates the differences between ex-ante and ex-post tracking errors. The study reveals that shrinkage methods significantly outperform the sample covariance matrix and naive value-weighting, resulting in much lower tracking-error forecasting errors. Not only do these methods perform better on average, but they also exhibit a tightly condensed distribution of forecasting errors near zero. Furthermore, the violin plots highlight the well-known issue of in-sample optimism associated with the sample covariance matrix, where ex-ante tracking error is consistently lower than ex-post tracking error. Robust covariance matrix estimation via shrinkage methods nearly eliminates this in-sample optimism, a critical advantage for both active and passive managers.

Thanks to the enhanced ex-ante vs. ex-post tracking error management (reduction of forecasting error and in-sample optimism), shrinkage and multivariate GARCH models consistently and significantly reduce ex-post tracking errors, as evidenced by Figures 2 and 3.

The clear takeaway is that portfolio managers who seek to minimize or control tracking error—whether they are passive or active managers tied to a benchmark—must move away from relying on the sample covariance matrix or simple value-weighting techniques. Failing to adopt more sophisticated techniques like shrinkage estimators and multivariate GARCH models could raise questions about whether they have neglected their fiduciary duty to adhere to best practices.