Asset pre-selection for a cardinality constrained index tracking portfolio with optional enhancement

Technical Summary: The Impact of Asset Pre-selection for a Cardinality Constrained Index Tracking Portfolio with Optional Enhancement

Problem Definition
The paper addresses the Cardinality Constrained Index Tracking Portfolio (CCITP) problem, which involves selecting a subset of $n$ assets from a larger universe of $M$ index constituents to replicate the risk and return of a market index (specifically the S&P 500). While full replication is accurate, it incurs high management costs; conversely, selecting a small subset reduces costs but increases tracking error. The problem is computationally difficult because it requires simultaneously choosing the asset set and optimizing their weights, a task that historically relied on heuristics for large cardinalities. Furthermore, the paper distinguishes between pure index tracking (minimizing tracking error) and enhanced index tracking (seeking returns exceeding the index while minimizing tracking error).

Methodology
The authors propose a novel two-stage approach that separates asset selection from weight estimation, contrasting with literature that often treats them as a joint optimization problem.

1. Stage 1: Asset Pre-selection: The authors employ statistical regression procedures to identify the optimal set of $n$ assets before weight optimization. They evaluate eight distinct pre-selection procedures based on two directions (Forward Selection and Backward Elimination) and two regression objectives (Ordinary Least Squares [OLS] and Least Absolute Deviation [LAD]), each tested with and without a regression constant.

   • Forward Selection: Iteratively adds the asset that maximizes explanatory power (adjusted $R^2$ for OLS, minimized Mean Absolute Deviation for LAD) of the index return or residual variation.
   • Backward Elimination: Starts with the full universe and iteratively removes the asset with the least explanatory power (lowest t-value for OLS, smallest absolute coefficient for LAD).
   • Enhancement: A target enhancement ($\lambda$) can be added to the index return during selection to facilitate enhanced tracking.

2. Stage 2: Weight Optimization: Once the asset set is fixed, weights are optimized to minimize a specific objective function over an in-sample period ($T_1$). The objective function minimizes the squared Root Mean Square Tracking Error (RMSE) plus a penalty term for the difference in mean returns between the portfolio and the index. A quasi-Newton method (BCONF) is used for this non-linear optimization.

3. Evaluation Framework: The study utilizes a moving time-window approach on daily S&P 500 data from January 3, 2005, to December 29, 2023. The process involves selecting a portfolio over an in-sample period ($T_1$) and evaluating its out-of-sample performance over a subsequent period ($T_2$). Transaction volume is approximated by the sum of absolute changes in asset weights between rebalancing periods.

Key Results

• Pre-selection Performance: Among the eight procedures, Backward Elimination using OLS without a constant (BE-OLS(n)) consistently ranked highest across in-sample tracking error, out-of-sample tracking error, and transaction volume. Forward selection procedures generally yielded higher tracking errors, and LAD-based procedures performed poorly, particularly in backward elimination, likely due to degeneracy in linear programming with large asset sets.
• Cardinality and Tracking Error: The study experimentally demonstrates that out-of-sample tracking error decreases according to the inverse square root of the cardinality ($1/\sqrt{cardinality}$). Specifically, the relationship is modeled as $TE \approx \theta n^{-\omega}$, where $\omega \approx 0.58$ for out-of-sample errors, supporting the approximation that error is proportional to $1/\sqrt{n}$.
• Volatility Dependence: Out-of-sample tracking error is strongly correlated with index volatility. However, as cardinality increases, the portfolio's dependence on index volatility decreases, leading to greater stability.
• Enhanced Tracking: For portfolios seeking enhanced returns, the results differ significantly from pure tracking. While larger cardinalities reduce tracking error, they do not necessarily maximize excess returns. The study finds that cardinalities in the range of 10 to 20 are most effective for generating enhanced returns.
• Sensitivity Analysis: Longer in-sample periods ($T_1$) generally reduce tracking error and transaction volume. Shorter out-of-sample evaluation periods ($T_2$) reduce tracking error but increase transaction volume due to more frequent rebalancing. For enhanced tracking, a 3-year in-sample period and 6-month out-of-sample period yielded the best balance.

Significance and Claims
The paper claims a distinct contribution to the literature by proposing a separation of the asset selection and weight estimation stages. The authors argue that this approach removes the computational difficulty associated with jointly optimizing asset sets and weights, allowing for the analysis of larger cardinalities (up to 100 assets) without relying on complex heuristics.

The study provides empirical evidence that:

1. Asset pre-selection via statistical regression is a viable and effective alternative to joint optimization heuristics.
2. The relationship between tracking error and cardinality follows a predictable power law ($1/\sqrt{n}$), offering a practical rule of thumb for portfolio construction.
3. There is a trade-off between tracking accuracy and return enhancement; while large portfolios minimize tracking error, smaller portfolios (cardinality 10–20) are more effective for generating excess returns.

The authors remain modest regarding the universality of their findings, noting that the effectiveness of pre-selection depends on the specific statistical properties of asset returns (e.g., fat tails) and that the "optimal" portfolio is not unique, as different high-performing procedures often select significantly different asset constituents.