Weighting scheme design: what investors need to know

Posted by TEBI on April 1, 2022

Weighting scheme design: what investors need to know





As explained in my April 13, 2016, article, not all systematically managed funds are created equal. In a related January 31, 2019, article, I showed that this is true even if they are in the same asset class. And it would be true even for funds that target the same factors (such as size, value, momentum and profitability/quality). How a fund designs its eligible universe, what metrics (such as price-to-book, price-to-earnings, price-to-cash flow and price-to-sales for the value factor) are used, how often it rebalances and how it trades can have significant impact on results and tax efficiency. Strategies that are designed poorly may not be able to translate research insights about the factor premiums into real-world value-adds. 

In their February 2022 study Weighting for the Right One: Weighting Scheme Design for Systematic Equity Portfolios, Wei Dai, Namiko Saito and Gigi Wang examined various  weighting schemes frequently used in the construction of systematic equity strategies to determine their impact on turnover costs, tracking variance and returns. They began by noting: “A key aspect of a well-designed weighting scheme is the link between security weights and market prices. Current market prices reflect the latest news and aggregate expectations of market participants, providing real-time information about expected return differences across securities. By closely tying weights to prices, investors can effectively consider up-to-date information and target higher expected returns. … Without a close tie to market prices, continuous price movements can drive up turnover and the costs of maintaining desired security weights. Price-based weighting schemes help limit those costs by continuously and gradually adjusting security weights as prices change.” 

They examined five non-price-based weighting schemes and their performance:

  • Equal weighting: All firms were held at the same weight.
  • Rank weighting: Firms were ranked from 1 to N, with N being the total number of firms, based on independent univariate sorts on their market capitalisation (from large to small caps), relative price (from growth to value) and profitability (from low to high)—smaller, deeper value or more profitable firms had higher ranks, while larger, growthier or less profitable firms had lower ranks. They re-ranked the firms according to their average ranks across those three independent sorts, which made for gradual changes in the weights across firms and avoided abrupt changes. Firms were then weighted in proportion to their final ranks.
  • Z-score weighting: They first calculated each firm’s market capitalisation, relative price and profitability z-scores, defined for each dimension as the corresponding characteristic’s raw value minus its cross-sectional average divided by its cross-sectional standard deviation. The z-scores were then transformed into a value between 0 and 1 using the cumulative distribution function of the standard normal distribution. The transformed z-scores were bigger (closer to 1) for smaller, deeper value or more profitable firms, and smaller (closer to 0) for larger, growthier or less profitable firms. Finally, firms were weighted in proportion to the product of their transformed z-scores, which still ranged from 0 to 1.
  • Inverse volatility weighting: Firms were weighted in proportion to the inverse of their return volatility, where volatility was calculated as the standard deviation of daily returns over the trailing 60 trading days (with a minimum of 20 trading days)—firms with lower recent volatility were held at larger weights, while more volatile names were assigned smaller weights.
  • Fundamental weighting: Firms were weighted in proportion to their “economic footprint,” measured as the sum of book equity, sales and cash flow in the latest fiscal year.

Their sample period was July 1974 to December 2019. Following is a summary of their findings:

  • All simulations outperformed the market, with annualised excess returns ranging from 1.7 percent for fundamental weighting to 6.0 percent for z-score weighting. The annualised volatility of the fundamental weighted and the inverse volatility weighted simulations was about 16 percent, or slightly higher than that of the market (15.5 percent), while the volatility of the other three weighting schemes was around 20 percent. The range of tracking error against the market was wide: over 10 percent annualised for equal weighting, rank weighting and z-score weighting, followed by 7.6 percent for inverse volatility weighting, with the lowest at 4.5 percent for fundamental weighting.
  • All weighting schemes tilted toward small cap and value securities.
  • Breaking the link between market prices and security weights can lead to extreme outcomes, with the smallest stocks dominating portfolios, having significant implications for turnover and trading costs—both of which are higher in smaller stocks. For example, for every dollar invested in the z-score weighted strategy, more than 90 cents were in stocks in the bottom 4 percent of the market capitalisation. For rank weighting, the allocation to micro caps was similarly high, above 80 percent, followed by around 65 percent for equal weighting and slightly above 50 percent for inverse volatility weighting. Similarly, value stocks, defined as the bottom half by market capitalisation when ranked on price-to-book, made up more than 90 percent of the rank weighted and z-score weighted simulations, and around 75 percent of the other three simulations.
  • Rank and z-score weighting schemes had negative exposure to the profitability factor, meaningfully lower than that of the market despite an explicit focus on the profitability premium—smaller cap and deeper value firms tend to have lower profitability. Thus, a naive combination of factor ranks or z-scores does not effectively account for such interactions between premiums and may lead to offsetting tilts.
  • Fundamental weighted strategies outperformed the market due to their indirect exposure to securities with higher expected returns but did not add value over their tilts to the premiums.
  • While excess returns were positive for the other weighting schemes and reliably so for rank weighting, z-score weighting and inverse volatility weighting, their reliable alphas disappeared once microcaps were excluded from the eligible universe—suggesting that the paper performance of those strategies is unlikely to survive in the real world once trading costs are accounted for.
  • At the overall portfolio level, fundamental weighting had the lowest one-way turnover of 20 percent per year due to the positive correlation between market capitalisation and company fundamentals. The average annual one-way turnover for other weighting schemes was meaningfully higher, ranging from 33 percent for equal weighting to 54 percent for z-score weighting. The problem of higher turnover is compounded by the greater allocation to microcap stocks and their higher trading costs.


As Dai, Saito and Wang noted, because of their significant overweighting to microcap stocks, “in periods of market stress and heightened volatility, non-price-based weighting schemes may face even more challenges in getting the trades done at a reasonable price within an acceptable time period.” They added: “While excluding micro caps has little impact on long-term market returns, it meaningfully reduces the historical performance of all weighting schemes considered thus far over the same sample period.”

Dai, Saito and Wang also examined how various market-cap weighting schemes impacted exposures to the size, value and profitability factors often targeted by investors.

  • Rank times market cap: Firms were weighted in proportion to their final ranks (defined the same as above for rank weighting) multiplied by their market capitalisation.
  • Z-score times market cap: Firms were weighted in proportion to the product of their transformed z-scores (defined the same as above for z-score weighting) multiplied by their market capitalisation.
  • Integrated core: Firms were independently sorted on their market capitalisation, relative price and profitability. The intersections of this three-way sort formed groups of firms with similar characteristics. For example, mega cap firms with higher relative price and higher profitability formed one group, while mega cap firms with higher relative price but lower profitability formed another group. Within each group, firms were held in proportion to their market capitalisations. Each group was weighted in proportion to its total market capitalisation times a multiplier, where the multiplier effectively controlled the group’s over- or underweight relative to the market and gradually increased as they moved from groups with lower expected returns to those with higher expected returns. The multipliers used in the study were designed to achieve a moderate and balanced emphasis on the size, value and profitability premiums while accounting for the interactions among them.

They found that “tying security weights to market prices helps to avoid extreme and uncontrolled deviations from the market, but to a varying degree. Among the price-based weighting schemes, integrated core leads to a more balanced emphasis on the premiums and more measured over- and underweights across holdings, while rank-times-market cap is a close second on these metrics.” In addition, the integrated approach reduced turnover and therefore trading costs. And by better accounting for interactions among premiums, the integrated approach allowed for better management of exposure to different premiums. They added: “Using the integrated core approach, we can assign multipliers with more moderate gradients across small cap groups, thus reducing the expected turnover and costs associated with maintaining the desired weights as small cap names migrate across groups.”


Investor takeaways

Dai, Saito and Wang’s findings demonstrate the importance that should be attached to looking “under the hood” of a factor-based strategy before investing, being careful to look beyond simulated performance. Weighting scheme design is among the many aspects to consider when assessing systematic strategies. Due diligence should include a deep dive into the fund construction rules chosen by a manager to gain exposure to the desired factors in order to discover if the strategy will be able to deliver the premiums that academic research has identified after consideration has been given to real-world trading costs. Designs that do not include market-cap weighting can lead to extreme outcomes in terms of design, with significant overweight to microcap stocks, which has significant implications for trading costs. In addition, an integrated approach provides a more effective way to manage exposures to the desired factors. And finally, a patient trading strategy that accepts random tracking error (pure indexing strategies demand liquidity as the price paid for minimising tracking variance) can minimise trading costs. 


For informational and educational purposes only and should not be construed as specific investment, accounting, legal, or tax advice.  Certain information is based upon third party data which may become outdated or otherwise superseded without notice.  Third party information is deemed to be reliable, however its accuracy and completeness cannot be guaranteed. By clicking on any of the links above, you acknowledge that they are solely for your convenience, and do not necessarily imply any affiliations, sponsorships, endorsements or representations whatsoever by us regarding third-party websites. We are not responsible for the content, availability or privacy policies of these sites, and shall not be responsible or liable for any information, opinions, advice, products or services available on or through them. The opinions expressed by featured authors are their own and may not accurately reflect those of the Buckingham Strategic Wealth® or Buckingham Strategic Partners®, collectively Buckingham Wealth Partners. Neither the Securities and Exchange Commission (SEC) nor any other federal or state agency have approved, determined the accuracy, or confirmed the accuracy of this article. 


LARRY SWEDROE is Chief Research Officer at Buckingham Strategic Wealth and the author of numerous books on investing.



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