TMA: I am pleased to introduce Dorian (Randy) Young as a contributor. Randy is an independent investment professional living in the Bay Area.
Power-Assisted Diversification: Not Too Concentrated and Not Too Diversified
Dorian (Randy) Young, CFA, CAIA
A number of years ago, as someone proficient with benchmark methodologies, I was asked by an investment measurement firm to consider a challenge they were having with their custom benchmarks. For their purposes, they found one set of their benchmarks too concentrated and another set too diversified. They had not found the “just right” benchmarks.
This is a story of how I developed a methodology that I entitled Power-Assisted Diversification, and how this methodology can be applied to other situations today. And like the story of Goldilocks, this one also has a happy ending.
The Challenge
The firm was using two common types of equity benchmarks. One was weighted by each stock’s market capitalization—thus, capitalization weighted (CW). In the other, all stocks were weighted the same—thus, equally-weighted (EW). They were using these custom benchmarks as part of a larger analytical package measuring hypothetical portfolios of stock recommendations, and from their perspective, many of their CW benchmarks were too concentrated (not diversified enough), while their EW benchmarks were too diversified.
For example, a large stock like General Electric (GE) and a stock 1/100th the size of GE would have the same weight in an EW benchmark, while in a CW benchmark, GE would be 100 times the weight of the smaller stock. Their perspective was that because GE was so much larger, it was more important than the small company, and so it should have more weight—but 100 times the weight was too much for their purposes.
In fact, in their CW benchmarks, the ratio of the largest stock weight to the smallest stock weight (L/S Weight Ratio) was commonly above 100 and in some cases exceeding 1000. The firm was more comfortable when the L/S Weight Ratios were in the single digits, a signal which served as a target metric.
The Solution
A weighting methodology that produced benchmarks between the too-diversified EW benchmarks and the not-diversified-enough CW benchmarks would seem to satisfy their objectives, especially since other common benchmark design characteristics (e.g. return, risk, turnover) did not need consideration in their design phase given their specific purposes. As there are an unlimited number of mathematical solutions to this situation, the key challenge was to find an elegant one that would be easy to communicate to their clients.
One mathematically elegant property that existed in both their EW and CW benchmarks was a size-weight ratio consistency property. Given the size ratio of two stocks, the benchmark weight ratio was always explicit and independent of the benchmark. For example, if the size ratio of two stocks was 10, then the CW benchmark weight ratio was always 10, the EW benchmark weight ratio was always 1, and the new benchmark weight ratio would always be some number N, where N was between 1 and 10. The desired N would produce L/S Weight Ratios generally in the single digits and certainly less than 20.
At that time, their CW benchmarks contained stocks that ranged in size from less than 
100 million, a ratio of 4000. This ratio served as an upper limit for existing L/S Weight Ratios and was used to calculate an upper limit of N. To keep the new L/S Weight Ratios below 20, the upper limit of N was calculated from the equation
An N was now needed that was greater than 1, less than 2.297, and would serve to make the communication as elegant and as understandable as possible. There was also motivation to choose an N that was not too close to 1, because as N approached 1, the result would be approaching the too-diversified EW benchmarks.
Given these mathematical constraints and design objectives, the obvious choice was N=2. This choice led to the following simple explanation: “If a stock is 10 times the size of another stock, then its benchmark weight will be just 2 times the weight of the other stock—for all stocks in the benchmark, adjusted proportionately.”
Revisiting the General Electric example above, where the two stocks had a size ratio of 100, their benchmark weight ratio would now be 4—not too large and not too small.
The firm found this solution attractive and went forward with this specific methodology.
The Specific Methodology
To implement the size-weight ratio consistency, the methodology takes each stock’s market capitalization (MC) and raises it the power of x, which is then used to weight the stocks.
As noted above, both the CW and EW benchmarks have this consistency: CW benchmarks are generated when x=1, and EW benchmarks are generated x=0; two special cases.
For our N=2 case,
and so weighting the stocks in the benchmarks by MC raised to the power of 0.30103 generated the desired result for the firm.
The General Methodology
Beyond the specific applications of x=0, x=1, or x=0.30103, there is a more general methodology where x is any number between 0 and 1. In this more general case, the weighting by MC raised to the power of x produces a benchmark more diversified than the CW benchmark but not as diversified as the EW benchmark. Hence, Power-Assisted Diversification.
Power-Assisted Diversification can be further generalized in that it not need be based on a stock’s market cap, but rather Power-Assisted Diversification can be applied to any benchmark weighting methodology, including fundamental weighting, GDP-weighting, etc.
In fact, not only can Power-Assisted Diversification be applied to a benchmark’s weight factor, but it can also be directly applied to any benchmark’s actual weights. Thus, it can be applied without the knowledge of the benchmark’s underlying weighting methodology.
Case Study: Power-Assisted Diversification Applied to the S&P 500 Index
To explore how Power-Assisted Diversification can impact a benchmark today, a case study of the S&P 500 Index as of November 30, 2012 is considered.
By sorting the stocks in the index by their weights from largest to smallest, these 500 stocks can be segmented into 10 groups with the same number of stocks in each group (50), thus forming 10 deciles. Decile 1 contains the 50 largest stocks, while Decile 10 contains the 50 smallest stocks. One simple measure of benchmark concentration is the total stock weight contained in Decile 1.
TABLE 1 contains the decile weights for the S&P 500 Index after Power-Assisted Diversification has been applied using several values for x. In the table, the first column lists the deciles, and the next five columns display the decile weights for these five values of x:
- x = 1, which is identical to the CW and the S&P 500 Index itself
- x = 0.5, which is identical to taking the square root of the MC
- x = 0.33333, which is identical to taking the cube root of the MC
- x = 0.30103, which was used for the firm above
- x = 0, which is identical to the EW and the S&P 500 Equal-Weighted Index
The top decile of the S&P 500 Index contains over 50% of the entire benchmark weight, and when this concentration is considered too high, Power-Assisted Diversification can lessen it. In the case of x = 1/2, the top decile concentration decreases to just over 25%, and when x = 1/3, this concentration is just under 20%. Using x = 0.30103 from above, this concentration is about 18%, and when x = 0, the result is the EW benchmark where the 10% concentration is at its lowest.
As part of the Power-Assisted Diversification application, a decision is required regarding which value of x to use. There are two approaches to choosing x: first, a specific value of x (e.g. 0.5) can be used at every rebalancing; second, a varying value of x can be used to achieve a specific result, such as a specific concentration of the top decile—requiring the value of x be backed into. An example of this second approach would be a requirement of the top decile to contain exactly 25% of the total index weight; the final column in TABLE 1 shows the backed-in value of x = 0.48358 in this example does produce the desired 25% concentration.
TABLE 2 displays how these values of x affect the stock weights for the five largest and five smallest stocks (top and bottom percentile) and impact the L/S Weight Ratio. The S&P 500 Index has a L/S Weight Ratio of 352 which declines to 19 when x = 1/2 and declines further to 7 when x = 1/3. This decline would be even more pronounced for all-cap benchmarks.
Many other measures of benchmark concentration can be calculated in this manner, and one of particular interest is the Concentration Coefficient as this calculation includes the weight of every stock in the benchmark. Mathematically, a benchmark’s Concentration Coefficient is the inverse of the sum of all weights squared. Conceptually, a benchmark’s Concentration Coefficient is the number of stocks in a concentration-equivalent EW benchmark. Thus, an EW benchmark’s Concentration Coefficient will equal the number of stocks in the benchmark; and as benchmarks move away from EW and become more concentrated, the Concentration Coefficient will decline.
For example, in TABLE 3 the Concentration Coefficients for the CW (x=1) and the EW (x=0) S&P 500 Index are 121 and 500, respectively. Thus, the CW S&P 500 Index has the same amount of concentration as an EW benchmark with 121 stocks, while the EW S&P 500 Index has the same amount of concentration an EW benchmark with 500 stocks—itself.
Power-Assisted Diversification produces Concentration Coefficients between the CW and EW values. For example, when x = 1/2, it is 351; when x = 1/3, it is 433.
Applications
While CW benchmarks are the most widely used benchmarks by investors today, the 21st century has seen a huge growth in non-CW benchmarks—“alternative equity index strategies”—being created and used, particularly in the proliferation of exchange traded funds (ETFs). These alternative index weightings (also known as “alternative beta”, “smart beta” and “better beta”) include EW, fundamental weighting, and many specific to individual ETF managers as more benchmark designers have become aware that the benchmark weighting methodology will have a meaningful impact on important benchmark characteristics, including return, risk, turnover, concentration, and exposure to size, value, and rebalancing premia.
How these characteristics could be impacted by Power-Assisted Diversification would be best understood by historical simulation research, although a rough expectation would be that they would generate results between any benchmark and an EW version of that benchmark. A more refined expectation could come from a 2011 research paper by Research Affiliates. In that paper, several benchmark weightings are compared, include CW, EW, and a method named Diversity-Weighted Equity Indexing that—like Power-Assisted Diversification—seeks a weighting between CW and EW. The Diversity-Weighted Equity Indexing methodology was described in 1995 and 1998 papers and developed by Dr. Robert Fernholz at INTECH using stochastic calculus—which would have rendered this approach too complex for the firm in this story. However, analysis of Diversity-Weighted Equity Index versus CW and EW in the paper can offer a hint as to how Power-Assisted Diversification may impact benchmark characteristics, including return and risk.
Benchmark designers, especially for ETFs, can use Power-Assisted Diversification to help shape the concentration, diversification, and other characteristics of their benchmarks. Additionally, any benchmark that has a high concentration of weight in a country, region, sector, industry, etc., can also have the Power-Assisted Diversification applied to lessen these concentrations.
Summary
Power-Assisted Diversification is a simple methodology that can be applied to any non-EW benchmark to shape its diversification and concentration profile and perhaps improve other characteristics, including return and risk. As more benchmarks with alternative weightings are being designed for the ETF industry, benchmark designers have the opportunity to use Power-Assisted Diversification in the development of these new indexes.
In the case of the firm in this story, Power-Assisted Diversification with x = 0.30103 was so attractive to them that they named me their honorary employee of the month—a happy ending.
Appendix A: Interactive Worksheet Tool
Table 4 reproduces Table 1 from above in interactive format where you can change the value of X (in the yellow-shaded cell) and view the resulting decile weights.