# Diversification

– Weighted

Performance

Evaluation

Investors hold diversification, long a bedrock of investor prudence, in high esteem. This is curious, as diversification is both ubiquitous and misunderstood. Diversification is typically associated with asset allocation, but fundamentally, diversification is a weighting strategy. We have formalized Diversification Weighted Performance Evaluation.

## DIVERSIFICATION– WEIGHTED PERFORMANCE EVALUATION

**A new approach to building a portfolio**

Using the components of the S&P 500 Index, we investigate an alternative, process-driven methodology for index construction and optimization called Diversification Weighting. We find that when reconstituting the S&P 500 universe from January 1996 through May 2014, using Diversification Optimization, we achieved annualized outperformance of 427 basis points against the capital-weighted benchmark, and achieved favorable results when compared against other weighted approaches applied to the same investments universe. We also introduce diversification measurement for the purpose of context and clarity underpinning Diversification Optimization.

**2023 Update and Revision:**

In May 2023, this research was updated through April 4th 2023. We have left all of the text the same, including any references to performance in the text. However, the majority of the charts and graphics have been updated. Notes on the update and revision are available on request. This brings the analysis to 27 years and the performance of the Diversification Weighted Strategy outperforming the Capitalization weighted strategy by 3.83% per year, net estimated fees and expenses.

**Overview**

In addition, we show portfolio diversification to provide material performance improvements to both risk and return. Part of the attraction to smart-beta euphoria is an intuition that repeatable investment performance is systematic. Investors prefer a repeatable process and especially a process rooted in things that are rather simple and easily understood.

In essence, diversification increases for every equally weighted, uncorrelated asset added to a portfolio. When we put the portfolio in a geometric space, such additions add a dimension, which may measure the diversification therein. When we weight assets in a manner consistent with maximizing the portfolio dimensionality, perhaps subject to a utility function, the portfolio is said to be diversification weighted.

**table of Contents**

**Defining Diversification: The Framework**

For Measurement And Optimization

For Measurement And Optimization

For readers interested in diversification or how the results are obtained, we need to set some foundations in the geometric modeling. The ensuing geometric modeling framework provides the background for both diversification measurement and optimization.

Asset correlations are measured and assets are projected to a space that maps the measured correlations to the angles separating one asset from another, so that assets with higher correlations have a more acute angle separating the vectors. Assets sharing zero correlation are mapped to a 90° angle and assets with a -1 correlation are mapped as diametric opposites having a 180° angle.

Imagine a three-asset portfolio; each asset has zero correlation to the other two assets. Each asset literally takes the portfolio in a new direction. Note that all available dimensions are used as degrees of freedom to convey the relativity of the assets. It is correlation in every dimension. We refer to this as having a relativity domain space. Imagine that we introduce a fourth uncorrelated asset. Where would it fit? It cannot fit in 3-D without creating some distortion. There is not enough dimensionality to hold it. We need to introduce an additional dimension, which may accurately represent the fourth uncorrelated asset.

In mathematics, there is no limit to dimensionality. So it is with portfolios. Diversification has no upper boundary; there can always be more. Any portfolio has an ambient dimensionality equal to the number of assets in it. The ambient dimension compacts to the spanning dimension after ringing out all of the redundant information caused by disparate weighting and positive correlations. The spanning dimension thus contains all of the portfolio’s information. It is a bit like a .jpeg at 100 percent quality. The dimension of the portfolio can be further reduced when associated with a confidence interval, such that it captures X percent of the portfolio. That dimensionality is called an “effective dimension.” It is like viewing a .jpeg at 90 percent quality; it captures the essence of the image, but does not require same amount of data. Thus, the lowest dimensions 1,2,3 … are the most pervasive and measure systematic diversification and risk exposure.

This framework explains how portfolios sprawl across multiple dimensions. The primary dimensions give the systematic exposure on the left and it integrates with the idiosyncratic exposures on the right. Each new dimension acts as a source of diversification unto the portfolio. The blue shadow line gives the investor an understanding of what perfect diversification (given the investment quantity) would look like. The portfolio would mimic

this line if each asset were equally weighted and uncorrelated. The more exponential the portfolio graph, the greater the risk, as such exponential graphs are indicative of superficial diversification in asset quantity only.

We apply this diversification framework to weighted as well as unweighted portfolios. When the portfolio is already weighted, the framework gives the measurements of diversification as well as the visualization.

When the framework is applied to a weighted portfolio, the given weights for each asset are modeled as the vector lengths for each asset. Intuitively, the weight extends the assets in its given direction (Figure 2).

In essence, it is like a multidimensional pie chart containing relativity data.

The resulting level of symmetry induced by the model structured by the assets’ weights and relationships provides a visual explanation of portfolio diversification. Portfolio balance is manifest as three-dimensional balance in the True Diversification model framework. True Diversification accounts for the diversification directly and captures both the systematic and nonsystematic perspectives (Figure 3). This process can be applied to any asset or any mix of assets, including disparate assets classes, or mixing of various instruments. We recycle this geometric modeling process to optimize diversification.

**Diversification Optimization:**

The Theoretical Foundation

The Theoretical Foundation

The portfolio thus has two elementary inputs: the relationship of each asset to every other asset in the portfolio, as measured by its correlation matrix and modeled by the relative angles; and the utility function for every asset measuring the relative attractiveness of each asset and modeling the magnitude of each vector. These are the only inputted asset variables.

As Figure 5 shows, there are two types of assets: those that combine to create the frame (a) and those trapped inside (b). Efficient assets grow the portfolio in their respective directions. The interior assets are doing an inefficient job of growing the portfolios in their respective direction and are dominated by the assets comprising the diversification frontier. Interior assets are inefficient and normally discarded unless subject to a constraint.

While the 3-D graphics are useful for understanding the structure of the portfolio, the allocation mathematics may take place in a higher-dimensional space. Geometric principles of angles, lengths, convex hulls and volumes work in any dimension. As a result, the higher the dimensionality of the space, the more inclusive the allocation to marginally efficient assets.

When we model the asset vectors with all-equal lengths, the resulting portfolio is maximizing diversification. Every asset is assured a spot on the efficient frame, and the asset’s weight is exactly proportionate to the asset’s relative uniqueness.

Now that the portfolio has a tangible body, we compute the volume and assign weights to every asset based on the pro rata volume contribution. The equation is simple: Volume = Money (Figure 6).

Highly correlated assets crowd one another out fighting for the space in the portfolio. Such highly correlated assets are also—intuitively— increasingly sensitive to small differences in their utility function, whereas a more uncorrelated asset in an otherwise-correlated portfolio will be rather insensitive to changes in that asset’s utility function. If one could find a historically negatively correlated asset, then it too would be relatively insensitive to the magnitude of its utility, if it had positive utility.

**Diversification Weighted 500:**

Performance Refinements

Performance Refinements

Collect the historical index component data that were in the index (S&P 500) at the start of the observation period. The data source includes delisted assets to eliminate survivorship bias.

**Investment Policies**

For each optimization’s anniversary date on Aug. 1, we would select two sample periods for the historical data. The first sample period was the previous one year; the second sample period was the previous 10-year period. The baseline risk, return and correlation values are calculated for every asset as the average of these two periods. We believe these two periods merge a positive predictive momentum signal arising from the one-year data with the stock’s performance over at least one full market cycle in the long sample. We tested staggered multiple optimization intervals and found no material effect. We applied the sampling procedure uniformly over all sectors and years.

We designate the optimization dimension for each portfolio in an effort to attain a comfortable balancing point between systematic and idiosyncratic diversification. We were also cognizant to set the value high enough that it would more fully allocate to the natural diversification derived from the process and to minimize the application of the constraints. Optimizations taking place in lower dimensions better diversify the portfolio from a perspective of systematic influence, and optimizations taking place in higher-dimensional space more effectively diversify the portfolio from an idiosyncratic perspective.

We apply constraints to all assets. An asset constraint was set at 3 * 1/N (N = Asset Count). For a sector with 20 stocks, N is 5 percent, and the maximum constraint is 15 percent. We applied a similar minimum constraint. While these constraints denied the performance test, the benefit of the diversification/utility efficiency logic, the constraints ensured an allocation to all 500 stocks, creating a more relevant comparison focused on the weighting approach and less on the logic of diversification efficiency.

**Contrast To Other**

Weighting Approaches

Weighting Approaches

Diversification Weighted

Equal Weighted

Fundamental Weighted

Market Cap

Revenue Weighted

Earnings Weighted

Volatility Weighted

Funds of this ilk have seen recent success both in performance and garnering new assets. Now, 16 ETFs collectively manage $16 billion in funds from a strategy that did not exist before 2004. As the nature of indexation becomes more complex, the ETF market is now more accepting of smart-beta strategies. This will likely continue, but when put in context with the $2 trillion of global capital Markowitz estimated as being managed in accord with mean-variance optimization (Markowitz, private conversation, 2010), one can see the ETF space is nascent in its adoption of more complex systematic index products.