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.

**Overview**

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 as a weighting approach.

We believe Diversification Weighting is both unique and often superior to conventional weighting approaches. Importantly, it is not constrained to a particular market, style or even asset class. 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.

Diversification, as a concept, has been around for a long time, but only in vague or indefinite terms. This paper circumscribes various inventions for diversification measurement, visualization and optimization, but a full treatment of all of these topics is beyond the scope of this article. The question we mean to address here is, does diversification work?

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.

**Defining Diversification: The Framework For Measurement And Optimization**

For something to be optimized, it must have a clear and definable objective function. Diversification, while ethereal by its nature, can be defined with precision and definiteness.

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.