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1 February 2013 CFA Institute Journal Review

The Diversification Delta: A Higher-Moment Measure for Portfolio Diversification (Digest Summary)

  1. Florian B. Schweizer

Portfolio diversification is typically represented by a correlation matrix and thus does not account for higher moments of the distribution. The inclusion of variance only slightly improves the situation. The authors present a new measure of diversification that is based on all moments of the distribution while still being of practical use.

What’s Inside?

The authors introduce the diversification delta as a new measure of diversification. Their measure is based on the concept of information entropy, which is a measure of uncertainty in a distribution. They compare the diversification delta with the correlation coefficient and find that the diversification delta is a strong diversification analysis tool.

How Is This Research Useful to Practitioners?

The authors apply the concept of information entropy in a portfolio management context. Developed by the physicist Claude Shannon and first applied to communication problems, it is a measure of how much uncertainty can be attributed to a whole statistical distribution. Thus, the concept goes beyond traditional finance theory and may be more reflective of reality.

"Diversification delta" is the authors’ alternative diversification measure, and they define it as the reduction in entropy (i.e., uncertainty) on a relative basis when a number of single assets are combined in a portfolio. It can take values between 0 and 1. The authors illustrate that the diversification delta will lead to a different portfolio structure for active management.

How Did the Authors Conduct This Research?

First, the authors compare the diversification delta with the correlation measure. They find that using the diversification delta results in more weight being given to uncorrelated stocks.

Second, they calculate the diversification delta for a case in which wealth is spread across assets. They find that most of the diversification gains can be harvested with the first 30 assets. As more assets are added, the diversification delta slowly increases to 1. This increase is explained by the fact that in adding more assets, idiosyncratic risk can be diversified away, leaving only systematic risk.

Finally, the authors compare the diversification delta with the correlation of infrastructure investments with varying sensitivities to the business cycle. The calculations indicate that after the Lehman Brothers bankruptcy, the diversification delta and the correlation coefficient tell very different stories. In particular, the diversification delta shows a decrease in diversification during a bust while the correlation measure remains flat. This finding indicates that the actual level of diversification is misstated by the correlation measure.

The examples are based on randomly generated Gaussian datasets. The authors determine the correlation coefficient ex ante. For each value of the correlation coefficient, they calculate the diversification delta. They conduct 1,000 iterations to avoid selection bias and present averaged data. Data for the infrastructure investments are from the Dow Jones Brookfield Global Infrastructure Index for the period 2002–2010.

Abstractor’s Viewpoint

Information entropy has turned out to be a powerful concept, and its application to finance seems to be promising. The authors clearly show how the simple correlation measure can lead to misleading conclusions in a dynamic environment. Thus, it is important to look for other indicators on which to base decisions. Although they are not as easy to understand, information entropy and its derivative, the diversification delta, reveal how to deal with the situation more accurately.

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