Diversification and the Distribution of Portfolio Variance, Part 2: Volatility Stability as a Measure of Diversification

Abstract

We introduce a new framework for understanding portfolio diversification that provides a coherent basis for comparing methodologies and offers a new approach to portfolio construction. The primary argument is that measures of diversification based only on a covariance matrix are ambiguous because in such a risk setting only the overall portfolio variance is of any import. To resolve this we propose that the purpose of diversification is most helpfully viewed as reducing the variance of portfolio variance itself, which in turn is only meaningful when one accounts for excess kurtosis. Connecting diversification and the variance of variance provides a natural extension to the ubiquitous mean-variance approach. Examples are provided to demonstrate the intuitive nature of portfolios that maximize diversification through minimizing kurtosis. Furthermore, we introduce portfolio dimensionality as a transformation of kurtosis that allows us to interpret diversification in terms of an equivalent number of assets with independent and identically distributed (IID) returns.