# I Dream of Gini

The Gini Coefficient is used widely by economists as a standardized measure to compare the degree of income inequality across different countries. It is an alternative measure of standard deviation because it better captures the degree of imbalance caused by significant outliers. The Gini Coefficient ranges between 0 and 1, where 0 would represent perfect equality, and 1 would represent a situation where the entire range of values is concentrated in one measurement.The statistic is calculated by measuring the area between the Lorenz Curve of the cumulative distribution and the line of perfect equality. The greater the area, the higher the Gini Coefficient.

More recently, the Gini Coefficient has been used as a measure of portfolio concentration or diversification. Typically, the practice has been to look at the distribution of portfolio weights in percentage terms versus an equal weight portfolio. This makes some conceptual sense but presumes that an equal weight portfolio represents the greatest form of diversification. Roncalli takes a more nuanced approach and suggests using an equal risk contribution (ERC)- or risk parity- portfolio as the benchmark for calculating the Gini. A portfolio with equal risk contributions (ERC) has the benefits of efficient risk weighting and also the use of all available portfolio assets. An ERC portfolio has a risk that lies somewhere between the Global Minimum Variance Portfolio and an equal weight portfolio. This is an attractive benchmark for comparison and makes conceptual sense for the Gini measurement. The calculation would entail creating a cumulative distribution of marginal risks and comparing to an ERC as the perfect equality line.

The Gini can also be used as a measurement of outliers in return data, and also as an alternative to standard deviation in a mean-variance framework. Shalit et al. use the Gini as the basis for a new “Mean-Gini” portfolio framework with the justification that it is more robust to the non-normality of financial markets than a mean-variance framework. The applications of the Gini also span into the areas of machine learning and data mining. As a unique measure of dispersion and an intuitive framework for evaluation, the Gini is worthwhile adding to anyone’s quantitative toolbox.

Thank you David for this analysis. For people who are interested by a generalization of the ERC portfolio, they could look at this paper :

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2009778

Regards,

Thierry