The use of the Lorenz curve, Gini index and related measures of relative inequality and uniformity in securities law

摘要

During the stock market "boom" in 1999-2000 substantial sums were made by investors who were able to obtain shares in "high tech" companies at their initial public offering (IPO) price and then sold the shares when they began trading on the stock exchanges. Some investment firms established profit sharing arrangements with a group of customers who promised to return a portion, e.g. one-third, of their profits from those shares in increased commission business with the investment firm. A regulatory body accused one firm of such an arrangement with a group of customers whose commission business increased on days they received IPO shares. As commissions from other customers also increased on those days, it is appropriate to consider the ratio of the commissions to the customers profit on IPO days. If a 'profit sharing' agreement was operating one would expect those ratios to be concentrated about the agreed upon share. The use of measures of relative variability; especially the Gini Index, Lorenz curve and Coefficient of Dispersion to analyze such data are described. The values of the Gini Index and Coefficient of Dispersion on nearly all the data in the case showed substantial inequality or non-uniformity contrary to what one would expect if a 'profit sharing' system were in effect. Because these measures are typically used on much larger data sets, e.g. household surveys, the applicability of both large sample theory and the percentile bootstrap to the data was explored subsequently. Formal statistical inference, however, from the available sample sizes is questionable as neither asymptotic theory nor the percentile bootstrap confidence intervals were sufficiently reliable for a variety of important distributions, e.g. the Pareto.