Summarizing Insurance Scores Using a Gini Index

Individuals, corporations and government entities regularly exchange financial risks y at prices Π. Comparing distributions of risks and prices can be difficult, particularly when the financial risk distribution is complex. For example, with insurance, it is not uncommon for a risk distribution to be a mixture of 0's (corresponding to no claims) and a right-skewed distribution with thick tails (the claims distribution). However, analysts do not work in a vacuum, and in the case of insurance they use insurance scores relative to prices, called "relativities," that point to areas of potential discrepancies between risk and price distributions. Ordering both risks and prices based on relativities, in this article we introduce what we call an "ordered" Lorenz curve for comparing distributions. This curve extends the classical Lorenz curve in two ways, through the ordering of risks and prices and by allowing prices to vary by observation. We summarize the ordered Lorenz curve in the same way as the classic Lorenz curve using a Gini index, defined as twice the area between the curve and the 45-degree line. For a given ordering, a large Gini index signals a large difference between price and risk distributions. We show that the ordered Lorenz curve has desirable properties. It can be expressed in terms of weighted distributions functions. In special cases, curves can be ranked through a partial ordering. We show how to estimate the Gini index and give pointwise consistency and asymptotic normality results. A simulation study and an example using homeowners insurance underscore the potential applications of these methods.