From Concentration Profiles to Concentration Maps. New tools for the study of loss distributions

Abstract

a b s t r a c tWe introduce a novel approach to risk management, based on the study of concentration measures of the loss distribution. We show that indices like the Gini index, especially when restricted to the tails by conditioning and truncation, give us an accurate way of assessing the variability of the larger losses -the most relevant ones -and the reliability of common risk management measures like the Expected Shortfall. We first present the Concentration Profile, which is formed by a sequence of truncated Gini indices, to characterize the loss distribution, providing interesting information about tail risk. By combining Concentration Profiles and standard results from utility theory, we develop the Concentration Map, which can be used to assess the risk attached to potential losses on the basis of the risk profile of a user, her beliefs and historical data. Finally, with a sequence of truncated Gini indices as weights for the Expected Shortfall, we define the Concentration Adjusted Expected Shortfall, a measure able to capture additional features of tail risk. Empirical examples and codes for the computation of all the tools are provided.

Lorenz-generated bivariate Archimedean copulas

“…Notice that, for all exponentials, the (mirrored) Lorenz curve does not depend on λ, i.e. all exponentials share the same Lorenz curve, as observed in [15]. For g < the random variable X is shifted away from zero by a factor equal to ( − g)λ.…”

“…As observed in [51], all inequality indices are nothing more than generalizations and improvements of some common measures of variability like the variance or the standard deviation. This justi es the rising interest for their application outside inequality studies, in elds like biostatistics and nance, see for example [15] and references therein.…”

Gini estimation under infinite variance

“…In particular, we can approximate the distribution of the deviations of the estimator from the true value g of the Gini index for finite sample sizes, by using Equations (14) and (15). Figure 2 shows how the deviations around the mean of the two different types of estimators are distributed and how these distributions change as the number of observations increases.…”

“…The literature concerning the estimation of the Gini index is wide and comprehensive (e.g. [9,30] for a review), however, strangely enough, almost no attention has been paid to its behavior in presence of fat tails, and this is curious if we consider that: 1) fat tails are ubiquitous in the empirical distributions of income and wealth [19,23], and 2) the Gini index itself can be seen as a measure of variability and fat-tailedness [8,10,11,15].…”

“…The relationship between regularlyvarying random variables and the Fréchet class thus allows us to deal with a very large family of random variables (and empirical data), and allows us to show how the Gini index is highly influenced by maxima, i.e. extreme wealth, as clearly suggested by intuition [15,19], especially under infinite variance. Again, this recommends some caution when discussing economic inequality under fat tails.…”