From concentration profiles to concentration maps. New tools for the study of loss distributions

Andrea Fontanari, Pasquale Cirillo, Cornelis W. Oosterlee: In this work we introduce a novel approach to risk management, based on the study of concentration measures of the loss distribution. In particular we show that indices like the Gini index, especially when restricted to the tails by conditioning and truncation, represent an accurate way of assessing the variability of the larger losses–the most relevant ones–and the precision of common risk management measures like the Expected Shortfall. We then introduce the Concentration Profile, that is a sequence of truncated Gini indices that, we show, is able to characterize the loss distribution, providing interesting information about tail risks. Combining concentration profiles and standard results from utility theory, we then develop a Concentration Map, which can be used to assess the risk attached to potential losses on the basis of the risk profile of the user, her beliefs and historical data. Finally, we use the sequence of truncated Gini indices as weights for the expected shortfall, defining the so-called Concentration Adjusted Expected Shortfall, a measure able to capture interesting additional features of tail risk. All tools are applied to empirical data to show how to use them in practice.