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.

# ESR6 Andrea Fontanari

Andrea Fontanari, Pasquale Cirillo, Cornelis W. Oosterlee: We 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.

Andrea Fontanari, Nassim Nicholas Taleb, Pasquale Cirillo: We study the problems related to the estimation of the Gini index in presence of a fattailed data generating process, i.e. one in the stable distribution class with finite mean but infinite variance (i.e. with tail index α ∈ (1, 2)). We show that, in such a case, the Gini coefficient cannot be reliably estimated using conventional nonparametric methods, because of a downward bias that emerges under fat tails. This has important implications for the ongoing discussion about economic inequality. We start by discussing how the nonparametric estimator of the Gini index undergoes a phase transition in the symmetry structure of its asymptotic distribution, as the data distribution shifts from the domain of attraction of a light-tailed distribution to that of a fat-tailed one, especially in the case of infinite variance. We also show how the nonparametric Gini bias increases with lower values of α. We then prove that maximum likelihood estimation outperforms nonparametric methods, requiring a much smaller sample size to reach efficiency. Finally, for fat-tailed data, we provide a simple correction mechanism to the small sample bias of the nonparametric estimator based on the distance between the mode and the mean of its asymptotic distribution.

Andrea Fontanari, Iddo Eliazar, Pasquale Cirillo, Cornelis W. Oosterlee: We propose quantum majorization as a way of comparing and ranking correlation matrices, with the aim of assessing portfolio risk in a unified framework. Quantum majorization is a partial order in the space of correlation matrices, which are evaluated through their spectra. We discuss the connections between quantum majorization and an important class of risk functionals, and we define two new risk measures able to capture interesting characteristics of portfolio risk.