We present a general framework for a comparative theory of variability measures, with a particular focus on the recently introduced one-parameter families of inter-Expected Shortfall differences and inter-expectile differences, that are explored in detail and compared with the widely known and applied inter-quantile differences. From the mathematical point of view, our main result is a characterization of symmetric and comonotonic variability measures as mixtures of inter-Expected Shortfall differences, under a few additional technical conditions. Further, we study the stochastic orders induced by the pointwise comparison of inter-Expected Shortfall and inter-expectile differences, and discuss their relationship with the dilation order. From the statistical point of view, we establish asymptotic consistency and normality of the natural estimators and provide a rule of the thumb for cross-comparisons. Finally, we study the empirical behavior of the considered classes of variability measures on the S&P 500 Index under various economic regimes, and explore the comparability of different time series according to the introduced stochastic orders.

Bellini, F., Fadina, T., Wang, R., & Wei, Y. (2022). Parametric measures of variability induced by risk measures. INSURANCE MATHEMATICS & ECONOMICS, 106(September 2022), 270-284 [10.1016/j.insmatheco.2022.07.009].

Parametric measures of variability induced by risk measures

Bellini, Fabio
;
2022

Abstract

We present a general framework for a comparative theory of variability measures, with a particular focus on the recently introduced one-parameter families of inter-Expected Shortfall differences and inter-expectile differences, that are explored in detail and compared with the widely known and applied inter-quantile differences. From the mathematical point of view, our main result is a characterization of symmetric and comonotonic variability measures as mixtures of inter-Expected Shortfall differences, under a few additional technical conditions. Further, we study the stochastic orders induced by the pointwise comparison of inter-Expected Shortfall and inter-expectile differences, and discuss their relationship with the dilation order. From the statistical point of view, we establish asymptotic consistency and normality of the natural estimators and provide a rule of the thumb for cross-comparisons. Finally, we study the empirical behavior of the considered classes of variability measures on the S&P 500 Index under various economic regimes, and explore the comparability of different time series according to the introduced stochastic orders.
Articolo in rivista - Articolo scientifico
Scientifica
Expected shortfall; Expectiles; Risk management; Stochastic orders; Variability measures;
English
Bellini, F., Fadina, T., Wang, R., & Wei, Y. (2022). Parametric measures of variability induced by risk measures. INSURANCE MATHEMATICS & ECONOMICS, 106(September 2022), 270-284 [10.1016/j.insmatheco.2022.07.009].
Bellini, F; Fadina, T; Wang, R; Wei, Y
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10281/388205
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