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
Expected shortfall; Expectiles; Risk management; Stochastic orders; Variability measures;
English
13-lug-2022
2022
106
September 2022
270
284
none
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].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/388205
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