Parametric measures of variability induced by risk measures

TL;DR

This paper develops a unified framework for comparing variability measures induced by risk measures, introduces new parametric families, and analyzes their mathematical properties, statistical estimators, and empirical behavior on financial data.

Contribution

It characterizes symmetric and comonotonic variability measures as mixtures of inter-Expected Shortfall differences and studies their stochastic orders and empirical performance.

Findings

Inter-Expected Shortfall differences effectively compare risk variability.

Symmetric variability measures are characterized as mixtures of inter-Expected Shortfall.

Empirical analysis on S&P 500 shows variability measures distinguish economic regimes.

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 behaviour 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.