Time consistency of dynamic risk measures and dynamic performance measures generated by distortion functions

TL;DR

This paper investigates dynamic risk measures and performance indices generated by distortion functions, establishing their properties, time consistency, and relationships within a discrete-time framework.

Contribution

It introduces and characterizes dynamic coherent risk measures and acceptability indices generated by distortion functions, including their time consistency properties and new representations.

Findings

Conditional Choquet integrals are dynamic coherent risk measures.

DCRMs generated by distortion functions are sub-martingale time consistent.

These risk measures are not super-martingale or weakly acceptance time consistent.

Abstract

The aim of this work is to study risk measures generated by distortion functions in a dynamic discrete time setup, and to investigate the corresponding dynamic coherent acceptability indices (DCAIs) generated by families of such risk measures. First we show that conditional version of Choquet integrals indeed are dynamic coherent risk measures (DCRMs), and also introduce the class of dynamic weighted value at risk measures. We prove that these two classes of risk measures coincides. In the spirit of robust representations theorem for DCAIs, we establish some relevant properties of families of DCRMs generated by distortion functions, and then define and study the corresponding DCAIs. Second, we study the time consistency of DCRMs and DCAIs generated by distortion functions. In particular, we prove that such DCRMs are sub-martingale time consistent, but they are not super-martingale time consistent. We also show that DCRMs generated by distortion functions are not weakly acceptance time consistent. We also present several widely used classes of distortion functions and derive some new representations of these distortions.