On the Correspondence and the Risk Contribution for Conditional Coherent and Deviation Risk Measures

TL;DR

This paper establishes a theoretical link between conditional coherent risk measures and generalized deviation measures, introduces a continuous-time risk contribution framework, and interprets risk contributions via BSDE solutions.

Contribution

It provides an axiomatic framework for conditional deviation measures and shows the correspondence with coherent risk measures, including a microscopic interpretation of risk contributions.

Findings

Risk contributions of time-consistent measures remain time-consistent.

The second element of BSDE solutions represents risk contribution.

A new framework for continuous-time risk contribution is developed.

Abstract

We give an axiomatic framework for conditional generalized deviation measures. Under financially reasonable assumptions, we give the correspondence between conditional coherent risk measures and generalized deviation measures. Moreover, we establish the notion of continuous-time risk contribution for conditional coherent risk measures and generalized deviation measures. With the help of the correspondence between these two different types of risk measures, we give a microscopic interpretation of their risk contributions. Particularly, we show that the risk contributions of time-consistent risk measures are still time-consistent. We also demonstrate that the second element of the BSDE solution $(Y, Z)$ associated with $g$-expectation has the meaning of risk contribution.