New Risk Measures “<i>VaR</i> to the Power of <i>t</i>” and “<i>ES</i> to the Power of <i>t</i>” and Distortion Risk Measures

Abstract

Distortion risk measures have been popular in financial and insurance applications in recent years due to their attractive properties. The aim of the article is to investigate whether risk measures “VaR in the power of t”, introduced by the author, belong to the class of distortion risk measures, as well as to describe the corresponding distortion functions. The author introduces a new class of risk measures “ES to the power of t” and investigates whether it belongs to distortion risk measures, and also describes the corresponding distortion functions. The author used the composite method to design new distortion functions and corresponding distortion risk measures, to prove that risk measures “VaR to the power of t” and “ES to the power of t” belong to the class of distortion risk measures. The paper presents examples to illustrate the relevant concepts and results that show the importance of risk measures “VaR to the power of t” and “ES to the power of t” as subsets of distortion risk measures that allow identifying various financial catastrophic risks. The author concludes that risk measures “VaR to the power of t” and “ES to the power of t” can be used in risk management of companies when assessing remote, highly catastrophic risks.

Keywords

• catastrophic risks
• distortion risk measures
• distortion functions
• composite method
• coherent risk measures
• risk measures “var to the power of t”
• risk measures “es to the power of t”