Probability Equivalent Level of Value at Risk and Expected Shortfall

Abstract

Probability Equivalent Level of Value at Risk and Expected Shortfall (PELVE) is a novel risk measure proposed and explored by Li and Wang (2020), in [16]. It is motivated by the transition from 99% Value-at-Risk (VaR) to 97.5% Expected Shortfall (ES) for banking regulatory purposes made by The Basel Committee on Banking Supervision (BCBS) in the revised Fundamental Review of Trading Book (FRTB) in 2015. A numerical method of estimating PELVE and many desired properties as a risk measure have been demonstrated in [16] as well. In this dissertation, we propose a method to obtain the distribution of a loss random variable based on its PELVE. In Chapter 1, the motivation and structure of this dissertation is stated. In Chapter 2, we provide the definition of PELVE, prove some properties of PELVE, and show some examples of calculating PELVE by given random variable’s distribution. In Chapter 3, we discuss the situation when PELVE is assumed to be a constant for different levels of VaR. In Chapter 4, we further assume PELVE as a step function with two distinctive values. The importance of this work is that it provides a method to gain more distributional information by the values of PELVE. To our best knowledge, this is the first work on obtaining distributional information based on PELVE.