Tail Risk Equivalent Level Transition and Its Application for Estimating Extreme $L_p$-quantiles

TL;DR

This paper introduces the Tail Risk Equivalent Level Transition (TRELT) to analyze tail risk changes between $L_p$-quantiles, providing new theoretical insights and estimation methods for extreme value analysis in risk management.

Contribution

It develops the TRELT concept for $L_p$-quantiles, investigates its theoretical properties, and proposes novel inference methods for extreme quantile estimation.

Findings

TRELT exists uniquely under certain conditions.

Asymptotic properties of TRELT estimators are established.

Simulation and real data show the effectiveness of the proposed methods.

Abstract

$L_p$-quantile has recently been receiving growing attention in risk management since it has desirable properties as a risk measure and is a generalization of two widely applied risk measures, Value-at-Risk and Expectile. The statistical methodology for $L_p$-quantile is not only feasible but also straightforward to implement as it represents a specific form of M-quantile using $p$-power loss function. In this paper, we introduce the concept of Tail Risk Equivalent Level Transition (TRELT) to capture changes in tail risk when we make a risk transition between two $L_p$-quantiles. TRELT is motivated by PELVE in Li and Wang (2023) but for tail risk. As it remains unknown in theory how this transition works, we investigate the existence, uniqueness, and asymptotic properties of TRELT (as well as dual TRELT) for $L_p$-quantiles. In addition, we study the inference methods for TRELT and extreme $L_p$-quantiles by using this risk transition, which turns out to be a novel extrapolation method in extreme value theory. The asymptotic properties of the proposed estimators are established, and both simulation studies and real data analysis are conducted to demonstrate their empirical performance.