Time-consistency of risk measures with GARCH volatilities and their estimation

TL;DR

This paper develops methods to construct time-consistent risk measures for GARCH(1,1) modeled returns, providing analytical formulas for VaR, bounds for AVaR, and incorporating EVT for tail analysis, with real data application.

Contribution

It introduces a novel construction of time-consistent risk measures for GARCH models, including explicit formulas and bounds for VaR and AVaR, and integrates EVT for tail risk assessment.

Findings

Analytical formula for time-consistent VaR derived.

Bounds for time-consistent AVaR established.

Tail analysis enhanced with EVT techniques.

Abstract

In this paper we study time-consistent risk measures for returns that are given by a GARCH(1,1) model. We present a construction of risk measures based on their static counterparts that overcomes the lack of time-consistency. We then study in detail our construction for the risk measures Value-at-Risk (VaR) and Average Value-at-Risk (AVaR). While in the VaR case we can derive an analytical formula for its time-consistent counterpart, in the AVaR case we derive lower and upper bounds to its time-consistent version. Furthermore, we incorporate techniques from Extreme Value Theory (EVT) to allow for a more tail-geared statistical analysis of the corresponding risk measures. We conclude with an application of our results to a data set of stock prices.