Value at Risk Estimation Using the GARCH-EVT Approach with Optimal Tail Selection

Krzysztof Echaust,Małgorzata Just

doi:10.3390/math8010114

Abstract

A conditional Extreme Value Theory (GARCH-EVT) approach is a two-stage hybrid method that combines a Generalized Autoregressive Conditional Heteroskedasticity (GARCH) filter with the Extreme Value Theory (EVT). The approach requires pre-specification of a threshold separating distribution tails from its middle part. The appropriate choice of a threshold level is a demanding task. In this paper we use four different optimal tail selection algorithms, i.e., the path stability method, the automated Eye-Ball method, the minimization of asymptotic mean squared error method and the distance metric method with a mean absolute penalty function, to estimate out-of-sample Value at Risk (VaR) forecasts and compare them to the fixed threshold approach. Unlike other studies, we update the optimal fraction of the tail for each rolling window of the returns. The research objective is to verify to what extent optimization procedures can improve VaR estimates compared to the fixed threshold approach. Results are presented for a long and a short position applying 10 world stock indices in the period from 2000 to June 2019. Although each approach generates different threshold levels, the GARCH-EVT model produces similar Value at Risk estimates. Therefore, no improvement of VaR accuracy may be observed relative to the conservative approach taking the 95th quantile of returns as a threshold.

Highlights

Value at Risk (VaR) is the most widely known measure of market risk
This paper provides an empirical study of conditional Extreme Value Theory (EVT)
We propose to use the Generalized Autoregressive Conditional Heteroskedasticity (GARCH)-EVT model with optimal tail selection

Summary

Introduction

Value at Risk (VaR) is the most widely known measure of market risk. VaR indicates how big the maximum loss of an asset or a portfolio of assets is over the target horizon so that there is a low, pre-specified probability q that the actual loss will be greater. VaR can be considered in the context of returns. We denote by rt the return on assets at time t. The one-day-ahead Value at Risk for a long trading position at a q significance level, noted VaRq (rt ), anticipated conditionally to an information set, Ft , available at time t is defined by the formula:. This definition shows that VaR is a qth conditional quantile of the returns distribution. For a short trading position VaR is a 1–qth conditional quantile of the returns distribution:

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Methods

Results