Modelling conditional heteroskedasticity in JSE stock returns using the Generalised Pareto Distribution

Abstract

Extreme equity market returns demand the use of specialised techniques for standardised treatment that focuses exclusively on rare tail events. Extreme Value Theory (EVT) is used in this article to model heteroskedastic stock returns of the All Share Index (ALSI) at the Johannesburg Stock exchange (JSE). Daily data of the ALSI at the JSE over the period 2002–2011 is used. A two-stage modelling framework is proposed. In stage one we fi t an Autoregressive Moving Average–Generalised Autoregressive Conditional Heteroskedastic (ARMA-GARCH) model to the stock return series. In stage two we filter the residuals from the ARMA-GARCH model. We then fi t a Generalised Pareto Distribution (GPD) to the upper tail of the residual series, and refer to this hybrid as the ARMA-GARCH-GPD model. The threshold is estimated using a Pareto quantile plot. Empirical results show that the Weibull class of distributions can be used to model daily returns data. The ARMA-GARCHGPD model produces more accurate estimates of extreme returns than the ARMA-GARCH model. These results are important to risk managers and investors.

Keywords: Extreme value theory; GARCH; Generalized Pareto Distribution; risk management