Review of extreme value theory with application to the Johannesburg Stock Exchange financial market data

Abstract

The financial sector is vital to the economy as it encourages economic growth. Therefore, modelling the extremes and risk management of financial stock returns and losses is vital to economic survival. However, with limited amount of information it can be difficult to estimate the parameters of the distributions due to the enormous variance and heavy tails of the financial stock returns. The problem of limited amount of information has inspired the present study to examine and model the extreme value behaviour of the Johannesburg Stock Exchange (JSE) financial market data using the extreme value theory (EVT). The study employed secondary data acquired from the JSE in South Africa, which comprises the daily total return index of the All-Share Index (ALSI) as well as the daily USD/ZAR exchange rates. The generalised extreme value distribution (GEVD), r-largest order statistics GEVD (rGEVD), generalised Pareto distribution (GPD), and the Poisson point process along with the newly proposed alternative for GEVD called blended GEVD (bGEVD) models were applied to the five-year daily JSE financial market data. The parent distribution of the maximum daily JSE financial market data was investigated and the Gamma distribution was found to be the optimal parent distribution. The block maxima method was employed in the study to fit the EVT models. The GEVD models for the USD/ZAR exchange rate and the All Share Total Return Index (ALSTRI) were developed using the weekly and monthly block maxima method. Both results of weekly and monthly maxima GEVD and the monthly rGEVD models for the ALSTRI and the USD/ZAR exchange rate can be modelled by the Weibull and/or Gumbel family. The 100-year return levels of the monthly GEVD, bGEVD, and rGEVD models are almost equal to the maximum observations of the financial markets, revealing that the ALSTRI and USD/ZAR exchange rates will exceed 10802 and R18.89 respectively, at least once in 100 years. The Poisson point process return level estimates are quite comparable with the GPD estimates, indicating that the ALSTRI and USD/ZAR exchange rates will surpass 17501.63 and R23.72 respectively, at least once in 100 years. This implies that the investors will experience higher gains in the total returns of the ALSI. The USD/ZAR exchange rate return levels suggest that the Rand will become more unstable in the long run. Instead of focusing merely on the traditional methods of block maxima, the use of advanced extreme value methods that accommodate even small datasets such as GPD, r-largest order statistics, bGEVD, and Poisson point process are encouraged. The researcher discovered that there are no studies conducted on bGEVD in the field of finance or financial markets. In the future, more studies on bGEVD, vine copulas, and r-largest order bGEVD can be conducted on the financial markets and/or finance sector. Therefore, the present study will add value to the literature and knowledge of statistics and econometrics.