Tail Risk Modelling in Foreign Exchange Markets: Evidence from Botswana Pula, Bitcoin, and South African Rand

Abstract

The volatility of an event often indicates the probability of it occurring within a designated timeframe. The aim of this paper was to assess the efficacy of the Generalised Extreme Value (GEV) and Generalised Pareto Distribution (GPD) models in accurately representing extreme quantiles of return risks for the BWP/USD, BTC/USD, and ZAR/USD exchange rates. The weekly maxima and minima from the return series were used to analyse tail behaviour. Thresholds for the GPD were established between the 90th and 99th percentiles through the analysis of mean residual life and parameter stability plots of potential thresholds. Parameters were estimated using maximum likelihood, and $95\%$ confidence intervals were utilised to evaluate significance. Model performance was assessed through information criteria (AIC and BIC) and goodness-of-fit tests, specifically the Kolmogorov–Smirnov, Cramér–von Mises, and Anderson–Darling tests. The findings demonstrate that BTC/USD shows relatively constrained extremes under the Generalised Pareto Distribution, while ZAR/USD and BWP/USD exhibit heavy-tailed characteristics, especially in the upper tails, indicating a heightened vulnerability to significant positive shocks. The GPD demonstrated a strong fit in all tests, establishing it as the preferred model for both minima and maxima of BWP/USD returns. The GPD effectively captures extreme quantiles for all three currencies, except BTC/USD maxima. The findings illustrate the importance of suitable model selection in risk management, especially concerning Value-at-Risk (VaR) and Expected Shortfall (ES). Future research may expand this analysis to include multivariate extremes and integrate regime-switching frameworks to improve the comprehension of dynamic exchange rate risks.