Tail risk modelling in foreign exchange markets: evidence from Botswana Pula, Bitcoin, and South African Rand

Abstract

We use data for each currency pair (Bitcoin vs USD; South African Rand vs USD; Botswana Pula vs USD) to model the extreme quantiles of their weekly returns using two distributional frameworks: the Generalised Extreme Value (GEV) and Generalised Pareto Distribution (GPD). In this study, two methodological approaches are employed to estimate these distributions: the block maxima/minima approach and the peaks-over-threshold (POT) approach. To ensure an optimal estimation of the parameters of the distributions, we employed a stability-prediction threshold optimisation algorithm. The predictive error metrics (mean absolute error, mean squared error, root mean squared error) as well as the maximum likelihood estimation (MLE) are utilised to provide an unbiased estimation of the parameters of the distributions. In addition, goodness-of-fit diagnostics and the information criteria (Akaike Information Criterion and Bayesian Information Criterion) were employed to evaluate the appropriateness of each distributional framework for modelling the tails of the three currencies’ distributions. Our empirical evidence shows important differences among the three currencies in how they exhibit tail behaviour. Specifically, we find that Bitcoin/USD has pronounced heavy-tailed behaviour relative to both ZAR/USD and BWP/USD. Furthermore, the GEV was found to accurately model the extreme gains experienced by investors holding Bitcoin, whereas the GPD better captures extreme losses on this asset. On the other hand, we find that the GPD is a much better model than the GEV for capturing moderate tail risk in ZAR/USD, especially for extreme depreciations against the US dollar. Finally, the BWP/USD exhibits very little heavy-tailed behaviour, i.e., thin tails, likely due to its more stable currency regime compared to either ZAR/USD or BTC/USD. Overall, our empirical results show that the GPD is a much more reliable predictor of both upside and downside extreme events for all three currency pairs examined in this paper.