Comparing riskiness of exchange rate volatility using the Value at Risk and Expected Shortfall methods

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This paper uses theValue at Risk (VaR) and the Expected Shortfall (ES) to compare the riskiness of the two currency exchange rate volatility, namely BitCoin against the US dollar (BTC/USD) and the South African Rand against the US dollar (ZAR/USD). The risks calculated are tail-related measures, so the Extreme Value Theory is used to capture extreme risk more accurately. The Generalized Pareto distribution (GPD) is assumed under Extreme Value Theory (EVT). The family of Generalized Autoregressive Conditional Heteroscedasticity (GARCH) models was used to model the volatility-clustering feature. The Maximum Likelihood Estimation (MLE) method was used in parameter estimation. Results obtained from the GPD are compared using two underlying distributions for the errors, namely: the Normal and the Student-t distributions. The findings show that the tail VaR on the BitCoin averaging 1.6 and 2.8 is riskier than on South Africa’s Rand that averages 1.5 and 2.3 at 95% and 99%, respectively. The same conclusion is made about tail ES, the BitCoin average of 2.3 and 3.6 is higher (riskier) than the South African Rand averages at 2.1 and 2.9 at 95% and 99%, respectively. The backtesting results confirm the model adequacy of the GARCH-GPD in the estimation of VaR and ES, since all p-values are above 0.05.

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    • Figure 1. Time series plot of one-day BTC/USD prices (left) and one-day log returns (right)
    • Figure 2. Time series plot of ZAR/USD prices (left) and one-day log returns (right)
    • Figure 3. Mean excess function for BTC/USD returns
    • Figure 4. Mean excess function for ZAR/USD returns
    • Table 1. Descriptive statistics of exchange rate returns
    • Table 2. Estimated GARCH parameters for both BTC/USD and ZAR/USD
    • Table 3. Maximum likelihood estimates (MLE) of GARCH (1,1) residuals from BTC/USD to the Generalized Pareto Distribution
    • Table 4. Maximum likelihood estimates (MLE) of GARCH (1,1) residuals from ZAR/USD to the Generalized Pareto Distribution
    • Table 5. VaR estimates using fitted hybrid GARCH(1,1)-GPD
    • Table 6. ES estimates using fitted hybrid GARCH(1,1)-GPD
    • Table 7. Backtest results for VaR estimates using fitted GARCH(1,1)-GPD
    • Conceptualization
      Thabani Ndlovu, Delson Chikobvu
    • Data curation
      Thabani Ndlovu
    • Formal Analysis
      Thabani Ndlovu, Delson Chikobvu
    • Investigation
      Thabani Ndlovu
    • Methodology
      Thabani Ndlovu
    • Software
      Thabani Ndlovu
    • Validation
      Thabani Ndlovu
    • Writing – original draft
      Thabani Ndlovu
    • Writing – review & editing
      Thabani Ndlovu, Delson Chikobvu
    • Project administration
      Delson Chikobvu
    • Resources
      Delson Chikobvu
    • Supervision
      Delson Chikobvu