Estimating the value-at-risk of JSE indices and South African exchange rate with Generalized Pareto and stable distributions

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South Africa’s economy has faced many downturns in the previous decade, and to curb the spread of the novel SARS-CoV-2, the lockdown brought South African financial markets to an abrupt halt. Therefore, the implementation of risk mitigation approaches is becoming a matter of urgency in volatile markets in these unprecedented times. In this study, a hybrid generalized autoregressive conditional heteroscedasticity (GARCH)-type model combined with heavy-tailed distributions, namely the Generalized Pareto Distribution (GPD) and the Nolan’s S0-parameterization stable distribution (SD), were fitted to the returns of three FTSE/JSE indices, namely All Share Index (ALSI), Banks Index and Mining Index, as well as the daily closing prices of the US dollar against the South African rand exchange rate (USD/ZAR exchange rate). VaR values were estimated and back-tested using the Kupiec likelihood ratio test. The results of this study show that for FTSE/JSE ALSI returns, the hybrid exponential GARCH (1,1) model with SD model (EGARCH(1,1)-SD) outperforms the GARCH-GPD model at the 2.5% VaR level. At VaR levels of 95% and 97.5%, the fitted GARCH (1,1)-SD model for FTSE/JSE Banks Index returns performs better than the GARCH (1,1)-GPD. The fitted GARCH (1,1)-SD model for FTSE/JSE Mining Index returns is better than the GARCH (1,1)-GPD at 5% and 97.5% VaR levels. Thus, this study suggests that the GARCH (1,1)-SD model is a good alternative to the VaR robust model for modeling financial returns. This study provides salient results for persons interested in reducing losses or obtaining a better understanding of the South African financial industry.

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    • Figure 1. Time series plot of FTSE/JSE All Share Index prices (left) and one-day returns (right)
    • Figure 2. Time series plot of FTSE/JSE Banks Index prices (left) and one-day returns (right)
    • Figure 3. Time series plot of FTSE/JSE Mining Index prices (left) and one-day returns (right)
    • Figure 4. Time series plot of USD/ZAR exchange rate prices (left) and one-day returns (right)
    • Figure 5. FTSE/JSE ALSI Returns Pareto quantile plot of positive (left) and negative (right) standardized residuals
    • Figure 6. FTSE/JSE Banks Index Returns Pareto quantile plot of positive (left) and negative (right) standardized residuals
    • Figure 7. FTSE/JSE Mining Index Returns Pareto quantile plot of positive (left) and negative (right) standardized residuals
    • Figure 8. USD/ZAR Exchange Rate Returns Pareto quantile plot of positive (left) and negative (right) standardized residuals
    • Figure 9. GPD positive ALSI residuals
    • Figure 10. GPD negative ALSI residuals
    • Figure 11. Stable density plots of return series
    • Table 1. Descriptive statistics of financial stock market indices and exchange rate price returns
    • Table 2. ML parameter estimates for GARCH (1,1) model with normal innovations and corresponding goodness-of-fit statistics for Financial stock returns
    • Table 3. Sign bias test of return series
    • Table 4. ML parameter estimates for asymmetric GARCH-type models with normal innovations on ALSI returns
    • Table 5. ML parameter estimates of hybrid GARCH-type-GPD model
    • Table 6. ML parameter estimates of the hybrid GARCH-type-SD model
    • Table 7. VaR estimates of the financial market indices and exchange rate price returns using fitted hybrid GARCH-GPD and GARCH-SD model
    • Table 8. p-values of the Kupiec likelihood ratio test for financial indices and exchange rate returns
    • Table 9. Most appropriate hybrid GARCH-type model selected for financial indices and USD/ZAR exchange rate returns at different VaR levels
    • Conceptualization
      Kimera Naradh
    • Formal Analysis
      Kimera Naradh, Knowledge Chinhamu, Retius Chifurira
    • Investigation
      Kimera Naradh
    • Methodology
      Kimera Naradh
    • Validation
      Kimera Naradh, Retius Chifurira
    • Writing – original draft
      Kimera Naradh
    • Software
      Knowledge Chinhamu
    • Supervision
      Knowledge Chinhamu, Retius Chifurira
    • Writing – review & editing
      Knowledge Chinhamu, Retius Chifurira