Estimating the value-at-risk of JSE indices and South African exchange rate with Generalized Pareto and stable distributions
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Author(s)Kimera NaradhLink to ORCID Index: https://orcid.org/0000-0003-1381-2748
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Knowledge ChinhamuLink to ORCID Index: https://orcid.org/0000-0003-4296-7677
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Retius ChifuriraLink to ORCID Index: https://orcid.org/0000-0001-7889-3417
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DOIhttp://dx.doi.org/10.21511/imfi.18(3).2021.14
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Article InfoVolume 18 2021, Issue #3, pp. 151-165
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- 671 Views
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277 Downloads
This work is licensed under a
Creative Commons Attribution 4.0 International License
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.
- Keywords
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JEL Classification (Paper profile tab)C22, C58, G32
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References38
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Tables9
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Figures11
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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
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- 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
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- Ali, G. (2013). EGARCH, GJR-GARCH, TGARCH, AVGARCH, NGARCH, IGARCH and APARCH models for pathogens at marine recreational sites. Journal of Statistical and Econometric Methods, 2(3), 57-73.
- Balkema, A. A., & De Haan, L. (1974). Residual life time at great age. The Annals of Probability, 2(5), 792-804.
- Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307-327.
- Borak, S., Härdle, W., & Weron, R. (2005). Stable distributions (SFB 649 Discussion Paper No. 2005-008). Berlin.
- Byström, H. N. (2005). Extreme value theory and extremely large electricity price changes. International Review of Economics & Finance, 14(1), 41-55.
- Campbell, S. (2005). A review of backtesting and backtesting procedures (FEDS Staff Working Papers).
- Chifurira, R., & Chinhamu, K. (2017). Using the generalized Pareto and Pearson type-iv distributions to measure value-at-risk for the daily South African mining index. Studies in Economics and Econometrics, 41(1), 33-54.
- Chifurira, R., & Chinhamu, K. (2019). Evaluating South Africa’s market risk using asymmetric power auto-regressive conditional heteroscedastic model under heavy-tailed distributions. Journal of Economic and Financial Sciences, 12(1), 475.
- Coles, S. (2001). An Introduction to Statistical Modelling of Extreme Values. London: Springer-Verlag.
- Ding, Z., Granger, C. W., & Engle, R. F. (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance, 1(1), 83-106.
- Dwarika, N., Moores-Pitt, P., & Chifurira, R. (2021). Volatility dynamics and the risk-return relationship in South Africa: A GARCH Approach. Investment Management and Financial Innovations, 18(2), 106-117.
- Embrechts, P., Kluppel, C., & Mikosh, T. (1997). Modelling Extremal Events: For Insurance and Finance. In: Pricing Insurance Derivatives: The Case of CAT Futures (Chapter 3). Berlin: Springer-Verlag.
- Embrechts, P., Resnick, S., & Samorodnitsky, G. (1999). Extreme value theory as a risk management tool. North American Actuarial Journal, 3(2), 30-41.
- Escanciano, J., & Olmo, J., (2010). Backtesing parametric value-at-risk with estimation risk. Journal of Business& Economic Statistics, 28(1), 36-51.
- Fama, E. (1965). The behavior of stock market prices. Journal of Business, 38(1), 34-105.
- Glosten, L., Jagannathan, R., & Runkle, D. (1993). On the relation between the expected value and the volatility of the nominal excess return on stocks. The Journal of Finance, 48(5), 1779-1801.
- Ilupeju, Y. E. (2016). Modelling South Africa’s market risk using the APARCH model and heavy-tailed distributions (Master’s Thesis). University of KwaZulu-Natal. Durban. South Africa.
- Jafari, G., Bahraminasab, A., & Norouzzadeh, P. (2007). Why does the Standard GARCH (1, 1) model work well? International Journal of Modern Physics C, 18(7), 1223-1230.
- Katsenga, G. (2013). Value at risk (VaR) backtesting’ Evidence from a South African market portfolio’ (Doctoral Thesis).
- Kupiec, P. (1995). Techniques for verifying the accuracy of risk measurement models. The Journal of Derivatives, 3(2), 73-84.
- Levy, P. (1925). Calcul des Probabilites. Gauthier-Villars et Cie.
- Mandelbrot, B. B. (1963). The variation of certain speculative prices. Journal of Business, 36(4), 394-419.
- McNeil, A., & Frey, R. (2000). Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. Journal of Empirical Finance, 7(3-4), 271-300.
- McNeil, A., Frey, R. & Embrechts, P. (2015). Quantitative risk management: concepts, techniques and tools-revised edition. Princeton university press.
- Mzamane, T. P., Achia, T., & Mwambi, H. (2013). Garch modelling of volatility in the Johannesburg Stock Exchange index (Master’s Thesis). University of KwaZulu-Natal. Durban. South Africa.
- Naradh, K., Chinhamu, K., Chifurira, R., & Hammujuddy, M. (2016). Multivariate Elliptically Contoured Stable Distributions with Applications to BRICS Financial Data (Master’s Thesis). University of KwaZulu-Natal. Durban.
- Nel, C., Chapman, M., & Garnett-Bennett, J. (2020). Financial Market Impacts of COVID-19. PwC Inc, South Africa.
- Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59(2), 347-370.
- Nguyen, T., & Sampson, A. (1991). A note on characterizations of multivariate stable distributions. Annals of the Institute of Statistical Mathematics, 43(4), 793-801.
- Nolan, J. (2001). Maximum likelihood estimation and diagnostics for stable distributions. In Lévy processes (pp. 379-400). Boston, MA: Birkhäuser.
- Nolan, J. (2003). Modeling financial data with stable distributions. In Handbook of heavy tailed distributions in finance (pp. 105-130). North-Holland.
- Nolan, J. (2013). Financial modeling with heavy-tailed stable distributions. WIREs Computational Statistics, 6(1), 45-55.
- Nolan, J. (2020). Univariate Stable Distributions: Models for Heavy Tailed Data. Springer Nature.
- Pickands, J. (1975). Statistical inference using extreme order statistics. Annals of Statistics, 3(1), 119-131.
- Ren, F., & Giles, D. E. (2010). Extreme value analysis of daily Canadian crude oil prices. Applied Financial Economics, 20(12), 941-954.
- Tian, G. (2016). Parameter Estimation for Stable Distribution: Spacing based and Indirect Inference (Doctoral Thesis). UC Santa Barbara.
- Tsay, R. S. (2013). Multivariate time series analysis: with R and financial applications. John Wiley & Sons.
- Zakoian, J. (1994). Threshold heteroscedasticity models. Journal of Economic Dynamics, 18(5), 931-955.
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Value-at-risk (VAR) estimation and backtesting during COVID-19: Empirical analysis based on BRICS and US stock markets
Muneer Shaik 
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Lakshmi Padmakumari
doi: http://dx.doi.org/10.21511/imfi.19(1).2022.04
Investment Management and Financial Innovations Volume 19, 2022 Issue #1 pp. 51-63 Views: 1532 Downloads: 892 TO CITE АНОТАЦІЯValue-at-risk (VaR) is the most common and widely used risk measure that enterprises, particularly major banking corporations and investment bank firms employ in their risk mitigation processes. The purpose of this study is to investigate the value-at-risk (VaR) estimation models and their predictive performance by applying a series of backtesting methods on BRICS (Brazil, Russia, India, China, South Africa) and US stock market indices. The study employs three different VaR estimation models, namely normal (N), historical (HS), exponential weighted moving average (EMWA) procedures, and eight backtesting models. The empirical analysis is conducted during three different periods: overall period (2006–2021), global financial crisis (GFC) period (2008–2009), and COVID-19 period (2020–2021). The results show that the EMWA model performs better compared to N and HS estimation models for all the six stock market indices during overall and crisis sample periods. The results found that VaR models perform poorly during crisis periods like GFC and COVID-19 compared to the overall sample period. Furthermore, the study result shows that the predictive accuracy of VaR methods is weak during the COVID-19 era when compared to the GFC period.
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Comparing riskiness of exchange rate volatility using the Value at Risk and Expected Shortfall methods
Thabani Ndlovu ,
Delson Chikobvu
doi: http://dx.doi.org/10.21511/imfi.19(2).2022.31
Investment Management and Financial Innovations Volume 19, 2022 Issue #2 pp. 360-371 Views: 531 Downloads: 166 TO CITE АНОТАЦІЯThis 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.
