# The Generalised Pareto Distribution Model Approach to Comparing Extreme Risk in the Exchange Rate Risk of BitCoin/US Dollar and South African Rand/US Dollar Returns

^{*}

## Abstract

**:**

## 1. Introduction

#### Literature Review

## 2. Methodology

#### 2.1. The Generalised Pareto Distribution (GPD)

#### 2.1.1. Parameter Estimation of GPD

#### 2.1.2. Excess Distribution

#### 2.2. Risk Measures

#### 2.3. Model Adequacy

_{0}: E$\left[\frac{{x}^{p}}{N}\right]=p$, i.e., (the expected proportion of violations is equal to $p$).

## 3. Results

#### 3.1. Descriptive Statistics

#### 3.2. Data Analysis

#### 3.2.1. Analysing the BTC/USD Returns

#### 3.2.2. Analysing ZAR/USD Returns

#### 3.2.3. Model Diagnostics for the ZAR/USD Returns

#### 3.3. Parameter Estimations

#### 3.4. Risk Measures

#### 3.5. Model Adequacy

## 4. Discussion

#### Limitations and Further Related Studies

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Almeida, José, and Tiago Cruz Gonçalves. 2023. Portfolio Diversification, Hedge and Safe-Haven Properties in Cryptocurrency Investments and Financial Economics: A Systematic Literature Review. Journal of Risk and Financial Management 16: 3. [Google Scholar] [CrossRef]
- Artzner, Philippe, Freddy Delbaen, Eber Jean-Marc, and David D. Heath. 1999. Coherent Measures of Risk. Mathematical Finance 9: 203–28. [Google Scholar] [CrossRef]
- Bader, Brian, and Jun Yan. 2020. eva: Extreme Value Analysis with Goodness-of-Fit Testing. R Package Version 0.2.6. Available online: https://cran.r-project.org/web/packages/eva/eva.pdf (accessed on 10 December 2022).
- Balkema, August A., and Laurens de Haan. 1974. Residual lifetime at great age. Annals of Probability 2: 792–804. [Google Scholar] [CrossRef]
- Beirlant, Jan, Petra Vynckier, and Jozef L. Teugels. 1996. Tail index estimation, Pareto quantile plots, and regression diagnostics. Journal of American Statistical Association 91: 1659–67. [Google Scholar]
- Beirlant, Jan, Goedele Dierckx, and Armelle Guillou. 2005. Estimation of the extreme-value index and generalized quantile plots. Bernoulli 11: 949–70. [Google Scholar] [CrossRef]
- Bouri, Elie, Peter Molnár, Georges Azzi, David Roubaud, and Lars Ivar Hagfors. 2017. On the hedge and safe haven properties of BitCoin: Is it really more than a diversifier? Finance Research Letters 20: 192–98. [Google Scholar] [CrossRef]
- Caeiro, Frederico, Maria Ivette Gomes, and Dinis Pestana. 2005. Direct reduction of bias of the classical hill estimator. Revstat 3: 113–36. [Google Scholar]
- Cai, Juan-Juan, Laurens de Haan, and Chen Zhou. 2013. Bias correction in extreme value statistics with index around zero. Extremes 16: 173–201. [Google Scholar] [CrossRef]
- Chen, James Ming. 2018. On Exactitude in Financial Regulation: Value-at-Risk, Expected Shortfall, and Expectiles. Risks 6: 61. [Google Scholar] [CrossRef]
- Chikobvu, Delson, and Thabani Ndlovu. 2023. The Generalised Extreme Value Distribution Approach to Comparing the Riskiness of BitCoin/US Dollar and South African Rand/US Dollar Returns. Journal of Risk and Financial Management 16: 253. [Google Scholar] [CrossRef]
- Chou, Heng-Chih, and David K. Wang. 2014. Estimation of Tail-Related Value-at-Risk Measures: Range Based Extreme Value Approach. Quantitative Finance 14: 293–304. [Google Scholar] [CrossRef]
- Danielsson, Jon. 2011. Financial Risk Forecasting. London: Wiley. [Google Scholar]
- Davies, Martyn. 2017. Is South Africa the Next Brazil? Emerging Market Insights. Available online: https://www2.deloitte.com/content/dam/Deloitte/za/Documents/africa/DeloitteZA_Is_South_Africa_the_next_Brazil_Sep2017.pdf (accessed on 2 May 2023).
- Dekkers, Arnold L. M., John H. J. Einmahl, and Laurens de Haan. 1989. A moment estimator for the index of an extreme-value distribution. Annals of Statistics 17: 1833–55. [Google Scholar] [CrossRef]
- Dyhrberg, Anne Haubo. 2016. BitCoin, gold and the dollar—A Garch volatility analysis. Finance Research Letters 16: 85–92. [Google Scholar] [CrossRef]
- Fisher, Ronals, and Leonard Tippett. 1928. Limiting forms of the frequency distribution of the largest or smallest member of a sample. Mathematical Proceedings of the Cambridge Philosophical Society 24: 180–90. [Google Scholar] [CrossRef]
- Fratzscher, Marcel. 2002. On Currency Crises and Contagion. Working Paper Series 139; Franfurt am Main: European Central Bank. [Google Scholar]
- Ghalanos, Alexios. 2020. rugarch: Univariate GARCH Models. R Package Version 1.4-4. Available online: https://cran.r-project.org/web/packages/rugarch/index.html (accessed on 10 December 2022).
- Gneiting, Tilmann. 2011. Making and evaluating point forecasts. Journal of the American Statistical Association 106: 746–62. [Google Scholar] [CrossRef]
- Grable, John. 2000. Financial risk tolerance and additional factors that affect risk taking in everyday money matters. Journal of Business and Psychology 14: 625–30. [Google Scholar] [CrossRef]
- Heffernan, Janet E., and Alec G. Stephenson. 2018. ismev: An Introduction to Statistical Modeling of Extreme Values. R Package Version 1.42. Available online: https://cran.r-project.org/web/packages/ismev/ismev.pdf (accessed on 10 December 2022).
- Hull, John. 2006. Risk Management and Financial Institutions, 1st ed. Hoboken: Prentice Hall. [Google Scholar]
- Ibrahim, Mohamed, Walid Emam, Yusra Tashkandy, Mir Masoom Ali, and Haitham M. Yousof. 2023. Bayesian and Non-Bayesian Risk Analysis and Assessment under Left-Skewed Insurance Data and a Novel Compound Reciprocal Rayleigh Extension. Mathematics 11: 1593. [Google Scholar] [CrossRef]
- Joale, Dan. 2011. Analyzing the Effect of Exchange Rate Volatility on South Africa’s Exports to the US—Theory and Evidence. SSRN Electronic Journal. [Google Scholar] [CrossRef]
- Kaseke, Forbes, Shaun Ramroop, and Henry Mwambi. 2021. A Comparison of the Stylised Facts of BitCoin, Ethereum and the JSE Stock Returns. African Finance Journal 23: 50–64. [Google Scholar]
- Kupiec, Paul H. 1995. Techniques for verifying the accuracy of risk management models. Journal of Derivatives 3: 73–84. [Google Scholar] [CrossRef]
- Markowitz, Harry. 1959. Portfolio Selection: Efficient Diversification of Investments. New York: John Wiley & Sons. [Google Scholar]
- Penalva, Helena, Sandra Nunes, and Manuela M. Neves. 2016. Extreme Value Analysis—A Brief Overview With an Application to Flow Discharge Rate Data in A Hydrometric Station In The North Of Portugal. REVSTAT—Statistical Journal Volume 14: 193–215. [Google Scholar]
- Pflug, Georg Ch. 2000. Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk. Probabilistic Constrained Optimization. Boston: Springer, pp. 272–81. [Google Scholar]
- Pickands, James, III. 1975. Statistical inference using extreme order statistics. Annals of Statistics 3: 119–31. [Google Scholar] [CrossRef]
- Pretorius, Anmar, and Jesse De Beer. 2002. Financial Contagion in Africa: South Africa and a Troubled Neighbour, Zimbabwe. Paper presented at the 7th Annual Conference of the African Econometrics Society, Kruger National Park, South Africa, June 19–23. [Google Scholar]
- R Core Team. 2021. R: A Language and Environment for Statistical Computing. Vienna: R Foundation for Statistical Computing. Available online: https://www.R-project.org/ (accessed on 1 December 2021).
- Rached, Imen, and Elisabeth Larsson. 2019. Tail Distribution and Extreme Quantile Estimation Using Non-parametric Approaches. In High-Performance Modelling and Simulation for Big Data Applications. Cham: Springer, vol. 11400. [Google Scholar] [CrossRef]
- Rockafellar, Ralph Tyrrell, and Stan Uryasev. 2002. Conditional value-at-risk for general loss distributions. Journal of Banking & Finance 26: 1443–71. [Google Scholar] [CrossRef]
- RStudio Team. 2022. RStudio: Integrated Development Environment for R. Boston: RStudio, PBC. Available online: http://www.rstudio.com/ (accessed on 10 December 2022).
- Scarrott, Carl, and Anna MacDonald. 2012. A Review of Extreme Value Threshold Estimation and Uncertainty Quantification. REVSTAT-Statistical Journal 10: 33–60. [Google Scholar]
- Shanaev, Savva, and Binam Ghimire. 2021. A fitting return to fitting returns: Cryptocurrency distributions revisited. SSRN Electronic Journal 1: 1–33. [Google Scholar] [CrossRef]
- Smith, Richard L. 1987. Estimating Tails of Probability Distributions. The Annals of Statistics 15: 1174–207. [Google Scholar] [CrossRef]
- Takaishi, Tetsuya. 2018. Statistical properties and multifractality of BitCoin. Physica A: Statistical Mechanics and Its Applications 506: 507–19. [Google Scholar] [CrossRef]
- Tretina, Kat. 2023. Top 10 Cryptocurrencies of 2023. Forbes Advisor. Available online: https://www.forbes.com/advisor/investing/cryptocurrency/top-10-cryptocurrencies/ (accessed on 7 March 2023).
- Van Der Merwe, E. 1996. Exchange Rate Management Policies in South Africa: Recent Experience and Prospects. South African Reserve Bank Occasional Paper No. 8, June 1995. Pretoria: South African Reserve Bank Occasional. [Google Scholar]
- Yamai, Yasuhiro, and Toshinao Yoshiba. 2002. Comparative Analysis of Expected Shortfall and Value at Risk: Their Estimation Error, Decomposition, and Optimization. Monetary and Economic Studies 20: 87–121. [Google Scholar]
- Yousof, Haitham M., Walid Emam, Yusra Tashkandy, Mir Masoom Ali, Richard Minkah, and Mohamed Ibrahim. 2023a. A Novel Model for Quantitative Risk Assessment under Claim-Size Data with Bimodal and Symmetric Data Modeling. Mathematics 11: 1284. [Google Scholar] [CrossRef]
- Yousof, Haitham M., Yusra Tashkandy, Walid Emam, Mir Masoom Ali, and Mohamed Ibrahim. 2023b. A New Reciprocal Weibull Extension for Modeling Extreme Values with Risk Analysis under Insurance Data. Mathematics 11: 966. [Google Scholar] [CrossRef]
- Zhang, Yuanyuan, and Saralees Nadarajah. 2017. A review of backtesting for value at risk. Communications in Statistics—Theory and Methods 47: 3616–39. [Google Scholar] [CrossRef]

**Figure 1.**Plot of BTC/USD prices (

**left**) and one-day log returns (

**right**). Reproduced with permission from Chikobvu and Ndlovu (2023).

**Figure 2.**Plot of ZAR/USD prices (

**left**) and one-day log returns (

**right**). Reproduced with permission from Chikobvu and Ndlovu (2023).

**Table 1.**Descriptive statistics of exchange rate price returns. Reproduced with permission from Chikobvu and Ndlovu (2023).

Observations | Mean | Median | Maximum | Minimum | Skewness | Kurtosis | |

BTC/USD | 2370 | 0.001990 | 0.001757 | 0.237220 | −0.480904 | −0.994382 | 16.15451 |

ZAR/USD | 1694 | −0.000125 | 0.000000 | 0.049546 | −0.048252 | −0.264130 | 4.121644 |

Test for normality, autocorrelation, and heteroscedasticity | |||||||

BTC/USD | ZAR/USD | ||||||

TEST | Statistic | p-value | Statistic | p-value | |||

Jarque–Bera | 17,478.40 | 0.000000 | 108.4967 | 0.000000 | |||

Ljung–Box | 11.7 | 0.0006249 | 0.40504 | 0.5245 | |||

ARCH LM Test | 52.87 | 4.345 × 10^{−7} | 70.789 | 2.28 × 10^{1} | |||

Test for unit root and stationarity | |||||||

BTC/USD | ZAR/USD | ||||||

Unit Root Test | Statistic | p-value | Statistic | p-value | |||

ADF Test | −52.20130 | 0.0001 | −40.47263 | 0.0000 | |||

PP Test | −52.10963 | 0.0001 | −40.47011 | 0.0000 | |||

KPSS Test | 0.092067 | 0.347000 | 0.090747 | 0.347000 |

Model | Number of Exceedances | $\widehat{\mathit{\xi}}$ | $\mathit{S}\mathit{e}\left(\widehat{\mathit{\xi}}\right)$ | $\widehat{\mathit{\beta}}$ | $\mathit{S}\mathit{e}\left(\widehat{\mathit{\beta}}\right)$ |
---|---|---|---|---|---|

BTC/USD Gains | 396 | 0.0302 | 0.0527 | 0.0284 | 0.0021 |

BTC/USD Losses | 319 | 0.1096 | 0.0535 | 0.0311 | 0.0024 |

ZAR/USD Gains | 229 | −0.0164 | 0.0650 | 0.0005 | 0.0064 |

ZAR/USD Losses | 243 | −0.0844 | 0.0313 | 0.0065 | 0.0003 |

BTC/USD | ZAR/USD | |||
---|---|---|---|---|

Losses | Gains | Losses | Gains | |

90% | 0.07 | 0.06 | 0.02 | 0.02 |

95% | 0.09 | 0.08 | 0.02 | 0.03 |

99% | 0.16 | 0.13 | 0.03 | 0.03 |

BTC/USD | ZAR/USD | |||
---|---|---|---|---|

Losses | Gains | Losses | Gains | |

90% | 0.11 | 0.09 | 0.02 | 0.02 |

95% | 0.13 | 0.11 | 0.02 | 0.03 |

99% | 0.21 | 0.17 | 0.03 | 0.04 |

BTC/USD | ZAR/USD | |||
---|---|---|---|---|

Losses | Gains | Losses | Gains | |

90% | 0.9901 | 0.96278 | 0.7140 | 0.9811 |

95% | 0.6725 | 0.4288 | 0.420 | 0.5514 |

99% | 0.3694 | 0.4212 | 0.8316 | 0.7375 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ndlovu, T.; Chikobvu, D.
The Generalised Pareto Distribution Model Approach to Comparing Extreme Risk in the Exchange Rate Risk of BitCoin/US Dollar and South African Rand/US Dollar Returns. *Risks* **2023**, *11*, 100.
https://doi.org/10.3390/risks11060100

**AMA Style**

Ndlovu T, Chikobvu D.
The Generalised Pareto Distribution Model Approach to Comparing Extreme Risk in the Exchange Rate Risk of BitCoin/US Dollar and South African Rand/US Dollar Returns. *Risks*. 2023; 11(6):100.
https://doi.org/10.3390/risks11060100

**Chicago/Turabian Style**

Ndlovu, Thabani, and Delson Chikobvu.
2023. "The Generalised Pareto Distribution Model Approach to Comparing Extreme Risk in the Exchange Rate Risk of BitCoin/US Dollar and South African Rand/US Dollar Returns" *Risks* 11, no. 6: 100.
https://doi.org/10.3390/risks11060100