# Extreme Value Theory Modelling of the Behaviour of Johannesburg Stock Exchange Financial Market Data

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### Rationale

## 3. Research Methodology and Analytical Procedures

#### 3.1. Data Source and Study Area

#### 3.2. Methods

#### 3.2.1. Generalised Extreme Value Distribution (GEVD)

#### 3.2.2. Blended GEVD (bGEVD)

#### 3.2.3. bGEVD Models

#### 3.2.4. r-Largest-Order GEVD (GEVD${}_{r}$)

#### 3.2.5. Generalised Pareto Distribution (GPD)

**Theorem**

**1.**

#### 3.2.6. Poisson Point Process

**Theorem**

**2.**

## 4. Results and Discussion

#### 4.1. Exploratory Data Analysis

#### 4.2. EVT Results

#### 4.3. GEVD Models

#### Diagnostic Plots

#### 4.4. bGEVD Models

#### 4.5. GEVD${}_{r}$ Models

#### 4.5.1. Entropy Difference Test Results

#### 4.5.2. Diagnostic Plots

#### 4.6. GPD Models

#### 4.7. Poisson Point Process Models

#### 4.8. Return Levels

## 5. Conclusions and Recommendations

#### 5.1. Recommendations

#### 5.2. Future Studies

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ALSTRI | All-Share Total Return Index |

bGEVD | Blended Generalised Extreme Value Distribution |

COVID-19 | Coronavirus of 2019 |

EVT | Extreme Value Theory |

FTSE/JSE-ALSI | Financial Time Series Exchange/Johannesburg Stock Exchange–All-Share index |

GEVD | Generalised Extreme Value Distribution |

GEVD${}_{r}$ | r-largest Generalised Extreme Value Distribution |

GPD | Generalised Pareto Distribution |

i.i.d | independent and identically distributed |

INLA | Integrated Nested Laplace Approximation |

JSE | Johannesburg Stock Exchange |

MCMC | Markov Chain Monte Carlo |

MLE | Maximum Likelihood Estimation |

NICD | National Institute for Communicable Diseases |

POT | Peak-Over-Threshold |

P-P plot | Probability–Probability plot |

p-value | Probability Value |

Q-Q plot | Quantile–Quantile plot |

SAWS | South African Weather Services |

USD–ZAR | United States Dollar against the South African Rand |

ZAR | South African Rand |

## References

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**Figure 1.**Time series plot of the total return index of the ALSTRI from 1 February 2016 to 26 April 2021.

Min | 1Q | Median | Mean | 3Q | Max | kurt | Skew | |
---|---|---|---|---|---|---|---|---|

ALSTRI | 5802 | 7336 | 8162 | 8068 | 8518 | 10,860 | 4.175 | 0.7622 |

Log Returns | −0.102 | −0.005 | 0.0008 | 0.0003 | 0.006 | 0.0726 | 11.108 | −0.955 |

USD–ZAR | 11.58 | 13.49 | 14.31 | 14.43 | 15.05 | 18.99 | 0.716 | 0.719 |

Log Returns | −0.0599 | −0.0069 | −0.0008 | 0.0000886 | 0.0064 | 0.0485 | 2.099 | 0.246 |

**Key:**ALSTRI = all-share total return index; 1Q = 1st quantile; 3Q = 3rd quantile; Kurt = kurtosis; Min = minimum; Max = maximum; USD–ZAR = USD–ZAR exchange rate; Skew = skewness.

Location $\left(\mathit{\mu}\right)$ | Scale $\left(\mathit{\sigma}\right)$ | Shape $\left(\mathit{\xi}\right)$ | 95% CI of $\left(\mathit{\xi}\right)$ | NLL $\left(\mathit{\lambda}\right)$ | |
---|---|---|---|---|---|

ALSTRI | $7960.54\left(104.56\right)$ | 743.68 (75.43) | −0.083 (0.084) | (−3.02, 0.007) | 503.0 |

USD–ZAR | $14.27\left(0.179\right)$ | 1.292 (0.124) | −0.115 (0.077) | (−0.267, 0.037) | 111.25 |

**Key:**ALSTRI = all-share total return index; USD–ZAR = USD–ZAR exchange rate; CI = confidence intervals; NLL = negative log-likelihood.

Location $\left(\mathit{\mu}\right)$ | log.Scale $\left(\mathit{\sigma}\right)$ | Shape $\left(\mathit{\xi}\right)$ | 95% CI of $\left(\mathit{\xi}\right)$ | PM of $\left(\mathit{\xi}\right)$ | |
---|---|---|---|---|---|

ALSTRI | $3.9385$ | 7187.27 | 0.0089 | (−0.2137, 0.1498) | −0.0545 |

USD–ZAR | $0.0553$ | 0.0270 | 0.0097 | (−0.266, 0.111) | −0.093 |

**Key:**ALSTRI = all-share total return index; CI = confidence intervals; PM = posterior mean; USD–ZAR = USD–ZAR exchange rate.

**Table 4.**Posterior means, standard deviation (SD (${\mu}_{q}$)), and quantiles for the estimated linear regression coefficients ${\beta}_{q}$, and ${\beta}_{s}$, and hyperparameters of the bGEVD INLA fits for the monthly maxima of the JSE financial market data.

Explanatory Variable | Mean | Standard Deviation | 2.5% Q | 50% Q | 97.5% | |
---|---|---|---|---|---|---|

ALSTRI | bGEVD intercept | 0.947 | 0.088 | 0.765 | 0.95 | 1.112 |

Monthly ALSTRI maxima | 0.387 | 0.212 | −0.008 | 0.38 | 0.823 | |

${s}_{\beta}$ | 1.143 | 0.121 | 0.925 | 1.135 | 1.402 | |

$\xi $ | 0.060 | 0.053 | 0.002 | 0.044 | 0.190 | |

${\beta}_{q}$ | 0.105 | 0.046 | 0..14 | 0.105 | 0.197 | |

${\beta}_{s}$ | 0.005 | 0.058 | −0.109 | 0.005 | 0.119 | |

USD–ZAR | bGEVD intercept | 1.162 | 0.116 | 0.954 | 1.155 | 1.408 |

Monthly USD–ZAR maxima | 0.392 | 0.258 | −0.189 | 0.421 | 0.815 | |

${s}_{\beta}$ | 1.380 | 0.155 | 1.105 | 1.370 | 1.715 | |

$\xi $ | 0.112 | 0.028 | 0.067 | 0.108 | 0.176 | |

${\beta}_{q}$ | 0.067 | 0.048 | −0.030 | 0.068 | 0.160 | |

${\beta}_{s}$ | 0.008 | 0.058 | −0.106 | 0.008 | 0.122 |

**Key:**Q = quantile; ${s}_{\beta}$ = spread for bGEVD observations; $\xi $ = tail for bGEVD observations; ${\beta}_{q}$ = estimated linear regression coefficients (spread) for bGEVD observations; ${\beta}_{s}$ = estimated linear regression coefficients (tail) for bGEVD observations.

**Table 5.**MLE parameter estimates and standard errors (in parantheses) for the GEVD${}_{r}$ monthly maxima.

r | $\widehat{\mathit{\mu}}$ | $\widehat{\mathit{\sigma}}$ | $\widehat{\mathit{\xi}}$ | 95% CI of $\widehat{\mathit{\xi}}$ | ${\widehat{\mathit{\lambda}}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|

ALSTRI | 1 | 7960.54 (104.56) | 743.68 (75.43) | −0.083 (0.084) | (−3.02, 0.007) | 503.70 |

2 | 7907.17 (76.30) | 771.05 (56.70) | 0.128 (0.059) | (−0.25, 8.15) | 503.697 | |

3 | 7921.10 (59.16) | 722.34 (41.37) | −0.074 (0.048) | (−0.16, 0.0198) | 1509.62 | |

4 | 7909.55 (51.26) | 724.97 (36.34) | −0.073 (0.042) | (−0.16, 8.23) | 2012.28 | |

5 | 7895.81 (45.68) | 721.93 (32.38) | 0.071 (0.03) | (−0.14, 0.0028) | 2514.90 | |

6 | 7881.46 (41.65) | 720.20 (29.53) | −0.068 (0.03) | (−0.13, −0.0005) | 3017.98 | |

7 | 7871.31 (38.49) | 717.21 (27.15) | −0.066 (0.03) | (−0.13, −0.0035) | 3520.82 | |

8 | 7855.15 (36.01) | 718.95 (25.76) | −0.06 (0.03) | (−0.12, −0.004) | 4023.53 | |

9 | 7848.58 (33.96) | 716.84 (24.06) | −0.064 (0.03) | (−0.12, −0.007) | 4527.09 | |

10 | 7836.89 (32.27) | 718.14 (22.92) | −0.064 (0.027) | (−0.12, −0.01) | 5030.87 | |

USD–ZAR | 1 | 14.27 (0.179) | 1.292 (0.124) | −0.115 (0.077) | (−0.267, 0.037) | 111.25 |

2 | 14.23 (0.126) | 1.285 (0.087) | −0.112 (0.055) | (−0.221, −0.005) | 221.983 | |

3 | 14.20 (0.11) | 1.28 (0.07) | −0.107 (0.04) | (−0.20, −0.02) | 332.21 | |

4 | 14.17 (0.09) | 1.27 (0.06) | −0.11 (0.03) | (−0.18, −0.03) | 442.12 | |

5 | 14.15 (0.08) | 1.27 (0.05) | −0.105 (0.03) | (−0.17, −0.04) | 552.13 | |

6 | 14.12 (0.072) | 1.267 (0.049) | −0.1048 (0.031) | (−0.1668, −0.0427) | 661.95 | |

7 | 14.10 (0.066) | 1.264 (0.046) | −0.1038 (0.029) | (−0.1614, −0.0464) | 771.54 | |

8 | 14.08 (0.062) | 1.260 (0.0428) | −0.1027 (0.0275) | (−0.157, −0.049) | 880.83 | |

9 | 14.06 (0.058) | 1.258 (0.040) | −0.1017 (0.0259) | (−0.153, −0.051) | 990.02 | |

10 | 14.04 (0.055) | 1.254 (0.038) | −0.101 (0.024) | (−0.149, −0.052) | 1098.61 |

**Key:**ALSTRI = all-share total return index; USD–ZAR = USD–ZAR exchange rate; CI = confidence intervals.

**Table 6.**Bayesian MCMC parameter estimates for GEVD${}_{r}$ monthly maxima with r = 1, … , r = 10, posterior means and confidence intervals of the shape parameter with lower limit (2.5%) and upper limit (97.5%).

r | $\widehat{\mathit{\mu}}$ | log−$\widehat{\mathit{\sigma}}$ | $\widehat{\mathit{\xi}}$ | CI of $\widehat{\mathit{\xi}}$ | Posterior Mean of $\widehat{\mathit{\xi}}$ | |
---|---|---|---|---|---|---|

ALSTRI | 1 | 3.9385 | 7187.27 | 0.0089 | (−0.2137, 0.1498) | −0.0545 |

2 | 3.5876 | 3510.50 | 0.0047 | (−0.1851, 0.08219) | −0.0680 | |

3 | 5.1359 | 2164.15 | 0.0027 | (−0.1651, 0.0369) | −0.0692 | |

4 | 12.5749 | 1516.26 | 0.0019 | (−0.1550, 0.0167) | −0.0734 | |

5 | 3.1170 | 1189.84 | 0.0015 | (−0.1448, 0.008247) | −0.0728 | |

6 | 3.2037 | 982.35 | 0.0013 | (−0.1387, 0.004181) | −0.0702 | |

7 | 5.2620 | 854.73 | 0.0010 | (−0.1309, −0.003485) | −0.0705 | |

8 | 6.1561 | 798.04 | 0.0009 | (−0.1256, −0.009892) | −0.0694 | |

9 | 32.413 | 665.25 | 0.0009 | (−0.1238, −0.0049) | −0.0653 | |

10 | 11.1110 | 552.88 | 0.0008 | (−0.1196, −0.00919) | −0.0671 | |

USD–ZAR | 1 | 0.0553 | 0.0270 | 0.0097 | (−0.266, 0.111) | −0.093 |

2 | 0.0227 | 0.0116 | 0.0045 | (−0.2286, 0.0389) | −0.1049 | |

3 | 0.0139 | 0.0065 | 0.0027 | (−0.1961, 0.0032) | −0.1035 | |

4 | 0.0101 | 0.0047 | 0.0019 | (−0.1833, −0.0142) | −0.1038 | |

5 | 0.0078 | 0.0039 | 0.0015 | (−0.175, −0.0252) | −0.1029 | |

6 | 0.0063 | 0.0030 | 0.0012 | (−0.1662, −0.0327) | −0.1001 | |

7 | 0.0052 | 0.0027 | 0.0011 | (−0.1620, −0.0329) | −0.1008 | |

8 | 0.0048 | 0.0021 | 0.0009 | (−0.1594, −0.0408) | −0.1014 | |

9 | 0.0039 | 0.0019 | 0.0008 | (−0.1564, −0.0431) | −0.1000 | |

10 | 0.0035 | 0.0017 | 0.0007 | (−0.1489, −0.0445) | −0.0993 |

**Key:**ALSTRI = all-share total return index; USD–ZAR = USD–ZAR exchange rate; CI = confidence intervals.

r | p-Value | ForwardStop | StrongStop | Statistic | $\widehat{\mathit{\mu}}$ | $\widehat{\mathit{\sigma}}$ | $\widehat{\mathit{\xi}}$ | |
---|---|---|---|---|---|---|---|---|

ALSTRI | 2 | < 0.001 | 0.1022 | 0.2011 | 5.0119 | 37.8250 | 13.8850 | −0.4403 |

3 | < 0.001 | 0.1159 | 0.0585 | 4.2664 | 42.9786 | 11.6131 | −0.4402 | |

4 | 0.0007 | 0.1314 | 0.0239 | 3.3734 | 46.1198 | 10.2285 | −0.4405 | |

5 | 0.0400 | 0.1531 | 0.0107 | 2.0533 | 48.2916 | 9.2697 | −0.4404 | |

USD–ZAR | 2 | 0.0039 | 0.1510 | 0.5396 | 2.8879 | 38.4297 | 14.1052 | −0.4405 |

3 | 0.0604 | 0.1694 | 0.4273 | 1.8780 | 43.6701 | 11.8000 | −0.4403 | |

4 | 0.0108 | 0.1847 | 0.2557 | 2.5498 | 46.8554 | 10.3947 | −0.4404 | |

5 | 0.0067 | 0.2137 | 0.1296 | 2.7106 | 49.0684 | 9.4226 | −0.4403 |

**Key:**ALSTRI = all-share total return index; USD–ZAR = USD–ZAR exchange rate.

Scale $\left(\mathit{\sigma}\right)$ | Shape $\left(\mathit{\xi}\right)$ | 95% CI of $\left(\mathit{\xi}\right)$ | NLL $\left(\mathit{\lambda}\right)$ | |
---|---|---|---|---|

ALSTRI | 478.528 (28.774) | 0.153 (0.046) | (0.064, 0.242) | 4800.515 |

USD–ZAR | 1.2363 (0.0774) | −0.05358 (0.049) | (−0.1511, 0.0439) | 795.886 |

**Key:**ASTRI = all-share total return index; USD–ZAR = USD–ZAR exchange rate; CI = confidence intervals; NLL = negative log-likelihood.

Location $\left(\mathit{\mu}\right)$ | Scale $\left(\mathit{\sigma}\right)$ | Shape $\left(\mathit{\xi}\right)$ | 95% CI of $\left(\mathit{\xi}\right)$ | NLL $\left(\mathit{\lambda}\right)$ | |
---|---|---|---|---|---|

ALSTRI | 11,714.95 (309.14) | 929.65 (159.75) | 0.125 (0.039) | (0.048, 0.203) | 2128.33 |

USD–ZAR | 19.915 (0.462) | 0.933 (0.198) | −0.053 (0.049) | (−0.150, 0.043) | −2128.96 |

**Key:**ASTRI = all-share total return index; USD–ZAR = USD–ZAR exchange rate; CI = confidence intervals; NLL = negative log-likelihood.

**Table 10.**Monthly block maxima return levels for GEVD, bGEVD, GEVD${}_{r=4}$, GPD, and the Poisson point process.

Model | Financial Market | 5 Years | 10 Years | 20 Years | 50 Years | 100 Years |
---|---|---|---|---|---|---|

GEVD | ALSTRI | 9075.05 | 9595.61 | 10,076.19 | 10,672.10 | 11,100.08 |

USD–ZAR | 16.13 | 16.97 | 17.71 | 18.61 | 19.23 | |

bGEVD | ALSTRI | 8688.45 | 9886.61 | 10,834.82 | 10,860 | 10,860 |

USD–ZAR | 16.83 | 17.63 | 18.96 | 18.99 | 18.99 | |

GEVD${}_{r=4}$ | ALSTRI | 8967.12 | 9447.35 | 9884.68 | 10,418.73 | 10,796.50 |

USD–ZAR | 15.95 | 16.75 | 17.45 | 18.29 | 18.87 | |

GPD | ALSTRI | 13,917.65 | 14,913.01 | 16,019.93 | 17,675.85 | 19,092.42 |

USD–ZAR | 21.35 | 21.93 | 22.50 | 23.21 | 23.72 | |

PPP | ALSTRI | 13,249.21 | 14,132.22 | 15,061.02 | 16,394.82 | 17,501.63 |

USD–ZAR | 21.26 | 21.89 | 22.47 | 23.20 | 23.72 |

**Key:**ALSTRI = all-share total return index; PPP = Poisson point process; USD–ZAR = USD–ZAR exchange rate.

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**MDPI and ACS Style**

Metwane, M.K.; Maposa, D.
Extreme Value Theory Modelling of the Behaviour of Johannesburg Stock Exchange Financial Market Data. *Int. J. Financial Stud.* **2023**, *11*, 130.
https://doi.org/10.3390/ijfs11040130

**AMA Style**

Metwane MK, Maposa D.
Extreme Value Theory Modelling of the Behaviour of Johannesburg Stock Exchange Financial Market Data. *International Journal of Financial Studies*. 2023; 11(4):130.
https://doi.org/10.3390/ijfs11040130

**Chicago/Turabian Style**

Metwane, Maashele Kholofelo, and Daniel Maposa.
2023. "Extreme Value Theory Modelling of the Behaviour of Johannesburg Stock Exchange Financial Market Data" *International Journal of Financial Studies* 11, no. 4: 130.
https://doi.org/10.3390/ijfs11040130