# The Asymptotic Decision Scenarios of an Emerging Stock Exchange Market: Extreme Value Theory and Artificial Neural Network

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## Abstract

**:**

## 1. Introduction

- The estimates of the generalised extreme value and generalised Pareto distribution;
- The estimates of risk measures such as return level and value-at-risk (VaR);
- The backtesting procedure for validation of parameters’ estimate of EVT;
- The predictions of the stock exchange within a five-month trading period by the ANN model.

## 2. Materials and Methods

#### 2.1. Theoretical Review

#### 2.2. Extreme Value Theory (EVT)

**,**the Fischer–Tippet theorem in written as: $Pr\left\{{M}_{n}<m\right\}=Pr\left\{\left({W}_{n}-{\sigma}_{n}\right)/{\sigma}_{n}<m\right\}$.

#### 2.3. Return Levels

#### 2.4. The Generalized Pareto Distribution (GPD)

#### Peak-Over-Threshold (POT)

_{u}/n), where n denotes the overall size of the sample and N

_{u}represents the number of observations that exceed the predetermined threshold. Hence, the tail estimator represented in Equation (7) as:

#### 2.5. Expected Shortfall (ES)/Conditional Value-at-Risk (CVaR)

#### 2.6. Backtesting Procedure

_{ind}also has a chi-squared distribution with 1 degree of freedom. The conditional coverage test statistic is $L{R}_{cc}=L{R}_{ind}+L{R}_{unc}$ and has a ${\chi}^{2}\left(1\right)$ distribution. The study performed both tests at a 5% significance level. The critical values for the test are 3.841 for ${\chi}^{2}\left(1\right)$ distribution and 5.991 for a ${\chi}^{2}\left(2\right)$ distribution. If the model, when applied to an index, does not produce any violations, the test statistics cannot be calculated and consequently there is a rejection of the model (Vee et al. 2014).

#### 2.7. Neural Network Architect

#### 2.7.1. The Perceptron Model

#### 2.7.2. The Network Training Rules

#### 2.8. Methodology

_{1}: 0 whether maximum gain present month < 2.1203 or 1 if maximum gain current month > 2.1203.

_{2}= 0 when the maximum increase in the present month < 2.2260; conversely, T

_{2}= 1 when the maximum increase in the present month > 2.2260. The comparable significance of the network estimators for increases (${x}_{11}$, ${x}_{12}$, ${x}_{13}$, ${x}_{14}$, ${x}_{15}$) and decreases (${x}_{21}$, ${x}_{22}$, ${x}_{23}$, ${x}_{24}$, ${x}_{25}$) can thus be established. The last section in the methodology focuses on the risk measures related with GSE like the VaR and expected shortfall techniques founded on the peak-over threshold decision approach of GEV. These techniques are utilised to quantify the risk of increase and decreases at confidence levels of 95%, 98%, 99% and above.

## 3. Results

#### 3.1. Backtesting Results

_{ind}of 2.5662 is another indication that the violations produced are ungrouped. We illustrate the relative performance of the EVT model of the GSE index by producing graphs of the predicted VaR figure along with the actual returns as shown in Figure 3. The graphs are used to understand the scores of the loss function produced.

#### 3.2. Network Training Results

#### 3.3. Estimates of Expected Shortfall (ES)

#### 3.4. Risk Indicators for the Five-Month Trading Period

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Plots of value-at-risk (VaR) forecasts and log returns for Ghana Stock Exchange (GSE) index where extreme value theory (EVT) model is not rejected.

Series | Mean | Median | Max. | Min. | Std. Dev | Skew | Kurtosis | Jarque-Bera | n |
---|---|---|---|---|---|---|---|---|---|

Gains | 1.61 | 0.83 | 49.92 | −0.52 | 3.19 | 10.91 | 159.19 | 343,003 | 331 |

Losses | −1.26 | −0.71 | 2.69 | −48.44 | 3.00 | −12.08 | 186.16 | 470,749.3 | 331 |

Daily Returns | 0.11 | 0.02 | 49.92 | −48.44 | 1.35 | 1.15 | 689.02 | 0.0000 | 5239 |

GEV | Maximum Monthly Gains | ||
---|---|---|---|

Location | Scale | Shape | |

Estimates | $\mu $ = 0.6753 | $\sigma $ = 0.7177 | $\gamma $ = 0.5935 |

95% CI (normal app) | (0.5911, 0.7594) | (0.4914, 0.7094) | (0.6424, 0.7929) |

Standard error | 0.04293 | 0.03842 | 0.03719 |

t-ratios | 15.7303 | 18.6804 | 15.9586 |

Negative log-likelihood | 484.5558 | ||

Estimated return levels | 95% lower | Estimate | 95% upper |

2-month | 0.8514812 | 0.9572381 | 1.062995 |

3-month | 1.2909609 | 1.4463045 | 1.601648 |

4-month | 1.6124254 | 1.8150757 | 2.017726 |

5-month | 1.8724199 | 2.1203239 | 2.368228 |

6-month | 2.0934523 | 2.3847389 | 2.676025 |

7-month | 2.2871712 | 2.6201563 | 2.953141 |

8-month | 2.4604967 | 2.8336700 | 3.206843 |

9-month | 2.6179176 | 3.0299219 | 3.441926 |

GEV | Maximum Monthly Falls | ||
---|---|---|---|

Location | Scale | Shape | |

Estimates | $\mu $ = 0.5313 | $\sigma $ = 1.0725 | $\gamma $ = 0.0689 |

95% CI (normal app) | (0.4090, 0.6537) | (0.9903, 1.1546) | (0.0382, 0.0996) |

Standard error | 0.0624 | 0.0419 | 0.0157 |

t-ratios | 8.5144 | 25.5967 | 4.3885 |

Negative log-likelihood | 542.2874 | ||

Estimated return levels | 95% lower | Estimate | 95% upper |

2-month | 0.7932829 | 0.9294033 | 1.065524 |

3-month | 1.3626743 | 1.5301971 | 1.697720 |

4-month | 1.7321571 | 1.9265252 | 2.120893 |

5-month | 2.0084402 | 2.2260053 | 2.443570 |

6-month | 2.2299253 | 2.4679581 | 2.705991 |

7-month | 2.4151242 | 2.6715300 | 2.927936 |

8-month | 2.5744494 | 2.8475710 | 3.120693 |

9-month | 2.7143614 | 3.0028526 | 3.291344 |

Series | Violation | LR_{unc} | LR_{ind} | LR_{CC} | QL | AL | ASL | QUL |
---|---|---|---|---|---|---|---|---|

GSE Index | 7 | 1.5383 | 2.5662 | 4.1044 | 0.00195 | 0.0686 | 0.1570 | 0.0082 |

Maximum Monthly Returns | Model Summary | MSE | Percent Incorrect Predictions |
---|---|---|---|

Maximum monthly gains | Training | 15.620 | 4.2% |

Testing | 1.162 | 0.0% | |

Maximum monthly fall | Training | 4.191 | 0.4% |

Testing | 0.195 | 0.0% |

Maximum Monthly Gains | Maximum Monthly Falls | |||||
---|---|---|---|---|---|---|

N | Percent | N | Percent | |||

Sample | Training | 236 | 71.3% | Training | 225 | 68.0% |

Testing | 68 | 20.5% | Testing | 69 | 20.8% | |

Holdout | 27 | 8.2% | Holdout | 37 | 11.2% | |

Valid | 331 | 100.0% | 331 | 100.0% | ||

Total | 331 | 331 |

Maximum Monthly Gains | Maximum Monthly Falls | ||||
---|---|---|---|---|---|

Sample | Observed | Percent Correct | Sample | Observed | Percent Correct |

Training | 0 | 88.9% | Training | 0 | 90.0% |

1 | 83.9% | 1 | 87.1% | ||

Overall Percent | 87.8% | Overall Percent | 89.6% | ||

Testing | 0 | 89.1% | Testing | 0 | 90.0% |

1 | 88.9% | 1 | 91.0% | ||

Overall Percent | 90.7% | Overall Percent | 92.5% | ||

Holdout | 0 | 90.0% | Holdout | 0 | 90.0% |

1 | 90.0% | 1 | 90.0% | ||

Overall Percent | 90.0% | Overall Percent | 90.0% |

Maximum Monthly Gains | Maximum Monthly Fall | ||||
---|---|---|---|---|---|

Importance | Normalised Importance | Importance | Normalised Importance | ||

x_{11} | 0.190 | 85.8% | x_{21} | 0.154 | 72.5% |

x_{12} | 0.196 | 88.5% | x_{22} | 0.208 | 97.9% |

x_{13} | 0.192 | 86.7% | x_{23} | 0.213 | 99.9% |

x_{14} | 0.199 | 89.7% | x_{24} | 0.214 | 100.0% |

x_{15} | 0.222 | 100.0% | x_{25} | 0.212 | 99.6% |

**Table 9.**Neural network predictions for maximum gains and fall for 2016, 2017 and 2018 (January–May).

Month | Observed Maximum Gain | Actual Return Level Passes * (1/0) | Predicted by Network | Observed Maximum Fall | Actual Return Level Passes * (1/0) | Predicted by Network |
---|---|---|---|---|---|---|

January-2016 | 0.488747 | 0 | 0 | 0.574227 | 0 | 0 |

February-2016 | 0.789811 | 0 | 0 | 0.597471 | 0 | 0 |

March-2016 | 0.420338 | 0 | 0 | 0.858001 | 0 | 0 |

April-2016 | 0.248504 | 0 | 0 | 0.924067 | 0 | 0 |

May-2016 | 0.564332 | 0 | 0 | 1.34669 | 0 | 0 |

June-2016 | 0.688893 | 0 | 0 | 0.384274 | 0 | 0 |

July-2016 | 0.518302 | 0 | 0 | 0.435149 | 0 | 0 |

August-2016 | 0.724784 | 0 | 0 | 0.744693 | 0 | 0 |

September-2016 | 0.437602 | 0 | 0 | 0.495978 | 0 | 0 |

October-2016 | 1.029356 | 0 | 0 | 1.613268 | 0 | 0 |

November-2016 | 2.146811 | 1 | 0 | 2.758307 | 1 | 0 |

December-2016 | 1.781095 | 0 | 0 | 1.074789 | 0 | 0 |

January-2017 | 0.890714 | 0 | 0 | 0.255778 | 0 | 0 |

February-2017 | 1.963109 | 0 | 1 | 1.704333 | 0 | 0 |

March-2017 | 0.710071 | 0 | 0 | 0.760428 | 0 | 0 |

April-2017 | 0.84596 | 0 | 0 | 0.68419 | 0 | 0 |

May-2017 | 0.568504 | 0 | 0 | 0.748083 | 0 | 0 |

June-2017 | 0.652319 | 0 | 0 | 0.266952 | 0 | 0 |

July-2017 | 2.377966 | 1 | 1 | 0.158834 | 0 | 0 |

August-2017 | 0.905768 | 0 | 0 | 0.298297 | 0 | 0 |

September-2017 | 2.466185 | 1 | 1 | 2.914361 | 1 | 1 |

October-2017 | 1.326011 | 0 | 0 | 1.557715 | 0 | 0 |

November-2017 | 2.014369 | 0 | 0 | 0.685622 | 0 | 0 |

December-2017 | 0.758078 | 0 | 0 | 0.415567 | 0 | 0 |

January-2018 | 2.670898 | 1 | 1 | 0.181759 | 0 | 0 |

February-2018 | 2.055273 | 0 | 1 | 0.196484 | 0 | 0 |

March-2018 | 1.021322 | 0 | 0 | 0.994395 | 0 | 0 |

April-2018 | 0.534966 | 0 | 0 | 0.097597 | 0 | 0 |

May-2018 | 0.223925 | 0 | 0 | 1.724581 | 0 | 0 |

Parameters | Positive Returns | Negative Returns | ||
---|---|---|---|---|

Shape | Scale | Scale | Scale | |

Estimates | 0.8822 | 0.3303 | 0.9164 | 0.3193 |

Standard error | 0.07801 | 0.07072 | 0.10113 | 0.08481 |

t-ratio | 11.3088 | 4.6705 | 9.0616 | 3.1834 |

p-value | *** | *** | *** | *** |

Threshold call | 1.1867 | 1.2487 | ||

Proportion above | 0.0997 | 0.1003 | ||

No. above threshold | 316 | 180 |

Positive Returns (%) | Negative Returns (%) | |||
---|---|---|---|---|

Probability | Value-at-Risk | Expected Shortfall | Value-at-Risk | Expected Shortfall |

0.9500 | 1.871509 | 3.526073 | 1.955632 | 3.660353 |

0.9800 | 3.057482 | 5.296363 | 3.073298 | 5.556875 |

0.9900 | 4.225673 | 7.040111 | 4.246940 | 7.548380 |

0.9990 | 10.728086 | 16.746204 | 11.705815 | 20.205042 |

0.9999 | 24.632205 | 37.500756 | 30.907941 | 52.788352 |

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**MDPI and ACS Style**

Ibn Musah, A.-A.; Du, J.; Ud din Khan, H.S.; Abdul-Rasheed Akeji, A.A.
The Asymptotic Decision Scenarios of an Emerging Stock Exchange Market: Extreme Value Theory and Artificial Neural Network. *Risks* **2018**, *6*, 132.
https://doi.org/10.3390/risks6040132

**AMA Style**

Ibn Musah A-A, Du J, Ud din Khan HS, Abdul-Rasheed Akeji AA.
The Asymptotic Decision Scenarios of an Emerging Stock Exchange Market: Extreme Value Theory and Artificial Neural Network. *Risks*. 2018; 6(4):132.
https://doi.org/10.3390/risks6040132

**Chicago/Turabian Style**

Ibn Musah, Abdul-Aziz, Jianguo Du, Hira Salah Ud din Khan, and Alhassan Alolo Abdul-Rasheed Akeji.
2018. "The Asymptotic Decision Scenarios of an Emerging Stock Exchange Market: Extreme Value Theory and Artificial Neural Network" *Risks* 6, no. 4: 132.
https://doi.org/10.3390/risks6040132