# Estimating Conditional Value at Risk in the Tehran Stock Exchange Based on the Extreme Value Theory Using GARCH Models

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## Abstract

**:**

## 1. Introduction

## 2. Research Background

## 3. Research Hypothesis

## 4. Research Data and the Software Used

#### Citation of the Software Used

## 5. Research Methodology

#### 5.1. Estimating the Expected Shortfall

#### 5.2. Models for Predicting Mean and Volatility

#### 5.3. Estimating the α Percentile

_{s}that is big enough, the Peak Over Threshold function can be approximated by the generalized Pareto Distribution. Generalized Pareto Distribution is stated as follows:

_{u}. The methods of estimating the threshold include evaluating the Quantile–Quantile plot, the Mean Excess Function (MEF), and the Hill-plot. Based on the McNeil and Frey’s (2000) suggestion, it was best to choose the threshold in a manner that gave about a 100 observations for the regression of the Pareto Distribution. In this study, for the estimation of the threshold, the method proposed by Danielsson and De Vries (2000) was used. This method was based on the Monte Carlo Simulation of the Hill method and it could determine the amount of threshold for each of the models. In the present study, the available codes in the R software were used. It must be noted that when the data in the model were extensive, due to the market conditions and admission of new data, the threshold value needs to be revised from time to time, which makes research more difficult. In this research, re-estimation of the threshold, due to the admission of new data, was not done.

#### 5.4. Assessment of the Methodology Used

## 6. Data Analyses

- Step 1:
- Pre-Estimation Analyses
- Step 2:
- Estimation of Parameters
- Step 3:
- Post-Estimation Analyses

#### Pre-Estimation Analyses

- Analysis of the autocorrelation of the structure of returns and their squares.
- Analysis of the distribution of returns.

## 7. Conditional Extreme Value Theory

## 8. Backtesting Models

#### 8.1. Value at Risk Models Backtesting

_{1}is one of the VaR Breaks and the ratio is compared to distribution percentile, ${\chi}^{2}(1)$.

#### 8.2. Expected Shortfall Models Backtesting

## 9. Research Findings

## 10. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 9.**Partial Autocorrelation of squares of residual values for the AR(1)–GJR–GARCH (1,1) Model.

The Number of In-Sample Data (n) | 472 |

Degree of Freedom for the T-student distribution | 4 |

Number of Bootstrap Samples (Nr) | 10,000 |

**Table 2.**Maximum Likelihood Estimates (MLE) of the parameters of Generalized Pareto Distribution (GPD).

Tail | $\widehat{\mathit{\xi}}$ | $\mathit{S}\mathit{e}(\mathit{\xi})$ | $\widehat{\mathit{\sigma}}$ | $\mathit{s}\mathit{e}(\mathit{\sigma})$ |
---|---|---|---|---|

Tehran Stock Exchange | 0.22 | 0.108 | 1.5 | 0.371 |

Lag | ADF | p Value |
---|---|---|

No Drift, no trend | ||

0 | −3.76 | 0.0100 |

1 | −2.65 | 0.0100 |

2 | −2.61 | 0.0100 |

3 | −2.55 | 0.0115 |

4 | −2.39 | 0.0182 |

5 | −2.47 | 0.0147 |

With Drift, no trend | ||

0 | −4.81 | 0.0100 |

1 | −3.43 | 0.0108 |

2 | −3.45 | 0.0101 |

3 | −3.36 | 0.0141 |

4 | −3.26 | 0.0191 |

5 | −3.33 | 0.0159 |

With drift and trend | ||

0 | −6.09 | 0.0100 |

1 | −4.37 | 0.0100 |

2 | −4.37 | 0.0100 |

3 | −4.39 | 0.0100 |

4 | −4.18 | 0.0100 |

5 | −4.48 | 0.0100 |

Data | C_{t}^{P} | C_{t} | C_{n}^{P} | C_{n} | G_{t}^{P} | G_{t} | G_{n}^{P} | G_{n} | S_{t}^{P} | S_{t} | S_{n}^{P} | S_{n} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

q = 0.95 | |||||||||||||

Overall | 0.053 | 1.000 | 0.109 | 0.043 | 0.050 | 1.000 | 0.062 | 0.051 | 0.102 | 1.000 | 0.047 | 0.045 | |

Free Float | 0.086 | 0.766 | 0.015 | 0.086 | 0.213 | 0.848 | 0.049 | 0.077 | 0.192 | 0.771 | 0.047 | 0.078 | |

Industry | 0.218 | 0.834 | 0.153 | 0.087 | 0.225 | 0.877 | 0.171 | 0.055 | 0.241 | 0.870 | 0.072 | 0.068 | |

Top 50 | 0.592 | 0.772 | 0.185 | 0.033 | 0.619 | 0.764 | 0.132 | 0.040 | 0.537 | 0.827 | 0.065 | 0.035 | |

q = 0.975 | |||||||||||||

Overall | 0.184 | 1.000 | 0.250 | 0.001 | 0.229 | 1.000 | 0.160 | 0.001 | 0.312 | 1.000 | 0.110 | 0.001 | |

Free Float | 0.213 | 0.797 | 0.125 | 0.001 | 0.217 | 0.678 | 0.178 | 0.002 | 0.226 | 0.796 | 0.046 | 0.001 | |

Industry | 0.326 | 0.937 | 0.267 | 0.007 | 0.304 | 0.877 | 0.206 | 0.009 | 0.360 | 0.861 | 0.202 | 0.003 | |

Top 50 | 0.546 | 0.637 | 0.248 | 0.010 | 0.483 | 0.774 | 0.208 | 0.008 | 0.499 | 0.718 | 0.262 | 0.011 | |

q = 0.99 | |||||||||||||

Overall | 0.165 | 1.000 | 0.049 | 0.001 | 0.059 | 1.000 | 0.056 | 0.001 | 0.053 | 1.000 | 0.051 | 0.001 | |

Free Float | 0.065 | 0.654 | 0.011 | 0.001 | 0.029 | 0.423 | 0.025 | 0.001 | 0.033 | 0.424 | 0.047 | 0.001 | |

Industry | 0.061 | 0.719 | 0.024 | 0.001 | 0.081 | 0.757 | 0.047 | 0.002 | 0.046 | 0.738 | 0.041 | 0.001 | |

Top 50 | 0.034 | 0.245 | 0.033 | 0.001 | 0.142 | 0.398 | 0.072 | 0.001 | 0.083 | 0.352 | 0.061 | 0.001 | |

q = 0.995 | |||||||||||||

Overall | 0.043 | 0.300 | 0.045 | 0.001 | 0.073 | 0.635 | 0.066 | 0.001 | 0.093 | 0.536 | 0.086 | 0.001 | |

Free Float | 0.035 | 0.121 | 0.030 | 0.001 | 0.014 | 0.190 | 0.006 | 0.001 | 0.004 | 0.094 | 0.006 | 0.001 | |

Industry | 0.068 | 0.684 | 0.057 | 0.001 | 0.061 | 0.815 | 0.044 | 0.001 | 0.049 | 0.777 | 0.039 | 0.001 | |

Top 50 | 0.055 | 1.000 | 0.036 | 0.001 | 0.112 | 1.000 | 0.011 | 0.001 | 0.071 | 1.000 | 0.009 | 0.001 |

Rating | Description | Value at Risk | Expected Shortfall |
---|---|---|---|

2 | Peak Over Threshold GARCH with a T-student distribution for residual values | Green | Green |

4 | GARCH with a T-student distribution for residual values | Green | Green |

6 | Peak Over Threshold GARCH with a normal distribution for residual values | Green | White |

8 | GARCH with a normal distribution for residual values | Red | Red |

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**MDPI and ACS Style**

Tabasi, H.; Yousefi, V.; Tamošaitienė, J.; Ghasemi, F. Estimating Conditional Value at Risk in the Tehran Stock Exchange Based on the Extreme Value Theory Using GARCH Models. *Adm. Sci.* **2019**, *9*, 40.
https://doi.org/10.3390/admsci9020040

**AMA Style**

Tabasi H, Yousefi V, Tamošaitienė J, Ghasemi F. Estimating Conditional Value at Risk in the Tehran Stock Exchange Based on the Extreme Value Theory Using GARCH Models. *Administrative Sciences*. 2019; 9(2):40.
https://doi.org/10.3390/admsci9020040

**Chicago/Turabian Style**

Tabasi, Hamed, Vahidreza Yousefi, Jolanta Tamošaitienė, and Foroogh Ghasemi. 2019. "Estimating Conditional Value at Risk in the Tehran Stock Exchange Based on the Extreme Value Theory Using GARCH Models" *Administrative Sciences* 9, no. 2: 40.
https://doi.org/10.3390/admsci9020040