# Estimating and Forecasting Conditional Risk Measures with Extreme Value Theory: A Review

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## Abstract

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## 1. Introduction

**Volatility-EVT**. This class of models proposes a two step procedure that pre-whitens the returns with a model for the volatility and then applies a model based on EVT to the tails of the estimated residuals (Bee et al. 2016; McNeil and Frey 2000).**Quantile-EVT**. This class proposes using time-varying quantile models to obtain a dynamic threshold for the extremes. An extreme value model can then be applied to the exceedances over this threshold (Bee et al. 2018; Engle and Manganelli 2004).**Time-varying EVT**. This class models the returns exceeding a high constant threshold, letting the parameters of the extreme value model to be time-varying to account for the dependence in the exceedancees (Bee et al. 2015, Chavez-Demoulin et al. 2005, 2014).

## 2. Extreme Value Theory

#### 2.1. Main Results

- Let $\Phi \left(m\right)={e}^{-{m}^{-1/\xi}}$ with $\xi >0$ be the cumulative distribution function (cdf) of the Frechét distribution. As $x\to \infty $,$$F\in MDA(\Phi )\iff \overline{F}\left(x\right)={x}^{-\frac{1}{\xi}}L\left(x\right),$$
- Let $\Lambda \left(m\right)={e}^{-{e}^{-m}}$ being the cdf of the Gumbel distribution. As $x\to \infty $,$$F\in MDA(\Lambda )\iff \overline{F}\left(x\right)={e}^{-x}.$$
- Let $\Psi \left(m\right)={e}^{-{(-m)}^{-1/\xi}}$ with $\xi <0$ be the cdf of the Weibull distribution. As $x\to \infty $,$$F\in MDA(\Psi )\iff \overline{F}({\omega}_{F}-{x}^{-1})={x}^{\frac{1}{\xi}}L\left(x\right),$$

#### 2.2. The Peaks over Threshold Method

## 3. Estimating Conditional Risk Measures with EVT

#### 3.1. Volatility-EVT

#### 3.2. Quantile-EVT

#### 3.3. Time-Varying EVT

## 4. Discussion

#### 4.1. Volatility-EVT

#### 4.2. Quantile-EVT

#### 4.3. Time-Varying EVT

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Time series of the S&P500 returns (Upper-left), empirical kernel density of the returns (Upper-right), autocorrelation of the squared returns (Lower-left), and extremogram computed on both the upper (grey bullets) and lower tail (white bullets) of the returns, and their differences in red (Lower-right).

**Figure 3.**Predicted VaR (red) and ES (blue) at the $\alpha =0.01$ level for the different volatility models, and losses of the validation sample (grey).

**Figure 5.**Predicted VaR (red) and ES (blue) at the $\alpha =0.01$ level for the CAViaR and Realized CAViaR models, and losses of the validation sample (grey).

**Figure 6.**Predicted VaR (red) and ES (blue) at the $\alpha =0.01$ level for the DPOT and RPOT models, and losses of the validation sample (grey).

**Table 1.**p-values of the Unconditional Coverage (UC), Independent violations (IND) and Conditional Coverage (CC) tests of Christoffersen (1998) on the VaR predictions, and of the bootstrap test (BOOT) on the ES predictions. Rejections at the 5% level are in bold.

GARCH | levGARCH | HAR | levHAR | HEAVY | levHEAVY | |
---|---|---|---|---|---|---|

UC | 0.71 | 0.09 | 0.20 | 0.09 | 0.20 | 0.09 |

IND | 0.52 | 0.39 | 0.44 | 0.39 | 0.43 | 0.39 |

CC | 0.75 | 0.16 | 0.32 | 0.16 | 0.32 | 0.16 |

BOOT | 0.62 | 0.96 | 0.99 | 1.00 | 0.95 | 1.00 |

**Table 2.**p-values of the UC, IND and CC tests of Christoffersen (1998) on the VaR predictions, and of the bootstrap test (BOOT) on the ES predictions. Rejections at the 5% level are in bold.

CAViaR | RealCAVIAR | |
---|---|---|

UC | 0.41 | 0.09 |

IND | 0.47 | 0.40 |

CC | 0.54 | 0.16 |

BOOT | 0.81 | 1.00 |

**Table 3.**p-values of the UC, IND and CC tests of Christoffersen (1998) on the VaR predictions, and of the bootstrap test (BOOT) on the ES predictions. Rejections at the 5% level are in bold.

DPOT | RPOT | |
---|---|---|

UC | 0.41 | 0.71 |

IND | 0.02 | 0.52 |

CC | 0.06 | 0.75 |

BOOT | 0.70 | 0.94 |

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Bee, M.; Trapin, L.
Estimating and Forecasting Conditional Risk Measures with Extreme Value Theory: A Review. *Risks* **2018**, *6*, 45.
https://doi.org/10.3390/risks6020045

**AMA Style**

Bee M, Trapin L.
Estimating and Forecasting Conditional Risk Measures with Extreme Value Theory: A Review. *Risks*. 2018; 6(2):45.
https://doi.org/10.3390/risks6020045

**Chicago/Turabian Style**

Bee, Marco, and Luca Trapin.
2018. "Estimating and Forecasting Conditional Risk Measures with Extreme Value Theory: A Review" *Risks* 6, no. 2: 45.
https://doi.org/10.3390/risks6020045