We see both for observed data and analytically , that GARCH ES(97.5%) and VaR(99%) are almost exactly the same.

Below, we further examine these results by means of Monte Carlo simulations. We pick two distributions that are representative of actual distribution assumptions by industry, the normal and the and the Student-t(3). The former, is probably the most common in industry, perhaps in its conditional forms, like GARCH, while the parametric Student-t sees some applications. The main value in making use of the Student-t in the Monte Carlo experiments is that itâ€™s tail thickness resembles that of observed asset returns and therefore provides a realistic experiment for historical simulation estimation.

We consider two sample sizes, 10^{4} and 300. The former aims at getting at asymptotic results while the latter is more related to actual sample sizes used by industry. While one could increase the lower sample size, the results should be qualitatively similar, sitting between the small and large sample results.

The number of simulations is 10^{5}.

In the case of the Student-t we estimate the degrees of freedom simultaneously with the standard deviation with a maximum likelihood procedure.

HS stands for historical simulation and parametric for a procedure whereby we estimate the standard error of the Monte Carlo samples and in the case of the Student, the degrees of freedom as well.<p>

We report the both the mean and the standard error of the VaR and ES across the simulations.<p>

The probabilities are 99% and 97.%.

There are four tables reported below, for the Gaussian and the Student for both probabilities. The means should correspond almost exactly to those in the analytical results up to Monte Carlo error and problems caused by the sample size not being infinite.

HS VaR | HS ES | Parametric VaR | Parametric ES | |
---|---|---|---|---|

mean | 23.28 | 26.63 | 23.26 | 26.65 |

se | 0.37 | 0.46 | 0.16 | 0.19 |

HS VaR | HS ES | Parametric VaR | Parametric ES | |
---|---|---|---|---|

mean | 19.61 | 23.37 | 19.60 | 23.38 |

se | 0.27 | 0.32 | 0.14 | 0.17 |

HS VaR | HS ES | Parametric VaR | Parametric ES | |
---|---|---|---|---|

mean | 45.52 | 69.93 | 45.41 | 70.05 |

se | 1.69 | 4.83 | 0.95 | 2.43 |

HS VaR | HS ES | Parametric VaR | Parametric ES | |
---|---|---|---|---|

mean | 31.85 | 50.36 | 31.82 | 50.40 |

se | 0.81 | 2.27 | 0.46 | 1.34 |

For this small sample size, we expect some anomalies to emerge, especially for the Student. The results still broadly correspond to the large sample upon and the analytical results , but the effects of the small sample become visible, especially in the much larger standard error of the estimates. This is what we would expect.

HS VaR | HS ES | Parametric VaR | Parametric ES | |
---|---|---|---|---|

mean | 23.85 | 26.08 | 23.24 | 26.63 |

se | 2.27 | 2.58 | 0.95 | 1.09 |

HS VaR | HS ES | Parametric VaR | Parametric ES | |
---|---|---|---|---|

mean | 20.16 | 23.36 | 19.58 | 23.36 |

se | 1.61 | 1.88 | 0.80 | 0.96 |

HS VaR | HS ES | Parametric VaR | Parametric ES | |
---|---|---|---|---|

mean | 49.61 | 67.37 | 45.40 | 70.86 |

se | 12.30 | 26.92 | 5.57 | 14.92 |

HS VaR | HS ES | Parametric VaR | Parametric ES | |
---|---|---|---|---|

mean | 34.03 | 50.79 | 31.78 | 50.71 |

se | 5.43 | 13.66 | 2.69 | 8.12 |

In order to further investigate the relative relationship between the VaR and ES, we calculate the the ratio of the 99% VaR to 97.5% ES, obtained from the table above.

As expected, the parametric methods, give the same results as the analytic results in here, but what this table adds is showing that the same result obtains almost exactly for historical simulation for the large sample, for both distributions, and to a lesser extent also for the small sample.

HS VaR p0.01/ES0.025 | Parametric VaR p0.01/ES0.025 | |
---|---|---|

Gaussian | 0.996 | 0.995 |

Student-t(3) | 0.904 | 0.901 |

HS VaR p0.01/ES0.025 | Parametric VaR p0.01/ES0.025 | |
---|---|---|

Gaussian | 1.021 | 0.995 |

Student-t(3) | 0.977 | 0.895 |

Finally, we are interested in the relative volatility of 97.5% ES compared to the 99% VaR. To this end, we report the ratio of the standard errors, obtained from above. These results suggest that ES 97.5% is forecasted less precisely than VaR 99% in all cases except nonparametric Gaussian.

HS VaR p0.01/ES0.025 | Parametric VaR p0.01/ES0.025 | |
---|---|---|

Gaussian | 1.171 | 0.995 |

Student-t(3) | 0.743 | 0.705 |

HS VaR p0.01/ES0.025 | Parametric VaR p0.01/ES0.025 | |
---|---|---|

Gaussian | 1.212 | 0.995 |

Student-t(3) | 0.900 | 0.686 |

The results reported here show no big surprises, they confirm both the observed data and analytical results, that ES(97.5%) and VaR(99%) are almost exactly the same, across both the distributions and estimation methods considered here.

There is weak evidence that the ES(97.5%) are slightly less accurate and more volatile than VaR(99%), leading to the conclusion that VaR(99%) is preferred.