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Probability-unbiased Value-at-Risk estimators

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  • Ivo Francioni
  • Florian Herzog

Abstract

The aim of this paper is to introduce a new property for good quantile or Value-at-Risk (VaR) estimators. We define probability unbiasedness for the α-quantile estimator from a finite amount of data ( ) for the distribution function F θ with parameter θ such that the estimator also has this probabilistic ‘threshold property’ in expectation of a F θ distributed random variable X for all θ, i.e. . Probability unbiasedness means that the α-quantile estimated from a finite amount of data is only exceeded by probability α for the next observation from the same distribution. Moreover, we show that plug-in estimators for estimating quantiles at a given probability are not unbiased with respect to the probability unbiasedness property, i.e. using a Maximum Likelihood Estimator for the parameters and plugging in the estimated values into the distribution function to obtain the quantile values. Therefore, estimating a quantile needs to be corrected for observing only finitely many samples with a distortion function to obtain a probability-unbiased estimator. In the case of estimating the VaR (quantile) of a normally distributed random variable, the distortion function is calculated via the distribution of the VaR/quantile. Using the distribution derived for the VaR estimate, we also quantify the approximate probability-unbiased confidence bands of the VaR for a finite amount of data. In the last part, the new VaR estimator is tested on a time series. It outperforms the other models examined, all of which are much more complex.

Suggested Citation

  • Ivo Francioni & Florian Herzog, 2012. "Probability-unbiased Value-at-Risk estimators," Quantitative Finance, Taylor & Francis Journals, vol. 12(5), pages 755-768, November.
  • Handle: RePEc:taf:quantf:v:12:y:2012:i:5:p:755-768
    DOI: 10.1080/14697681003687569
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    Citations

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    Cited by:

    1. Pitera, Marcin & Schmidt, Thorsten, 2018. "Unbiased estimation of risk," Journal of Banking & Finance, Elsevier, vol. 91(C), pages 133-145.
    2. Marcin Pitera & Thorsten Schmidt, 2016. "Unbiased estimation of risk," Papers 1603.02615, arXiv.org, revised Aug 2017.
    3. Dominique Guegan & Bertrand K. Hassani & Kehan Li, 2016. "Measuring risks in the extreme tail: The extreme VaR and its confidence interval," Documents de travail du Centre d'Economie de la Sorbonne 16034rr, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne, revised Jan 2017.
    4. Dominique Guegan & Bertrand Hassani & Kehan Li, 2017. "Measuring risks in the extreme tail: The extreme VaR and its confidence interval," Post-Print halshs-01317391, HAL.

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