Probability-unbiased Value-at-Risk estimators

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.