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The conditional value-at-risk (CVaR) is a useful risk measure in fields suchas machine learning, finance, insurance, energy, etc. When measuring veryextreme risk, the commonly used CVaR estimation method of sample averaging doesnot work well due to limited data above the value-at-risk (VaR), the quantilecorresponding to the CVaR level. To mitigate this problem, the CVaR can beestimated by extrapolating above a lower threshold than the VaR using ageneralized Pareto distribution (GPD), which is often referred to as thepeaks-over-threshold (POT) approach. This method often requires a very highthreshold to fit well, leading to high variance in estimation, and can inducesignificant bias if the threshold is chosen too low. In this paper, we derive anew expression for the GPD approximation error of the CVaR, a bias term inducedby the choice of threshold, as well as a bias correction method for theestimated GPD parameters. This leads to the derivation of a new estimator forthe CVaR that we prove to be asymptotically unbiased. In a practical setting,we show through experiments that our estimator provides a significantperformance improvement compared with competing CVaR estimators in finitesamples. As a consequence of our bias correction method, it is also shown thata much lower threshold can be selected without introducing significant bias.This allows a larger portion of data to be be used in CVaR estimation comparedwith the typical POT approach, leading to more stable estimates. As secondaryresults, a new estimator for a second-order parameter of heavy-taileddistributions is derived, as well as a confidence interval for the CVaR whichenables quantifying the level of variability in our estimator.

Bias-Corrected Peaks-Over-Threshold Estimation of the CVaR