  The conditional value-at-risk (CVaR) is a useful risk measure in fields such
as machine learning, finance, insurance, energy, etc. When measuring very
extreme risk, the commonly used CVaR estimation method of sample averaging does
not work well due to limited data above the value-at-risk (VaR), the quantile
corresponding to the CVaR level. To mitigate this problem, the CVaR can be
estimated by extrapolating above a lower threshold than the VaR using a
generalized Pareto distribution (GPD), which is often referred to as the
peaks-over-threshold (POT) approach. This method often requires a very high
threshold to fit well, leading to high variance in estimation, and can induce
significant bias if the threshold is chosen too low. In this paper, we derive a
new expression for the GPD approximation error of the CVaR, a bias term induced
by the choice of threshold, as well as a bias correction method for the
estimated GPD parameters. This leads to the derivation of a new estimator for
the CVaR that we prove to be asymptotically unbiased. In a practical setting,
we show through experiments that our estimator provides a significant
performance improvement compared with competing CVaR estimators in finite
samples. As a consequence of our bias correction method, it is also shown that
a much lower threshold can be selected without introducing significant bias.
This allows a larger portion of data to be be used in CVaR estimation compared
with the typical POT approach, leading to more stable estimates. As secondary
results, a new estimator for a second-order parameter of heavy-tailed
distributions is derived, as well as a confidence interval for the CVaR which
enables quantifying the level of variability in our estimator.
