Bias-corrected peaks-over-threshold estimation of the CVaR

Dylan Troop, Frédéric Godin, Jia Yuan Yu
Proceedings of the Thirty-Seventh Conference on Uncertainty in Artificial Intelligence, PMLR 161:1809-1818, 2021.

Abstract

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 address this bias-variance tradeoff by deriving 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 CVaR estimator that is asymptotically unbiased and less sensitive to lower thresholds being used. An asymptotic confidence interval for the estimator is also constructed. In a practical setting, we show through experiments that our estimator provides a significant performance improvement compared with competing CVaR estimators in finite samples from heavy-tailed distributions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v161-troop21a, title = {Bias-corrected peaks-over-threshold estimation of the CVaR}, author = {Troop, Dylan and Godin, Fr\'ed\'eric and Yu, Jia Yuan}, booktitle = {Proceedings of the Thirty-Seventh Conference on Uncertainty in Artificial Intelligence}, pages = {1809--1818}, year = {2021}, editor = {de Campos, Cassio and Maathuis, Marloes H.}, volume = {161}, series = {Proceedings of Machine Learning Research}, month = {27--30 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v161/troop21a/troop21a.pdf}, url = {https://proceedings.mlr.press/v161/troop21a.html}, abstract = {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 address this bias-variance tradeoff by deriving 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 CVaR estimator that is asymptotically unbiased and less sensitive to lower thresholds being used. An asymptotic confidence interval for the estimator is also constructed. In a practical setting, we show through experiments that our estimator provides a significant performance improvement compared with competing CVaR estimators in finite samples from heavy-tailed distributions.} }
Endnote
%0 Conference Paper %T Bias-corrected peaks-over-threshold estimation of the CVaR %A Dylan Troop %A Frédéric Godin %A Jia Yuan Yu %B Proceedings of the Thirty-Seventh Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2021 %E Cassio de Campos %E Marloes H. Maathuis %F pmlr-v161-troop21a %I PMLR %P 1809--1818 %U https://proceedings.mlr.press/v161/troop21a.html %V 161 %X 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 address this bias-variance tradeoff by deriving 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 CVaR estimator that is asymptotically unbiased and less sensitive to lower thresholds being used. An asymptotic confidence interval for the estimator is also constructed. In a practical setting, we show through experiments that our estimator provides a significant performance improvement compared with competing CVaR estimators in finite samples from heavy-tailed distributions.
APA
Troop, D., Godin, F. & Yu, J.Y.. (2021). Bias-corrected peaks-over-threshold estimation of the CVaR. Proceedings of the Thirty-Seventh Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 161:1809-1818 Available from https://proceedings.mlr.press/v161/troop21a.html.

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