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SIMULTANEOUS CONFIDENCE BANDS FOR CONDITIONAL VALUE-AT-RISK AND EXPECTED SHORTFALL

Published online by Cambridge University Press:  03 August 2022

Shuo Li ,
Liuhua Peng  and
Xiaojun Song
Shuo Li
Affiliation:
Tianjin University of Finance and Economics
Liuhua Peng*
Affiliation:
The University of Melbourne
Xiaojun Song
Affiliation:
Peking University
*
Address correspondence to Liuhua Peng, School of Mathematics and Statistics, The University of Melbourne, Parkville, Victoria 3010, Australia; e-mail: liuhua.peng@unimelb.edu.au.
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Abstract

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Conditional value-at-risk (CVaR) and conditional expected shortfall (CES) are widely adopted risk measures which help monitor potential tail risk while adapting to evolving market information. In this paper, we propose an approach to constructing simultaneous confidence bands (SCBs) for tail risk as measured by CVaR and CES, with the confidence bands uniformly valid for a set of tail levels. We consider one-sided tail risk (downside or upside tail risk) as well as relative tail risk (the ratio of upside to downside tail risk). A general class of location-scale models with heavy-tailed innovations is employed to filter out the return dynamics. Then, CVaR and CES are estimated with the aid of extreme value theory. In the asymptotic theory, we consider two scenarios: (i) the extreme scenario that allows for extrapolation beyond the range of the available data and (ii) the intermediate scenario that works exclusively in the case where the available data are adequate relative to the tail level. For finite-sample implementation, we propose a novel bootstrap procedure to circumvent the slow convergence rates of the SCBs as well as infeasibility of approximating the limiting distributions. A series of Monte Carlo simulations confirm that our approach works well in finite samples.

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Type
ARTICLES
Information
Econometric Theory , Volume 39 , Issue 5 , October 2023 , pp. 1009 - 1043
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Footnotes

We thank the Editor (Professor Peter Phillips), the Co-Editor (Professor Dennis Kristensen), and two anonymous referees for constructive comments which led to great improvements of the paper. Li acknowledges financial support from the National Natural Science Foundation of China (Grant No. 11801399). Song acknowledges financial support from the National Science Foundation of China (Grant No. 71973005). Li dedicates this paper to his beloved wife Zhang Xian and their new son Li Guxin.

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