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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.

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.

References

REFERENCES

Bai, J. and Ng, S. (2001) A consistent test for conditional symmetry in time series models. Journal of Econometrics 103, 225–258.CrossRefGoogle Scholar
Barberis, N. and Huang, M. (2008) Stocks as lotteries: The implications of probability weighting for security prices. American Economic Review 98, 2066–2100.CrossRefGoogle Scholar
Bollerslev, T. (1987) A conditionally heteroskedastic time series model for speculative prices and rates of return. Review of Economics and Statistics 69, 542–547.CrossRefGoogle Scholar
Chan, N.H., Deng, S.-J., Peng, L., and Xia, Z. (2007) Interval estimation of value-at-risk based on GARCH models with heavy-tailed innovations. Journal of Econometrics 137, 556–576.CrossRefGoogle Scholar
Danielsson, J., Ergun, L.M., de Haan, L., and de Vries, C.G. (2019) Tail Index Estimation: Quantile Driven Threshold Selection. Discussion paper, Bank of Canada.Google Scholar
Daouia, A., Girard, S., and Stupfler, G. (2017) Estimation of tail risk based on extreme Expectiles. Journal of the Royal Statistical Society Series B 80, 263–292.CrossRefGoogle Scholar
de Haan, L. and Ferreira, A. (2006) Extreme Value Theory: An Introduction . Springer Series in Operations Research and Financial Engineering. Springer.CrossRefGoogle Scholar
Einmahl, J.H.J., de Haan, L., and Zhou, C. (2016) Statistics of heteroscedastic extremes. Journal of the Royal Statistical Society Series B (Methodological) 78, 31–51.CrossRefGoogle Scholar
Engle, R.F., Lilien, D.M., and Robins, R.P. (1987) Estimating time varying risk premia in the term structure: The arch-M model. Econometrica 55, 391–407.CrossRefGoogle Scholar
Fernández, C. and Steel, M.F.J. (1998) On Bayesian modeling of fat tails and skewness. Journal of the American Statistical Association 93(441), 359–371.Google Scholar
Francq, C. and Zakoïan, J.-M. (2010) GARCH Models: Structure, Statistical Inference and Financial Applications . Wiley.CrossRefGoogle Scholar
Francq, C. and Zakoïan, J.-M. (2016) Looking for efficient QML estimation of conditional VaRs at multiple risk levels. Annals of Economics and Statistics 123, 9–28.CrossRefGoogle Scholar
Gao, F. and Song, F. (2008) Estimation risk in GARCH VaR and ES estimates. Econometric Theory 24(5), 1404–1424.CrossRefGoogle Scholar
Glosten, L.R., Jagannathan, R., and Runkle, D.E. (1993) On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance 48, 1779–1801.CrossRefGoogle Scholar
Hall, P. (1982) On some simple estimates of an exponent of regular variation. Journal of the Royal Statistical Society. Series B 44, 37–42.Google Scholar
Hamilton, J. (1994) Time Series Analysis . Princeton University Press.CrossRefGoogle Scholar
Hill, B.M. (1975) A simple general approach to inference about the tail of a distribution. Annals of Statistics 3, 1163–1174.CrossRefGoogle Scholar
Hill, J.B. (2015) Robust estimation and inference for heavy tailed GARCH. Bernoulli 21, 1629–1669.CrossRefGoogle Scholar
Hoga, Y. (2019a) Confidence intervals for conditional tail risk measures in ARMA–GARCH models. Journal of Business & Economic Statistics 37(4), 613–624.CrossRefGoogle Scholar
Hoga, Y. (2019b) Extreme conditional tail moment estimation under serial dependence. Journal of Financial Econometrics 17, 587–615.CrossRefGoogle Scholar
Hoga, Y. (2022) Limit theory for forecasts of extreme distortion risk measures and expectiles. Journal of Financial Econometrics 20, 18–44.CrossRefGoogle Scholar
Martins-Filho, C., Yao, F., and Torero, M. (2018) Nonparametric estimation of conditional value-at-risk and expected shortfall based on extreme value theory. Econometric Theory 34(1), 23–67.CrossRefGoogle Scholar
McNeil, A.J. and Frey, R. (2000) Estimation of tail-related risk measures for heteroscedastic financial time series: An extreme value approach. Journal of Empirical Finance 7, 271–300.10.1016/S0927-5398(00)00012-8CrossRefGoogle Scholar
Mikosch, T. and Straumann, D. (2006) Stable limits of martingale transforms with application to the estimation of GARCH parameters. Annals of Statistics 34, 493–592.10.1214/009053605000000840CrossRefGoogle Scholar
Pan, X., Leng, X., and Hu, T. (2013) The second-order version of Karamata’s theorem with applications. Statistics & Probability Letters 83, 1397–1403.CrossRefGoogle Scholar
Romano, J., Shaikh, A., and Wolf, M. (2018) Multiple testing. In Durlauf, S. and Blume, L. (eds.), The New Palgrave Dictionary of Economics , 3rd Edition, pp. 9185–9189. Palgrave Macmillan.CrossRefGoogle Scholar
Spierdijk, L. (2016) Confidence intervals for ARMA–GARCH value-at-risk: The case of heavy tails and skewness. Computational Statistics and Data Analysis 100, 545–559.CrossRefGoogle Scholar
Theodossiou, P. and Savva, C.S. (2015) Skewness and the relation between risk and return. Management Science 62(6), 1598–1609.CrossRefGoogle Scholar
Tsay, R.S. (2010) Analysis of Financial Time Series Analysis . Wiley Series in Probability and Statistics. Wiley.10.1002/9780470644560CrossRefGoogle Scholar
Tversky, A. and Kahneman, D. (1992) Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty 5, 297–323.CrossRefGoogle Scholar
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