Improved Estimator of the Conditional Tail Expectation in the case of heavy-tailed losses

  • Mohamed Laidi National High School of Technology, Algiers, Algeria.LRDSI laboratory, Blida 1 university, Blida, Algeria
  • Abdelaziz Rassoul High School of Hydraulics, Blida, Algeria
  • Hamid Ould Rouis University of Blida 1, Blida, Algeria
Keywords: Risk measure, Conditional tail expectation, Bias reduction, extreme quantile, order statistic, heavy-tailed distribution.

Abstract

In this paper, we investigate the extreme-value methodology, to propose an improved estimator of the conditional tail expectation (CTE) for a loss distribution with a finite mean but infinite variance.The present work introduces a new estimator of the CTE based on the bias-reduced estimators of high quantile for heavy-tailed distributions. The asymptotic normality of the proposed estimator is established and checked, in a simulation study. Moreover, we compare, in terms of bias and mean squared error, our estimator with the known old estimator.

Author Biographies

Mohamed Laidi, National High School of Technology, Algiers, Algeria.LRDSI laboratory, Blida 1 university, Blida, Algeria
National High School of Technology, Algiers, Algeria; LRDSI laboratory, Blida 1 university, Blida, Algeria  
Abdelaziz Rassoul, High School of Hydraulics, Blida, Algeria
High School of Hydraulics, Blida, Algeria
Hamid Ould Rouis, University of Blida 1, Blida, Algeria
University of Blida 1, Blida, Algeria

References

C. Acerbi. Coherent representations of subjective risk-aversion. Risk measures for the 21st century, pages 147–207, 2004.

C. Acerbi and D. Tasche. Expected shortfall: a natural coherent alternative to value at risk. Economic notes, 31(2):379–388, 2002.

C. Acerbi and D. Tasche. On the coherence of expected shortfall. Journal of Banking & Finance, 26(7):1487–1503, 2002.

M. F. Alves, M. I. Gomes, L. de Haan, and C. Neves. A note on second order conditions in extreme value theory: linking general and heavy tail conditions. REVSTAT Statistical Journal, 5(3):285–304, 2007.

P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. Coherent measures of risk. Mathematical finance, 9(3):203–228, 1999.

J. Beirlant and G. Matthys. Extreme quantile estimation for heavy-tailed distributions. Rapport technique, Department of Mathematics, KU Leuven, page 23, 2001.

B. Brahimi, D. Meraghni, A. Necir, and D. Yahia. A bias-reduced estimator for the mean of a heavy-tailed distribution with an infinite second moment. Journal of Statistical Planning and Inference, 143(6):1064–1081, 2013.

V. Brazauskas, B. L. Jones, M. L. Puri, and R. Zitikis. Estimating conditional tail expectation with actuarial applications in view.Journal of Statistical Planning and Inference, 138(11):3590–3604, 2008.

J. Cai and K. S. Tan. Optimal retention for a stop-loss reinsurance under the var and cte risk measures. ASTIN Bulletin: The Journal of the IAA, 37(1):93–112, 2007.

J.-J. Cai, J. H. Einmahl, L. Haan, and C. Zhou. Estimation of the marginal expected shortfall: the mean when a related variable is extreme. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 77(2):417–442, 2015.

L. de Haan. De (1970) on regular variation and its application to the weak convergence of sample extremes. Mathematical Centre Tracts, 32, 1970.

L. de Haan and A. Ferreira. Extreme value theory: An introduction, 2006.

L. de Haan and U. Stadtm¨uller. Generalized regular variation of second order. Journal of the Australian Mathematical Society (Series A), 61(03):381–395, 1996.

L. d. De Haan and L. Peng. Comparison of tail index estimators. Statistica Neerlandica, 52(1):60–70, 1998.

J. Dhaene, L. Henrard, Z. Landsman, A. Vandendorpe, and S. Vanduffel. Some results on the cte-based capital allocation rule.Insurance: Mathematics and Economics, 42(2):855–863, 2008.

A. El Attar, M. El Hachlougi, and Z. E. A. Guennoun. Optimality of reinsurance treaties under a mean-ruin probability criterion. Statistics, Optimization & Information Computing, 7(2):383–393, 2019.

B. Gnedenko. Sur la distribution limite du terme maximum d’une serie aleatoire. Annals of mathematics, pages 423–453, 1943.

O. Hakim. Pot approach for estimation of extreme risk measures of eur/usd returns. Statistics, Optimization & Information Computing, 6(2):240–247, 2018.

P. Hall. On some simple estimates of an exponent of regular variation. Journal of the Royal Statistical Society. Series B

(Methodological), pages 37–42, 1982.

P. Hall, A. Welsh, et al. Adaptive estimates of parameters of regular variation. The Annals of Statistics, 13(1):331–341, 1985.

B. M. Hill et al. A simple general approach to inference about the tail of a distribution. The annals of statistics, 3(5):1163–1174,1975.

P. Jorion. Value at risk: the new benchmark for controlling market risk. Irwin Professional Pub., 1997.

R. Kaas, M. Goovaerts, J. Dhaene, and M. Denuit. Modern actuarial risk theory: using R, volume 128. Springer Science & Business Media, 2008.

Z. M. Landsman and E. A. Valdez. Tail conditional expectations for elliptical distributions. North American Actuarial Journal, 7(4):55–71, 2003.

D. Li, L. Peng, and J. Yang. Bias reduction for high quantiles. Journal of Statistical Planning and Inference, 140(9):2433–2441, 2010.

D. M. Mason. Laws of large numbers for sums of extreme values. The Annals of Probability, pages 754–764, 1982.

J. Morgan. Creditmetrics-technical document. JP Morgan, New York, 1997.

A. Necir, A. Rassoul, and R. Zitikis. Estimating the conditional tail expectation in the case of heavy-tailed losses. Journal of Probability and Statistics, 2010.

L. Peng. Asymptotically unbiased estimators for the extreme-value index. Statistics & Probability Letters, 38(2):107–115, 1998.

L. Peng and Y. Qi. Estimating the first-and second-order parameters of a heavy-tailed distribution. Australian & New Zealand Journal of Statistics, 46(2):305–312, 2004.

S. I. Resnick. Heavy-tail phenomena: probabilistic and statistical modeling. Springer Science & Business Media, 2007.

R. T. Rockafellar and S. Uryasev. Conditional value-at-risk for general loss distributions. Journal of banking & finance, 26(7):1443–1471, 2002.

R. T. Rockafellar, S. Uryasev, et al. Optimization of conditional value-at-risk. Journal of risk, 2:21–42, 2000.

K. S. Tan, C. Weng, and Y. Zhang. Optimality of general reinsurance contracts under cte risk measure. Insurance: Mathematics and Economics, 49(2):175–187, 2011.

D. Tasche. Expected shortfall and beyond. Journal of Banking & Finance, 26(7):1519–1533, 2002.

I. Weissman. Estimation of parameters and large quantiles based on the k largest observations. Journal of the American Statistical Association, 73(364):812–815, 1978.

J. L.Wirch and M. R. Hardy. A synthesis of risk measures for capital adequacy. Insurance: mathematics and Economics, 25(3):337–347, 1999.

Y. Yamai, T. Yoshiba, et al. Comparative analyses of expected shortfall and value-at-risk: their estimation error, decomposition, and optimization. Monetary and economic studies, 20(1):87–121, 2002.

Published
2020-02-17
How to Cite
Laidi, M., Rassoul, A., & Ould Rouis, H. (2020). Improved Estimator of the Conditional Tail Expectation in the case of heavy-tailed losses. Statistics, Optimization & Information Computing, 8(1), 98-109. https://doi.org/10.19139/soic-2310-5070-665
Section
Research Articles