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Asymptotics for the conditional higher moment coherent risk measure with weak contagion

Published online by Cambridge University Press:  20 December 2024

Jiajun Liu  and
Qingxin Yi Open the ORCID record for Qingxin Yi [Opens in a new window]
Jiajun Liu
Affiliation:
Department of Financial and Actuarial Mathematics Xi’an Jiaotong-Liverpool University, Suzhou, China
Qingxin Yi*
Affiliation:
Department of Financial and Actuarial Mathematics Xi’an Jiaotong-Liverpool University, Suzhou, China
*
Corresponding author: Qingxin Yi; Email: Qingxin.Yi18@student.xjtlu.edu.cn
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Abstract

Various measures have been introduced in the existing literature to evaluate extreme risk exposure under the effect of an observable factor. Due to the nice properties of the higher-moment (HM) coherent risk measure, we propose a conditional version of the HM (CoHM) risk measure by incorporating the information of an observable factor. We conduct an asymptotic analysis of this measure in the presence of extreme risks under the weak contagion at a high confidence level, which is further applied to the special case of the conditional Haezendonck–Goovaerts risk measure (CoHG). Numerical illustrations are also provided to examine the accuracy of the asymptotic formulas and to analyze the sensitivity of the risk contribution of the CoHG. Based on the asymptotic result in the Fréchet case, we propose an estimator for the CoHM via an extrapolation, supported by a simulation study.

Keywords

Conditional higher-moment risk measureConditional Haezendonck–Goovaerts risk measureExtreme value theoryRapid variationRegular variation

Information

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 55 , Issue 1 , January 2025 , pp. 121 - 143
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The International Actuarial Association

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References

Abduraimova, K. (2022) Contagion and tail risk in complex financial networks. Journal of Banking & Finance, 143, 106560.CrossRefGoogle Scholar
Acerbi, C. (2002) Spectral measures of risk: A coherent representation of subjective risk aversion. Journal of Banking & Finance, 26(7), 1505–1518.CrossRefGoogle Scholar
Acerbi, C. and Tasche, D. (2002) On the coherence of expected shortfall. Journal of Banking & Finance, 26(7), 1487–1503.CrossRefGoogle Scholar
Adrian, T. and Brunnermeier, M.K. (2016) CoVaR. The American Economic Review, 106(7), 1705.CrossRefGoogle Scholar
Ahn, J.Y. and Shyamalkumar, N.D. (2014) Asymptotic theory for the empirical Haezendonck-Goovaerts risk measure. Insurance: Mathematics and Economics, 55, 78–90.Google Scholar
Artzner, P. (1999) Application of coherent risk measures to capital requirements in insurance. North American Actuarial Journal, 3(2), 11–25.CrossRefGoogle Scholar
Asimit, A.V. and Badescu, A.L. (2010) Extremes on the discounted aggregate claims in a time dependent risk model. Scandinavian Actuarial Journal, 2010(2), 93–104.CrossRefGoogle Scholar
Asimit, A.V. and Li, J. (2018a) Measuring the tail risk: An asymptotic approach. Journal of Mathematical Analysis and Applications, 463(1), 176–197.CrossRefGoogle Scholar
Asimit, A.V. and Li, J. (2018b) Systemic risk: An asymptotic evaluation. ASTIN Bulletin, 48(2), 673–698.CrossRefGoogle Scholar
Balkema, A. and de Haan, L. (1972) On R. von mises’ condition for the domain of attraction of $exp({-}e^{-x})$ . The Annals of Mathematical Statistics, 43, 1352–1354.CrossRefGoogle Scholar
Balla, E., Ergen, I. and Migueis, M. (2014) Tail dependence and indicators of systemic risk for large us depositories. Journal of Financial Stability, 15, 195–209.CrossRefGoogle Scholar
Bandt, O.D., Hartmann, P. and Peydró, J.J. (2012). Systemic risk in banking: An update. The Oxford Handbook of Banking. Oxford University Press.Google Scholar
Bauer, D. and Zanjani, G. (2016) The marginal cost of risk, risk measures, and capital allocation. Management Science, 62(5), 1431–1457.CrossRefGoogle Scholar
Bellini, F., Laeven, R.J. and Rosazza Gianin, E. (2021) Dynamic robust Orlicz premia and Haezendonck–Goovaerts risk measures. European Journal of Operational Research, 291(2), 438–446.CrossRefGoogle Scholar
Bellini, F. and Rosazza Gianin, E. (2008) On Haezendonck risk measures. Journal of Banking & Finance, 32(6), 986–994.CrossRefGoogle Scholar
Bellini, F. and Rosazza Gianin, E. (2012) Haezendonck-Goovaerts risk measures and Orlicz quantiles. Insurance: Mathematics and Economics, 51(1), 107–114.Google Scholar
Ben-Tal, A. and Teboulle, M. (2007) An old-new concept of convex risk measures: The optimized certainty equivalent. Mathematical Finance, 17(3), 449–476.CrossRefGoogle Scholar
Bingham, N.H., Goldie, C.M., Teugels, J.L. and Teugels, J. (1989) Regular Variation. Cambridge University Press.Google Scholar
Chen, Y. and Liu, J. (2022) An asymptotic study of systemic expected shortfall and marginal expected shortfall. Insurance: Mathematics and Economics, 105, 238–251.Google Scholar
Chen, Y. and Yuen, K.C. (2012) Precise large deviations of aggregate claims in a size-dependent renewal risk model. Insurance: Mathematics and Economics, 51(2), 457–461.Google Scholar
Chen, Z., Hu, Q. and Lin, R. (2016) Performance ratio-based coherent risk measure and its application. Quantitative Finance, 16(5), 681–693.CrossRefGoogle Scholar
Chen, Z., Zhang, F. and Yang, L. (2011) Postoptimality for mean-risk stochastic mixed-integer programs and its application. Mathematical Methods of Operations Research, 74, 445–465.CrossRefGoogle Scholar
Davis, R. and Resnick, S. (1988) Extremes of moving averages of random variables from the domain of attraction of the double exponential distribution. Stochastic Processes and their Applications, 30(1), 41–68.CrossRefGoogle Scholar
de Haan, L. and Ferreira, A. (2006) Extreme Value Theory: An Introduction, Vol. 3. Springer.CrossRefGoogle Scholar
Dentcheva, D., Penev, S. and Ruszczyński, A. (2010) Kusuoka representation of higher order dual risk measures. Annals of Operations Research, 181, 325–335.CrossRefGoogle Scholar
Dentcheva, D., Penev, S. and Ruszczyński, A. (2017) Statistical estimation of composite risk functionals and risk optimization problems. Annals of the Institute of Statistical Mathematics, 69, 737–760.CrossRefGoogle Scholar
Embrechts, P., Mikosch, T. and Klüppelberg, C. (1997) Modelling Extremal Events for Insurance and Finance. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Fang, R. and Li, X. (2018) Some results on measures of interaction between paired risks. Risks, 6, 88.CrossRefGoogle Scholar
Fisher, R.A. and Tippett, L.H.C. (1928) Limiting forms of the frequency distribution of the largest or smallest member of a sample. Mathematical Proceedings of the Cambridge Philosophical Society, 24(2), 180–190.CrossRefGoogle Scholar
Föllmer, H. and Knispel, T. (2011) Entropic risk measures: Coherence vs. convexity, model ambiguity, and robust large deviations. Stochastics and Dynamics, 11, 333–351.CrossRefGoogle Scholar
Furman, E., Wang, R. and Zitikis, R. (2017) Gini-type measures of risk and variability: Gini shortfall, capital allocations, and heavy-tailed risks. Journal of Banking & Finance, 83, 70–84.CrossRefGoogle Scholar
Gao, N., Munari, C. and Xanthos, F. (2020) Stability properties of Haezendonck-Goovaerts premium principles. Insurance: Mathematics and Economics, 94, 94–99.Google Scholar
Gómez, F., Tang, Q. and Tong, Z. (2022) The gradient allocation principle based on the higher moment risk measure. Journal of Banking & Finance, 143, 106544.CrossRefGoogle Scholar
Gnedenko, B. (1943) Sur la distribution limite du terme maximum d’une serie aleatoire (french). Annals of Mathematics, 44(2), 423–453.CrossRefGoogle Scholar
Goovaerts, M., Kaas, R., Dhaene, J. and Tang, Q. (2004) Some new classes of consistent risk measures. Insurance: Mathematics and Economics, 34(3), 505–516.Google Scholar
Haezendonck, J. and Goovaerts, M. (1982) A new premium calculation principle based on Orlicz norms. Insurance: Mathematics and Economics, 1(1), 41–53.Google Scholar
Idier, J., Lamé, G. and Mésonnier, J.-S. (2014) How useful is the marginal expected shortfall for the measurement of systemic exposure? A practical assessment. Journal of Banking & Finance, 47, 134–146.CrossRefGoogle Scholar
Kiriliouk, A., Segers, J. and Warchoł, M. (2016) Nonparametric estimation of extremal dependence. Extreme Value Modeling and Risk Analysis: Methods and Applications, pp. 353–375. CRC Press.Google Scholar
Kouri, D. (2019) Higher-moment buffered probability. Optimization Letters, 13, 1223–1237.CrossRefGoogle Scholar
Krokhmal, P.A. (2007) Higher moment coherent risk measures. Quantitative Finance, 7(4), 373–387.CrossRefGoogle Scholar
Krokhmal, P.A. and Soberanis, P. (2010). Risk optimization with p-order conic constraints: A linear programming approach. European Journal of Operational Research, 201(3), 653–671.CrossRefGoogle Scholar
Krokhmal, P., Zabarankin, M. and Uryasev, S. (2011) Modelling and optimization of risk. Surveys in Operations Research and Management Science, 16(2), 49–66.CrossRefGoogle Scholar
Li, J. (2022) Asymptotic results on marginal expected shortfalls for dependent risks. Insurance: Mathematics and Economics, 102, 146–168.Google Scholar
Ling, C. and Liu, J. (2022) Extremes for a general contagion risk measure. European Actuarial Journal, 12(2), 579–605.CrossRefGoogle Scholar
Liu, J. and Yang, Y. (2021) Asymptotics for systemic risk with dependent heavy-tailed losses. ASTIN Bulletin, 51(2), 571–605.CrossRefGoogle Scholar
Mainik, G. and Schaanning, E. (2014) On dependence consistency of CoVaR and some other systemic risk measures. Statistics & Risk Modelling, 31, 49–77.CrossRefGoogle Scholar
Matmoura, Y. and Penev, S. (2013) Multistage optimization of option portfolio using higher order coherent risk measures. European Journal of Operational Research, 227, 190–198.CrossRefGoogle Scholar
McNeil, A., Frey, R. and Embrechts, P. (2015) Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton University Press.Google Scholar
Nelsen, R. B. (2007) An Introduction to Copulas. Springer Science & Business Media.Google Scholar
Nolde, N., Zhou, C. and Zhou, M. (2022) An extreme value approach to CoVaR estimation. arXiv preprint arXiv:2201.00892.Google Scholar
Resnick, S.I. (1987) Extreme Values, Regular Variation, and Point Processes. New York: Springer-Verlag.CrossRefGoogle Scholar
Rockafellar, R. and Uryasev, S. (2002) Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, 26(7), 1443–1471.CrossRefGoogle Scholar
Sordo, M.A., Bello, A.J. and Suárez-Llorens, A. (2018) Stochastic orders and co-risk measures under positive dependence. Insurance: Mathematics and Economics, 78, 105–113.Google Scholar
Tang, Q. and Yang, F. (2012) On the Haezendonck-Goovaerts risk measure for extreme risks. Insurance: Mathematics and Economics, 50(1), 217–227.Google Scholar
Tang, Q. and Yang, F. (2014) Extreme value analysis of the Haezendonck-Goovaerts risk measure with a general young function. Insurance: Mathematics and Economics, 59, 311–320.Google Scholar
Vinel, A. and Krokhmal, P.A. (2017) Certainty equivalent measures of risk. Annals of Operations Research, 249(1), 75–95.CrossRefGoogle Scholar
Wang, S.S., Young, V.R. and Panjer, H.H. (1997) Axiomatic characterization of insurance prices. Insurance: Mathematics and Economics, 21(2), 173–183.Google Scholar
Xun, L., Jiang, R. and Guo, J. (2021) The conditional Haezendonck-Goovaerts risk measure. Statistics & Probability Letters, 169, 108968.CrossRefGoogle Scholar
Yang, H., Gao, W. and Li, J. (2016) Asymptotic ruin probabilities for a discrete-time risk model with dependent insurance and financial risks. Scandinavian Actuarial Journal, 2016(1), 1–17.CrossRefGoogle Scholar
Yang, Y. and Konstantinides, D.G. (2015) Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks. Scandinavian Actuarial Journal, 2015(8), 641–659.CrossRefGoogle Scholar