Chaoubi, Ihsan; Cossette, Hélène; Gadoury, Simon-Pierre; Marceau, Etienne On sums of two counter-monotonic risks. (English) Zbl 1445.91050 Insur. Math. Econ. 92, 47-60 (2020). Summary: In risk management, capital requirements are most often based on risk measurements of the aggregation of individual risks treated as random variables. The dependence structure between such random variables has a strong impact on the behavior of the aggregate loss. One finds an extensive literature on the study of the sum of comonotonic risks but less, in comparison, has been done regarding the sum of counter-monotonic risks. A crucial result for comonotonic risks is that the value-at-risk and the tail value-at-risk of their sum correspond respectively to the sum of the value-at-risk and tail value-at-risk of the individual risks. In this paper, our main objective is to derive such simple results for the sum of counter-monotonic risks. To do so, we examine separately different contexts in the class of bivariate strictly continuous distributions for which we obtain closed-form expressions for the value-at-risk and tail value-at-risk of the sum of two counter-monotonic risks. The expressions for the subadditive tail value-at risk allow us to quantify the maximal diversification benefit. Also, our findings allow us to analyze the tail of the distribution of the sum of two identically subexponentially distributed counter-monotonic random variables. Cited in 1 ReviewCited in 4 Documents MSC: 91G05 Actuarial mathematics 91G70 Statistical methods; risk measures Keywords:counter-monotonicity; extreme negative dependence; risk measures; diversification benefit; subexponential distributions Software:QRM PDFBibTeX XMLCite \textit{I. Chaoubi} et al., Insur. Math. Econ. 92, 47--60 (2020; Zbl 1445.91050) Full Text: DOI References: [1] Acerbi, C.; Tasche, D., On the coherence of expected shortfall, J. Bank. Financ., 26, 7, 1487-1503 (2002) [2] Albrecher, H.; Teugels, J. L.; Beirlant, J., Reinsurance: Actuarial and Statistical Aspects (2017), John Wiley & Sons · Zbl 1376.91004 [3] Alink, S.; Löwe, M.; Wüthrich, M. V., Diversification of aggregate dependent risks, Insurance Math. Econom., 35, 1, 77-95 (2004) · Zbl 1052.62105 [4] Barbe, P.; Fougeres, A.-L.; Genest, C., On the tail behavior of sums of dependent risks, ASTIN Bull.: J. IAA, 36, 2, 361-373 (2006) · Zbl 1162.91395 [5] Bürgi, R.; Dacorogna, M. M.; Iles, R., Risk Aggregation, Dependence Structure and Diversification Benefit (2008), Stress testing for financial institutions [6] Cheung, K. C.; Dhaene, J.; Lo, A.; Tang, Q., Reducing risk by merging counter-monotonic risks, Insurance Math. Econom., 54, 58-65 (2014) · Zbl 1291.91098 [7] Cheung, K. C.; Lo, A., Characterizations of counter-monotonicity and upper comonotonicity by (tail) convex order, Insurance Math. Econom., 53, 2, 334-342 (2013) · Zbl 1304.60025 [8] Cheung, K. C.; Lo, A., General lower bounds on convex functionals of aggregate sums, Insurance Math. Econom., 53, 3, 884-896 (2013) · Zbl 1290.91081 [9] Cheung, K. C.; Lo, A., Characterizing mutual exclusivity as the strongest negative multivariate dependence structure, Insurance Math. Econom., 55, 180-190 (2014) · Zbl 1296.60037 [10] Cossette, H.; Côté, M.-P.; Mailhot, M.; Marceau, E., A note on the computation of sharp numerical bounds for the distribution of the sum, product or ratio of dependent risks, J. Multivariate Anal., 130, 1-20 (2014) · Zbl 1292.62077 [11] Cossette, H.; Marceau, E.; Nguyen, Q. H.; Robert, C. Y., Tail approximations for sums of dependent regularly varying random variables under Archimedean copula models, Methodol. Comput. Appl. Probab., 21, 2, 461-490 (2019) · Zbl 1480.60140 [12] Dacorogna, M. M.; Elbahtouri, L.; Kratz, M., Explicit diversification benefit for dependent risks, SCOR Pap., 38 (2016) [13] Denuit, M.; Dhaene, J.; Goovaerts, M.; Kaas, R., Actuarial Theory for Dependent Risks: Measures, Orders and Models (2006), John Wiley & Sons [14] Dhaene, J.; Denuit, M., The safest dependence structure among risks, Insurance Math. Econom., 25, 1, 11-21 (1999) · Zbl 1072.62651 [15] Dhaene, J.; Denuit, M.; Goovaerts, M. J.; Kaas, R.; Vyncke, D., The concept of comonotonicity in actuarial science and finance: Theory, Insurance Math. Econom., 31, 1, 3-33 (2002) · Zbl 1051.62107 [16] Dhaene, J.; Goovaerts, M. J., Dependency of risks and stop-loss order, ASTIN Bull.: J. IAA, 26, 2, 201-212 (1996) [17] Dhaene, J.; Linders, D.; Schoutens, W.; Vyncke, D., The herd behavior index: A new measure for the implied degree of co-movement in stock markets, Insurance Math. Econom., 50, 3, 357-370 (2012) · Zbl 1237.91237 [18] Dhaene, J.; Vanduffel, S.; Goovaerts, M. J.; Kaas, R.; Tang, Q.; Vyncke, D., Risk measures and comonotonicity: A review, Stoch. Models, 22, 4, 573-606 (2006) · Zbl 1159.91403 [19] Dhaene, J.; Wang, S.; Young, V. R.; Goovaerts, M., Comonotonicity and maximal stop-loss premiums, Bull. Swiss Assoc. Actuar., 2, 99-113 (2000) · Zbl 1187.91099 [20] Embrechts, P.; Klüppelberg, C.; Mikosch, T., Modelling Extremal Events: for Insurance and Finance, Vol. 33 (2013), Springer Science & Business Media [21] Embrechts, P.; Lambrigger, D. D.; Wüthrich, M. V., Multivariate extremes and the aggregation of dependent risks: Examples and counter-examples, Extremes, 12, 2, 107-127 (2009) · Zbl 1224.91057 [22] Embrechts, P.; Nešlehová, J.; Wüthrich, M. V., Additivity properties for value-at-risk under Archimedean dependence and heavy-tailedness, Insurance Math. Econom., 44, 2, 164-169 (2009) · Zbl 1163.91431 [23] Gaffke, N.; Rüscherndorf, L., On a class of extremal problems in statistics, Math. Oper.forsch. Stat. Ser. Optim., 12, 1, 123-135 (1981) · Zbl 0467.60004 [24] Hobson, D.; Laurence, P.; Wang, T.-H., Static-arbitrage optimal subreplicating strategies for sasket options, Insurance Math. Econom., 37, 3, 553-572 (2005) · Zbl 1129.62424 [25] Hobson, D.; Laurence, P.; Wang, T.-H., Static-arbitrage upper bounds for the prices of basket options, Quant. Finance, 5, 4, 329-342 (2005) · Zbl 1134.91425 [26] Inui, K.; Kijima, M., On the significance of expected shortfall as a coherent risk measure, J. Bank. Financ., 29, 4, 853-864 (2005) [27] Jeon, J.; Kochar, S.; Park, C. G., Dispersive ordering – some applications and examples, Statist. Papers, 47, 2, 227-247 (2006) · Zbl 1105.62014 [28] Joe, H., Multivariate Models and Multivariate Dependence Concepts (1997), CRC Press · Zbl 0990.62517 [29] Laurence, P.; Wang, T.-H., Sharp upper and lower bounds for basket options, Appl. Math. Finance, 12, 3, 253-282 (2005) · Zbl 1138.91457 [30] Lee, W.; Ahn, J. Y., On the multidimensional extension of countermonotonicity and its applications, Insurance Math. Econom., 56, 68-79 (2014) · Zbl 1304.62086 [31] Lee, W.; Cheung, K. C.; Ahn, J. Y., Multivariate countermonotonicity and the minimal copulas, J. Comput. Appl. Math., 317, 589-602 (2017) · Zbl 1359.62168 [32] McNeil, A. J.; Frey, R.; Embrechts, P., Quantitative Risk Management: Concepts, Techniques and Tools (2015), Princeton University Press · Zbl 1337.91003 [33] Müller, A.; Stoyan, D., Comparison Methods for Stochastic Models and Risks, Vol. 389 (2002), Wiley New York [34] Puccetti, G.; Wang, B.; Wang, R., Complete mixability and asymptotic equivalence of worst-possible VaR and ES estimates, Insurance Math. Econom., 53, 3, 821-828 (2013) · Zbl 1290.62019 [35] Puccetti, G.; Wang, R., Extremal dependence concepts, Statist. Sci., 30, 4, 485-517 (2015) · Zbl 1426.62156 [36] Shaked, M.; Shanthikumar, J. G., Stochastic Orders (2007), Springer Science & Business Media [37] Tchen, A. H., Inequalities for distributions with given marginals, Ann. Probab., 8, 4, 814-827 (1980) · Zbl 0459.62010 [38] Wang, R.; Peng, L.; Yang, J., Bounds for the sum of dependent risks and worst Value-at-Risk with monotone marginal densities, Finance Stoch., 17, 2, 395-417 (2013) · Zbl 1266.91038 [39] Wang, B.; Wang, R., The complete mixability and convex minimization problems with monotone marginal densities, J. Multivariate Anal., 102, 10, 1344-1360 (2011) · Zbl 1229.60019 [40] Wang, B.; Wang, R., Joint mixability, Math. Oper. Res., 41, 3, 808-826 (2016) · Zbl 1382.60022 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.