TY - JOUR

T1 - Value-at-risk, tail value-at-risk and upper tail transform of the sum of two counter-monotonic random variables

AU - Hanbali, Hamza

AU - Linders, Daniël

AU - Dhaene, Jan

N1 - Publisher Copyright:
© 2022 Informa UK Limited, trading as Taylor & Francis Group.

PY - 2023

Y1 - 2023

N2 - The Value-at-Risk (VaR) of comonotonic sums can be decomposed into marginal VaRs at the same level. This additivity property allows to derive useful decompositions for other risk measures. In particular, the Tail Value-at-Risk (TVaR) and the upper tail transform of comonotonic sums can be written as the sum of their corresponding marginal risk measures. The other extreme dependence situation, involving the sum of two arbitrary counter-monotonic random variables, presents a certain number of challenges. One of them is that it is not straightforward to express the VaR of a counter-monotonic sum in terms of the VaRs of the marginal components of the sum. This paper generalizes the results derived in [Chaoubi, I., Cossette, H., Gadoury, S.-P. & Marceau, E. (2020). On sums of two counter-monotonic risks. Insurance: Mathematics and Economics92, 47–60.] by providing decomposition formulas for the VaR, TVaR and the stop-loss transform of the sum of two arbitrary counter-monotonic random variables.

AB - The Value-at-Risk (VaR) of comonotonic sums can be decomposed into marginal VaRs at the same level. This additivity property allows to derive useful decompositions for other risk measures. In particular, the Tail Value-at-Risk (TVaR) and the upper tail transform of comonotonic sums can be written as the sum of their corresponding marginal risk measures. The other extreme dependence situation, involving the sum of two arbitrary counter-monotonic random variables, presents a certain number of challenges. One of them is that it is not straightforward to express the VaR of a counter-monotonic sum in terms of the VaRs of the marginal components of the sum. This paper generalizes the results derived in [Chaoubi, I., Cossette, H., Gadoury, S.-P. & Marceau, E. (2020). On sums of two counter-monotonic risks. Insurance: Mathematics and Economics92, 47–60.] by providing decomposition formulas for the VaR, TVaR and the stop-loss transform of the sum of two arbitrary counter-monotonic random variables.

KW - Counter-monotonicity

KW - decomposition formulas

KW - extreme negative dependence

KW - stop-loss transform

KW - Tail Value-at-Risk

UR - http://www.scopus.com/inward/record.url?scp=85133295286&partnerID=8YFLogxK

U2 - 10.1080/03461238.2022.2092419

DO - 10.1080/03461238.2022.2092419

M3 - Article

AN - SCOPUS:85133295286

SN - 0346-1238

VL - 2023

SP - 219

EP - 243

JO - Scandinavian Actuarial Journal

JF - Scandinavian Actuarial Journal

IS - 3

ER -