# On sums of two counter-monotonic risks

*Ihsan Chaoubi*,
*Hélène Cossette*,
*Simon-Pierre Gadoury* and
*Etienne Marceau*

*Insurance: Mathematics and Economics*, 2020, vol. 92, issue C, 47-60

**Abstract:**
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.

**Keywords:** Counter-monotonicity; Extreme negative dependence; Risk measures; Diversification benefit; Subexponential distributions (search for similar items in EconPapers)

**Date:** 2020

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**Persistent link:** https://EconPapers.repec.org/RePEc:eee:insuma:v:92:y:2020:i:c:p:47-60

**DOI:** 10.1016/j.insmatheco.2020.02.010

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