On sums of two counter-monotonic risks

Value-at-Risk, Tail Value-at-Risk and upper tail transform of the sum of two counter-monotonic random variables

“…One of them is that it is not straightforward to express the VaR of a counter-monotonic sum in terms of the VaR's of the marginal components of the sum. This paper generalizes the results derived in Chaoubi et al (2020) by providing decomposition formulas for the VaR, TVaR and the stop-loss transform of the sum of two arbitrary counter-monotonic random variables.…”

“…The present paper contributes to the body of literature devoted to the decomposition of risk measures of sums of two random variables (Dhaene and Denuit, 1999, Cheung and Lo, 2013, Cheung et al, 2017. The paper extends the work of Laurence and Wang (2009) and Chaoubi et al (2020) by deriving explicit decompositions of the VaR, TVaR and stop-loss premium of counter-monotonic sums with general marginal distributions. Whereas the decomposition of the VaR is a natural result that follows from the basic definitions, the decomposition of TVaR's and stop-loss premiums are more challenging.…”

“…, where the W i 's are symmetric and identically distributed random variables, which implies that the function g is always monotone, and the sign of its derivative depends upon the sign of b 1 − b 2 . Chaoubi et al (2020) derive decompositions of the VaR and TVaR of countermonotonic sums in the case of non-decreasing functions g (Proposition 1), as well as in the more challenging case of concave/convex functions g (Propositions 2 and 3). They also show that the function g can be either non-decreasing or concave/convex for several combinations of marginal distributions F X 1 and F X 2 .…”

“…For the equivalent problem of comonotonic differences, Laurence and Wang (2009) derive a decomposition for stop-loss premiums in the case of two continuous random variables, but they do not provide closed-form expressions for the corresponding retentions. Chaoubi et al (2020) also restrict their analysis to continuous random variables with a specific choice of marginal distributions, including symmetric or unimodal distributions. When the random variables follow such marginal distributions, the function defined as the sum of quantiles of counter-monotonic random variables have appealing properties.…”

“…Whereas the decomposition of the VaR is a natural result that follows from the basic definitions, the decomposition of TVaR's and stop-loss premiums are more challenging. This is due to the fact that the sum of quantiles of two counter-monotonic random variables may exhibit complex features, such as jumps (and hence, does not fall in the framework considered in Laurence and Wang (2009) and Chaoubi et al (2020) with continuous random variables) or multiple inflection points (and hence, for VaR and TVaR, violates Chaoubi et al (2020)'s convex/concave assumptions).…”