Comonotonicity

Abstract

In an actuarial or financial context it is often necessary to calculate the distribution function or risk measures for random variables of the type where the marginal distributions of the X i are known but their dependence structure is too difficult to work with. Comonotonicity, which is an extremal form of positive dependence, can be useful to determine easily computable and accurate upper and lower bounds for the distribution of S , and hence also for risk measures related to S .

Stop-Loss Reinsurance Threshold for Dependent Risks

“…Positive dependence means that individual i X 's move in the same direction, such that when there is perfect positive dependence then distribution of S is said to be comonotonic with cumulative distribution is defined in (3). Comonotonic sum results into the riskiest portfolio in any combination of risks i X 's, and has been studied by many such as [23] [24] [25] and [26],…”

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

“…The fact that the stop-loss premium of a comonotonic sum can be expressed as a sum of the marginal stop-loss premiums is a long-established result (see Meileison and Nádas (1979)), and is valid regardless of the number of components of the comonotonic sum S u . The explicit expression (2.12) for the retentions x i was derived in Dhaene et al (2000); see also Hobson et al (2005), Chen et al ( 2008), Chen et al (2015b) and Linders et al (2012).…”

“…Under the assumption of extreme dependence structures, the TVaR and the upper tail transform can be used to assess the diversification benefit from combining different risks or investment opportunities (Embrechts et al, 2009. Further, since even the most hardened modelers are not exempt from model risk, it is prudent to use sharp comonotonic and counter-monotonic bounds which are consistent with the available information on the univariate distributions (Dhaene et al, 2000, Kaas et al, 2000. These bounds are also informative on the extent of dependence model risk, which can be quantified using the dependence uncertainty spread, i.e.…”

Comonotic Approximations for a Generalized Provisioning Problem with Application to Optimal Portfolio Selection

“…For more characterizations and an overview of the theory of comonotonicity and its many applications in actuarial science and …nance we refer to Dhaene et al (2002a,b) and Dhaene et al (2008).…”

“…(25) with U 1 uniformly distributed on the unit interval. The correlation coe¢ cients r i can be determined using (10). Using the additivity property explained in Theorem 1, the quantiles of S l 1 (z) can be determined as:…”