The Concept of Comonotonicity in Actuarial Science and Finance: Theory

Abstract

In an insurance context, one is often interested in the distribution function of a sum of random variables. Such a sum appears when considering the aggregate claims of an insurance portfolio over a certain reference period. It also appears when considering discounted payments related to a single policy or a portfolio at different future points in time. The assumption of mutual independence between the components of the sum is very convenient from a computational point of view, but sometimes not realistic. We will determine approximations for sums of random variables, when the distributions of the terms are known, but the stochastic dependence structure between them is unknown or too cumbersome to work with. In this paper, the theoretical aspects are considered. Applications of this theory are considered in a subsequent paper. Both papers are to a large extent an overview of recent research results obtained by the authors, but also new theoretical and practical results are presented.

Conditional comonotonicity

“…Characterizations and properties of comonotonic random variables can be found in Denneberg (1994) or Dhaene et al (2002a). In particular, if two random variables x and y are such that there exists a nondecreasing function ϕ for which x can be written in the form x = ϕ(y) (or if y can be written in the form y = ϕ(x)), then x and y are comonotonic.…”

A Measure of Monotonicity of two Random Variables

“…1), except a countable or finite set of points The concepts of comonotonicity and countermonotonicity were studied by Bauerle and Muller (1998); Denuit and Dhaene (2003); Dhaene et al (2002);Dempster (2002);Rachev (2003). Comonotonicity of a pair of random variables X and Y means their monotone increasing dependence, i.e., their values change in the same direction.…”

“…Countermonotonicity of the pair of X and Y means their monotone decreasing dependence, i.e., their values change in opposite directions. Dhaene et al (2002) give a mathematically accurate definition of comonotonicity for n random variables, which we reproduce here for n = 2. 1) A set A ⊆ R 2 is called comonotonic if for any of its elements <x 1 , y 1 > and <x 2 , y 2 >: either (x 1 ≤ x 2 and y 1 ≤ y 2 ) or (x 1 ≥ x 2 and y 1 ≥ y 2 ) holds.…”

“…Three criteria for comonotonicity were proven in (Dhaene et al, 2002). In case n = 2 they have the following form.…”

On Tail Value-at-Risk for sums of non-independent random variables with a generalized Pareto distribution

“…Recently in actuarial literature several authors have derived lower and upper bounds in the sense of convex order for sums of random variables when marginal distributions are given, but their joint distribution is unknown (Dhaene and Denuit 1999;Dhaene et al 2002a;Kaas et al 2000).…”