Risk Bounds, Worst Case Dependence, and Optimal Claims and Contracts

Some classical results on risk bounds as the Fréchet bounds, the Hoeffding–Fréchet bounds and the extremal risk property of the comonotonicity dependence structure are used to describe worst case dependence structures for portfolios of real risks. An extension of the worst case dependence structure to portfolios of risk vectors is given. The bounds are used to (re-)derive and extend some results on optimal contingent claims and an optimal (re-)insurance contracts. 1. Risk bounds and comonotonicity For a risk vector X = (X1, . . . , Xn) of risks Xi with distributions Pi resp. distribution functions Fi it is a classical problem to determine (sharp) bounds for a risk functional of the form EΨ(X) induced by dependence between the components Xi of X . The class of all possible dependence structures is given by the Fréchet classM(P1, . . . , Pn) of joint distributions with marginals Pi. For the case of real risks one can consider equivalently the class F(F1, . . . , Fn) of joint distribution functions with marginal distribution functions Fi ∼ Pi. The sharp upper and lower dependence bounds for the risk function Ψ are given by M(Ψ) = sup {∫ ΨdP ;P ∈M(P1, . . . , Pn) } and m(Ψ) = inf {∫ ΨdP ;P ∈M(P1, . . . , Pn) } . (1) They are called (generalized) upper resp. lower Fréchet bounds. Typical risk functionals of interest are in the case of real risks Xi risk functionals of the joint portfolio like ( ∑n i=1Xi −K)+, 1[t,∞)( ∑n i=1 Xi) or maxiXi leading to bounds for the excess of loss, for the value at risk and for the maximal risk of the joint portfolio.