We derive lower and upper bounds for the Value-at-Risk of a portfolio of losses when the marginal distributions are known and independence among (some) subgroups of the marginal components is assumed. We provide several actuarial examples showing that the newly proposed bounds strongly improve...

Based on a novel extension of classical Hoeffding-Fréchet bounds, we provide an upper VaR bound for joint risk portfolios with fixed marginal distributions and positive dependence information. The positive dependence information can be assumed to hold in the tails, in some central part, or on a...

The problem of establishing reliable estimates or bounds for the (T)VaR of a joint risk portfolio is a relevant subject in connection with the computation of total economic capital in the Basel regulatory framework for the finance sector as well as with the Solvency regulations for the insurance...

We study conditional expectiles, defined as a natural generalisation of conditional expectations by means of the minimisation of an asymmetric quadratic loss function. We show that conditional expectiles can be equivalently characterised by a conditional first order condition and we derive their...

Optimal transportation w.r.t. the Kantorovich metric l1 (resp. the Wasser- stein metric W1) between two absolutely continuous measures is known since the basic papers of Kantorovich and Rubinstein (1957) and Sudakov (1979) to occur on rays induced by a decomposition of the basic space, which is...

We introduce the concepts of φ-complete mixability and φ-joint mixability and we investigate some necessary and sufficient conditions to the φ-mixability of a set of distribution functions for some supermodular functions φ. We give examples and numerical verifications which confirm our findings

We give analytical bounds on the Value-at-Risk and on convex risk measures for a portfolio of random variables with fixed marginal distributions under an additional positive dependence structure. We show that assuming positive dependence information in our model leads to reduced dependence...

We introduce the Mixability Detection Procedure (MDP) to check whether a set of d distribution functions is jointly mixable at a given confidence level. The procedure is based on newly established results regarding the convergence rate of the minimal variance problem within the class of joint...

The probabilistic characterization of the relationship between two or more random variables calls for a notion of dependence. Dependence modeling leads to mathematical and statistical challenges; recent developments in extremal dependence concepts have drawn a lot of attention in probability and...