Bounds for joint portfolios of dependent risks

In this paper, we survey, extend and improve several bounds for the distri- bution function and the tail probabilities of portfolios, where the dependence structure within the portfolio is completely unknown or only partially known. We present various methods for obtaining bounds based on rearrangements, duality theory, conditional moments and reduction techniques. In particular, we consider the case where only the simple marginal distributions are known, the general overlapping marginals case where certain joint distributions are known and the case of additional restrictions on the dependence structure, as, for example, the restriction to positive dependence. Some of the bounds pose a considerable numerical challenge. We discuss the quality of the bounds and numerical aspects in some examples.