Overcoming dimensional dependence of worst case scenarios and maximum loss

Thomas Breuer

Abstract

ABSTRACT

Maximum Loss (MaxLoss) over admissibility domains with a specified probability shows a peculiar kind of dimensional dependence. For a fixed portfolio and fixed probability of the admissibility domain, the inclusion of additional risk factors increases MaxLoss – even if the additional risk factors are irrelevant for the portfolio or if they are highly correlated to other risk factors already included in the description. We propose a method to avoid this counter-intuitive effect: if we characterize the admissibility domain by its Mahalanobis radius instead of its probability, the inclusion of irrelevant risk factors does not affect MaxLoss. This result is also formulated for non-normal risk factor distributions. Furthermore, we present the problem of coordinate dependence of MaxLoss: MaxLoss over admissibility domains of point scenarios depends on the choice of coordinates. This problem is shown to be an artefact of considering point scenarios. Maximum expected loss over admissibility domains of generalized scenarios with I -divergence below some threshold is invariant under coordinate transformations.