Maximum Spectral Measures of Risk with given Risk Factor Marginal Distributions

TL;DR

This paper introduces the Maximum Spectral Measure (MSP), a method to compute worst-case spectral risk measures for losses depending on two factors with known marginals but unknown joint distribution, using an optimization framework related to optimal transport.

Contribution

It formulates MSP as an optimization problem akin to optimal transport with a general objective, extending duality results and analyzing stability and asymptotic properties.

Findings

MSP provides a worst-case spectral risk bound under marginal constraints.

The formulation generalizes optimal transport with a broader objective.

Asymptotic analysis characterizes the distribution of the MSP under sampling.

Abstract

We consider the problem of determining an upper bound for the value of a spectral risk measure of a loss that is a general nonlinear function of two factors whose marginal distributions are known, but whose joint distribution is unknown. The factors may take values in complete separable metric spaces. We introduce the notion of Maximum Spectral Measure (MSP), as a worst-case spectral risk measure of the loss with respect to the dependence between the factors. The MSP admits a formulation as a solution to an optimization problem that has the same constraint set as the optimal transport problem, but with a more general objective function. We present results analogous to the Kantorovich duality, and we investigate the continuity properties of the optimal value function and optimal solution set with respect to perturbation of the marginal distributions. Additionally, we provide an asymptotic result characterizing the limiting distribution of the optimal value function when the factor distributions are simulated from finite sample spaces. The special case of Expected Shortfall and the resulting Maximum Expected Shortfall is also examined.