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Maximum Spectral Measures of Risk with Given Risk Factor Marginal Distributions

Author

Listed:
  • Mario Ghossoub

    (Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada)

  • Jesse Hall

    (Credit Risk Technology, Scotiabank, Toronto, Ontario M5H 1H1, Canada)

  • David Saunders

    (Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada)

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 (MSM), as a worst-case spectral risk measure of the loss with respect to the dependence between the factors. The MSM 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.

Suggested Citation

  • Mario Ghossoub & Jesse Hall & David Saunders, 2023. "Maximum Spectral Measures of Risk with Given Risk Factor Marginal Distributions," Mathematics of Operations Research, INFORMS, vol. 48(2), pages 1158-1182, May.
    • Handle: RePEc:inm:ormoor:v:48:y:2023:i:2:p:1158-1182
    DOI: 10.1287/moor.2022.1299
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