Asymptotic Distribution of the Optimal Value in Random Linear Programs: Application to Maximum Expected Shortfall

Abstract

The properties of risk measures are of fundamental concern in quantitative finance, particularly in times of uncertainty. We study the behaviour of the asymptotic distribution of the maximum expected shortfall of a portfolio that has both market and credit risk, where the marginal distributions of the risk factors are known but their joint distribution is unknown. We study the limiting behaviour of linear programs, as the maximum expected shortfall has a form similar to an optimal transport problem with stochastic cost function, and derive a result for the asymptotic distribution of the optimal solution similar to the central limit theorem. We then present simulations of maximum expected shortfall for a portfolio consisting of two counterparties subject to credit and market risk using the Basel IRB approach and a Merton single-factor copula model for portfolio losses. We observe that the histogram of maximum expected shortfall is well-described by a generalized extreme value distribution with a negative shape parameter.