Abstract: The only class of non-degenerate limiting distributions for component-wise maxima is the class of max-stable distributions. This theoretical result has lead to their advocation for use in the statistical modeling of multivariate extremes which play an important role in various applications including insurance, finance, weather and climate. However, there remains contentious issues with respect to the practicality of max-stable models. In general, they lack tractable likelihoods, which hampers statistical inference. Furthermore, component-wise maxima do not necessarily occur in reality, leading to inference about an artificial quantity while simultaneously discarding a vast amount of data. This implies that a great amount of care must be taken with respect to the use of max-stable models in practice. When dealing with extremes, there are many cases where one would like to be conservative. For example, with flooding one would like to error on the safe side when specifying construction heights for a system of levees. In statistical inference this conservatism equates to preserving stochastic ordering. Thus, obtaining good stochastic upper bounds on models for dependent extremes has great utility in practice. In this work we present a principled methodology for estimating stochastic upper bounds of extremes by exploiting unique properties of max-stable distributions that preserve both ordering and dependence structure, while simultaneously making inference tractable for moderately high dimension. Through various simulations, we examine the quality of our estimation and explore the proximity of the upper bound to the truth. We apply our method to estimate upper bounds on extreme value at risk for a portfolio of assets.