<p>We explore a comprehensive class of distributionally robust optimization models that maximize the probability of exceedance defined as a ratio of stochastic functions. We develop a general reformulation framework that encompasses two forms of ambiguity (moment and Wasserstein), two types of distributional support (uncertain probabilities and continuum of realizations), and multiple functional forms for the probability of exceedance which we successively express as a ratio of linear-to-linear, linear-to-quadratic, quadratic-to-linear, quadratic-to-quadratic, and linear-to-quadratic norm functions. For each case, we first construct the hypographical formulation taking the form of a semi-infinite optimization problem with a distributionally robust chance constraint in which the probability level is a random variable, and derive then a finite-dimensional and computationally tractable reformulation. For the continuum of realizations support, the reformulations are biconvex problems for which we design a customized algorithm that converges finitely and is general enough to handle all (integer and continuous) biconvex reformulations. The numerical experiments demonstrate the computational efficiency of the proposed reformulations and solution method, and the benefits of adopting a distributionally robust approach for the probability of exceedance. The models with moment-based ambiguity sets outperform those with Wasserstein ambiguity in terms of out-of-sample Sharpe ratio and cumulative return when the asset universes’ size is small. In contrast, Wasserstein-based models exhibit superior performance and scale better as the size of the asset universe increases.</p>