This research proposes and analyzes new models for a stochastic resource allocation problem that arises in a variety of operations contexts. One of the primary contributions of the paper lies in providing a succinct, robust, and general model that can address a range of different risk-based objectives and cost assumptions under uncertainty. Although the model expression is relatively simple, it embeds a reasonably high degree of underlying complexity, as the analysis shows. In addition, in-depth analysis of the model, both in its general form and under various specific risk measures, uncovers some interesting and powerful insights regarding the problem trade-offs. Furthermore, this analysis leads to a highly efficient class of heuristic algorithms for solving the problem, which we demonstrate via numerical experimentation to provide close-to-optimal solutions. This computational benefit is a critical element for solving a class of broadly applicable larger problems for which our problem arises as a subproblem that requires repeated solution.