In this Thesis, we develop computational models for line failure risk and cascading power failures on bulk power systems. General reliability issues cost society billions of dollars and large blackouts from cascading power failures have an extreme impact on society and can lead to loss of life. We start by building an analytic formulation of the cascading simulation found in literature, formulating the problem as a multi-stage stochastic program with integer variables to model the decision-dependent uncertainty. This allows for the flexibility to include the cascading simulation in as a subproblem for optimization in long-term design problems. We then use the Monte Carlo simulation of the cascading process and large computational resources of HTCondor to optimize transmission capacity expansion to reduce the expected value of load shed for a set of initial contingencies. The second half of the thesis focuses on the five minute dispatch market for power systems. We develop a system risk measure that is a joint chance constraint on the probability that one or more lines fail, which we constrain to be a small number. We model the probability that one line fails as a piecewise linear function. Additionally, we model uncertainty in generation and demand as a multivariate Gaussian and form a linear approximation of our system risk measure under uncertainty. We solve this large, convex problem using cutting planes. In order to account for exogenous contingencies, we model the N-1 contingencies and apply our risk measure under uncertainty. Finally, we find weights to measure the impact of losing a particular transmission line using the cascading simulation to evaluate rare event stress. We use these weights in a new weighted version of a joint chance constraint with N-1 contingency and solve with cutting planes. These computational models explore rare event stress through the cascade simulation. They incorporate uncertainty in demand, the initial contingencies that start cascades, and the uncertainty in cascade evolution. The first two models look at long term design problems given a budget to improve infrastructure and the last two models look at the five minute market to reduce rare event risk.