This technical note investigates the effect of stability on the probability of correctly predicting level-crossings for the output of a linear dynamical system driven by Gaussian noise. It is found that decreasing the stability margin has a favorable effect on predictive capability. The insight from this finding was derived from a parametric analysis of a given measure of predictive capability, represented as an explicit function of the spectral radius. The formulae used to characterize this relationship were derived under strict technical conditions in previous work. However, as a result of the closed-form nature of these expressions, using these formulae to gain insight on the influence of stability was much more computationally efficient than would otherwise be possible under more relaxed technical conditions. These findings are also extended to determine how system dynamics are concurrently influenced, as quantified by both the natural frequency and damping ratio of a second order linear dynamical system. It is found that for an optimal predictor, as the stability margin is decreased infinitesimally, system dynamics have no adverse influence on one-step zero-crossing predictive capability. However, for two suboptimal predictors investigated it is found that under the same conditions the system dynamics influence predictive capability adversely as the natural frequency approach critical values. For zero-crossing predictive capability with a prediction horizon consisting of a single step, these facts will be rigorously proven for any ARMA(2,1) process.