We investigate the stability properties of two different classes of metabolic cycles using a combination of analytical and computational methods. Using principles from structural kinetic modeling (SKM), we show that the stability of metabolic networks with certain structural regularities can be studied using a combination of analytical and computational techniques. We then apply these techniques to a class of single input, single output metabolic cycles, and find that the cycles are stable under all conditions tested. Next, we extend our analysis to a small autocatalytic cycle, and determine parameter regimes within which the cycle is very likely to be stable. We demonstrate that analytical methods can be used to understand the relationship between kinetic parameters and stability, and that results from these analytical methods can be confirmed with computational experiments. In addition, our results suggest that elevated metabolite concentrations and certain crucial saturation parameters can strongly affect the stability of the entire metabolic cycle. We discuss our results in light of the possibility that evolutionary forces may select for metabolic network topologies with a high intrinsic probability of being stable. Furthermore, our conclusions support the hypothesis that certain types of metabolic cycles may have played a role in the development of primitive metabolism despite the absence of regulatory mechanisms.