We analyze a class of custom-mode laser cavities with aspheric mirrors that become either asymptotically flat or hemispheric away from the optical axis. Previous work classifies the modes of such cavities as either bound or unbound. We develop an analytic approximation for the losses and frequencies of such modes. The bound modes have losses that diminish exponentially as the size of the cavity mirror increases and have frequencies that become independent of the mirror size. On the other hand, unbound modes have losses that diminish asymptotically as the inverse third power of the mirror width and frequencies that converge toward those of the unperturbed (flat or hemispheric) cavity with increasing mirror width. Finally, we show good agreement between our model and numeric cavity eigenvalue calculations.