For a series of randomly discounted terms we give an integral criterion to distinguishbetween almost-sure absolute convergence and divergence in probability to $\infty$, these being the only possible forms of asymptotic behavior. This solves the existence problem for a one-dimensional perpetuity that remains from a 1979 study by Vervaat, and yields a complete characterization of the existence of distributional fixed points of a random affine map in dimension one.