We consider the task of assessing the right-hand tail of an insurer's loss distribution for some specified period, such as a year. We present and analyze six different approaches: four upper bounds and two approximations. We examine these approaches under a variety of conditions, using a large event loss table for US hurricanes. For its combination of tightness and computational speed, we favor the moment bound. We also consider the appropriate size of Monte Carlo simulations and the imposition of a cap on single-event losses.We strongly favor the Gamma distribution as a flexible model for single-event losses, because of its tractable form in all of the methods we analyze, its generalizability and the ease with which a cap on losses can be incorporated into it.