  We consider the task of assessing the righthand tail of an insurer's loss
distribution for some specified period, such as a year. We present and analyse
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
favour 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
favour the Gamma distribution as a flexible model for single event losses, for
its tractable form in all of the methods we analyse, its generalisability, and
because of the ease with which a cap on losses can be incorporated.
