representing the maximum insured loss. The moment generating function of a Beta distribution is an untabulated function with an infinite series representation, and so will be more expensive to compute accurately; this will affect the Chernoff bound. There are no difficulties with the moments. However, we would question the suitability of using a Beta distribution here. The insurer’s loss from an event is capped at the maximum insured loss. This implies an atom of probability at the maximum insured loss: if f$_{i}$ is the original loss distribution for event i and u is the maximum insured loss, then where p$_{i}$ := ∫ u 0 f$_{i}$ ( x ) d x and δ is the Dirac delta function, as before. A Beta distribution scaled to [0 , u ] would be quite different, having no atom at u . The Gamma distribution for f$_{i}$ is tractable with a cap on losses. If f$_{i}$ is a Gamma distribution then the MGF is where γ is the incomplete Gamma function, and f$_{i}$ ( x ; u ) = f$_{i}$ ( x ) 1$_{x<u}$ + (1 − p$_{i}$ ) δ ( x − u ) M$_{i}$ ( v ; u ) = ( β$_{i}$ β$_{i}$ − v$^{)}$$^{α$_{i}$}$ γ ( α$_{i}$, ( β$_{i}$ − v ) u ) Γ( α$_{i}$ ) + (1 − p$_{i}$ ) e $^{vu}$, γ ( s, u ) := ∫ u 0 x s − $^{1}$e − $^{x}$d x, p$_{i}$ := γ ( α$_{i}$, β$_{i}$u ) Γ( α$_{i}$ ) . 15