Excess Loss and Stop-Loss Reinsurance for Exam STAM

Excess of Loss Reinsurance

Per-occurrence excess of loss reinsurance covers individual claims above a retention (attachment point) M, up to a limit u. The reinsurer pays min(max(X - M, 0), u) for each claim. The expected reinsurance cost per claim is E[min(max(X - M, 0), u)] = E[X wedge (M+u)] minus E[X wedge M]. Pricing requires knowledge of the severity distribution's limited expected values. Higher retentions reduce reinsurance cost but increase the cedant's retained risk.

Stop-Loss (Aggregate Excess) Reinsurance

Stop-loss reinsurance covers aggregate losses exceeding a threshold. If S is aggregate losses, the reinsurer pays max(S - d, 0), possibly subject to a limit. The expected stop-loss premium is E[max(S - d, 0)] = E[S] minus E[S wedge d]. Computing this requires the aggregate loss distribution, often obtained via convolution, recursion (Panjer), or simulation. Exam STAM tests your ability to price both types of reinsurance, calculate expected costs using distributional properties, and understand how reinsurance structures manage insurer risk.