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Halliwell, Leigh J. 2012. “The Mathematics of Excess Losses.” Variance 6 (1): 32–47.
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  • Figure 1. Excess-loss function of an Exponential(θ) loss
  • Figure 2. Layer function


After laying a fairly rigorous foundation for the mathematical treatment of excess losses, this paper shows that the excess-loss function is akin to the probability distribution of its loss. All the moments of the loss can be reclaimed from the excess-loss function, the variance being especially simple. Excess-loss mathematics is a powerful tool for pricing loss layers, as in reinsurance. In some settings it may be more powerful than standard probability techniques. An example featuring the mixed exponential distribution demonstrates this. Two appendices deal with Stieltjes integrals and with proofs of two findings about layered losses that are commonly known among reinsurance actuaries.