Computing the mean and the variance of the cedent's share for largest claims reinsurance covers

We present mathematical results allowing one to evaluate the moments of order 1 and 2 of the cedent's share in the framework of reinsurance treaties based on ordered claim sizes. These results consist of closed analytical formulas that do not involve any approximation procedure. This is illustrated by numerical examples when the claim number has the Poisson or the negative binomial distribution, and the claim cost has the exponential or the Pareto distribution.