Pareto Type Distributions and Excess-of-loss Reinsurance

To be consistent with Extreme Value Theory the pricing of excess-of-loss reinsurance contracts should be based on Pareto type distributions. In this context, two recent Pareto type distributions are considered and compared. The state of a geometric Brownian motion after an exponentially distributed random time with log-normally distributed initial state generates a four-parameter Pareto type distribution, called double Pareto lognormal distribution. It has been introduced by Reed and Jorgensen and exhibits two-sided power-law behaviour. On the other hand, the mixture of a truncated exponential distribution in the lower-tail with a Pareto distribution in the upper-tail generates another analytical simple four-parameter Pareto type distribution, called exponential Pareto distribution. The statistical fitting of these two models is illustrated with a large claims data set and its application to the pricing of excess-ofloss reinsurance in actuarial science is illustrated. As a main result, it is shown that the selected best fitted righttailed Pareto lognormal distribution with minimum Cramér-von Mises K-statistics yields the safest mean and standard deviation of excess-of-loss reinsurance layer risks. The corresponding method is useful for conservative excess-of-loss reinsurance pricing.