Economic Risk Capital and Reinsurance: an Extreme Value Theory's Application to Fire Claims of an Insurance Company

The viability of an insurance company depends critically on the size and frequency of large claims. An accurate modelling of the distribution of large claims contributes to correct pricing’s and reserving’s decisions while maintaining through reinsurance an acceptable level of the unexpected fluctuations in the results. We present an application to the fire claims of an insurance company providing a model for large losses that we evaluate through simulations based on both a traditional and a Peaks over Threshold’s approach. Under the first one we estimate separately loss frequency, according to Negative Binomial distribution studying a claims number development triangle, and loss severity, according to Generalized Pareto distribution. A Peaks over Threshold’s approach is then developped estimating jointly frequency and severity distribution and considering the time dependence of data. We calculate the economic risk capital as the difference between the expected loss, defined as the expected annual claims amount, and the 99 .93 th quantile of the total cost distribution corresponding to a Standard & Poor’s A rating; we then simulate the impact of a quota share and an excess of loss reinsurance structure on the distribution of total cost amount and on economic risk capital. We provide a tool to price alternative programs and investigate how they can a .ect economic risk capital and explain the rationale of the choice of the optimal reinsurance programmes to smooth economic results.

Keywords: Economic Risk Capital; Peaks over Thresholds’ Approach; Homogenous and Inhomogenous Poisson Process; Modelling Trends; Negative Binomial Distribution