The Extreme Value Theory and Risk Reserve in non-life insurance companies

Many studies have been carried out on the application of the Extreme Value Theory (EVT) in the insurance field, especially in order to model the distribution of large claims. In this paper attention is focused on the Peaks Over Threshold (POT) approach which represents one of the two related ways to the EVT. It is emphasized how the POT method can be applied in order to determine accurate and correct premiums. In particular we are interested in investigating the portfolio represented by the “1991-1992 SOA Group of Medical Insurance Large Claims”. By taking into account the results concerning the price of portfolio and of excess of loss reinsurance obtained in Cerchiara and Esposito [2005] and achieved by using POT we show the excess of loss reinsurance impact on solvency ratio with a simplified approach. The analysis has been undertaken by modelling a Risk Reserve with particular reference to the effects of an excess of loss reinsurance treaty in a short time horizon (one year). This type of reinsurance treaty presents a natural link with the POT approach. Especially with reference to the Solvency II project attention to capital requirements is growing and not only on the long term but also over the short period. Therefore the simplified approach provided may be a useful tool for internal risk models evaluating the solvency in the short term. The paper is organised as follows. In section 2 we describe the basic theory concerning the POT method; in section 3 we provide the key issues concerning the modelling of Risk Reserve; in section 4 we analyse the 1992 data in order to highlight their long tailed nature; in section 5 we show the effects of excess of loss reinsurance treaty on solvency ratio and finally we focus our attention on some concluding remarks and future developments.