Estimating Reinsurance Premiums Using Pareto Conjugate Priors and Extreme Value Methods: Studies Case of Fire Insurance Claims in Denmark

Loss distributions in insurance are typically right-skewed with heavy tails. As a result, modelling such distributions often involves the use of heavy-tailed distributions, such as the Pareto family, Cauchy, Student-t, and mixture distributions. This study employs the Generalized Inverse Gaussian (GIG) distribution as a conjugate prior to the Pareto distribution. The GIG distribution is characterized by three parameters and includes the modified Bessel function of the third kind in its density, which makes parameter estimation using the likelihood method challenging. Therefore, a Bayesian estimation approach is adopted, utilizing two prior distributions from the GIG family: the Inverse Gaussian and the Reciprocal Inverse Gaussian. The modelling is carried out within the framework of Extreme Value Theory (EVT), focusing on excess values over a specified threshold and the probability of claims exceeding that threshold. The results obtained from this analysis can be used to derive a premium estimation formula that insurance companies can apply when reinsuring their claims with a reinsurance company