A novel two‐step extrapolation‐insertion risk model based on the Expectile under the Pareto‐type distribution

The catastrophe loss model developed is a challenging problem in the insurance industry. In the context of Pareto‐type distribution, measuring risk at the extreme right tail has become a major focus for academic research. The quantile and Expectile of distribution are found to be useful descriptors of its tail, in the same way as the median and mean are related to its central behavior. In this article, a novel two‐step extrapolation‐insertion method is introduced and proved its advantages of less bias and variance theoretically through asymptotic normality by modifying the existing far‐right tail numerical model using the risk measures of Expectile and Expected Shortfall (ES). In addition, another solution to obtain the ES is proposed based on the fitted extreme distribution, which is demonstrated to have superior unbiased statistical properties. Uniting these two methods provides the numerical interval upper and lower bounds for capturing the real quantile‐based ES commonly used in insurance. The numerical simulation and the empirical analysis results of Danish reinsurance claim data indicate that these methods offer high prediction accuracy in the applications of catastrophe risk management.