Evolutionary credibility risk premium

Abstract

This article provides the first systematic study on the risk premium calibration under the celebrated evolutionary credibility models which had been studied in Jones and Gerber (1975) and Albrecht (1985) but only for net premium, while our work now simultaneously estimates the process variance and the hypothetical mean. Our objective is to minimize the mean square deviation of the empirical estimates from the respective theoretical mean and process variance, which leads to extending the set of classical normal equations. Despite that no more closed-form solutions of the normal equations can be obtained, we obtain an effective numerical scheme featuring a novel recursive LU algorithm for the progressively enlarging coefficient matrices, and we shall also demonstrate its effectiveness through several common time series models, namely ARMA. Our proposed method can also be viewed as a robust extension of the recent SURE estimator used in statistics literature, which assumes the underlying data being i.i.d. with the Normal-Inverse-Wishart structure, while we allow a temporal dependence structure among the data without specifying the probability model.