Some results on quadratic credibility premium using the balanced loss function

Abstract

Purpose – This paper generalizes the quadratic framework introduced by Le Courtois (2016) and Sumpf (2018), to obtain new credibility premiums in the balanced case, i.e. under the balanced squared error loss function. More precisely, the authors construct a quadratic credibility framework under the net quadratic loss function where premiums are estimated based on the values of past observations and of past squared observations under the parametric and the non-parametric approaches, this framework is useful for the practitioner who wants to explicitly take into account higher order (cross) moments of past data. Design/methodology/approach – In the actuarial field, credibility theory is an empirical model used to calculate the premium. One of the crucial tasks of the actuary in the insurance company is to design a tariff structure that will fairly distribute the burden of claims among insureds. In this work, the authors use the weighted balanced loss function (WBLF, henceforth) to obtain new credibility premiums, and WBLF is a generalized loss function introduced by Zellner (1994) (see Gupta and Berger (1994), pp. 371-390) which appears also in Dey et al. (1999) and Farsipour and Asgharzadhe (2004). Findings – The authors declare that there is no conflict of interest and the funding information is not applicable. Research limitations/implications – This work is motivated by the following: quadratic credibility premium under the balanced loss function is useful for the practitioner who wants to explicitly take into account higher order (cross) moments and new effects such as the clustering effect to finding a premium more credible and more precise, which arranges both parts: the insurer and the insured. Also, it is easy to apply for parametric and non-parametric approaches. In addition, the formulas of the parametric (Poisson–gamma case) and the non-parametric approach are simple in form and may be used to find a more flexible premium in many special cases. On the other hand, this work neglects the semi-parametric approach because it is rarely used by practitioners. Practical implications – There are several examples of actuarial science (credibility). Originality/value – In this paper, the authors used the WBLF and a quadratic adjustment to obtain new credibility premiums. More precisely, the authors construct a quadratic credibility framework under the net quadratic loss function where premiums are estimated based on the values of past observations and of past squared observations under the parametric and the non-parametric approaches, this framework is useful for the practitioner who wants to explicitly take into account higher order (cross) moments of past data.

Keywords

• Quadratic credibility
• Loss function
• Credibility premium
• Parametric approach
• Non-parametric approach