Probability Transforms with Elliptical Generators∗

This paper introduces the notion of elliptical transformations for possible applications in constructing insurance premium principles. It is a common notion in actuarial science to transform or sometimes said, to distort the real probability measure of a risk X and then calculate premiums based on the expectation of the transformed distribution. Elliptical transformations are a way to distort the probability distribution of a risk and it is based on the relative ratio of a density generator of a member of the class of elliptical distributions to the density generator of the Normal distribution which is also well-known to belong to the class of elliptical distributions. The class of elliptical distribution models consists of distributions considered symmetric and provides a generalization of the class of Normal loss distribution models. We examine the premium principle implied by this elliptical transformation. We find that the elliptical transforms lead us to, as special cases, the Wang premium principle, the Wang Student-t distortion principle, as well as the Esscher premium principle. In several cases, the resulting premium principle has a replicating feature of the standard deviation premium principle. A numerical example has been elaborated to illustrate the practical implementation of this elliptical transformation. ∗Keywords: probability transform, distortion, risk measures, premium principles, elliptical distri-butions, elliptical transforms