Abstract
Distortion (Denneberg in ASTIN Bull 20(2):181–190, 1990) is a well known premium calculation principle for insurance contracts. In this paper, we study sensitivity properties of distortion functionals w.r.t. the assumptions for risk aversion as well as robustness w.r.t. ambiguity of the loss distribution. Ambiguity is measured by the Wasserstein distance. We study variances of distances for probability models and identify some worst case distributions. In addition to the direct problem we also investigate the inverse problem, that is how to identify the distortion density on the basis of observations of insurance premia.
Highlights
The function of the insurance business is to carry the risk of a loss of the customer for a fixed amount, called the premium
After some introduction about general premium principles we propose generalizations of the distortion premium
The smoothness properties are important for robustness aspects, it is well known that a quite smooth direct function makes the inverse problem difficult
Summary
The function of the insurance business is to carry the risk of a loss of the customer for a fixed amount, called the premium. The premium has to be larger than the expected loss, otherwise the insurance company faces ruin with probability one. There are several principles, from which an insurance premium is calculated on the basis of the loss distribution. An insurance premium is a functional, π : {X ≥ 0 defined on (Ω, F, P)} → R≥0. We will work with functionals that depend only on the distribution of the loss random variable (sometimes called law-invariance or version-independence property, Young 2014). If X has distribution function F we use the notation π(F) for the pertaining insurance premium, and E(F) for the expectation of F. – The distortion principle (Denneberg 1990). – The certainty equivalence principle (Von Neumann and Morgenstern 1947). – The ambiguity principle (Gilboa and Schmeidler 1989). – Combinations of the previous (for instance Luan 2001)
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