The distortion principle for insurance pricing: properties, identification and robustness
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The distortion principle for insurance pricing: properties, identification and robustness Debora Daniela Escobar1
· Georg Ch. Pflug2
Published online: 20 December 2018 © The Author(s) 2018
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. Keywords Ambiguity · Distortion premium · Dual representation · Premium principles · Risk measures · Wasserstein distance
1 Introduction 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. The difference between the premium and the expectation is called the risk premium. There are several principles, from which an insurance premium is calculated on the basis of the loss distribution. Let X be a (non-negative) random loss variable. Traditionally, 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.
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Debora Daniela Escobar [email protected] Georg Ch. Pflug [email protected]
1
Department of Statistics and Operations Research (ISOR), University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
2
ISOR and International Institute for Applied Systems Analysis (IIASA), Laxenburg, Austria
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Annals of Operations Research (2020) 292:771–794
We use alternatively the notation π(F) or π(X ), resp. E(F) or E(X ) whenever it is more convenient. To the extent of the paper, a more specific notation is used for particular cases of the premium. We consider the following basic pricing principles: – – – –
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).
1.1 The distortion principle The distortion principle is related to the idea of stress testing. The original distribution function F is modified (distorted) and the premium is the expectation of the modified distribution. If g : [0, 1] → R is a concave monotonically increasing function with the property g(0) = 0, g(1) = 1, then the distorted d
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