Convex risk measures for the aggregation of multiple information sources and applications in insurance

Abstract

We propose a novel class of convex risk measures, based on the concept of the Fréchet mean, designed in order to handle uncertainty which arises from multiple information sources regarding the risk factors of interest. The proposed risk measures robustly characterize the exposure of the firm, by filtering out appropriately the partial information available in individual sources into an aggregate model for the risk factors of interest. Importantly, the proposed risks can be expressed in closed analytic forms allowing for interesting qualitative interpretations as well as comparative statics and thus facilitate their use in the everyday risk management process of the insurance firms. The potential use of the proposed risk measures in insurance is illustrated by two concrete applications, capital risk allocation and premia calculation under uncertainty.

Robust policy selection and harvest risk quantification for natural resources management under model uncertainty

“…Then, substituting (34) to ( 22) we obtain the stated result (19) in Proposition 1. Also, combining equations ( 25) and ( 34) and substituting to (21) are obtained the optimal controls stated in (20) in Proposition 1.…”

“…Clearly, on the discussed context, this framework is not applicable since the provided information consists of a collection of models M and not just one model. This multi-prior setting is robustly handled by the framework of Fréchet risk measures [20], which are a natural extension of the one prior case to the multi-prior setting, where the notion of the Fréchet mean [14] is employed to determine an aggregate prior model using the concept of barycenter to condense and robustly represent the multiple information by a single model. The distance of a probability measure Q from the prior set M is measured in terms of the Fréchet function…”

“…The framework of convex risk measures [13,15] provides a worst-case scenario risk perspective, with respect to the factors that affect randomly the optimal control problem and allows the decision maker to provide his/hers risk aversion preferences and appropriately quantifying the related risk output. Recently, Papayiannis and Yannacopoulos [20] proposed an extension of the convex risk measures framework that is compatible with cases of model uncertainty as described above, providing risk calculations on the rationale of subjective worst-case scenario but robust to model ambiguity.…”

“…where d is an appropriate metric in the space of probability measures on R d , P(R d ) e.g., Wasserstein distance [20,21,22,25], and η > 0 is the maximum desired deviation level from model Q 0 in terms of the metric sense that is used. The optimization problem in (4) can be equivalently expressed in a multiplier-preferences formulation instead of the current constraint-preferences representation by…”

“…for an appropriate choice of the aversion parameter γ > 0 (quantifying the manager's aversion preferences from Q 0 ). This concept is further generalized and extended to the very interesting case where a multitude of models is provided for describing the behaviour of random variable Z possibly divergent and with varying probabilities of realization [20,21]. In optimal harvesting problems, different probability models could identify different scenarios about the conditions in the spatio-temporal domain that the natural resources optimal management problem is considered.…”