Robust and Pareto Optimality of Insurance Contracts

Abstract

a b s t r a c tThe optimal insurance problem represents a fast growing topic that explains the most efficient contract that an insurance player may get. The classical problem investigates the ideal contract under the assumption that the underlying risk distribution is known, i.e. by ignoring the parameter and model risks. Taking these sources of risk into account, the decision-maker aims to identify a robust optimal contract that is not sensitive to the chosen risk distribution. We focus on Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR)-based decisions, but further extensions for other risk measures are easily possible. The Worst-case scenario and Worst-case regret robust models are discussed in this paper, which have been already used in robust optimisation literature related to the investment portfolio problem. Closed-form solutions are obtained for the VaR Worst-case scenario case, while Linear Programming (LP) formulations are provided for all other cases. A caveat of robust optimisation is that the optimal solution may not be unique, and therefore, it may not be economically acceptable, i.e. Pareto optimal. This issue is numerically addressed and simple numerical methods are found for constructing insurance contracts that are Pareto and robust optimal. Our numerical illustrations show weak evidence in favour of our robust solutions for VaR-decisions, while our robust methods are clearly preferred for CVaR-based decisions.

Optimal Robust Insurance with a Finite Uncertainty Set

“…This approach is well-known in finance and insurance applications, where stochastic models are constructed as potential candidates to represent the unknown "true" model; for example, see Fukushima (2009), Huang et al (2010) and Asimit et al (2017 and, where one should note that the first two papers considered a convex hull of the candidate models. Note that this approach produces a large uncertainty set that may be affect the robust optimal decision and therefore, it would be better to initiate a non-convex uncertainty set that is only composed of the possible models as explained in Asimit et al (2017).…”

“…Note that this approach produces a large uncertainty set that may be affect the robust optimal decision and therefore, it would be better to initiate a non-convex uncertainty set that is only composed of the possible models as explained in Asimit et al (2017).…”

“…This is no longer the case when the main aim is to find the best strategy within an optimisation problem, where finding the "best" model does not guarantee a robust decision, which is the main aim of the modelling process. A standard way to achieve this is to use the method of robust optimisation; comprehensive surveys could be found, for example, in Ben- Tal literature could be found in Balbás et al (2015) and Asimit et al (2017).…”

“…That is, the modeler does not know which probability model is appropriate and the optimal decision is produced by incorporating the risk measurements under all (but in a finite number) of the possible probability models. That is, the uncertainty set is constructed over a finite number of models as in Zhu and Fukushima (2009), Huang et al (2010) and Asimit et al (2017), where the first two papers considered a convex hull of the candidate models. This approach leads to a large uncertainty set that may be detrimental to the robust optimal decision and therefore, it would be better to consider a non-convex uncertainty set that is purely composed of the possible models as explained in Asimit et al (2017).…”

“…That is, the uncertainty set is constructed over a finite number of models as in Zhu and Fukushima (2009), Huang et al (2010) and Asimit et al (2017), where the first two papers considered a convex hull of the candidate models. This approach leads to a large uncertainty set that may be detrimental to the robust optimal decision and therefore, it would be better to consider a non-convex uncertainty set that is purely composed of the possible models as explained in Asimit et al (2017). We extend this idea by investigating various robust optimisation formulations and try to understand the effect over the robustness of the optimal decision, which is in fact the main aim of robust optimisation.…”