Abstract
The 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.
Highlights
Finding the optimal insurance contract has represented a topic of interest in the actuarial science and insurance literature for more than 50 years
Extensive research has been made when the preferences are made via coherent risk measures and VaR; for example, see Cai and Tan (2007), Balbás, Balbás, and Heras (2009);
The optimal insurance contract problem under parameter/model uncertainty has been investigated only by Balbás, Balbás, Balbás, and Heras (2015), where only the worst-case is investigated for a large class of risk measures that includes Conditional Value-at-Risk (CVaR), but not VaR, and a particular choice of the uncertainty set of probability measures
Summary
Finding the optimal insurance contract has represented a topic of interest in the actuarial science and insurance literature for more than 50 years. Whenever the model and parameter risks are present, it is prudent to consider insurance contracts that are optimal under a set of plausible models and this is precisely what robust optimisation does It is a vast area of research with applications in various fields and a standard reference is Ben-Tal, El Ghaoui, and Nemirovski (2009), while comprehensive surveys can be found in Ben-Tal and Nemirovski (2008), Bertsimas, Brown, and Caramanis (2011) and Gabrel, Murat, and Thiele (2014).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
