Optimal reinsurance under general law-invariant risk measures

• Published in: Scandinavian actuarial journal Vol. 2014; no. 1; pp. 72 - 91
• Main Authors: Cheung, K.C., Sung, K.C.J., Yam, S.C.P., Yung, S.P.
• Format: Journal Article
• Language: English
• Published: Stockholm Taylor & Francis Group 01-02-2014; Taylor & Francis Ltd
• Subjects: Actuarial science; convex risk measures; Decision analysis; Expected utility; geometric approach; Insurance industry; insurance layer; Mathematical problems; optimal reinsurance; Optimization techniques; Reinsurance; Risk assessment; stop-loss insurance; Studies
• ISSN: 0346-1238; 1651-2030
• DOI: 10.1080/03461238.2011.636880

In recent years, general risk measures play an important role in risk management in both finance and insurance industry. As a consequence, there is an increasing number of research on optimal reinsurance decision problems using risk measures beyond the classical expected utility framework. In this paper, we first show that the stop-loss reinsurance is an optimal contract under law-invariant convex risk measures via a new simple geometric argument. A similar approach is then used to tackle the same optimal reinsurance problem under Value at Risk and Conditional Tail Expectation; it is interesting to note that, instead of stop-loss reinsurances, insurance layers serve as the optimal solution. These two results highlight that law-invariant convex risk measure is better and more robust, in the sense that the corresponding optimal reinsurance still provides the protection coverage against extreme loss irrespective to the potential increment of its probability of occurrence, to expected larger claim than Value at Risk and Conditional Tail Expectation which are more commonly used. Several illustrative examples will be provided.