| Summary: | Optimal reinsurance problems under the risk measures, such as Value-at-Risk (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>VaR</mi></semantics></math></inline-formula>) and Tail-Value-at-Risk (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>TVaR</mi></semantics></math></inline-formula>), have been studied in recent literature. However, losses based on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>VaR</mi></semantics></math></inline-formula> may be underestimated and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>TVaR</mi></semantics></math></inline-formula> allows us to account better for catastrophic losses. In this paper, we propose a new family of flexible risk measures denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>LVaR</mi></semantics></math></inline-formula>, which is a weighted combination of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>VaR</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>TVaR</mi></semantics></math></inline-formula>. Based on the new risk measures, we deal with the optimal reinsurance problem by minimizing the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>LVaR</mi></semantics></math></inline-formula> of the total risks of an insurer when two types of constraints for reinsurer’s risk exposure are considered. The results indicate that the two-layer reinsurance is always an optimal reinsurance policy with both types of constraints. Also, we find that the optimal reinsurance policy depends on the confidence level, the weight coefficient, the safety loading, the tolerance level, as well as the relations between them. Finally, we illustrate the results by numerical examples and compare them with the results in Lu et al.
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