JEL classification: G22 G11
MSC: 91B30 91B28 90C48
Keywords:Optimal reinsurance Risk measure and deviation measure Optimality conditions a b s t r a c t This paper studies the optimal reinsurance problem when risk is measured by a general risk measure. Necessary and sufficient optimality conditions are given for a wide family of risk measures, including deviation measures, expectation bounded risk measures and coherent measures of risk. The optimality conditions are used to verify whether the classical reinsurance contracts (quota-share, stop-loss) are optimal essentially, regardless of the risk measure used. The paper ends by particularizing the findings, so as to study in detail two deviation measures and the conditional value at risk.
Abstract. This paper deals with the optimal reinsurance problem if both insurer and reinsurer are facing risk and uncertainty, though the classical uncertainty free case is also included. The insurer and reinsurer degrees of uncertainty do not have to be identical. The decision variable is not the retained (or ceded) risk, but its sensitivity with respect to the total claims. Thus, if one imposes strictly positive lower bounds for this variable, the reinsurer moral hazard is totally eliminated.Three main contributions seem to be reached. Firstly, necessary and su¢ cient optimality conditions are given. Secondly, the optimal contract is often a bang-bang solution, i:e:, the sensitivity between the retained risk and the total claims saturates the imposed constraints. For some special cases the optimal contract might not be bang-bang, but there is always a bang-bang contract as close as desired to the optimal one. Thirdly, the optimal reinsurance problem is equivalent to other linear programming problem, despite the fact that risk, uncertainty, and many premium principles are not linear. This may be important because linear problems are easy to solve in practice, since there are very e¢ cient algorithms.
The purpose of this paper is to show how linear programming methodology can help us to design Bonus-Malus premium scales with some interesting theoretical and practical attributes. Examples of these properties are the financial equilibrium of the system, the monotonicity and proper variability of the premium scale, and the improvement of some efficiency measures such as the RSAL and the elasticity of the system. We will conclude that the use of the linear programming methodology makes possible a high degree of interaction between the designer and the mathematical model.
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