An Optimality Approach to the Application Ratio for the Matching Adjustment in the Solvency II Regime

We refer to the Technical Specifications provided by EIOPA to implement the package of long-term guarantees measures which shall be included into the Solvency II Framework Directive. One of these regulatory measures concernes the Application Ratio, a coefficient defining what portion of the Maximum Matching Adjustment an insurance company can apply to the risk-free rates for discounting her obligations, given the matching properties of the assigned asset portfolio. In this paper we propose an optimization algorithm providing a reliable assessment of the Application Ratio. The Application Ratio provided by the algorithm is optimal in the sense that it has the maximum value given the structure %matching properties of the asset-liability portfolio. This value corresponds to the minimum attainable level for the losses incurred from forced sales of defaultable bonds with mispriced market value. We show that under natural assumptions this optimality problem has the form of a linear programming problem, which can be easily solved using standard numerical procedures. A matching criterion defined in stronger form can also be applied by imposing appropriate run-off constraints in the linear programming problem. The value of the optimal Application Ratio can be used by a Supervisor as an objective benchmark for checking the appropriateness of the Application Ratio adopted by the undertaking. The optimal liquidation policy provided by the algorithm can also be used by an insurance undertaking which want to apply conservative management actions to her asset-liability portfolio.