Capital allocation and RORAC optimization under solvency 2 standard formula

Solvency II Directive 2009/138/EC requires an insurance and reinsurance undertakings assessment of a Solvency Capital Requirement by means of the so-called “Standard Formula” or by means of partial or full internal models. Focusing on the first approach, the bottom-up aggregation formula proposed by the regulator allows for a capital reduction due to the diversification effect, according to the typical subadditivity property of risk measures. However, once the overall capital has been assessed no specific allocation formula is provided or required in order to evaluate the contribution of each risk source on the overall Solvency Capital Requirement. The aim of this paper is twofold. First, we provide a closed formula for capital allocation fully compliant with the Solvency II Capital Requirement assessed by means of the Standard Formula. The solution enables a top-down approach to assess the allocated Solvency Capital Requirement among the risks considered in the Solvency II multilevel aggregation scheme; we demonstrate that the allocation formula adopted is consistent with the Euler allocation principle. Second, a solution is found as a result of an optimum capital allocation problem based on a Return On Risk Adjusted Capital measure; we establish the equivalence between the Return On Risk Adjusted Capital optimization, when the risk adjusted capital is calculated according to the Standard Formula, and the Markowitz mean-variance optimization.