Matrix Regularization Procedure within the Aggregation Process of Solvency II Internal Models

In recent years, the insurance market has undergone great transformations, mainly due to the increasing complexity of the insurance products. This fact has generated the need to develop a new directive for the regulation of the insurance industry. To meet this necessity, the so called Solvency II directive was adopted by the Council of the European Union and Parliament in November 2009 and came into effect on 1 January 2016. One of the main objectives of this new directive is to enhance the measurement of the risk faced by an undertaking and, consequently, of the capital required to guarantee that an insurance company remains solvent. The directive provides a standard method to calculate the so called Solvency Capital Requirement, but it also allows the companies to use an internal model, developed on the basis on their specific risk profile. In both cases, the quantification of the capital requirement starts from an evaluation of the single risks affecting the company and then requires a process of aggregation in order to obtain the overall Solvency Capital Requirement. Within this aggregation process, the dependency structure between the risks will be specified through their correlation matrix. Moreover, the aggregation methodologies require this correlation matrix to be positive semi-definite. This assumption is not always satisfied by the correlation matrices calibrated by the insurance companies and, consequently, a regularization procedure has to be applied to the matrix before it can be used. The values of the correlation parameters can have great influence in the resulting capital requirement, because they represent the level of diversification that can be achieved between the risks considered. For this reason, the regularization procedure should search for the positive semi-definite correlation matrix which is the ¿closest¿ to the given (not positive semi-definite) correlation matrix. In this paper we will deal with some of the most widespread regularization algorithms present in the literature. Moreover we will present the results of some tests performed in order to understand the impact of the matrix regularization procedure on alternative correlation structures. The algorithms used for those tests have been selected among the ones presented.