Capital requirements and optimal investment with solvency probability constraints
IMA Journal of Management Mathematics
This is the accepted version of the paper. This version of the publication may differ from the final published version. Permanent repository link: http://openaccess.city.ac.uk/13146/ Link to published version: http://dx. Abstract Quantifying the economic capital and optimally allocating it into portfolios of financial instruments are two key topics in the Asset/Liability Management (ALM) of an insurance company. In general these problems are studied in the literature by minimizing standard risk
... izing standard risk measures such as the Value at Risk (VaR) and the Conditional Value at Risk (CVaR). Motivated by Solvency II regulations, we introduce a novel optimization problem to solve for the optimal required capital and the portfolio structure simultaneously, when the ruin probability is used as an insurance solvency constraint. Besides the generic optimal required capital and portfolio problem formulation, we propose a two-model hierarchy of optimization models, where both models admit the so-called secondorder conic reformulation, in turn making them particularly well suited for numerics. The first model, albeit naively asserting the normality of the returns on assets and liabilities, under minor further simplifications admits a closed form solution -a set of formulas, which may be used as simple decision-making guidelines in the analysis of more complex scenarios. A potentially more realistic second model aims to represent the "heavy-tailed" nature of the insurer's liabilities more accurately, while also allowing arbitrary distributions of asset returns via a semi-parametric approach. Extensive numerical simulations illustrate the sensitivity and robustness of the proposed approach relative to model's parameters. In addition, we explore the potential of insurance risk diversification and discuss if combining several liabilities into a single insurance portfolio may always be beneficial for the insurer. Finally, we propose an extension of the model with an expected return on capital constraint added.