Piecewise Linear Dynamic Systems for Own Risk Solvency Assessment

Solvency II represents a complex project for reforming the present vigilance system of solvency for European insurance companies. In this context many innovative elements arise, such as the formal introduction of risk management techniques also in the insurance sector. This allows to correctly assess risks and their interdependences and to take opportunities for example in terms of new insurance products, whose impact on company’s solvency is estimated beforehand. In this framework there is a growing need to develop so-called internal risk models to get accurate estimates of liabilities. In the context of non-life insurance, it is crucial to correctly assess risk from different sources, such as underwriting risk with particular reference to premium, reserving and catastrophe risks. Besides the underwriting cycle provides an additional volatility to liabilities distribution and so it could increase the solvency capital requirement or affect negatively the profitability of insurance companies and so it could be included inside an internal risk model. The aim of this paper is to improve the modelling of underwriting cycle for non-life insurance companies, also taking into account its effect on the solvency ratio. Starting from Collective Risk Theory, a dynamic control policy is defined to specify the relationship between solvency ratio and safety loading, to model the underwriting cycle. The corresponding dynamic equation for the solvency ratio, under some assumptions, assumes the form of a one dimensional piecewise linear map. The model could be easily extended to the definition of a correct distribution policy of dividends, in order to have a dynamic control of profitability, taking also into account the solvency requirements. This approach is developed both on a short and a long time horizon. Numerical analysis and stochastic assessments of the model conclude the work.