Calculation of solvency reserves for biometrical risks in life insurance
Zusammenfassung der Projektergebnisse
In 2000 the European Commission decided to fundamentally change the supervisory system of the insurance industry in Europe and started the project 'Solvency II'. The central change was the call for a prospective and risk-oriented approach, which should reflect the true risks of an insurance company. In order to meet that high expectations, the Commission stipulated that future solvency capital requirements shall be based on modern stochastic risk measuring techniques. However, because of the absence of adequate mathematical concepts, the current proposal for a standard formula does not completely live up to expectations. The framework directive constitutes the use of the stochastic risk measure Value-at-Risk, but the standard formula does not really use probabilistic modeling. The key objective of the project was to develop a 'true' stochastic approach for the calculation of solvency reserves for life insurance business that fits into the basic concept of Solvency II. In a first step we considered the classical 3-state Markov model for disability insurance policies and fitted the multivariate Hyndman and Ullah generalization of the Lee-Carter model. As a result, we obtained a joint stochastic model for mortality, disability, and reactivation rates that also allows for possible correlations between the different transitions. We used the new model to calculate joint confidence bands for mortality, disability, and reactivation rates. In a second step we combined our confidence bands with analytical worst-case concepts in order to obtain upper bounds for biometrical solvency reserves based on the Value-at-Risk measure. We developed several worst-case calculation approaches and ended up with an approach that overestimated the Value-at-Risk by less than 5% in all the examples that we studied. In order to obtain such sharp bounds, not only the choice of the worst-case concept was important but also the shape of confidence bands. Using limited expansion around a best-estimate we could define approximately optimal shapes. The result is an alternative formula for the calculation of biometrical solvency capital requirements that still fits to the basic concept of Solvency II, is truly probabilistic (in contrast to the standard formula), and takes respect of the different dependency structures of life insurance portfolios. It can help to calibrate the parameters of the Solvency II standard formula or even serve as an internal model for an insurance company. In a third step we had a deeper look at the way that the standard formula aggregates risks, the so-called square root formula. Again using limited expansion around best estimates, we were able to theoretically substantiate the square root formula for the life insurance undenwriting risk. Numerical illustrations performed on the basis of our German data suggested that the Solvency II QIS correlation matrix was not appropriate and that the correlations greatly varied from one product to another. We learned that the problem is not so much with the square-root formula itself and the normality assumption but well in using the same correlation values for all types of products. Our alternative model from step two can help here to improve the QIS correlation assumptions. Additionally to this main results we used the worst-case techniques mentioned above in order to study which combinations of different policy types have a strong netting effect provided by the natural hedge between payments that are due when sojourning in a state and when leaving a state. The results can help to reduce risks already at the stage of the designing of insurance contracts.
