During the last decade the idea of forward interest rates has been transferred to the mortality framework, and meanwhile forward rates have become a helpful tool for the securitization of mortality and longevity risks. Up to now the market for mortality-linked assets is relatively small, but under Solvency II it becomes more and more important for two reasons. First, such assets can help the insurance companies to lower their Solvency Capital Requirement by outsourcing some risks to the financial market. Second, the calculation of the Solvency Capital Requirement under Solvency II fundamentally bases on market values. Therefore, models for the construction of market values are needed, including all biometrical risk factors.

While there exists a lot of literature on forward mortality rates, multi-state insurance products are hardly discussed. A first attempt to define general forward transition rates was made by Norberg (Insurance: Mathematics and Economics, 2010, 47(2), 105-112), who discusses different approaches to define forward transition rates, but concludes that in general there is no clear answer.

Our paper has three major objectives: a sound definition of forward rates that features some desirable properties, uniqueness of that forward rate definition, and necessary conditions for the existence.

First, we theoretically discuss how forward transition rates can and should be defined. In particular, we follow the substitution concept and stress the notion that forward rates should be invariant with respect to some set of derivatives. These sets should include all common benefits (i.e., for staying in one state and for the transition into another state) and some standardized products (i.e., a forward representation of each single hazard rate). We give examples where this invariance property is fulfilled.

Second, we discuss the uniqueness of our forward rate definition. We consider cycle-free multi-state models where unique definitions can be obtain from the Kolmogorov forward equations. Most multi-state models can at least be approximated by a cycle-free model. In particular, we show the link between uniqueness of forward rates and the set of derivatives that are included into the model.

At last, we show under some weak requirements what kind of dependency structure is necessary to obtain the invariance property. The result bases on a fixed class of stochastic processes that includes most of the forward mortality rate models that can be found in the literature.

*Awarded Life Track Prize