Forward Transition Rates in Multi-State Models
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