On the forward rate concept in multi-state life insurance

Similarly to the notion of modeling credit risk by using forward credit default spread rates, mortality risk in life insurance contracts is nowadays often modeled by using forward mortality (spread) rates. Since recently this concept is also discussed for more complex life insurances that include multiple lifes or intermediate states that correspond to the health status of the insured. For consistency purposes and for technical reasons most authors assume that the underlying financial and demographic events are stochastically independent. In the present paper we study sufficient and necessary conditions under which general transition forward rates are indeed consistent with respect to the relevant insurance claims. This shows the theoretical limits of the forward rate concept in life insurance. Our study is based on a model where the underlying financial and demographical developments are diffusion processes driven by a multivariate Brownian motion. This allows us to investigate independence properties by analyzing the asymptotic behavior of mixed (conditional) moments. In particular, we obtain that for joint life and disability insurance policies some specific demographic events need to be dependent in order to ensure consistency. JEL classification: G12; G22 AMS Subject Classification (2010): 60J60; 91B30; 91G30; 91G40