The Policyholder’s static and dynamic decision making of life insurance and pension payments

We deal with the introduction of life insurance and pension decisions in the personal financial problem of optimal lifetime consumption of lifetime income. We introduce in Sect. 2 the classical notion of reserves and present well-known differential equations characterizing these. We start with the survival model and discuss also the case where pension saving takes place in a bank. We then analyze the disability model and the multistate model that are generalizations of the survival model. This structure is repeated in all sections of the paper. In Sect. 3 we introduce the notion of utility reserves that makes it possible to compare the different contracts offered by the insurance company, and present differential equations characterizing these. The utility reserve is the basis for static optimization of payment streams in Sect. 4 and for dynamic optimization of payment streams in Sect. 5. In particular in the case of dynamic optimization, the differential equation characterizing the utility reserve plays a crucial role since the so-called Hamilton–Jacobi–Bellman equation characterizing the optimal solution is based on it. Sections 2–5 are ended by a continued numerical example illustrating our findings for the survival model. We conclude by further remarks on generalizations and applications.