We treat three topics in projections of multi-state life insurance contracts, and two topics in utility theory using the equilibrium approach. We derive a system of forward differential equations for the retrospective reserve of a with-profit insurance contract, where the dynamics of the reserve are affine. To reduce the sheer size of the system of differential equations required for a projection of an entire insurance business, we reduce the state space of insurance contracts through a transformation of the transition intensities and payment streams, resulting in a smaller, approximating system of differential equations. We derive a system of infinite partial differential equations for the moment-generating function of retrospective reserves with polynomial dynamics.
We truncate the infinite partial differential equations to produce numerically feasible procedures, applicable for the projection of retrospective reserves. Using an equilibrium approach, we study how to dynamically approximate utility functions by polynomials so that there is a small difference in the corresponding optimal controls. Finally we derive a fixed-point equation for the equilibrium control of an investor with a prospect-theoretic utility function.