Time consistent portfolio management

This paper considers the portfolio management problem of optimal investment, consumption and life insurance. We are concerned with time inconsistency of optimal strategies. Natural assumptions, like different discount rates for consumption and life insurance, or a time varying aggregation rate lead to time inconsistency. As a consequence, the optimal strategies are not implementable. We focus on hyperbolic discounting, which has received much attention lately, especially in the area of behavioural finance. Following [10], we consider the resulting problem as a leader-follower game between successive selves, each of whom can commit for an infinitesimally small amount of time. We then define policies as subgame perfect equilibrium strategies. Policies are characterized by an integral equation which is shown to have a solution. Although we work on CRRA preference paradigm, our results can be extended for more general preferences as long as the equations admit solutions. Numerical simulations reveal that for the Merton problem with hyperbolic discounting, the consumption increases up to a certain time, after which it decreases; this pattern does not occur in the case of exponential discounting, and is therefore known in the litterature as the "consumption puzzle". Other numerical experiments explore the effect of time varying aggregation rate on the insurance premium.