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.
