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Abstract
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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. 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. |
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