Optimal Investment and Consumption Decisions when Time-Horizon is Uncertain

Abstract

Many investors do not know with certainty when their portfolio will be liquidated. Should their portfolio selection be influenced by the uncertainty of exit time? In order to answer that question, we consider a suitable extension of the familiar optimal investment problem of Merton (1971), where we allow the conditional distribution function of an agent's time-horizon to be stochastic and correlated to returns on risky securities. In contrast to existing literature, which has focused on an independent time-horizon, we show that the portfolio decision is affected. For CRRA preferences, we also show that a solution formally similar to the one obtained in the case of a constant time-horizon can be recovered at the cost of a suitable adjustment to the drift process of the risky assets.