Survival and Growth with a Liability: Optimal Portfolio Strategies in Continuous Time

We study the optimal behavior of an investor who is forced to withdraw funds continuously at a fixed rate per unit time e.g., to pay for a liability, to consume, or to pay dividends. The investor is allowed to invest in any or all of a given number of risky stocks, whose prices follow geometric Brownian motion, as well as in a riskless asset which has a constant rate of return. The fact that the withdrawal is continuously enforced, regardless of the wealth level, ensures that there is a region where there is a positive probability of ruin. In the complementary region ruin can be avoided with certainty. Call the former region the danger-zone and the latter region the safe-region. We first consider the problem of maximizing the probability that the safe-region is reached before bankruptcy, which we call the survival problem. While we show, among other results, that an optimal policy does not exist for this problem, we are able to construct explicit e-optimal policies, for any e > 0. In the safe-region, where ultimate survival is assured, we turn our attention to growth. Among other results, we find the optimal growth policy for the investor, i.e., the policy which reaches another higher valued goal as quickly as possible. Other variants of both the survival problem as well as the growth problem are also discussed. Our results for the latter are intimately related to the theory of Constant Proportions Portfolio Insurance.