Minimizing the Lifetime Shortfall or Shortfall at Death

Abstract

We find the optimal investment strategy for an individual who seeks to minimize one of four objectives: (1) the probability that his wealth reaches a specified ruin level {\it before} death, (2) the probability that his wealth reaches that level {\it at} death, (3) the expectation of how low his wealth drops below a specified level {\it before} death, and (4) the expectation of how low his wealth drops below a specified level {\it at} death. Young (2004) showed that under criterion (1), the optimal investment strategy is a heavily leveraged position in the risky asset for low wealth. In this paper, we introduce the other three criteria in order to reduce the leveraging observed by Young (2004). We discovered that surprisingly the optimal investment strategy for criterion (3) is {\it identical} to the one for (1) and that the strategies for (2) and (4) are {\it more} leveraged than the one for (1) at low wealth. Because these criteria do not reduce leveraging, we completely remove it by considering problems (1) and (3) under the restriction that the individual cannot borrow to invest in the risky asset.