Minimizing the lifetime shortfall or shortfall at death

We find the optimal investment strategy for an individual who seeks to minimize one of four objectives: (1) the probability that his/her wealth reaches a specified ruin level before death, (2) the probability that his/her wealth reaches that level at death, (3) the expectation of how low his/her wealth drops below a specified level before death, and (4) the expectation of how low his/her wealth drops below a specified level at death. Young [Young, V.R., 2004. Optimal investment strategy to minimize the probability of lifetime ruin. N. Am. Actua. J. 8 (4), 105–126] showed that under criterion (1), the optimal investment strategy is a heavily leveraged position in the risky asset for low wealth. s paper, we introduce the other three criteria in order to reduce the leveraging observed by Young, the above mentioned reference. We discovered that surprisingly the optimal investment strategy for criterion (3) is identical to the one for (1) and that the strategies for (2) and (4) are 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.