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Byline: Erhan Bayraktar, Virginia R. Young
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
In this 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.
Department of Mathematics, University of Michigan, Ann Arbor, MI
48109, United States
Received 11 January 2007; Revised 6 January 2009; Accepted 7