Optimally investing to reach a bequest goal

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• 作者：Erhan Bayraktar ; Virginia R. Young ; ^{vryoung@umich.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：C61 ; G02 ; G11
• 刊名：Insurance: Mathematics and Economics
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：70
• 期：Complete
• 页码：1-10
• 全文大小：513 K

文摘

We determine the optimal strategy for investing in a Black–Scholes market in order to maximize the probability that wealth at death meets a bequest goal 90c526a292c1476" title="Click to view the MathML source">b$b$, a type of goal-seeking problem, as pioneered by Dubins and Savage (1965, 1976). The individual consumes at a constant rate c$c$, so the level of wealth required for risklessly meeting consumption equals c/r$c/r$, in which r$r$ is the rate of return of the riskless asset.

Our problem is related to, but different from, the goal-reaching problems of Browne (1997). First, Browne (1997, Section 3.1) maximizes the probability that wealth reaches b<c/r$b<c/r$ before it reaches a<b$a<b$. Browne’s game ends when wealth reaches 90c526a292c1476" title="Click to view the MathML source">b$b$. By contrast, for the problem we consider, the game continues until the individual dies or until wealth reaches 0; reaching 90c526a292c1476" title="Click to view the MathML source">b$b$ and then falling below it before death does not count.

Second, Browne (1997, Section 4.2) maximizes the expected discounted reward of reaching b>c/r$b>c/r$ before wealth reaches c/r$c/r$. If one interprets his discount rate as a hazard rate, then our two problems are mathematically   equivalent for the special case for which b>c/r$b>c/r$, with ruin level c/r$c/r$. However, we obtain different results because we set the ruin level at 0, thereby allowing the game to continue when wealth falls below c/r$c/r$.