Survival and Growth with a Liability: Optimal Portfolio Strategies in Continuous Time
Survival and Growth with a Liability: Optimal Portfolio Strategies in Continuous Time
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DOI：
10.2139/ssrn.57909
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发表时间：
1996-07
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影响因子：
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通讯作者：
S. Browne
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作者：
S. Browne
我们研究了投资者在单位时间内被迫以固定利率连续提取资金的最优行为（例如，支付债务、消费或支付股息）。投资者可以投资于任何或所有给定数量的风险股票，这些股票的价格遵循几何布朗运动，也可以投资于回报率恒定的无风险资产。无论财富水平如何，撤资都是持续执行的，这一事实确保了有一个地区存在破产的积极可能性。在互补区域，破坏是可以避免的。将前一个区域称为危险区，将后一个区域称为安全区。我们首先考虑在破产前到达安全区域的概率最大化问题，我们称之为生存问题。虽然我们在其他结果中表明，对于这个问题不存在最优策略，但我们能够为任何e b>构建显式的e-最优策略。在最终生存得到保证的安全区域，我们把注意力转向增长。在其他结果中，我们找到了投资者的最优增长政策，即尽快达到另一个（更高价值）目标的政策。还讨论了生存问题和生长问题的其他变体。我们对后者的研究结果与恒比例投资组合保险理论密切相关。
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.