This paper studies a mean-risk portfolio choice problem for log-returns in a continuous-time, complete market. It is a growth-optimal portfolio choice problem under risk control. The risk of log-returns is measured by weighted Value-at-Risk (WVaR), which is a generalization of Value-at-Risk (VaR) and Expected Shortfall (ES). We characterize the optimal terminal wealth and obtain analytical expressions when risk is measured by VaR or ES. We demonstrate that using VaR increases losses while ES reduces losses during market downturns. Moreover, the efficient frontier is a concave curve that connects the minimum-risk portfolio with the growth optimal portfolio, as opposed to the vertical line when WVaR is used on terminal wealth, and thus allows for a meaningful characterization of the risk-return trade-off and aids investors in setting reasonable investment targets. We also apply our model to benchmarking and illustrate how investors with benchmarking may overperform/underperform the market depending on economic conditions.

中文翻译：

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动态增长——风险控制下的最优投资组合选择


本文研究了连续时间完整市场中对数回报的平均风险投资组合选择问题。这是一个在风险控制下的增长最优投资组合选择问题。对数回报的风险是通过加权风险值 （WVaR） 来衡量的，WVaR 是风险值 （VaR） 和预期短缺 （ES） 的概括。我们描述最佳终端财富，并在通过 VaR 或 ES 衡量风险时获得分析表达式。我们证明，在市场低迷期间，使用 VaR 会增加损失，而 ES 可以减少损失。此外，有效边界是一条凹曲线，它将最低风险投资组合与增长最优投资组合连接起来，而不是在终端财富上使用 WVaR 时的垂直线，因此可以对风险-回报权衡进行有意义的描述，并帮助投资者设定合理的投资目标。我们还将我们的模型应用于基准分析，并说明具有基准分析的投资者如何根据经济状况跑赢/跑输市场。