Managing the intertemporal risk of optimally constructed multifactor portfolios adds to performance. The increases in Sharpe ratios are in addition to the utility that investors gain from controlling how much active risk they are exposed to over time. We derive a simple closed-form formula for security weights in optimal multifactor portfolios with an active-risk target. We test the risk control of five well-known factors—value, momentum, small size, low beta, and profitability—and the optimal multifactor portfolio. Our empirical research was carried out on the large-capitalization US equity market for 1966 through 2019. We conclude that for the equity market, more active factors are better than fewer if each subportfolio is “pure” as to factor, anchored to the benchmark, and combined on the basis of forecastable risks. Our portfolio construction methodology allows for transparent performance attribution and replication of the process in other markets and time periods. Disclosure: The authors report no conflicts of interest. Editor’s Note This article was externally reviewed using our double-blind peer-review process. When the article was accepted for publication, the authors thanked the two anonymous reviewers in their acknowledgments. Submitted 18 November 2019 Accepted 9 April 2020 by Stephen J. Brown.

中文翻译：

---

风险管理与股票市场因素的优化组合

管理优化构建的多因子投资组合的跨期风险会增加绩效。夏普比率的增加是对投资者通过控制他们随着时间的推移所面临的主动风险程度而获得的效用的补充。我们为具有主动风险目标的最优多因子投资组合中的证券权重推导出一个简单的封闭式公式。我们测试了五个众所周知的因素——价值、动量、小规模、低贝塔和盈利能力——以及最优的多因素投资组合的风险控制。我们的实证研究是对 1966 年至 2019 年的大市值美国股票市场进行的。我们得出的结论是，对于股票市场，如果每个子投资组合的因子“纯”（锚定于基准），活跃因子越多越好，并根据可预测的风险进行组合。我们的投资组合构建方法允许在其他市场和时间段进行透明的绩效归因和过程复制。披露：作者报告没有利益冲突。编者注 本文使用我们的双盲同行评审流程进行了外部评审。当文章被接受发表时，作者在致谢中感谢了两位匿名审稿人。2019 年 11 月 18 日提交，Stephen J. Brown 于 2020 年 4 月 9 日接受。作者在致谢中感谢两位匿名审稿人。2019 年 11 月 18 日提交，Stephen J. Brown 于 2020 年 4 月 9 日接受。作者在致谢中感谢两位匿名审稿人。2019 年 11 月 18 日提交，Stephen J. Brown 于 2020 年 4 月 9 日接受。