Studies on Design of Index Fund and its Properties

研究了一种传统的Markowitz类型的资产配置，并提出了一种有效的算法来设计一个给定证券数量的指数基金，使基准投资组合和指数基金之间的跟踪误差局部最小化。当基准投资组合处于前沿时，我们得到了一些关于跟踪误差最小化的指数基金性质的有趣的发现。这些发现通过简单的数字例子得到了说明。此外，基准投资组合(例如，市场投资组合和日经225指数基金)与其指数基金之间的一些横截面关系在投资组合理论和资产配置中起着核心作用。重点研究了这些有效前沿投资组合在前沿曲线上的一些分析性质，跟踪误差定义为投资组合收益的均方误差。这些关系和性质与传统相关跟踪误差下已有的关系和性质相似。为CAPM和指数基金建设的实证研究提供了有用的信息，并为未来的研究留下了许多问题。特别是，澄清跟踪误差的实际影响，并投资于各种成本与指数基金所含证券数量之间的关系是重要的领域。

This study is concerned with a traditional asset allocation of the Markowitz type and develops an efficient algorithm to design an index fund with given number of securities that locally minimizes the tracking error between the benchmark portfolio and the index fund. When the benchmark portfolio is on the frontier, some interesting findings on the properties of the index fund with minimizing tracking error are derived. These findings are illustrated by simple numerical examples.Moreover, some cross-sectional relationships between the benchmark portfolio (for examples, market portfolio and Nikkei 225) and its index fund which play central roles in portfolio theory and asset allocation. Emphases are on some analytical properties of these efficient frontier portfolios on the frontier curve with the tracking error being defined as the mean square error of portfolio return. Those relations and properties are similar to the existing ones under the traditional correlation tracking error. They provide useful information for empirical studies on CAPM and index fund construction.Many problems are left for future research. In particular, clarifying the practical implications of the tracking error and investing the relationship between various costs and the number of securities included in the index fund are important fields.