Studies on Design of Index Fund and its Properties

本文研究了传统的马科维茨型资产配置，并开发了一种有效的算法来设计给定证券数量的指数基金，使基准投资组合与指数基金之间的局部跟踪误差最小化。当基准投资组合处于前沿时，对跟踪误差最小的指数基金的特性进行了一些有趣的研究。通过简单的数值算例说明了这些发现。此外，基准投资组合（例如市场投资组合和日经225指数）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。重点讨论这些有效前沿投资组合在前沿曲线上的一些分析性质，并将跟踪误差定义为投资组合收益的均方误差。在传统的相关跟踪误差下，这些关系和属性与现有的关系和属性相似。这为实证研究CAPM和指数基金建设提供了有益的信息。许多问题有待未来的研究。特别是，明确跟踪误差的实际含义和投资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.