Abstract: Accurate measurement of the risks and benefits of securities is important not only for investment management, but also for financial theoretical research and even for transformation of theoretical results into practical application. Based on the fractal theory, this paper constructs two fractal statistical measures, such as fractal expectation and fractal variance, in order to overcome the shortcomings of non-fractal statistical measures in terms of risk income. On this basis, the portfolio model is constructed by using the fractal statistical measure. Also, the analytic solution to the fractal model is given; then, the effectiveness of fractal statistic measurement in the portfolio application is verified by empirical analysis. The innovation of this paper is that two fractal statistical measures are constructed, which are fractal expectation and fractal variance, based on the reality of fractal characteristics of securities price. Moreover, the investment portfolio model is constructed under the fractal statistical measure, and this paper takes the fractal characteristics present in the securities price into the portfolio research.

Key words: fractal expectation, fractal variance, fractal statistics measure, portfolio

摘要： 准确测量证券的风险和收益无论是对投资管理,还是对金融理论研究,甚至对理论成果向实践应用转化都至关重要。本文在证券价格具有分形特征的现实背景下,基于分形理论构建了分形期望和分形方差两个分形统计测度,以克服非分形统计测度在风险收益方面测不准或不可测的缺陷。在此基础上,应用分形统计测度构建了投资组合模型,给出了分形组合模型的解析解;随后,利用实证分析验证了分形统计测度在投资组合应用中的有效性。本文创新之处在于针对证券价格具有分形特征的现实背景构建了分形期望和分形方差两个分形统计测度;并基于分形统计测度构建了投资组合模型,将证券价格普遍存在的分形特征纳入投资组合的研究框架。

关键词: 分形期望, 分形方差, 分形统计测度, 投资组合