An effective portfolio selection model is constructed on the premise of measuring accurately the risk and return on assets. According to the reality that the tail of returns on assets obey power-law distribution, this paper firstly builds two fractal statistical measures, fractal expectation and fractal variance, to measure the asset returns and risks, inspired by the method of measuring curve length in the fractal theory. Then, by incorporating the fractal statistical measure into the return-risk criterion, a portfolio selection model based on fractal statistical measure is established, namely the fractal portfolio selection model, and the closed-form solution of the model is given. Finally, through empirical analysis we find that the fractal portfolio selection model is effective and can improve investment performance.

有效的投资组合选择模型是在准确衡量资产风险和收益的前提下构建的。针对资产收益率的尾部服从幂律分布的现实，本文首先借鉴分形期望和分形方差两个分形统计测度来度量资产收益和风险，借鉴了资产收益率曲线长度的度量方法。分形理论。然后，通过将分形统计测度纳入收益风险准则，建立了基于分形统计测度的投资组合选择模型，即分形投资组合选择模型，并给出了该模型的闭式解。最后，通过实证分析，我们发现分形投资组合选择模型是有效的，可以提高投资绩效。