Fractal statistical measure and portfolio model optimization under power-law distribution

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.