Many investors employ portfolio selection models to calculate the expected return on their investments. This expected return follows an asymmetric power-law distribution, and investors employ statistical measures of expectation and variance to find the best place for their money.

However, when it comes to investing with the principle of “maximum return-minimum risk,” traditional linear models are far from foolproof given that the expected risks and returns may be inaccurate or infinite.

Feng et al. constructed an alternative portfolio selection model based on fractal statistical measures. Their model effectively accounts for nonlinear aspects of financial systems, improving portfolio performance and providing investor decision-making support. The study references all six industry indexes from the Shanghai Stock Exchange as samples over an 18-year period beginning in 2004.

“Empirical evidence suggests that the fractal portfolio selection model can effectively improve the portfolio performance for investors,” said author Xu Wu.

In building their model, the researchers created fractal statistical measures informed by two principle ideas: that the infinite problem is transformed into a finite problem, and that the measurement result is replaced by the measurement process itself. They then incorporated the fractal statistical measures into the maximum return-minimum risk criterion to build the fractal portfolio selection model.

“We hope this method can be used as a reference and as inspiration for other fields to deal with asymmetric power-law distribution,” said Wu.

Source: “Research on portfolio optimization under asymmetric power-law distribution of return tail,” by Qian-Ying Feng, Xu Wu, Lin-Lin Zhang, and Jia Li, Chaos (2023). The article can be accessed at