The Fractal Portfolio Selection Model With Transaction Costs

The process of the portfolio management mainly includes the valuating of each property risk and return, the optimization of the portfolio selection, the scaling of the portfolio efficiency. In these three processes, the valuating and the optimization have big theoretic meanings. They are the core process in the portfolio management research. But the method and meaning of the valuation differs with the market efficiency dissimilarity. Therefore we should examine the market efficiency before the valuation.The accurate measurement of the capital market become possible under EMH, various theoretic models emerge with the tide of the times. From the Modern Portfolio Theory of Markowitz to the capital equilibrium pricing theory of Sharpe, Lintner and Mossin, and the following Option Pricing Theory of Black- Scholes, and the Arbitrage Pricing Theory of Ross, all is under EMH or closely-related with it.However, Before EMH coming into being, someone have already discovered that the return of financial capital market did not agree with the normal distribution assumption, and the return is not independent neither, meanwhile there are some obvious limitations in EMH. Owing to that, people began to lead the nonlinear science to the financial research. Mandelbrot emphasized the 'fat tail and high peak' characteristic of the return sequence on the foundation of Pareto's study. He suggests to fit the fat tail characteristic in fractal distribution, these fat-tail-distributions usually reveal nonlinear stochastic process. Hence many scholars start to study the portfolio theory in fractal distribution.The parameter A is used to describe the discrete degree in fractal distribution. Fama and Samuleson have ever used this parameter to describe the risk and construct portfolio. They supposed all the stocks have the same peak and fat tail characteristic parameter, in this way the portfolio risk can be denoted by the risk and weight of each stock. However this is not realistic, because the empirical analysis denotes that different stocks express different values. It's regretted that the multi-extremum theory of portfolio selection with different fat tails is still a problem that needs to be resolved.At the same time, measuring securities risk with this parameter still exist the similar problem as that of the deviation method , that is, the equal management to the lose and earning is not agree with the investor's mental feeling to the securities risk.In this paper, the downside risk method is used to measure the securities risk and the market friction factor is considered , while the portfolio has different characteristic parameters in fractal distribution. Since overselling risk property and debiting un-venture capital is prohibited in many securities markets( such as Chinese market), these factors are still considered in this portfolio model.