Stable Distribution And Portfolio Research

Right investment decision requires reliable predictions of return and risk, andreliable predictions can only be obtained if the underlying statistical model rests onrealistic assumptions. In the traditional portfolio researching, for example theMarkowitz portfolio model, the normal distribution was mostly applied to fit theprobability distribution of the security's return rate. But in many practical researching,the financial data time serials represent the phenomenon of the heavy tail and excesskurtosis, which the normal distribution can't describe. Stable laws are able to capturethe two main empirical characteristics that heavy-tailed and volatility clustering.Therefore, by means of John Nolan's stable distribution analysis software—the stableprogram, some problems are discussed, which include portfolio with stabledistribution, the optimal portfolio selection of probability criterion, the VaR of stockreturns of China with stable distribution, etc.The main contents and results are as follows:Firstly, the basic theories of univariate stable distribution and multivariate stabledistributions are introduced.Secondly, the mean-absolute deviation model based on the stable distribution isbuilt and we modify the model and research its solution under the stock market ofChina.Thirdly, the optimal portfolio selection of probability criterion with normaldistribution is considered, the analytic representation of the optimal portfolio withshort selling allowed is obtained and the method for solving the optimal portfolio withno short selling allowed is given;The optimal portfolio selection of probabilitycriterion based on stable distribution for risky assets is considered and thesimultaneous perturbation algorithm for stochastic optimization is implemented tosolve the model. Regarded the Shenzhen Stock Sub-index (SZSI) and Shanghai StockComposite Index (SHCI) as risky assets, the portfolio model of probability criterion isbuilt and the steps of numerical method are given.Lastly, compare with the VaR with normal distribution, the VaR of stock returnsof China based on the stable distribution is discussed. The results show that thecalculation of the VaR is feasible and effective based on the stable distribution.