Portfolio Optimization Based On Regularization

Portfolio optimization has always been a core issue in the financial field. Markowitz used simple mathematical ideas to establish mean-variance model under the assumption of complete series in1952, creating the modern portfolio theory. Investors no longer just rely on experience to carry out investment decisions after that. However, the value of mean-variance model has too many tiny positions, that investor can’t invest even one hand; and the value of the largest weight is too big, that a stock with the largest weight will make a greater affect on portfolio. Therefore, this model is not suitable for actual operation because the tiny weights in the portfolio could not be invested and the biggest weight will have a big effect on it.In order to make a suitable portfolio, this paper added l1+l2-norm in the Markowitz mean-variance model. l1-norm regularization added a penalty proportional to the sum of the absolute values of the weights. l1-norm regularization putted the most penalty on tiny weight, then the weights, which equal to zero, increased to make the answer sparse. l2-norm regularization added a penalty term proportional to the sum of the square of the weights, the biggest value of the weight will become small, than the answer will be smooth.This paper used statistics of daily yield from200stocks listed in Shanghai Stock Exchange in2013to do empirical test. With the comparisons of l1-norm, l2-norm, l1+l2-norm regularization model and Markowitz mean-variance model, the results show that:(1) compared with mean-variance model,the value of l1-norm model can make weight sparse. Due to the effect of some maximum value, the risk of portfolio will be changed a lot.(2) model l2-norm can make the value smooth. Like the mean-variance model, it selects all the stock to the portfolio.(3) l1+l2-norm model combines advantages of those former two models efficiently, offsets the disadvantages of those single models,solves the problem of the mean-variance model, consisting too large of the value of the bigger weight ang too tiny of the small weight. Then make the result more practical for investors to apply to the actual operation.