Abstract:

The core of portfolio optimization problem is how to balance return and risk in the process of investment. The mean-variance model is a classical model to solve this problem. However， with the deepening of research， the variance as a risk measure cannot adapt to the real market situation， and the mean lower partial moment model came into being. At the same time， the mean-risk model often confronted the problem of parameter estimation error. Parameter uncertainty and parameter estimation error will have a negative impact on investment decision-making. Therefore， solving this problem has become a hot research direction.It is refered to Ban et al. ［Management Science， 2016， 63， 1136-1154］ that proposed performance-based regularization （PBR）. By adding PBR constraints to the mean variance model and the mean CVaR model， they reduced the parameter estimation error and improved the out of sample performance of the model. The core idea of PBR method is to improve the performance of investment strategy outside the sample by punishing the variance term of risk measurement and expected return estimate.In this paper， their PBR constraints are extended to a more general model， which is mean-lower partial moment （LPM） model. Specifically， the convexity of the mean-LPM model with PBR constraints is proved， so as to ensure that the solution of the optimization problem is globally optimal. Furthermore， the asymptotic consistency of the solution of the optimization problem is proved. In addition， the standard k-fold cross test method is modified， and the optimal parameter values of PBR constraint conditions in real time are updated by using machine learning method and linear backtracking method. Finally， the model in the downward market is put for empirical test.As a result， it is found out that compared with the traditional mean-LPM model， the mean-LPM model with two PBR constraints has higher Sharpe ratio， Sortino ratio， Omega ratio， average rate of return and cumulative rate of return， so as to achieve a better balance between return and risk. In addition， the mean-LPM model combined with PBR constraint performs better than the mean variance model combined with PBR constraint in Ban et al. （2016）.In this paper， the PBR method is extended to the general mean-LPM portfolio model， and the effectiveness of the PBR method for solving the parameter estimation error is confirmed. Regularization method has a wide range of applications， and the research of PBR method still has broad prospects. In the next step， the PBR method can be applied to the index tracking problem， or further the nature and other applicable fields of the PBR method can be explored.

Key words: mean-lower partial moment, k-fold cross test, linear back tracking method, performance-based regularization