A Global Optimal Theory of Portfolio beyond R-$\sigma$ Model

The deviation of the efficient market hypothesis (EMH) for the practical economic system allows us gain the arbitrary or risk premium in finance markets. We propose the triplet $(R,H,\sigma)$ theory to give the local and global optimal portfolio, which eneralize from the $(R,\sigma)$ model. We present the formulation of the triplet $(R,H,\sigma)$ model and give the Pareto optimal solution as well as comparing it with the numerical investigations for the Chinese stock market. We define the local optimal weights of the triplet $(\mathbf{w}_{R},\mathbf{w}_{H},\mathbf{w}_{\sigma})$, which constructs the triangle of the quasi-optimal investing subspace such that we further define the centroid of the triangle or the incenter of the triangle as the optimal investing weights, which optimizes the mean return, the arbitrary or risk premium and the volatility risk. By investigating numerically the Chinese stock market as an example we demonstrate the validity of the formulation and obtain the global optimal strategy and quasi-optimal investing subspace. The theory provides an efficient way to design the portfolio for different style investors, conservative or aggressive investors, in finance market to maximize the mean return and arbitrary or risk premium with a small volatility risk.