Standard Error Adaptive Moment Estimation for Mean-Value-at-Risk Portfolio Optimization Problems by Sampling

In this paper, an improvement of the adaptive moment estimation (Adam) method equipped with standard error (SE), namely the AdamSE algorithm, is proposed. Our aims are to improve the convergence rate of the Adam algorithm and to explore the utility of the AdamSE algorithm for solving mean-value-at-risk (mean-VaR) portfolio optimization problems. For this, 10 stocks from the top 30 equity holdings list released by the Employees Provident Fund (EPF) have a weak correlation among them. The weekly stock prices of these stocks are selected for the period from 2015 to 2019, and then the mean, covariance and required rate of return are calculated to build a mean-VaR portfolio optimization model. In this way, the Adam and AdamSE algorithms are used to solve the model, and their results are compared. During the calculation, the stochastic gradients of the model are simulated through sampling, and nine samples are taken into consideration. With this sampling, the standard error of each sample is computed and the optimal weight for each sample is determined using the AdamSE algorithm. After convergence is achieved, the results show that different sample sizes could provide a satisfactory outcome for the portfolio concerned and from these nine samples, the lowest and highest iteration numbers were obtained to guarantee a robust optimal solution to the model constructed. Hence, we concluded that the AdamSE algorithm through sampling reveals its computational capability for handling the mean-VaR portfolio optimization problem.