Probabilistic approach for optimal portfolio selection using a hybrid Monte Carlo simulation and Markowitz model

Abstract

In this paper, a probabilistic form of the portfolio selection problem is established in which the uncertainty of risky assets is considered through a probabilistic optimization problem. To this end, by taking seven portfolios of Shanghai stock as a case study, the mean and standard deviation of daily return values are calculated based on five years of real data. The optimal values corresponding to each random case were then stored as a comprehensive database of system responses. Then, by sorting the resulting optimal values from best to worst, the exceedance probabilities of return and risk rankings were calculated for each portfolio and presented in the form of probabilistic pie charts. The results showed that the portfolio with the highest deterministic rate of return has the highest probability of getting the best return ranking as well. However, since the probability of risk in all cases was calculated, the probability of each portfolio to place in the lower rankings (i.e., ranking 2–7) could also be discussed. Additionally, to check the convergence of the model, the probability values calculated by the Monte Carlo method against the sample size were plotted to ensure the accuracy of the final answer. Eventually, by generating Gaussian random noise and importing it into the model input, probability changes were calculated to assess the robustness of the proposed algorithm.

Keywords

• portfolio optimization
• Markowitz mean–variance theory
• Monte Carlo sampling method
• Exceedance probability
• robustness