Constrained Portfolio Optimization: Markowitz Model and Index Model

School of Mathematics, Shandong University, Jinan, China

^* Corresponding author: 202000130006@mail.sdu.edu.cn

Abstract

Portfolio optimization is a crucial aspect of contemporary finance, facilitating the efficient balancing of risk and return for investors. In light of the intricate nature of actual investments, constrained portfolio optimization has assumed greater significance as a means of reflecting the practical realities of investment scenarios. Based on historical data, this study employs the Markowitz and Index models to optimize portfolios utilizing 24 years (2000-2024) of daily total return from 10 stocks across five sectors as well as the S&P 500 index. Several financial metrics are calculated, including annualized return, standard deviation, alpha, and beta. Five distinct constraints are applied to determine the minimum variance frontier, efficient frontier, and maximum Sharpe ratio for both models. The results indicate that an effective investment portfolio can reduce investment risk. Furthermore, constrained optimization yields portfolios with balanced risk and return characteristics, offering valuable insights for investors seeking practical strategies to manage risk within real-world limitations.