Construction of the efficient frontier for portfolios combining risky and risk-free assets: an MPT-Based optimization model and visualization analysis

The volatility of financial markets has driven the diversification of investment instruments, encouraging investors to keep improving their portfolio-picking techniques. While numerous studies based on Modern Portfolio Theory (MPT) have developed accepted methods for determining optimal portfolios, not enough research has been done on the visual graphical analysis of risk preferences to accommodate diverse investors. Furthermore, by including risk-free assets in the analysis, this study presents an innovative methodology. This study's main goal is to find and analyze the risk portfolio frontier while examining the complementing of risk-free investments. Under idealized assumptions, portfolio returns and risks are formulated and solved as equality-constrained optimization problems, yielding frontier portfolios. The risk-efficient frontier and preference levels are depicted graphically, with explicit discussion of unfettered short-selling possibilities. The inclusion of risk-free assets further broadens the models practical applicability. The framework's viability is empirically validated using historical data from twelve stocks. These results illustrate investors can use the efficient frontier as a foundation to match portfolio selections to their own risk tolerances, providing practical guidance for adaptable wealth management.