Geometric Mean Maximization: A Note on Expected, Observed, and Simulated Performance

Abstract

This note discusses why the maximization of any portfolio optimization model cannot be isolated from the investor’s risk perception. To allow for differing risk preferences among investors, the efficient frontiers of Sharpe ratio maximization (SRM) and geometric mean maximization (GMM) are the appropriate metrics for making a comparison. The authors demonstrate that for a given level of risk the two optimization techniques will choose the same portfolio asset weights. Although GMM provides investors with a different way to approach portfolio optimization (maximizing terminal wealth), it is not a competing portfolio optimization technique to mean–variance maximization but rather a complementary one.

TOPICS: Portfolio construction, quantitative methods, statistical methods, performance measurement

Key Findings

• ▪ Comparison of portfolio optimization techniques require a predetermined risk level.

• ▪ Geometric mean and Sharpe ratio maximization lead to the same portfolio weights for a given level of risk.

• ▪ Geometric mean maximization expands the opportunity set of the mean–variance space.