Mean-Variance Efficiency of Optimal Power and Logarithmic Utility Portfolios

TL;DR

This paper derives closed-form solutions for optimal portfolios under power and logarithmic utilities assuming log-normal returns, showing their mean-variance efficiency and analyzing their behavior in stock markets.

Contribution

It provides new analytical results linking utility-based optimal portfolios to mean-variance efficiency under log-normal return assumptions.

Findings

Optimal portfolios are mean-variance feasible and efficient.

Closed-form expressions for portfolio weights are derived.

Behavior of portfolios varies with risk aversion levels.

Abstract

We derive new results related to the portfolio choice problem for power and logarithmic utilities. Assuming that the portfolio returns follow an approximate log-normal distribution, the closed-form expressions of the optimal portfolio weights are obtained for both utility functions. Moreover, we prove that both optimal portfolios belong to the set of mean-variance feasible portfolios and establish necessary and sufficient conditions such that they are mean-variance efficient. Furthermore, an application to the stock market is presented and the behavior of the optimal portfolio is discussed for different values of the relative risk aversion coefficient. It turns out that the assumption of log-normality does not seem to be a strong restriction.