Actively Managed Mean-Variance Portfolios

Abstract

Analytical solutions are presented to the mean-variance portfolio optimization problem of trading off active return against tracking error relative to a prespecified benchmark subject to a budget constraint and a beta constraint. Imposing a constraint on the beta of the portfolio makes the benchmark relevant to the portfolio problem. I provide an intuitive interpretation of the best portfolio in the general mean-variance case, as well as a detailed expression for the optimal weights when returns follow a single-factor model. An economic interpretation of the value-added function for the best portfolio is presented as the best possible increase in the certainty equivalent utility gain of the investor relative to the certainty equivalent return of the benchmark. An empirical investigation into the out-of-sample performance of the best portfolio is presented for two equity universes using industry portfolios and individual stocks.

TOPICS: Wealth management, portfolio theory, portfolio construction

Key Findings

• • The value-added of active management is the incremental certainty equivalent utility over that delivered by the benchmark.

• • In the mean-variance framework the incremental certainty equivalent utility obtains as a trade-off of active return and active risk weighted by the investor’s risk aversion.

• • The optimal portfolio is a (very particular) linear combination of well-known portfolios like the minimum variance portfolio, the maximum correlation portfolio and the unconstrained optimal portfolio.