On the optimal combination of naive and mean-variance portfolio strategies

A disheartening fact in portfolio choice is that the naive equally weighted portfoliooften outperforms the estimated optimal mean-variance portfolio out of sample. In an influential paper, Tu and Zhou (2011) reaffirm the value of portfolio theory by combining the two portfolios to optimize out-of-sample performance. They achieve this under a seemingly natural convexity constraint: the two combination coefficients must sum to one. We show that this constraint is unnecessary in theory and has several undesirable consequences relative to the unconstrained portfolio combination we derive. In particular, it leads to an overinvestment in the sample mean-variance portfolio, and a worse performance than the risk-free asset for sufficiently risk-averse investors. However, although wrong in theory, we demonstrate that the convexity constraint acts as a bound constraint on combination coefficients and thus can help improve performance when they are estimated. Our empirical analysis shows that the Tu and Zhou rule performs well for investors with small risk aversion, but quickly deteriorates as risk aversion increases. In contrast, our portfolio rules perform consistently well. Finally, we show theoretically and empirically that there are larger out-of-sample diversification gains from combining the sample mean-variance portfolio with the equally weighted portfolio instead of the minimum-variance portfolio.