Optimal Portfolio Selection with and without Risk-Free Asset

Abstract

In this paper, we consider optimal portfolio problems with and without risk-free asset, taking into account estimation risk. For the case with a risk-free asset, we derive the exact distribution of out-of-sample returns of various optimal portfolio rules, including the two-fund and three-fund rules suggested by Kan and Zhou (2007), and compare their out-of-sample performance with the equally weighted portfolio (i.e., 1/N rule). We find that the dominance of the 1/N rule over various optimal portfolio rules as documented by DeMiguel, Garlappi, and Uppal (2009) was due in part to the exclusion of risk-free asset in their construction of optimal portfolios, even though those optimal portfolio rules were designed to include the risk-free asset. In order to have a direct comparison with the 1/N rule of risky assets only, we also consider an optimal portfolio problem without risk-free asset and develop a new portfolio rule that is designed to mitigate estimation risk. We show that our new portfolio rule performs well relative to the 1/N rule in both calibrations and real datasets. Overall, our results reaffirm the value of portfolio optimization for the case with and without risk-free asset.