Portfolio Selection with Parameter and Model Uncertainty ∗

In this paper, we extend the mean-variance portfolio model to explicitly account for uncertainty about the estimated expected returns and/or the underlying return-generating model. We do this by first imposing an additional constraint on the mean-variance portfolio optimization program that restricts each parameter to lie within a specified confidence interval of its estimated value, and then by minimizing over the choice of parameters and/or models subject to this constraint, in addition to the maximization over portfolio weights. Our model has several attractive features. One, it is firmly grounded in decision theory. Two, it is flexible enough to allow for uncertainty about expected returns estimated jointly for all assets or different levels of uncertainty about expected returns for different subsets of the assets. The estimation may be undertaken using classical estimators based on sample moments or a factor model or using Bayesian methods. Three, we show that the max-min problem can be reduced to just a maximization that has the same complexity as the classical mean-variance problem. We demonstrate our approach using MSCI data on equity index returns for eight countries. We find that the out-of-sample returns generated by our portfolio model have a higher mean and lower volatility compared to those from a standard mean-variance model, and to those from Bayesian models.