Portfolio selection with parameter uncertainty

In the mean-variance portfolio model (Markowitz (1952, 1959), Sharpe (1970)) expected returns, variances, and covariances are estimated with error. But classical mean-variance portfolio optimization ignores the estimation error, and consequently, the mean-variance portfolio formed using sample moments has extreme portfolio weights that fluctuate substantially over time and the out-of-sample performance of such a portfolio is quite poor.

In this paper, we extend the mean-variance portfolio model where expected returns are obtained using maximum likelihood estimation to explicitly account for uncertainty about the estimated expected returns. In contrast to the Bayesian approach to estimation error, where there is only a single prior and the investor is neutral to uncertainty, we allow for multiple priors and aversion to uncertainty.

The multi-prior model has several attractive features:

a)    just like the Bayesian model, the multi-prior model is firmly grounded in decision theory;

b)    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;

c)   in several special cases of the multi-prior model one can obtain closed-form expressions for the optimal portfolio, which can be interpreted as a shrinkage of the mean-variance portfolio towards either the risk-free asset or the minimum variance portfolio.

We illustrate how to implement the multi-prior model using both international and domestic data. Our analysis suggests that allowing for parameter uncertainty reduces the fluctuation of portfolio weights over time and, for the data set considered, improves the out-of sample performance.