Portfolio Selection with Parameter and Model Uncertainty: A Multi-Prior Approach

In this paper, we show how an investor can incorporate uncertainty about expected returns when choosing a mean-variance optimal portfolio. In contrast to the Bayesian approach to estimation error, where there is only a single prior and the investor is neutral to uncertainty, we consider the case where the investor has multiple priors and is averse to uncertainty. We characterize the multiple priors with a confidence interval around the estimated value of expected returns and we model aversion to uncertainty via a minimization over the set of priors. The multi-prior model has several attractive features: One, just like the Bayesian model, it is firmly grounded in decision theory. Two, it is flexible enough to allow for different degrees of uncertainty about expected returns for different subsets of assets, and also about the underlying asset-pricing model generating returns. Three, for several formulations of the multi-prior model we obtain closed-form expressions for the optimal portfolio, and in one special case we prove that the portfolio from the multi-prior model is equivalent to a ‘shrinkage’ portfolio based on the mean-variance and minimum-variance portfolios, which allows for a transparent comparison with Bayesian portfolios. Finally, we illustrate how to implement the multi-prior model for a fund manager allocating wealth across eight international equity indices; our empirical analysis suggests that allowing for parameter and model uncertainty reduces the fluctuation of portfolio weights over time and improves the out-of sample performance relative to the mean-variance and Bayesian models.