The Risk of Expected Utility under Parameter Uncertainty

We derive analytical expressions for the risk of an investor's expected utility under parameter uncertainty. In particular, our analysis focuses on characterizing the out-of-sample utility variance of three portfolios: the classic mean-variance portfolio, the minimum-variance portfolio, and a shrinkage portfolio that combines both. We then use our analytical expressions to study a robustness measure that balances out-of-sample utility mean and volatility. We show that neither the sample mean-variance portfolio nor the sample minimum-variance portfolio exhibits maximal robustness individually, and one needs to combine both to optimize portfolio robustness. Accordingly, we introduce a robust shrinkage portfolio that delivers an optimal tradeoff between out-ofsample utility mean and volatility and is more resilient to estimation errors. Our results highlight the importance of considering out-of-sample performance risk in designing and evaluating investment strategies and stochastic discount factor models.