Sensitivity to measurement errors of the distance to the efficient frontier

This paper builds confidence intervals for the distance in the mean-variance plan between any portfolio and the Markowitz efficient frontier. The distance can be calculated in any risk-return direction chosen by the investor. To do so, we introduce random variations of inputs and outputs and estimate the frontier. We then use subsampling approximations to derive confidence intervals around the distance of portfolios to the efficient frontier. This methodology offers a novel statistical approach to mean-variance portfolio choice, which is key for asset management. We apply this approach to show that the distance between the S&P 500 index and the efficient frontier made up of all the shares in the index is significantly different from zero in all testable directions. This result adds robustness to the still controversial Roll critique of the Capital Asset Pricing Model (CAPM). In the general setup of production theory, our paper addresses the sensitivity of the estimated efficiency scores to random variations in the original inputs-outputs.