Asymptotic analysis and consistent estimation of high-dimensional Markowitz portfolios

We study the consistency of large-dimensional minimum variance portfolios that are estimated on the basis of weighted sampling and shrinkage. In an asymptotic setting where the number of assets remains comparable in magnitude to the sample size, we characterize the convergence of the out-of-sample or realized risk of the estimated portfolio in terms of the underlying investment scenario. The previous characterization represents a means of quantifying the effects of estimation risk in the portfolio performance. As it is well-known for naive portfolio implementations based on the sample covariance matrix, these effects can lead in practice, if not corrected, to inaccurate and overly optimistic investment decisions. Our results are based on recent contributions in the field of random matrix theory. Along with the asymptotic analysis, we also provide estimators of the optimal sampling weights and shrinkage coefficients that are consistent in the high dimensional observation regime.