DISCUSSION PAPERS IN STATISTICS AND ECONOMETRICS SEMINAR OF ECONOMIC AND SOCIAL STATISTICS UNIVERSITY OF COLOGNE No. 1/07 Linear Statistical Inference for Global and Local Minimum Variance Portfolios

Traditional portfolio optimization has often been criticized for not taking estimation risk into account. Estimation risk is mainly driven by the parameter uncertainty regarding the expected asset returns rather than their variances and covariances. The global minimum variance portfolio has been advocated by many authors as an appropriate alternative to the classical mean-variance optimal portfolio. This is because there are no expected asset returns which have to be estimated and thus the impact of estimation errors can be substantially reduced. However, in many practical situations an investor is not willing to choose the global minimum variance portfolio but he wants to minimize the variance of the portfolio return under specific constraints for the portfolio weights. Such a portfolio is called local minimum variance portfolio. Standard small-sample hypothesis tests for global and local minimum variance portfolios are derived and the exact distributions of the estimated portfolio weights are calculated. Unbiased estimators for the expected portfolio returns and their first two moments are presented. Further, the out-of-sample means and variances of global and local minimum variance portfolio returns as well as unbiased estimators for all the corresponding quantities are given analytically.