Estimation of optimal portfolio compositions for Gaussian returns

We consider the expected return and the variance of the expected quadratic utility portfolio and the tangency portfolio. The expected returns on the individual assets and their covariance matrix are estimated by the sample mean and the sample covariance matrix. Replacing the unknown parameters by these estimators in the portfolio characteristics estimators of the expected portfolio return and the portfolio variance are obtained. In this paper we calculate the densities of these estimators assuming independent and multivariate normally distributed returns. Because the densities can be computed by using standard mathematical software packages these representations are very useful. These results can be applied to construct tests and confidence intervals for the parameters of the efficient frontier.